INVESTIGATIONS OF THE
MOVEMENT AND
STRUCTURE OF
D-REGION
IONOSPHERIC
IRREGUI.ARITIES
By
I^I.K. HOCKING' B. Sc. (Hons)
A Thesls presented for the degree of
DOCTOR OF PHILOSOPHY at the
UNIVERSITY OF ADELAIDE
(Physics DeParEment)
FEBRUARY I98I
VOLUTIE TÏ Page CONTENTS
VOLUME rI
ON SIGNAI 292 CTIAPTER VI THE EFFECTS OF NOISE THE
293 6.1 Introductlon 294 6.2 Notation and AssumPtions 294 6 .2.L NotaEion 298 6.3 Complex Data 6.3.1 DetermínaÈlon of nolse usíng comPlex data 298 301 6.4 Ampli tude-onlY S arnPling 6.4.L Effect of noise on the data series 301 6.4.2 Effect of noise on the autocovariance 303 ' a. Introduction 303 b. Power (IntensiÈY) analYsis 304 c. AnPliEude-onlY analYsis 309 (f) Noise estimation 309 . (fi) Autocovariance distortion 312 6.5 Experimental Results 318 322 6.6 Conclusions
323 CHAPTER VII OBSERVATIONS USING COMPLEX DATA
324 7 1 Intro CHAPTER VII Continued 7.4 Experimental data 347 347 7 .4.L DoPPler r¿inds 7 .4.2 Spectral widths using the vertical beam 349 beam 354 7 .4.3 SpecÈral wídEhs using Èhe tilted 7 .4.4 Theoretical spectra for anisotropic turbulence 3s7 7 .4.5 Interpretatlon of specEral widths using tilEed beam 362 I'Javes 7 ,4.6 The relaEive roles of gravity and Èurbulence in the mesosPhere 373 7.5 Deconvolution 380 7.6 Conclusions 382 PROFILES CHAPTER VIII COMPARISON OF PARTIAL REFLECTION AND ROCKET MBASUREMENTS OF BLECTRON DENSITY 384 8.1 Introduction 385 8.2 ReflecÈi-on of a radío pulse in a horizontally straËified ionosPhere 386 8.2.1 TheorY 386 a. Pulse convoluÈion 386 b. Fourier Procedure 388 8.2.2 Some SimPle APPlications 39r B 3 Analysis of sirnultaneous rocket and partial ref lection me4sur emenÉs 393 8.3.1 Introduction 393 8.3.2 Experimental deLalls 394 a. Technique 394 b. Results 396 8.3.3 CompuEer símulaEion of partial reflection height Profile 400 8.3.4 Some extra observations of interest 404 a. Diffuse fluctuations in electron densitY measurements 404 b. Structure below 80 kn 405 8.4 Conclusions 407 Page CONTENTS Continued. .. . OF CHAPTER IX INVESTIGATION INTO THE GENERATION THB HF SCATTERERS 408 9.1 IntroducÈion 409 9.2 Scat.terer correlations with winds 410 g.2.L Correlations between mean wind profiles and scaÈÈering LaYers 410 a. BxPerfmenÈal observations 410 b. Results from other references 413 g.2.2 Gravlty wave effects ín producing scaÈtering l-aYers 415 9.3 The relative roles of turbulence and specular scatter 420 9.3.1 Turbulence 420 g.3.2 Expected specular to isotropí-c scatLer ratios 423 9.4 Concluslons 428 429 CHAPTER X INVESTIGATIONS AT 6 lúlz 43L t_0 t_ Introduction 433 10 2 Experimental Results LO.2.L Power proflles and temporal varÍations 433 LO.2.2 Fading times and power sPectra 438 443 10. 3 TheoreÈícal InterPreËatíons 10.3.1 Theory and InterPretation 443 10 .4 Conclusions 449 450 CHAPTER XI DISCUSSION AND CONCLUSIONS 11 .1 IntroducEion 45r LL.2 Summary of Facts 4s2 11.3 Dis cussion 4s7 11.3.1 Discussion of doubÈful poínts 457 a. Product.ion of turbulence 457 b. Reasons for sPecular scatter 4s9 c. Reasons for Preferred heíghts 463 d. Seasonal varÍations 46s Page CONTENTS Continued CHAPTER XI Continued 11.3.2 The Authorrs Vi-ew 466 11.4 Inconclusive resulLs, future Projects and schematíc summarY 468 APPENDICES A The Neutral Atmosphererand Èhe Ionosphere above 100 km the B Propagation of radío waves through Ionosphere c. The Dynamical equatíons governlng the AÈmosphere D. Angular and Temporal CharacËerísÈics of Partial Reilectlon from Èhe D-Region of Ehe Ionosphere (A paper presented ín Journal of Geophysical neseârch, 84, 845, (1979) ) E Computer Program SPecPo F Computer Program ScaËPr G CompuEer Program Volsca H Co-oråinates of some ImporÈant Middle- ALmosphere Observatorj-es (Past and Present) BIBLIOGRAPHY 292, CHAPTER VI THE S OF NOISE ON THE SIGNAL 6.1 Introduction 6.2 Notation and AssumPtíons 6,2.L Notation 6.3 Complex Data 6.3.1 Determination of noíse using complex data 6.4 Arnplitude-on1y Sampling 6,4,L Effect of noise on the data seríes 6.4.2 Effect of noise on the auËocovarfance â.¿ Introduction b. Power (IntensitY) analYsis c. Amplitude-onlY analYsis (i) Noise estímation (ii) Autocovariance dístortion 6.5 Experlmental Results 6.6 Conclusíons 293. 6.1 Introduction It is of course desírable that the effects of noise be minimized in any experiment. Any noise whích is recorded should at ,least be monítored, so an idea of its mean level can be gaíned. Averaging sets of successive points can help to reduce the effective noise level. This ís a very useful technique íf complex data is used, and the averagíng length is less than about one quarter of the shortest period at which the signal varies (the procedure ís generally called "coherent l-ntegration".) However, if arnplitude-only data is recorded, great care must be taken v¡hen averaging sets of successive points. such a procedure does reduce the noise - buË also has the effect of distorting Ehe signal. The purPose of this chapter is Lo show just \'ühen averaging is allorvable, and what happens to the data if averaging is applied when Ít should not be. It is shown that the presence of noise in a signal which is sampled with amplitude-only distorts the autocovaTiance of the sígnal, but sampling with powers (intensíty) leaves the autocovariance looking similar to thaÈ for the signal alone' save for a noise spike at zelo time lag. Formulae are derived from whích the RMS noíse ca.n be calculated using the autocovariance of the íntensity of Èhe signal plus noise. The simplificatíons ín obtaining such noíse esËimates l^7hen data is recorded as complex data are also pointed out. 294. 6 2 NotaÈíon and Ass tions 6.2.I Notation In thís chapËer, the following notation will be adhered to as closely as possíble. Before detection, Èhe total received signal wíl-l be considered t,o consist of 3 contributíons. i(, (r) v (6.2.r.L) y(r) = s(t) + r(t) + e(t) 1=Y(t)e ) where (t) i0 s (6 .z.t.z¡ S(t) = S(t)e s= lql j 0"( t) (6.2.L,3) r(t) = r(t)e. El i0t (t) (6.2.L.4) e(t) = e(t)e "= lel are three complex signals. (The notation x will represent a complex vector, and x will represent its magnitude) ' s(r) sha11 be called a "specular component", though it may vary slowly in time. T(t) shall be called a "ïandom scattelil component. This is assumed to satisfy Rayleígh assumptíons, wíth RMS value kr. However, each successíve poinË of r(t) ís not assumed to be uncorrelated. IÈ is assumed that the autocorrelation of r(t) has finite r¡idth. For example, r(t) might be the signal produced by scatter from turbulence. Let jóo(t) (6. 2. 1. s) D(t) = ¡(t)e =S*r : D(t) lql S*r Thís wíll be called the "data component" of the total received signal (or the informatíon or message). IfSisaconstant' lo(t) | i" ni.. distributed. 295. The term e(t) shall be called the "external noise". It represents the unwanted component. Thís is also assumêd to satisfy Rayleigh assumpËíons (as was r(t)), with RMS value ku' (The ineorning noise on the antenna is wide-band, but becomes band-limited by the effect of the receiver. Impulsíve noise such as atmospherics ís excluded. ) Let iö (t) (6.2.I.6) I(t)e-'n' = r * e. Here n(t) = lr + el This represents the total random conlponefit of J.. It consísts of a random component carrying useful data, r(t), and an external (unwanted) noise comPonent e(t). n(t) has RMS value k'={k"2+kr2)\ (see Chapter V). Thus this situation represents the type of signal which may be received from the ionosphere, with a specular contribution, a random contributíon (e.g. turbulent scatÈer) and an unwanted external noíse contribution. The features are distinguished by Èheír fadíng times (i.e. the time for the. magnitude of their complex autocorrelation to fall to 0.5) . The external noise has a fading time defined by the receiver bandr¡ídth - if the frequency response is Gaussian and the bandwidth at halfpower Ls 25kíz (typical of the receivers used in this thesis), then the fading time is .441 (05IIO3) seconds = lBUs. At the PRFs used in thís thesís (=2OHz), the autocorrelations of this external noise will be uncorrelated after 1 shift. The random scaÈter component produces fadíng times in the region l-3s. (e.g, beam broadening, turbulent fluctuations etc) ' The specular components have much longer fading tímes' After detection an amplíÈude-on1y signal, which will be denoted as y(t) (= l¿(t) l), tirr result. 296. If there had been no external noíse, the amplitude detected would have been D(t) = lS * rl. Notice in all these definitions, a line under the symbol (e.g. such a líne A(t)) implies a complex vector, whereas the lack of means the arnplitude only (e.g. y(t)) is being consídered. The followíng parameters will also prove useful: o = Rice Parameter of total received signaf y(t) It will be seen that the autocovariance of the amplitude-only signal looks like the following' f¡ P tf, t be denoted Uy p--(u) pry'"' @): The autocovariance of y(t) will . y ' f¡ = Autocovariance of y(t) at r=O (the "full" autocovariance) ' autocovariance at zexo shíft - i.e. the narro\^t f i = ínterpolated spíke due to the external noise is interpolated acloss. Likewise, an autocovariance of the powers (intensity) can also be formed. p of the total po\¡rer (intensity) y2(t) 'vP G) = autocovariance P- = autocovaríance of the po\rùer y2G) at t=O 'fp p. of the intensity autocovariance ,tp = interpolated autocovariance at t=0 . A running mean of length S on the data y(t) will also be consídered at times, to procluce a functíon m(t). Occasionally y(r) witl be written as y(t)=rn(t)+ern(t), e*(t) being zero mean noise. The follor,ring will also be used: n(yt) = expectation value for yt fot the Rice distributed series y(r). 297, data lnto inËervals (see later). ô j = Jth interval used ín splitting in ô tDl j = consËant value assumed for D(t) j rh no(rn), = exPectatlon value in the j interval assuming a constant data sígna1- [D]j' Plus noise. yp(t) = (Y(t))2 ¡u--P'-' (t) = running* mean of y-(È).-P < > lndicates overall average < >, lndlca,tes average Ín interval i. At one stage, it ís assumed D(t) is Ríce dístributed, with specular componenE 3 and RMs scatter comPonent k*. Rlce parameter oD, fn(t) = autocovariance of information signal arnplituderhad there been no noise. = f " fo(o) P¿{o) G, a (, J¿rrnø 3W, T. Fig. 6.1 illustration of the magnítude of the complex âutocovariance of the signal y(È) descríbed in Section 6.2. The Section (a) represents the autocovariance due to the noise' and -Oiagramatícbecomes uncorrelated extremely quickly. Then there is a more slowly falling component (b) due to the random scatÈer contríbution, and fínally a very slowly falling componerrt (.) due to the specular part of the signal. Fe*un C f, o. Eie. 6.2 l.WW --- Power spectrum of Y(t). In actual fact, an indívidual Power spectrum would show consíderable fluctuation between successíve pàittt" (e.g. see Chapter VII for examples), but this is the result àft.r smoothing. The part (a) is the noise component, and would eventually fall to zero at large positive and negatíve frequencies (since rhe spíke (a) in Fig. 6L. has non-zero width). The part (b) is due to random scatter, and (c) is due to the specular component, just as in Fig. 6.1. 298. 6.3 Complex Data 6.3.1 Determínation o f noíse using complex data consider the complex autocovariance of the time series ¡(t) described ín section 6.2. IÈ wíll look like Fíg' 6'1' Let the * + Then the autocovariance íÈh poínt recorded b. å = 9i å å' at zero lag is I--N ¿ (6.3.1.1) r (o) =- ) (yi - l) (ri r) v N:.r-=l_ N I 2 T N i=1 + and sínce = D. The latter follows because since *l-v. --LD. _Ie. , X. the mean ',r"r,rå of the external noíse, ã , is zero' Thus N (o) I - r*) + ¡ ( (Dri D ) + e-.)2), P N I -rt- 'y_ l-=I.r.{({ro. "*r)' where jDri, * j -- /-L. % = Dni_ + gi =.Rí "Ii, i Expanding N P(o) 1 t {(o*, - l*)2 ¡ (D1í D I)2\ 'y N í=1 N N 1 2 + (e2 +e2 T(D Dn) t Ri Ii )+ N Ri "u,- N í=1 i=1 N 2 + x(D - lr) N Ii i=l- "ri ThefírsttermistheauEocovariancewhichwouldhavebeen obtaínedhadtherebeennonoise.Thesecondtermissimplythe (ignoring mean squared noíse, U"'. The last two terms are zelo' statistical fluctuations) since (OOt Dn) and eR are uncorrelated and each inas zero mean. Likewise (lr' - ut) and e la 299. Thus (6.3.L.2) f"(") = Autocovariance due to the information + k^2' (data) comPonent of the signal e If Èhe autocovariance at a s-atl lag is considered, it is possíble, in a simílar r^7ay to the above' Èo sho\n7 that the value obtainecl ís simply the value whích would have been obtained had there been no noise, and of course this is nearly the autocovariance due to the information component of the signal at 0 lag' Infact,iftheautocovarianceisinterpolatedacrossthe in Fig . 6.L, then ís equal Ëo spíke to produce 'Iinterp/., /.,.¿interp the auÈocovariance at O had there been no noise' This and 6.3.1.2 means that +k 2 (6.3.1.3) = fu e f,,(o)¿_ ¿ interp Hence if the compl x autocovariance function is obtained, obtained, Ëhe mean square e*t"""al (/"(t)), and /"(o) .td fli'rerp noise (noise porrer or intensity) can be easily obtained as (6.3.1.4) n"'= P"{o) - f"rnr"r, It ís instrucÈive to consider this derivatíon from another point of view. Since S(t), r(t) and e(t) are all uncorrelated' the power spectrum of ¿(t) = S * r * e is simply the sum of the indívidual power spectra of S, r and e' The resultant is a funcEion as shown irr Fíg. 6.2. Now the Fourier transform of the sum of several functions is the sum of the respective índivídual Fourier transforms, and the Fourier transform of the Po\^7er Spectrum is the autocovaríance. Thus the autocovariance of ¿(t) is the sum of the individual autocovariances of S, r and e' 300 Ilence we again see an autocovariance like Fig. 6.1 is produced. The spike (a) has a length equal to the area under part (c), of Fig. 6.2, which is of course Èhe mean po\¡ler, U"'. Thus it ís again clear that 6.311.4 is valíd (ignoring statistical fluctuatíons) . I \.$ ç \{ \- ( s Fíg. 6.3a 1 Typical amplitude-only data series y(t). The slowly varying broken line rePr esents the amplitude of the specular component' and the faster varying b roken line represents the amplitude which would have resulted if there had been no external noise, - that ís, the ínformation parË of the signal, D(t)=l-q*f l . .-$ - { $ D(¿) \JË * 1ñt3, t Fíe. 6 3b section ð in Fíg. 6.3a. The solíd líne An expanded view of the ttspecular .C represents the average sígnal, and { is the effective component" for thís interval. I I t m(t) --.-$ I I \-- \ D(¿) rt'+ I $ \ I \- 5 I Ê L \ \3 Jr>rrrorC. Fig. 6. 3c View of a section, several tímes ô ín length, of Ëhe Èotal sígnal, after a running mean of length ô has been applied. This is represented by m(t). The broken line shows the signal which v¡ou1d have resulted had there been no external noise, D(È). 301. 6.4 Arnplltude-Onl v Samplins 6.4.I Effect of noise on the data series Consider the time series of the amplitude-only seríes y(t), as shown in Fig. 6.3a. Now consíder a length of data, ô, from this time series, sufficiently short that the ínformafion part of Èhe signal (q(t)) does nor change significantly in this period. Then this section is Rice-distributed, wiÈh "steady component" Ç G actually beíng due to the specular sígnal S and the random scatter signal r), and RMS noise k.. The distribution is gíven by equation 5.2,L.15 in Chapter V, with k+ke and S+E' Suppose norir an experimenter decides to reduce the effect of noise by averaging in blocks of lengËh ô. Then the mean signal will be somewhat more than {, as shown in FÍg. 6.3b. The rnean is ín fact given by equatíon 5.2.2.7, Chapter V; i'e' (6.4.1.1) lo=Eo (v) - ke- t(3/2) rFr Gr";7 i-& /2) f2\lk t(3/2) /l/2. , trIhere a = e , = and VO, UU(V) refer to the mean in the interval 6' The imporranr rhing is thaÈ (au(v)-6) is a function of 6 (see Fíg. 6.4). Thus if l¡re \^7ere to imagine a runníng mean of length 6 being applied to the data, the resultant would be as illustrated by the solíd line in Fig. 6.3c. For large n(t), the mean is only slightly greater than the true D(t); for small D(t)' the mean is sígnif icantly greater than D (t) . ttlhen D (t) ís zeto, the mean is /l/Z k". A brief descrÍption as to why noise is least important for large D(t) ís given in Fig.6.4b. It is quíte clear that this averaged signal is NOT the same in form as the arnplitude whích would have resulted had there been no external noise. The data series has been distorted. ryI (MEAN- 3) 0.5 00 1.0 2e8 Fig.6.4a n Value - specular component E) for a Ríce dístríbuted sample r^/íth RMS noise k. = 1.0, as a function of the specular component component, the mean is that of the noíse {. NoticeÊrz€ror specular l* alone - í.e. aet . ¿ t Fíe. 6 .4b The mean value for a large specular component plus weak noise is approximately the specular componenË. Thís can be seen in this diagram. Here, k represents a typícal noise vector, and iË can have any ori.entation. The resultant.c = g + the projection of k on { (the "ín phase" component, say). The projection on Ç ls taken as positive if it is in the same dírection as !, and negative in the case shor^rn here. The projection of k on E has a mean whích is close to zero for Iarge l, so the *""tt of l!,1 [ends to lg-l . 302 !ühat if we had averaged po!'/ers (intensíty) with our runníng mean? Then (6.4.L.2) nr(v2) = ç2 + k"2 (chapter V, equation 5'2'2'l.o) = D(t)' * u"' Clearly the average is once again greater than that which would have resulted had there been no external noise (i.e. D2(t)) but at least Ëhe average is exactly Èhe same in form as Èhe series D2(È) ít has Just been shifted up by a constant k.'' Thus here lies Ëhe first warnlng - it is not valíd Èo símply average blocks of amplitude-only data to reduce noíse; it is' however, valíd with íntensitY. 303. 6.4.2 Effect of no íse on the autocovariance 6.4 2a Introduction In Section 6.3, noíse t/as seen to add a narro!'I spike at zero lag to the cornplex autocovariance and it was possible to use this spike to estimate the RMS noise. In the case of amplitude-only data, a narro\¡r spike is again produced on the autocovariance. Again Èhe noíse can be deduced from the heíght of this spíke - but how? The problem ís no longer sírnple. Further, Ís ít noÈ possible that the noise may affect more than just the zeto Lag value of the autocovariance? trnle have already seen that averaging the data distorËs it. Perhaps noíse could distort the , autocovariance? Are there any advantages in usíng powers (íntensíÈy) rather than amplitudes? The next few sections will ans\¡rer these questions. An understandíng of these ef f ects ís essential when dealing with amplítude-only data. It is the authorrs belíef that most of the following work is original. No references to similar material have been located. The autocovariance of the real data series y(t), of length T' ís (v(t +r¡-l¡ dt (6.4.2.r) PrG) = + J"(v(t)-l) (Ú Then at zero Lag = 0) ' (6.4.2.2) Pr(O) = Now suppose, as discussed in section 6.4.L, a running mean of length ô (ô being much less than the correlation time of Ehe scatter component r(t)) is applied, to produce the function m(t) in Fig. 6.3c. Recall D(t) in rhat díagram is the function whích would have been recorded had there been no external noise (i.e. it ís the inforrnation) . IrIe can wríte (6:4.2.3) y(t) = m(t) + e*(t) r where e (t) ís a noise comPonent wíth zero mean (by definition of m m(t)). Ir should be clear that er(t) ís not equal to e(t). The autocovaríance of y(t) is then that of m(t) plus that of e (t). This follows by sírnilar arguments to sectíon 6.3.1. The m crucial reason why this can be stated is that er(t) has zero mean' Thus the interpolated autocovariance at zeTo 1ag (i.e. ignoring the spike) is (6.4.2.4) Pi = 6.4.2b Power (Intensitv) analysis Because it is easíer, let us firsÈly discuss the case where íntensíty is recorded, as distinct from amplitude. (The íntensity can of course be obtained from the arnplítude símply by squaríng) ' Let (6 .4 .2.5) yn(t) = (Y(t))2 be the intensity. Let urn(t) be the running mean of yp(t), ín a símilar v¡ay to that in which m(t) was the running mean of y(t). (f) autocovariance of y- (t) and let ""-LeE tP yp'' ¡e the -P ' 305 (6.4.2.6) P- = Then the mean Po\Árer ín any interval ô. is (6.4.2.8) ñ..* = [, (Y2) , Pr- ôi the expectation of y2 for a Rice distribuÈed series, where Uôi í" given by equation 6.4.L.2- Thus (6.4.2.9) h-* = (D*)2 * u"', D. being the amplitude of the pa r- information signal in this interval. Then ít follows that (6.4 (6 10b) <^ ,2 ='(<¡2> + ke2) - Also, by 6. 4 .2.9 , .^ 2> = .(t2(t) + k 2)2>, so thaÈ P e 306 (6 .4 .2.LL) .^ ?> = (6 Then (6.4.2.L4) Hence (6.4.2.L5) .rr"r, = oiu + 4 ke2D'2 + 2 k"4, and so .y 2u = <( (ø.t+.2.:-7) q-ip lft - =2k2 o (6 . k2- 2, cqn be readilY found, so an All the terms .rnr, .!n I fp and P, of k 2 can be made. But should the fUu + or -? estimate e Consider some examPles. (i) y(t) = e(t) (aII noise). Then D(t) = Q' p. (in principle - in practice, statistical Thus tLp = o by 6.4.2.12. fluctuations may mean this is not quite true). Turther by 6.4.2,I7, 2\ 2+0. o = k 4. So 6.4.2.18 implies U"u = U"" = k - Pfu ti-p e " e e' e í. e. the formula makes sense. (ii) Consider the case of no noise, y(t) = D(t), and e(t) = Q. Then the mean square noise estimated by 6'4'2'L8 Ls Then f íp Itn 4y¡* should be (v) (6.4.2.Le) k2 = Then the technique to obtain the mean square external noise from intensíty data ís: (i) Find the autocovariance of the intensity (íi) Find the autocovariance at f=0, i.e. P+ (iii) Find the ínterpolated autocovaríance at'Ç=O. To do this, use several points either side of f=0 and fit (say) a splíne to these points, ignoring the value at'iî=0. This spline function is used to interpolate the value at zeto, i.e. fn. (Y (iv) Find the mean square signal,' 'p ) G.y2r). Note that thj.s ís done wíthout removing the mean first (although the mean is removed to form Èhe autocovariance). (v) Use 6.4.2.19 to estimaÈe k.2. Some final comments are also worthwhile. Notice by 6.4.2-L2 that ,Lpp. = 6.4.2c Amplitude-only analvsis 6.4.2c(i) Noise estímation Now let us return to the problem of obtaining the autocorrelation usíng ampliÈudes. Again, consider a running mean m(t) of length 6, and write (6 .4.2.20) y(t) = m(r) + e*(t) , where er(t) has zero mean. (see equation 6.4.2.3). Then (6 .4.2.2L) Pr= tY" - !'Ie also know (6.4.2.24) Thus we now seek .*2>. Again consider successive blocks of length ô. Then .*2> = <(e"(y))2t- ò' I,tlhere A (V) is mean of the Ríce disEributed series in ínterval ô, . By 6.4.1.I, ^.2 (6.4.2.2s) uor(v) = k"f (3/2) ¿r(-La;l;- ï), where a= f2D./lr" . Thus @ _a2 n @ ç'4) (-1) " (6.4.2.26) eur(v) n u2n ='4/i I 2 '4/l v n=0 n=0 zî ç-,.r.¡z where (")r, = a.(a+1)(a+2).... (a+n-l) and (")o=1. Thís is clearly a complicated function, and cannot readily be squared. Thus evaluation of To make some pïogress wíth this problem, the following assumption wíll be made. ASSUMPTION: It will be assumed that the ínformation signal D(t) = ls(t)+s(t)l is Rice distributed itself, wiÈh a constant specular component 3, and a "random component" wiÈh rms value k*. It must be pointed out that 3 is not equal to S(t), unless S(t) is constanÈ. The assumption really amounts to assuming S(t) is eíÈher constant or Rice distributed (sínce r(t) is Rayleígh distributed) although the phase variatíons are much slower than those for r(t) and e(t). This assumption may not always be valid (although it frequently is in the case of D regíon scaËter, as \¡Ie have seen), but ít at least allows us to examine the problem in some detaíl ín this specific case. Let the probability that the information signal has an arnplitude in the range D to D*dD be (6.4.2.27) Rs(D)dD, where (6 .4 .2.28) Rs(D) = 2 ElkN2 e '",#, dD with Rice parameter oD = õ !/kR. These stat.ements follow from the Rice dístribution formula (5.2.f.15) . Then the series y(t) is the sum of the Rice distributed series O(t) and the Rayleigh distributed noise e(t), so is Ríce distributed (see with specular component 3 and mean square noÍse = h' * ku' the last paragraph of Sectíon 5.2.L, Chapter V). Thus (6.4.2.29) < 311. _d2 (6.4.2.30) (y) = (m> (by 6.4.2.23) = 4/-*{,rr (-};1; +)} where'neK cr = /1 zl ttr4æ + and k = ñTeR kT Inlhat of .^2>? In the interval ô., = (:|;r; -r1), (6.4.2.3r) m, = Eu,(v) \/l u" rtsr 2 where a . = {2D ./ke, as stated in 6. 4.2.25. ' Let. the probability of a mean value between m and m*dm be P dm. m Then I (6.4,2.32) .^'> = dm , Jm2r*(m) BuË the probabilíty of a mean value between m and m*dm is also the probability of a D value between the corresponding D and D+dD' sínce the RMS external noise is constant, so D and m have a I to 1 relationship. Thus (6.4.2.33) r*(m)dm = Rr(D)dD Hence by 6.4.2.32, .*2, = l*2 nr{l)at J _1 2) where Fr -D2 lk . m=\{lk e1 Ç;1; e Thus, finally' *-JÁ 'P¡ *-* (6.4.2.34) = .tzr- Fig. 6.5 shows a plot-* of m vs k" for-rrr a value-* D=1'0'-* Fig' 6'6 shows a rypical RS(D) plot (3=1.0, op=O.5), and also a plot of LO ( filÆnn S ) 1.0 5tu 1, zo ttt' 9-:l-- ^" v,ñ ',D D -,"2-as a functíon of. lT cr"ph of . o . t'l ,-L, ,;L; \) - This is thus a graph of the mean value for a Rice disÈributed data set as a functíon of the RMS noise k^, hoth the mean and k being with respect to the component S. e normalízed "p."rrÎtt 03 r¿ -- R(' ß;R E =1.0, fto 0.5 d åu=0'6 b OJ 0 1 2 3 4 5 67 lfr m& D fre f \ \ t rrt, Þ. Fi 6.6 Graphs of P*(m) vs m, and R"(D) vs D. Here D(t) is the ínformaÈion amplit.ude, and it is assumed to have a Rician distributíon with "specular component" 3=1.0 and Rice parameter cr'=0.5. The function n(t) ís the running mean of the information signal D(t) plus an external noise component with RMS value k"=0.6. f*(m) and R"(D) are the probability distribution functions of m(t) and D(t). (note- do not confuse m(t) with the overall mean of the data sample. IÈ ís NOT It can be seen the mean never falls below {l U-, the mean this). Ze' for noise alone (although there will be statistícal fluctuatíons). The smaller graph shows a P distributíon for a very large k. value. m 3L2. P (n) vsm (P* (n) = n"(D) ,tUJr, with external noíse k" = 0.6. m t Note also that ft - Pi = tYzu:t^2'' so 2 (6 .4 .2.36) = ( (D) dD) f ¡-fi-.vz> f*'u" where m = \G k" rFl <7rrrÊr. So by solving 6.4,2.36, it is possible to find k"' That is (i) Obtaín the Rice parameter onfor Ëhe total sample y(t) this wíl1 equal õz/ET+*z (ii¡ Then 12 = .!2, {t + Z/azn)-I (by equation 5 .2.4.1, Chapter V). (iii) Solve 6.4.2.36 for thís 32 value, subject to kO2+k.2=at22-a2 (by 6.4.2.29).. This final step is possibly best done by graphical means. That is, plot (lr2 n.(D)dD)Z as a functíon of k", using JÞ k*r=.trr-Ur-U"r. Then fínd that k. value where this expression equals the appropríate f¡- fr-.y2r - It can hence be seen that iË is also possíble to estimate the true external noise usíng the sPike on the amplitude-onlY autocorrelation. However, the problem is more difficult than that descríbed by the intensity equation 6.4.2.L9' and in this thesís, the power (íntensity) autocorrelation was always used to estimaÈe the noise. Equation 6.4.2.19 also has the advantage that ít ís applicable for any Èype of specular contribution S(t)' whereas 6.4.2.36 assuned S(t) was either constant or Ríce distríbuted. Having estimated k.r thís can then be used to get ideas of the stgnal to noise ratio and the true Rice parametcr (Chapter Vt equatíons 5.2.3.2 and 5.2.3.3). 6.4.zc(Lí) Autocovaríance dis tortion In sectíon 6.4.2b, it v¡as mentioned that apart from producing the spike aL zero, Rayleigh distributed noise does not alter the 313. shape of the autocovariance of the intensity. This staÈement ignores statistical effects. Some computer exPeriments were done to check ít. Data recorded at Townsville in complex form were used, and Èhe autocovaríance of the po\,¡ers was found. Then a specular componerit r^zas added to Èhe complex data, and the autocovaríance funclion again found. The two autocQvaríance functions should be identical, since adding a specular componen¡ should símp1y result in the addítíon to the po\4lers of the square of this specular component (on average). The two autocovaríance functions wefe quite símílar in form, but not eompletely identical. This may indicate that Ëhe theory developed is not entirely accurate. However, by adding oTr this specular component, the distribution of the powers (after removal of the square of the specular component) will differ from the original distríbution, due simply to sÈatistical fluctuation, and the dífference in the tvro autocovaríance functions may simply reflect this statistical fluctuation. The dísagreement r¡ras worsË when the fadíng of the information signal was fastest. Presumably, rapid fadíng may mean there were insuffícient points recolrded in an interval of constant ínformation- signal for equations like .!'rí - .O'r, * k"' to hold exact.ly. (l may Changes-e ín k-2 during the records minute blocks were used) also distort the autocovaríance. If the noise \¡las not Rayleígh dístributed, the formulae may also break down. Noise esÈimated by equation 6.4.2.19 were also performed. In prínciple, they should be the same in each case. Thís l¡las not quite true, but agreement was reasonable (better than 502). However, overall it ís fair Eo say that noise does not dístort the autocovariance of the pohrer too greatly. 3r4. But r^rhat abouË the autocovaríance of the amplitudes? If there had been no noíse e(t), then y(t) = D(t), and the autocovariance of the information signal, PDQ), would have been formed. Suppose thaÈ the aurocovariance at zeÍo lag is (6 .4 .2.37) fD (o) = f . " assume D(t) is Rice How is Jo related. Eo f .1. As previously, dístributed. Then These lr"a.*"rras follow because .-2 -iÐ r = \{l k Ft Ç};tt is the mean for a Rice distributed fzr seríes of specular component 6 anc Rice parameter cr = É' ancl tj;t;Ê) is a runction onlv or a ' rhus î = ,O ,*, rFr In the second term Prlzz is dependenÈ only on oD. 6'4'2'35, divided by Z2 is clearly dependent only on o.= /-zz/(t"2+k*2)à = tf;#r' = (o. * or-')\, e\ and so dependent only on cr. and cl'' trnlhat about 2 in 6. 4.2'35? rhis etuals (år R (D) dD. *,!') [t'*"(D)dDs J S uow * ís a function only of /-2Dlke. Also, Rs(D) dD 2E/kR2 .'p G(+. ro(oo. = þ-rÌ SO (6 .4 .2 .3e) Rs(D)uo*;.ro2 e"p G(+ " *)Ì lo(*ocxo) dcro' where a* = /zllç. 1.0 0.5 08 D(D- 0ó 0,4 0,2 00 0.0 1.0 2D Fig. 6 .7 a Plots of fi/ fr, where fi = interpolated value of the autocovariance at T=0 for the total signal, and, fs = the auËocovariance at A=O which r¿ould have resulted had the information signal alone been recorded, plotted as a functíon of the[external RMS noíse k. divided by the "specular component" of the Rice distributed informaËion signal]. Only values which are felt to be accurate (for the computer program used) are plotted. The parameter cxD is the Ríce parameter for Ehe information sígnal (ttris signal is assumed to be Rice distributed). 1 08 xD- 2.0 Nr25 0ó f,4, OA o.2 tte.--9.¿þ --jiTps (as described in Fig. 6.7a) ploÈted as a function of thetRl"lS external noíse k. divided by the mean information signa! (a form of noise:signal ratio). 0.0 0.0 1.0 fr 2A l 315 Thus (m/g)2 is a function only of fzo/u"=t/-zo/u*) (kR/ke)=cxo.o./ao (*/g)2 i" function only of ao, o., oDl R"(l)dl is a Í.e. ^ function of only co and aO. Thus thís integration may be performed, and the only dependent variable left will be cl", oD. The íntegration may be done either by using the change of variables suggested by 6.4.2.3g, or simply ín the form Þ j."*r(D)dD,uslng any particular values for 3, ke, kR. The rerm P"/fi: (Ps/z\/(Pr/=z) ís clearly dependent only on cl. and cx'r sínce we have just shown thaË both the numerator and denominator are dependent only on these terms. Fíg. 6.7a shows pJ-ots of f Pi as a funcriorr of k"/E ç= õlu) for various o¿D. "/ Larger values of cl' ímply shallower fadíng for the "specular component" S. Notice ttrat Ps The irnporÈant poínt is that /" * f . and agreement gets r,Torse for larser"e k values. ,o (g). il;mø Å/"tt,T FÍe. 6.8 DiagramaËic comparíson of autocovaríances of the total signal y(t) (Pì and of the information sígnal D(t) ("pt). Notice in at f=0. particular the decrease of the lnterpolated value "t f, 316. Is it possible to explain why fi. fr? Indeed it is. Examine Fíg. 6.3c again. Recall, too, that Ehe autocovariance away from f=0 is definecl by the running mean m(t). trle see that m(t) is sígnificanEly larger than Èhe informatíon sígnal amplitudes n(t) when the latter are sma]l, but m(t) becomes closer to the D(t) at large arnplitudes. Thus m(t) is t'bunched up" closer to its mean than D(t) was, as illustrated below. êfr¡ê This reduces the covariance, sirrce fluctuatíons from the mean are less However, it is reasonable to expecE Èhat the autocovariance of m(r) (and hence y(t)) falls to zeTo at a simílar point to that at r¿hich D(È) fell to zero. Thus if r,/e compare 4rG) and fD(C), the picture is as illustrated in Fig. 6.8. The re-normalized autocorrelation of fn and f, (spike removed) will thus be dífferent . The width of the autocorrelaÈion may also be alÈered, although the form of this alteraÈion is uncertain. I-tg 6.8 suggests a widening (larger :rr2) and this wili Ue so if Ëhe autocorrelatíon near Èhe minÍmum üãkes a similar form to the oríginal, buf this is not necessarily so. The author expects a widening (variations in signal strength are less rapid: sharp peaks and dips on amplilude vs. time ploËs become smoother) but this has not been proven. What about the effect of noise on cross-correlations? If two receivers record ionospheric data, and the records are cross-correlated, it rnight be exPecÈed that (i) the indívidual autocorrelations witl be distorted, and (if) if Èhe receivers record different noíse levels' one runníng mean m(t) may be dístorted more compared to iÈs true information 3L7. signal- than the oÈher, decreasing Ëhe degree of correlation of the two signals. From these conments, then, tt ís quite clear that data analysls shoul-d be done using intensity (Powers) rather than amplÍtude, since intensity suffers less distoÏtíon. This is why' in this Èhesís, Pohlers were used for forming autocorrelatíon funcÈÍons, and in any averaging procedures. c. 1 S R 5 R 5 a ß LLE, 977. I I I t .7 I I t \ I ^/rN.Õ I fl- $,s S.4 rn)rll z2õ'' z1Ll) ú. 30ó o I¿þ00 0000 tooo 2000 0600^ t 600 Fig. 6.9 LOCAL TIME(hr) The s olid line represents noise estimaÈes at Townsville for days 77/304-306, using equation 6.4.2.19. Autocovariances were formed with one mín data blocks at all 10 heights recorded, and then these noise estímates I^Iere averaged in groups of about 30 mins to t hour. However, data was noË ah,^/ays recorded during the whole period - generally only 3 mins of data was collected every 5-10 mins. The error bars are sample standard deviations, and thus give some idea of Ëhe variability of indivídual noíse measurements. Only typical error bars in each period are given. These tend to be smallest in daylight hours. The variability in individual noíse measurements is partly a reflection of real noise fluctuatíons, and partly an indication of the statístical fluctuatíons inherent ín the method. Noise estímates v/ere converted to pV and then divided by the receiver bandwidth (=25k1z) to give the RMS noise in uV/kHz' No coherent íntegration \¡Ias used. Also shown are some actual measuremenEs of the noise, made with the tïansmitter turned off (black dots). The broken line indicates behavíour which sometimes appears to exist, with noise being quite high until uridnight and then falling. It can be seen that the error bars are much larger before rnidnight than after, Índicatíng much more variability of noise before 0000hrs. The vertical lines refer to the times 0600 (denoted SR, or sunrise) and 1800 (denoted SS, or sunset). The vertical arrows denote rnidnight. 318 6.5 Exper imental results tr{e have now developed the theory for estimation of noise from the total signal. Equation 6.4.2.19 was used in this thesis when amplitude-only data was recorded. If complex data was avaílab1e, equat.ion 6.3.1.4 could be used. But hovr does the theory l^/ork out in practice? Fig. 6.9 shows the first índícatíon. The data v/as recorded at Townsville at a central frequency of 1.94MH2, and the receiver had a half-power bandwidth of about 25kl/-z. Results are presented in units of vY /kli^z at the ïeceiver input (50Q impedance) Agreement between experimental observatl-ons and calculal-ions using equation 6.4.2.19 seems to be excellent in places where a comparison can be made. The diagram also illustrates well the diurnal variation in noise. A rapid rise can be seen at sunset, as D region absorption decreases, allowing noise to Propagate over much longer distances. The increase in noise may also perhaps be partly due to people switchíng on applíances (In such cases, the Rayleígh assumptíons may not be valid, whích may explain any errors in estimates of the RMS noise.) The noíse is quite clearly high all night, and then around sunrise decreases dramatícal1y, as the D region gror¡rs ín electron density, absorbing out much of the noise arriving from large distances. There was also at times indícations that the noise level iollowed the broken líne in Fíg . 6.9, with noíse quiLe l-rigl"r until midníght, and then falling to a lower value, (possibly due to people going to bed?). Even in Iig. 6.9 ít can be seen that the maximum noíse levels were largest in the períod of sunset to O0O0 hrs, since the error bars extend to quite high values, suggesting large variabílity ín noíse at thís time. The \ -\ rJJ x ô 5 f -\ t-- a \ H -.Þ --\ o_-J \- z \. a TÏME-* Fig. 6.10 Illustration of ðígiLízation noíse. The smooth solid curve represents the true temporal variation A(t), and the dots represent These dots have been joined by a broken line the digítized values. t'noise" to make their fluctuations more obvious. Such is not really governed by Rice statistícs, but has perhaps a Poísson distribution about the signal. However, when the signal to noise is large, both Poisson and Rice distrj-butions tend Èo Gaussian distributions, so the theory in this chapter ís relevant for this type of noise a1so. 319. minimum ín noise at 2000 hrs on both days 304 and 305 is an interesting feature, although it is unclear r¿hether this ís a regular characteristic. . OfËen, in dealing with noise, the signal to noise ratio is a more ímportant parameÈer than the actual noise level itself. In such cases, estimaÈion of noise by the procedures outlined in this chapter is acÈually better than by direct measurement of the noise level with the transmitter turned off. The first reason concerns receiver-generated noise. In Chapter III ít was mentioned that often an atbenuêLor was used before the signal entered the receiver, and this attenuation factor was taken into account after the data '$ras recorded. If there ís receíver noise míxed in wíth the signal, then this noise is also multíplíed up by the attenuatíon factor, even Èhough it occurred aft.er the attenuator. However, generally when actual noise measurements we1e made, the attenuation utas set to zero dB, so the noise measured by this method, whilst being a true measure of the noise, ís not the best measure to use insofar as estÍmating signal to noise ratios is concerned, sínce ít has not been mult.íplied up by the attenuation factor. However, generally Èhis ís only a mínor consideration whenever non-receíver noise dominates. Thís is usually the case' particularly at night. In thedayLimerhoweverrreceivernoísecanbeímportant' The second reason concerns dígitization. DigiËization can be regarded as a form of noise, as illustrated in Fig. 6.10. The recorded signal clearly has fluctuations which \¡Iere not evident in the real data. Estimation of noíse by the methods described in this chapter takes such effect.s into account. In fact, some real data recorded at Townsville, with Èhe receiver gain alternating frorn high to low on successive minutes' íllustrated thís effect 320. quite well. Noise estinates made with low gain always exceeded those made on high-gain. This would be expected if digitization "noise" 1nras important. Agaín, this is an effect which should be considered for proper sígnal to noise ratio estimates. However, actual tneasurement of noise is more accurate if it is the actual lgi-qg which is sought. such measurement also considers the effects of special interference, which may not be uncorrelated after I shift. One point which did come out of these experímental investigatíons was that noise estimates were only really useful whenever the fading times (noíse spike removed) of the pov/ers were greaLer than about L.5-2s. This was because for shorter fading times, ínterpolation of the autocovariance at zeTo often produced an p.. was to reject all noise over-estimate of /l Thus it usual measurements made with fadíng times of less than 1.5-2s. Dat.a which saturated, or for which the gain ra/as not high enough, was also rej ected. The installation of equípment cpapble of recording the complex signal at Townsville (see Chapter III) considerably ímproved the signal to noise levels. Data vlas recorded at 20H2, and then an I poinÈ coherent integratíon was done. In prÍnciple' thÍs reduces the noise po\¡/ers by B tímes. This fol1oh7s as a direct result of the cenËral límit theorem, whích staÈes in part Chat by averaging n estimates of a parameter' the variance for the mean is reduced by n tímes compared to the variance of a single measuïement. (e.g. see- Huntsberger and Billíngsley, 1973, 96.5). However, the variance of the in phase and quadrature componenÈs 2/2, as seen in Chapter V (equatíon 5.2.2.3), k^2 ís símp1y ke' e 32L beíng Èhe mean noise pohler. Hence this is reduced by n times when coherent integration over n points ís done. Indeed, this coherent integration did produce a significant reductíon in noíse. As seen ín Fig. 6.9, the Rl"lS day time noise before coherent integration rilas e.f - .2 Vylk]F.z. Aft.er coherent integration, measurements taken on d,ay 79/082 gave RMS noise values of about .04 to .08 UV/kIlz. Thus 2 k"ee' is reduced by about 2.5 Èimes' or k by about 6 tirnes. The installatíon of this equipment allowed another test of the theory presented in this chapter. Noise estimates could be made by both equations 6.3.L.4 and 6.4.2.19 for the same data. This was done using 1 minute data blocks for all helghts used at any time' and Èhese results were then averaged in 10 minute blocks. Then noise estimates rnade by the two techníques \^rere plotted on a correlatíon graph. The results are presented in Fig. 6.lla and b. Generally, the t!üo sets of measurements (i.e. by 6.3.L.4 and 6.4.2.I9) appear to agree wel1. The smal1 offset in 6.11a has not been explained. Notice the pohTers are greater in 6.11b, ín showÍng the effecÈ of digitiza|ion noíse discussed earlier. d^. 7 @l å 6 È DAY 79/062 TSVLLE. a .N 5 a 1535 -1810 hr, C MODE ì a 4 a a "T 4 3 a ñ r' a a /oî 2 I a { t.tt b. 'þ # È 1 2. d 0 4Ð cl ru o0 1.0 2Ð 30 a$| e! Po,,r*n tfrt %. TSVLLE O MODE S> 7e/082/næv810) R a 20 a a I a a i a a 10 I ,a d11¡ 3 \s¿ I a e:(Pnrr*"r 0 fwf 1 20 322. 6.6 Conclusions The results of this chapter suggest the following' (í) The effect of uncorrelated noise on a signal recorded as complex data (I(t)) is to add a spike to the autocovariance at f=0 of height k 2 - mean Po\4ler of the uncorrelated noise' e ((v(t))2 Èhe (ii) If Èhe signal is sampled wíth po\^/ers only "ty) l=O ef f ect is to add a spíke to the autocovariance of the po\^Iers aL of heíght 2 The form of the autocovaríance, apart from the noise spike, is unchanged (in princíple) by the noise' y(t) not only is a (iií) If rhe sígnal is sampled with amplítude ' spike produced at T=o lag by the uncorrelated noise, but the inÈerpolaËed value of the autocovariance at l=0 ís less than it would have been had there been no uncorrelated noise. Thus the autocovariance with spíke removed is díffertnt bo that which vrould have been obtaíned had there been no uncorrelated noise' (iv) Experimental investigations suggest that the methods developed in this chapter give reasonable estimat es of the uncorrelated noíse po\,fers (correlated noise such as interference from another transmítter is another problem), provided adequate rejection criteria are used (rejection of saturating da1a, or data recorded with the gains too 1ow, and rejectíon of data with fading tímes less than about 1.5-2s), and provided a reasonable degree of ave-raging is used' 323. CHAPTER VII OBSERVAT IONS USING COMPLEX DATA 7.L Introduction 7.2 Some simPle observations 7.2.L Short duratíon data (I-2 minuÈes) 7.2.2 Interpretation of short term daËa 7.3 Interpretatíon of íonospherlc scatter data 7.3.1 SimulaÈion of beam broadening 7.3.2 More accurate est.ímaLes of expected specl-ral wídths 7 .3.3 Two unequal spectral peaks, and their effect on the auto-correlations. (OR the phase of the sum of Ewo sine waves beating together) 7.4 Experímental data 7 .4.L Doppler winds 7 :4.2 Spectral r¿idths using the verÈical beam 7 .4.3 Spectral widths usíng the tilted beam 7 .4.4 Theoretícal spectra for anisotropic turbulence 7 .4.s Interpretatíon of spectral wídths using tilted beam 7 .4.6 The relative roles of gravity r¡raves and turbulence in the mesosphere 7.5 Deconvolution 7 .6 Conclusions. 324 CHAPTER VII Obs ervations usíng Complex Data 7.1 Introduction In Chapter IV, it was pointed out that although amplitude-only data was quíte adequate for observing atmospheric mot.ions on scales of minutes and morercomplex data was desj-rable for exami-nation of shorter term Processes. The ease of dealing with complex data has also been discussed. This chapter will concentraÈe on the results of observations made using such complex data. By recording ín such a way, all the ínformaÈion contained l-n the received signal is utilized. The chapter begins with some typícal observations. Some features of this data may appear to be a little strange, and so some computer símulatíons will be made Èo help clarify ínterpreÈatíon of this informatíon. The discussion of these símulaÈíons is rather lengthy, but it is felt that its inclusion is essential. Follorving this, some observatíons are discussed. The discussion wíIl centre almost exclusively on observations made on day 8OlO72, sínce that day sho¡,¡s most of the main features Lo be discussed. Interpretation of thís data vtill be shown to lead to estimates of parameters such as the turbulent energy depositíon rate and eddy diffusion coefficient. It will also be shown that Doppler measurements of r^¡ind velocíties and estimates by Partial Reflectíon Drífts, are in excellenL agreement. Some space wíll also be devoted to a discussion of the relative roles of turbulence and gravitY waves. Finally, a bríef discussion on the possíbílities and meríts of deconvolution will be undertaken. IN E rilo l URE HF -J o_ 0.0 < o 1 2 3 4 5 6 7 B 910 TIME (MINS) Fig 7.1 Example of typical in-phase and quadrature components received in an ionospheric D-region HF scatter experiment frour heights above BO km. The Ëime series are actually formed by a computer, and repiesent the data series corresponding to a Gaussian power spectrum of f,"ff-porãr-ha1f-wídth of .O7 Hz, with random phase (see text). However, the essential features are the same as those for real scaËÈer data, and so these are adequaËe for illustrative purposes. The half-width of the corresponding auto-correlatÍon function is 3.1 seconds' B.P NARROIA BEAM O" C MODE 80/072/(138-9) 1.98 VIHZ g4Krvl (Ot) -Qhs LDJ zË's LLJ È ô-"srs\s I t l*-, 0.0 -ß6 0 A 1,. t4¡l f"l,^, ø--- \ \ \ \ \ 05 Ø -TT, Co., 2 40 L6 J¿rmz 4, ) Fie 7_.2 (a) Por¿er Spectrum, and (b) Complex auto- correlation functíon, for a typical 1 minute series of data obtained at Buckland Park for heights above 80 km, usíng a vertícal narrow beam. Notice the existence of a few quite narro\,ü spectral peaks (much less than Ëhe expected beam broadening - see text), and the corresponding oscillations in the auto-correlation. The spectral peaks are not reliable and neither is aO.5(".. Èext) . 325. 7.2 Some simple observations 7.2.1 Short duration data ( 1-2 mínutes) Fig 7.1 shows a typical 10 mínute series of data which could be produced by íonospheric scatter. The data was actually produced by a computer simulatíon (see Sect.íon 7.3) , but is quiLe simílar to real data ín form. Thís has been verifíed by visual inspecEion. Thís data has a complex fadíng time of about 3 seconds. Experimental data fades more slowly at heighEs below 80 krn, and of course the overall mean amplitude varies more. For example, the existence of Latger bursts of scatter lasting 2-3 minutes from below 80 kn has already been díscussed, HotTever, the diagram is adequate for Íllustrative pulposes. Figs 7.2a and 7.2b show a quite typícal po\^rer spectrum and the corresponding auto-correlaËion function for a recording of 1 minute of acÈual data. NoÈe the exístence of quite narror¡r spectral peaks in the pohrer spectrum. These might be interpreted as being due to a few specular scatterers, but are in fact misleading, and r¿ill be considered shortly. In Chapter IV, fading tímes (tr) using amplitude-only data were presented. Wíth Èhe ínstallation of equípment to record complex data, ít is only natural that fading times be taken from the magniËude of the complex autocorrelatíon. These wíl1 be denoted c, (or ao.s). How do and ,, '% measurements compare ? Fig 7.3 provides part of the anshTer to thís questÍon. It shows calculations of C, andt, using the same data, ploÈted on a scatter plot. It can be seen that in general larger ar" values imply larger C, values. and více versa. However, there ís quite a scatter of points. 4 It can also be seen that ,, is generally larger Lhan ,h, which is not surprising ín view of the discussion in Chapter IV, section 4.3.lb. For TSVLLE. O M ODE. NTDE BEAM 'ø,{¿ 2/(s35-1820) BBKM C 92 KM cy;s) 10 I lo ?t, t0 / ¡l I ,t t/ / / 't. ,,, I , cÁà 0 A4 KM 7BKYl 20 20 // C,, / t0 / t0 / / / I / t/I I (s) 0 Crf, 7óKM ç,tn@ ó8KM 20 20 / / / i / / ,r. / / .i to lo fs) (s) 0 o 7tz 326 turbulenÈ scaÈter, it ís Eo be expected that fr ,r, (by equations "rr= 2.2.3.11 and 2.2.3,12, Chapter II; thís ígnores beam broadeníng, but in facÈ Cr- .,D t, is also expected if bearn broadening acts, too). Thís qrould appear to be the case at BB km, and perhaps 84 km. At 92 krn, the echo is largely due to a low sporadic E-type structure. In this case it appears Cr- 2 TU. The reflection was quite specular. Processes such as those discussed for specular scatterers in Chapter IV, Section 4.3.L may be lmportant. At heíghts below B0 krn Ëhe relation is possibly more 1:1. However, it ís difficult to be sure, as there is considerable scatter. This is parÈly due Èo Èhe ilbursÈinessrr of these echoe-s. On thís day the amplítude of echoes could'burst up and die wíthin 20 seconds or less aÈ times, and Ëhís process largely defined the fading tímes. Fig 7.4 shows typical height profiles of ,r" and ,% for day BO/072. Again, ít is clear that ,r, is larger than TL4. The 1:1 relation at heights below B0 km is not always apparent. It occurs only for rapidly bursting short-lived echoes. For steadíer echoes, iË ís often true that ,r" is considerably larger than ,r, (often C%- 2 Tr). Thís is shown quite well Ín Fíg 7.4 where ít can be seen that at heights below BO km, is consíderably larger than TZ. This "ta situatíon is more typical at Buckland Park, and for those more stable Tol¡nsvil-le echoes. However, the relation between ,% and ,r, is generally approximaËely monotonic, which is useful. In the regíon 85-92 krn, CU- some turbulent scatter may be ímportant. This supports t/-Í. rr_¡'4- so type of prevíous evidence pointing to an important Èurbulence contríbutíon from above BO krn. (e.g. see Chapters IV and V). o. BP NARRON BEAM. O o MODE. BO/A72/ (1130-1300) 19BV|HZ. ,f t- ,J # __ Ur^ I cy?. È-. gB0 -Þ- 5 I-Ll \t 70 Frr, Z *tr t ó0 1 2 FADINC TIME ( seca ) FLg 7..L Typícal profiles of ,h and ,r" for the vertical narror^r beam at Buckland Park usíng l-minuLe data blocks and averaging. (Cr_ and TL are lesi for an 11.6o tilt of the beam). rn. -" d^E^ is-2 for Bo/072 (March, 1980). The fading times above 92 l 7.2.2 Inter etat.lon of short-term data The magnitude of the autocorrelation function for the full 10 mlns of data in FÍ-gure 7.1 has a Gaussian form, wiËh a co.s value of 3.1 seconds. This means that the power spectrum ls also Gaussian, with a half-power-half-wldth of -l '' (7 .2 .2.1) î = Q,nL/r¡) ^' 0 . 22lcr2 ,4 "r, Thus in this case r ,4 should equal o.o7 Hz. Yet the indlvidual po\¡Ier spectral peaks in Fig 7.2a are much narrower than thÍs. In facÈ, several peaks occur withín a frequency band of ,1-4 Hz. Ihe question arises then, as to how these narro\^r peaks are to be 1nËerpreted? Further, how are they related to the form of the autocorrelaÈion ? 328. 7.3 InËerÞretaEion of íonospheríc scatter data 7 .3.L Simulation of beam broadenín Imagine a horizontal wÍnd, of speed V, blowíng horízontally over an aerial array at D-region heights. Then a range of Doppler shifted frequencies wíl| be received from scatterers moving with this wínd, sínce scatter comes from a range of angles. (beam broadening) . The range of frequencies will be defined by the combined polar diagram effects of the axy.ay and the scatterers. If the scatterers are isotropíc, this po1-ar díagrarn will be essentíally that of the array. The resultant power sPectrum w111 peak at 0 Hz, and fal-l a\^ray on either síde. Let us assume ít is Gaussían ín form. The half-power-half-vrídth of the povrer spectrum \À7í11- be approximately' (7.3.1.1) f r4 = + 'r, (see Chapterlll, section 3.2e) where v,u is the radial velocíty"2 observed at an angle 0y from the zenith, Ura beíng the angle aÈ which the effective polar diagram falls to half power. For isotropíc scaÈter moniÈored with the narrov¡ beam at Buckland Park, 0r, - 4.5" (Chapter III) . Thus -2 (7.3.L.2) t'4 Vsin0 ,4 À will be given shortly, but for now Ëhís A more accurate value for î % equation ís adequate. As an example, if À = 151.5m, V = 75 DS-l , and or= 4'5" then r x .08 llz. ' '4 Thus a beam broadened spectrum produced at Buckland Park should have such a widEh. BuË as discussed ín the last section, l minute po\der spectra do not produce such forms. Do the narrow peaks perhaps refer to discrete scatterers ? 329. To investigate thís, the following computer simulation r¿as performed. A Gaussian power spectrum was formed, with a half-power-half- width of .O7 Hz. Phases r^rere assigned randomly to each point of the povrer spectrum. These por¡rers and phases hrere used to form the complex spectruru, and this was then Fourier transformed to produce a time seríes of complex data. The range of frequencíes, and frequency spacing of successive points of the porÁrer spectrum were chosen so that thís final time seríes \¡ras a sj-mulation of a l0 minute record of data recorded at Buckland Park, wíth a data acquisitíon rate of 1 point every 0.8 seconds. Eig 7.1 shows this fína1 time seríes. Having obtained this time seríes, pov/er spectra were then formed, using varíous lengths of this data. One minute povrer spectra did índeed look like Ei-g 7.2a. Looking at Fig 7.1, it is easy to see why. One minute of data only contains a few cycles of oscíllation. Often, "regular oscillatíons" can be seen (with periods - BCZ, i.e. - 2O-3O seconds) maíntaíning themselves over 2 ox 3 cycles. A Fourier spectrum of such short lengths of data will ínevítably only pick a few frequencies. Bearing in mínd that this (1) the existence of a t'::egular oscillation'r maintaining Ítself over 2 or even 3 cycles ís not immedíate evídence of a genuine oscillation. We have seen here that such oscíllations can occur from random data. (2) Power spectra of one mínute samples are inadequate for describing the statistics of this situation. Fie 7.5 RunnÍng 1 minut.e por^rer spectra formed on the data in FÍg 7.1. See text for details. For elarity each pohrer specÈrum has been shÍfted vertlcally from the previous one. The times indieated are the sÈart times of each I minut,e block. Each successive spectrum corresponds to a L2s shíft 1n Èhe data block used. A 3 point smooth (4, '4, tò has been applied to Ëhe spectra âs well. I I I I I I I i i I I : ú)z H 4 LLI : FH t- E rt- Õ 1 -01 0. .07 5 FREAUENCY ft12) Flg 7.6a Power spectrum produced from the first 8 minutes of the daËa shown in Fig 7.1. The points are for the actual direct porrrer spectrum, with no suroothing. Notice also the presence of a large spike (indíeated by rrarrowtt) . Fíe. 7 .6b Magnitude and phase of the autocorrelation of the fÍrst B urinutes of data in Fig 7.J-. a \ a (ol a a aa a I a ta a ' a a a ? t ,01 a a a a a aa a o o o aa !t aa. a aa a aa aa rl la aa ta a aa aa aa ra aa a a a ta O o a ! aa a a l¡oll a a t . I I Ot ! a o ao a a o t la f a a oa a a aa I a, aa a a o al t a aa aa I a !a aa a aa a ,a ' a at t a a a I lt at a I a aa. o r tto¡ t t t tt a a a l o ¡ o a aa a t aala aaao.at a aoa a, a t a aa aa O a aara a a a a 1 o (r) rT, 0 ø f ø 0 0 10 20 0 Jrirrru Ð ) Fig 7.6c This power spectrum is simply Fí-g 7.6a, but with : frequencies averaged in blocks. The averages have been presented j-n a histogram form. A running mean may perhaps have been better, but this ís adequate. A smooth Gaussian-shaped curve has been fitted Èhrough the spectrum by eye, and the half-power-ha1f-width ís measured as .065 Hz. rlg 7.6d This ís also an average of Fig 7.6a, but points with values great,er than .OL4 have not been used. This procedure is used to reject very large spectral peaks in the ra¡.r spectrum which may bias the averages (see text). Notice ín all these diagrams (7.6 a-d), the mean \^ras fírst removed before performing the po\^/er spectra. This is why the averaged power specÈra have dips near O Hz. (c) Po*tn Ð4 .004, -1 0 I Frequen.yHâ Poupr Ðrt*rç @t C 1 1 ¡requency (Hz) 330 Fíg 7.5 shows the inadequacy of 1 minute power spectra quiÈe well. It is a running l minute po\¡/er spectrum. That is, Èhe po$Ier speetrum of the first minute was formed. Then, the power spectrum of the data block from 12s to 72s was formed. Then, the power spectrum of the data block from 24s Eo B4s was formed, and so on' shífting the data block by 12s on each occasion. Each of these po\^rer spectra \^ras smoothed by a sirnple 3 poínt runníng mean, and then plotted on Fig 7.5, one "behind" the other, wi'th a successive upward 'offseË. All Èhe spectra look líke Fig 7.2a, and it is quite clear the narrow peaks in Fig 7.2a cannot be ínterpreted as discrete scatterers wlthín the bcam. They have no phyei,rli meanÍng, aparL from the fact Ëhat they are the po\¡¡er spectra of the short data seríes. Notice in Fig 7'5 Èhat successive spectra do bear some resemblancerbut thís is not surprísíng consÍdering consecutíve spectra have 4Bs of data in common. A similariËy between spectra 5 shÍfts apart would be surprísing however, and does not appear Ëo occur. If it did occur in some real data, it could indicate a discrete scatterer producíng a narro\^t spectral peak' hlhat íf longer lengths of daÈa are used to forrn the Por^/er spectra ? Fj:g 7.6a shows the power sPectrum for the first 8 minutes of Fíg 7.1' A smooth Gaussian-type envelope is evident, even though each point shows large scarter. The half-width of thís envelope is roughly .I4 Hz. Fig 7'6b of shows the corresponding autocorrelation function. To remove the scatter of 20 points, poÍnts ín Fíg 7 .6a, the power sPectrum \^7as averaged in blocks and Fig 7.6c shor¿s the result. Thís is begínning to look like the original than poT^rer specËrum from which the data was derived. It is a little narrower be Èhe oríginal. The reason for this can be seen in Fig 7 .6a, where it will ttraw noted there are some values in this power spectrumtt, close to zeto Hz' sígnificantly larger than the rest. These are due to statistical fluctuaÈions' If they are noË used ín the averaging process, Fig 7.6d result, rvhich has a half-widLh close to .O75 Hz. It may seem to unfairly bias Èhe data by Fis 7.7a Dírect por^rer specÈrum of the first 5 mÍnuÈes of the daËa shown in FÍg 7.1. Fj's 7.7b Fig 7.7a, wÍth spectral points averaged in bl-ocks of 20. The smooth curve shows an approximately Gaussían curve fiÈted by eye to the data. In a real experíment, a least squares fitting procedure would be desirable, but 1n this case the diagrams are principally í1lustr:ative. Notice also the large "dpike'r in this spectrum. This arises from purely random effects. NoÈíce also that Ehe columns of this histogram are wider than those ín Fig 7.6. This ís sirnply because there was only 5 mínutes of data, so the frequency spacing between successíve frequencies j-s larger 1n this case. v 0 ;- q.*,*G wcl ( t) .¡ V 0 v It a ta a I , aa¡tta a aa | a ,at a ' a ' t a.a la a a a aa aaa a a a a a a a a t aa a I a t a aa ? a a a aa a a a 'a 2o' a a (o) 4.,*G*cJ 331. lgnoríng these large "spikes" near o Hz, but this is not really so. The ,'average'r is not really a good sÈatistic; the Presence of one abnormally large value can unfairly bias Lhe average. Thus in averaging these blocks àt ,O frequencies, it is ín fact faírer to leave abnormally large values out. The dífficulty líes in deciding what consÈítutes "abnormally large". However, the important point is that wíth adequate data lengths and sufficient care, the original power spectrum can be reproduced' (fhe poürer spec¡rum of the full 10 minutes reproduces the oríginal po\ÀIer spectrum exactly of course.) The process outlined above üras repeated using 5 minutes of data. It was again no""tUt" to approxímately reproduce Ëhe original power spectrum, and this is shown in Figs 7.7a andb. Notice in thís case that the large ttsmoothedrr spikes near O Hz are quíte severe, and even show ín the po\¡7er spectrum. It was found Èhat for data lengths greater than about 3 ot 4 rninutes, it was possible to reproduce Êhe oríginal po\ÁIer spectrum. Thus, it appears that a data length of (7.3.1.3) T I (90 -à 120) "r. is necessary to form a useful Povler spectrum. This is an ímportant resulL (tt also has repercussíons in other fíelds, too - for example, the analysis of winds. The width of the por¡rer spectrum of gravíty-\¡/ave - produced winds is less than about 0.2 min-t ( - the minimum period of oscíllation is about 5 rnin above 80 krn) , so the "fadíng time" would be greater than .22/ .2 - 1 rnin (assurning a roughly Gaussian po!üer spectrum) . Thus, at least 100 míns of data is necessary Ëo form anyEhing líke a reliable wínd spectrum' Further, only those frequencíes which can fit at least about 100/8 (i.e. - L2) cycles into the data length have reliable power densíties') 332. It will be noticed that only one autocorrelation function is shown in Fígs 7.6. Yet autocorrelations could'be formed as the Fouríer t.ransforms of both Figs 7.6c and 7.6d. However, these do not differ much from Fig 7.6b. the maÍn difference is an attenuation in the oscÍllatíons at large lags. This ís not surprising. Averaging the pornler spectra in blocks is equivalent to doing a running mean on them, with a box function of length 20 frequency spacings. This is also a convolution, as the box function is symmetric. Thus ín the Fourier transform space, the origínal Fouríer transform is simply multiplíed by the Fourier transform of this box function. SÍnce the box functíon ís quite narrorlr, itrs Fourier transform is wide, and so the original .autocorrelatíon is not greatly modifíed by it. AnoÈher \^ray to look at it ís that averagíng removes the rapíd fluctuations ín the power spectrum, and Èhese rapid fluctuations correspond to the larger lags of the autocorrelatíon. It may seem better Ëo use the fading times as estimated from the autocorrelaËions Èo estimate the spectral wídth. However, investigatíons of this showed simílar diffículties Èo those using Èhe power spectra. One minute autocorrelations frequently showed oscillatory character, reflecÈíng the dominance of only a few spectral peaks ín the associaËed power spectrum. (Also see Awe, L964). rft ta o.tt c4 / (t;ne- log) As discussed ín Chapter IV, the rate of decay of the envelope is related to the width of the índividual spectral peaks. However, ín this case the wídth of the individual spectral peaks is defíned purely by the data length, and has no meaning. The indivíduaI Crn values are not really 333. meaningful, since they simply reflect the frequency differences bet¡¡een the -2 to 4 peaks of the poÌÀ7er specÈrum. However, the 1 mínuËe spectral peaks do generally lie \rithin the expected spectrum, and averages of many 1 minute por¡rer spectra generally reproduce the original spect.rum. In a símílar way, a group of estimates should carry ínformation abouË Èhe original "r" spectrum and autocorrelation function. Various procedures hrere trÍed, based on simplistic assumptions. For example, one resultíng procedure was to find average of all the ,ra = ( >(Cr, í)-t )-t. Surprisingl-y, the simple ^u. 'r, values seemed Èo more nearly reproduce the expected value (3.fs) for "r, thc whole sample. However, thís was not testerl í,n a great deal of detaíl . Five mínute po\,üer spectra showed slow oscillations upon the general expected form, as illustrated below. t.o bserred tPl expecled ? flit',e- tog)' This tended to make estimation of ,ra diffícult. These oscillations are due Èo the presence of ttspikestt in the power spectrum, as shown in, saY, E1-.g 7.7a. It is easíer to remove these effects from the po\^ler spectrum than from the autocorrelatíon function. The general results of this sectíon would, thenrappear to be thaË. íÈ is possible to simulate the tttrue" po\,üer specÈrum of an ionospherically scattered radíowave data sample, províded data lengths of at least L}O C, are used (if smaller lengths are used (e.g. I minute), individual spectra must be averaged together), and provided sufficíent smoothing ís applíed to the po\4rer spectrum. The error in estÍmation of spectral widths apPears to be 334. of the order of L07., or better Íf sufficÍent care is Èaken. These results are not just vaIíd for data produced by beam broadening either. They are valíd for any type of Gaussian po\^Ier specÈrum' and probably (in perhaps slightly modified form) for any PoI¡ler sPectrum r¡hich decays smoothly with increasing frequency' The possibility of spurious "spikes" in Ëhe power spectrum also exists. These should be removed before obtainíng Èhe true Pohrer specËrum. It ís not claímed thaË this work has revealed any propertíes of power spectra which are not r¿e1l lcnown to professional sÈatísticíans ' However, at tíme's it does appear that some of these points are noL ful1y apprecíated by practícal workers. 33s. 7.3.2 More accurate estimates of expected spectral widths In equatíon 7.3.L.2, it was assumed that the half-power-half-width of the spectrum produced by beam broadening was símply related to the half- power-half-width of the polar diagram. However, the approxímation used ís -not completely valid, although íÈ does provide a useful first orderestimate. A power spectrum of íonospheric scatter data is produced by several effects (i) beam broadening (í1) vertical motíons (ííi) turbulent fluctuations (1v) shear broadening (for off-verËical Èilted beams only). One objective of this chapter is to make estimates of the energy dissipation rates for turbulence. To do thís, the spectrr:m must be measured, and Èhen the other effects removed. Term (ív) wíll be díscussed later, and term (ii) will be essentially ignored; vertícal moÈions generally oscíllate with períods greaÈer than about 5 minutes, and sirnply produce a shifÈ in the peak of the spectrum from O Hz (provided the length of data used ís less than about t to '< of the shortest oscillation period). To effectively remove tern (i), the expected half-power-half-wídth of the specÈrum produced exclusively be beam broadening must be known. This section will produce thisrand some interesting effects will also be seen for the case of a wíde beamed receiver. The problem can be expressed as follows:- rrFind the power spectrum received (as a function of virtual range)wíth a coherenË radarrfor the case of some scatterers, with backscatter cross- section of o-(f), f beíng the vecËor to the scatterer, moving in a wind of speed v in a direction defíned by 00, given the radar polar diagramr!' 7 7 a -* *t -J æ Fig 7. Ba DiagrammaÈíc representation of first problem discussed ín Sectíor. 7.3.2. 7 +'ll€, uto¿, -1J{- \r n0dø e æ Fie 7.Bb Contours of constanÈ radial veloeity at heighÈ H 336. FÌ-g 7.8a illustrates the problem. If í, j and & are unit vectors ín the x, Y and z direcËíons, v = vcos ge i + vsin ørj +Ok, and the componenÈ of velocity observed fon scatter from a scatterer atris and equals (7 .3.2.t) vr v . r rwhere r ís a unit vecËor (sin 0 cos Q)í * (sin 0 sin 0)j+ (cos 0)k. Define a normalized radial veloeity by (7.3.2.2) V = vr/v . Then by (7 .3.2.L) , (7.3.2.3) V = cos Qg sin 0 cos $ f sin Sg sin 0 sin $' AfËer squaring this, and applyíng appropriate manipulation, it becomes clear that (7.3.2.4) V = sín 0. {'4 ( 1+ cos 2(00 - 0)) }h The square root is taken as positive if Q lies betl¡leen þo + r/2 and þg + 3Tf/2, and negative oÈherwíse. This is to say a velocity is considered positive if t, is towards the receiver' Thus equation (7.3.2.4) defines the surface of constant V values fn the e, þ space. In Fíg 7.8b, the surface producing V value-s beÈween V and V + dV is shown, for a given d0. 337. IÈ wíll be noticed that Ëhere are lirniting values of 0, . These are defined as the points where (2.s.2.4) becomes ó.'ml_n and 0max imaginary l-. e ' t< . cos-l (2 v" 1) (7 .3.2.s) ó'm]-n þo ! - maX rfv is positive' use ( Örir, + T ). max The surface shown ín Fig 7'Bb, gíven dV and dQt can be found as follows. using equation (7.3,2.4) rsquaring and differentiaÈíng' one obtains (7.3.2.6) 2 sín 0 cos 0 d 0 (2vlrK)dv- 2(P lxt)sin 2(Qo-0)¿0 where X = r4G + cos 2(Qo-0)) Thus thís defines d0, given d0 and dV' The solid angle defined by d0 and dV is (7,3.2.7) df,) = sín 0d0d0 (fíg 7'Bb) (2vlx)dv. (dþ/(2 cos 0)) - 2N2lxz)sin 2(00-0)/ 2 (2 cos 0).(d0) by (7 .3.2.6) I¡Ienowseekthepo\¡TerscatteredfromsolídangledCI.Letthe backscatteríng cross-section at (r, 0, 0) per unít volume into unit steradian be o(r, 0, o). Let the polar diagram for the radar be P(0' 0)' R, Then the pov/er received from solíd angle da and at a vírtual range ís proportíonal to (7.3.2.8) T(R, 0, 0)dA o(r, 0) (r2¿n) I e(r) ] P(0, 0). t roo, 338 Ilere, g(r). defínes the pulse transmítted ínto the atmosphere. It was seen in Appendix B (equation 8.26b) that the scattered sígnal was a convolution between the pulse and Èhe scattering function. The t-4 term arises because of an t2 fa1l off with distance from the transmítter, and a further t2 fall off after scatter. (lhese effects \'7ere noÈ considered ín Appendix B). The r2¿Q term aríses due to the scatÈering area. In thís formula, absorptíon has been ignored. A better forrnula wíll be deríved later in Chapter VIII, but for present purposes absorption can be ignored. Equations (7.3.2.7) and (7.3.2.8) can nohl be used to estimate the scattered povüer received in the velociÈy range V to V + dV by integrating (7.3.2.8) over all acceptable 0, 0. Thus, for any vírtual range R' a pohrer spectrum can be Produced. For compuÈer application, a layer of ísotropic scatterers, with base height H, and Ëhickness d << H' I^Ias considered. (the program used is shown in Appendix E ("Specpo1") .) H Let 2 (7.3.2.9) r(V, n) [0rr* P(0, 0) L olr O e(t) ldCI, ) 0-ö'm1n are defined by equation (7.3.2.5), 0 is given by (7.3'2.4), where 0mln max and V(dv )dó X=L¿ (1 *cos2(00-ô)) df-¿ Xcos0 where (ignoríng the second term ín (7.3.2.7), whích is a second order effect, as it involves (¿O)2). Here, !-jg 7.9a Expectecl l¡eam broadened po\,r'er spectra for Èhe vertical narro\^/ Buckland park beam aÈ 1.98 Mttz (transmitter polar diagram included) for a layer of isolropic scatterers extending from B0 to 82 km and of infinite horizontal exEent. The numbers on the graphs refer to the ranges (in km) of the data used to form the spectrum. The "normalized velocity" is equal to the radial velocity divided by the horizontal wind velocity. The power pulse used r^tas of the form (h/8.5 x n) between h = 1 4.25 km, and zero elsewhere. "o"4 The broken curves sirnply refer to ranges greater than the range of maximum pors/er. The po!üer units are rather arbitrary, but can be compared to Fig 7 .9b. The graph in the inset shows Èhe half-po\{er- half-widEh of the varÍous graphs. Notice this ís a function of the range. The reason for thÍs can be seen in the diagram (i) below (not to scale), where i-t can be seen that significant scatter comes from a wider range of angles at ranges RZ and R¡ than aÈ range Rt so larger beam broadening is to be expecLetl. (i) (t¡) A 3 ß¿ R3 R3 p, Tx Notice in the t'*3" case, Èhere is no scatter from directly overhead, so it might seem that no scatter wíth zero Doppler shÍft should occur. However, it should be borne ín rnind thaE fig (i) is a 2-dimensional slice through a 3-dimensional system, and ftg (11) lllustrates this. Scatter from the point B would in fact produce zero doppler shift so there would still be a sígnificanÈ zero frequency contribution to the spectrum. However, fig (i) does illustrate why the spectra are broader and the side lobes are larger at 83 and 84 km range in Fig 1.9a. This 3-dimensional nature should also be borne in mind when consideríng Figs 7.9b and c. 1 Pe idt t a TJ 83 Ronge tr I'tJ 0 C o- \ \ \ / \ / \ \ B I B I -1 0 tt u IIil u tt t¡ It I a3 \ t \ r 0 t s3 -20 -o.2 0J 0.2 0,3 NORMALIZED VELOCITY Fig 7.9b Expected 6s¿m-broadened power spectra for the narro!/ Buckland Park Uôãm-"t L.9B l{ùlz, tilted at 11.6' in the dírection of the wind, for a layer of fsotropic scaEterers extendÍng frorn 80 to 82 km and of ínfinite horízont,al extent. The numbers on the graphs refer to the ranges (in km) of the data used to form Èhe povrer spectrum. The pulse used was Ëhe same as Èhat for Fíg 7 "9a. The power units are rather arbitrary, but. can be compared directly to Fig 7.9a. The graph in the inset shows the half-power-half-r¿ídth of the spectra, and Èhe offsets of the peak, as a function of range. Notice after maxirnum scat,tered po\Àrer is reached, the half-width falls. The reason for this can be seen in the diagram - it can be seen the layer thíckness itself límits the effective r¿idth of the polar diagram. Notice also thaË only at the peak of power is Èhe offset equal Eo R¡ the expected normalized velociËy of - 0.2. This is thus a warning for Doppler measuremenEs of the - the spectrum used must rrespond to the range of maxímum scatÈered po\¡/er, or else erroneous numbers rvill result. (The "nornalized velocíËyr' ís equal to the radlal velocity divided by the horizont-al wínd. Radial velocities ar.ray from the receiver are taken as negaÈive). .o5 I ea o 0 n9"( 6 {0 e3 -20 19 8r -A ,3 a -1 0 I NORM ALTZED VELCCITY Flg 7.9c Expected beam broadened po\,üer spectra for an isotropíc transmitter - receiver system for a layer of isotropic scat.terers exËending from 80 to 82 krn and of infinite horízontal extent. The numbers on the graphs correspond to the ranges (in km) of the data used to form the spectra. The pulse used was the same as that ín Figs 7.9a, b. The po\¡ter units are arbitraty, and cannot be compared to Figs 7.9a, b. Notíce that at large ranges the spectrum splits. The reason for thís can be seen in the following díagram. Qange A similar effect acÈed in Fig 7.9a, but in that case the polar díagram was quíte narror^/ and double peaked spectra were thus not produced. Notice, however, that significant powers sti11 occur at zero frequency, and this is due Èo the 3-dimensional nature of the problem, as discussed in Fig 7.9a. co -a o o_ 4 8s 2 7A -4 ;3 --2 -1 0.1 .2 3 4 NORMAUZED VELOCITY 0o = oc þo = tt/zc n"(tn) ,U 0r(des) nr(tcur) ur" 0r(dee) 2l*Tz beam g1 4 pointing 81 .o72 4.L3" .072 .13" vertically 6 MHz beam 1.43' pointing 81 .o25 1.43' B1- .025 vertically 2 MHz beam pointing 11.6" 82.7 .062 3.55" 82 .7 .075 4 . 30" off vertical I Y, errors tyPically t .003 I ( Vr, dirnensionless) TABLE 7.I 2 km deep and effective 0r- values (to be used ín equatíon 7.3.L.2) Íor a v,'4 base height at B0 km, for a ranse t" maxímum lãyer- wíth fu ""11:-"9311t:9 reach the scaÈtered po\^rer (ttris is approximately the distå'nce required to midpoinr of the i"y"t in ti-tã direction of tilt of the polar diagram) ' The pulse i" defined in FiguresT.g- Various polar díagrams and transmitted "" of the wind directions Qo are given. uere, 0o can be taken as lhe angle from the planã of tilt of the diagram. (The verËical polar wínd vector most díagram is not quite ãy**"tti. in aximuth but is close enough for purposes.) particularly, notice the different u4 values aË 11.6o. This is not really unexpected, consideríng the different confíguration' 'tl c Qo=nzz- ø" o' t for = 0 the peak of Èhe pov/er spectrum has an offset of Notice 0o the peak' y = - O.2 whilst for þo = r/2, there is no offset of The abo.re diagra^s i lf usLr¿Le why this is so' 339 . olr 2 a e(r) r olt2 g (R-r) dr, and o = I wíthin Èhe layeï and zero outside' Then F(V, R) was calculated for various R values, and plotÈed as a function of V. This was done for varíous polar díagrams - (t) the narrow beam at Buckland Park looking vertically (2 and 6 lülz) (ii) Èhe narrow beam aÈ Buckland Park tilÈed to 11.6" off zeníÈh (2 and 6 lü12), and (iíi) an isotroPíc radar. Various layer thicknesses d, and various wínd direcÈíons Qo r \¡Iere used. Figs. 7.9 show typical spectra. The full polar diagram, as described in Chapter III' \^Iere used for these calculations' A large amount of informatíon is contained in these spectra. It ís not possíble to present all of this here, and the spectra have stíll not been fully exploited' As an example, ít can be seen in Fig 7.9b that the peak of the pol^7er spectra (whích líes change as the range changes. only aÈ the Ïange of peak po!¡er near the midpoint of the layer) ís the measured peak at Èhe correct V shift" (Thís is a warning - clearly Doppler wind measurements will be erroneous if thls range is not. carefully selected). If these spectral peaks are plotted as a function of range, a near-straíght líne ís produced. Thís slope is a function of the layer depth, and provides a possible !üay to measure the depÈhs of scatteríng layers at B0-95 km. However, the matËer ís more complícated if a wind shear exists wíthín the layer. Símilarly, the half- power-half-width of these spectra are a funcÈíon of range' The situation is certainly far more complicated than equation (7.3.L.2) suggesÈs. Table 7.1 shows ,ra values (half-power-half-r¿idth) and effective "a values which could be used to make Èhe approxímaÈion (7.3.1.2) valid, for various polar diagrams and wind directions ' 340. The spectral power spectrum half-width for any wind vecÈor can be found from tables líke 7.1 as (7.3.2.10) f ,4 = 2/^ { ( uru(r) ur)' + (ur.(r) ur)2 }" where the wind vector has been resolved ínto a component parallel to the direction of tilt of the beam, tl (with the corresponding Urr(t) beíng found in Table 7.L), and a component uZ perpendicular to thís tílt. This statement is not totally obvious, but can be proved as follows. Let the por^7er specÈrum due to each component be exp {- $/f á)2} and .*p {- Glfì }2. The resultant po\^7er spectrum fs the convolutl-on at these two, which is exp { 12/(r^2 + fb2) }. Thus, if the two orígínal spectra had half -¡^ridths of ft : f aW and half-width is (|L2 * t..22)\. Hence follows f z = fotÑ, the resultant (7.3.2.L0) (by usíng the equatíon ¡ =(2/À).v¡aa from Chapterlll, Section 3.2e). Notice for a verËícal beam, Ura(r) = V>rç2¡, ,o f Z/\ v, teot, ttoa being the total r¿ind vector. U = However, it musE be borne ín mind that Table 7.I is only applicable for the range of maxímum scattered poI^rer. The assumption that the resultant pohrer spectrum ís a convolution between the power spectra produced by the 2 orthogonal wind components is also an approximation. It reIíes on the assumption that all parallel cross- sections through the polar diagram are constant in form. That is, if the polar diagram, looking from above, 34L ô b 4, C here, then cross-secÈlons a and b musÈ be has contours as 111-ustrated same' Thls is not really Èhe same ín form, and c and d must also be the the síde l-obes true, particularly if the slde l-obes are cbnsldered' Ilowever, outlíned above are close make a negligible contríbuËíon, and the assumptions enough for most work. 342. 7.3.3 Two unequal spectral peaks, and their effect on the autocorrelation (On "the phase of the sum of two sínet/aves beatÍng together" ) In the preceding discussion, the relevance of narro!ü spectral peaks in the po\^rer spectrum has been consídered. It was found that if suffícíent data was not taken, these peaks may noË be meaningful. But suppose sufficient data is used, and two spectral peaks stíll result ? Fig 7.9c shows examples of such cases. Then r¿haÈ affect does this have on the autocorrelation ? In particul-ar, what effect do these peaks have on Èhe phase of Èhe auto- correlaÈion funcÈion ? IÈ is lmportanË to understand this problem, because often the rate of change of phase of the autocorrelation function aE zero 1ag ís used to deduce Doppler velocíÈíes (e.g. Woodman and Guillen, Ig74). If perhaps two specular scatterers contribut,e to the po\^rer spêctrum, then how is this rate of change of phase related to the respectíve Dopple-r velocities of the scatterers ? Rastogi and Bowhill (1976b) have also considered thís problem, and have proposed a very complex formula. This tends to hide the physícs of the problem, so the problern will be díscussed here from a píctoríal viewpoint. Pøwcç nsiÌy A frcqency f, fz Consider a po\^/er spectrum function like that shown by the broken line above. Thís spectrum is taken as the sum of the two Gaussian-like solíd curves. The Gaussian-like curves are assumed to have the same form. This spectrum may be considered as the convolution of two narro\r peaks at ft and fZ convolved with the Gaussian-líke curve, ï.e. 343 I B C Êowcn dcns ¡ry' Povtc? on sily. a t fz frrgrenc¡. ,/snc} The three por^rer spectra drawn above will be denoted A, B and C, as labelled. Then the auÈocorrelation function corresponding to fígure A (i.e. the Fourier transform of A) is the Fouríer transform of spectrum B multíplied by the Fourier transform of speetrum C. Thís follows from the convolutíon theorem of Fourier analysis (e.g. see Champeney, 1973, p. 73). The effect of the Fourier transform of spectrum C is simply to make the envelope of the Fouríer transform of spectrum B fall avüay more rapidly with increasing lag; it does not affect the phase. Thus the phase varíatíon of the required autocorrelatíon is the same as the phase variatlon of the Fouríer transform of specÈrum B. This considerably simplífíes the problem. Further, the funcËions defined by spectrum B are just two sinusoÍdal waves of frequencíes ft and f.Z. Thus the problem becomes just that of examiníng the amplitude and phase varíations of tr¡o sinusoidal waves, with angular frequencies ,l and ,2, beating together. The easíesE way to consider this problem is to consider two waves of equal frequency, but with one gradually changing in phase. i,e. cos ûJ t=cos (rrt where 0 (o ûJ )t. 2 - 0), 1 2 Let, these r¿aves be Arcos tr,lrt and Arcos(rrt - 0). The resultant is Arcos trlrÈ + Arcos(rft - 0) ed *. iff 5 .0 ec 18d Ø¿,çç Fie 7.10a Graphs of the resultant phase Qr." of the sum of two sine hraves (frequencies ,1, ,2 and amplitudes 41, A2) of different frequencies plotted as a function of the phase ãirtåt"""" Our* between the two sine waves; 0¿rrr( = (or-urr) t) . The results are ínÈuítívely reasonable ' The numbers on the graphs are AZ/ÃL values. For small LZ/^L values, 0r." - 0, and the ,1 wave dominates; for large A./AL values, 0r"" * O¿rtt and the 62 wave dominates' of the resulËant For A2 = 41, . 0r." = þAift/2. The amplítude maximizes aÈ OUrr, = O, and falls to a mínimum at O¿iff = 180" Fí 7 10b c (schernaÈic only) Graphs of the phase and amplitude of the sum of Ewo si-ne waves of different frequencies ploÈÈed as a function of time. Case (b) gives the phase with respect ro that, of one of the sÍne r^/aves (angular frequency o., ) and Case (c) gives Èhe phase with respecÈ to some third sine wave ¡ - angular frequency o*. Case (d) shows Èhe situation r¿hen Al = 42. Notíce in all these cases, 02 > 01 and both are greater than uJ-. This is not always necessary, of course, and in cases with ,1, t2 a t*, 0r." decreases with Eirne. The important Èhing is thaÈ the phase fluctuates around the phase variati-on which the frequency of largest amplítude would have. (The broken lines shor¿ that phase varlatlons of the pure sine curves). Phase changes occur mosÈ rlpidly at Èhe minima in arnplitude. In Ehe case (d), (41 = Ar), a 2T- ambiguity means the phase can be regarded as flucÈuating around elther the phase of tl or that of u2. círl (ütut s A2 Q,rt A,>A ê + R ¿e '".Pl -¡ ¡a,-A.l tÍ e 4"s ( rf)¿ Qr*, A A c t tdt A, Q*" tJ¿ 2 e ul r-6"" rC =(t1-t{aþ ú c 344 (7 . 3.3.1) Ao cos(ttt 0r.") (7.3.3.2> where .^t oru" = (4, sln o)/(et + A, cos Q) (7.3.'3.3) and A* 2 or'*2^L^zcosQ*or' functíon of for varíous Az/Ll values, If Ô'res _ is plotted as a Q curves resul-t like those in Fig 7.1-0a. Herer 0r." i" the phase of the resultant nornr r^7e use = (ot and plot 0r." as with respect to cos uJrÞ If 0 - ^ìË a functíon of t, and also, Ao as'a function of- tr' curves result, like those in FXg 7.10b. Two curves are plotted. Curve I resulËs if A1 u LZ, and Curve 2 if A2 t A1. The l-íne Qr"" = (tl, - tlr)t Ís the phase of cos 0J2l- with ao rrt. Thus Ëhe phase oscillates about the line of phase of the respect "o" frequency with dominant amplítude. Note also that the magniÈude of the phase fluctuation depends on the raÈío At:At. , Now, in a real experiment ín which two Doppler frequencies f L and receíver, they are usually converted down to a 1ow f Z are receíved.by a frequency by beatíng wíth a third frequency f*. In Èhe cases dealt with in this thesis, fik is just the transmitted frequency. Then, let us finally suppose that we consider the phase with respecÈ to a third frequency trt*' Then the amplitude and phase are as shown in Fig 7.10c' The lines (t'tt - t'^t*)t and (trl, - ur*)t represent the phase of the índivídual frequencies wlth respect to trl*. The phase of the resultant oscillaÈes around the phase line of the frequency of largesË àmplitude. I have drarnm the phase as 0 at t=0. For an autocorrelatíon functíon the phase ís always zeto at zero Lag. Consider Fig 7.10c as an auÈo- correlat.ion. Then the important poínt to note is that the slope of the phase related to or u2. It depends on the relaËíve at zero lag ís NOT símply '1 amplitude of the two. This is important, because, as discussed, in much 345 Doppler work, the slope of the phase of the autocorrelation at zeto is used to obÈaín the frequency shift, and hence the velocity (e.g. Woodman and Guillen, Lg74). Clearly, in the case presented here, erroneous results would be obtained. To get the correct frequencíes, fit a straight líne through Èhe phase fluctuations. This líne will give the frequency of the dominant amplitude, f say. The other frequency fb can Ëhen be found by ^ (f is using the fact that the beat frequency is a - f b). I^Ihether ft the resultant greater or less Èhan f a can easily be found by seeíng whether phase curve has a steeper or lesser slope than the phase of the domínant frequency at zero !ag. (tfrts descriptiorr also explains the maín features of the autocorrelation function in Fig 1 .2b). ' - . '-- Thd case of equal amplitudêé' A1 and A2 Ís i-nteresting. The phase varíes as sho\,/n ín Fig 7.10d. The phase can be made to appear to oscillate about either the (ur, - o*) t line or the (ur, - ur*) t line by utilizíng the Z¡c shift which occurs at Èhe poinË of rapid phase change. Another interesÈing case related to this theory concerns observations of atmospheric tides. OfÈen a minimum in arnplitude of tt'e 24 hour tíde can be observed at 90 km wíth a correspondíng large phase jump (e.g. see Elford and Roper, Ig6L; Stubbs and Vincent, L973; Elford and Craíg, 1980). This could easíly be explained as belng due to the fact thaÈ the tíde is the sum of 2 modes, wíth different vertícal wavelengths. The amplitude and phase would vary with height in a símí1ar manner to the variatíon in tlme sho¡,m in Fíg 7.10c. A r:apíd phase change would occur aË minima ín strength. However, the major purpose of this section has been to show the possible errors ín Doppler measurements which could resul.t when 2 scatterers exist with different Doppler shifts, íf the slope of the phase at 0 lag ís used "blindly" to esÈimate the Doppler velocity. Notice also, with regard to 346. Ftg 7.2b, that not only is there insufficient data, resulË1ng ln false peaks, buÈ also evaluatlon of do/at at T=0 .gíves a slope dependíng on all the frequency peaks 1n Fig 7.2a. Thus even íf the peaks were rea1, evaldaÈion of dQ/¿t would not help get the correct Doppler velocitíes. Fie 7.11 Plots of po\4rer and horizont.al wínds as a function of height for day 80/072. The mean por{rer profile (0 mode, vertical narrolrr beam) was obbained between 1133 and LL42 hours above 80 km, and between L22O and L229 hours below. The mean noise has been removed. Layers were presenÈ at heights of abour 66-68 km, 72-75 km, 78-80 krn and 86-88 km on this day. (The layer at - 74 km does noÈ show very clearly, so has been indicated by exaggerating the minimum at 76 krn (broken line). The layer could easily be seen r¡hen waEching the temporal variation of Èhe echo profile). The rise in echo poÌirer above 94 km is due to the E-region, and that below 62 km is due to a ground echo. With regards to the winds, the unlabelled error bars denote Partial Reflection Drift (PRD) measurements. Síngle points denote that only one acceptable measurement was obtained. E denotes eastward, N denotes northward. beam aÈ Buckland Park is tilted towards the it hlhen the narror^r "west", rrthe actually poínts 4o souËh of west. In this chapter, the statement beam was tilted 1l .6o to the wesE" means the beam I^Ias tilted at a zenith angle of 11.6o in the verÈi-cal plane 4" south of west. The tríangles in this díagrarn represent the r¡índ vector component (deduced from the partíal- reflection drífEs) parallel to this t1lþ direction of the narrow beam. The error bars with a square are Doppler measurements made with the bean tilÈed at 11 .6o to the "\./est". They should be compared to the triangles. The pRD measurements are the averages over the períod 1310 to 1340 hours. The Doppler measuremenLs were taken in the periods f146-1153, I233-L24I and I4L3-L42I hours. Al1 error bars denote sÈandard devíations for the mean. The broken línes show the best fit wind profiles (eye-fít only) using all data. Note that Doppler measurements may be under-esEimates below B0 km, as described ín the text, and inclicated by the inseË díagram. The wlnd profíle is probably dominated by tidal wínds above B0 km. BP ao/072. o PR D # L .-r.._._ i4, 90 ì* \ N \ ef¿ \ v t(+t r-'J use$ul PRD a nd fô e0 I t-- \ I ! , r fcr F& ,E Effec{io" Btq" d iogrqrn lH I 70 + ioI ram L / o! Pourer. o ht diogrort Gø of pÞ; ) o. oÇ scoltererS -50 0 50 NIND SPEED('''s') 347 . 7.4 Experimental DaÈa 7 .4.L Doppler l^linds As discussed in chapLer I, there has been some debate over recent years as to the validity of Partial Reflection Drift measurements' Briggs (1980/81) has shovm thaÈ Doppler measurements and PRD measurements effectively measure the same thíng, and comparisons with other techniques (rockets, meLeors - see Chapter I) suggest PRD measuremenEs do measure the neutral wind. Fig 7.11 shows further support for these statements' It Park' shows PRD and Dopplcr wind measurements macle on day 80/072 at Buckland Doppler wlnd vatrues are the means of l0 mín sets of data' and r¡ere calculated from the phase of Ëhe autocorrelation function, after consideration of the effects discussed in Sectíon 7.3.3. Agreement would apPear to be good at heights lrhere both techniques produced measuremenÈs ' In fact' the were measuremenËs complemenÈ each other. Below 76 km, few PRD measurements obÈaíned (this is rather unusual, however) and Doppler measurements give an indicatíon of winds here. Above 92 km, Doppler measurements gave velocíties of the order of 150-200 *"-t . It turned out thís was due to leakage from the strong 86-88 km layer through a side lobe of the polar díagram at about 30o from the zenith. lühen this was considered, these Doppler measurements thus gave further esËimates of the wínds at 86-88 km, and agreed well wíth those presenÈed on Fig 7.LL. (These latter wind measurements have not been plotted). Thus Doppler measurements could noÈ gíve wínds above 92 km' whereas PRD measurements díd. IÈ is also worth commentíng on the accuracy of Doppler measurements belor¿ 76 km. These are very likely under-estimaÈes. Thís is because the scatter from these heíghts is highly aspect sensitl-ve (e.g. see chapter Ïv; also see later in this chapter). Thus, as shovm in the ínset of Fig 7'11 348. the effective polar díagram is not tilted at the same angle as the beam. In all Doppler calculatlons, ít 1s normaL to assume thaÈ scatter is strongest from the dLrection in which the bearn is tílted. If this 1s not true, Doppler estl-mates will be fn error. PRD measurements do not suffer from thfs problem. Thís poi-nt has also been discussed by nõutger (1-980) and VincenË and nottger (1980). Fte 7.L2 Running porrrer spectra, with a 3 polnt running rnean ( \, h, \) , at 12s steps, for d,ay 8O/072 at 86 km range uslng the narrow beam polnting vertically. Positlve frequencies mean a Doppler component towards the receiver. Compare this Èo Fig 7.5. There appears llttle evidence of any one spectral peak maintaining Ítself for several mlnutes. SpecÈral peaks do ofÈen appear to perslst for up to a minute; but thls is not. surprising, since specLra with less Èhan a I mlnute separation have common data. This point r^ras al-so discussed ín Seetlon 7.3.1. o a 8M a a 6 1139 113 113 113 r.rl 11 35 Hl- t-- 1134 É # J 11 33 a1 -.05 "05 ,1 FRECI.UENCY (HZ) 349. beam 7 .4.2 Spec tral vrid ths using the vertical ondayso/oT2therewereprincipalscatteringlayersaËheights data near 65 km, 74 km, 78 km and 86-88 km. Thus most ínvestigations of has already using the vertical beam r^rill be done at these ranges, since it been seen that incorrect choice of the range can bías Èhe results. Fíg 7.12 shows a running polfer spectrum (1 mínute blocks) for day runníng mean' There 80 072. Each spectrum has been smoothed with a 3-poínt over more does not appear to be any evidence of one peak being sustained (as discussed in than about 5 specEra, suggesting little specular scatter minute can be the capÈion, some Persistency over time shifLs of up to 1 expected) . -Ehe term po\^Ier spectrumrs Hence the only important contributions to long pol^Ier spectrum wldth are beam broadening and turbulence. The beam broadened can be assumed to be 2 O¿ Ln 26) / (2/N.v. v )2 (7 .4.2.L) B(f ) "xp { - \ Where v is the horizontal wind velocity' This follows because the (2/\\str. half-width in Ëerms of radial velocities ís tr = vV,4 and f = u The turbulence spectrum can be wrítÈen as f 2 X7))" v*r, )' (7 .4 .2.2) T(J) cx e I to since it can be assumed the component of velocities of eddies due turbulence along the line'of sight is distríbuted as ,f Tv- e RM3 with v is the RMS velocity along the líne of síghÈ assocíated Here Rt\/s Ff e 7.13 (a) Nlne minute power specËrum (after averagíng in frequency blocks) for the vertical narro\âI beam at Buckland Park, aË 1-.98 MIIz, for a range of 88 km, on day 8O/072 (Lzth March, 1980), taken beËween 1133 hours and 1142 hours. Positive frequencies irrply movement towardg the receiver. Al-so shown ís an approximate GaussÍan fft to the spectrum (fitted þy eye only). (b) The corresponding autocorrelation funct,Íon for Fig 7.13(a). NARRCN BEAM:O": C M ODE: 88 KM. ) Power Densiþ 80/072/ffis3-1142) fç.or -1 0 I F.uqr.n.yû-1J c Ø 0" 0.5 -Tt" 0.0 0 10 20 30 TIME SHIFT(s) 350. scales xlz metïes, and it has been assumed that the turbulence ís isotropic, so the situatíon looks the same ín all dírecti-ons. If this were not so, the turbulence spectrum would vary dependíng on the angle aÈ which the turbulence was viewed. If both turbulence and beam-broadening are active' Ëhe resultant por.rrer specËrum is a convolutíon of B and T (ttris is not proved here; however ít seems intuitívely reasonable and r^rí11 be proved to be true laÈer (see equation (7 .4.4.14)) and can be written as (7 .4.2.3) P(f ) exp{- (v v4)2 /9,n 2 + 2v**r2 )-r ], so the half-power-ha1f-widÈh is ,4 2 (7 *2v Ln2l .4.2.4) f t- 2/^lvvr.)2 R tvîS Consequently if f can be measured, v can be found as a RIUS' ,4 ,'r4)2 (7.4.2.s) v i Q/2 r (v I lQ tn D \ RNIS t %)'z Fíg 7.13 sho¡¿s the smoothed power spectrum of 9 minutes of data for day 80/072 usíng the narrow beam at Buckland Park at 1.98 MHz. (The raw power spectrum looked símilar to Fíg 7.6a). The half-power-ha1f-widÈh ís about .o75 Hz. Now, Y4= .072, so íf the mean wind is taken as that at 86 km, (- ls *"-t), then u Yra: 5'4 ms-l ' rf the mean wind is that at 87 krn' u yr"= 4.3 rns-I . These provide upper and lower limíts to u Ur". Thus the expected half-power-ha1f-width due Èo beam broadening alone is 2/x (5.4 + 4.3 ms-l¡ = .071 + .057 Hz. This is comparable to the and suggests there is very little turbulent contributíon. observed f '4r, There must clearly be some error j-n the estímate of f ,4- perhaps ! L07". In fact, the 86 km pov/er spectrum had a half-wídth close to ,055 + .06 H'2. However, an upper limit could be placed on the width, and hence on v r rj-e 7.L4 Running povüer spectra wíth a 3-poinÈ running mean (N, lr, ,<) at 12s sËeps, for day BO/012 at 74 km range using the narrow beam poinÈíng vertically. Positive frequencíes mean a Doppler component Èowards the receíver. BÙO72 BP d 1.98MH 74 KM. 122 1226 122 1 224 Lrj H F- l- 1 22 E. b J 1 222 \ O I 9 1 221 1220 -.05 0 05 FREGUENCYlF,Z) 351. and lt hras found thaÈ v 2-3 ms-r (for scales of 75 metres). RlvÉ, S (It should perhaps also be poinÈed out ín connectíon with these Po!üer spectra thaÈ a noise level also exísts (e.g. see chapter vI). This should be removed before any fítting procedures are applied, although in most cases it was a very small effect at 2 l*Iz. AÈ 6 lúlz, noíse became more important) ' 'Notice also the narro\¡/ spikes which occur ín the Po\^Ier spectrurn ín Fig 7.13. To some exÈent Ëhese are to be expeqted, due to staÈistícal fluctuation, as poinÈed out in Section 7.3.L. However, Rättger (1980) has al-so observed such spíkes ín some of his poI^Ier sPecÈra from trospospheric and sËratospheric data, and inÈerPrets some of the very large ones as being due to specular scatter. Such specular scatterers would produce a very narro\^r effective polar diagram, and thus a narror¡¡ width to the spectral lines they produce. The spíkes ín Fíg 7.L3a are probably not due to specular scatter, particularly in viernr of the runníng poI^7er spectra' which did not shovl any susËained frequency. However, some other large spikes did occur at tímes whích could indicate specular scatter. However, these wí1l not be discussed in detail, sínce their interpretation is not certain. (fhe possíbility also exísts that slowly varying specular scatter could enhance the powers at low frequencies, artifícially narrowing the specÈrum. This shoul-d be borne in mind. Determination of Rice parameters can help determine íf this is happeníng). ríg 7.14 shows a running power sPecÈrum for day Bo/072 at 74 km range, 2 lúlz, 0o with the narro\r beam aE Buckland Park. In thís case' there may well be frequency trends whích persist for a long time. The two arrows ín as E1g 7.14 show Èhe movement of the spectral peaks and could be ínterpreted BUCKLAND PARK NARRON BEAM,O": O MODE z 74KM BO/072 D I OWer D"uily I 0 I Fre P 1.0 05 00 1 20 30 40 TIME SHIFT (s) Fíg 7.15 (a) Níne minute po\^Ier spectrum f or the vertical ;ãr", beam at Buckland Park, at l.9B MIIz, for a range of 74 km, on day Bo/072, Èaken between :*220-1229 hours. No averaging of the spectrum in frequency bands has been applied' positive frequencíes ímply movement Èowards the receíver. (b) The coriesponding autocorrelation function for Fig 7.L5a. 352. indÍcative of scatterers moving into the beam, moving overhead, and then a\^ray. Note Ëhat the origínal power spectra had the zero frequency po\'¡er set to zeTo. The non-zero values result due to Èhe smoothíng applied, but are not representative of the true zero frequency por¡/ers. Thís is part of the reason for the apparent "doubly peaked" spectra at about L223-I224. Measurements of the half-power-half-wídth of 5-10 min power spectra from rangès below BO krn r¿ere also obtaíned. The widths were Quirte símilar Èo those expected due to beam broadening - perhaps even a little less, as might be expected due to the aspect sensitíve naÈure of the scatterers. (ttris would narro\^r the effective polar diagram). For example, takíng a wind speed of 23 ms-l at 74 kn gave an expected spectrum half-width of .022 Hz. The measured half-width was of the order of .Ol5 Hz. This narro\^l scattering spectrum is also evj-dent ín Fig 7.14 where it can be seen that few echoes occur wíth frequencies less than -.O2 Hz or greater than I .02 Hz f1g 7.15 shows a typical 9 mínute spectrum and correspondíng auto- correlation functíon for ranges less than about B0 km. The spectrum is quite narro1r, and the autocorrelation correspondingly wide. As discussed, this ís partly due to the anisotropy of the scaLterers, and partly due to the 1o¡¿ wínd speeds on this day at these heights. Consequently the beam-broadened spectrum is quite narrovt. In connection with speeular scatter, an interestíng problem aríses. I{e have already seen (Sectíon 7.4) that anísoÈropie scatter can distort Doppler measurements of horizontal r¿ind. Could Èhe form of the specular scatterer introduce extra complications ? Imagine the case of a specular scatterer in Èhe form of the cap of a sphere; or a cylinder, movíng ín a direction perpendícular to the cylinder axis.(Such a form could result from, say' the perturbation of an electron density gradíent by a gravity \^/ave. One half ( r V h r H Fie 7.16 DÍagramrnatíc íl-lustration of a partícular type of ttspecular reflector" (see text). 353. cycle of the gravity \¡Iave Perturbatíon could have such a form. In this case, the horizontal speed of the wave would be measured') Such a situatíon ís íllustrated in Fíg ' 7 'L6 ' Suppose the reflector is moving horizontally at velocíty v. Then what would the measured horízontal velocity be ? The phase at time È, as illustraÈed ín Fíg 7.16, would be 0 2¡r /xl( +r + )-rl So the rate of change of Phase is gE 2tt/), H*r + d,r = fuC AS ¡ ís a consËant. Thís ís precisely the rate of change of phase for a poínt scatterer at poínt P, r.ríth velociÈy v. Thus in this case the Doppler velocíty measurements would give the correct horizonËal veloeiÈy. The point of reflectíon on Èhe cap changes in jusË a \^Iay so as to look like a point scatterer at a hígher height. I find this an interesting result. However, there are certainly many cases where thíngs are not so simple. A flaË reflector could not have its horizontal velociÈy measured by Doppler measurements, for exarnple.. (probably in most cases the reflectors are quite flat, but ít ís sÈill ínterestíng to consider oËher cases). Indeed Doppler the measurements are not partícular1y good for measurements of r'rínds then scaLterers are strongly aspect-sensíÈive refl-ectors ' @,1 (i ') Rur rllo nt \ I \ I \ \ I \ H \ I \ Po r rO I I \ / \ / / eg u encJ 0 ú,t lr "rt (i ) rmo9 e of ob VE / / / 4 \ \. I \ R*u, Itonf. I \ \ I \ H I \ or r03 ro \ I \ I \ I V I I ,\ \ I I \ / f \ lr" nT lue T'ie 7.I7 Illustration of the effecË of wínd shear broadening (see teixt for details). Positíve frequencies imply movement towards Èhe receiver. Arrows in the lagrs indicate wind vectors. 354 7 .4.3 Spectr al width usins the Èilted beam The most obvious reason Èo Ëilt the beam ís Èo make Doppler veloeíty measurements. As seen in relatíon to Fig 7.9b, the measured Doppler velocity ís a funcÈion of the range, even if the wínd speed is constant with height' due Èo the finite thickness of the scatteríng layer. Plots of this Doppler shift as a function of range were made for the data of day 80/072, and agreed closely with thaÈ expected for a 1-4 km thick layer. However, the effect of the wind shear \n/as not fully considered, so this result may be coíncÍdence. Let us now consider Ehe processes contribuÈíng to the width of the po\,Jer specËrum for a tílted beam. Beam broadening acts' of course. But íf the wind vector i" ,rot constant with height across a scatÈeríng layer, wind- shear broadeníng can also be important. To consider the effect of wind-shear-broadening examine Fíg 7.L7. Consider a layer based at height H, depth D (Fig 7.17a(i)). Suppose inítially Èhat the wind speed is constant wíth height. Consíder the po\^Ier sPectrum observed for a range Ro. Thís is also the sum of the power spectra produced by the separate layers 1-7. Now the po!üer spectrum due to layer 4 is that for a layer of depth d, based at heighË H+3d. The power spectrum for layer 3 ís that for a layer of depth d, based at height H + 2d - cl1. ít can be approxímately regarded as that for a layer based at H + 3d' like layer 4, and a range"R"{fi#}' Thus the fínal spectrum is approximately the sum of the spectra for different R values for a single layer of depth d at heíght H * 3d, with "*o" values o varying from (H + 3-d)/H . * Ro(rrue) ro (H + 3d)/(H + 6-d) x Ro(true). Fig 7.9b shows such power spectra, so the sum ís the sum of spectra lilce those shown in Fig 7.17a(ii), plotted as a functíon of Doppler frequency, f rnot as a function of V. The resultant is the envelope. Now ímagíne rhe wind profile is as shown in Fig 7.L7b(i) . Then the individual power spectra are as shown ín f'íg 7.L7b(ií) . Notice (1) has shifted (7) has shifted to the negaËive considerably, and wídened, whilst spectrum 355. tor^rards zeto, and narrowed. The result is a v¡íder spectrum wíth some asymmeÈrf Notíce in Fíg 7.L7b that curve 1 suffers a larger absolute shíft in frequency. llhy ? I.f. the wind increases by x tines in layer 1, it becomes *-1 of íts former value in layer 7 (compared to Fíg 7.17a(í)). Thus the peak of 7 occurs at a frequency *-1., , and thaÈ of I occurs at ú¡ , f1- and f, beíng the original frequencíes of the peaks of I and 7. Thus the shift in frequency of curve 1is f1.(x-1), and for curve 7 Ís fl'(J-L/x) =¡filx)'(x-l)' > the latter term is less than Ëhe former for all x > 1 (and indeed for x h/L )' heíght' If x is much less Èhan 1, ít means the ¡¡ind velocity increases with ín which case reverse as)¡rnmeËry may be expected' This illustrates wínd-shear- broadenlng. However, notice Ëhat íf the wind shear had been such Ëhat wínd (7) and (1) would speed increased with height, then in Fíg 7.176(ii), graphs (The move toÍrard spectrum (4) and the resultant po\^7er sPectrum would .ry. for fading rate woul-d decrease and thus the fading tirne would be larger than lì the narrow beam) Thus wind-shear-bro adeníns" is a mísleading term. Notice that the major contributor to the change in spectral shape is the a vertical shifË of Èhe contríbuting spectral peaks. Thls does not occur for tilted beams' beam, so wind-shear "broadening'r is generalJ-y only important for l{índ-shear broadening ís noÈ easíly dealt with by approximate methods ' treat Thus the program "specpol" (SecËion 7.3.2" Appendix E) was modífied to pol4Ter spectra' a layer as the sum of several narro\^I layers, and sum of all the spectra' to produce a good estimaËe of the beam-broadened and shear-broadened QuítegeneralwindprofilesthroughthelayerhTerepossible. A 2 kn thick layer \^ras assumed between 86 and BB km, and a wind profile appropriate to that shovm ín Fig 7.11 (day 8o/o72) assumed. The resultant po!üer were por^rer spectra for ranges corresponding to a peak of the scattered Hz' (Cornpare then formed. For a vertícal beam, it was found that f 4 = 'O6 this to the approxímation of r 14- .058 to .o7]- ÍIz made previously and see the díscussíon of Fig 7.L3, following equation (7.4.2.5)). However, this írnproved Fig 7 .1-8 Expected po\¡Ier s pectra for Èhe narro\^r beam at Buckland Park poínted at 11.6" to the l^Iest (using rnodified version of program SPECPOL, as dÍscussed ín the text), for the following cases: (") a layer of ísotropic scatterers between 86 and 90 km, with a wÍnd profile approximately given by that in Fig 7.IL. (rEW = 66.s + (h - 86) x (53-66.5)/)88-86)) (t", - 28 + (h - 86) x (22.s + 28)/(88-86)) and (b) a layer of isotropíc scatÈerers bet¡¿een 86 and BB km, wíth a wínd profile approxirnately given by Èhat in Fig 7.IL, as above. Each spectrum corresponds to a range r¿hích ís approximately Èhe range of peak power. (90 km for (a), 89 km for (b)). > Úrc4 ool L q) 3 d 0 I 2 .3 Fre n") (H=) luu NARROIA BEAM :116 I^I ÉST > 'õ 80/072/(146-1154) : I B KM c 0) ô L o \ ) I t I \ d I \ I \ \ ì I \ I \ \ I I \ I / \ I I \ I I \ / \ / \ \ 0 I .2 .3 ñ"1r"n.y (Hr) Ej_e 7.19 Observed pol^rer spectrum usíng B mins of data for the narrow beam ;;1;;à-;.' 11.6' Èo the west on day BO/072. The broken curve centred at 0.14 Hz shows the expected spectrum for a 4 km thick layer at 86-90 kn (see Fig 7.1S), with no turbulence' PosiÈíve frequencies imply motion towards the receíver. It will be noÈiced is considerably wider than the expected that Ëhe observed "p""ÈJrr* non-turbulent one. of the curve suggests the layer is greater than 2 krn thick' The asyrnmetry (see since a 2 km thick layer woulã-produce a faíxLy symmetríc spectrum If rhe layer r"." gt"tter than 4 km thick, this could Fig 7.r8b). :/ Thus it be seen by a widening åf the meãn poüIer profile for this layer ' appears tire layer was between 2 and 4 krn thick' peak The small peak at o Hz is inÈerestíng. Fíg 7.18b suggests such a should occur due to side lobes of the polar diagramrbut not as large as For a 4 km thick layer, the peaí< should be smeared out (Fig 7'l8a)' thís. the However, the presence of Èhis peak suggests Ëhat the sj-de lobes of polar diagram of the a:.nay are larger than Ëhose estimated by function component of specular BPRES in program "SPECPOL;', and/or that there is some from these heights, enhancing the vertical scatter componenË' scatter from Probably both factors contríbute. Certainly it ís known that scatter km is not entirely isotropic - as seeri in Chapter IV' A specular - 80-90 there component r"y be surprisíng. It was also seen in Chapter V that ís oft.en a specular'õ component of scatÈer aÈ - 86 km' 356. did not change the conclusions reached. For a beam estimatíon of r , tilted at 11.6' towards the west and a 2 km thick layer, it was found t'broadening" \,vas expected to be .055 Hz (1.e. wínd shear actually Í,'4 narro\¡rs Èhe spectrum). Fig 7.18b shows Èhe expected po$ler spectrum for a 2 km thíck layer. Fig 7.18a shows the expecÈed spectrum for a 4 k¡n thick Hz' As suggested in the caption to layer, and in this case f ho '062 Fig 7.18, the 1-ayer was probably between 2 and 4 km thÍck - possibl-y nearer .055 and Hz' 4 km. Thus f ,'4 is beËween '062 Eig 7.18 shows an B minute pol^ter spectrum for day 8O/O72 aÈ 88 kn' The hal-f-power-half-wídth ís about .o95 Hz (! LO% ?) - definitely much larger have than Èhe preaict.d .055 - .06 Hz. Since all factors except turbulence Yet the been considered, this suggests Ëhe extra wídth is due Èo turbulence' the verÈícal beam showed no such indicaÈion of turbulence, so this means turbulence must be highly anisotropic, with eddies oscillating primaríly in the horízontal. This ís surprising, because scales of )¡/2 = 75 m ate close to the Kolmogoroff microscale at 80-90 kn (rie 1.9a) and so rnight be expecËed to be associated with isotropic turbulence. This enígma wíll resolve ítseIf shortly. For the present, however' v/e must devise a means to esÈimate the horízont.al t** values. This ís done ín the following secEion' The explanation of thís wídened spectra at off-vertícal Lilts proves to be quite simple, but it is bel-ieved that this is the fírst time that this explanation has been recognízed. 357 . 7 .4.4 TheoreÈical Spectra for anísotropic turbulence + I z, I I +_r_ P ß 7n Rx. consider the spectrum produced by a poínÈ P in space, at which turbulence is active. tr{hat ís the received power speetrum ? Let a scatterer have vertical velocity Yz, and horizontal velocity tr' (assume isotropy ín the horizontal p1-ane). The component of vel-ocity observed at the receiver ís (7.4.4.L) vrad = vrsin 0 + vrcos 0 Let v,- have a probabílity dístribution functíon (7 P (v dv K. exp{ ur'/ ( }utÍ) Ì¿v .4,4.2) t r) r - r v have a probability dístributíon function and z (7 P (v )dv K" exp{- u / {ruu') Ìav .4.4.3) z' z' z rt z Then rH is Ëhe RMS velocíty along any direcÈion in the horízonËal' and RInfS velocity in the vertical directíon. vv ís the Then the probabílíty of observing a radial velocity vrad, from the point P, is equal to the probability of a horizontal velocity vrr times the probabílity of a vertical velocity ,ir 25 Horizontol RmS uelo"ity) 1 1 5 I 5 locity) 0 0 2 4 %rs(11.ó') Fis. 7.2O Plots of the expected observed RMS velocíty for a beam tilted at 11.6o, when the scattering eddies have horízontal RMS velocity tH, and vertical RMS velocÍty \. 358. (7.4.4.4) v (v . v sín 0)/cos 0 (by ( 7 .4 .4 .1)) z -rad - x and integrated over all acceptable vr thus (vr"¿, R, o, ( dvr"u)( dv) (7.4.4.s) Pv 0) rad 2 2 (rr"d O)/cos uuz -v r 2uH - vrsin eT/z KlK2dV e dvr e all vr x d((vr"U - vrsin O)/cos 0) 2 where dV represents a volume of space = R sín OdOdQdR. Ignoring terms involving (dv.)', thís becomes, upon evaluation' P (vr"¿, R, 0, (7.4.4.6) vrad 0)dt.r¿dv dv. exp{- ur^uz } = C. dvrad /¡u**') where C is a constant, and 2 2o tH S l_n 2 2 2 -1 (7 .4,4.7) v v cos 0 [1- 22^¿ 2o l RN/lS v vv cos u + vH san Notice v** is the root mean square velocity observed for turbulent scatter from the Polnt P. Notice if vr, = v'r then t*^- = tH = tro' Thus the observed RMS velocíty is the true RMS velocity along any direction for isotropic tH 0 turbulence. It can also be shown that tr* = \cos 0 when = and v*,* = v"sin 0 when t*, = 0. Fíg 7.20 shows plots of t*n- t"'tH values. for various vv happens Now the above formulae are purely for turbulence at P' I{hat final poller when we integrate the effects over a full region ? lJhat is the spectrum when turbulence and beam broadening are considered ? 359 Thls is best done by building on equation (7.3.2.8), which is repeated below as (7 .4.4. B); 2 (7.4.4.8) T(R, e, O)dCI = P(0, 0) t 4}p (r dCI) @ g(r) l ¡fhe scattered pohrer for a radial velocíÈy vrad t (due to turbulence effects alone) observed at range R, from angles 0, 0 l_s o(r, 0, 0) (R 0 P(0, (7.4.4.e) P t t 0, rr"d .)df2 = 0) t ;z P (v 0, o g(r) vrad rad t, r, 0) ldf¿ Then the pov/er observed wí th. toËal velocíty tT l-s dfl (7 .4.4 .10) P. (R, o , 0, vr"¿ ¡) all 0, Q where the íntegration is performed over the regíon r¿here (7 0 vl{, (1 * cos 0(00-ö))}% (bv (7 -3.2.4)) .4 .4 .11) "ir,-l and where g = (vr - tr"d /uho.ix t) , vr beíng the horizontal wind speed blowíng in the direction 0o . ho_.-,_ r¡z Thus the fínal resulL is a complicated type of convolutíon. Now consíder the case for a scattering layer ín whích the turbulent condítions are independent of posit.ion within Èhe layer. Then O covers Èhe radial dependence of trr"d in the convolutíon in (7 '4'4'9), so ttrrd can be regarded as a function of vrad t, 0 and Q only, and can be taken outsíde of the convolutíon. This símplifies the problem, because no\^l (7 .4 .4. 10) b ecomes ( (7.4.4.L2) IP ( PM(R, o, dtr"d ) vrad v."d r' e o ) Ôrr"atrvra¿ t)dQ t' all al-1 vradt accetable e, 0 360. where tNf is just the integrand in (7.3.2.9)r and where 0 is defined by (7.4.4.11). P is also lndependent of 0 and then , Notice that if vrad 0, (7 .4.4.L2> becomes (v PNT (R,O,Q, vo)dCI dvo (7 .4.4.13) Pv rad rad t) i, atr1 all vrad t acceptable 0, 0 to, and (7 .4.4.l-1) holds . where vrao t rT - This Ís símply (7 .4.4.L4) Pv rad(v."u) a F',r(vt"u' R) F',r(vtru R) F(V, R) in equation (7 .3.2.9> , and where ' vhoriz being the horizontal wind sPeed. v-=V.v..racl norLz of 0, (í.e', the direction of Hence if P., :ad is independent Q vÍewing) and is índependent of its posiÈion wíthín the scattering region, then Ëhe resultant sp ectrum ís a convolution of the turbulence spectrum and the non- turbulence-p roduced spectrum. This was assumed in derivíng equation (7.4.2.5), whích is valid for isotropíe turbulence' Thus .(7.4.4.L4) ís valíd for isoÈropíc turbulence, and any polar díagram. It is also approximately valid for anisotropic turbulence observed wiÈh a narro\^r beam ti1Ëed sígníficantly from the vertical, since, as a fírst 0 and r¿ithín order approxímation, P.,, ,"d is roughly índependent of Ó the maín lobe of the beam, and contribuËions from outside this lobe are small. For a vertical beam obsetrrirrg anisotropic turbulence (7 .4.4J'4) ís strictly not valíd. However, if t., ís much 1-arger than v"sin 0¿, U, beíng 361. the half-power-ha1f-wídth of the polar dfagram, then only vertical- componenÈs are important, and (7.4.4.L4) 1s again valid. In the work presented in the fo11-owlng section, (7.4.4.L4) wíll be assumed va1ld for the narrorir beam at Buckland Park tilted to 1l-.6o from the zenfth. Thus, the t** value due to turbulence alone can be calculated by equatíon (7.4.2.5), vLz, (7.4.4.1s) ù*^- = {t( (À/2)f ù'- (rrr>'l^, Ln 2\}\ and then (7.4.4.7) can be used to estímate tH, provlded t, 1s knov¡n. Here, ls the hal-f-power-hal-f-wídth of Èhe specËrum (ploËted as "r; a functlon of radíal velocj.ty) expected due to non-turbulent processes ls the measured half-wldth (beam broadenlng, shear "broadenlng"), and f b of the specÈrum. 362. 7.4.5 Interpretation of spectral widËh us tilÈed beam seen'' Hz' YeÈ the Let us no!ìt reconsider Fig 7'19' As f '"= 'O95 shornm Èo be .055- Hz in expected f ,, for non-turbulenÈ scatt.er was '06 at 11'6o to the tr'Iest' 5ection 7.4.3 for day BO/O72, using a beam tílted This sectíon ís primarily for illustraÈive purposes, so take 1 r"= '055' Tf but none ¡r"=.06 is used, some of the numbers produced wi1-l vary slightly' (7'4'4'15) of the general conclusíons will be changed. Thus by equation 2 2 ( (. oss) ) l/2 e"n 2 vRMS t(ry(.oes))2- v 4.5 ms-r ) Eiot v 5.0 ms-l . (rf f ,4 .06 Hz, Rlv6 = RfVE using the vertical beam, that vv.S 3 *s-t. (rnis Now we h"í" "."rr, derívation, ín Section 7.4.2, effectívely assumed the resultant specÈrum I'/as a in the convolutíon of Ëhe turbulent and non-turbulent spectra' As seen in the last secEion, this is not strícËly valíd for a vertícal beam observing be anisotropíc turbulence, but ís close enough if v.,, ì v"sín (4'5'), and wíil assumed here). ms-l Then, from Fíg 7.20, wíth t*,* = 5 ms-l and t'o between 0 and 3 ' hTeSeev,-2L_26ms-l.Thisisalargevalue,anditseemsunreasonable that scatterers with scales of the order of 75 rn could be associated wíth suchanRMS velocity ( e= Uu3lL; if 9,- 75m, v-25 fls-r, e- 200\^lkg-1 , (2.2'3'3b) it and even if k - 10-2, r - 2l^trkg-l . But in Chapter II, equatíon was shown k - 1O-r ). speed is However, consider the possibílíty that the horizontal wind about, not constant over the observíng period. Then the spectrum wíll move could cause such smear out, and hence be significantly broadened. what then wínd flucËuations on scales of the order of B mins ? At least trvo But in this possibilíties exist, these beíng gravity l^/aves, and turbulence. are not those case, the RMS velocítíes assocíated with the turbulence of the order of uT' associated with the scattering eddies, but with scales 363. I beíng the mean wind speed, and T the observing time. This sÈaÈement follows Taylorrs frozen turbulence hypothesís (e.g. Gage, L979), which assumes that turbulence can be regarded as being "ftozen" in the background wind. This would also explain the anÍsotropy, since at scales of thÍs size ([ - TOms-l, T- Smin+0-33kn), turbulence is decídedly two dimensional. (In actual fact, the observed tH is the íntegrated effect of all scales from the smallest up to about [t, but this will be considered later) . If the spectral width is due Èo turbulent fluctuaÈions of winds on as suggested, then it can be expected that if the duraËion scales of about ,UT, of observatíon T is changed, so should tH change. A 5 minute po\¡Ier spectrumrusíng the fírst 5 minutes of data used in preparíng FÍ-g 7.19, was generated, and thís had a half-power-ha1f-width of .085 Hz, somewhat less than the .095 Hz produced by B rninutes of data. A 3-minute por^7er spectrum gave These then, would appear to support the above l, -4 = .O75 Hz. results, explanation of the spectral widths observed. This explanaÈion does not appear to have been presented before in any literatuïe. Fukao et al (1980b) have also noted a dependence of the spectral width upon the length of observaÈion at VIIF, but only recorded for data lengths up to 100 seconds. Further, Èhey only tilted their beam by about 3o off-vertícal, so the effect they observed may not be the same as the above - they may not have used suffícíent observation Èimes and beam Ëílt to see the above effecÈ. The author feels thís process offers an important facility for estimation of turbulent energy dissipatíon rate" (0d), and this r¿i1l be íllustrated below. Let us assume that the fluctuating velocities are due to turbulence alone. (The role of graviEy rlaves will be discussed later). Then ü7e may expect that 364. -1 3 (7 e æ T* tH ll,, f,-uT , .4.5. la) d in a símilar way to equarion (2.2.3.rJt) it chapter II. Ilowever, derived for eddy equation (2.2.3.3b) may be more aPpropriaËe' as thaÈ was sizes 9" . Then 3 (7 e æ llLO tH /9,, .1, = uT. .4.5.lb) d Chapter II but The derivatíon of this equation l^¡as not given in , it may be fruitful Èo consider it here. The structure function ,4 2 2 o < tu,(t) ulr+r) l >] (í = componenÈ) (7 .4 .5.2) T { related Èo the overall RMS velocity vR by the relatíon is û' 2 Gage, lg79) (7 3) o 2 uRz (1 - R(r)) (e.g. .4. s. T wind fluctuations' where n(t) IS the autocorrelation functíon for the is quíte símilar The paramet.er tH measured is not v*, however, but to o But T . 2 2 2 5 3 (7 o A. ed L .4.5.4) T l_ Here' Ai is a constant; where .Q, = -r.{t, by Taylorts transformatíon' and Aa to the Ëransverse Ãy refers to the longitudinal wind component ' (Ai here is equivalent to components. As seen in chapter II, Section 2.1, ísotropíc Èurbulence' A*ri2 there) , LL 1t-75 to 2, and Ãt- 2'35' for Gage(1979)poíntsoutLt=4/3^Lforaisotropicturbulence,andA.=5/3 A,Q,for2-dimensionalturbulence.Thusfor2-dimensionalturbul-ence,At= 2.9 ro 3.3. Then assuming v" = OTr (7.4.5.4) gíves 365 z 3 (7.4.s.s) e æ A. 3 tH / L where l, = uT. d L This derivaríon is similar to that of Lloyd et al (1972), excepÈ Èhat those authors did noÈ look at the- component form of 6T, but rather the full vector structure function, (hence tH ís not quiÈe relevant to theír forurula). Thus they derived equation (7.4.5.1-b). Thus (7.4.5.5) suggests that 3 (7.4.5.6a) ,d N ,ro-t + where lZO - 2.3 to 2.8 f.or the longitudinal component' T2n- 4.9 to 6 for Ëhe transverse e-omponent. So both (7.4.5.1b) and (7.4.5.6a) suggest that T líes beÈween about 2.3 and f0.0. 2D It was poínted out above that the measured value tH is in fact an integrated effecÈ of all- scales less than about Uf. Do the above formulae for rd consider this effect ? In fact they do, and this is besË seen by examining the turbulence spectra. Let us assume the value rH observed ís due to the effect of all scales less than .1,, = If . In actual fact, larger scales will have some effect, but the larger the scale the less the effect wíll be. These effects will be ignored. Assume tH ís a longitudinal or transverse component. Then --, zt 5/3 k dk tH EÍ(ki) dki c[.r-d e. kr ktr 1 where kt = 2tt/9", I Recall that two normalizatíons can be used; 2 (í) E.(k.)dk. u' and I- a' t-- l- k, 366 (rr) E.(k.)dk.- u'2 /2. t- l-- l_ ktr where u'' Ís the fluctuating component. I shal1 use case (i), as the formulae derived will ínvolve longítudinal and transverse spectra, and the og, ot values given in Table 2.1, Chapter II, assumed this norrnalization.l Thus; N eul3.{zn or -.22ur' cr. /9.,r_)' , tH 3 -1 (.,H2) 3/2 o 3 (7 .4 . e -L assuming tH s.6b) d 2D 9"L Thus we see a form líke (7.4.5.6a) again results, and Ít can be seen that formula does consider all scal-es. In this case, 3/2 T az 11.5 o 2D L NoËice by Tatarski (1961 ¡equatíon 2.22), that (7.4.5.6c) oi N .2488 A. (after appropriate change of notation, and compensation for the fact that the normalizatlon dk. 17, is used in Tatarskí T'i(kí) 312 Ilence, T2l L.4271^. It 1s ínteresting to compare Ëhis with (7.4.5.6a), where 1t can be seen that T2D = Ort/'. The two different approaches gíve a difference in T of a factor of L.4. 2D - Thus for the longitudi-nal component, with ÃU = 2.0, TZD æ 4.0 and for Èhe transverse component in 2-D turbulence, using At ^' 3.3, TzD o 8.55. Thus these values also líe betr"reen 2.3 and 10.0, as proposed tn (7.4.5.6a). 3ol . I Notice that if the non-component form of the spectrum is used, and the ffnormalizarton,, f uCOlOn = F/, 1s used, then cr - 1.4 to 1.6 (Chapter Ir), ) and TZO becomes ZZ.O a3/2 x 42. This can be compared wíth a value of 10 produced fn the structure-function derivatlon (see 7.4.5.lb)). However, it should perhaps be realízed that Èhe constanÈ in equation (7.4.5.lb) (taken as 10 here) actually assumed 0 = I (see Chapter II, equation 2.2.3.3b). If ln fact ct 1s taken as - L.4 Èo 1.6 the constant would be - 20, and it is thls which should reaIIy be compared to the value T2D o 42 ). So lt seems that equatíon (7.4.5.6a) 1s qulte valid. Let us try and puË it Ínto practice. For Èhe 5 mínute, sPectrum' 2 2 t-r v x { [ r.oes¡ -(.oss) 1ft2 !,n 2'I\ (75.75) x 4.2 ms-r R ÑIS¡ v-- DS-r, Fig 7.20 implles tH - I6-2L r"-t . Also, 9" - 5 x 60 x For v = 0-3 60 metres, assuming a mean wind speed of 60 ms-1 -. .-1 -1 T2D t (t6' * x ro3) -m (.ZZ .5l)l^Ikg Thus, ,d = zt3)/(r' 2D - for the 3 mÍnut.e spectrum, v * 3.3 fls-r , gívíng tH - 9-L6 fls-!, and so Rlvls -1 e -'t ('07 '' . 38) t^Ikg - d -T 2D -1 -1 For the 8 mÍn specfrum, VRMS æ5ms so Fig 7.20 inplies v" - 2L-26 ms -t Hence e. = Tzo-1 (. 32 -> .6r) w kg, so all estimations are reasonably cons is tu$ t . Notlce the reglon of common overlap for these ur3/n estimatesls .32+.38 *3"-1¡ so íf we take .35 as the true value, Ëhen for the I mín specÈrum, v"= 22 *s-1, for the 5 mln spectrum, v, = 18.5 t"-1, and for the 3 min spectrum, v, = 15.6 *"-1. Thus using these tH values, and Èo make estímaEes of v--. v'RtvGt values as measured, FÍ.g 7.20- can be used \/ For the B min spectrum, r., o 2 *"-1. For the 5 mln spectrum, vv- 1.5 - 2.2 ,n"-1 . For the 3 mín spectrum' t., - 0.8 - 1.5 *"-1 . Nocíce for 368. -1 -1 I 10' .035 wkg Thus ít appears v.*, - 1-2 ms This 2D 'd the verÈical RMS component is also the RMS velocíty associated l^rith scatteríng eddíes' so rnle exPect, by equation (2 '2'3'3c) ' that + As shown in ChapÈer II, ,d - To-1 çt3 - Z3¡¡ ( 1# ) - To-1 (0.083 .66). T* ís a different value to T2D, since it refers Èo radio l¡Iave scatter. Wtg-l. This is somewhat Using T* ^, 15T (see 2.2.3.3¿'1, eU - .002 - .015 less than the esËimate using v" (i.e. .035 l^Ikg-l¡, bt-'a is not inconsistent hrith thaÈ value, partícularly íf t., - 2 *"-1 is assumed' given the ' uncerLaínty of the T* and TrO values) ' Thepossibilityalsoexístsforestímationofeddydiffusion coefficients. If iÈ can be assumed that the wínd profile given ín Fíg 7'11 ís the true one, and there exíst no smaller scale fluctuatíon, then Èhe wind shear around 86-88 km is about 25 ms-lkm-l. The vaLidity of thís assumption thaÈ we have sufficient resol-ution to observe the true wind profile is questíonable , however. Certainly in the troposphere and stratosphere, a resolution of 2'4 km would not resolve the true wind profile; there are fíner scale fluctuations. This has been pointed out by van Zandt et al (1978) andcrane (1980), and has been díscussed ín chapter II, SecËíon 2.3.2' (The outer However, 1et us assume sufficient, resolution has been achieved' scale aË these heights is around 300 + 600 m (Fig 1.9a) so fluctuations turbulence of the mean wind on a scale fíner than this are unlikely, since shear woul-d smear them out). If it can be assumed the turbulence is wind g. giving 1.5625x 1o-4 rad s-t, generared, rhen =uu2/, H r2 = 0.25, -u'= or TB = 8.4 minutes' TB is the Brunt-Vaisala period' d But .B 2 ò dT + f^ I' T ð'" d dT km-1 as the temperature gradient. The mean Ëemperature giving d" = -6.45K to gradient at 30" S in March at 80-85 km is --1K krn-l accordíng 369 in March to be the same as 30oN in CIRA 72, part 2 Table 18b (assuning 305 Fig 1'0' However' temperaÈure SepÈember), and aecordíng to CIRA 72' patt 1' CIRA 72' part 1' Fíg 10' gradienËs of - -5K kn-l are noÈ unconmon (e'g' GmedíarJ'curve), so Èhís result 1s not unreasonable' Then K = C1 ,a ,¡-2 is Ëhe eddy diffusíon coeffícíent hTith use C, = 2'O' Cl - 1-3 (see Chapter II, Section 2'2'4)' I shall gíven e¿and K' Then K = 64 - IzB*2"-1 for ,d = .02 -.04 l'{ ttg-l' Then, the outer scale ís L = (K3/eu)U t a-1, or Lo = 3Bo-530 metrcs' Theenergysupplyrate,¿,andbouyancydíssipationrate'areroughly1.4eu cL 2 ín equation (2 ' e respectívely, using cz = c3 = 1' = '2'4'L2) and 0.4 d layer at 86-88 Thus a picture emerges of a shear-generated turbulent of 75 m of - 2 ms-l' and with kn wíÈh RMS velocities associaLed wíth scales largerscalehorízontalfluctuationsinÈhe¡^¡ind.IÈappearsthated-.02 15n' produce reasonably to .04 I^I t g-1 . use of rro = 10, and ro = explaín the observed power consístenÈ results. This picture can adequately provides an spectra. Furthermore, analysís of power specÈra ín this way rniddle atmosphere excellent way of evaluating turbulence parameËers in the greatly to the wind above B0 km - províded gravity htaves do not conl-ríbuÈe fluctuations. Beforedlscussingtheroleofgravítywaves,however'onemore II, Section 2.2.3, that point deserves commenL. IE was mentioned ín Chapter nith the f,adíng due if the fadíng time for ionospheríc scaÈter could be found mean wínd removed then this Ëo the movemerit of the írregularítíes I^Iíth the ' by equation (2'2'3'11) fadíng t.íme could be used to estimate V*on=, as índicated t*r" could be found usíng full and ( 2.2.3.L2) - It was shovm that such a correlatíonanalysis.However,thequestionnowarísesasÈohowthis'** 370. estimaEe should be inÈerpreted. s.M. Ball (1981) has made estimates of t** _1-. for Buckland Park winds data and fínds t*,* - 4-5 ms In light of the previous díscussion, it would seem reasonable that this t** is related to tH, the horízontal RMS turbulent velocity. However, several authors have FCA technÍ-que and assumed that this v is calculated v_..-RMs usíng this = nr's Èhe velociÈy assocíated with scales of the order of ),/2 metres. If this observed t** is related Eo tH, as the author feels, then estimates of eU produòed by the FCA technique are hrrong. For example, Manson and Meek (1980) have done thís. Thus it would seem faír to dísmiss their results, calculated usÍng their equatíon (2). Even when it ís realized that t** 1s related to tH, a conversion needs to be found to obtain vH. This is considerably rnore difficult than for the case of a tílted beam discussed earlier. The scales associated wíth tH are of the order of < U T), where u is the mean wind and T is the length of time used to obtaín each drift estimate (generaLLy L-2 minutes). If U and T are approxinately constant, then e = vH3/L ís ? proportional to t"' and if r** is linearly related to tH (see Fíg 7.20; .' this l-s true for an 11.6o Èílt for large v"), then the formula e - k 'tt*n*'/À may work, because k wíll be constant. But Èhe whole thing is an extremely crude analysis, and cannot be relíed on. The value k r^rould have to be adjusted seasonally as u varíed. Interestingly, I,üríght and Pittaway (1978) have invesÈígated the case of scatËerers oscillatíng in the horizontal in a mean wind, and found that the parameter "U." whích is produced in Full Correlation Analysis is approxímately equal to the RMS fluctuating velocity of these scatterers. Thus ít may seem that V" ís equívalent to tH descrlbed above, where in thís case rH ís the Rì,lS wind velocíty assocíated wíth scales correspondíng to about one minute of data (1 minute generally being the data lengths used for FcA). If this l¡Iere so, turbulence estimates would be possible. However, Stubbs (Ig77) has shown that V" is typically - 4q to 80 ms-l, which ís much 37L. larger than the rH values expected to be associated with scales correspondíng to abouË one minute of data. (It may be expected that 1 N v" - (TrO .U I. 60 secs)i - 12 r"-l for e - .04 W tg-l, E 7O rns-l). It thus appears that the scheme proposed by I'lright and ?ítteway ís not quite scale the same as that for scaEtering eddíes being moved around by large turbulence, or that some other process not consídered by wríght and PÍtteway also contríbutes to the experímental Vc' (1980) a commenË while discussing the paper by Manson and Meek ' on their equatíon (3) may be worthwhíle' They use an equation e r d b fb being the Brunt-Vaísala frequency in Hz. Thís seems a strange relatíon, v--- ís a funcÈion of the scatteríng scale, and fb ís a constant' slnce R NrS suggestíng that ad is a func.tion of the scatteríng scale! It would apPear there ís an error here. Possibly t*,* ín this case should be the velocity associated with the Kolmogoroff mi-croscale. But agaín, the t^r" measured by the system is not so much related to turbulent velocities associated with scatteríng scale, but rather to vH. Thus this formula appears to be inapplicable Eo the data of Manson and Meek, and again their results may be in error. schlegel et al (1973) have produced e estímates which are even less reliable than those of Manson and Meek. These authors simply used the fading tímes rneasured directly from their data, r¡íthout even an attempt to remove the effect of movement of the irregularíties with the mean wfnd' consequently, a Large contribution to probably coil]¿s from beam "4 broadening, and their data is possibly unreliable. The "seasonal dependence" of. ,d that Èhey see ís most likely due to seasonal variations of the mean fyz and thus wind, thus varying the "beam broadening" effect, and so varying BUCKLAND PARK >. NARRON BEAM o c 11.6" N ESI oo 80/072/(233-1242) L 70 KM o d3 4 01 2 Freluenc) Hz) PLe-.L.zI Typical- po\trer specÈrum obtained with Lhe tilted narro\^r beam ,at Buekland Park for ranges below about 76-80 kn. 372. t' v ,, . Unfortunately, due to the uncertaínty with k ín the relatíon RTúS ? ,d = k v-/L, authors are able to adjust their results to givettreasonablett t'acceptable". ,d values and so gíve thelr results an artificíal tag of Llttle has been said regarding 11.6" pohter spectra below 80 km. The reason for this was given ín connection r¡ith Fig 7.11. That ís, the anisotropy of scatEer means that Èhe spectra are unreliable for estimatíng wlnds. The vel-ocíties obtaíned ín Fig 7.11 are averages of the velocitÍes índlcated by - 1-3 mín spectra. However, rig 7.2I shows a full 9 mín spectrum aL 70 km and 11 .6o. It is clearly narror^r, and somer¿hat similar to thaÈ-for 0o suggesting that much of the scatter is leakage from the vertícal' It is in fact possíble that the small, appalently secondary peak at - 0.]-Hz may in facË be the contríbution from Èhe off vertical (and even then, the bulk of ít may be from angles less than tl.6'). If ít \^/ere assumed the scatÈer was from 11.6" for this "sidebbe", it would suggest a wínd speed of - 40 *s-1' However, the wínd direcÈion is opposite to that índicated in Fig 7 'tL' Thus this peak may be ¿,r. to some o¡her effect (e.g. a strong, short burst leakíng into Èhe bearn from the opposite síde to that of the bearn), or the valueSín Fig 7.11 nay be wrong. More investigation of such effects is necessary' 373. 7 .4.6 The rel-ative roles of eravíty \^raves and turbulence in the mesosphere Before Hínes (1960) showed the importance of gravíÈy Í/aves Ín the mesosphere, most non-tídal fluctuations r¡rere assumed Èo be due to turbulence (e.g. the ,d estímates of Booker and Cohen (1956) of - 25 I,l kg-l lteÏe !/rong for thís reason) . After Hines t paper, the pendulum appears to have swung the other way, and the effects of turbulence wíth É1me scales of mínutes and tens of mínutes appear, to have been largely ígnored. The rnajority of oscillatíons at these periods have been assumed to be due to gravÍty r^raves. (At small-er scales, the role of turbulence has been appreciated, and most measuïements of turbulence parameters have been done at short scales (e.g. rocket measurements (e.g. Rees et al L972)tmeteor measurements (e.g. Elford and Roper, 1967)). It is worthwhile to briefly reconsider the relative roles of gravity r^raves and turbulencerparticularly in víew of the rd estímates made ín the previous section. In the troposphere, wind fluctuations wiÈh scales less than about t) 2-4 km (and hence time scales uP to - 4-B minutes for wind speeds - IO ns appear to be prirnarily due to turbulence (e.g. Kaimal et al L972; Doviak and Berger, 19BO). Even at scales of 8-16 km, (time scales - L3-25 mins) turbulence makes a major contribution to the r¡índ spectra' although some spectral peaks due to coherent hTaves can occur (e.g. see Doviak and Berger, 1980, Fig 1O). Even at temporal scales of 10 to 200 minutes, turbulence makes the major conÈríbution to wind fluctuations (Gage, L979). To properly investígate the effects of Èurbulence in the mesosphere, structure functíons should be formed for the longítudinal and Èransverse rrtwo components and a fít to the thírds" lawattempted (e.g. see Gage, L979). The constants A.C and Aa described in equation (7.4.5.4) should be found, and if they are ín the ratio 3'.5, this would atrso support a turbulence 1 Ê L 1 ({ ,(l) (a.l "E \ =cú) o10 ô ú ) L o 3 [L^o \ 1 \ \ Freg u e ncy ( hr-') 1 .05 I .5 1 10 20 Fís. 7 .22 "Typical" (see text) power specËrum of winds measured by the PRD technique for Buckland Park for June, 1978 and for a height of BB kur. Section (a) was formed using about 10 days of data, and section (b) is the average of a few 3 hour blocks. Separate spectra for the North-South and East-trIest wind components have been averaged t.ogether. (Also shown (broken lines) are slopes of - 5/3). The sPectrum is from Ball (1981). Note "power density" means pol¡ler per unÍt frequency band. 374. contribution. A slíghtly inferior, but still useful technique (see TaËarski, 1961, for a discussíon of the advanËages of sÈructure function) is to plot porÂrer spectra. These should ideally be plotted as a function of k = 2r u-!, f being the frequency and U the total mean wínd. The sPectra should a form E(k) o. assuming Taylorrs "frozen then follor¡ = 'u3tt-5/3, turbulence" hypoÈhesis (e.g. see Gage, LgTg). If the k-513 law does appear to ho1d, Èhen thís suggests that the winds may be due to turbulence. However' iÈ is not conclusive. For example, suppose gravíty \raves vlere generated in the troposphere by turbulence effects, and propagated to the mesosphere. If no filtering of these rÁraves occurred (e.g. see Pitteway and Hines 1965; Hines and Reddy, Lg67), then it míght not be entírely unreasonable to suppose Èhat the gravity r¡rave spectrum would be similar t.o that of the source region - _., I and so a rr¡-)/3 lar" could occur. Thus other tests should also be conducted - for example, are og, and ot (see equation (7.4.5.6a to 7.4'5'6b) in the expected ratio ? The mean wind in the source region and ín the regíon of observation are unlikely to be the same in directíon, so íf Ot and oL were ín the ratio (5:3), this would be suggestive of turbulence. Cospectra (e.g. see Kaimal et al L972) could also be obtained and compared to turbulence theories. Energy dissipatíon rates td could also be cal-culaÈed - if these r^rere consístenÈ with accepted turbulence values, this would also support a turbulence hypothesís. No spectra plotted as a function of k for the mesosphere could be found ín the líterature, and shortage of time has prevented the author from doíng this. However, Ball- (1931) has plotted spectra as a function of frequency for Adelaide and Townsvílle mesospheric winds. If u is constanÈ for Ëhe duraÈíon of the daÈa interval used, then it can be expected that rhe energy E( f,) uÍ-5/3 for turbulence. Fíg 7.22 shows a log-1og 375. spectral ploÈ due to Ball (1981), using winds measured by partíal reflection drifts, and several facts can be noticed. A slope of - 5/3 oecurs for períods of - 15 min to t hour. Thls may indicate much of the wind f,luctuatíon could be due Èo turbulence in this períod range, but the reservations discussed above should be borne in mínd. For periods between 15 mins and - 5 mins, the - 5/3 slope is not followed, indicating that gravity wave effecÈs are almost certainly important. At períods less than about 5 mínutes no gravity r^rave activity is expected, this being less than the Brunt-Vaisala period. The graph ín Fig 7.22 does shor¡ that at periods t'peakst' less than 5 minutes a - 5/3 slope may again be evídent. The present may be due to acoustic hraves, but recall that the vertical scale ís a 1og plot, and these "peaks" at low pol^rers could well be statistícal fluctuations. Personal experience suggests that the fluctuatíons are statistíca1, and acoustic r¿aves do not make a significant contribution to rn¡ind f luctuations at these scales. Notice thaÈ the line of slope - 513 for periods less than 5 mínutes ís dísplaced from that for periods of L5-20 mins to I hour. This is perhaps not surprising since gravity r¡/aves r¿ould supply an extra source of turbulence for these smaller scales, and so increase the energy díssipation rate for these scales. Many other spectra have been produced for Adelaíde and Tor,msvílle (Ba1l, 1981), and the above description appears to be generally applicable, although the role of gravity \Á/aves varíes. The case presented here is one in whích gravíty rnraves \^rere particularly active. The fit of the - 513 slope for periods of 15 - 60 minutes ís particularly good in this case' too; not all cases fit quite so we1l. (fo be faír, I have been a 1itt1e selectíve ín choosing this graph, to emphasíze tine point. However, it ís not aËypical). It ís instructÍve to estímat.e the energy dissipaÈíon rate" ed 376. associaËed with the - 513 slopes mentioned above. If a "normalízationil of the spectra _t (as used bY Ball), I"' (t<,)a t. ní is used was 2 oi td^3 k. -s/3 then Eí(ki) = l_ where o.C component ín isotropic turbulence, cla =(4 lZ)aU (e'g' Taüarski' L96]-' chapter 2; Kaimal et al, L972; Gage, lrgTg). For two-dimensional turbulence (e'g' however, (which is undoubtedly applicable here), ot = 5/3 aU see equatíon (7.,4.5.6c) (crr,oc Ai), and Gage, L979 (Ar = 5/3 A[)) ' (longítudinal) or Then if a frequency spectrum Srt0 i) (í = l' t(transverse)) is generated such that -2 f'irff i)ff i u¡ , it follows that 2/g 2/z -s/ 3 s. .) cÍ. e. (u/2rr) Í.r r-r-ff l- ad Since the specÈrum presented is an average of the NS and EInl components' c)¿a 0'5 there is some doubt as to the appropriate c[.. I shall use both = æ (typical for 88 km and ar= 513 x 0.5 = .83. Taking u 20 to 50 ms-1 at Adelaíde during thís observation períod (June, 1978)) gives, for Èhe spectrum in Fig 7.22i e. .085 +.5 I,I kg-l ín the periods 15 míns to t hour, d = and td 1.2'> 6.3 I^I kg-l for periods less than 5 minutes' model 2 The estimates of ,d presented ín the captíon to Fíg 1.9a for < (ea .01 - .5 lI kg-l) wer. generally made at fairly small scales I krns), ís to so should correspond Ëo the latter set of td estímates (recall that ít 377 . beexpectedthat,dbelessinthe15mín_lhourperíodrange'asexplained above).ItEhusappearsthatthese'destimatesareslightlytoolarge, However' also bear in which mighr irnply bome non-turbulenË conEributíon' in the inertial range of mind that this region of ínvestigation is NOT (Also the effect of Èurbulence, so the above oi may noÈ be relevanÈ. than the Erue energíes noise may have pushed the spectrum to higher values at low powers) - Recall from aÈ these scares. This could be quite importanÈ ,first range" had a k-5/3 chapter II, Fig 2.2, that Intreinstockts bouyancy for the inertial range' dependence, but the constant c[ differed from that scales being In a símilar way' di = 0'5 may not be relevant for the observedinFigT.22-alargeroimaybeneeded'Forexamplc'an rt thus apPears increase of oí'by x Èimes reduces e by *3/2 ti,n""' are necessary at such some more experimental estimates of this constant may not be relevant large scales. Likewíse, the oi used by Gage (L979) the TZO values at. his scales. Of course, íf oi musÈ be changed' then c¡¿' 0'5 s/as estimated earlier may also need to be changed' A value = If in fact associated wíth TZt = 4. (see equation (7 '4'5'6b) ' aS Suggested ín Section 7.4.5, then cx. ^,0.9 T2D ^, l0 is more relevant' 4.3/2 equation (7 and this reduces the (usíng Tzo = rt.s - "tt '4'5'6b) ' ,destimaEesbyabout2'5times'ItaPpearsthatmorer¿orkisnecessary in deriving these cr values at large scales' in the 0i, and The author feels, gíven the uncerËainties involved gíventhatnoisemayhaveplayedanimPortantroleatthelowpovTers,that the,dvalueassociatedwithperíodss5mínsis.consistent\^'iÈhthe due to turbulence'' speculatíon that all fluctuations at these scales are power spectra, and Manson and Meek (1980) have also presented such data, although those similar conclusions appear valid for some of their For example, Fig 2 of that authors do noÈ discuss the role of turbulence. 378. -sl 3 reference, for the NS components at 82-85 km, shows a reasonable k fit for periods of 15 to 90 minuÈes, and the oÈher graphs ín that figure show similar Èrends. Whatrepercussions,then,doesthishavefortheestimatesof'd power spectrrxn' periods made prevíously (Section 7.4.5) ? For the B-minute and periods up to - 16 minutes wíll up Èo B minutes contribute to 'r', in B make a lesser contïíbutíon (these complete more than a half-cycle minutes). l,Ie maY write -a 2 tH vtt + -.vn ( tur¡ ) (cr¿) has already where GI^l t.f"t" to gravity üIaves and acoustic waves' It to been stat.ed that it is felt that acoustic \^Iaves rnake little contribution period range the specÈra. Gravity hTaves may make some contribution in the equal 5 to 10 minules, however. suppose gravity \¡Iaves and turbulence make contributíons. Then 2 tH 24 (turb)' too so the ,d esÈimate in section 7.4.5, may be perhaps 23/2 - 2.8 tímes large. However, it is felt that the ad estimaÈes made in section 7 '4'5' assumed Tzo are unlikely Èo be out by more than a factor of about 4. The increase the td was taken as 10 - it could be as low as 2.5, which would estimatesby-4times;andconsíderationofÈhegravity\^Iaveeffectaboveis gravíty wave unlikely Lo decrease td by more than - 4 times. The above effect may also explain why td estímates made using an I minute Pol¡/er minuËe or 3 specErum ín section 7.4.5. were larger Èhan those using 5 u"' mínute spectra - gravity \^laves were making a larger contribution to for the g minute spectrum. This íllusËrates that to properly utilíze thé about 5-B minutes Lechnique developed in Sectíon 7.4.5, spectra of less than of data should be used - otherwise gravity \,,/aves will make too much contribution. 379. Thls discussion brings into question another techníque for estimating energy dissípation rates. VincenÈ and Stubbs (1977) and Manson and Meek (1980 , equarion 4) used a formula v2vg ew h o to..est,ímate the energy loss rate of gravity l¡Iaves. Here, V2 ís the mean square horizontal perÈurbation velocity of internal gravity \^7aves withín some specified frequency band, ho is the scale height of the energy t densíty po Y', and U, is a typical gravity wave vertical group velocity. Thls formula canrroL be applied without some care. It must be established that in the frequency band being consídered, gravity \¡raves dominate the -1 spectrum, (e.g. 5 - 15 minute periods ín Fíg 7.22). Othen^7ise, both V- t h^ wíll be affected by turbulence contributíons to V-. Víncent and and o Stubbs used períods in the range l--4 hours, and obtaíned ew values in reasonable agreement r^rith turbulent energy dissipation rates estímated by. rocket and meteor measurements (e.g. Rees et a1 (L972); Elford and Roper, f967). Thís could perhaps be taken as evidence Èhat gravity \^laves do dominate the spectrum ín the range t hour to 4 hours. However, the author feels that further investigation of the spectrum ín the períod range 15 minutes to t hour may be necessary to establish the contributíons from turbulence in that region. Gravity rlraves domínate at periods of - 5 - 20 minutes, and turbulence appears to dominate at períods ( 5 mínutes. 380. 7.5 DeconvoluÈíon The recording of complex data allows deconvolution of the scattered sígnal to be achieved. This offers Èhe possibility of improving Èhe height resolutíon of the system. However, ín practice this is a procedure which rnust be treated carefully. närtger and Schmidt Q979) discussed the piEfalls and requirements involved in deconvoluËíon procedures. One important point is Èhat if a resolution of x metres is required, then data must be recorded at least at intervals of12 metres. There is no poínt in recording data at 2 km steps, and expectíng Eo get a resolutíon better than 4 km. A second rnajor point concerned r¿ith deconvolutíon concerns the effects of noise,. For reliable deconvoluEíon, the high wavenumber comPonents of the por4/er spectra of the complex amplitude vs. height profile must be recorded accurately. These are usually the weakest signals. Suppose that the críEical wavenumber k" is reached where Èhe signals are so weak that they are comparable to noise levels and that aÈ larger k" the signal 1s even weaker. Then no informat.ion can be obtained at resolutions less than abou¡ (2ttlk"). Digitízation, as díscussed in Chapter VI' rePresents one type of limiting noise, and estimations show that for the pulse used at Buckland Park, and the receivers used and the digitízalLon employed, resolutions of better Èhan about I km are not really possible. The major ís the fact that límítation to deconvolution in dealing with ionospheric data must be used the signal is contínually varyíng, so an instantaneous profile todothedeconvoluÈion(orratbest,aprofileproducedfromcoherent Èo reduce integration of about ls of data). Little averaging is possíble does not the noise. This contrasÈs with cases where Èhe signal structure to reduce the noíse, change with time. In such Cases averagíng can be used and deconvolution becomes a useful' technique' For these reasons outlined above, then, no deconvolution has been such as watching atÈempted ín the work for thi-s thesis. Rather, procedures IV, proved the variation of the height of echoes, as discussed i'n chapter 381 ttlayer'r more useful for fnvestlgations of widths and structures. However, crude deconvoluÈions of the mean po\¡¡er can be attempted. For example, on day BO/O72, the transmitted pulse had a half-power-ha1f- wldth of 3 km. The measured half-porter-hal-f-widËh of the 10 minuÈe mean proflle was about 3.5-4 kn (¡'ig 7.11). If 1t is assumed ÈhaË the height varíation of the layer is Gaussian in form, with half-power-ha1f- width equal tô d krn, then 2 d2 +3 3.s2 * 42 so this lmplles d - 1.8 + 2.6 km. Of course, this is crude, but is not inconsistent wíth previous estimates of a layer of width - 2-4 km (although ín those cases a sharp edged layer was assumed). 382. 7.6 Conclusions concerned One of the more important early discussíons in this chapter the signifícance of spectra formed I^Iíth complex data and it was found Ëhat at least these r4rere unïe1iable unless they were formed wíÈh time ínÈervals from Èhe complex - L}O t, in length, Tk beíng the fading time derived discussed' correlation function. The use of running Po\ÁIer spectra was also Èímes Some space had been devoted earlier to comparisons of fading uslng amplitude-only data and complex data, for data lengths of the order of I minuÈe. rt was found that the parameters were approxímately proportional, points but considerable scatter exisËs in the rel-ation between individual ' po\¡/er specËra The reasons for this became clear r¿hen it was realízed that formed with 1 mínute of data \^tere somewhat unreliable' Another important experíment \^Ias to look for the effects of turbulence using a vertical beam. To do Ëhis, beam broadening effects had to be removed' and considerable care \^7as taken to estimaÈe such effects. It was found since Ëhat turbulence esEimates using the verÈícal beam could noE be made, beam broadening was the domínanÈ contributor to the spectra. and Some comparisons of partíal reflection drifts (pru) measurements Doppter measurements of wind speeds were made. Agreement t"s t*cåll"nt in the region where both Èechníques could be used' Thespectratakenusingatildedbeamprovedtobeextremely interestíng, being considerably wí-der than the expected beam-broadening and' to interpret wínd-shear-"broadeningrr effecËs would predict' It was possible this as due to turbulence effects associated with scales of the order of E f, T beíng the observíng períod and U the mean wínd speed. A procedure wíth was developed by means of which horizontal RMS velocities associated 383 scales of the order of ü r could be calculated, and ít was Èhen shor^m that energy dissípation rates could be calculated. This was one of the major original contríbutions of this chapter' However, ít was poÍnted ouÈ that there \¡Ias some uncertaÍnty in such ,d estimates, due to the uncertaín contribution from gravity waves' If there are no gravity wave effects, then 12 c= (u "d ,rr-' vH'/ T), VH being the horizontal.RMS velociËy meásured from the spectrum producerJ by the tilted beam. It was decided that T2l lay between 2'3 and for 10, and a value of abouË 10 seemed to províde the greatest consístency the data presented. The possibility of using root mean square velocítíes deduced from Full Correlation Analysís of partial reflection drifts llas dfscussed' It have was shoum that often formulae which the author feel-s are erroneous been used to estimate td values. 384. CHÀPTER VIII COMPARISON OF PARTIAL REFLECTION ?ROFILES AND ROCKET MEASUREMENT S OF ELECTRON DENSITY 8.1 Introduetion 8.2 Reflection of a radio pulse in a horizontally stratífied ionosphere 8.2.L Theory a. Pulse convolution b. Fourier Procedure 8.2.2 Some simple aPPlications 8.3 Analysis of sÍmultaneous rockeÈ and partial reflection measurements 8.3.1 Introduction 8.3.2 Experimental details a Technique b Results 8.3.3 Computer símulaËíon of partial reflection heíght profile 8.3.4 Some extra observations of interest a. Diffuse flucÈuations in electron densíty measurements b. SÈructure below B0 km. 8.4. Conclusions 38s. Chapter VIII Comparíson of partíal reflection profiles and rocket measurements of electron densiÈv 8.1 Introduction AlÈhough considerable information concerníng the scatter characteristics of D-region scatt,erers has been presented in this thesis, it has noÈ been possíble Èo positíve1-y define the actual forms of the scatterers ' One technique which can be used to obtain a better description is to actually measure the electron density as a function of heíghË usíng sensitíve probes posit.íoned on rockets. However, no simultaneous observations of partial refl-ecÈion scaËter profiles and rockel- ltleasurements seem to havc appcared in the líterature. This chapter discusses perhaps the first such near- simultaneous observations ever made. Before presenting this daÈa, however, some theory is necessary. If the elect.ron density profile is known, it is desírable that an accurate expected HF scatter profíle can be computed from it. For this reason, the next sectíon is devoted to a discussion of the proPagatíon of an HF radio pulse through the D-region. Some simple results are also presented' Having produced accurate formulae for these processes, the rocket data is then analysed in the following section' and imporÈanÈ insights into the nat,ure of these D-region scatterers are thus obtaíned. Di corre9PondinS to time lo A¿. 3 Fie 8.1 Illustration of a pulse at 0 tÍme, with symbols appropriaÈe to the discussion in the text. The peak of the pulse is aÈ z = O at zero time. The point S will be at z = O aÈ t = At. 386 8. 2 Reflection of a radio pulse in a horizontally stratified ionosphere 8.2.L Theory Already in this thesis iÈ has been mentioned that, the height-profile (of iomplex anplitude) produced by HF scatter from the D-region ís approxímately a convolution between the pulse shape and the reflection coefflcient profile (e.g. Appendix B equation B26b; Chapter 4 equation 4.5.2.4; and also in a 3-dimensional form in chapËer 7, e.g. equatíon 7.4.4.8). However, let us develop a more sophÍsticated formula. 8.2.La Pulse Convolution Consider a \^rave-packet composed of spherícal wavef ronts ' If we consider only the vertically propagating part of the \¡Iavefronts' then the time and space description at Èime È and height z, is gíven approxímately by (e.g. see Appendix B, equation B'17) :j E-, d= jû)È (8.2.1.1) Eo i e. (t r e e o (Ilenceforth, let Eo=l) Here T ís the mean (complex) refractive index of the frequencies contributing to Èhe wavepacket, and U*t is related to the group refractive index (see Appendix B). For most purposes in this chapter urt can be taken as approximately equal to the group refractíve index Ug. fte means ttreal part of", and c denoÈes Ehe speed of light in a vacuum. Note that. this formula is not actually precise, for example, Ehe '=I dependence should also have been associated with an p-%effect, to properly preserve flux. Also, at a frequency of 2MIIz this ray theory is only valid í:1l the D-region; as u approaches zero (e.g. E-regíon), a full wave theory is necessary (see APPendix B). LeÈ us now imagine that the pulse is reflected from a weak reflector at heíght z, (ÍJ:e 8.1). The following assumptions wíll be made. 387. I Fresnel Zone in ( ¡ ) The reflecÈor will be assumed to be at least horízontal extent, so that a reflection coefficient can be evaluated. In fact, only Ëhe vertical sPace co-ordínate is considered, so in effect the ionosphere is assumed t,o be horízontally stratified. (ii) The Born approximation will- be used. That is, after the pulse is forward weakly reflected at some heighL, Ít wíll be assumed that the remaíning propagatíng radiation has sÈrength equal to that whích íL would have had íf there had been no reflection. Thís is generally valÍd as the reflection coefficients are usually less than LO-z ' (iii) It wÍll be assumed that the ray travels in a straight líne, i'ê' suffers no signí'ficant bending of its paÈh' The reflected pulse returns to the ground' and the echo sÈrength at time t, assuming the reflection coeffícient vras R(zr), is (2. Re(þo) ju)(t - , ¿") l"'1.c ( R(21) , ds). e )o (8.2.r.2) /zzr).e.(r - )' -ã* o The arnplitude recorded due to reflection of part s of the pulse in fíg 8.1 from height ,L at time (8.2.1.3) r ,l"t*('=lo" +^t = Jc o is thus, substítuting ín (8. 2.L 2), j "tl ds) tl(,-T c e . (8.2.1.4) s.(Àt) o rz$.x<,r> Now assume there are reflectors at alf heights. tle seek the total signal received at Èime T. Then we sírnply sum (8.2.I.4) over all ,L values, for fixed T. Thus for a given Tr each dÍfferenÈ poínt of the pulse was reflected from a slightly differenË reflection height, so all arrive at the 388. ground aÈ this time T. Let us also use the vírtual height' SO (8.2.1.s) T 2zo/ c (erg. see APPendix B) Let R(z) = r(z)dz. Then inÈegrating (8.2.1.4) gives the returned amplitude at vi rt sal height z AS iu2z o c (8.2.1.s) A ( z ) =e Qo - } o r",{ Ï [1ße(pr)d{) o 'L=o 2ju "tv dt 1 c 'x{ e I a,, n("t) 2z o I 1 Ã(z is complex. This can be written as o ) 2iuz c e { erQ) a s"f*('i.) Ì (zo) where er(zo) e"( z) and z* - ce(u*) dE is Èhe optícal ! l"r group path length ,L o and where ( u dE ,:^ c ) s", (zr) (zt)(lâ"f e I i-s an"effective s.¡,t ("x) = ' o scatt.er funcÈion.tt Thus, t(zo) is a convolution, but not as simple as suggested by equation 8.26b in Appendix B. That form is only valid if U = Ug = 1.0 (ín -1'dependence. partícular, it assumes zero absorption), and there is no (2z r) In that case' (8.2.1.6) equals 826b. This formula, then, gives some insíght inÈo a formula for the refl-ecËed pulse. However, for comPuter operations, it ís sti1l a little inaccurate' A more precíse procedure is descríbed in the followíng secÈion. 8.2 . lb Fouríer Procedure The greatest weakness of the above formulae is that they all assume that the shape of the pulse envelope does not change as it propagates. If the phase refractive index changes'too rapidly, and Èoo non-linearly, as a 389. function of frequency, then this assumptíon is not valid, and the group refractive index becomes a meaningless term. (In fact, often if U* is calculated under such círcumstances (see Appendix B, equation 8.21)) U, can be less than 1.0, suggesting informatíon can propagate at, speeds greater than c. tr{hat this really means though, is that the envelope changes form as iÈ propagates. To properly consider these effects, the following procedure was adopÈed. (i) Ëhe transmitted pulse was first Fourier analyzed on a digital computer ínto its various specÈral components' (ii) thus each spectral component corresponds to a continuous \^/ave' and (8.2.I.'6) can be used wíth e"G*) = constanË. Thus, the echo strength at angular frequency o and vírtual heíght ,o is, by (8.2.1.6) ' 1u2z (8.2.t.7) fu(zo) = n*(ur) "+ J s"rcrrl dz. o ,I u ¿r t- -2ju -9c where S ( z ) r(z ) 'rê ef 1 1 2 ,T o R (o) is the frequency response of the receiver, (this term \^7as and where x noÈ included in equation 8.2.1.6). Here, u(l) Ís the actual refractive índex at frequency (t, and is quite exact. This ís another advanÈage over equat,ion 8.2.L.6; the U used rgas something of an approximat,ion, but here, because r¡le are only dealing with one frequency, it is precisely U for that frequency. Thus, this term (8.2.L.7) is calculated for each frequency 0J obtaíned in the Fouríer Èransform in step (í)r 390. (iil)Finally,theterms(s.2.I.7>aremultípliedbytheFourier a componenË deduced in step (i) and re-added. The resulEant ís complex number r¿hich gives Èhe anplitude and phase of the received signal. (tt i.s usually necessary to rrmix the signal down" to O Hz before the received form produced in a real experiment is properly slrnulated) . Thls procedure thus, in some l¡Iays, simulates NaÈure's own approach to Ëhe problem. It has the following advant'ages (a) r(2,) can be calculated at each frequency, rather than usÍng an average value for all Èhe frequencies coneerned (see Appendix B, equaElon 8.25); z*dz (z) dz r vþ+dz)+vQ) (b) absorpt,ion effects are calculated independently at each frequency; (c)problernsinvolvedwiththepulseenvelopechangingarenot importanÈ - in fact, if the envelope does change, Èhís procedure wíll show it ; (d) Èhe approach also considers Èhe receí-ver frequency resPonse. |'SCATPRF". The program used is given in Appendíx F; Program It wíll be noticed Ehat Èhe sen-tr^lyller equat.ions are used to obËaín refractive indices. 391. 8.2.2 Some Slmpl e Applicatíons Havlng produced program "SCATPRF", ít \iras a sirnple procedure to put in model electron density profiles and examine the reflected portter. It wil-l be recalled from Chapter III, that the parameter used to measure the scattering effect of scattering during the observations made in this thesis r¡ras an effective reflection coefficient, which r,\ras exactly the amplitude received divíded by the amplitude which would have been received had there been total TTSCATPRF" reflection and no absorptÍon. If the anplitude produced by is urultlplled by twice the heíght of scatter, exactly the same Parameter is produced by this numerÍca1 simulation. This made it posslble to examíne more clearly the types of scattering structures necessary to produce the observed recelved Powers. As an example, it was found that with a typical D-region profile, but with a sharp 5% ehange in electron density at 86 krn, an effective 0 n¡ode reflection coefficient of 2 x LO-4 was produced. Likewlse, a sharp 1OZ change ín elecËron density aË 73 krn produced an effective 0 mode reflectíon coefficient of about 4 x 10-5. (The "typical electron densÍ-ty profiles" used were those shown ín figs. 8.4 and 8.5 and small artifícial steps in electron denslty r¿ere then puÈ in those profiles for these investigations). These are reasonable values - typical reflectÍon coefficients are about L-4x10-4 ar TOktr and 2-6x10-4 at 86km. Thecollision frequencies, magnetic fields etc., used in these calculations can be found in AppendÍx B, fig B.1, or in Section 8.3.3. of this chaPter). One point does deserve some conment. In all the above calculations, and fndeed in those to follow, abrupt changes in electron densities were used. The collision frequency rvas assumed to vary slowly. The possibility that abrupt changes ín collision frequency can cause RF reflectlons exísts, but it ís generally assumed Èhat such sharp changes do not occur. Piggott and Thrane (1966) and Lindner (1972) have pointed out that changes to the Differentíal Absorption theory (see Chapter I) would be necessaty if collision 392. frequency changes are important. The colllsion frequency is proportional to the pressure, and sharp steps in pressures are generally assumed not to occur. However, iÈ should be noÈed that sharp sÈeps 1n pressure do occur 1n the troposphere (e.g. see Merríll, L977). In the work presenÈed in this chapter, it will be assumed that a sharp change in eleetron density has no associated colltsion frequency change. Thls approximation is usually assr¡med for Differential Absorptlon measurements (e.g. see Thrane 3!1!, 1968) and some experimental data supportÍng this assumption has been produced by Belrose et al (L972). j I t I t- I I i I 393. 8.3 Analysls o f slmultaneous rocke t and particle ref lection measurements 8.3.1 InÈroducÈ1on A revlerr of stratification of HF partial reflection profiles and the relaÈion Èo structures in hígh resolution electron density profíles has been presented in Chapter I., section 1.4.ld. There is little poínt in re-producing that, here, excePÈ to remind the reader that the few analyses so far carried out have been of a staùisÈical nature (e.g. Manson, Merry and Vincent, 1969) ' It l-s also useful to recall the work presented in Chaptet !V: section 4.5.2,, where Ít was shonm that to achieve significant scatterr the reflecting sÈructure should consist of a change in refractive index which takes place within a thickness of less Ëhan about one quarter of the probing wavelength. However, if an extremely large refractíve índex change occurs, it ts stíll possible that significant scatter can occur even if the changes occur over several wavelengths thickness. But generally, for changes in (e.g' elecËron densiÈy of l-ess Èhan abou| LOi| (which most changes seem to be see Manson, Merry and víncent 1969)), the change should be eompleted wlthin about a quarter wavelength to be lmportant' The followlng section presenÈs some measuremenÈs of electron density and partial reflection profiles made at i,loomera, Australia (31oS, 136on). The measurements rrere very nearly simultaneous in space and time, and Ít ls believed that they were the first of their type' 394. 8.3.2 Experimental details 8r3.2a Techníque Two rockets hrere flown at l,loonera, Australia (31oS, 1360¡), and Partial reflections at 1.98 MIIz were monitored close to and during these fllghts. Aboard the rockets were Langmuír probes, whích were capable of quite high resoluÈÍon measurement of electron density along the rocket paËh. The voltage outpuÈs of the Langmuir probe l¡Iere recorded on photographic film, and later digitized for conputer calculations. The partial reflection experíment was locaÈed about 30 kn from the rocket launch siÈe (see VincenË et al (Lg77) for derails of equiprnent and location wíÈh respect to the launch area and the roclet range). The first flighÈ was denoted AP6/6 , and was fired at 1230 hrs Australian Central St,andard Tine (ACST) on 27t]n l"lÃy, ]r976. The approprlate solar and geomagneÈic parameÈers at the Èime wer (1) solar zeníth angle X = 54o (ti) sunsPot number Rr= 0, and monthly mean F = 13 o (fii) three-hourly Kn index È '1, and mean *n O (for (ii) and (iÍi) see Lincoln, L976) This was the main flight artalyzed. The second flight was denoËed AV5/1 and was fired aÈ 1106 hours ACST on 30th June 1976. Unfortunately, Ëhe data from this flíght was digitized aE only a quite coarse resolution, and then the filn was lost before the author obtaj_n it. Hence a detailed analysís of this data could noÈ be carried out. This was extremely unforËunate, because it apPears that 1t woul-d have been better data to analYze. The partÍal reflection data $rere recorded ín a simíIar way to the procedures descríbed in Chapter III. AnpliÈudes \¡rere recorded at 2 km steps 395. from 82 to 100 km, each range being recorded 5 times per second. The primary purpose of these rockeË flights was for comparisons of partíal reflection drift measuremenÈs and neuËral winds measured by rocket released chemlcals, and the main height regime of inÈerest was between 80 and 100 kn. Consequently, few measurements of partial reflection echoes below 80 km were obtained, a¡rd this limited the region where detailed comparisons of Partlal reflectíon profiles and Langmuir probe measurements could be made. Some 62 to 80 kn data r¿ere recorded about an hour after the flight on the 30th June, but ín Èhis chapter \^re will concenÈrate on the 82-100 km region. It should also be pointed out that none of Ëhe partial reflectíon amplitudes were callbrated so no absolute measurements of reflection coefficients were possible. rypfcal reflection coeffícients for the echoes at lrloomera are known, however. The experiment suffered one other problem. Although Langmuír probes show fine scale fluctuaÈions ín electron density very well, the current produced by them does not bear a constant relationship to the electron density. The ratio between Èhe mean electron density and the Langmuir probe current, usually changes slowly with height (e.g. see Bennett et al, L972, fig. 9), and it is generally necessary to adjust the fíne scale Langmuir probe measurements to fit an elecÈron density profile nade by some other procedure which, while noË producíng the high resolutíon of the Langmuir probe, does at least produce a relÍable elecËron density profile. In the experlments performed for this chapter, no such coarse-resolution measurementss were made. They had been planned, but never eventuaÈed due to pracÈical difficultÍes. The procedure adopted $ras Ëo fit the Langmuir probe measurements Èo a typical quÍet-day electron density proflle - namely, one due to Mechtly eË al, (L972, table 5;"Quiet Sun") . This procedure Ís quite adequate for the purpose of thís chapter, since the fine scale fluctuations are Èhe important structures Noom era 27 Ylay ,1976. O MCDE. Un broted "ål¡ Por"/JB) E 90 -l( 2I t6 t2 o ( ô o 09 TO Locol Time ( hts Fis 8.2 power plotted Smoothed contour plot of 5 min means of and time, for the period leading-up to againsÈ range Notíce the rocket ffight (fZ¡ó hours) on 27th May' L976' oi a 90 km echo, and a strorig blanketíng it" p..""t". and sporadíc-E above lO0 km' The data ís uncalibrated por¡rer units arbiÈrary ' ECHO OCCURRENCE ECHO OCCURRENCE 10 00 27 ¡"1AY 1976 30 JUNE 1976 1230 C ST 1106 CST X = 54o X= 55" ? : UJ 3go 90 80 80 -7.0 - 6.5 -6.0 - 5.5 - 5.0 -6.5 -6.0 -5.5 -5.0 LOG ( CURRTNT ) LOG ( CURNEruT ) assumed that height g.3 of frequency of occurrence of echoes vs. height (1t can be Fig HÍstograms alongside Langmuir probe and range are appro*inåtely equivalent - see text). Plotted currents (solfd lfnes) ¡ ' 396. to be examined, and the true electron densiÈy profile is unlikely to have differed Ëoo much from that of Mechtly et al, since the solar and geomagnetíc parameters matched theirs quite well' 8.3.2b ResulÈs Figure 8.2 shows a contour plot of 5 mínute means of the receíved porüer plotted as a function of range and time for 27th NIay, 1976' Data \fere recorded up to 1200 hrs, half an hour before the rocket flighC' Notice the presence of a layer at abouÈ 90 kn. It will be noted that the mean power varl-es as a function of time, as ís to be expected (see Chapter IV). The (8") strong povrers above 94 to 96 km was due to a strong sporadic E layer at 105-110 lm. This resulted ín total reflection of the incídent radío pulses, with subsequent large pohTers aÈ heights several kllometres either side of the peak (due to Èhe pulse wídÈh) hiding the much weaker partial reflections from above 94-96 kn. By using 5 rninute means the E, echo dominates. However, 1f the profile ls observed on scales of the order of seconds, the fading of the echoes can D-region at t,imes weaken the E" effecÈ at, these heights above 92-94 km, and partÍ4l reflections can at tímes show. For thís reason' the lnstantaneous pr.ofiles of amplÍtude as a function of range l^rere analyzed, and the peaks of the amplitude profiles found. These were then tabulated as a function of range, and the resulËs are shown in Fig 8.3. (the histogram shows the vertÍcal scale as altiÈude. In actual fact Ít. should be range, but on average the two are equivalent. Part of the widËh of each peak may be due to oblique scatter, but the peaks of the hlstogram can be taken to indicaËe scaËÈering layers close to the heÍght índicated). Also shown ín fig 8.3 are the upleg Langmuir probe currenËs, plott,ed at a relatively coarse resolutíon' The histograms for 27th May were taken from randomly selected instant ampliÈude profíles during the period 1018 hours to L2O4 hours ACST' The histogram for 397 . 30th June is for the períod 1105-1108 hours, which surrounded the fÍring tine (1106) of the rocket. A vlsr.¡al inspection of this data alone shows i (i) a very steep ledge Ín currenÈ occurs at around 84 kmrand this was undoubtedly associated with a large lncrease in electron densiÈy; (ii) echoes occurred preferentially at heíghts of about 84-88 km' Just at Èhe top of the sËrong ledge jusË mentioned. (tlote that on 27t:n vtay, the rockeÈ rüas fired half an hour afÈer the lasÈ parÈial reflection echo was recorded, so the slíght disagreement beÈween the 84 kn layer and the echoes at 86-88 km must be a result of thís. The partial reflectÍon results and rocket elecËron densíÈy profíles were sirnultaneous on June 30th, and the 84-86 lun ledge and parËial reflections agreed well then) ¡ and, (íií) echoes also occurred preferentially aÈ heights of 90-94 km, and may be relaÈed to irregular electron density fluctuations at that heighÈ, PartÍcularlY on 27l.}l l{aY. Echoes a1-so appear to occur preferentially at abouË 98 km' but ít is not possible to ascertain how much of this is due to interference effects associated r¿ith the E and E" echoes. Therefore these wÍll not be discussed 1n detail. A peak in scatÈered power was also seen to occur at about 30-81 krn, which does not show on the diagram, since partial reflectíon data was only recorded above 82 krn. Thís nay be associated with Ëhe small ledge at 81 kn on 27th May. Ilaving noted Ëhese correlations, the film fron flíght AP6/6 (27th l4ay Lg76) was digitLzed aE a higher resoluËíon. A resolution of better than 30-40 metres was produced by thís digitizatíon. These Langmuir probe currenËs r^rere then compared Èo Ëhe Mechtly et al (L972) "quiet Sun" profile díscussed r.¿OOMERA (APó/óì zl MAY ]Êfü. f23O C,5T. X=54', R=13. &-0 Ç=o ç =1 TJPLEG- DOh¡l.lLEC-{-¡ MECHTLY ¡L o[(1972. a 3v t- ¡¡ I o trr Ò ô â a ro -? ELECTRON DENSITY ( cM') Fie 8.4 Electron density vs;.height profiles for 2TttlNIay L976. Both the upleg and downleg are shown, as well as the values from Mechtly et al (1972). The díp above 70 kur on the downleg is a result of the rocket tipping over' and so ís noË a real effect. The Mechtly profile has been shifted down about 1.2 kn from rhat presented in table 5 of MechÈly et aJ Q972). 39 8. earlierò Henceforth this will bq called the rrMechtly profilerr. This Mechtly profile was shifted down in height by 1.2 km in order thaÈ the large ledge at 84 km matched the LangmuÍr probe ledge. It ls not surprlsing that such a shift rras necessary; Trost (L979) has shown that this ledge, which is a common feature of many electron density profíles, varÍes consíderably in height over even a few hours. It was found that above about 83 lm, the mean Langmuir probe currenË a¡rd the Mechtly profile bore a constariÈ ratio, R, say. Below 80 kn, the Langmuir profile current to I¡gchtly electron density ratio was about 1.6R on the upleg, and 2.8R on the dovrnleg. The fact that this ratio differed on the up and down legs uray .- indicate that the elecËron densÍÈies differed on Ëhe two legs; or j-t rnay be- símply an instruinental effect or perhaps a r¿ake effecÈ of the rocket. Iùhatever the reason, both sets of data have been adjusÈed to the Mechtly values. Between 80 and 83 km, the probe current Èo Mechtly densfty ratio rüas assr¡med to change linearly from R at 83 km to 1.6R or 2'.8R (depending on the leg beÍng consídered) at B0 km. Thís had the effect of-changing the shape of the 80 to 83 kn section in fig 8.3, but the main features are still present. Figure 8.4 shows Ëhe electron densitÍes calculated in this way for both Èhe up and down legs on 27th May, 1976. The Mechtly profile is also shown. NoËice that the main feaÈures are essenÈially unchanged when compared Èo flg8.3and are Èhe same on both 1egs. Ilowever, notÍ-ce that the downleg has some írregular structure at about, 85 kn not present on the upleg. Thís nay inply Ëhe ledge is at times turbulenÈ, or at least unstable in some way. The ledge at 84 lsn was discussed in Chapter I, seetion 1.2.I. It was mentioned that Chakrabarty eÈ al (1978c) have claimed to be able to explain thc ledge and assocÍated dip at 81 km on chemícal grounds. They clained that it is associated with chemical effects due to the variations of the production 399 + 16 also useful to note that aÈ rates of OZ arnd NO+ wlth heighÈ. IË off rapidly, and Èhis height, the concenÈratÍon of hydronlum ions falls 1'2b) This is partly N0* becones the dominant ion (see Chapter I fig ' which slo'v¡s dor'in the rate due to the rlse in temperat,ure above the mesopaúse, oftransferofmolecularionstohydroniumions,andpartlyduetothe be perhaps speculaÈed decreaslng trend of total neutral density. It could density ratio at 80-83 Ehat the change of the Langmuir current to Mechtly ion dominance below 80 kn km nay be related to the transitíon from hydronium indlcate a real difference to NOt domlnance above. However, lt could also tr'loomera on 27th l4ay betr¡een the Mechtly electron denslties arrd those for Lg76, so this point has not been pursued' ECHO AMPL ITUDE 7 _L 5 10 10" r0 1 I I / t ( t , 95 t I \ t \ I I I t /- -/ 90 4 P I t ( \ \ - - ì E - , J t \ 185 -l (9 lrl :E ì 80 / / \ 75 a a a a a t a a a 2 l. 10 103 10 ELECTRON DENSITY (cm-3) Fíg 8.5 Elecrron density vs. heighË (-) for upleg on 27th l"lay L976. Afso pfotted are a sample true echo amplítude profíle vs. height at 1156 hrs, åo secs (-----), wiËh an effec:ive reflectíon coefficienÈof 10-3 at 92 km, and the calculated echo profile (by program SCATPRF - Appendix profile (-- F) for thís electron densi-ty -.). 400. profile 8.3. 3 Computer simulation of partial reflect ion height Having obtalned Èhe relatÍvely hlgh resolution electron density proflle the next step hras to use it with progr¿rm "SCATPRF" discussed in Section 8.2t to observe the exPected amplltude-height profile. However, bear in nind the assumption made, and ln parÈículax teaLí-ze that thls treatmenÈ does noÈ fully reproduce the echo produced by reflection from the E-reglon. Hence the observed and calculated profiles cannot be compared above about 95 to 100 kn, slnce 1n the real case the tail of the strong E reflection affected the profile at these helghts. (To properly produce the E-regfon echo (whtch was ln fact due to a 6poradic E layer), a full wave Lrealment ls necessary (see Appendix B))' The program also assumed horlzontal stratlfication, which may not be valíd. some,of the ,arr. at 85 km on the do$mleg may have For example, "at,-rcture horlzontal dimensions only of the order of their depth, lf they are due to t,urbulence. This ls much l-ess than one Fresnel zone (a Fresnel zone has a t- radlus of (ìrz)a = 3.6 km for a wavelengLh À of 151-.5 m and a helght z of 85 kn). calculations The followlng parameters r+ere used for the ' (1) the magnetÍc field was taken to be B = 6 x 10-5 x (1.0 + z1637o)-3 vJ"btt" *-2' z = height above ground 1n k:n' (11) the ray dírectíon was Èaken to be at 22.50 to the magnetlc f 1el-d. (iii)theelectroncollisl-onfrequencywithneutralpartlcleswas taken as (71kn 3.579 * 1010 exp{ - 217.47 ) z v 11 m 1.434 x 10 exp{ - 216,39 ) z 2z 71 km. (1v)thecontrolfrequencyofthepulsewas.takenatL.gSl*|z,with a half-amplitude full width of 28 Us ' 401. The results of the analyses are shown in fig 8.5 for the upleg flight of 27th May, Lg76. Also shor^m is a typical anplitude-height proflle for around 1200 hrs. This latter curve has been somewhat arbiErarily calibrated by assuming a reflection coefficient of 10-3 for the 90 km echo, which is quite typical (acÈual measurements were not, made on Èhis day , as discussed previously). Thus the theoretical and observed profÍles can be compared in arnpl-ttude to some degree. However, echo strengths vary conslderably with tlme (see fLg 8.2), so a detailed amplltude comparison is not warranged. sufflce to say that the amplitudes are of s1mlIar orders of magnltude ín the two cases, which is very promislng. More lmportantly, the observed and calculated profil-es do have similar sltapes, and peak at sirnilar altltudes' Thts would thus appear to verffy the prevfous sPeculaÈ1on that the causes of the echoes are (1) the snall- ledge at 81 km (li) the top part of the very large ledge at - 85 km, and (111) the lrregular structure at around 92 km. Notice that Ëhe 85 kn echo occurs due to a ledge much greater than a quarter wavelength ln vertlcal extent, but this 1s because of the large electron density change involved. To verify that the echo was indeed produced by this ledge, and not some small unseen step less than a quarter wavelength deep near 85 km, an analytlcal proflle siurllar to the real ledge was used as well, and Èhe echo was again reproduced. The assumed electron density proflle had a constanL electron density of ', 10'cm---2. below 83 kn, an exponential increase fron 83 km to 85.4 kn, and then a constant electron density of 103'36cn-3 this height, as lllustrated "bo*r" below{i). 402. (¡l ND g a5.4 dN z C J o ! t lse + PU 'e3 e3 to to &.Epulseo-pl¡tude.' N rz) d= ' (dillerentscoles) rhe effect of the assumed sharp change in slopes at 85.4 kn could be quesÈioned, but thís ís not expected to be important. The resultant scattered pulse is approximately a convolution between the pulse and g profile shown above(Xsee previously). Bounding off the corner of Èhe electron densiËy profile, as indÍcated by the broken líne in the first diagram above, modífies dN lÍne in the second diagram, and thÍs will not ä as shown by the broken affect the convolution significantly. The above díagrams also íllustrate why the scattered pulse matches the top of the electron densiËy ledge; - the electron densiÈy gradient is maximum there. It thus appears that specular reflection from thís ledge at least caused some part of the echo observed at 85 kn. lOf course it has been seen in 403. lsotropic, chapterlV that scatter from above 80 kn is generally somewhat This ls so there is almost cerEalnly anoÈher mechanism also contributíng' consistentwithobservaEionsofRiceparametersforthe-85kmecho Ehat there is a specular dlscussed in chapÈer v, ln r¡hLch 1t was determined scaÈter component' The reflection componenÈ, but also a significant random ledge at 84 kn and srnall dip assr.mptlon of horizontal stratíficatlon for Èhe aPpear on boEh legs. aÈ 81 km would appear Èo be valid, slnce both features quiÈe l1kely be turbulent The extra source of more isotropic scatter could ledge witl be dlscussed scatter. The expected turbulent scatter from such a The lrregular in chapter IX and lt will be seen that l-t is signifícant' possibly be structure at the top of the 85 km ledge on the downleg could suggestíve of such turbulence ' Theassumptlonofhorizontalstrat'lficationmaygivesomecausefor sÍm1lariËies srorry at 92 kn. The fíne structure at 92 km does show some (see so may not be entírely on both legs, but is not exactly the same fig 8.4), horlzcntallY sËratlfied' 9.llO l¡)iù i ii: u.) t, o O.)to >\ F{Ét{ooqJ O {J .r{ lJ ii o (d ¡Jt¡{ ÉÉ(Ilcú ï o .r{ t{ ooXlo o +Jt, -oþ(ÚÉ 9.t30 ' Èo o oal{Ja)oÈ o qt qt (n t{ trf30 {J{JUrÉ-d crÈ'r'l 'r{ 'F{ (ü+rÉÉö0- o o lJ 5 'r{ 'r{ rl ¡ r'l +J o I ÞelHq)o qr.G a ËÊBp(d o+r lrg¡0 ô-ôB øiÉ o l4O . (U r{-É((llJo.c trE t, 0 É (/) .r{É+r h'.1 ¡{ Fl(U -l F{Crll 9¡ o (dooo É(dÞt{. i¡froooO1JË{-ô d EO ' r{ O 5Fl o q¡.d É o.{J o -é]JÀ 0,5 J r+1 çi é O -dH'r{ öO E¡ -c-dBÉ...9iË É 0 t¡t 3 Ð lr J¿ 'c,o.rÚO I 1"03o I Z.Ot0 I ä .t I. t¡O e.e ' t; r E () Io t- .o o 5 o f Ir.L t 101 tt 1l il9 123 fírne l5) (kml -=> Alt¡tude (f) I c, o own e ô g c Lo o o IJ Fíg 8.7 plots of the full-width of the díffuseness of the Langmuir probevoltagelinesasafunctlonofheíghtforboththeup ånd down legs. The width of the fluctuations (dÍffuseness) has been approxímately converted to electron density units. (llote that iluctuatíon intensity increases dovmwards on the ù"ito* graph). There ís no díffuseness at all below 94.5 km' and onl] a small parih ar - LL6-L20 km above 112 km. Thís sÍmilarity in heights of the diffuseness is the most importanË poinÈ of itris dialram. Both plots also show peaks around on öS-gO k . (Much ót ttr" finer scale fluctuation is different The times shotm are time from rocket launcht the two 1_egs). (e.g. and so are l0 s different to Ëhose shornm on the film fig 8.6) . 404. 8.3.4 Some extra obs ervations of ínterest measuremenÈs 8. 3 .4a Diffuse fluctuations ín electron densi onepointofpossibleinËerestrelatedtotheLangmuirproberesults heights below concerns Ëhe form of the voltage output on the fílrn' Over all g4.5 km, the record of voltage outPut hTas slnply a line. BuÈ above about g4.5 krn, the l-íne sometínes became a diffuse smear. Fig 8.6 shows an used' example. Ïor dig:tizatíon purposes, the middle of this broad line was But ít was felt worthwhile to investigate the cause of the diffuseness' The width of the line was plotted as a function of height for both the up and goes from a down legs. The result can be seen in Fig 8.7. The voltage líne thin líne to a fully developed dlffuse line wíthin abouÈ 8OO metres aE 94'5 km' heíght on The diffuseness appears to be no accídent, since iÈ agrees well in (apart from both legs (see fig 8.7). There was also a cuËoff at about 112 km the volÈage a small paËch aE Il6-L20 km, not shown on fíg 7), and elsewhere altítude of 128 krn) ouËput riTas a straight líne (the rocket reached a maxímum ' It is conceivable that this diffuseness r¡Ias caused by very fine scale fluctuatíons of electron density, and Ëurbulence could perhaps be a cause' daytíme turboPause The cutoff at around l-12 km may supPort thís conjecture', the The small being somewhere around this helght (see chapter I, Section l-'3'3)' although amount of diffuseness ax LL6-L20 km may contradict this hypothesis, up to Rees eÈ al (Lg72) have found evidence for some degree of turbulence 130 kxn. Recall There ís one argument which possibly Ís agaínst this speculaËion' that the irregular fluctuations at 85 krn on the downleg vrere proPosed as possibly indicating turbulence. should this region not also show such densíties diffuseness ? However, it should be borne in mínd that Èhe electron be there are aroufid one tenth of Èhose aÈ 94 km, so the fluctuations would less. Further, the energy dissipatíon rate ls considerably less at 405. g5 lm than at 94 km (see fig 1.9c ( rS2 Í-s approxínately consÈant between 85 and 95 km)) by a factor of perhaps 3 times, whlch may also reduce the flucËuations. So perhaps Èhe fluctuations are sinply not rrÍsible at 85 km. At greater height, Èhe regÍon $7as perhaps laminar, until about 94.5 ktr' r¡here the region became turbulent. However, as a further argument agaínst the concePt, the 92 km echo appears related to írregular fluctuations around 92 km, as díscussed' Scatter from such heights is usually quasÍ-isotropic, so these fluctuations would probably also be assocÍated with turbulence. Yet no diffuseness shows at 92 km; but that region has sj-milar electron densíties (and presumably simílar energy dissipatior, r"a." to that at 94 km) so should show the effect. Iühatever the reason for the diffuseness, however, it does seem to be a real effecË. As an experlment, appropriate random fluctuations of electron program ITSCATPRF" density r^rere superimposed on Èhe mean profile used earlier, and re-.rtjn. A considerable increase irl. scaÈt"red power was produced. of course such fluctuations would hardly be horizontally sÈratífied, and the fluctuations would not be totally random (turbulence has a spatíal auto- correlatÍon function which has non-zero width), but this does illustraÈe that these random fluctuations would enhance Lhe scattered power' 8.3.4b Struc ture below 80 km In Section 8.2.2, typical electron densíty changes necessary to produce typícal reflecËion coefficients \Álere presented, and it was found that - 5-L07" fluctuations completed within less than a quarter wavelength produced reflectíon coefficienÈs of similar magnitudes (perhaps down by a factor of 3 (1969) or 4) to those observed. As poínted ouÈ by Manson, Merry and Víncent ' several such scatterers could easily explain the observed reflecÈíon coeffícíents ' 406. The Langmuir probe film was searched by eye for such changes. Some such rapid changes were found (e.g. fig 8.8) whtch would not have shown in the digitized daLa. Recall also that VIIF echoes occur from these heights' so the changes in electron density should really be compleËed within less than a few meÈres to cause VIIF scatÈer. However, it was difficulÈ to dlsÈÍnguish betq¡een instrumental and real effect. Without any partíal- reflectíons to compare with, there is liLtle point in any more díscussion, save to say that iÈ is possíble that some echo-eausing electron density changes have been seen below B0 km. 7 68 6 5KM Ïme(s)-'-+ 20OnL Fie 8.8 shown to be real). 407 . 8.4 Conclusions Studies of Langmuir- probe-produced electron density profíles and parËial reflections suggest that the echoes observed can'be related Ëo features on the electron density profile' (i) the 80-81 km echoes aPpears to be related to a small ledge of decreasing electron density at 81 kn' (ii) The 84-86 kn echoes aPPear to be related to a large Ledge of increasíng electron densíty at 84-85 kn, and it is possíble there may also be some related turbulence causíng scaËter of a more ÍsotroPíc nature (iíí) the 92-94 km echoes appear related to an irregular region of electron density fluctuations at around 92 km' One dímensíonal computer símulations support these concluslon, particularly with regard to the 81 and 84-86 km echoes It is also interestíng Ëo reca1l from chapter IVrsection 4'23, thaË an 86 kln echo discussed there was also felt to be assocíated wíth a rapld increase ín absorption (i.e. large electron density ledge). 408. CHAPTER IX INVESTIGATION INTO TTIE GENERATION OF TTIE HF SCATTERERS 9.L Introductíon 9.2 Scatterer correlations .with winds g.2.L Correlations between mean wind proflles and scatterlng laYers a. ExPerimental observatíons b. Results from other references 9.2.2 Gravltyr¡itave effects in producing scatËering layers 9.3 The relatíve roles of turbulence and specular scatter 9.3.1 Turbulence g.3.2 Expected specular to ísotropic scatter ratios 9.4 Conclusions 409. Chapter IX Investlga tíon into the generation of the HF scatterers 9.1 Introduction In Èhe preceding chapters, considerable space has been given to investigations of the nature of the scatteríng ProperËies at H.F. of the scaËterers in Ëhe D-region. Some speculation as to the actual cause of Ëhe scatterers has been presented, but l-ittle in the way of concreÈe.data has been given. The purpose of Èhis chapter is to investígaÈe the possíble causes of the scatÈerers. The chapter begíns by looking for correlation betr¡een mean wind shears, (vertical resolutíon - 4 km) and Ëhe presence of scatteríng layers. Havíng found that strong mean wind shears do noË appear to account for the najority of the layers, investigations into other processes are presented. It is found that gravity waves do appear to play some role, and sev,eral examples will be presented. Having presented these results, some lnvestigations into the scaÈtered por\¡ers associated wíth turbulence will be presented, and it will be shown that ísotropic turbulence is unlíkely to contríbute to the scatter below about 80 km. This is consístent with speculaÈioû presented so far. Then a simple nodel is presented, and it is shornm Èhat ít is possible to simulate the observed facts regarding the isotropy of scatter (see chapter IV) quíte well- using this model. Some bríef speculation regarding the reasons for preferred heights is presented, but the rnajority of such discussions will be left until the surmnarízing chapter (chapter XI) . 100 80 60 Y LrJo 40 f H 5 20 0 0 20 40 :l -l ( m6 km ) Fis-g¿ Minimum wínd shears necessary to produce turbulence for the Brunt-VaÍsala period profile given in Tig I'9a, Chapter I' Ilowever, thls profile should noË be regarded as final and fluctuations in and hence absolute - considerable local 'B ,dUr can occur (see text). I '-d.z I'mln . 410. 9.2 Scatterer correlati ons with winds and scatteri 9 .2,L Correlations b et$reen mean wind ofíles ers. 9 .2.La Experímental observations If turbulence is imporËant in causing the HF radío scatter observed unstable from the D-regíon, then thís requires eíther a hydrostatically requires that temperat,ure gradient, or high wind shears. More precisely, it 2 < 0.25 (e . 2.1.1) Ri = 0B'/( ål ) (sec Chapter II, equation 2.2.2.3) ,r' = ( * fg )1" Brunt-vaísala frequencv' where t Ë å; " 'n* Tbeíngthetempeïature'trtheadiabaticlapserate'zthe height, g the acceleration due to gravity and U the wind vecËor' with Èhe equípment available, ít was noË possible to-accurately äbout the deduce the temperature profile, but information ÍIas available wind velocíties as a function of heíght. Thus it seemed reasonable Èo attempt to find correlations beÈween wind shears and scattering layers ' produce Fíg 9.1 shows a plot of the minímr:m wÍnd shear necessary to turbulenc.(Ri=0.25)forËheBrunt-VaísalafrequencyprofileshownÍn of fig 1.9a ín chapter r. (i.e. låi lmÍn = 2ur").' rt is clear that shears the order of 20 to 40 *=-1k*-1 are necessary to produce turbulence' an average However, it should be borne in mind that this tB profíle is - less smaller vertical scale flucËuatíons of temperature could occur r¿íÈh much in stable temperature gradients, and the Èemperature gradient can change time. Fig 9.1 Ís thus a guide, but should not be taken as fÍnal' Vertical resolution is not only ímportanË with regards to temperature had a profiles. The wínd measurements used in the work for this thesis vertical resolution of about 4 km. Thus even if the measured wínd gradíent scale ) ís less than Ëhe values índicaÈed by fig 9.1, there may be finer 41r_. It ís unseen wlnd shears presenË which are capable of causing turbulence' unlikely Èhat these fíner scale fluctuations would occur on vertícal scales of less than the outer scale of turbulencer but this is certainly less than the 4 km resoluÈion. (Lo 300-600 m at about 85-90 kn; see II' chapter 1, fig 1.9a). Thís point has already been emphasized ín chapter (for (secÈion 2,3.2) and van zand.t et al (1978) have developed a Èheory the getting troposphere and stratosphere) which considers Èhe probabílÍty of pot,entially turbulent wínd shear fluctuatlons hídden in a mean wínd shear' have a one lmportant feature of the theory is that Laxgex mean wind shears larger degree of fluctuatíon of wind shears on fine-r scal-es than the a larger mean wind equipmenÈ resolqtíon, so for a gÍven Ëemperature profile, shear impl-ies more turbulence' to Although the resolutíon used in these experimenEs was Í-nadequate properly consider the generation of turbulence' it was decíded that it was still fruitful to look fot a possíble correlation between mean wind shears and scattering laYers. Perhaps the most strikíng example of such a correlation 1s that maxíma and shown in chapter Trfig 7.LL. The North-South velocity wj-nd-shear is the heights of the scattering l-avers apPear to be at simílar heíghts ' It case' an not clear however, ¡ühY the East-tr{esÈ wínds played no role' In this that attempt was made to estimate tU' for this day at 86 km, assuming scatter from that heíght was due to a mean wÍnd shear defined by fig 7.11' (chapter 7, section 7-4.5) IËturnsout,however,thatthiscorrelaLionofwindshearsand ,scattering layers is the exception rather than the rule. Many measurements for of wínd velocity as a function of heíght were produced duríng the work of this thesís, and these were plotted and compared to the heíghÈ profiles means to hourly scatter. The wind velocities used varied from 10 minute means.ExamplesaÏeshornminfigsg.2,g.3andg.4.Thecasesshownare Buckland belíeved to be quite typical' For example' hourl-y profiles for 1 BUCKLAN PAR l4I ++ t206 \ \ /t-I t, 't 6L -62 165 6 T6T6 No.th*otd. Eost..rord. \\\ Lo en 62 ' 1206 t+l6 ttþ+6 + lr- æ I 5 Loye" No Loy". o ht \ eg:68: Heig o 50 profiles (top 5) and matching hodographs (bottom 5) Fíe 9.2 Height (28th April for 5 x 10 minute'period= or, à"y 77/I:-B at Buckland Park given in the toP left hand corners are the start Lg77). The numbers all tímes of each lO rninãte block (local time). Notice Èhat not issing Points indicate that no easurements \,rere obtaíned ín the the rejection criteria used in see Stubbs' L976). The scale of ís 100 ms-l wíde. The vertical in kílometres. Northward and on the hodographs are EasÈward winds are taken as positíve. The numbers heighÈs ín kílometres. The heíghts of HF scatÈeríng layers are also indicated on all graPhs. 9.3@, 1500 t700 t900 oìo 70 \ \ illiñ-'*){ Eoslu/erd I I tt 1600 t700 t900 E 96 'L ao 2000 goîloooo Loyer ão --¡¡r... No LoYer o - ;ry, e.g. '6ano Herg ht l* o phs prePared from hourlY means for (c) 77 /306 f or Tor'msville top left hand corners are the start 11 heíghts are in km, and all Ie on these graPhs is dífferent to that for r.ig 9.2. Scattering layers are also índicated. The daytime echo appeared to be strongLst during the períod 305/f500 - 305/2000 -90 km were (fig b), when tidal effects werã quite weak, and total wind speeds 'somewhat less than those on days j04 and 306. However, the winds on day became important 305 were quite variable. On dãy 306, a quite clear tide period íere often quite smooth (e'g' fig c) ' The and hodographs for Ehis all (the echo strengths on day 306 seemed to be quite weak, íf present at questíon marks on E echo hid whatever g0 km echo vTas presenÈ - hence the the 90 km "laYern in fig c). 93&l ßoo r 500 7IO0 l 1r \ \ ì \ ¡50 âf) .5O 1600 t100 East,^rund ù\\iN,N\\ - \.S Loyer *lÍoo N t300 I 500 s6Pì F} t600 t700 Loyer 50 No Loyer o ùt e.9)ó8" Heightf krd' t too -too -NO5o 93tc¡ 6 il00 a I o I a 1 a \ 80 50 -50 0 50 t -50 0 I I -North*o- Eostword L oy"" 60 .50 o oo u00 1+ 0 92 90 ee ? ú 92 o4 e6 tl I + 9ì + 96 eol 98 'ge 94øt -LOO too 150o^ 1700 ., 200ro^ / -: J If ?- lI t: 220o- \+n \xtl -10 SS \ -50- -â'.o ¿{¿ -tí.ô 15o-0 17?o 200q ",#h:: ,t'#o 270o Loyer. o No Loyer. e.g,'68". Heightkrrn) o Fíg 9.4 (a) Height profiles.and hodographs-for prepared from hourly means days (a) llTzoe'^na (b) i7/3o7 guckland Park (November 1977). for (local Ëimes) The numbers in the top left-hand corners are the start times for each hour. Scattäring layers are also indícated. As a point of Ínterest, fíg g.4(a) at t7o0 shows quite a- good example of a tÍde (except for the 90 km and 98 krn points): \, ís around 24 km' so this could be a S\ mode. 9.4út Otr0O 07 00 o9oo .7 ) / I o. ðo .fO BO go ) Northword. ,10 I 70 - Fostwo"d. \\\\ L",yun N OrrO O Q70o, o9o.o E *98 9t6 Lo¡or o l,to Loyet øn."6ï"n Height (knl -t *50 0 4L2. Park were produced for 5 days in November, L977, and those diagrams shown in fíg 9.4 are typical of that period. The graphs are presented both ín the form of heíghÈ profiles, and hådographs. Examinatíon suggesËs' as ít did for most of the data examined, that the scatteríng layers do not always appear to be associaÈed wíÈh the largest wind shears. For some data' even hígher t.ernporal-resolutior- (I-2 minutes) was used. I,üith thís resolutÍon, ít agaín díd not appear that the scattering J-ayers \^rere associated wíÈh wind shears ' certaínly cases were found in r¿hich power bursts occurred durÍng a wind shear enhancement, but thís was by no means the rule; in fact, ít almost appeared to be the excePtion. The possibility exists that turbulence could form but produce no backscatter. AfEer all, radío wave backscatter requires a large potential refractíve index gradient as well as turbulence (chapter II, equations 2'3'2'LÐ' Thls ís a factor which could in part account for the lack of a correlatíon' Perhaps, too, the insufficient resol-ution may be the reason for a lack of correlation. The lack of temperaÈure lnformation also restrícts the inËerpretatj-on of the above result. On the other hand, perhaps the informatíon suggests Èhat rnrind-shear generated turbulence is not an ímportant cause of the scatterers. It can be stated, however, that measuïements with a vertícal resolutíon of the order of 4 km and t emporal resolution of mÍnutes to hours and do not show anY índicatíon of a s correl-at íon beËween HF scatterers wínd shears. trIind shears can often be found assocÍated wiÈh scattering layers, but stronger wind sheaIs on Èhe same heíght profile often occur wíth no associated scatterers. However, one poínt which may be importanË is that a strong wínd jet usually exists at 70 - 74 km (easEward in winter, westward in summer; also see fig 1.7, chapÈer 1), and HF scatterers often occur preferentially near this jet. It has not been possible Ëo determine unambíguously whether the two features are related however. 4L3. 9 .2.Lb ResulLs from other references Having discussed the above observations, 1t Ís useful to look at HF scatterers símilar work done by other authors. Few correlatíons between done w1Èh rockets and wind shears have been aÈtempted, but some work has been partÍcularly in the in comparíng wind profiles and layers of turbulence, may be region of B0 to 110 km. IL has already been proposed that turbulence to imPortant in producing IIF Scatter at these heights, so ít is useful consider these observations. Howeverr even rocket wind measurements have problem' vertical resolutíons of at least 1-2 km, so agaín resolution could be a As pointed,out in Chapter l, sectíon 1.3.3., turbulence is spatially turbulence and temporally intermíttent at heights of BO to 110 km. often thaË appears in horizontal bands. Blamont and Barat (L967) suggested at the maxima turbulence âppeared to be strongest at maxima i-n the winds' not important role in wind shears. They also felt that gravíty waves played an gravity in the generation of turbulence. Roper (1971) also has suggested that wind shears ín waves are more Ímportant in the generation of turbulence than generatíon of unstable the mean wind. Roperts explanation lnvolved the (L972) little temperature regimes due Èo gravity \¡/aves. Rees et al found the turbulence correlatíon between turbulence and wind shears and suggested that Anandarao et al (1978) may be associated wíËh unstable temperature gradients. but have also noted that turbulence is temporally and spatially intermittent, and turbulence' in cl-aímed to find a strong correlaÈion between wind shears (1976) and Sidi and contrast Èo the previous authors. Teitelbaum and sidi may be important Teitel-baum (1978) have also proposed that strong wind shears in the generation of turbulence, but the wind shears they proposed I^rere induced by non-línear interaction between tides and gravity waves' Lloyd, Low and víncent (1973) have also found correlations between turbulence and high wind wind shears) play shears and suggested Èhat shear \¡Iaves (waves índuced by strong 4L4. an lmportant rol-e 1n the generation of turbulence' Thus it appears some authors see wind-shears associated q¡ith strong turbulence, buÈ that this is not always the case. other processes are al-so important for the generation of turbulent 1-amínae, and one of the princípal suggesËions ís that of graviÈy waves. The nexË section of this chapter wfll exanine gravity r^raves and their relation to IIF scaÈterers. ai) 1æO l3+o laso Bucl.fond Par k. t t I ¡ t 77/151 , à ,, f æ (' æ Northwa .d -TããÎîo"d ì Heightkhl 6o a -50 50 VE e). Chapt (a) (i) Contour díagrams of powers at Buckland Park on day 77/L5L. (Also see fig 4.Lc, Fíg 9.5 the winds at 72, 74 and with associa ted North-South wind variaÈions at 74 km. (the points are averages of between heights). A correlation beËr¿een 76 km; all these heíghts shor¿ed the oscillatíon, wíth no phase shift appears to exist., Typical heíght Profíle of 10 m mean wínds, rnaximum NorËhward wínds and power bursts çÐ top left shown, for Ëimes jusË bef ore the tirne Period shown in (i) . The nurnbers in the with scatËering layers q Èypical the start times of the 10 mín b lock used to obtain the profile. The profiles are uite hand corners are shown in (i) and day. In particular, the strong easÈhtard j et hras PromÍnent during the Èime inÈerval for this the fíles are typical, none are shown Èhe structure above 76 km was quíte írregular during this period. Since Pro near 74 km appear to be above or belorv for the period 1445-1530. For these profiles , the layer of scatËerers w shears of the jet. However, the Èhe peak jet velocity, and could be taken to be associated wíth maximum ind author feels thaÈ this ís not always so, and is coincidental in Lhese cases. 1530 1450 1510 15?0 (¡) N S WIND 71, km , 31 MAY 1977 10 2A I o E -20 -L0 X MODE , NARROW BEAM AT 0.0' 80 /ó 76 1l jE 72 r! (9 70 z. É. 66 64 b/. 145 0 1500 1510 1520 1/rJf t 72 km 7*km 76 km a a' a Oa aa rr a I -__ ___ a - _J a - t .-.åb --to ¡t t t#- --;'- I - a ¡ --- - a 7-t ' 1A km .gO kn Btþn a \ a \ \ o \ '\-\\. a \ \' I a ¡ a a a, { a a t a aa ll a a \ I a a a \ a \ \ \ \ 50 8¡tkm 8ó km BB km I tal a a a o¡ a a"a O - a a 'ì.':': ,+r<{__ a a =-- a r-êea a -2-ttT' I a a a a I a a -50 E -',0 \\-ì=--+- Maon V¡nd polodzotron Fíg e.s(b) Fluctuating components of the wínds on day 78/067 at Buckland Park. The arrows show Ëhe mean wind duríng the period 1100-1600, and the dots represent 5 min mean winds, with the mean for the full period 1100-1600 subtracted. The broken lines índicaÈe best fít lines, Of partícular interest is the polarized form of these fluctuating components at B0 and 82 km. Layers of strong HF radio \^rave scatter occurred at - 7O-74 km and - 84-86 krn. The graphs are due to S. Ba1l, (1981) . (o) ¿l 00 1,1716-17n 2172A726 3Ê rcq, f, :Ë 70 ltìO+ - 7627, =< ./ J 2A 0 20 -40 -l Veloci ty(m5 ) Northwatd Eostword \li\-ils Loye" wínds on day 76/259 (15th SepÈember, n blocks of data are shown for heíghts m is the mean Profile for the Períod .ayers are also indicated' The inset rproximaLely to scale) for the 90 km may be related to echo. NoËíce the strong wind shear at 16 to 78 km, which i"y.t aÈ that height. Also notice that the scatËeríng layer the scatterj-ng jet. There seems Ëo be at 66-78 km is ãgáin assocíated with a strong wínd Ì¡rave actíviÈy present on this day, judging by the regular consíderable gravity further wind oscillarions with height in ihis figure. Fig 9.6(b) shows evidence f or gravity \^rave activíty on thís day ' Fíe 9.6(b) Temporal variations of the po\^/er and winds for A"y øTfSg ar 66-68 km. (a scatterÍng layer occured at this height). 1he pohrers are uncalibrated and are 3 minute means for 68 fm,. The poínts are plotted at the mídd1e of each 3 min block' The 1 minuÈe wind means for heights of 66 to 70 km are joined by the broken lines, and an aPpÏoxímate 5 min running mean is råpresented by the solid líne. The squares on these solíd línes are 5 min means for the períod - 2% mins to either síde (l'lthough at - 1455, the Latge northward vel0ciËy 'rspikes'r have been ígnored) . The vertical lines at the base of graph (i) show tímes when large po\¡Ier bursts begin, and these correlate reasonably well with mãxima in Ëhe Northward wínd velocity. These fluctuations have a period of abouÈ 7 to 10 minutes ' &) (¡) mtn mêong. 16 68 KM ú, t+ ¡fic h L to P -oç I o tr 6 t.lj 3 4 o -4 0_ E¡ o (ttJ I ll Norfh-orJ ll Ë il J n B o \o \o i(,) E l--- H O o Eå,rh/o n I J I (t lì UJ t I I t \ 20 v u' tt, o I v z I t0 I I I 3 0 1s05 1605 LOCAL TIME(h"s) îtrÍ, É 92 KM. E osf*ord t¡l ¡nd o o 0 ) -5 0905 1005 1105 1205 Fíg 9.7 Eastward winds aE 92 km for üloomera on 30Ëh June, ffi6, pLotted as a function of time' The points are 10 min me"ns, and the error bars are sample standard deviation' Single poirrt" indicate that only 1 point hlas recorded during tne]0-minute interval. The shaded areas represerit strong bursts of power which occurred in a layer at 90-94 km' 4L5 9.2.2 Gravity wave effects in Producíng siattering lavers Some índications of gravity $Iave effects associated wíth radio frequency scatterers in the D-region have already been presented in Chapter IV. For example, f.ig 4.8 showed a possíble graviÈy wave induced variation ofi power and fading Èimes. The po\ÁIer and f ading times \ÁIere possibly antí- correlated. Fíg 4.9 showed evidence of downward moving patches of scattel:' reproduced at quasi-regular intervals, and thís was also tentatively interpreÈed as a graviÈy wave effect' Fig 9.5(a) shows a further example of possible gravity \^tave effects' appears A sÈrong correlation between povler bursts and maximum northward winds to exist, wíth the northward wj-nds showing a quasí-regular oscillation whích could be indicative of a gravíty k7ave. Also shown aie typical 10 mín mean the wind height profiles near this tíme, and the sínilarity of the heights of scatÈering layer and the 74 km wínd jet can again be seen' Fig 9.5(b) shows evidence of a gravity wave at 80-82 km. The graphs are due to Ball (1981). At heíghts below 78 km, little evídence of poLaxízation shows, buË at BO-82 km polarizaËiorl is evident. This may (e'g' índicate eíther some filtering of upward propagating T¡Iaves below B0 km (perhaps, an see Hines and Reddy, :1967), amplificatíon of the wave say, via over-reflection process (e.g. Líndzen and Tung, (1978)), or generation of a gravíty wave in-siÈu. I,trave ducting ís another possíbility. The wave does not appear Ëo exist above 84 km. This could suggest a crítícal level absorption or over-reflection' or could suggest that the wave breaks ' Layers and ít ís not of HF scatÈerers occurred at 70-74 km and 84-88 km on this day, impossíble that the B4-BB km layer \^las associated with the gravíty wave may be mentioned. tr{e have seen in Chapter VIII Èhat the 84-88 kur layer associated wíth a Laxge elect,ron densíty ledge, and it wíll be seen shortly 416. required such a that productÍon of signíficant scaËter by turbulence also gradíent.However,ifagravitywavewerealsotogenerateturbulence'for examplerthenthiswouldenhancethescatterrandmayhelpexplaínthe temporal variabilitY. Fígs9.6and9.7aLsoíllustratepossiblegravitywaveeffects.In fíg 9.6, as in fig 9.5(a)' a correlation wiÈlr NorLh-South oscíllations 76-78 km can also be appears to exist. Evidence of a wind-shear effect at quite isotropic on this seen in fig 9.6. The scatterers at 76-78 km were To be falr' the day, as discussed in Chapter IV, section 4'2' and fíg 4'5' being oDly one cyclc' date presented ín fíg g.7 is far from conclusive, Èhere and fíg 4'8' However, it is included because it contrasts wíth fíg9'5(a) the fígs 9'5(a) and 9'6(b) since por^rer bursts occur twice per cycle, whereas and4.Bshowburstsofpo\feronlyoncepercycle. in whích a gravíty The examples gíven above are rather special cases to observe separate wave ís clearly vlsíble. It is not always possible if gravity waves were graviËy \^Iaves, particularly íf many are Present. However, layers would be more ímportant, ít might at least be expected that scattering more variable' some evídenL on days when the wind motions apPeared to be to exist' but no fírm evidence for this at Tournsville (November, Lg77) seemed conclusions have been reached' (e'g' see fig 9'3)' powers can other examples of gravíty wave-induced effects on scatterlng (1977) have be found ín the literature. For example, Harper and Inloodman VIIF radar, and found observed VHF scatter from - 70 krn wíth a powerful frequency of gravity that power bursts occurred at. approximaÈely Èwíce the (most such graviËy $raves which appeared to be associated wj-th the layer They also found that these gravíty vTaves had periods of around 10 minutes). a few kílomeËres above wave oscillations did not appear to extend more than 4L7 . many of the or below the scatteríng layer. This is also consistent with fig 9.5(a) observatíons made for this thesis (e.g. the gravity wave seen in Harper and coulcl not be seen clearly above 78 km or below about 70 kn) ' I¡7aves \¡rere generated at I,{oodman felt that this indícated that either the and these levels and were evanescent, or that the waves had been trapped ducted. ì4i11er et al (1978) have found evidence of a correlation between 60-80 km' breaking gravity \^7aves and strong bursts of vlIF scatter at In the Èroposphere, too, correlaÈions between gravity \^7aves and power of power at bursts have been found. van zanð,t et al, (1979) have found bursts gravity wave frequencies VIIF with freque4eies approximately equal to associated ' Merrill and Grant (Lg7g), using a radio-wavelength of about l0 cm, have encounter presented quite a detaíled account of a gravity-wave-critical-level is ín the atmospheric boundary l:ayet. In their model, a gravity wave the wínd- critícally absorbed at a wind shear, and thís process steepens shear. This generates turbulence, which smoothes out the wínd shear' Thus then begins to Ëhe turbulence ceases, upon which the gravíty v/ave absorption the wavelength steepen the wínd shear agaín, and the Process repeats' However, of the gravity v/aves was less than a few hundred metres, and such r¿aves D-region gravity would not be observed in the D-region by present radars ' The dírectly' but vraves mentioned thus far may not have produced this effect higher could perhaps generate these smaller scale gravity r¡/aves in the atmosphere (e.g. by non-linear effects, or by breakíng)' is The reasons why gravíty \¡laves can generaËe radío I^Iave scalterers that critícal- another problem. Several papers have been presented which show steepen absorpËion of gravity waves at a wínd shear can quiËe signíficantly This can the wind shear (e.g. Jones and Houghton, L97L; r'rÍ-tts L978, L979) ' below lead to Kelvin-Helmholtz ínstabilities (ín the oceans and atmosphere 418. 130 km), and these may then induce layers of turbulence (e.g. Fnitts, L97B Ig79). Hodges (1967) has also poínted out that gravity !ìIaves can produce unstable ËemperaÈure gradients, particularly íf the background atmosphere is close to unstable. It is interesting Ëo noÈe that in thís model, turbulence is expected to occur only once per gravity wave cycle. There thus appears to be quite a few models which can explain the generation of turbulence by gravity $Iaves. But it should be recalled that scatter from below about B0 km is usually quite specular. The usual concepts of turbulence do not appear capable of expl-aining this feature' Thís suggests that either the scaËterers are not relaÈed to ísotropic turbulence' or else turbulence can somehow produce sÈratification. Some evidence that the l-atter statement may be true has appeared in recent literature' For example' Bolgíano (1963) has proposed that intense turbulence could mix up a potentíal refracÈive index gradient, producing a consLant mean value across the turbulent 1ayer, rnriËh steps eíÈher síde, as illustrated ín the followíng diagram. Thís will be referred to frequently during this chapter, and wíll be called the Bolgiano model. It will be used prímarily for illustratíve purposes. This does not mean the author has accepted it. 0 G.C turbuleni loyer. N .9 !) originol n(z) .,(zi ofier turbulence begf ns. refroc*íve índex, n (z). Bolg1ano (1968) has also mentioned that turbulence in strong wind shears can be quíte anisotropic. Other experiments have appeared ¡^rhich suggest that 4L9. seÈtle out once the generatíng source has been removed, turbulent reglons fnto horixontal 1-ayers. llowever, dlbcussíon of these effects will be left untíl- ChaPter XI. play At present,, 1t is sufflcient to say Ëhat gravlty ülaves do appear to a slgnífícant rol-e in the generation of radio hTave scatterers in the D-region' Fig 9.8 AsshornminChapterllrequations2'3'2'LOaarrd2'3'2'11' the refractive index sÈructure constanÈ cnt is given by 4/3 2 2 c a L where n o l" l2 dl,nN -4 M'= ân --Ã;- (1.4 x 10 )Ì ãñ *{+tf}+r"> âì ðn m + -T; ¡rm and L ís the outer scale, T is the absolute temperat.ure' f 1s the adlabatic lapse rate, z Ls the height, n is tne refracti$e índex, N ís thã electron density, and l* is the electrc¡n collisíon frequency. The electron density profile of Mechtly et al (L972, fLg 5, quiet sun) (also see Chapter VIII) was adoptedr.and ."q calculated ior this profile, using tiie full Sen-Inlyller equations foi calculatíons of the refractíve índex. The profiles are for L'98 lü12' Figg.BashowstherelativeconÈributionsoftheterms dT d .Cn N -4 +f ) B and C 1.4 x 10 A = +, dz a dz Notice the large effect of qåJ at 85 km, due to the large electron density ledge at that height (see chapter vIII). The term 1 ( dT + f Ís only approximaËe, but is adequate for the purpose T d" a ) for whích it is used. a\) ân m In Fíg 9.Bb' l"nl' and l' are shown. It can be âì.m T; the seen that the latter term does not make a large contribuÈion to potential refractíve index graaTent' excePt at the minima (and in these ä"".", ít will be seen thaË turbulert scatter could not be detected anyway). (0 b \0 ALT]TUDE(KM ) ALTITUDE(KM) s CD --s, \o g Or O. I I O q l-0 \ \ \ J\/-¿ \ I t lr. f + o ll o o o ll tl .-\ -ùr CXÐ> o 's ÐJA- rp = xH I p I 0 x + t- G) 't a o --sol Ff ¿ J. + il 3 lr }' }' Fie 9.8(c) Plots of the assumed electron density profíle, Ëhe potent.ial refractive index profile, and the refracÈive index structure constant C2- proiíl. (using outer scales as given in Fig 1.9a, ChapÈer I, àhd assuming , 4/3 o c_¿no-e' = L^ . l¡r l') 2 as a function of height, at 1.98 lüIz. Also shown are the C' values required to produce effective reflectíon coefficients of R = 10-5 , I0-4, and 10-3 (see equatíon 9 .3.I.2), assuming turbulence fills the radar vol-ume (solid vertícal lines). Effective reflection coefficients of about 10-5 are just detectable with the Buckland Parlc array. The ( shows proflle whÍch would be broken llne --- ) the C-2n produced if the elpctron density gradient term *dfrnN _q\ dz = dz r^ras not important. Notice thaÈ at 85 km, an electron density ledge similar to that which occurs in this profíle is necessary to produce signifícant turbulent scatter. ) The C_- values may be increased by íncreasing the energy t/'), dissipaËion rÏa" ,d (Lo - e4 ,u- rnr.n may occur in localized patches, thus producing the temporal variations of echo strength received. If turbulence does not fíl1 the radar volume, this wíll reduce the observed effective reflection coefficient. The ímportant points to notice are (i) turbulence wíll cause observable HF scatter above B0 krn under normal conditions. (ii) turbulence will not cause observable HF scatter belor¿ B0 km under normal conditions - and certainly cannot _L account for reflection coeffícients of greater Ëhan 10 j An íncrease in td of greaÈer than 100 tímes is necessary to produce this, and if the layers are only a few hundred metres thick (e.g. Chapter IV; secÈion 4.5.3), even larger eà values are necessary to explaín the observed effective rëflection coeff ici-ents . l¡ EI ron enei t (c m 9.8fc) 1 / elecfron I densi ( llcchtl" et ot. , ? f"Ho s, fviet 5r¡n Y8 .':.t:'id:_ -oo .':t tJ r:j.'l JE 7 ".;r ¡ r.lli ( ó0 F 5 3 1 0 I 1 0 -'18 -14 -10 -6 tog,J c; ¡,';?s¡,ltål ('"-'ü 420. 9.3 The relatíve roles of turbulence and sP ecular scatter 9 .3.,1_ Turbulence Turbul-ence has been regularly dj-scussed in thís thesis as a mechanísm for producing the radio klave scaÈter observed. However, it is inËeresting to calculate the expected effective reflection coefficients for such scatter as a function of height, for typical atmospheric conditions. This can readily be done, by utilizing equations 2.3.2.10a', 2.3-2.LL and 2.4.L.5 from Chapter II, and equation 3.3.2.1-9, Chapter III' Thus, assumíng the turbulence fills the radar volume, - rl3 2 ( c (9.3.1.1) R2 uT, .38) ). n , and for the narrow beam aË Buckland Park, Ëhe half por¡rer half width of the Èhe pulse length L ís typically 4 to 6 km polar diagram in 0r-'4 = *4.5o, and the wavelength À is 151.5 rn at 2 þffi2. So, 2 4l3 t (e C (where c2 M. (, by equation .3.r.2) *z- 2.4 n n =Lol e (2.3.2.rr) (aæ1) for the Buckland Park array (narrow beam) and a pulse length of 6 krn. This equation was also presented in Chapter IV, sectíon 4.2.L. " IÈ has been assumed that the effective radar volume ís (no 0r)2 nL , Ro beíng the dístance t,o the scatteríng region. Tn Chapter X, this approximation wíll be improved upon, but. for these PurPoses ít is quíte adequate. In Chapter IV, section 4.2.L, typical ;2 values for turbulence were found, but a detaíled height profile had not yet been prepared. In the 2-t work presented in this section, profiles of ,n' , and hence Ro , have been calculated for typical electron densÍty profiles and typical outer sc'ales ' 42L. The ouÈer scal-es were taken frorn Fig 1.9a, Chapter I, and the electron density profile used was that due to Mechtly et al (L972, table 5, "quieÈ sunt'). Recall this profile was also used ín Chapter VIII. It was produced from several individual profiles, so any small scale irregular fluctuations have been smoothed out. The results of this analysís are shown in Fig 9.8. perhaps the most ímportant conclusíon is that above B0 km, Ëurbulent scatter is capable of causing reflection coeffícients comparabl-e (equati on 2.2.4.2à) so aft to those observed. Recal.l Ëhat Lo .u% ^u-t/2 , 2 increase of td by 4 times can increase C' by 2.5 times. Thus a faírlY small increase in the energy díssípation rate can vary the amount of HF scaLter at 1.98 lú12. It should be noted that ín these calculations it has been assumed thaL turbulence fills the radar volume. If this were not so, and the turbulent layer \¡rere say one outer scale (Lo) in vertical extent, then in equaËion 9.3.1.1, L wouldbe replaced by Lo. This would mean that ;2 would have an ,o'/' dependence on Lor noË an dependence as suggested by "oo/t equation 2.3.2.LL. If, however, the layer thickness ere Lo, equation 9.3.1.2, would have to be modifíed, and Ëhe constant 2.4 would be reduced. Bear Ín mind, though, that Lo need not necessaríly be the thickness of turbulence. RaËher, Lo is the scale over whích turbulent fluctuatíons become comparable to changes in the mean values. The expected scaËter due to turbulence at the B5 km ledge is comparable to the specular scatter due to the ledge itself, (as calculated in Chapter VIII). Thís is consistent with results from Chapter V - boËh specular and ísotropíc scatter play roles for the 86 km echo. If the ledge does not exist, turbulent scatter Í-s quite Inleak' as is the specular reflection. A second imporEant conclusion ís that dírect turbulenË scatter does not appear to be likely to be the reason for scatter from heíghts bel-ow B0 km. 422. The powers produced by this mechanísm are much ¡¿eaker than those observed. This also supports conclusíons reached previously. This finding does not preclude turbul-ence as a cause of the scaÈteï - if turbulence could produce sÈratified steps, perhaps as proposed by Bolgiano (see earlier discussíon) the scatËer could índeed be associated wíth turbulence - but not in the usual sense of turbulent scaLter. If in fact an extremely large electron density gradient could be produced at these lower heíghts, then turbulent scaÈter could make a contributíon, but if stratified scatter dominates when normal el-ectron density gradients are present' then it coul-d still be expected to domínate with high gradients. Haug et-al Q977) have proposed just. such a Bolgiano-type descríption Ëo explaín very strong reflection frorn 60 km (in the auroral region) with an effective reflection coefficient of between lg-J-? and LO-¿.-) They found this very strong scatter occurred duríng a períod of. high electron precipítaËion, and proposed that the assocÍated large electron density increase in the D-region produced the large elecËron density gradient necessary to produce this large effective reflection coeffícient. A similar strong echo from 70 km (n - ro-3) aL Torn¡nsville on 22nd Januaryr l9B0, has already been mentioned ín Chapter IV, section 4'2'L' and presumably a similar l-arge electron densíty gradient r^ras assocíated wíth it. Llhether the Bolgiano model' or some other mechanism, produces the electron density sËep producing this scatter is debatable' However, the díscussion in this sectíon does show Èhat (i) turbulent scatËer makes a signífícanË conLribut,ion above B0 km (ii) turbulent scatter does noL make a contríbution from below 75 to B0 km, unless veïy unusual conditions prevail. Even if such conditions do prevail, íÈ ís possíble Èhat the specular processes causing scatter rnay still dominate. This will be díscussed in the followíng sectíon. 423. 9 .3.2 ExpecËed s pe cular to ísotropic scatter ratios I,tre have seen that horizontal strata, and turbulence, both occur in the D-region. The expected turbulent scatter ís (assuming turbulence fills Èhe scaËtering volume) -L/3 413 (e Nv (.38) À L .3.2. 1) ;z turb eff o lr.l' ,4 _3/ 2 of scatter, t; e ,B ís the outer Vef f is Ëhe effective volume s and M is the potentíal refractive index gradíent. cale, e Now consider the following very simple model. Let us assume that in the atmosphere there exists patches of t.urbulence, and also horizontal-l-y stratifíed steps'in electron density, of the order of a Kolmogoroff microscale (n) in vertical extenË (e.g. the edges of the layer produced by the Bolgiano model). These stePs, and paËches of turbulencer may or may noÈ be related. All they need do ís exíst. Asstime that the size of these steps is proportional to the electron densíty gradient. In the Bolgiano case, ít would also be proportional to the layer wídth. Then the effective reflecËíon coeffícient 1s (e R *, . EQ), (r = refractíve index) .3.2 .2) spec = # where E(z) describes Èhe efficiency of scatter of the sÈep, and z is the height, and K, is an unknor^m constant. Then -1 2 -4/ 3 (e .3.2.3) R2 /R2 (v ) E(z) L spec turb À eff o Here, it has been assumed that dn N ân dN g let-l where N = electron densíty. dz AN dz An This approximaÈion wíll be discussed shortly. ( T can be taken as approximat.ely constant; e.g. see Chapter II' equation 2.4.L.8). Lo has 424. very \,üeak variatíon with height (e.g. fig L.9a, Chapter I), and can be assumed to be constant for the purposes of this calculatíon. This wíll be seen to be justified afÈer the calculations, as it will be shown that E(z) is a much more rapídly varyíng term. !ühat is E(z) ? Let us assume that the steps mentioned have a Gaussian form, as íllustrated below. Here, n(z) = refracLive índex, t(z)dz = width of . effecrive reflecËion coefficíenÈ, and d is the e-l full 4:d,z ¡ :t(zl (?) d N 1 I o) (, T1 z,) * -rtù- (Notice that this form ís different to Èhe form assumed ín Chapter II' section 2.2.3, but the two forms both illustrate the same important points). Then by equation 4.5.2.6, Chapter IV, (and noting Ëhe slíghtly dif ferent definítl-on of d) . tr2 aG)2 ),-2 } (9.3. 2.4a) E(z) = "xp { - Let us assume d(z) = Y n(z). Then (9.¡.2.4b) E(z) exp { - n2 y2 n2(r) x-2 } The profile of n shown in fig L.9a, Chapter I was chosen for those calculatíons. Then (e .3.2.5) n i-oß to0' ( ct .0444, ß = - 2.532 for 60 km( z B0 km where z 100 km t cl .0477 , ß = - 2.796 for B0 km( 5 Here, z is ín kilometres' and n in metres. 425. -*' vlas E(z) was calculated as a function of z, and then ."/ita*o pl-otted as a function of heíght in fig 9.9, using equaËíon (9.3.2.3). The term (KÀV.tt . r¿as taken to be constant' and r^ras obtaíned "o-O/t) by comparing expected Èurbulent power aÈ 70 km from fig 9'8 observed for 70 km scatter (*' - 10-8). (*'.rrib 2 '4 * 10-13) to poú/ers has This assumes that all the scatter from 70 km is indeed specular, but this been verífíed ín the precedÍng chapters ' Thus 4 2 (s ;2 -t N (4x10 ) B (z) (Buckland Parlc). .3.2.6) turb ls-s pec trconstantrr 4 It should be understood that the 4 x 1-0 could be wrong bY as much as an order of magnítude. t'Bp, ttBP, plots The curves "Bp, y=ltr y=11 lt, y=2¡rr irt fig 9.9 show of this ratlo at 1.98 MHz, for the Buckland Park array. Notfce that for \=2T, the specular scatter becomes qulte small at 80-85 km' The I'malnly transition from "malnly speculart' to turbulent" is quite sharp, and change this even a variation of the coristant 4 x LO4 of 10 Èlmes does not translÈion height greatly. Thus thls model simulates the observed rapid kn transi-Èion from hlghly specular scatter to more ísotropíc scatter above 85 quite well - províded ^( = 21 1s val-ld ' ' If turbulence did not f111 the radar volume, thís would lncrease the -t 3 g' This would only be imporÈant above 85 km' and the ratio Rz- spec' / turb. ,-1. adjustment would not be too gÏeat, as Èurbulence should at least have a vertical depth of the order of the ouÈer scale \ ( - 300 - 600 rn above B0 km). Hence veff may be perhaps 10 tirnes less than the value used, buE we have seen Ehat such a change does not have a great effect on the transition equal height . T.-t is also interesting to note thaE d = zt71 ís approxlmately to nn-t-1 (meÈres) in Fig 1.9a - í.e. the l-ower limlt of the Tchen range' A major weakness of this model is the assumption Ëhat the refractive index assumed in equation change across the step l-s proporti-onal to åä, as (9.3.2.2). FurÈher, the assumption that #"# ís quiÈe legltimate' 426. but # and l".l are only proportlonal if the temperature gradient and total electron denslty Eerms ln equation (2.3.2.10a) are small. (The term ò\)* òn j----r:i hac hos¡ shown be as discussed in Fig 9.8). As -:ðrt' òz to negligible, m shor¿n in Fig 9.8, this ís only approximately valid above 85 km, but for the accuracies ínvolved in these eomputations it is adequate. Having done all Èhis at 1.98 MHz, ít seemed useful to repeat the calculations for the Jicamatca array at 50 MHz, ín the líghE of Èhe data presented in Fig 4.24, which showed the speculariEy of VHF scatter below 70 km . This required the calculatlon of a new consÈanÈ for the equation (9.3.2,6), as well as a re-evaluaËion of E(r). Both lM.l and \0". involve a Cerm %U r¿hich descrlbes the change in thelr values as the of R2 I problng wavelength changes, so thls term cancels in estimation spec' 4-- -,R'r,rrb. Itence rhe constant ín (g.3.2.6) ls simply (4 x 10") [V.rr(l) /Veff(BP) .(À(J)/À(BP)- I/3)-I, where J refer to Jicamarca, and BP refers to Buckland Park. À(J) = 6 ln, À(BP) = 151.5 rn. For the Jlcamarca array, the half-power-half-wídth 1s Ur"- ," and the pulse length ís about 4 km. Hence the constant ís about 4 x 105. Thus _t 2 N 4 x 105 nr2{r) (Jícamarca) R turb /s'spec This was calculated and plotted on fig 9.9 also. (For fi-g 9.9, the constant used was I x 105, but this does not change Èhe results significantly). In thÍs case' for \:2T, turbulent Scatter always dominate above 60 kn. For y=fl, the transition height is about 62 km, and for Y=l ls about 73 krn. Thls suggests Y - 1 ís best, by comparison with ftg 4.24, Chapter IV' Thus, the two Y-estimates differ at Buckland Park and Jicamarca' Thls lllustrates inadequacies in the model. The true turbulent scaEEering volume may only have a depth of lOO m or so (e.g. see nättger et al , L979). This would increase the constant above by perhaps (2 km/100 m) - 20 times, but thís will not alter the transition heights greatly. 100 BBð= 9 B 70 -4 -2 0 2 4 rt2 ñ.çpeo. "3 Rtr"b. Fíg 9._Z Expected specular scatter to turbulenË scatter po\^7er ratio as a f"ncti"n of heíght, where it has been assumed that the atmosphere is fíl1ed wíth both turbulence and hor-ízontally stratífied steps of wídth d ='Yn, n beíng the Kolmogoroff microscale. IË has been assumed that all wavelengths receive scatter from scales in the inertíal range, and that the turbulence fill-s the radar volume. The B.P. case corresponds to the narrow beam aÈ Buckland Park at 1.98 MHz (0r- = 4.5', pulse length = 4 to 6 km)^ and the Jicamarca case refers to the Sicamatca aTray at 50 MHz (0r- = 1-, pulse iengtn=4km) '2 Many factors could change the shape of these curves, as díscussed ín the Ëext, but the important point ís that specular reflectíon becomes extremely inefficient with increasing height. 427. One important poínt which will affect the Jicamarca curves ís that scatter from heights above 65 km will be from scales in the viscous range of turbulence (see fig 1.9a, Chapter I). This will signÍficantly alter the curves ín f ig 9.9 f.or heíghts above about 60 to 65 km' and the true specular to turbulenË scatter ratio wíll increase. Thus Y=Tr may perhaps be more valíd at 50 MHz. The widths calcul-ated at 50 MIIz and 2 lülz can perhaps be adjusted to agree better by usíng a dífferent profíle for the step than that used. Assumption (9,3.2.2) may also need revisíon. However, although there are clearly ínadequacíes ín the model, it does show that specular scat.ter becomes inefficient at greater heights, and the transition from dominantly specular to dominantly turbulenË scatter ís quite sharp, just as is observed in Ëhe real atmosphere. It thus appears that both stratifíed steps and turbulent layers may occur aË al-I heights in the atmosphere, and Ëhese scatter radiation with dífferent efficíencíes, thus resultíng in the angular spectra observed for scatter from these irregularítíes. As a fína1 point, ít should be mentioned that this discussion has not considered specular scatter from large electron density ledges, as discussed in Chapter VIII. ThaÈ such specular scatËer is ímportant has already been íl1ustrated. 428. 9.4 Conclusíons The data presenLed in this chapter suggests that the fol-lowing conclusiorìs are valid' (í) There does not appear to be a strong permanent correlation betr^¡een HF scattering layers and wind shear measurements made with a vertical resolution of - 4 km , and temporal resolutíon of between 1 mínute and t hour. Ilowever, on occasíons bursts of power and scatterj-ng Layet, can be associaÈed with wind shears. The wind jet at - 70-74 km rnay be related to scatterers aË thís heíght. (Íi) There .'do appear to be strong gravity wave effects on these scatterers. (ii) some, if not most, of the scatter from above B0 km is due to- lrregularities produced by turbulence. (iv) Very líttle, if any, of the scatter frou below 75-80 km ís due to isotropíc turbulent scatter. (v) It is quite possíble Ëhat turbulence and horizontally stratified steps of the order of the Kolmogoroff microscal-e ín vertícal extent exist at all- heights in Ëhe atmosphere. Inlhether these are caused by relaËed mechanisms, or índependent mechanisms, Ís unknown. However, such a picture is quíte capable of causing the observed scatter characteris tlcs . The reasons for Ëhe thin steps in electron density are not known, and the problem stíl1 remains as to how they remaín stable (they could perhaps be expected to díffuse out in a time of t - H2 /K (K = eddy diffusion coefficienË , H = depth), or È- 1-10 s at 70 km (see Chapter I, section 1.4.1e). That they do exisË, however' seems 429. fairly well established. Some clues as to the reasons for their existence will be gíven in Chapter XI. (ví) The importance of an electron densíty ledge at - 85 krn ín causing the 85 km echo was again noted, and it was shov¡n that with the ledge, quiÈe standard atmospheric turbulence can cause the observed ísotropíc component of the scatter, on toP of the specular scatter discussed in chapter vIII. ltrithouÈ the ledge, neíther sígnificant specular or turbulent scatter generally occurs. At thís height, steps in the refractíve index of the order of 1 Kolurogoroff microscale thick could possibly producc scatter just comparable to the Lurbulent scaËter' accordi-ng to fig 9.9, \=2t¡. However, the transitíon height of dominant specular to dominant turbulent scaÈËer is generally found to be 76-80 km (see Chapter IV), so this is an unlikely reflection mechanism at 85 km. Thís suggests Y may be slightly larger than 2t¡ in fig 9.9. 430. CHAPTER X INVEST IGATIONS AT 6MHz 10.1 Introduction 10.2 Experimental- Results 10.2.1 Power profiles and temporal variations LO.2.2 IadÍng tímes and power specÈra 10.'3 Theoretical interpreÈations 10.3.1 TheorY and ÍnterPretaËíon 10.4 Conclusions 4 31. Chapter X Investisations at 6Wlz 10.1 Introductíon some space Ín t.hís thesis has been devoted to general comparisons of D-regíon scatter experiments usÍng dífferent frequencies. However, simultaneous comparisons of scAtÈer at t\^ro separate frequencies are desirable, since this could greatly ímprove understanding of the scatterels. Dierningen(1952) observed the scatterels wíth a sensitive ionosonde, and detected them over a frequency range of 1.6 to 4WIz (also see Chapter I, Section 1.4.la). However, Ëhe sensitíviËy of this sysËem was quite poor relative,to the sysÈems used in this thesís. Few other simultaneous observations of these scatterers aË several frequencfes have appeared in the literature, and those whích have been published have not compared the scattered powers quanÈitatively. Belrose and Burke (L964) used frequencies of 2.66 and 6.271fr12, and found that scaÈtered Powers for the second case rl,'Ìere about 20dB down, compared to 2.66þtlz, but no great accuracy was achieved. Llttle detaíled comparison of powers for Índividual layers was presenÈed. Títheridge (L962a) also compared powers aÈ .72WIz and I.421ü12, but again only general features \¡Iere ínvesti-gated, and the angular spectra of the scatterers \ÀIeïe not considered properly. The receíving array at Buckland Parlc is resonant at 1.9BMHz' but can also receive radiatÍon efficienËly aÈ the third harmonic, whích is close to 6:l1fflz. A transmitter operating at 5.995MI2 was buÍlt at Buckland Park, and began operation in late 1979 (see chapter III for details). It Ëhus became possible to do simultaneous 2 and 6lülz experiments at Buckland Park, and detailed comparisons of polger could be made once the systems had been correctly calibrated (see Chapter III). 432. This chapter presenËs results of such nearly símultaneous comparisons. It i6 also believed that this ís the fÍrst Lime such comparisons have been made on 2 frequencies using equipment r¿hích could record both ampJ-ítude and phase. ThÍs allowed coherent fntegration, and also Doppler measurements of vertical moËions. Ilowever, the facillty to swíng the narrow beam of the attay has not yet been utilízed at.6MHz. 2-l4Hz;02 6Af4z;d Tx Tx off off 90 ¿ 90 eo. e0ã I T I I I I , I I I I s T I I I I I I I t t t I I I I I t t I I I t I t I I I ¡ I I I t t I I , I I , I \ I I I I I w l l 70 t l I I I I t I \ I \ I I t I , I t I I I I l I I t I I I , t I t I I I B 20dÊ I t. t t I ¡ t I 60 od I 60_ r¡ tt+l 0 6 F\Hz; A" 2 PlHz; A" (f M 20 dB l---+ ff 80 e0 Tx I 10 10 60 60 t20g t217 1220 tLZ9 Power profiles at Buckland Park for day Bo/072, ligj__10.1 one minure mean porrer usi"g beams. Noíse has not been removed. Noise levels can be seen by examíning"a;row the "Tx off" curves. The reference levels for each set of graphs .t" ãhorn in the bottom left-hand corners, and each graph ís shifted i¿S-to the ríght from the previous one. The tímes indícated are the sÈart times of each minute. The peak aÈ 62km for 2MHz is a ground echo effect. The Zlltlz pov/ers-"t.-¿S" of irrPl")2, (sometimes denoted ¡per")2) and 6NÍÍLz po\,üers are in dBs of (yPr.r) 2. The large po\,¡ers at ranges of greater than >94kn 94km at 2wtlz are due to E-region total reflectíon. The weak peaks at at 6lüIz may be due to, a weak scatteling layer, or obliques (see text) ' 433. t0.2 Experimen tal results IO .2.1 Power orofi-les and t emporal variat.ions The received signal was digitized as in-phase and quadrature componenÈs, aÈ a frequency of IOHz per range sËe-P, and 10 ranges were recorded símultaneously at steps of 21cm. Coherent integration over sets of B points ltas then used, to gíve an effecEive data raËe of I point every 0.8 seconds at each range' The general procedure adopted was to take 10 mínute- seÈs of data, at (i) Z:MÊ;z,O" ; (ii) 2l{112,11.6o and (iíi) 6MHz,0" ' It was not possible to do all these simultaneously, but ín the future thís should be possible. Duríng each 10 mínute run, the transmitter r^ras turnecl off for 1mínute, to give a measule of the noise level' At some stage during the day the system \^las set up to record partial reflection drífts as well, and Èhus wínds \^Iere recorded for at least half an hour, close Èo the periods of observation. These data were then analyzed on a computer' generally in one mínute blocks. Mean noise Pov/ers for each 10 mínute period were calculated by using the minute of data when the transmitter \^/as turned off, and also by utilizing the spíkes at zero 1ag of the autocorrel_ation function, as described in chapter vI. A1l estimates showed reasonable agreement, although at times a signifícant non-random noise component occurred. The mean pov/ers for each minute were then calculated, and plotted as a functíon of height. Fig.10.1 shor¿s some examples. It will be noticed that the heights of enhanced scatter at 2lrÍHz and 6MHz are ín good agreement, and the usual temporal variatíon of scattered power will be noted, particularly below B0km, where bursts of scatter can be seen surrounded by periods of little scatter' PARTIAL REFLECTION STRENGTHS AT 2&6MH2,17 OCT 1979 ZMHz , Oo 6 MHz , 0o 2 MHz , 11- 6o r400-1409 1t-18 - 1tr26 1433 -1r-/+1 100 E J oUJ f 90 J 80 5 10 15 20 -20 -15 -10 0 5 10 15 POWER (dB) Fig. lO.2 Typical po\,rer profiles for day 79/290 at Buckland Park using the narrow beam and 0 Tode polarízaLion, for g0-98ktn. The noise has been subtracted. The powers are in (pPrr)2 î.or 2Þß2, and (pPr.r)2 for 6MLlz- To convert to effecËive reflection coefficients, see text. Noise levels were Èypically = - ISdB ax 2WIz, and = - 26 dB ar 6MHz. [These noise levels are also in unÍts of (pPrr)2 and lper6r)2J PARTIAL REFLECTION STREr'¡G HS AT 2&6MH2,17 OCT1979 ?MHz , 11 .6o 2 MHz , Oo 6 MHz , Oo ltr/+6 -11-51+ 1 59-1508 r510-1519 80 E J UJ o 70 l ¡- 60 -r5 -10 -50 -5 0 5 -30 -25 -20 PO ER ( dB) beam and 0 mode Fig. io. 3 Typical poürer profiles for day 79/29O at Buckland Park using the narrow poLarLzai'Lorrr- ior 60-78 km range. Noise has been subtracted. Noise levels r¡ere the same as those indicated Ín Fig. i0.2. 434. It should be remembered Ëhat the powers aE 2 and 6MHz cannot be compared directly. However, if effective root mean square reflectíon coef ficients are found I i'e' (cn 3.3.2.14)l (10. 2. r. 1) (P)t=K*H/Power rtt, eluh. ' then these nay be cornpared. Here, H is the range of scatter' and Ko is equal to'K*ZM for 2lú72 and KoU" tor 6ÙftIz' AccordÍng Ëo table 3.1, Chapter III' K*2M = .8t'4, and KnUt = 3'Btl'6' For more accurate cornparísons of poller, these one minute profiles were averaged and Ehen the noise was subtracted, to leave a profíle of signal only. By averaging the full 10 rnin' set of profiles (except the noise profile), temporal varíations in po\^/er were reduced. Tigs.l0.2and10.3showtypicalpower-heíghtprofiles.In each case, the noise has been subtracted' Noíse levels are indícated in the captions, and refer to the levels after coherenË integraËion. It is immediately evident that the ranges of (0") enhanced scatter are similar at 2 and 61fr12. The 2Wz echoes peak aÈ about B5km, 76.5kn and 68km, and those fot 6MIlz peak at about B4krn and 75.5km. It is not clear whether the slight 1 km \^7as due to díf f erenc e ío 2 and 6MHz heights of pref erred scatter the fact that the records r¡rere taken at díffererit times, whether it was due to an erroï in settíng the range markers, or whether ít r¿as due to an error ín settíng the range markers, or whetlrer íÈ was a real effect. However, the consistent 1 krn difference suggests that the reason was ínstrumental. Ifne transmitter pulses at 2 delay and 6MHz were adjusted as nearly as possíble to have zero with respect t,c, the o km range marker, buË thís could only be done with certainty to about a I km accuracy. Furtherr the 2 anò 6MHz 43s. receivers could have had slightly different delay tímes, and this could also account for the 1 km range difference.] The strong echo at 98 |¡rn at 2l{Ílz was due to the leading edge of the E region totally reflected echo. The echo evident above 98kn at 6MHz is quíte ínÈerestíng. It could be due to a low sporadic E at around 100krn (6MHz is not totally reflected from the regular E regíon) ' However, recall from Fig. 3.I2b that the array has 4 graÈíng lobes at.about 33.5' from the zenith when used at If Èhe 'lfrIz. scatter from the - B5km echo were quasi-isotropic, radiatíon scattered by this layer aÈ angles of around 33" would be received sËrongly through these grating lobes and would apPear as a peak in power at around 1o2km range (This effect is íllustrated in Fíg. 10.10). Thus the rise in power around 100krn at 6MHz could indicate Èhat scatter from the B4km layer at 6þIHz could contain a signífícant quasi-isotropíc component. More vTork needs to be done to ascertain if thís is so. At 2Mf:1z, the scatter from the - B5km certainly contains a quasi-isotropic comPonent. This can be seen by the por¡rers received at 2ÌtP.z, 11.6o, which are only about 5dB dovrn on those received at Oo. Further, the range of the echo t'peak" is closer to BB-9Okm at 11.6o, also Índicatíng off- vertical scaÈter. Thís contrasts wíth the - 68krn echo, where i't can be seen that the 11.6o signal at 2NfHz is down Ín strength by s 10dB compared to the 0o signal. For the 76kn echo, the 11.6o signal is down by = 6dg compared to the 0" signal at 2Wz' thìs Hovrever, notice Èhaütf'eo"andtl6"rangos in the latten case. aresimilar,and suggests that the off-vertical scatter is not very large. The fact that the powers are only = 6dB different may be a result of temporal variati-ons - stïonger scatter may have occurred during the 11.6" period. Strong temporal variations are a more serious problem below B0km than above. Fig. 10.4 Typícal po\,rer profÍle (= 10 min means) for day 80/072 (I2tt. Marçh, 1980) at Buckland Park using the narrow beam and 0 mode polarization. Noise pohrers have been subtracted, but Ehe noíse levels are also índicated. Powers are in dBs of (UPrr*)2 fot 2WLz arrð. dBs of (¡rPrU")2 f.or 6Wz. The numbers associated wíth each graph are the frequency used, the angle of tilt of the beam from the vertical, and the st.art Eime of each = I0 nin block. The 2 and 6MHz profíles look similar, although the profiles at 1208 anð l22O are a'líttle different. Thís discrepancy occurs because the 1208 to 1218 profile was dominated by a very strong burst at 1208 to L2O9 (see Fig. 10.1) which biased the profile. t00 Hz 2 N¡12 o" o" 90 12 I t o JE q)(, c o Ve g É. oi se l,l flz 2 n¡12 2 l{Hz Hz oo oo Il.6' o" t1 o n 32 t1 il 6 -30d8' odB o¿B -s}dï €odB o¿B 2 î\Hz 2 Ì4Hz AHz 2 l,t Hz 2 llHz l!Hz oo 71.6' t d t 1.6" 12 2. 12 t 7 T l E J q"q) c o É. -LA¿B -fodB 2c-d PONER 436. These qualitative results are all quite consistent h7íth conclusions reached in Chapter IV. Of more qrrantitative importance are the ratios of 2 and 6NIHz effective reflectíon coefficienÈs' According Èo Table 3.1, Chapter III, Ko2M/Ko6M ís equivalent to -13.5d8 I 9 dB. Then the ratios of effective RMS reflection coefficients (RT.r*rr/€øtqtr)U tot day 79/2-90 were: (10. 2 .r.2a) l4dB 1 9dB at B5km, ancl lldB t 9dB at 76km. However, notice from Table 3.1 that the neaÏest Koo" value to day 7g/2g} was taken on day 79/248, when KoUr= 2'4' If this value were used, the effective reflection coefficients would have had ratios, for day 79/29O, of (10. 2 .L.2b) 18dB at B5km, and 15dB aÈ 76km. similar profiles are shown in Fig. 10.4 for day Bo/072. Echo peaks occurred at about B6km, 7}-74km and 66krn befòre 1300 hours. Later in the day, an echo from around TBkm became noticeable. The profiles shown ín Fig. 10.4 have had the noise removed, and the noise levels are also indicated. The noise levels aË both 2 and 6þÍÍlz are similar to those for day 79/2gO, but notice that the signal 1eve1s are weaker. The signal to noíse ratíos ax 6MHz reach maximum values of only 3 to 4dB. Usíng *oZ*/*o6, equal to - 13.5dB + 9dB, gives (*t^""/*t*t r)h vaLu"s ol' (10. 2. 1.3a) 18.5d8 at B6kln 10dB at T8krn 18.5dB at 66-68kri 437 . ThevaluefortheTo-lhkmechoísnotgiven'asaveryStrongburst of scatter occurred duríng the 6MHz run, and thís would badly bias the results. The value aÈ 66-68km ís álso considered unreliable, signal to (the period 1208-1230 was used for this estimate) ' as the ín noise ratío \47as no better than -BdB at 6MHz, so small errors subtraction of the noise po\^ters could severely bias Èhe resulÈs' Notice that from Table 3.1, K*6ÌI = 2'7 on day 801072' and if thi's value is used for comparÍng the data, **Zt/*o6t is -10'6dB' Thus (otr*/,*Z*,) is equivalent to (10. 2.1.3b) 21,5d8 at 86knr, and 13.5dB at TBkm One last point regarding the similarity of the heights of scatter at both frequencies ís worthy of comment' It has been a assumed all through this thesis that the scattering is due to structure contaíning a wide range of Bragg scales (e.g. turbulence, or steps in refractive index) ' However, a structure whích consisted of horizontally stratified sinusoidal oscillations ín refractíve index in the vertical direction with a Bragg scale of at 1'9BMIIz' 75.75m could equally cause the specular scatter observed such a structure v¡ould only cause scatter at l.9BMLlz, however, and the fact that scatter does occur simultaneously aÈ these two frequencies from Ëhe same height helps Èo dísmiss such models' 438. LO:2.2 Fadi times and ÏS ectra AsseeninChapterVll,beambroadeníngisthedominant' contributor to the spectral widths at Zl{Hz (and hence the width of Ëhe autocorrelation function and the fading period). The half- width due to beam broadening is: (10.2 .2.L) 1r" = e /x) ' uu, ar¿ (e.g. see Chapter VII, equation 7 '3'I'2) r 0r, being the half-power half-width of the polar diagram produced by the combined effects of the array and the scatterers, I the probing wavelength, and v" the lrorizontal r,¡ind velocity. For Ísotropíc sc'atter, Ur"=4'5o at I'98MHz used and 0, at,6MHz when the whole aTray at Buckland Park is 4=I.4o j.f for recept'ion (see Chapter III, }.igs. 3.I2). Thus the Scatterers are reasonably ísotropic ouË Ëo zenith angles of about 10o, and turbulence is not ímportanÈ, then the specËral widths should be similar aE 2 and 6MHz, sínce Or-/À is approximately the same at each frequency. However,otherfactorscouldaffectthisresult.Ifthe scatterers ín a layer .1,', are totally ísotropíc' scatter will be received through tne 6MHz gratj:rlg lobes. Thís will not affect the g-¡, fading ïates due to the near-vertical scatter from layer since it wíll be from a much greaËer range, but it could affect apParent fading rates and powers in a higher layer' There is thus a possibilíty of mixing the effects of Ëwo layers' A second' more important factor concerns the effective polar diagram. If the scaËterers are not isotropic, this will alËer the effective polar diagram; the values of }rmay no longer be proportional to )" A third point concerns the motíons of the scattelels themselves' It was seen in chapter vII that for turbulence at about B6km, the o BP NARRON BEAM. O O MODE. BO/072/ (1130-1300 ) F+-{1-. ,f 6 MHZ I 2 MHZ. 180 5 lrj \t 70 Z # E. ó0 1 2 FADINC TIME (secs) Fis.10.5 Typícal fadíng times for day Bo/072. The fadíng times were obtained from one-tit,tt" complex autocorrelation functions' The 2lülz (1.9BMHz) results were also presented in Fig. 7.4, Chapter VII. All points are averages of sets of about B to 10 one-minute fading times. Error bars are standard deviations for the mean. The data for 2MHz, Bo-g8kfD,was for the period 1132-1142 lnts, and for ?ÌftI2, 60-T8kmrwas for the period l245-I254hrs. At 6MHz, the was BO-gBl,xn data was for the períod 1156-l206, and the 60-78km data for rhe period L2OB-L2L7. The 6ItHz data belor¡ 80km could be affecÈed by contarnínant signals. 439. RMS velocities associated with Bragg scales for 1.9Bl"1IIz backscatter r^rere not a major contributor for v" values of =70ms-1 . Upper limits (*3rn"-1) could be placed on the values, but no more. (The possibility of obtaining tRMs totld be improved for smaller v" values). Turbulence alone would produce a half-power half-width of the spectrum of (10.2 .2.2a) fr=2/\ {2*--n2 v*, /3, 7 . Since v =(e L)L L= being (Chapter VII, equation .4.2.2) Rl"fS d L the Bragg scale (see ChaPter II, equatíon 2.2.3.3a), then ft,o ),-2/2. (Lo.2.2.2b)' 4 The.beam-broadeníng effect fol isotropic scatter is fairly independent of frequency if the same array is used (equation LO'2'2'I), so Èurbulence clearly should make a slightly larger contributíon at higher probing frequencies. trühether it ís a sígnificant contribution or not will be seen shortly' Fig. 10.5 shovls a profile of mean 1 minute fading times, usíng the complex autocorrelation with the spike at zeTo lag removed. clearly the 6MHz values are generally less than the 2MHz values' Some care must be taken before interpreting this data, however, noise particularLy f.ox 6l"1Hz below Bokm. Recall that the signal to ratio was quite low (often less than OdB) and some of the noise was atEimesnon-random.Partsofthísnoisemayhavebeenleakage from other parts of the receíving system (e'g' the clock)' and some was external. The frequency 5.995MH2 is close to a short \nlave broadcast band, and there I^Ias sporadíc interference from other short vTavesources.Theexístenceofthisnolsecouldaffectthe fadingtimes.Thereasonswhythefadingtimesareastheyare Fis. 10.6 Raw Power spectrum of 9 minutes of data rêcorded at Buckland Park using Èhe narror¡I beam at 6lüIz, for the period índicaÊed. This figure may be compared to Figs. 7.6a, 7.7a. Notice also the four very large values close to OHz (arrowed). These could indicate specular scatter' but ur,ay also be pureiy statistical effects (see Chapter VII, secÈÍon 7.3.1). Fie. L0 .7 Fig. 10.6, averaged ín frequency blocks of about 20 points. The four very large points ín Fig. 10.6 have been Ígnored, in accordance with Ëhe procedure outlined in Chapter VIIt Section 7.3.L, since such points unfairly bias the means of their frequency b1ock. The shaded area represents the noise level, and the smooth solid 1íne is an eye-fit to the data. The half-po\¡rer half-width appears Ëo be about O.O6Hz. Notice this half-width is similar to that for 2NÍHz (=.O75Hz; see Fig.7.I3, Chapter VII)' as suggested by equation LO.2,2-L (i.e. À decreases by 3 times, as does 0o, so the two effects cancel) . ' to.6 t Buck lot d Po' k ó lll'lz: O': O Hode. 'õ N orrow Beo m. ç o eolol2lunbfi^o). â L I q,, t 3 ñ .oI o -2 -1 0 I .¿ .+ to.7 '06 Hz L fa= (¡) Ì ce ¡se Leu"l o. -2 -1 0 I FrequencJ a o a tÍ il27 tn J'^ ú) L t --c \-, G) E Ë I L (] +- U) o -lt22U o J ttal {5 o .o5 .t -.1 Ftul"ency (l-lz) Fig . 10.8 Running pov/er spectïa with a 3 polnt runníng mean (\r'"rt<) on each spectrum. n"ãn corresponds to a one minute data block shifted 1'2 sec along frorn-"p."trum the prevíous data bl ock. The spectra are for day 8OlO72 at 861cm range ,t"ittg the narrow bear¡ pointing vertically, for a frlqrrency of 5.9951¡H.z and O mode polarizatíon. Positive frequencies r."., . Dáppler component towards the receíver. Sirnilar spectra were presented i.n Chaptàr VII, Fig. 7.I2, for 2NlLIz. Recall that the àppu"r"r"e of -5 peaks in succession is not necessarily physically significant. 440. wíll not be discussed in much more detail using Fig. 10.5; I¡Ie sa\^7 in Chapter VII that although one mínute sets of data can give a rough guíde to fading characËeristics, they are not reliable for detailed analyses. Meaningless peaks occur in the power sPectra' and so forth. Admittedly the values in Fig. 10.5 are means, but it was felt that it would be more useful to examine the pohler spectra of 5-10 mín sets of data. ' Fig. 10.6 shows the ra\^r power spectrum' and Fíg. 10.7 shows the smoothed por^rer spectrum (compare these to Figs. 7.6c, d, and 7.L3, for example), fox BO/072, 1121-1130 hrs. The observed half width is about 0.0611z (error is probably greater than 10%) compared to the expected wí.dth due to beam broadening alone (calculated by program Specpol ) of about .O7ÍIz. Thus once again, as for 1.9BMIIz, it is very difficult to detect a signíficant Ëurbulent scatter conÈríbuËion. By placing upper limíts on the width of the po\À7er spectrum, it was possible to say that Èhe RMS velocity of the 1, scatterers at 6ffi2 \^Ias less than lms br'r, little more can be staËed wj-th certaínty. As a different way of viewing the data, Fíg. l0.B shows the running po\¡rer sPectrum for the 6lü12 data aE B6krn for the period 1121-1130 hrs. The closeness of the observed and expected spectral widths aË B6km suggests thaË scatter at is reasonably constant in 'lfrIz strength out to angles of a few degrees. The wídths of the observed spectra offers a possibility of deËermining some information about the nature of Ëhe scatterers - if the scaÈterers were quite specular, the observed spectrum should be narrower than that estimated by equation 10.2.2.L, (or, bett,er, by Program specpol). However, the narrow nature of the 6MHz polar diagram makes Ëhís a very diffícult experimenÈ, and no useful results were obtaÍned' \IY nl (J) L -c 12t2 o Ewt l- +- IL U) o U o J I I -.1 -.@ o .o5 .l ro.g Eie. (Þu,%,'<) each Runníng Por¡rer sPectra with a 3 Po int running mean on spectrum. Each specËrum corresponds to a one minute data block shifted 12 secs along from the previous data b lock. The spectra are for daY BO/O72 at 74km range, usíng the narrow beam pointing verticallY and for a frequency of 5.9951úlz with 0 mode po larizatíon. Posítíve frequencies mean a Doppler component towards the receiver. Sirnilar spectra hTere presented ín Chapter VII, Fig. 7.L4, for 2lú12. The black arrol^rs índicate possibly significant sÈríngs of spectral peaks. 44L. It r'rould be rur¡ch easier to do with an array with a wide polar diagram.Thisisonemajorexperimentat6MHzwhichhasnotyet the been performed - direct determination of Èhe height profile of angqlar spectrum of the scaËterer s at 6WIz' Some indirect information will be seen shorÈly, and in sectíon 10'3' Such determínations r^¡ould be best done by tílÈing the array beam, but failingÈhis,calculatíonofthepowersandspectrawÍthboËha wide_beamedarrayandthenarro\¡Ibeamcouldgiveinformation. Isotropic scatteï would gíve a much wider spectrum with the wide beam. For specular scaÈter, the spectra on Èhe wide and narroT¡7 same' beams would be similar, and the receíved po\^Iers should be the af'Eexcompensationforthedifferentgainsofthearrays. Tig. 10.9 gíves some hÍnt as to the angular spectrum of the scatterers at 74kn. There appears to be evidence of some frequencypeaksmaíntainíngthemselvesover2and3mínutes,and thís could suggest that Ísolated specular reflectors may cause the sígnal. For example, a long, flat reflecËor gradually movíng down,andcausingmainlyverticalreflection,couldhavecaused thesÈringofspectralpeaksfroml20gtoL?.lz.Thehorízontal motion of such a reflector would not be seen in the spectra - the sígnal would cut in, and disappear' fairly sharply' wíth a long períod of constanÈ sígnal between appearance and disappearance. There is clejarly more work to be done at îlfrLz before a clear appreci-ation of the naËure of the scatterers aE Èhat frequency ís understood, although the ínformaËion previously presented for 2 and the 5OþÍIz, and the ínformatíon ín Fig' 1'9a, give some idea of expected resulËs. Turbulent scaËter should be weak above about range B5km at 6lüIz, the scattering scales beíng in the viscous (fig. l.9a). Thís ís supported by the fact that no strong echoes 442. above about 86km have yet been detected aE 6lúlz with the presenË equipmenÈ. Some further support of this statement can be found in Fig. 10.1. The powers at 6MHz fall away much mole rapidly with increasing height above B4-86km than those at Zwlz. (The rise in por{er at 100km at 6W1z has already been discussed) ' -70km, It is Èo be expected that below scatter should be fairly specular, since it Ís apparently so aË 50MHz (see Fig' 4'24' Chaþter IV). The analyses presented in Fig' 9'9 can be repeated aE 6lü12 to gain a better idea of the 6MHz specular to Ísotropic scatter transition height. This has been done, and results suggest ËhaE this height should be around 74 to Blkm for Y=2n to t. The following section shows some theoretícal atËempÈs to trequatíons" sÍmulate the results presented. ín 10.2.L.2 and 10.2'1'3' and the results of thís analysis allow better interpretatíon of the informaÈion. 443. 1-0.3 Theoretí ca1 Interpretatíon 10.3.1- Theo t:y and ínterpretatíon It has been discussed elsewhere (e'g' Appendix B' equation 8.25) that for reflectíon from a stratífied step less than a quarter wavelength thick in the ionosphere, the reflectÍon coefficíent is (10. 3. 1. 1) R(z) - '¿Ln, where an is Èhe change in refractive Índex across Èhe step. Thus (10.3 .t.Z> R(z) , 4 nX, (4.1. is a"uu^ud negl''gib1";see ffi ch.VIII,5 42.2) ) aN being the change in electron densÍty across the steP. Thus the ratio of reflectíon coefficients at 2 ar^d 6MIIz is (10.3.1.3) n2.)/x6Q) = ,F*l*,,r,F*å*,, ThÍs parameter has been evaluated at varíous heíghts in Èhe atmosphere for a typical D-regíon electron density profile. It has been found that this ratio is approximately 16dB at about 76km, and 16.5d8 ar B0 to 90km for a 10% step ín electron density. The ratio 1s relaÈÍve1_y independenË of the step síze so can be taken as a itequatíonst' constant. It is useful to comPare Èhese values to LO,2.1.2 and 10.2.1.3. It will be noÉiced that íf the Ko.on values measured closest Èo Èhe days of observation aTe used, then the ratios of reflecLion coefficíents at 74-79km are indeed about 14 to 15d8. This would thus appear to reinforce previous conclusions Èhat, scatter from below BOkrn ís largely by Fresnel reflectíon from sharp steps in elecËron density. (The actual measured reflectíon coefficients also are affected by absorption, but absorption ís very 444. weak up to =78km (=lds for O mode, 2lúlz, and =.3d8 for 0 mode, 6M}Iz)) Further, under thÍs assumptÍon, it is now possíble to place even betËer estÍmates on the rat'io of (K*2M/K*6M) ' By equation 10 '2'I'I' ( (10.3. 1. 4) (K*zMlK* R2 /*6 ) IT 6M "\ where P¡ are rhe received pornrers aL í=2 and 6MLIz. On day 79/290, p2/p6'= 24.7d8, and on day 80/072' P2/P6 = -23.5d8. The expected [2/n5 is about 16d8, as deduced, so *ozr/*o6* = -7.5 to + 8.7d8 (10.3.1.s) i.e. *o2*/*o6" ' .42 to .37 Of course, there ís some error in the P2/P6 ratio, since the P2 and quiÈe p6, measuremenEs !üere not simultaneous. But Ëhis is probably a good estímaËe of the ratio. Ilhat abouË the scatter from -Bskm? I'rIe have already seen this could contain an important turbulence component. I{hat would be the expect.ed effective reflection coefficient ratío for turbulent scatter? Recall from chapËer III, equations 3.3.2.L9 and 3.3.2.20, that the measured effective reflecËion coeffícient Rt can be related to Ëhe scatËering cross-section n and the effective volume of scatter by (10.3.1.6) [Ø= n Veff A L r¡ r¿here A = exp{-4 ãtr ds], s=o h being the range of scaÈter, ri the angular frequency, c the speed of light ín a vacuum, and n, the lrnaglnaÏy part of the refractíve index. The term A rePresents absorptíon' 445. Hencefort-h, Ëhe It t " on the R will not be used, and R will be taken as the effecËive reflectíon coefficient measured at the ground. Now 10.3.1.6 imPlies -2 -T (Y . Ã5) (10.3.1.7) n2lRo = (n2ln6) eff 2 lY eff o) @'2/ ,,2,, refers to 6MHz where refers to 2]lJfilz (1.98MFlz actual-ly), and "6tt (5.995M112 acÈually). The receíved po\¡ler aÈ a range f,o is proportional to v(t") = dç¿ Gr(o,'þ)GR(e,0){v o e(r)} (€o) (10.3.1.s) I J ' all space ) the where GT(0,0) is the transmiËter polar díagram, GR(e'0) is receiving arrayts polar diagram, T(r,0r0) ís an aspect sensÍtívity factor, and g describes the transmitted pulse' This equatíon is a sirnplificatíon of equation 7.3.2.g in chapter VII. The term o/t2 in equation 7.3.2.g has been approximated by \\/E? where {o is taken as the height of the layer. The Eo-2 term has been taken outside the integrals and ignored. The backscatter cross-section n has should be thus been Ëaken as r,l = y-Io. For isotropic turbulence, y (4n)-r (see Chapter III equatíon 3'3'2'15)' However' the values ís V(8") are used only for comparisons of 6 and 2MHz results' so ít onlynecessarythataconsistentconventionbeused.Thusyis It is taken as 1 for all angles ín the case of isotropíc scatter' can be for the same reason that the E;2 term discussed above ignored - it is the same at 2 and 6l{Hz' Hence V(g.) can be regarded as the effective volume of scatter. Also recall that 446. I C2 I 3 by equation 2.4.I.5, (r0 . 3. 1.9) n= .38 n r /s ís C2 = L\ uv equarion 2.3.2.11, where L Chapter II, and ^u luel, the outer scale of turbulence and a is a constant' , Combining these equatíons with 10'3'1'7 give (r0.3.1.r0) úG)/$ depend on Ëhe probing frequency') ' Finally' ð\r* än .N d0 dN N ât l- _ + Cq\* (10.3. r. r1) Me tT dz dz p dz'' âv* âz âN - (".g. Chapter II, equaÈion 2'3'2'10a') â"- (see The term ?u* usuallvut;udrrl is ne-gligible above BOkm % a, Chapter IX), and Èhe bracketed term is frequency independenE' Thus 10.3 .1.1 gives _t (À2lÀ5) / 3 {( ¡ tÞñ (v ett r(E) lv etto (6.) )' @2 / L6) n-fce .l /Ç{e .) = u\2 16'5dB lùe have already seen that the second term ({ }) is about wiÈh sen- at B5km, and (12/À6)-l/z = .7, or -1.6d8. calculations I.lyller formulae for a typical íonosphere show that, for reflection abouL and A5 is equivalent to from 85km, A2 is equivalent .Ëo -2ð8, about 0.5d8. Hence l3'4 o 2 --r0- ( 10.3.1.13) x?r¡,) /R;G") = 10 {Yeff 2(E") lvetto(6")} ' Veff being calculated by equation 10.3.r.8 A computer program hraswrittentoaccurat'elycalculatetheseeffectivevolumes.A Thus y' layer about 2km thick \fas assumed to be based at B5km. as insíde the aspecË sensitivity term in (fO'3't'8), was taken 1'0 1 \ \ \ I t / 100 é ftHz. 2 AHz. I 9 E \ -v \ Lovet: T o Jt , CD B / s, o x. I l. 2. t*, to. 20. 40. E f,fect¡ve Volum€. C=If G,G^Jrl.[Y e Fíe.10.10 il] Effective volume as a functíon of (virÈual) range Eo, as described by equatíon 10.3.1.8, for a layer of isotropic scaÈterers at 85 to BTkm' T1,,e 2WIz curve peaks about g.zdB above the 6MHz curve. Notice- also the profile peaks at -IO2 to 103km fox 6l{ÍIz. This ís due to leakage in through the grating lobes of the array. The assumed pulse had a form e"p{-¡z/52}, rhi"h is slíghtly wider than the pulse normally used at Buckland Parkrbut adequate for these purposes' 447 . the layer (ísotropic scaÈter), and zero outside, and the effective volume for this layer vras calculated as a function of range at 2 program is and 6W1z using progïam "Volscat". (see Appendíx G) . The símilar to program "Specpol", which was used ín Chapter VII, but ín this case it has not been necessary to calculate as a function of the Doppler frequencies (alternatively, it would have been possible to integrate the spectrum produced by program "Specpol" over all frequencies). The results of such calcul-atíons are shown in Fig. 10.f0. At Ëhe Peaks, (10 Veff lv ett 9.2d8 .3.1.14) 2 6 = Ilence, by equation 10.3.1.14, -2 -2 (10.3.1.ls) R2(6") /Re(E.) 22.6d8 for a 2km thick 1-aYer at about B0 to B5km. Thus the expected 2 and 6MHz effecÈive reflection coefficients wtll be in the ratio of abouÈ 23dB for turbulenÈ scatter. Thís can be compared Èo the values of lBdB and 2L.5dB for days 79 /29O anð' BO/072 respecrívely at -85km (see 10 .2.L.2b and lt}.2.1.3b). These results suggest that there is a turbulence contribution to the signals received from above B0km, since the observed ratios are gïea¡er than that expected for purely Fresnel reflection' (A step ín refractive índex which is not ínfinitely sharp could also produce -2 -2 an increased R;/R; value, since reflection would be more efficient at 2]MÍlz. However, such a step is likely to be about a Kolmogoroff microscale (see chapter IX) in vertical extent, and it was shown in Fig.9.9 that such a sÈep is quite inefficient as a reflector at of 86krn and would not be detected. The observed quasi-isotropy scatÉer also srrggests non-Fresnel reflectíon. The effect of a 448. large electron-densíty ledge at -86km should be considered, however, and Èhis will be done shortlY') ThisobservedratiothusSupporÈspreviousconclusions regarding turbulent scaÈter from above BOkm' If in fact (K*2MiK*6I4) ís taken as 0.4, as suggested by equation (10'3'1'5)' then the effective reflectíon coefficient ratíos at 2 and 6MIlz Íot the -85ktn echoes are abour 22dB on d.ay 79/290, and,24dB on day 80/072, (compared with equations LO'2'L'2 and LO'2'1'3) which gives even better agreement with the expected turbulence results' onemoretestcanbecarriedout.RecallthaËspecular reflectionfromthelargeelectr:ondensityledgeat-B5kmmakesa significantcontribuÈiontoÈhescatterfromthisheightat2l'{JJz. t'Scatprf" Is thís also true aE 6MHz2. Program was run for the frequency same ledge as that discussed ín Chapter VIII, but for a of6i]yiifrz.IÈwasfoundthatthemeasured6Mllzspecularreflection (tnis coeffícient should be down by 24ðB compared to that aE 2MHz includes the effecË of a5sorption). Thus this is also consistent it ís vriËh the observed experimenËal values ' It is felt that coincidental thaË the ratios of scatter stïengths for 2 and 6þfrlz forspecularandturbulentscatterarethesame.Itdoesmean' ledge however, that comparisons of, 2 and 6MIIz strengths from Èhis cannotbeusedtodisEinguishbetweenspecularandturbulent scatter, but oÈher methods available have already suggested that both processes are important (see Chapter VIII') ' 449. 10.4 Conclusíons Calibration and comparison of 1.98 (-Z)MHz and 5.995 ('')lfrLz íonospherically scaËtered radiation suggests that at heíghts below Sokm,6MllzscatteringStrengthsaredownby-l6dBwhencomparedto 2l,fÍlz, and above this height the rat,io ís closer to 22-24d8. Thís evidence supports previous claims that scatter from below about from 76-80km is Fresnel-like at. both frequencíes, and that scatter above 80km contains a sígníficant (possibly major) contríbutíon from t.urbulent scaËter. As pointed ouE i-n Section L0'2'2, the transiÈion height of domínantly specular to dominantly turbulent scatter is probably slightly lower for 6MLlz than for 2þfr12. It was also found that a large elec¡ron density ledge at -Bjkm, similar to that discussed in ChapËer VIII, is also capable of causíng a cômponent of specular scattef. at 6þlflz of similar magnitude to the turbulent scatteÏ. No f,luÍLlz scatter from above -85-90kur has been observed, except for possible weak sporadíc-E effects, and this supports information presented in Table L.9a, chapter I, which shows that the Kolmogoroff mícroscale reaches 25m at about 88 to 90km. This is also consistent wíth Fig ' 4'23, Chapter IV. I{ence little t,urbulent scatter would be expected from above these heíghts at 6þtrlz. 450. CHAPTER XI DISCUS SION AND CONCLUSIONS 11.1 IntroducÈion LT.2 Summary of Facts 11.3 Discussion 11.3.1 Díscussion of doubtful- points a. Productíon of turbulence b. Reasons for sPecular scatter c. Reasons for Preferred heights d. Seasonal variations l-1. 3. 2 ,T]ne author' s view LL.4 Inconclusive results, future projects and schematic suÍmary. I I ¡ I I ¡ t I 4sL. -, 11.1 Introduction It ís belíeved that during the course of thís thesís, a clearer pícture of D-region scatterers has emerged' Some conclusions have been reached which the author feels confident about, and the next section provi depenclence of the properties of the ecltoes' Some space ís devoted to a discussion of Èhese points ín sectíon 11.3, and some speculaËion is Presented' Fínally, possible experiments for Èhe future are discussed. 452. tL,2 Summary of Facts Most of the earlíer conclusíons reached by other authors regarding HF scattering in the D-region remain unchanged' These have been discussed ín Chapter I, Section I.4.L. In particular' the echoes occur between 60 and 1o0km, with reflectíon coefficíents íncreasing roughly exponentíal1y from l0-5 to 10-4 at 60 Èo 70km to about 10-3 (and often greater at night) at 90km' Preferred heíghts of scatter exist, especially at -65km, 7O-74km, -Bskm, and -90-95kn. Fading is slowest at the heights. below Bokm, and faster above this height. Scatter is also quasi-isotropic above 8Okm, but Fresnel-like be.low 75-80kn. The echoes are strongest in wínter, and almost non-existent ín summer, particularly those from below BOkm. Scatteríng layers can last from minutes to hours, and even daYs. However, the work presented ín thís thesis has clarified many points cotìcerníng these scatterers. The followíng results have been established. 1. A detailed formula for estímation of turbulent scatter sËrengths has been developed ín Chapter II, (e.g. equations 2'3'2'10a)' 2. The first direct ínvestígations at HF of D-regíon angular scatËering characteristics have been presented. Results support previous índirect measurements indicatíng specular scattel below -BOkn and quasi-isotropic scatter above' 3. Detailed plots of power as a function of height and time have shown quite conclusively that preferred heights of scatter do e-xist on any one day. These plots have also shown that strong temporal variations in power can occur, particularly at heights below Bokrn. Bursts of power -loclB stronger Èhan "norma1" levels often occur at these lower heíghts, and typícally last for periods of about 1-5 minutes. ¡\bove B0 km, bursËs are typically - 3 dB above "normaln'levels. 4s3. At all heights, bursts often have a quasÍ-periodiclty, typically of 5 - 15 minuÈes. 4. Height variations of echoes often show regular, possíbly gravity wave induced, fluctuations. The spread in heíght of scat.tered pulses also gives an ídea of the width of a scatteríng layer. Above 90km, layers can be quite thíck - up to 10km' At about 85km, thicknesses are typically a few kilometres. For heights of =70-74krn, scatter still often comes from spread of heights of a few kilometres Ehickness, but on occasions when only I reflector appears to be present (very slow fadíng) the reflector appears to be very well defined in height' It appears that scatter from these lower heights comes from discrete reflectors, ) sometimes alone and sometímes in groups' 5. The Rice parameter has been ínvestigated ín detail, and it has been shown that results using this parameter suPport the description of scatterers given above' The Ímportance of a specular contríbution from the =B5km echo was noted. The misleading conclusíons reached by other autholîs who have investigated the properties of D-region scatterers by using the Ríce parameter have been explained. 6. Some discussion regarding the data lengths necessary for relíable interpretatíons of fading tírnes and power spectra has been presented. It has also been pointed out that the polar diagram of the scatterers, and the related beam broadening effect, are the main causes of observed fading Límes , when using the Buckland Park array. Three-dimensional turbulent velocitíes associated wíth scales of. ),/2 are a second order effect. If such 3D turbulent velocities can be measured, Èurbulent energy (e, U.,. v3¡ '{¡, but ko is not yet dissipation rates can be derived 'd = :t well known. Thís point has been emphasízed in chapter II. 4s4. 7. The irnportance of 2-dimensional turbulence ín the mesosphere, particularly for scales correspondíng Ëo periods in the winds of less than 5 minutes, has been elucidated. A method for determining turbulent energy dissipation rates utilizing these scales has been described. It has also been pointed out (chapter vII) that fading times measured using full correlation analysis for the determination of wind velocities are not related Èo RlfS velocities associated with scales of ),/ 2 metres, and so esÈimations of energy díssipation rates by these procedures (such estimates have been presented in Èhe literature) are almost certainly erroneous' B. The results of what are believed to be the first partial ) reflection profiles obtained coincidentally in time and space with rocket measurements of electron densíty have been presented. The -85krn echo appears to be associated with Ëhe upper porlion of a Large elecËron density ledge at this heíght, and thís ledge has been shown to be capable of producing specular scatter cornparable to the observed strengths. The existence of the ledge has been explained ín terms of photo-chemícal effects by other authors (see Chapter I, Section L.2.L.). A weaker echo at about =80-B2km' below the stronger =B5km echo, may be associated with a small dip in electron density at ËhaË height' 9. Investigations díd not suggest a strong correlatíon between wind shears and layers of HF scatter. This is not conclusíve however, because the resolution used was about 4km. No temperature measuïements \^7eïe made, so calculation of Richardsonts numbers were not possible. Some (weak) evidence did suggest that the echoes at =70-74km may be associated with a wind jet aË that height' 10. There seems to be evidence that graviËy waves aÏe associated wiÈh the observed scattering layers. such \^7aves appear to partly 455. explain the quasi-periodicity of poI¡rer bursts. 11 . The role of turbulence \^7as discussed quantitatively. It was shor,rn that for typical atmospheric conditions, measurable scatter due to turbulence should occur aÈ HF from above 90km. If a strong electron densiLy ledge occurs at =85krn, turbulent scatÈer from this height should also be measurablerwiÈh a strength comparable Èo that producecl, by specular scatter from the ledge. Below =B0kmo turbulent scatter is unlikely to be observed. L2. It has been shown that if both turbulence, and steps of refractive índex \^liËh a vertical exÈent of about one Kolmogoroff microscale, occul ín the mesosphere, then the observed variation of the isotropy of scatter of the scatterers with height can be easily explained, and would ín fact be expected. Specular scatter should domínaËe below =76 to BOkm at 2\[Hz, and turbulenl- scatter above. 13. The first quantitaÈive comparisons of simultaneous measurements made aL =2 and 6MHz have been presented. The previous conclusions have been supported in fu1l. The expected transition height of speculaï to quasi-ísotropic scatter at 6MHz should occuï at most a few kilometres below ttre ZWIz transiÈíon heÍght. VHF experiments and observations by oËher authors have also helped in understanding these D-regíon scatterers. A detaíled sunnary has been presented ín chapter IV, Section 4.5. Such experiments have shor¡/n evidence of quasi-specular reflection (someÈimes called "diffuse reflecÈion" sínce the sca¡ter does not appear to be completely "mirror-líke"i see Chapter IV, Sectíon 4.5) from heights of around 70 to 74km at wavelengths of 6m. This 4s6. suggests that horizontally stratified reflecting irregularities with edges less than I to 2 met.res thick exist in the mesosphere' These are very likely the same irregularities as those causíng HF scatter. Experiments at VHF have sho\^In that these reflecÈing structures are often less than 100n thick' As well as the above points concerning D-region HF scatter' this thesis has also devoted chapÈer VI to the study of the effect of uncorrelated noise on a signal. It has been shown that if amplítude-only data ís used, then noise can distort the autocorïelation and autocovaríance funcËions. A formula has also been derived by means of which the root mean square noise level can be estimated from the autocorrelation frrnction. 457 . 11.3 Díscussion 11.3.r Discussion of doubtful points 11 .3.la Production of turbulence Although it was shown in Chapter IX that turbulence does cause scatter from above SOkm at HF, considerable doubt still exísts concerning the cause of this turbulence' If gravíty \taves are important, how do they generaÈe the turbulence? This sectíon will discuss some símPle ideas. In Chapter IX, it was seen that mean wind shears of =4km resolution do not appear to be ímportant. Miller et al. (l-978) have shown a case ín which gravity \^/aves appear to be breaking, and thus generatíng turbulent phenomena (e.g. Kelvín-Helmholtz instabilities). such a pïocess produces turbulence twice per gravÍ-ty wave cycle. Hodges (L967) has shown that temperature gradients are moclified by gravity I^Iaves and this can be particularly ímportant if Ëhe background temperature gradient is already close to unstable. Once per cycle, the gravity wave could induce unstable temperature gradients, and hence turbulence' as illustrated be1ow. l-ayer r¿h;ch is close to slafic instab¡l t+J \ t, ñ Palchos of turbulence. Ho.¡zonta! d;staneo evity wave osc¡llat¡ons \dï \d= \ 458. Possible examples of cases with power bursts twice per gravity wave cycle, and once per cycle, have been presented in Chapter IX' and the 1aËter case seemed to be more cofllmon' The suggestion of Hodges is interesting' as it offers the possibilíÈyofhorizontallystratifiedlayersofturbulence.For example, turbulence rvould occur more strongly in a regíon of near- unstable temperature gradient, and temperature profiles are generall.y hoiízontallY stratífied. Teírelbaum and sidi (Ig76) and Sidi and Teitelbaum (r978) have proposed that strong wind shears of short vertical extent (1-2km) can be generatecl by the non-linear interaction of gravíty non- r^raves and tides. It is interestíng Èhat processes ínvolving linear interaction of graviÈy \laves do not involve any requírements upon the background atmospheric state' so stratifíed turbulence (1978) may noË necessarily result. However, Sidi and Teitelbaum dosuggestthattheprocesseswhíchtheydescríbemaybemost ímportantíntheregíonlOOtollOkm,althoughtheycouldoperate to some extent down to 90km' AnotherprocesswhichmaybeimporËantiscriticallevel inËeractions between gravity v/aves and r¡ind jets' These can increase wínd speeds at the point of critical level absorption' but not immediately below, thus resulting in quíte strong wind shears' For example, Fritts (1978, IgTg) has mentíoned this process, and Jones and Houghton (1971) have also shown that strong wind shears can result from critical level Processes. Fritts (1979) has shown that these wind shears could generate Kelvin-Helmholtz instabilitíes' and can also radiate gravity waves. The critícal level is also expected to move towar:ds the gravity wave source' Thís might 459. explaín some of the downward motions of scatÈering layers which have been nored (e.g. see Fig. 4.9, Chapter IV). Fritts (1979) has also suggested that such critical-level interactions could exhíbit a quasi-oscillatory nature which could be related to the observed quasí-periodicity of power bursts. (A possibly similar process described by },lerrill and Grant (L979) for the Planetary Boundary Layer has been discussed in chapter IX, sectíon 9.2.2) Re-radiated gravity r¡Iaves could also cause critical-level interactions elsewhere. 11.3.lb Reasons for sDecular scatter One of Ëhe major problems not solved in the work for thís thesis concerns the nature of the specular reflectors. Are they steps in electron density - or do they take some other form - for narro\^7 strata? e. g. example, 7. st.¿tum.f -< too- n density. Further'aTethesharpelectrondensitychangesatracerof similar changes in Èhe neutral atmospheric density, or are they a separate process - do the steps occur only ín the electron density, due perhaps to some chemical process? Haug et a1. (Lg77) have made some simple calculations, and felt that chemícal processes could noÈ account for the rapid changes in electron density which have been observed. Specular reflection has been observed ín the troposphere and stÏatosphere aL VHF (see Chapter IV), and in this case the neutral atmosphere does cause the scatter. steps in refractive índex have been observed in the troposphere (e.8. see Crain, 1955) and the oceans (e'g'Simpson and hloods, 1970), and it appears that such sÈeps are noÈ uncommon' 460. It, seems reasonable to assume, then, that the specular reflections observed in the ionosphere are due to horizontally stratified stePs in the neutral atmospheric density, and thaÈ the electrons are tracers for the neutral densítY ' This still, however, does not explain why the steps occur' Several possibilities have been offered' The proposal due Èo Bolgíano (1963) has been mentÍoned in chapter IX, Section 9.2.2, in which intense turbulence generates a layer of turbulence with very sharp edges. Röttger(1980b)has also gíven.a review of some other possibilities. One of these whích may be important is "lateral convection". In this process, quasi-horizontaL displacement of , individual layers or "lenses" of air packeÈs are envisaged to occur when two differently stratified masses of ai-r are ín contact. Merrill (Lg77) has díscussed an ínteresting critical level encounter, and quíte sharp steps ín density \¡/ere produced (see his Fig.6)oflessthanlOrnverticalextent.Theeffectoftlre ground may be important for these structures, and the wavelength Ínvolved r^/as z30o-500m. Inlaves of this wavelength cannot exist strongly above =6Okm, being strongly damped due to viscosity (e.g.seeHínes,1960,Fí8.11).ThuswheÈhersuchadescription ís valid for the D-region is debatable, but it may be worth bearing in mínd. Recently'Someextremelyinterestingresultsduetol"lcEwan (1930) have appeared. McEwan produced two gravity waves ín a salt-stratified solution of water, and observed their interaction' Turbulence \^Ias produced, but the interesting point was what happened after the turbulence subsíded' Fie. 11.1a A breakíng event in a forced standing internal gravíty hlave fn a linearly salt-stratifíed vrater tanlc 500 mm wide and 250 mn deep. Shown are contours of constant net vertical density gradient derived from quantitative Schlieren photography. The circle of view is 300 mm in diamefer and the free uPper surface of the hlater is marked by a triangle. Contours are as : Black: statically unstable (p'>0); Heavy dots:0.7 på. pt< 0; Clear: 1.3 på. pt . O.ZOå; Líght dots: 2.0 p' . p' < 1.3 Oj; Dashed boundary: p' .2 på. The dark stippled region delineates the fine, three dimensíonal convoluted structures associaEed with mixíng. Also marked are representaÈive streak paths of neutrally buoyant particles for an interval of 0.5 N-i, where N = 1.26 s-L. The domínant elliptlcal component of the motion is due to Èhe forced primary internal wave mode. Shears are everywhere small' even in the mixing regíons. (í ) flrst appearance of well-defined static instability. (ii) 2N-1 later . (ti) 9N- I later . (frour McEwan, 1980) a I t/ \\ a \ l( \ tlf /{ \ v t I { I I ¡fl: I I \ì I Y- I til -T /, / I I I I / r) ¡ v/s rr v/s \ { -A_ 0- o) 0i, 0ii) Eís.11.rb model of mixing by convective overturn, Kinematical (linear) ( ¡ ) Particles of volume v are eichanged. The surrounding static densit-y profile is unaltered. (¡i) The particles extend as adjacent filaments and collapse towards a combíned equílibrium layer. The (exaggerated) effect on the surrounding profile is shown. (iii¡ After equitibrium, fluid has been permanently removed from levels 0 and zy and. deposited at z¡/2. There is a net gaín in static potential energy. S = horizontal area per unit parcel-exchange event, and pt = density gradíenE. (f rorn McEwan, (f980) ) 46L. During the mixing process, considerable interleaving of regions of similar density occurred, as illustrated in Fig. 11.la. Then, after the turbulence began Ëo subside' layers of approxinately cofistant density appeared in the hTater, wíth very narrow interface regions; e.g. 3 T.t:.." 1ën.':. z ì<5¡r-11 <1ñ1 [An approximate physical description for thís ís also illustrated in Fig. 11.1b (due to McEwan, 1980). The essential feature of the descríption is that displaced particles stretch and change form, rather than simply mix wíth Èhe environment' For a more deÈailed descríption, see McEwan, (1980) ' ] Eventually the weak residual turbulence died, leaving these straÈa. Since the water was hydrostatically stable the layers remaíned, molecular diffusion being the only process available to mix the layers. It thus appears that turbulence mixed the water, but that when the turbulence ceased, layers of separate fluid resulted. Once the turbulence had died, vertical mixing became quíte weak, although horizontal mixing could still occur to a suffícient extent to mj-x the layers horizontally' TheímportanËpoíntisthatLurbulencedoesaPPearto produce these strata. (Recall also from chapter IV, Section 4.6, that other experimenters (e.g. Baker, I97L; Calman, L977) have produced sËeps in the density of salt-stratifíed solutions, although ín those c-ases the process r^Tas related to differential rotaËion, so 462. the process may have been different.) Then assuming such strata do often result from turbulence, it could be envisaged that if turbulence occurred in a very stable region of the atmosphere, perhaps due to gravity wave effects, and then died, stable strata could occur, with interface regions of perhaps a Kolmogoroff microscale in vertical exËent. If the region were stable, these laminae could remain for some tíme. They would be detected by radio r¡/aves of wavelength Ì for as long as it took for the interface regions to become -À/B ín vertical extent, at whích stage "diffuse reflection" might occuï (e.g. see Röttger, 1980a; an interface with corrugation aÀ/B high would noÈ produce mírror- like reflection). This should be a duratíon of t=(O/B)-n)2 /v, and n being the initial thickness (=1 Kolmogoroff microscale) ' v the kínematic viscosiËy - at 75km, rì=2-3m (table 1'9a), and V-.3m2s-1, so for À = 150m, t ¿ 10 mj-ns. Hence in stable regions, these laminae would persist for some time. In regions with background turbulence, these laminae would quíckly díffuse so that the ínterface was > in vertícal extent, when little specular ^/4 scatter would occur. Thís could explaÍn the observations ín the lower atmosphere of Gage and Green (1978) and Vincent and Röttger (1980) that specular scatterers are associated with regions of large R. (high stability). (The Bolgiano model discussed ín Chapter IX would suggest specular scatteÏ comes from regions of low R..) Intermittent turbulence could be generated, and when the a turbulence dies, stable straËa result r¿hich remain stable if the atmosphere contaíns no background turbulence. The strong bursts ' of scatter observed would be related to thís íntermíttency of turbulence generatíon. It should perhaps be mentioned that other 463. explanatíons for the trburstiness" exist, such as focussing and de- focussing of the radiation (e.g. compared to Chapter III, Figs' 3.5a and c), but the author feels such processes are not the major cause of the intermittent bursts. Such focussing possibly does occur (due to spatíal oscillations of the reflecting edge height) butslnce the powers aïe either very strong or almost non-existent, it suggests that the scatterers are ej-ther present oI not present' respectively. It would also not be surprísing if at times there is some tilt of the reflectorg from the horizontal, but Èhese tilts have not been searched for in any detail as yet. Investigations of such tilts, and also examinations of gentle oscillations ín the surfaces of the reflectors, might be a useful future project. (For example, it might also be recalled from chapter v, Fig. 5.17c that a weak hint of specularily at ll.6o occurrerl for the B6km layer. This could perhaps have been due to gentle spatial oscillatíons of the height of the strong electron density whích was probably associated wíth the layer, gíving some off-vertical specular scatter). 11.3.lc Reasons for Preferr ed heiehts Some progress in explaining the reasons why echoes come from preferred heights has been achieved in this thesis, in that the =B5km echo has been exPlained' The suggesÈíon has been made that the =70-74km echo may be related to the wind "jet" aË that heíght (see chapter I, Fig. L'7 A peak in mean wind velocÍties occurs at 70-74km, and this has gravity ü/aves' , been called a jet here). For upward propagatíng thís jet would be the first time Ëhat wind speeds greater Èhan 464. 50m s-l t"t. encountered, so the regíon is a candidate for critícal- level effects. It is also interestíng that'schmidlin (L976) claims to have found that ËemPerature inversions can ofÈen occur at aro,und 70 to 75km. Such a temPerature inversion would produce a hydrostatically stable region, and so could produce the stable conditions required for the stratifíed layers discussed in the previous sectíon Èo maintaín themselves. If indeed such temperaEule ínversions do exist, they could be the reason for scatter from =70 to 74kn. one other major scatteríng regíon is Ëhat at about 90kn. I{e saw in chapter IX, Fíg. 9.8c, that if turbulence exists, l¡ observable scatter ( effective reflecÈion coeffícient ì lO-4) will be observed at 90-95krn. This is independent of the existence of steep electron density gradients - if such gradíents exist, horr/ever, the scatter will be even more enhanced. To produce effective. reflection coefficíents of " lO-3 probably requíres a layer of stronger than usual turbulence combíned with a faítl_y strong electron density gradient. llhy such layers of turbulence should occur is unclear, but they do appear to have been observed with rocket observaÈions (e.g. see Blamont and Barat, L967; Anandarao et aI., Teitelbaum (1978) 1978) ¡ Teírelbaum and sídi (1976), and sidi and ' have proposed that large wind-shears can be produced by non-linear interactions of gravity r^Iaves and tides, and these can produce turbulence. These authors feel that such wind shears occur preferrentíally in the region between 100 and 110km (maybe down to gOkm), and sínce D-region echoes cannot be observed above 95kms' (due to the leading edge of the E-regíon totally reflected pulse) nay some heíght preference for echoes to occur betr¡een 90 and 95km 465. perhaps not be surprising. At níght, echoes do indeed occur from 90 to about llokn. The large phase changes which sometimes occur in the tides around 85 to 90krn (e.g. Elford and Craig, 1980) would also be associated r¿ith large wind shears, and these could possibly induce some turbulence. However, the author feels Èhat these points do not fully explain Èhe echoes' One particularly interesÈing facet of these "90 km" echoes is the tendency for them to have, on occasion, strengths comparable to E-regíon total reflection, partícularly at Townsvílle (e'g' See Chapter III' se-ction 3.4). In such cases they appear Èo be close to critically reflecting, and perhaps a "wind-shear" mechanism may be acting (e.g. htrirehead L96L; L976). The wind shear mechanism ís usually considered to be ineffective at =90km, sínce the ions are supposedly controlled by neutral motions, but the author has not seen a convincíng discussion of this assumptíon, and the ions could possibly stilt have weakly independent motions. However, if wínd shears are important, they would be of short cluration (= mins) and would not be maíntained over more than one or two km in height, sínce the work in Chapter IX has shown that wind shears taken with a height resolution of =4km are not related to these echoes ín general. Sequential E" layers (see chapter III) have been noted to descend dor¡n to 9Okm, so this may also support a "wind shear mechanism" for the 90km laYer. 11.3.1d Sea sonal variaEions The author has lítt1e ne\n7 to offer to explain the observed seasonal variations of echo strengths. The suggestion by Vincent (Lg67) that gravity waves may be more conmon ín winter is very likely a valid comment. 466. ]L.3.2 The authorrs view AlËhough the ínformaÈion available is inadequate it ís difficulttoresistthetempËationtopresentaspeculative descríption of D-region scatter' The authorts present view of the D-region would be basically described by Fig' 11'3 ' (Perhaps the most doubtful point is how the Fresnel steps occur' In the following scenario, they are assumed to result after turbulence, but this ís bY no means certain') Intermittent turbulence occurs at all heights ín the mesosphere, produced prirnarily by gravíty r^/ave ef f ects ' This may occur in both stable and unstable regions. After the turbulence subsídes, laminae result, wíth interfacial layers of about a Kolmogoroff microscale thickness. If the regíon is normally stable, these for laminae can be quite stable, and produce specular HF reflection heights below about BOkm. Above Bokm, these interfacíal layers are too thick to produce signífícant specular reflectíon' If the region ís unstable, Ëhen these laminae are quickly dispersed' Turbulent scatÈer is too weak to be observed below Bokm (see Fig' 9'B)' but above 8Okm, turbulence is readily observed- Thus below B0km, reflection from stable layers occuTs. Above 90km' any Process which can produce turbulerrce, includíng Hodges (L967) rnechanism, windshears,andsoforth,willproducesomeobservableScatter. Turbulentscat.terwillalsobeseenat=B5kmíf.aLatgeelectron densityledgeexistshere.Thisledger¡illalsoproducea significantcomponentofspecularscatter'Gravity\'Taveeffects are very large at =90-l10krn, (e.g. Sidi and Teitelbaum effects, 1g7S), and tide and \^7aves can also break as Ëheir amplitudes become 1arge, so consíderable turbulence could occur above 90km' 467 . A full description of the quasi-períodicity of Po\^Ier bursts is not yet available. Hodges (1967) proposál may be relevant, slnce the wind oscillaÈions and pohter bursts often have equal per,iods.Anotherinterestingschemeisthatdiscussedin chapter IX, sectíon g.2.2, due to Merrill and Grant (L979), ín which an incident graviÈy \^Iave sÈeepens the wind shear and produces turbulence. Thís turbulence then decreases the wind shear, and so the turbulence dies. At this st,age, laminae may appear' producing specular scatter. The incident gravity wave Ëhen begins to steepen the wind shear (and also íncrease the wind speed at the top of the shear regíon?), upon which turbulence again is generated, , andsotheprocessrepeats.Thisshearregionmightalsobe expected to move down in time (e.g. see Fritt.s, 1978, L979). However, other processes such as breaking v/aves and gravity- wave induced wind shears may also be important' partícularly above 90kn. d1 a.-\ Ronge--9210-. rco ) o- off' E , -ro 'Àrur\,uw\o',1'. o 09 2 3 7 2 21 LOCAL T]ME 77/ 307. BUCKLAND PARK. X MODE. NIDE BEAM. Fig. 11.2 of a meteor observed with a wide beam at Buckland Park during November, L977. This can be recognized by the rapid-E""mple rise and exponenÈial decay. The transmitter \^ras only on for 2 mins in every 4. The rise time is approxímately the time for the meteor to move through l Fresnel zoîe, d; i.e. t' = d/v, 1, v = meteor velocity, =20-60km s and d = ,+r', R = distance Èo meteor, tr = 151.5 m ín this case' Thus t - .3s. The decaY tíme is tr2 1=-^ ' L6rzD, where D ís the diffusion coeffícient. Initial.ly the trail expands by molecular diffusíon so D- I m2s-1, (e'g' see USSA' 1962) but eventually turbulent díffusion becomes important (D =lOTn2"-1 at gOkm altitude for turbulent diffusion e.g. see al' 19-72' Chapter I, Fíg. 1.9a) . As an example, see Rees et ' rig. 6; turbulence becomes important afLer about 30 seconds in thãt case (rocket cloud release) ' For D -1n2s-1, r=2.5 mins i.e. a large decay time. The peak amplitude (50¡fr-.¡ corresponds to an effectíve reflection coefficient of abouEw (1.5) (90¡ (50 x 10-6) (see Chapter III, Table 3.1, ko2\^r = r.5) R 7 x 1O-3 or peak Themeteorprobablyoccurredatanoff_zeníxhangle.other¡¿ise ít would not gerìerâtty register, since the meteor trail mtrst be aligned perpendicularly to the line of sight, and meteor paths are seldom horízontal in the atmosphere. For Ëhis reason' meteors show more frequently with the wide beam than the narrow beam. The possibíliiy exists, however, that meteor trails could form, and be twisted by a wínd sl'rear to produce specular scatËer. Such scatter would be hard to recognize, however' 468. LL.4 Inconc lusive results, future proiects and schematic summarv There ís still a considerable amount of work to be done before D-regíon HF scatterers can be fully understood' . One phenomenum which was investigated very bríefly was the role of meteols. No doubt these contribute to some (if not most) of Èhe metallic ions observed at =90krn, and thís could be important at níght. Ilowever, thÍs was bríefty discussed in Chapter I, Sectíon L.2.I. and other processes such as ïesonant scattering of LCI and LU radiation by Èhe hydrogen geocorona, and electron precipítation, seemed to be major contributors to night-time electron densities' A search for direct-observaÈions of meteor trails was undertaken, and Fíg. 11.2 shows an example of a meteor trail observed at L.gBMHz using a wide beam. Meteors show much more clearly at 6MHz, however. Brown (1976) has also invesËigated meteors aÈ 2MfIz. The auÈhor does not feel that meteors make a major contríbution Eo 2MHz scatter, even at night. IË ís possible that meteor trails could have formed, however, and been Ëwísted by the wind to an angle at which specular reflection could occur. such a process could not be easily recognized - it would not have the exponential decay used to recognize the meteor in Fig ' LL'2' Other projecÈs which deserve consideration include the following (not in any special order:) 1. Experiments using widely spaced antennae with separatíon of several kilometres to determine more accurately the horizontal dimensions of scattering regions' 2. Better investigations of seasonal variatíons of varíous parameters (e.g. the height of transition from specular to quasi- lsotropic scatter (Lindner ,(L972; L975a,b) has done pr:e-limi-nary work on this) ) 469. (Chapter 3. Curves similar to those shown in Figs. 5.12 v) couldbepreparedforcasesotherÈhanRayleigh-distributeddata' so thaL Ínvestigations of specular to random scatter component ratios can be made more Èhoroughly' 4.Símu]-taneouspartialreflectíonexperimentsandrocket_ the measuremenLs of tempeÏature could be performed to test with assumption that the scatteÏers below =BOkm are assocíated stableregions.Símultaneouspartialreflectionandrocket well (e'g' experiments are desirable in many other contexts as observatÍons of turbul-encefPat1ia1 reflection correlations; more Ëhe =85km ledge). study the 5. The work in Chapter VII could be utilized to seasonal variations of e4 ' polar 6. Useful results could be obtaíned by narrowíng the díagram of the Buckland Park array, perhaps by interferometer techniques,sothaÈthemotionsassociatedwithturbulencescales of of )t/2 can be actually observed. Then accurate determinaËions Chapter II) will be necessary' k* (eU = to v3/1,; see prove 7. Searches for azimuthal assymmetÏy of scatter could inËeresting (e.g. c'f' Fig' 5, Harper and Gordon' f980) ' can be B. The programs Specpol and VolscaÈ (see appendices) thícknesses of utilized much more to investigate more closely the scattering layers, and also the isotropy of scatter' g.TheangularspectrumofScatterat6MHzshouldbeinvestígated more thoroughlY (see ChaPter X) ' l0.,,Imageforming''couldproveausefultechnique.Thatis' recordamplitudeandphaseatallaerialsoftheBucklandPark appropriate atr:ay separately, and then re-add these later' with 470. phase shifts, to simulate lookíng ín all directions of the sky' Thusa2-dimensionalpíctureofscaÈtercouldbebuiltup. ll.DAEexperimentscouldbeperformedwiththeviewtotesting the assumptíon that the =B5km echo ís always associated wíth a large electron densiÈy ledge. In the past, the DAE has been may unrelíable above =Bokm, but the use of coherent íntegratÍon possibly allow reliable X-mode measulements to be made for heights of'up to B5-90kn. such measurements may also be easier using 6MHz' L2.SimulÈaneousHFpartialreflectionandVHFexperimentsat VHF the same location are most desírable' Faíling this' Ëypical effectíve reflection coefficients should be calculated, so that comparisons with HF experiments can be made' Many assumptíons in have been made about the línk between HF and VHF scaÉterers thís thesís, (e.g. iÈ has been assumed the satn" scãÈterers cause both types of scatter), and these assumptions must be tested' Althoughthereísundoubtedlyalottobelearnedaboutthese of scatterers' an approximate picture has emerged duríng the course thisthesis,andFíg.ll.3isaschematicSummaryofthispícture. !üith a beÈter understanding of these scatterers, ínterpretation of much more many other current D-regíon experiments should become reliable. I Tv'hu lent Sp*c çJqr (R = effective reflection coefficient) qo @ R - ì- Quas i - X I 0-3 i sot rop ic: (up to Turbulent I 0-2 some - 5 - l0 km. Scatter nights cont i nuum of e (2, 6 ¡luz) scatterers. FADING TIME - I -3s, -2-6 specular 10- 4 - 2-6 kn. I weak strong eìectro n scatter density ledge / TRANSITION I lmportant for \ REGION. ? I J specular and ¡ turbulent ¡¡sÀ ) scatter. Mi rror-l ike t reflection - specular. 2-4 kn. c) ) k-r -4 - - Tu rbu I ence | ndi vi dual T x lO-a present but sca t te re rs not detectab e 5 100 m present, not FADING TIME thick detectabl e - 4-l 0s (horizontal sometimes extent present and up to 20s kn ? detectabl e - J-20 ) and more. - often Steps and turbulence at all seve ra ì heights. sca t te re rs in radar volume Frequ. of (lf e 6o Step thickness - Koìmogoroff only scatter microscale. 0ccu rrence height-stable and narrow pul se Fîg ll.3 Schematic summary of the important points regarding HF D-region scatterers. to0 Ge ne.otjgn Do¡lv essorul n3sc-'hgdsnt --J 90 km r layer present reaki ng waves, a ut umn gra wave effects. w to D spri ng. Day ê Nigh n sumfrer, - 3 dBrrbursts" nbe occur can o tal ly lasting - 5 min. .c- el ectron flecting density ledge Quasi-period i mportant for of 5-15 min. - specul ar and 8O I I echoes turbulence scatter t ronqest l,Jave ef fects al so utumn E important. inter E pring. akegt in uïrmer. choes below rom 70 Day Bo km in only - - 10 dB "Bq.rrsts articular often often occur. re less i n s t ronges t Quas i ommon Gravity waves imPortant. mi d- periodicitY umme r. - 70 km wind 'rjetrl a fte of - 5-15 al so important ? mins. Temp inversions ? 1 (on occasion, V. ht - stable 60 and steady ampl i tude echo) APPENDIX A THE NEUTRA]: ATMOSPHERE, AND THE IONOSPHERE ABOVE 1OOKM A. (i) The ouËer regÍons A. (ii) Classification of the atmosPhere A. (iii)a Partícle denslt.íes and structures above 100km A. (iii)b The Electron density profile above 100h A1 Appendix A: The Neutral Atmosphere, and the lonosphere Above 100kfi A. (í) The outer r íons The atmosphere is a large system, and the ínteracËions ínvolved are complex and intrícate. From the dense boundary layer, to the tennuous remnants thousands of miles ínto space, there are a wealth of physical processes both fascinatfng and Ímportant to all inhabitants of the Earth. At large distances free electrons and ions spiral freely along magnetic lines of force and ínËeract strongly with a wind of símilar parËicles flowíng from the sun. Thís t'plasma sphereil exËends far fnto space - up to aroûnd 25 to 30 Earth radii in places - and is a type of rrouter regiont' for the atmosphere. A revíernr of Èhis region can be found in J. Atmos. Terr. Phys., volume 40, number 3 (March) L978. Fig.4.1 shows a schematic diagram of the region. However, Ít ís regions much closer Èo the EarÈhrs surface whích are the subject of Ëhis thesj-s. This appendíx will gÍve a very brief account of Èhe maín regions of the a¡mosphere beÈween 100 to 500km, along the lines found in many texË books on Ëhe aÈmosphere' HEOS -2 AfH _---+ è MANfLE SOLAR WIND --+ RE 4 5MA -._-' BELI AND ------+ RING CURRENT POLAR WINO -----+ MAGNETOPAUSE --.1> __,> -----ù -_-.> FÍe 4.1 The EarËhts magnetopause and magnetosphere regíons' (From Bahnsen (1978) ). Also shown 1s the orbit of the staellíte HEOS-2, which was used for stuclies of thÍs region. The Bow shock occurs at about 15 Earth radíÍ on the sunr¿ard side. A2 A. (ii) Classification of the atmosphere Two of the most cofimon classíficatíon schemes used in the atmos- phere are based upon temperature structure, and elecËron density. Fig. 4.2 shows these classificátions' It j-s important to note Ëhat Ëhese sample profiles can only be approximaËe, sínce the ËemperaËuïes and electron densities can vary substanÈÍal1y with tjme and locaÈion. One inportanË regíon not presented on this diagram Ís the turbopause region. up to abouÈ 1OO-115kn turbuLence can play a major role in Lhe dynamícs of the atmosphere; but above. thís region, turbulence ís a relaËívely xare phenomenon. The reason is that the mean free path of partícles, and the íncrease in temperature, result ín large molecular diffusion rates, and most small scale transfers of heaE and partícles occur by such molecular transporË. The transiÈion reglon between the turbulent domínated and non-turbulent regimes is quiËe narro\^7' and is called the turbopause. This is discussed ín more detaíl elsewhere' It 1s useful to consider just why the Èemperature profíle follows the shape gíven. Below the tropopause, heating ís principally by re-radiation from the ground. The temperaËure falls off rvith íncreasing height, as less radiation penetraËes to greater heights. Convectíon and so forth also act, resultíng in a Ëemperature profile approxímatíng the adiabatic lapse rate. Of course many local processes also acÈ to produce loca1 deviaËion from thís pícÈure. This regíon wil-l noÈ be discussed greatly here. Above the tropopause, Ozone becomes important. AË heights greater than about 50km, there ís insufficient o, density to produce signifícant 03' but at around 20-40km, the o, density and incoming w radíation (less Èhan abou t 246nnù are in sufficient quantities to produce sig- Ftg. A.2 Typical daytime temperaEure profí1e for low sunspot acÈivity. T' is the neuÚral kinetic temperature, T, the. ion temperature, and T temperaEure. Tnr shows a. typical daytine e the electron neutral temperature profile during sunspot maxÍmum. At night, falls to abouÈ 600K at sunspot minimum and abouE 900K at T-n sunbpot maxiurum (King-He1e 1978). TemperaLure maximum occurs at about 1600 hours 1oca1 time, and minimum about 0400 hours, ln the themosphere (Kíng-Hele 1978). TTre classificaÈions of the various regions are also shown. The temPeratures below 100k¡rr are taken from Èhe U.S. Standard Atmosphere 1962, and above from Roble and Schmidtke (L979) (X = 55.4"), for data taken durlng 1973 (Sunspot mlnirnum). For sunspot maximum, Garrett and Forbes (1978) Fíg. 1 was used. The numbers can only be approxímate, as they vary latítudinally and wÍth season. Even the heights of the various regimes can change - for instance, the tropopause height varies borh latiÈudina1ly and with tíme of day. In the exosphere (about 500-1000km), kinetic temperature is not a meanÍngfu1 terrn slnce neutral atoms rarely collide (Klng-Hele L97B). Approxinate atmospheric pressures (from Houghton Lg77, AppendÍx 5) are denoted on the right hand verÈlcal axfs. Pressure follows an approximately exponentfal decrease r,¡ith height, as a result of the balance of gravíty and hydrostatic pressure (dp = -gpdzr p = ffi, * beÍng the mean mass of the atmospheríc molecules). Also shov¡n are "typlcalt' daytíme electron densities as a function of height, wlth the associated nomenclature. However, Èhese densíties can vary wíde1y wíth laÊitude, sunspoç condition, and a variety of other para¡nelers. (For example Ratcliffe L972, fÍg. 3.3 shows varíatlons of F region electron densíEies at different heights and lattitudes). The hatching gives some idea of the variations in electron densi-Ey which can occur. Data 1s taken from Craig (1965), figs. 9.11 and 9,I5, and RaÈcliffe (1972) fig. 3.3. Also shown are some typical E reglon níght time electron densities. The symbols D, E, F1 and F2 denote the ionospheric nomenclature. The E and F2 regíons are 1ocal peaks ín electron density, and Fl ís a local peak at t.ímes (particularly durlng Sumspot maximum summers). This thesis concentrates on the D region (60-100km), whích also covers the mesosphere and lower themosphere. E lect ro n density l | ( m-3 ) 10? los los lolo 1o 1ol2 1013 3 5 6 to7 (cm-3) 10 I lo 2to t6a t0 to s00 I Tn l I Te T¡ I I Tn' ; E /.0 0 ûr DAV L) r¡ UI 0, oL E I (J J ; I (¡, 30 0 \ _c.o (n .c o c'l F2 E õ E o û, .;ó o c E ô, x o 200 FI I o o o! o o o0, N IGHT E 0.0001 THERMOSPHER 100 0.0 0l MESOPAUSE 0.0 I D 0.1 Tt:ttjtï I -sTRATOPAUSE r0 STRATOSPHERE -TROPOPAUSE r00 r000 0 10.000 0 100 1000 Temperoture ( K ) A3 nificant Or. (The principal reactions can be found in Houghton (1977) 55.5), Mínor constiËuenËs such as NO, NO2, OH also play ÍmporËant roles, hor¿ever. UV radiatj-on of v¡avelengths less tha 1140nm, buË parËícularly less than 310nn, Ís al.so involved, being absorbed ín the photo díssociation of 6zone This production of O, absorbs out the IIV, and thus at heíghts belor,¡ 15-201sn, 1íttle of the radiatíon penetraËes. The result is a maxÍmum in the O, lrofile at around 25-30krn (Houghton 1977,95.5)' These O^ reactions produce molecular kineËic energy and hence atmospberíc 3 heating. Some of the heaÊ ís re-radíated by COz at infrared wavelengths, and the balance between O, heatlng and CO, losses produces a temperature peak at the stratopause (Houghton,94.7). At great-er heights, there ís 1íttle absorption of radiation and the temperature falls a\^7ay to a minimum aË about 90km. Coolíng by radiation of wavelengths around 15 microns bv co, also contribute signifícantly to this temperaËure decrease, e.g. see Al1en eË al. Qglg). Above this height, solar radiatíon capable of íonizing partÍcles ís of suffícient intensity to produce appreciable ionizaLior- - partícularly of NO, OZ, N, and O. Associated r,rith these reactions is an increase in kínetic energy, thus heatÍng Ëhe aËmosphere, resulting in a ríse Ín temperaËure. Above about lOOkm, neutral partíc1es, íons, and electrons no longer collide suffíciently often Ëo have Ëhe same Èemperature, and each therefore has a differenË temperaËure. The reason for Ëhe shape of the electron density profile will be discussed shortlY. A4 A. (íil) a. ParËicle densíties and sËructures above 100km Fig. 4.3 shows the parÈícle densities of the major neutral atmospheric constituents. N2 dominates below about 100km, buË above this height atomÍc Oxygen (O) Ís the main consËiËuent ín terms of number density. At even greater heights He domínates, since íË has a much larger scal e height than Ëhe heavíer O and consequently fa1ls off in densit.y more slowly. Hydrogen dominates aË even gleater heights, but thÍs species does not fo11ow a símple exponenËial fall off ln density. Thís arises because the rate of supply of. H, Ín the lower atmosphere, and Ëhe rate of loss at the top of the atmosphere' are quite rapid. (Houghton 1977, 55.3). Also shown is the electron densiËy distributíon with height. Notice even at the peak of electron density (82), electrons (and hence positive ions) are only abouË 1/300 to 1/1000 of the total neutral number density. This ís a point worÈh bearing ín mínd r,rhen consídering the íonosphere. Again, the sÈaËe of compositíon can vary considerably wiÊh varíatíons in the state of the sun (through a temperature dependence) (e.g. Ratcliffe L970, p. 131). The total atmospheríc density also varies significantly (King-Hele I}TB). During one day the densíty varies by abouÈ 1.5 times at 200km, and by abouË 6 times aÈ 60o1qn. Variation is maxjmum aÈ around 600km. Varíations over a sunspot cycle are even greater. There are also other types of density variations - semí-annual and Írregular being two of the others. Kíng-Ilele (1978) gíves a comprehensíve review of atmospheríc densitíes. Ionization of these various species is the reason for free electrons appearíng ín the atmosphere. The F region ís produced príncípal1y by radiation at 2Onm-Bonm ionÍzÍng N2, O2 and, particularly, O. The maxímum of íon production is at heights of about 150 t.o 170krn (Fl heíght). A5 Below thts height the intensity of this radiatÍon falls off, and above there are less partieles to be Lonized. (The electron densÍty however, continues to rise above this heighÈ because recombinaÈion is less rapid). The E region is Produced bY (i) Xrays (wavelengths 5 10nm) ionizing O, and Nt (il) W (wavelengths - L00nn) íonLzLtr.e Or' Fíg. 4.3 Typical densities of neutral atmospher:ic constituents (from lloughton 1977, fLg. 4.2; Ackerman 7979; and Roble and Schrnidtke 1979, fíg. 6). The NO densíty above 100km is from Roble and Schmidtke. The NO, CO2, H2O, 03 and OrrA (O2r^g) measurements came from Ackerman. All other values came from Houghton. These densitÍes fluctuate somer¡rhat as the temperature varies. Also shown, for comparison, are typical electron densit,ies (these are sho¡nm by Èhe broken line and shading. The shading gives some idea of the possible variatÍons. The long line represenËs daytime densíties, and the short sectíon to the left typical night time densities). IË is clear that, even aË 300km, the elecËron ntrmber densities are sma11 compared to the toËal neutral density. Only some of the more important minor gases are shown below 100km. For a more compleËe picture, see Ackerman (1979). In particular, CH4, N2O ancl CO have densities greater than or equal to the densit.y of NO, and HNO,, CH3C1, N02, HCL, SO2r CCI4, C1O, and HF have densities comparabl-e to that of NO (about 109m-3 at 20-4Okm). NO has been included because it, along with O' are the main t!üo const.ituents Ín the D region direcÈly ionízable by incoming radiation Above the D region, 0r, O and N, are the most. imporÈanË ionízab1e constituents up to 500-600km. At a thermospheric temperature of 700K, He be-comes more dominanË than 0 at about 500km, and H atoms take over from He at about 900km. At a ternperature of 900K, Ehe O-He Ëransition ís about 600km and He-H transíÈíon about 1800K (King-Hele I978, fí9. 2). Above about 2000km, ions become the major form of partícle, but below about 1000km, the domínant species are neutral. At ground level, the toÈal number density ís 2.5 x 102sm-3. The distribution (by number density) is 78.03% NZ, 20.957" OZ, .93% CO2 (Chemical Rubber Co. Handbook of Chemistry and Physics, 51sÈ ed., L97O, p, fI47). N umber density ( cm-3) l0 l2 11 10 I 1o 1o 1o 1 102 104 106 N He o Argon Oz 2 I 500 I Totot \ \ I I \ \t \ \t 400 \ \ \ E J \ \ ; 300 \ ol I o c, \ \ I .9 I \ c \ N2 (¡, .l o .\ He o- 200 I o it.tta\ o (9 Noi. ) 1¡ ¡: a \ lt' ..i o a 3 ao \ ñ 100 a a \. a a I 75 a a N01 a lo coz 50 aaaaa , art  ol al ) 25 a H 20 0 r6 ro t2 14 10 l ors 1o2o 106 1oo 10 10 10 Number density ( m-3) A6 A. (ilt)b. The Electron Processes. Ionizoble constituent Rodiqtion \ intensitY \ densit Y \ \ E cÞ o T / Number density & Rqdiotion intensitY Fig. 4.4 Tlreresultfsapeakíntheproductlonral-eofelecErons,as of the Íllusrrared 1n Fig. 4.4. (e litrle simÍlar to the formation Ozone laYer). Thereaxe2majorsuchpeaksínproducËlonrateintheupper F1 region (about 1-50- lonosphere - ar the B regíon (110-13okn) and Èhe 200km). L7 Ilowever, in looking aË the overal-l electron density, one must also consider electron loss rates. IË appears recombination is the dominanE elecËron loss Process Á..1 x*+.-+X*hv ' This is the reverse of the íonÍzing process ' A..2 Atiiachment, ff + e- + Ñ- i hv, appears not to be irnportant. (about However, 4.1 is a reactíon with a veÏy small rate coefficient 16-12"*3"-I¡ compared to the expected coefficien¡ of about tO-8, as judged from the decay rate of electron density after sunset' However, I reacËio.9t A.3 e* XY.+X+Y is a much faster since conservation of energy and momenËum are satisfíed more easily' Thus A"4 e-+Nr++N+N 4.5 and e- + Or+ + O + O ate 2 rnajor recombination reactions' For O+, 2 teactLons are invol-ved; 4.6 (i) o++Nr*No*+N (íi) e +No++N+o In this reaction pait, the rate is determined by the slowest of (i)and(ii).UptoabouË20Okm,(ií)isthes].owest.Thereactíon coeffícienr of (ií) is height índependent, and thus up to around 2001n' theelectrondensiEyprofileisroughlygivenbytheproductionrate. heíght, Hence E and F1 are generally Chapman-like Layers. But above this (í)isslowesË.ThisarisesbecausetheraÉeof(1)dependsonthe concentration of N2' which has falLen to quíEe 1ow values' Thus above Fl, both the productíon and loss rates fa11 off, as the concentratÍon N^ falls off. It so happens the loss raËe fal1s off faster, resultíng of ¿ in an increase in electron density above Fl. Sometímes the Fl layer this shows as a peak, before Ëhís rÍse, and sometimes iÈ is hídden by AB íncrease of electron densiËy. Fl is parËicularly prominent in the sunrner of solar maxÍmum Years. Then why does the elecÈron density fínally fal1 off at about 3001cn (F2)? (natctitfe 1979, p. II2). QuÍte simply, díffusion takes place very rapídly (due to the long mean free paths) and the electrons form into a densiËy distribuËíon governed by the hydrostatic equation dp = -pgdv. ) Thus Ëhe èlectron density símply decreases exponentially with height, with a scale height H (ín principle) much larger than thaL for the ions (U is about 30'0O0HI if the electrons were free to diffuse alone) . However, electrostatíc forces draw the íons and electrons together, and as a result the two specíes combine Ëo produce an almost cortrtron scale height I1 = 2Hr. APPENDIX B PROPA.3ATION OF RADIO I^IAVES THROUGH THE IONOSPIIERE B (iì General discussíon Groups B (íi) Ray Theory, Dífferential Absorption, and B1 AppendÍx B: Propagatíon of Radio ltaves Through the Ionosphere B. (í) General discussion The detailed examínaËion of the propagation of radio ldaves through the ionosphere is an extremely complex problem. The two classíc references on the subject are Ratcliffe (1959) and Budden (1966). However, some of the formulae - ParËicular1y the Appleton-Hartree formula - have been updated sínce Èhese earlier books (Sen and l^ly11er Lg6O, with corrections by Manchester 1965; Budden 1965). Fortunately, however' many simplifications can be made at frequencles above al¡out 2[flz in the D rcgÍon' and many of the more difficult aspects,of a fu1l treatment are not ímportant. The frequencíes used for investigatíons in this Èhesis all líe in the region close to and above 2MLIz, This shorË review wíll broadly consider Ehe equations governÍng the fulI solution, and simply mention the D regíon simplifi- catíons. A brief mention r¿i11 be made of crítÍcal levels, and cases where the ray theory breaks down - parË1-y Ëo give a feel for the condÍtions for whÍch ray Ëheory is relevanË, and partly because later, when graviÈy \^raves are consídered, similar complicaËíons arise whích in that case are Ímportant. In fact the passage of radio hraves through the ionosphere, and of gravíty \¡/aves movíng through an atmosphere, have many cofilnon features' Budden (1954) gíves a good overall surtrnary of Èhe derivation of the relevant formulae, and the maín poínts are repeated here. one begins by consídering the effects of an electromagnetíc (EM) Í/ave on free electrons. In Lhe Appleton approach (Appleton L932; Hartreers approach was some¡¿haË different (Hartree 1930; Budden 1954))' an ef f ectíve polar i-zatLon P of the medium l^7as assumed to occur due to the passage of an EM (ElecËromagnetic) r'lave' 8.1 P = Nerr where * = .,.".'"n densíEy, € = charge of an elecÈron, r = electron dísPlacement. B2 and magneÈic Maxwell t s equations \¡/ere used Èo relate P to the electríc fíelds E and H of the \^7ave' The electront s equaÈíon of motion htas assumed Èo be 8,2 -",Íå **.u,H e(Sx sl + eE, !ühere D^ = electron rest massr t = timer v* is the collision e E. frequency of an electron with neutral molecules, and B = 8.2 thus gíves the so-called "constiEutive relations" of Èhe medium' proportional to its The el-ectron has been assumed Lo have a dampíng force of thís assumpËÍon r.ras the basÍs of the velocitv S. Modífícation approachofsenandlnryller,whomadeuseofmorerecenËexperimentswhich constant' but showed that the electron collision frequency itself is not proportional to the square of the electron velocity. such modifícations wíll noÈ be considered here, although all work involving refractÍve indices in this thesis are calculaËed by the Sen-Wyller formulae' For a discussion of these modifications Budden (1965) is a useful reference' Maxwellts equations lead to equations 2.30 0f Budden(1966)- namely; 4 D, curl E = -ikl, curl H = to where H=Z^H,HbeíngthemagnetícfieldoftheelectromagneticU- permeability and radÍation, Zo = ,F5, and u0 and e0 are the permíttivityoffreespace.ThetermkisthewavenumberofËhe racliaËion,andDÍsthedisplacementcurlenË=eoE+P. (B'2) are All equations not ínvolving the consLitutive relatíons valid índependent of the modifications of Sen and ltryller' The polarizatíon is then related Ëo E by 8.3 P '0c tMl tiut where tMl is called the susceptibílity matrix, and is derived from considerations of the constitutíve relations' B3 Suppose one uses a cartesian co-ordinate system with z verËícal, and considers the sítuaËion of a plane wave with \'üave nornal in tlae x - z plane at angle 0 to the vertical, Íncídent upon the ionosphere. Manipulatíon of the formulae presented leads to four coupled l-ínear differentíal equations - equations 18.10 to l-8.13 of Budden (1966) viz - dEx 8.4 T ik dz I % 8.5 ik dz -Hx ð,H MM 1 x 8.6 (-My* * Lk dz ff"ì"* MM SM (c2+M yz zy * 'vv (1++,r'zz ))u, ç#>Hr, _ dll M 8.7 -Lv (1 dz *M**-ffiìt* -1=ik },T M SM + (Mxy - ffi))u, úH, Mxy Mxz Here, M = M M M and C=Cos0¡ yz vv yz MMM zx zy zz S = sin 0, 0 being the angle of íncídence (angle Èo the vertical) . H = zol, zo be.rngl ímpedance of free space. The parameter k is the ,þ= wave number in free space of Lhe radíation. In these equations, a tÍme. dependenc. .jtt, j = Æ, u = 2rf = Ëhe wave variation ín angular frequency, and a Ëerm "'-jkSx expressíng Lhe x directíon, have been removed. To obËain the fu1l soluËíon, the solutíon of 8.4 to n.7 should be multiplied by thcse terms. B4 In the case of an homogeneous aÈmosphere all Èhe coefficients of the of z, and the solut,Íon is of the term Hx'y s etc. are independenË "-ikcz. SubstÍtutÍon of Ëhís inËo 8.4 to 8.7 leads to a characÈeristÍc equation for the system of differential equatíons whích can be written g2¡y¡ SM zz zx 0 ---ñ Y I+M 1+Mzz zz 0 -q -1 0 MM SMyz ¡tl -q =0 8.8 -Myx -c2 - 1+M zz vv zz MM SM xz zx XZ 1+M M 0 - q xx ]+M - zy -]+M zz 7,2 so cal,led Booker quarÈic for a where I I = d.eterminant. Thís ís the homogeneous (though possibly anisotropÍc) ionosphere. Four solutions of q generally result, r.ríËh corresPondÍng solutÍons for E*, Er, H*, Hy' A typical solution might be -jker z n* nlt)"-jnot', H H(1)" 8.9 n*-x = Bjl)u-ikgrz,'x E.,y = u{t)"-jkctt,y = v v (1) 0f g1 being one root of the Booker quartic. E x etc. are constants. course the ionosphere is not homogeneous, but these equations give some insight ínto Èhe complexities of a ful1 theory. Now assume an ionosphere which varies \,¡íÈh height, but is horizontally stratified. LeË the densíty and collísíon frequency for electrons vary slowly with height, (the precise meaníng of t'slo\^rttwÍ11 be seen shortly), so the Booker quartic is stil1 roughly valid in any region. In each ttregiontt then, a solution -jker z 8.10 e is stil1 valid, but Ir varies wlth heíght, z B5 IÈ can be shown, at least for an isotropÍc ionosphere (ví2. no magnetic field), that q tf B.l-L = Ycos ' rf beíng the angle beÈween the wave normal and the vertical, and U the phase refractive índex, at heighË z (see Budden 1966, section 13.3). Then the solution to the dífferentlal equations 8.4 to 8.7 becomes of the form _lÀ B.L2 q--"exp{-it This ls one form of the Inlentzel , Krauers, BríllouÍn and Jeffreys approxímatÍon (theI^IKBJ, or sometimes símply called WKB, approxÍmation), axe small (thaÈ is' and ís valid provided terms like þrcr, ,#r/k2q term Èhe medíum is slowly varing, and q Ís noË srnall). The exponentíal Ís someËímes called the phase memory term' The ful-1 solutíon Ëhen becomes 4 sets of the form 3u lzrt'ð e io* tt' n*(z) ujl) (r)4 " 8.13 {,)a, "- "- ""' í = L, 2, 3, 4, SÍmílarly Er, H*, Hr- Each one of these sets corresponds to the normal concept of. a ray. If the q, are aLL unequal, the trv?s solution is val-id, and the 4 solutions correspond. to 2 upgoíng v/aves, and 2 downgoing lùaves. The I^7aves are denoted 0 (ordínary) and X or E (extraordínary), Ëhere being one upgoing and one downgoing O mode solutíon, and one upgoing and one downgoing X mode solutíon. Each is a plane wave solution, elliptically polarLzed' These solutions are the only waves which wÍl1 maínËaín their f'orm while prop¿gating through the partÍcu1ar quasi-homogeneous part of the ionosphere B6 under consideration. The solutions naturally vary aÈ different points in the non-homogeneous ionosphere. From the values of 9, U may also be found' p being the (complex) refraãtive inclex. llowever, I is the really ímporËant parameter, since it ís the tem whích aPPears in the solutÍon. In the case of an isotropíc ionosphere, U = l,fl (just as Ín the case of light passing through glass), where D = K.eoE = eE' That ís, the susceptibility matríx has all diagonal elements equal, and all other elements = 0, so P=to tMlE becomes a scalar set of equatíons, I = eo(Ke-l)E. The suscept.lbíliËy matrix has cliagonal elemenËs (Ke-l). In thís isotropíc case, Èhe soluËion of p Ëhen becomes particularly simp.le (for example, RatclLffe !972, chapËer B). The solution in Ëhe undamped case (v, = 0) ís v2=!-x., ,t? X- = o = angul-ar frequency of the radiaËion, and Qo = e +(5_ , plasma frequency = (tle2/eOm.)%, tl being the electron density, ethechangeofanelecËron'm"themassofanelectron. In this case, Èhe role of positive ions has been ignored, since theír much larger mass means they respond far more sluggíshly to the electric field of the radíation than the electrons. Ratcliffe (L972) discusses the effects of positÍve ions in his section B.5.3, but Ëhe case wíll not be dÍscussed here. In Ëhe case of radiation propagatíng vertÍcally' PP - *- J 8.10), u'-r=='r0*r'y' E = g (Raticl-if fe L972, equaÈion and this is valid even ín an anisotropÍc ionosphere. Ratcliffe uses this formula Èo discuss the simple cases of isotropic íonosphere (U = r{ c"tt B7 be seen to come ouÈ of Èhis asD=e u-E+? j' =Ke-E.sothate u-- P ' Some anisotropic cases are also 3= e'(f.-f)f and -å= Ke - 1). simple considered, but all r^rith zero damping. In the most general case of non vertical propagaËion, recourse t.o the Booker quarËic Ís generally the best approach. A refracËive index is real1y only relevanË if a l'trKB (ray) soluÈion is valid. Ratcliffe (Ig72, ch. s) assumes a I'IK3 (plane wave) solution, and real1y simply subsËiÈutes this form into Maxwellrs equations to derive U' Símilarly Budden (1965), adopts this assumptíon of a l'lKB solutíon ín his derÍvatíon of the sen-wyIler refracËive indices (section B). Hence he deríves the refractive indices given by his equation (82 ) (there are two solutions - one for Ëhe o mode and one for the x mode), with axial ratios giving of the ellipses gíven by equarions 76 , 77 . There is litËIe point ín the actual forms here. A computer program has been wriËten Èo calculate the sen-ltryller O and x rnode refractive indices, and aPpears in the appendices. The o and x modes correspond to identical ellíptical poLarízations, aË right angles to each other, with opposite senses of rotatlon. The !JI(B solutÍon, however, is not always valid. For most Purposes in this dissertation it is adequate, but a brief consj-deraËíon of some of its limiËs may be useful, again based on Buddenrs (1954) paper. tr{hen q gets small , Ëhe I^IKB solutíon breaks down, as \¡Ie have seen; g = 0 corresponds to Èhe case of crÍtical reflection - an upgoing WI(B ü¡ave',turns aroundtt and heads back to the ground. Such cases of small q can still be Ëreated in a relaÈÍve1y simple manner, untÍ1 q gets very close to zelo, however, fot an isoËropic atmosphere (Budden L954, equaÈion (12)). Because \¡7e are dealing wÍth equations of the form # -, uzqz(")E = o, BB and because q2 1s a llnear functíon of elecËron densiEy, ín cases of small q this differential equation can be reduced to d2y fi= BzE, ß = constanË, and thís can be solved (Budden 1954), vla Airy solutions (or, alternatívely, numerícally on a computer). In fact, for an isotropic (B = 0) íonosphere, the ray-Ëype I.IIKB soluËion is a reasonable approximation even near q = 0. For an anísotropic íonosphere, the I^lI(3 solutíon is stil1 usually valid províded no t\,¡o roots of the Booker quartic become nearly equal (Booker 1936). Airy functions can stíll be used close to these points, but there are ]imíts. These points where two q values are equal are call-ed branch poínËs. Another case where the trrlKB solution breaks dor,¡n is the case ¡¿here the refractive índex p becomes ínfiníte, or where one of Ëhe roots of the Booker quartic tends to infinÍty. It has been suggested that radio \^7aves are reflected at this 1eve1-, (Budden 7954) but trrrKB solutions are not valid, and a fu1l numerical solutÍon of the differential equations ís necessary. Ratcliff.e (L972, section 8.3) discusses ín a 1ittle more deËail the effects of zero and ínfinÍte refracËíve index. Thus far, we have regarded all solutions of the equations 8.4 to 8.7 as propag"atíng índependenËly. ThÍs ís valid in the cases where the trlKB approxÍmation is valid. Since terms tit The q solutÍons are * and - the appropriate 0 and X mode refractive indíces, sÍnce 0 = 0. For the isotropic case, Èhe system 8.4 to 8.7 reduces to t\^/o dif f erentíal equations, and there is only one ref racËive index for all plane \^Iaves. lrle thus see, once again, that Èhis is a much símp1er situatíon. Its símplified treaÈmqnt Ín RatclÍffe L972, }las already been mentíoned). Budden (1954) díscusses coupling ín a 11ttle more detail, and of course Budden (1966), has a more exËensive account. Physícally, Èhe O and X modes can be regarded as "ínËeracting'r to some degree r¡hen such coupling fs ímportant between 0 and X modes. Refl-ections, toor are a form of coupling - ín this case, ínteractlon between the upgoing and dornrngoíng modes occurs. 810 B. (Íi) Ray Theory, Díffer ential Absorpt ion and Groups As has been seen, Èhe I^IKB theory ís appropriate for many cases. In the D-region belor,r about 95 to 100kn and for frequencíes greater than 2Wz, it is generally applicable, sínce the phase refractive índex does not approach 0. The solution for these cases is a ray, as $/e have seen (equatíon B.t3) u'rt B.14 E*(2, t) = rj1) Cr> ,+"-iolïou'.-ju*'ino.j In this thesis, vertical propâgation ¡.+í11 be considered Ín most cases. Thus0=0rsoq=U. The following approximation w1l1 therefore be used. Generally point- like transmiËters are used, so spherícal wave fronts resulË 8.15a E(z,t)=¿;(1) (r)"-iÈI6o'€ i" where ds Ís an elemenË of the ray paËh, and U ís the complex phase refract,Íve index. fne # term has been dropped, as lUl is close to 1. The a.tr arises because we have s is the path distance travelled. Iù spherical radiatíon. ¡r is complex, = UR + jUI i = ,f-1 8.15a can also be wriËten U-z tro dsl -iY"tt"o i(l)t ¡-J Ouros 8.15b E(z) = þ(t) {,)" e" e f;/fiura" trlhen the uI Ëerm is -ve, thee term represenËs absorpËion. In the real ionosphere, p, Ís always negative. In some simple approxÍmations involving low absorptÍon, ulcr(-Nvn), N = electron densiÈy, v* = collision frequency of an electron vrith neuËra1 parÈicles. Hor¿ever, Ín the utore general case, this ís not valid. This subjecË is discussed to some degree 811 in Dyson and Bennett (1979), although that reference does not use the general Sen-I^Iy]ler formulae, but raEher a modlficatíon of Ëhe Appleton- Hortree formula whích approximates the Sen-Inlyller formula (e.g. see Budden 1965). However, the constant of proportíonalfty between UI and Nv is differenE for X and O modes 8.16 ur(ä) Nv (ä) m This relation forms the basis of the so cal1ed Dífferential Absorption Experiment. In the D-region of the Íonosphere, weak partÍal reflectfons occur from sma1l ðhanges Ín refractíve Índex with height. The reflectíons are noË critícal reflections, as the elecÈron densit.ies are too 1ow. Reflection coeffÍcíenËs are in the range 10-6 to 10-3. These reflections are discussed in detail Ín Ëhis thesis. If, then, the sËrengths of O and X modes parÈíal reflectioris are compared, and the expected reflection coefficient ratío of O and X modes is known, and k*, kO, and v are known, then Èhe elecÈron densíty N as a functlon of height in the D- region can be deduced. The method was first introduced by Gardner and Pawsey (1953), and has sj-nce been improved to some degree, (Flood 1968; Von BÍel I97O3 Belrose L97O; Coyne and Belrose 1973; Von BíeL 1977), although the essentíals remain the same as the orígínal procedures. perhaps the major modificaËions are Ëhe use of the Sen-I^Iyl1er refractive indÍces to get k*, k0 and the use of large arrays Ëo cut down scatter from the off-verËical. The more careful treatmenË of O and X modes is the subject of Von Bielts 1977 paper. Von Bíel et al-. (l-970) and Jones and Kopka (1978), discussed the use ofphasemeasurements to Ímprove the experÍment. There has also been some debate as to the rol-e of rapid changes in co1lísion frequency in causing reflections, as dísÈinct from Ëhe usual assumpËion that changes ín electron density are the principal BT2 cause of the refractive index changes (Piggott and Thrane L966; Jones and Kopka L}TB). GeneralLzation of B.16 is also desirable, as the assumption of línearity between Nv ancl U, ís noÈ reall-y valid (e.g. Dyson and Bennett LgTg). The relation (8.16) can be generalized (ur=f,..(Nrv*)), and the DAE theory generalÍzed around Ëhis' The objectíve' \v/ howeveå, remaíns the same - determÍnatíon of the electron density profile as a function of height, gíven the co1lisíon frequency profile. vr. does Belrose (1970) . Lindner (Ig72) gives a revíew of the DAn experiment ' as It is ímportant Ëo note that although the DAE experíment has been fairly successful, there is room for ímprovement. For example, iËs vertical resolutíon is only of the order of kilometres. Further, various assumptions have been made about the nature of the reflecting irregularitíes (some of which have already been mentioned) rvhich have not been fu1ly validated. A fuller understandíng of these scatterers ís necessary before the DAE results can be inl-erpreted unambiguously. The reflectíons are generally assumed Ëo be Fresnel refLections from:steps in electron density, and this has not yqt been confirmed fu11y. The angular sPectrum of scatter ís important, too - is there scatter only from the vertical, or ís there significant off-verLical scatter? If there is signíficant scatter from the off-vertical, some DAE resulÈs which use aerials with wíde beams will be invalid. In mosË íonospheric studies Ín thís dissertation, and indeed ín many D-regÍon studies, radio wave pulses are used. This is to gain an idea of Ëhe heíght of scaËter of the radío \^7aves, since the delay in tíme between transmission and receival of the pulses gives an estimate of thÍs quantity. DAE also uses pulses to get its estímates of heighËs' Hence a brief discussíon of wave groups Ín the ionosphere r¿il1 be gÍven here. 813 trIave groups may, of course, be regarded as a sua of many infinite sinusoidal plane hraves, each of v¡hose Ëlme and space history is given by an equatíon like (8.15b). The \^rave normal directions of these component \,ùaves vary, and their speeds also vary. The resultanE has an envelope movÍng at speed vr. Inside Ëhe envelope ls a continuous wave moving aE speed v, this belng approximaÈ.ely the mean phase speed of the component frequencíes. The field strengËh thus fol1oI47s the form -¡e"rþ.eg Re(u )ds j trt B.t7 B(z,t) = e where ¡¡s=t']{eal part of,', sG-Iä c þ(t) c,>" (u ,R" )ds whcrc g (t describes the envelope, and p Theformulae -/ g =".v 0c 6o for ul, v- will be dlscussed shorrly (8.18, 8.22), 'g' c A descríption of the form B.17 requl-res thaË the phase refracÈive index vary linearly as a functlon of frequency across the range of frequencies composing the group, and Ëhat Èhís range of frequencies isrrsmall". If Ehís is noË valid, a group representation is not possible. In such cases, the propagatíon of a pulse is best considered by treatíng each of Íts l'ourier components indlvidually, and re-suu¡ning these components upon their arrival at the recelver. BuÈ ín the case of a pulse which does propagate r¡ith an unchangíng envelope, the group (envelope) velocity (that is, the velocity along the ray direction) ís âtl I B.18 v = V, i,tr, g -K ãlk['""" " where Yn = 1# *j# * b#, and Lhe wave number vector of frequency f X yz 1s (kx, ky, kz). Here, a 1s the angle between the ray path and the wave normal of Ehe FourÍer component of angular frequencY o, and 8.19 fan0,=-- 1 âu u â0' 0 being the angle of the \4lave nonnal to Ehe verÈical . 814 There can be some confusion regarding group velocities. Some Íonospheric workers call aû) 8,20 tn=ry¡=vgcos ct the trgroup velocíËyrr, and call- v, the ray-group velocity, or hTave packet velocity. The v' is the velocity component of the group along the wave gB.B). normal of the Fourier component of frequency f (see RaËcliffe 1972 Theorists, on the other hand, often ca1-l v, the group velocity. I^Iorkers do, generatly, agïee on the defínition of group refracËive Índex, however; namely, àu B.2I u ô = ='f{'u) =u+ tt; õ #n The quanËíty denot"d uå in 1.2.2-L6 is c 8.22 u =-=UCOS 0 Þù vo 'g b RatclÍffe Ig72r 98.B, is a good reference for derívaÈion of some of the equatÍons 8.18 Èo 8.22. One can also define group and phase paths. These are given by B,z3 Phase path = .fPcoscrds, and 8.24 Group path = /ulds É where the integratíons are performed along the path of the ray (Dyson and BennetË 1979). As mentíoned, group represenËations are not always valid, particularly in cases r¿here p ís not sufficiently linear ín ttl over the range of frequencies consfdered. For example, for frequencíes close to the electron gyro frequency CI^ = I in the Íonosphere, the x mode refracËive index "ta varies rapidly and non linearly with frequency (see Fig. 8.1). In such Fig. 8.1 Real ancl ímaginary components of the Sen-lüyller refractive index p (note U < 1) for the electron densitíes and hei-ghEs shor,m. (t'tuttiply the vertical scale by the Pov¡er of 10 shovrn). The ímaginary part has been converEed to an absorption lerm, so amplitude up to heÍght z Ís ' a exp{- filea"}, where [ = -(o/c)*fm(p), trr Ís the angular frequency and c the speed of light in a vacuum. The refracEive indices were calculated using the Sen-l^lyl1er subroutines presented ln the appendlçes. The followlng assumptíons were made (i) Magnetic field B is given bY heigÞr (km) 3 -5 + Tesla, l¿l 6x10 /Q 637Okm ) (Íi) Co1lísíon frequency of an electron ls 5.2I7 * 1010.exp{-hetght(krn)/7 .2g7]} for height : 71km v 2.g}g * 1o11.exp{-heieht(lsn) /6.187 } for height > 71km (Ííi) The angle of lncídence to the B fíeld Ís assumed to be 22.5" Note in particular how the X mode refracÈlve index changes very rapidly near the gyrofrequency (-1.6Mtlz). 85 km 10cm-3 56 km 100 0 c m-3 18 18 t6 16 \ \ O mode - (x u, 1L - to-6) X mode ( x to-t ) 12 12 Ir10 1 0 t o X mode E B I ( x to-2 ) üc, 6 6 t, L X mode (x to-6) 2 2 0 mode (x I 0-2 ) 0 0 2.3 1.7 1.9 2.1 ?.3 1.7 1,9 2.1 -2 r.98 1 .98 MHz MHz 1L 1L X mode (x to-2) 1? 12 10 I 10 I I X mode ( x lo-g X mode ( x lo-B ) 6 6 ( t, O mode x to-8 ) l, O mode 2 2 (xlo-s) 0 0 2.3 2.5 1 5 1.7 1.9 2.1 1.5 1.7 1.9 2.1 2 .3 2.5 ( Frequency ( MHz) Frequency MHz) 815 to cases equaËÍon B.ZL can produce u, Í 1'O at 2MHz' This would appear imply Èhat the t'groupt' actually moves at fasLer than the speed of líght rrenvelope" ín a vacur:m! But r¿hat ít actually says is thaË the of the f,group,,changes form as it propagates. (Also see Jackson L975r PP. 302- 326). Then the "Fourier decompositíontr techníque discussed previously pulse. is necess ary to fu1ly investigate Ëhe Propågatíon of such a Gíven a ]nor:Izontally straËífÍed refracEíve index profile the reflectÍon coefficient for an incident radío wave arrivíng vertÍcally ís duo (,) X 8.25 Ro(z)dz = tuoG) X X This relation comes sÍmp1y from the reflection coefficient for light íncÍdent on the plane boundary of two slabs of opËícalIy Èransparent !t - ilz maËeríal with refracËive indíc-es il1 and pr. In such a case R = I;E;- f.or perpendícular.incldence (e.g. Jaclcson 1975, p. 28I), The relation for in non-perpendÍcular íncídence ís a little more complex, but can be found most optícs boolcs. R is usually given for Èwo orthogonal plane polar- izationt;, however, so ín the ionosphere the incídent radiation would have Ëo be broken up inËo two such plane \¡Iaves to calculate the reflected radiatíon. Then the returned echo sÈructure, given a transmítted pulse envelope g(t) at the transmitLer (z = 0) is, crudely, a convolution of the pulse and reflectíon coeffícÍenË sLructure' - víz' 8.26a E(t) = nO <Ér)ne(t), where @ represenÈs convolutíon, X (= {lRo(+).e(r-t)dt (champenev 1973, p' 66)) X ís the returned echo as a function of tíme t. The pulse has been 816 assumed to propagaÈe at sPeed c, so height z corÏesponds to time $, t beíng the time for the pulse to go to the heÍght z arLd return íf the pulse travelled at speed c. In most cases the delay time t is converted to an effective (or t'vírtualt') range CT 'r2o=- In fact, most recording systems use such virtual ranges' Iather than delay tíme r, as índicators of the pulse delay. Thus in this thesis, reference may be made lor sâY, ttËhe amplitude aÈ a heighÈ of 68km was ..."; this will really mean the "amplítude aË a vírÈua1 height of 68km"; but, for the D-region, virtual and real heíghts are very nearly equal and tlrere is rarely a need to discriminaËe beÈween the two. Then 8.26a becomes 8.26b er (zr) (z)øe tQ)* (= /l-no Q)eQr-z)ð,2) il) X CT rnrhere E E(r), ,r ,(z r) = 2 t g, (z) g(t), z = and = 2 A crude approxímation lÍke thís has been used by AusËÍn e!- 3-1. (1969) . Ilowever, a more acculate formula for the resultant echo structure wílI l¡e derived later, whÍch takes account of the spherical wavefronts of the probing radiaÈion, and does not assume a ïlave speed equal to Ëhe speed of light in a vacur:m. IË also considers Ëhe effects of absorption. In some cases in this thesÍs the even more accurate Fourier decomposítíon meËhod already discussed will be used. Of course all these formulae assume a one-dimenslonal atmosphere. ThÍs amounts to assuning the íonosphere is horizontally stratified, wÍth no off-vertícal scatter. In fact this is not always the case. In cases where horizontal stratification ís not valid, one must integrate the BT7 reËurned echo sËrengths over all dírectíons, and consider such effects as the polar diagram of the transmitting and receívíng arrays ' But t'Íntroductorytt, such considerations can no longer be regarded as and are consídered in more detail elsewhere in the thesís. APPENDIX C THB DYNAMICAL EOUATIONS GOVERNING THE ATI'ÍOSPHERE C1 Appendix C: The Dynanícal Equations Governing the Atmosphere To put the aËmospheric motíons on a basis from which understanding and prediction can be best achieved, it ís necessary to represent them mathemaËically. TheoreËícal solution of the resultíng dífferential equations can then be undertaken. usually some simplificatíons are necessary before this can be done. DÍfferent sirrplifications allow invesÈigations of different members of a whole farnily of solutions ' given Examples of these will be seen later. Only the basic equatÍons are below. The derivaËions can be found in mosl books on FluÍd Dynamics ' The basíc equations for a gencral fluid can be written, ín cartesÍan coordinates (xt, *2, *g) c.1 #+ j1", = o ContinuitY equatíon t_ ãv. âv, p(# + = oíj,j * Fí Equations of motíon (i = 1, 2, 3) c.Z "j#j) ðh. p# * orjårj Bnergv equation c.3 o Bå * = ,{ (A form of the firsË law of thermodynamics, E = Q f W' E = energy' system) Q = heat input, tr{ = worlc done by tn'e These equatíons are quiËe general. Repeated cornmon subscripts imply a sum (for examPle âv. ðv-. ðv* äv. * v3âxi) "jri = "tãf "'r--\+ Here, p is the fluid densitY t is tíme *irj-=l-r213referËo3carÈesíancoordinaËeaxes are StreSS tensors, and wi].l be described below' o,.IJ F. are the coordínaËes of the force applied externalLy per unit a volume c2 (rhe tine derivatÍve following the motion of a h- = h + vÊ, fluíd elemenÈ) rl, i = I, 2, 3 are fluid velocitíes in Èhe *L, xz, x, directíons, and are a function of (x, t) e ís the inËernal energy per unit volume h. ís a heat flux vector -a - if ds is a surface element, r^ríth uniÈ outward normal vector n, then t,he rate at which heaË is Ëransmitted across the surface ds AS hrnrds in the direction nr: and Ehe repeated i implíes a sum as usua1. The total heat ínPuÈ raËe is ah. _æ. .n = -IlI5;},. av At l"l- id, l_ e.. Ís the deformation Ëensor expressing particle displacement 1J x' (orjrj = 'i represents differentiation wíËh respect to ãï "ij) ,v. å.. is Èhe time derivaËÍve of the deformation tensor, = '4(uírJ * r_J J ,i) volume ( gx¡ Q is the gravitational potential energy per uniE = for a sysÈem røíth g, the acceleration due to gravity' = constant' and x, = verËícal dísplacement from a reference poínt xg = 0) Equations c.l to C.3 are 5 equations ín 12 unknor^ms:- p, e, Vi, 1r Oij. (The h, are functions of temperature T). There are only 12 unknowns here, r" oij is a synmetric matrix. This follows from consíderaËions of angular momenlum. Notice a1so tha! Èhese equatÍons are not strictl-y oijrj ís not a tensor' Seven more eçluatíons are tensor equations, "s necessary to close the system. Six come from so called constítutive relations, which relate o.. ao.rj and å.', and the seventh is the equatíon of state, which connects the mechanical and thermal properties of the media. The equation of staÈe can take several forms, generally based upon c3 the seconcl 1aw of thermodynamícst ds dQ = l;, where s 1s the entropy (ttdegree of disorderrr) of the system' and Q Ëhe heat input Appropriate manipulation, possibly with Ëhe first 1aw of thermodynamícs and a statisEical c.4 mechanical approach, can lead Ëhis to other forms, sueh as p = pk T (for an ideal gas) or p = (an C.*r)oT (ideal gas) where p = pressure, k* ís the gas consÈanÈ when P is ín mass per unit volume,'and Cn and C' are the specifíc heats at constant pressure and volume respectively - ví2, c c' rd Qr p tåÊ'"or,". p, \dT/const v In Èhese formulae p is strictly = ,#rconst E being the internal ", energy, V the volume. In a compressíble fluid aË rest¡ P maY be identífied with the usual mechanical pressure of a fluid. The consËituËive equatíons, just as Ín the case of radio wave propegatíon, depend on the medium. Theír derívation can be quite complÍ.caËed. In general, though, the o.. are function of P, T, the deformatÍon t,ensor e, and Èhe raÈe of deformation å. For solíds, orj is a functÍon of p, T and e. For fluids, oij ís a funcËion of p, T and å - forces o,, produce a movemenË e, rather than a distíncË measurable displace- aJ ment e. Only fluids will be considered here. (For a linear solid, the constiÈutive equations are a generalLza:uion of Hookers law:- uíj = Liit L"uL) Before consídering the constituÈive equatíons, some definit.lons are necessary. Consider firsË1y Fig. C.1. c4 x2 I v Ptote + Fluid Plote 2 x1 Fig. C.1 A fluld exlsts between two plates, 1 and 2. Plate 1 moves at I'Newtonj-an velocity v, plaCe 2 1s fíxed" Then for so cal1ed fluidsrt, Ehe veloclty as a funcÈ1on of x, fs l.ínear, increasing to veloclty v at plate 1. In such a case' + uz,r) úrz = 2vätz = u(vI ,2 where U ls the Dynamic (or shear, or flrst) Coeffícient of Vlscosíty dtt [a].ternatíve1Y as v, = 0, o12 o* being, the 'rdragt' force Per \Ç' unít area on the Platel. However, not a1t flu1ds are Ner,¡tonlan. There are generally 3 classes - I(leal (U = 0), NewLonian (p = const.), and fínal1y non-Nern¡tonian. dcnsíty, is called the lcinematíc viscosity] ' [The parameter v = Ë, O = Gases are Newtonj-an f luids. Most studies of f1uíds revolve arould Stokesian ftulds - that is, Ëhose in which o., are a conrinuous functlon of eUa only, (oij f(åk¿)), otj is índepenclent of posi.tion ín the'fluÍd (hornogeneous), the funcEion f o... ís fi-rdependenÈ of axÍs oricntatlon (isotroplc), and where = .nôrj' å.. = 0 when there Ls no deformation (ví2. the stress is l-rydrostatic). 1_f c5 As a first order approximation for SÈokesian f1uÍds' c.5 o ij (-p+Àåkk)6..+2päij' where À ís called the coeffícient of compressibility viscosÍty, or the vol-ume visco"iiy, or the second viscosiLy (as dístÍnct from the shear víscosiËy p, whích 1s so*àtímes ca1led the first viscosíty). In cases where 3À + 2¡r = 0, or åkk = 0, the mechanical pressure ,defn 1 . : Ê^ Ê1-^ e'.^*^r-..^-*.i^ ,âEr pn(="-^' - ä orr) is^ equal^---^1 Ëo the thermodynamíc pressure P = -(ãVr.. Fluíds which obey this relaÈion are said to obey Stokers condítion. The quantity (À + | u) = K ís sometimes called the bulk vÍscosity. Then equations C.1 Ëo C.5 define the behaviour of a 1ínear Stokesi-an (Newtonian) fluicl . Uot cases rvhere the fluicl j-s non-Newtonian, p, À and K do not real1y have meaníngs. Hínes (L977) Ín a Ëutoríal paper, discusses these concepts of víscosíty. Volume viscosÍty turns out to be a very poorly defined term, and l{ines feels it would have been better if it had never been defíned. He develops a ner^r formalism whÍch bypasses this defínition, and, rather, invokes a complex thermal capacitance which expresses the equatíons far more generally. However, the concepË of volume víscosity has been retained above, as the díscussion here ís purely descriptive, and the lack of generality of volume víscosity will not upset the results. Equations C.2 and C.5 can be combíned to give the Navíer-Stoke equations, Dv L C.6a p = p. * À'r.,t1i- u('r,ii * tj,ri) DË l- ä where [rki] means differentÍatíon ürrto xk' and then agaín wrto *Í For Íncompressible fluíds, * = O, by C.l, leading Èo a p + c.6b BË = I - Yp uv2y c6 OfÈen C.6a (or b) ls used as a replacemenË for C'2' Then the solutíon of these equatíons, subject Èo applied boundary conditions, and with appropriate forces, leads to the spaee-time evolution of the Bulerian descriptíon of the atmosphere. Generally, to símplífy the solution, one assl¡mes a background state, and assumes Ëhat the result of modifications to Èhe system produce only small perturbatíons. ThÍs is a parLicularly useful techníque for investÍgations of the propagation of hraves (that is, regular oscillations of pressure, temperature, velocíty and so forth m.oving through the atmosphere - for example sound consisEs of such waves). Thus pressure is r¡ríËten as p = P0 * pt, density 9s p = P0 * pt, and so on. Insertíon of these forms ínto the equatíons above, and elímination of ¡erms ínvolving multiplicaËíon of more than one "dashedtt Ëerm (e.g. O'#t), leads Ëo Ëhe linear approximatíon of the fluid equatíons. Many wave propagation sËudies can be done by such methods. Hí'nes (1960) is perhaps the cl-assic reference for studies of Ehe propagatíon of these \^/aves. His equatíons 6 to 9 are basically simplificatÍons of C'1, C'3' c.4, c.5 and C.6. vÍscosity is assumed. to be zero. The tenn c in YP" C2 is the speed of sound, assumÍng adiabatíc transport. Thís is in =::vpo effecË anoËher for¡n of the equatíon of state. Only gravity Ís considered as an external force, and Ít is assumeu orj = -pôÍj. For ease of reference, HÍners equaËions are reproduced here, but witn ft- = f * v.v used Ín â p. 1474. (A1so see Pitteway place of Ë, as discussed in Hines 1970, and Hines 1965, equaËions 1 to 4) c.7 pBl=ps-Yp Force equaÈíon c.B + y.ypo = .rrBå + v.loor AdÍabatíc equation fff of state poy.y o Continuity c.9 Bå+y.Ypo * = c7 YPo n2 c c.10 c2 =E_, g=9- Y= _-P. Yg cv P=POf9r eÈc. v is the total vel-ocity, = 9o + u,3; uO ís the background wind, u Ëhe perturbation velocitY. These equations do not look exactly Iíke c.l Èo C.6 but could be derived from Ëhose by suitable manipulation. Havíng díscussed these basic equatíons and their use, some results will briefly be discussed. However, there are numerous books discussÍng properties of gravity waves and other rel-aËed feaËures, and hence detail wiLl- be kept to a'minimum. Hínes (1960) begins by díscussing an isothermal atmosphere rviËh zero background wind, in which case the solutÍons ate oÍ. a complex Fourier form - namelY, UU c. 11 d- = .o5 = = n exp{j(ot - K*x - K"z} PPo f The wave ís assumed to have \^lave nollnal in the xz plane, z being Ëhe vertíca1 coordinate. SubstitutÍon in C.7 to C.10 relaËes the Complex constants P, R, 'aqd.-..4.1,r;. X, Z These are gi-ven by equatío1s 15 to 18 of llines (1960) . A relatíon between .r, K* and R, a1 so resul-ts - Èhe dispersÍon relation c.Lz ur4 -u2c2(**'* Kzz) + Q - L)e2**'* iygu'2Kr=0 c P where y = cv VarÍous solution-types result. GenerallV K* ís taken as rea1, = k" (thaf is, the wave propagates horízontally). K, may be purely imaginary' or of the type K, = k" + i#z = O, * h. CB Tf. Kz Ís purely imagínary (k, = 0) the wave propêgates horizontal'ly' but just fa1ls off (generally) in amplítude wiËh increasing height' It is called an evanescent, or surface, or external wave. Tf- Kz Ís of the propagate vertically as well as second form (k" * f"), the wave can horizonËally. IË increases in amplitude with increasing heíght as Thís is to conserve energy, as the atmospheríc density decreases ""pth\. exponentÍally røíth increasing height. There are t\nro branches, as far as frequency is concerned. Frequencies t! I'acousËic They with t¡ , ,^ = "ZL .r. members of the so called branch". behave much as sound behaves - in fact sound as heard by the ear is a part of this acoustic branch. The \^raves have ellípsoidal wave fronts, and near spherícal fronts at hígh frequencies. The angular frequency oa corresponds roughly Ëo a period ,^= = 4'4 minuËes for 1 = L'4' #a g = g.5ms-2, H = 6km (typical conditions inuthe lor¡er atmosphere)' Frequencies between oJ"r and rrr* = tt!}3, carinot exíst in the ttgravíty atmosphere. Frequencies wíth , . ,g are members of the so called long r^Tavert branch. They are largely affected by gravity, due Ëo theÍr periods, and behave quite differently to the acoustic branch. Particles oscillate nearly transverse to the phase velocity vector, ín contrast to the longiËudinal motíon of the acoustic branch. For the y, g and H parameters given above, u* corresponds to a period t, = 4'9 minutes ft= (Hines, 1960). In fact tr.r, ís also the Brunt-vaisala frequency trt, for an isothermal atmosphere - that is, the angul-ar frequency at whích a parcel of air would oscillate adÍabatically in Ëhe atmosphere. For a non- isothermal atmosphere, oU has addítíonal terms. For a more detailed discussion of these isothermal solutions, see Hines (1960). One ímportant point concerning graviËy \^raves relates to vTave groups. The group velocity and phase velocitíes in a group are generally in very dífferent direcËions. Downward phase propagaËiolì. generally implies upward grouP (and hence energy) proPagation. c9 A more complÍcated situation arises when the temperature and background wínd vary as a funcËíon of height. The coeffícíents in the resultant dífferential equations can at Ëimes become undefined (infiníte). For thís reason, the equaËíons C.7 to C.10 are usually modified, with new varíables. Two connnonly used variables are U z r. t- t (no-'n¡ c. 13 f, -_-_ (no'n¡ where X = Y.t, pO, í" the background pressure' and fl = uJ - k*U* is the Dóppler shifËed frequency of the gravíËy wave when viewed from the background r¿ind. (The background wind has velocity U*' in the x direction). (Recall the gravity !üave is propogating in Èhe xz plane. In a more general case' f) = û) - k.U). Then the differentíal- equations to be solved become (Vincen¡ L969; a different set of equations, though símí1ar technique , can be founcl in Pitter¡ay and Hines 1965):- dfr c.L4 tr + dz tf, 'tzf z df^ l;= ^ztftt ^r-rf, where "rr = -<þ * .#, c2k- 2 urz=t--æ-.,X a2 a2L (s2k*2)/G2n\ - þ a2z = -^rr These refined equations do not suffer from síngularitíes in Ëhe coefficíents The perËurbatlon parameters pt etc. can be then related ãtt,rJ various Qf, to f ,, f , (e.g. Pitteway and llÍnes 1965). c10 It turns out that when f,l = tB, reflection occurs, much as ín the case q = 0 for radÍo !üave propagation (Appendíx B). The gravity !üave turns around and heads back away from the Q = tdiulevel. If CI s 0, a criticál level occurs. The \{ave energy may ín facË be absorbed ínto rhe layer (e.g. Vidal-Madjar 1978). Jones and Houghton (1971) have done a more thorough ínvestigatlon of gravíty wave-critical layer couplíng, looking at non linear effects. In fact. many investígatíons of gravÍty lsaves lately have been looking at non lÍnear effects, and thís is necessary Èo examine some of the more intrícate aspects of grarrity \{ave cffccÈs (e.g. also see Teitelbar¡m and Sidi 1976; Frjtts 1978,1979). One method Èo solve the case of a hetght dependent temPerature is discussed in Hines (1960) - namelyËbetrlKB approximatíon. This can give some useful insight ínto the characterístics of gravity waves ín a real atmosphere. Einauril and Hlnes (1970) also show applícations of such an approxímation, as does Srníth (L977). More exact numerical techniques can be found in Hines and Reddy (L967); Klostermeyer (1972); and Bowman and Thomas (1976). Lthen reducing the equations of the atmosphere to workable form, two linear differential equaËions result, as seen above. These can be reduced to one second order differential equatíon in one variable. For example, íf the vari-able 0 = y.vexp{-Iþ} is used, an equatíon of the form -ð'o- .å#. results. Here, (a2-u 6o2 ^2) q2 = k*2 (;Ë- - tl + -vr- and oB2 = ((y-1) . #r# is the Brunt-vaisala frequency. c11 The wI(B solurÍon is then of the form (e.g. Hines 1960; Einaudíand Hines 1970) 0= lqnl-%""p 1-itqdzl much as in Appendix (This ís only va1íd if #/k^dzq terms eËc. are smal1 - B). Then the case q = 0 can occur. Analogously to radÍo r,^7ave PropagatÍon' this Ís assumed to resulÈ in reflection of the wave. InËerestingly, the heíght of critical reflecti-on by thís LIKB approximation varies depending on the varÍable used (0, or u, or whaÈever). However, the "titi."t reflection heights all lie r¿ithin the region where the I^IKB approxímation is no longer va1 id, so the use of the I^IIG approximaÈion is misleading (Pitteway and Hines Lg65, Eínaudi and Hínes 1970) near the reflectíon level. If q2 . O, the wave becomes evanescerit, and no longer propagates verticall-y. Gravity \^7aves can be generated by a variety of processes; these are díscussed in ChaPter 1. In the foregoing díscussíon, the Coríolis force due to the Earthrs roÈation, was ignored. If one Ëakes a coordinate system on the Earthrs surface, bodies movíng in Èhis frame do not move as they would ín an inertial frame since the frame is really rotaËing. The Coriolis force is a "pseudo-force" to some degree, Ín that a frame on Ehe Earthrs surface ls assumed to be inertíal, but al-1 bodies moving in this frame are taken to have this Coriolis force acting on them. In t.his way, Ëhe motíons can be described correcËIy from the point of vÍew of the revolving frame' (Houghton Lg77, sectíon 7.2). For periods greater than about t hour, Ëhis force becomes important (Klostermeyet L972). KlosÈermeyer also gives a suntrnary of the main forces actíng in the ionosphere, although below 100km íon drag forces and so forth are not ímportant. These forces are ínserted CI2 in equation C.l to C.6 wherev.t "I" occurs. The Coríol-is force on a body movíng over the Earthr s surface at speed v is given by -{F =2vxfl fl beÍng the angular velocity of the Earth (alígned along the spin axis) ' Consideration of this force in the atmospheric equations leads to sËudies of very long wavelength (L, tár 1/3 etc. of the Earthrs circumference) \^raves propAgating ín the Earthts aLmosphere. Such waves driven by the Coriolis force are called Rossby Waves. In general, such large scale r¡7aves are called PlaneÈary vlaves (Itoughton 1977, Ch. B). Proper invest- igaËions of these \^raves, both Ëheoretical-ly and experimentally, ar.e sLí1l in theÍr ear1Y "t.g.". !ühile planetary \¡Iaves and gravity \,faves are discussed separaËely above, note that they are all sol-utions of Ëhe same set of equations' It just so happens thaÊ the Coriolis force is unimportanÈ for gravíty r,raves of period less than about t hour, buË sÉrictly the force should be considered. AnoÈher member of this family of waves is the set of tides. In the atmosphere, strong regular heatíng of water vapour in the trop- osphere and straÈosphere, and Ozone ( about 30-50km), provides another Ëype of force in the atmospheric equations. (So1ar and Lunar gravíËaÈional effecËs have been considered for forcing atnospheric Ëídes, but are generally of second order (e.g. Lindzen and Hong 7974). This contrasts to the oceans where tides are almost entirely gravítatÍonal). Because the heating is not purely sinusoÍdal, but rather|ton" for 12 hours and "offt' f.or L2 hours, etc., oscillations with periods of 24, 12, B, 24/n hours (n an ínteger) result. These oscillations are knor¿n as atmospheric tides. perhaps one of the betËer earlier references on t.ides is Chapman and Líndzen 0970¡,, and another is Lindzen (l-974) ' Groves 8976)' gives a good review of experimental and Ëheoretícal progress concerning tídes' Various c13 modes of the tídes exist for each period, each having different latitudÍnal and longiÈudÍnal distributÍons, and differing vertj-cal wavelengths. The 12 hour tide seems to be basically forced by Ozone heating, and g¡e 24 hour tide below 100km forced mainly by water vaPour heating in the lower atmosphere (though Ozone heating is stiLl sf-gnificant) (Lindzen L974; Groves 7976). Some of the 24 hour tídes are in fact evanescent (i.e. do not propadate vertícally). These are pa::ticularly important poLeward of 45o, where propagating modes have dífficulty exísting (Groves L976, p, 443). The most ímportanË modes in the atnosphere are denoted Sl, s]r, Slr, tl, tT and Sl. (See Lindzen 1974 fot a description). The superscript denoËes n, where Ëhe wave perÍod is 24/n hours, and the subscrÍpts denote the varíous modes. Larger n values generally correspond Ëo a more complex latitudínal structure. Negative subscripts denote evanescent modes. Other modes apart from Èhese generally do not occur sígníficantly in the mesosphere, because they are hard to force (a1Èhough Groves 1976, consíders some quite hígh order 12 hour tides). Hígh order modes ofËen have very short vertical wavelengths, which is part of the reason they are hard to force (Lindzen I974). Tides occur in both the NS and EI^I wínd components. In many cases these componenËs have-roughly equal am.'litudog ahovo 30o latitude (see Groves L976, p. 450). (Forbes and Líndzen 1976a, shows graphs of the amplitude of EI^I and NS componenÈs of the varíous modes). The NLrI and El{ components a1 so have phase dífferences of about 90o ofÈen, too, and this results ín a rotating wínd vector (of constant amplÍtude if the NS and EI,tr componenÈs have equal amplitude). For upward propagatÍng energy (downward phase propagation), clockwise ïotation of the vector wíth increasing time generally results in the Northern hemísphere, and anÈiclockwise rotation cL4 in the Southern hemispherervhonviewed from above (e.g. see Craig (1965), p. 351; Elford and craíg (1980)). It should also be borne ín mind that lf no energy is lost whí1e propagating, the tidal amplitude will grow height as H beíng the scale height ' with ""pt ç+zn)), Tídes also fo1low the Sun westward (in theory), so the phases (two phases are defined; the hour of maxímum northward wínd, and hour of any maximum eastward wínd) ,Are ín principle the same at all locatíorts at one latiLude if expressed in local tÍme. Atmospheríc tides are presently the subject of some quite vigorous research. Experime.ntal observaÈions are sËartíng to give a pícture of the Barthrs tides, and theoretical investígaËions are havÍng some success in símulatíng these result.s. Much remains to be done, hou'ever' Some papers on theoretical investígatíons incLude Lindzen and Hong (I974) (in which the Ímportance of the background r¿ind is considered); Hong and Líndzen $976); Forbes and Líndzen Q976a, b); Garrett aud Forbes 0gzg) and Forbes and GarretÈ G97B). One partícu1-ar1y interestíng feature whÍch may have to be- considered is the asymmeËry of the Ozone distribution in the norËhern and southern hemispheres (tíetelbaum and Cot 1979). This may result in a larger contríbution to tide by asynrnetric modes than had been previously realísed (most emphasis to date has been on examination of rnodes which are symlnetric about the equator). Some recent resulÈs, particularly some obtaÍned by satellite observations, have illustrated that the northern and souEhern hemlspheres may not be as similar as would perhaps be íntuitivelY exPected. APPENDIX D Angular and Temporal Characteristícs of Partial ReflecÈionfromtheD-Regíonofthelonosphere by I^1. K. Hocking A paper presenËed in Journal of Geophysical Research' B4 845, G979), D1 MARCH I, 1979 vol. 84, N0. A3 JOURNAL OF GEOPI TSICAL RESEÂ'RCI{ ANGULARANDTEMP0RALCIIÀRÂCTERISTICSoFPARTIALREIÍLECTIoNS FROI'f THE D - REGION O}' THE IONOSPIIERE I,t. K, Hocklng DepartmentofPhysics,UnlversityofÀdelalde,Âdelalde'Australta5000 t.97 4; Ras AbstracE. Radio pulses have been used as a Jicanarca, tlgp¡þgg-g¡1!-çg!1-1-9n, çogi and Bor¡hill, fSZOUI and some cornparlsons wíEh are also díscussed. The". "b".."atlons Technlques with an approxlmately Gausslan envelope of tlme and helght. DisLlnct layers of stl.ong Pulses and half power wlclth - 25 Us \^tere tralìsml-tted at B repetltlon rate of 50 Hz from a square array of 4 half-wave folded dlpoles sltuated close Eo the receLvfng arr-ay. Efther 0(ordlnary) or X (extraordlnaiy) clrcularly po1-arlzed modes could be transmltEcd. Tlre receLving at:t:ay comprlsed a off-vertleal square grld of 89 palrs of crossed orthogonal scatter from a much larger range of perimeter, the reËurned frorn them var:íes hatf-t"v. dlpoles r+iEh a clrcular angl.es, and the Pof\rer by 0.6 wavelength and Ehe time than for the lower layers, whJ-ch di.poles belng separated leJs ln dtameter belng 900 m. The the theoretlcally predicted polar dlagram' Twelve of the north-souÈh allgned dlpoles of quall ta tively. This paper dlscusses direct observatlons rnade by swlnglng the bearn of the 900 m diameter Bucklanã Pãrk Aerial Array, near Adelaide, South Au"tr"lta (35's,138'E) tBrj€g" et al',1969] to an off-zenlth angle, and ãõ*[ã-res the results r¡lth echoes recelved r¡1th a vertical beam' Comparlsons are also made v¡íth observations taken wltir a snalle¡ array having a wlder beam' For ¿11 reflectfon coef f icÍenEs obEalned. Recently, D - reglon studies have been under- taken usfng hlgh power VHF scaEter technlques at Copyright 1979 by the Anerican Geophysical Union' 845 Paper nunber 840903 ' D2 tlocking: Brief Report 846 ft t)f uu 98 q6 dB 9¿ 7A 97 90 t6 88 !2 ^86 Ee¿ I - 8? tú80 L z <78 í 0 76 cq <0 7L \ 72 ?0 ,.! trU 'iit 66 I Er. 62 t/.00 11LA I 350 -110 CE NI RAL STANDARD T IME f power versus range and tirne taken ñ"""t are smoothed bY interPola- t" ittnttlttiot' The presencg of' a st ãbvíous feature' Strong bursts an be seen' 1-min means of echo po\^tert taken was computed from the 0.4s between successlve records' Fading steps of range' and then-smoothed by pol^ters over successive ;;-ã-k* iå.ãt"¿ by computing nean computer interpolatton lAkíma' .+2'l ]' of I mín' -- piFwrrr be noted aÈ -'--it"intervals beam i.t.t."""s of slgnal beam recelver and one wide 66 to 7.O k.1n' 71 km and narrovl of ."r,g." of approxlmately were calibrated before each set mode ls stronglv absorbed above t."åi.r"i here are 1n dB ;ö'^il: The x measuremenEs. nlf gi"phs presented the 90 km reflectlon coef flclents be compared áó il: t.,ã from ;:il call-brated iotlt, ãnd can ä..'-äåa"trry"o much larger than Ehey appear powers on the wlde beam have been 'valley-t can be dlrectly. The record these results. e ver| definite ín such t tãy-ttt"t both beams.would Tie srrong slgnal above 96 km adjusted a mirror noted at 80-82 krn' ;hå-;.*. value for reflection from ï;-ã;.-i. total reflection from the E - reglon However, íE should be noted that Ehe receivers' The tlme lnterval reflector. to v¡hich saturates the ;;;;i;; ãensitles in Figure 2 correspond ;;;;; r" Fígure 1 coincides.î:tl.t:; levels to thoãe 1n Figures 1 and 3' ffi'::o.;1" different \^Iere Sffectlve voltage reflection coefficients is laYer was short-l1ved' artt'otãh no allowance has been made relates malnlY to the two also obtain"d, coefflcients îJi-"i"otpaio,'' Thus these reflection 90 km. are underãstimates of the true reflection 85 to 95 krn is a connon oF the scatteríng irregularítles' thls laYer often Perslsts Iå"rii"i."as for on day pårii.,lr..ry above B0 km' No compensation day and night O!-tm layer hàs been lncluded throughout ' llre coefflclenE of varying angular ctrarácteristlcs 151 had a mínlmum O ã¿" reflection either. :á-""iõ-; rlslng to' 1.3 x 10-3 on some oä.""iá.,". The 74-k* 1"ytt x mode.reflection ."ti"d^ä'åiã-¿--* io-' to- Z 2-1 10.' coefflcient= an X mode reflection Res u1 ts and the 66-km layer reached i""rii.t... tt 5 x 1o-s ' 2 shows conEour diagrams of power on several days durlng Fig. Ehe perlod Observatlons utere made returned from 82 ;; fOG [m durlng f rom Mav 3I' 1977- (uhls being and wide b-eams' and ßlá--tsll' Data the 0915 to 1034 on both narrow day of the year, in early tll:"t in pair of-diac:9l? r"' th' the 151st illustrate ií-. ì"t" á similar will be used Eo .o so km rrom 1445 to 1604' observatlons r^tere conflned ;;;;";,-"il ãi "ätriátl-n.*i"pt'trt¡,Ehe maln features' t'tost iã"tr'rt. of both graphs is the^occurrence hours and on day 77lI5l covcred the powãr' Ehose 4t.90 knì to daylight (universal cime of strong bursts of pã.ìãä-oãôo-1700 hours local tíme 3 dB over che normal ler¡el- and rising by - I at each + 9% hours). km by fo ds' The burscs of power as a Ehose at 74 to 15 mln' figut. I ís a contour dlagram have a quasí-periodicity tt. using the nal:row beam helght lastl: less Èhan functlon of range and time For the 74-krn 1ayer, bursts often vercicãllY and usíng x mode- speeds were a maximum at 74 km ;;i;;i;; The values are Z"-t".- Wind iáfttfrãaf"n for transtnl-sslon' x r¡il-:E NÀRR )'V! ial NÂRRQW EEAM O ¡iODE BEÀM O MODE l¡l -----NARROW BIAM A r 00: RROW BEAM AÌ I1 6. -_ -,Nii¡ì'f,w a¿aM ¡ OFF -tc -9: z t'" _r¡ o $¿ o Ê. ô2 (b) WIOE BEAM r0c d iâ I P. o _¡ -" o o 4 z aì <¿È E ,¡2 CENTRAL SIANDARO IIME narrow range and tirne' taken with (a) a snoothed l-rnin power neans velsus Fí9. 2 contout diagran of have been-corrected to allot^I for differing b";;;-á;; iìltsf. m"'plot, krn with the and (b) a wide powers centred on 88-90 gains of the two antennae. .t"'la'.ãrrg"st 11 ".tayer powers on the wide at gg-92 kr;i;h ihe wide bean' The larger narrow bean, and strongest ù""t it pointed at 11'ð' õff-zenith' ard the increase i""!*g"-*t"l.thg layer' 0 beam, rhe irrdi""t""";;;; ,ignifi.-t ofi,-ver-tica1 scatter for this strong powers ar f i.O; ãtt absorption than x mode' lnlaximu¡n polari zatíoî*år-nt"ãl-rince trris-riir"tt less node or - 1'3 x 10-3 ' poh¡ers correspond tl""ã-ri"åtiãn co"rrici""it æ ! D4 Brief RePort 848 Hocking: o ó R 9 ! o q oV oX0i , EOO o É o.i>+r > -v u'9 ñ àh 3 6 Êii o+JqÀ :ié5!(HLr¡ Ët õ'¿nog 'Ô OU F dqrd.c o I ir : OPH9 q 'dóiiÉnE->L¡Jôu t r= É H 8o fl U dd ptJ OO 6 3 Ë o h" cü o Q 'drJo\o,CoÊ to È x .Prr E - É iD{JOHE 2 vodO..t"BIl,iã '.rt O {J b0Ø X U ¡j rr É d bô.¡ _g Ét tr.¡ ) > ot àô t¡ Ø+J .¡P,- {J 'd õ 0) b0 (É r f¡¡ Fr ô UI Þ a = r¡ ô lr É o cl 2 o (d ln q E 4 f{ É, o z 3 l¡l .J q ë , f¡¡o o e,= d,{ U o t : r! ì4 (I¡ :E l¡J b¡ c) .rl o. Þ t l¡J ú o z = 1 ô P.-'ÊF33ül; BPPÑÊRE3¿ü (urì (urt I f ÐNvu I :l9NVU D5 llocking: Brief Report 849 (prlnclpally eastward), attalnlng speeds of This ls again conslsEenÈ h'lËh Llndner [1975b]. 70 rn s-t for much of the day. It ls fnterestlng Prellmlnary analysis of the fadlng statlsLlcs that thls peak coíncldes wlLh the layer of sÈrong has shown that layers above 80 krn exhlbit more reflectlon at 74 km. rapld fading Eharr those below thls helght. F1g. 3 shows Ehat the 74-km layer backscatters Also amplltude probab1llËy dlstrlbutlons show of sErong scaÈter very ilttle Power at 11.6o relaÈlve to that that well-deflned layers at 0.0". There ts a sllght increase in range of usually produce fading w1Èh a signlflcant thls layer at 11.6' at this time of measurement' specular comporfenL, especlally at Èhe lower This would be expecEed lf chere q'ere some helght. lCha.n_dra and Vlncent,1_977f , from 11.6' slnce reflecËlons from It is useful t.o examlne Èhe helghts of the reflectlon , of that angle would have their range lncreased by peak powers of the I mln means as a functlon a facEor sec(11.6"). However, the wlde beam shows tinre. Although such plots exhlblt some degree change 1n range on thís occaslon, of random fluctuatlonr at tlmes it is possible a sírnilar perlods suggesEing thaE thls increase is due to an to see quasl-pertoclic osclllatlons with áctual Lncrease of rhe layer helght witir Ëime, t0 to 120 mlns and arnplítude - 15 krn. It 1s ratlÌer than belng a result of t1-lclng the beam' posslbJ,e Èhat these are evldence of lnEernal Observatlons ai ocher times show l1cEle or no gravfty waves. Of course Èhe acÈual helght of tlre systenì ls much greater than lncrease in range at 11'6". Hence 1t seems resolution tpulser to assume that the najority of tl-re ! k. and any 1 rnin average of scattererl reasonable range of power: recorded at 11.6' 1s reálly leaklng ln pov¡er may have contrlbutl-ons from a irom the near vert1ca1., through the edge of the helghts. The Presence of helght osclllatlons mafn 1obe, and through the first sfde-lobe, of the may lndlcate elther that one thln oscfllatlng' polar dlagram. layer produces most of the reflectlon or LhaL 1s assumed to obe¡-a the ¡:elatlve contrlbutlons ¡¡l.thln t.he reglon If the power reflecled varlaEfon. functlon proportlonal to exp{-(stnO/sfnOo)' }, follow a regular patterri of helght 0 belng the angle frorn the vertical, then lt can be concludecl that 0¡1 2o -3' for this Dls cus s lon layer Ic.f, Llndner, 1975b]. Only an upper be-plaied on 0s, since mu-ch of the llmit can have dlscuseed the povrer apparently received frorn 11.6' ls Brless and Vlncent [f973] clls t rib utlon probably from the verEical. These sna11 values relation betweerr e.lecLron denslty of g¡ lndicaEe thåt the 74-km layer ls and angular reflecElon characterlsLics and have alnosË a specular reflector. concluded that cl-ouds of electrons wlth a In contiast to the 74-km layer, the higher horlzontal extent elgnlflcantly gleater Èhan ^ values of 90-km layer clearly shows strong scatter ac 11'6- thefr depth procluce scatter v¡lth small off-zenlth (Flgure 2). Furtherr the received 0or whl1st nore lsotroplc lrregul-aritles angles and por./er on the wlde beam ls - 6 dB larger than on produce scatter from a wlde range of Thls suggests the ma{n beant, when allowance ls made for the hence have larger values öf 00. dlfferent gains, and this also suggests the that the 90-km layer o¡ 77|LSI may have exlstence of strong scacter from off-vert1ca1 conslsted of approxlmately {sotroplc lrregular- ansles. An lncrcase 1n the range of the layer J.tJ.es, v¡hllst l.rregularitles at 74 km were much of - 2 km at 11,6" compared Èo 0.0o can also anisotroplc, wlth horlzontal dlmenslons be seen, further suggestlng slgníflcant off- larger than thefr vertLcal dimenslon. vertical scatter. Slrnilarly' the nean range of Another polnt of lnterest ls the 1'arge v¿1th the wlde beam ls greater varlatlon ln power withtn a short space of tllne, the layer observed s1mllar than the range measured uslng Ehe narrow beam' partlcularly at 74 l 850 Hocking: Brief Report reported lìere aE 74 km show similar duratiotls Acknowledp,enrcjrìts. The auLhor would like Èo to those observed aE 50 IlHz. Wlud ureasurements rhank B. II. Briggs and R. A. Vincent for helpful show speeds - 70 m s-r at 74 km, so possibly discussion artd comments during the course of the nìost reasonable suggesEion Èo explain the these observations and during the vJrlting duraLlon of t-he echoes would be thac it is the of the paper. The work was supported by the tlme for a region of high scatEer Eo ûìove AusLralian Research GranEs Comrnittee. througlì Ehe array beam, raElter Èhan the life The Eclitor thanl Hocking: Brief Report 851 D - regfon of the lonosphere by the use of observatlons of winds and curbulence Ln Èhe partfal reflecÈed radio l¡aves, J. r\tmos. straEosphere and mesosphere, J. Âtnros. Sc1. Terr-Phys_., 35, 909, 1973. 31, 493, 1974. Vlncent, R. À., and J. S, Belrose, The angular dlstrlbutlon of radlo \raves parËIally reflecÈed from the lower lonosphere, J. Recelved l4ay 23, L97B Atrnos . Terr. Phys. , 40, 35, L978. Revtsed August L6, L978 Woodman, R, F., and A. Guillen, Radar Accepted AugusÈ L7, L978, APPENDIX E Computer Program SPecPol- See Chapter VII for explanatíon of use ThepoJ-ardiagramisdescribedbyfunctlonBPRES.Tlrecoordinatesof (noÈ Èhe aerials are entered as data cards shown) ' E1 SPECPOL 2 (I r OUTPUT ) PKoGRÉìH SPECPOL. NPUT S PEC POL 3 EX]'F.fiNAL PH] NT SPEC POL A ct)ñrloN./T/l HET,q S PEC POL 5 cot-il1ÜN/ P/ PH I r ATTEN r RADX (.?-, r PHI X (2) SPEC POL 6 c0Hr'10N/u/ uA\rE[-1 HT F: Etì r llT T X, D I PLF-N 7 ' Rfr D(99) S PEC POL ct)l'1MtlN.' f'itl t'll[]. 9'l ).N AËR: SPEC POL B DEP I H TEPSTROTPI COHHON/ H/ H'T , UID S PEC POL I COTlION/DEL/T)E.L SPEC POL 10 CiIIfIITJN,/ T / I PLJL 9'I-P SPEC PÖL 11 U ]- ND I COHHOI,I,/J / PIi ] VvPHÉrS:Ë'PHREÍ- S PEC POL !l;: (. 100) r l: UNtl'l I 0(l' I DllfËN!.lIDN '. qE s PEC POL_ t3 C T F'IND POUE rt'ði;È.órrriiN As A FUNcTIoN VIErllô!-rlANGE FIOG}1AiI TII A SPEcIFTED PcTLAR DrAc s PEC POL t4 c FOR AN IONO$PHERIC êcÃïiÉrrïi¡d r-ä:ltn'eñb s PEC POL 15 XX=1 . O ( *:1 s PEC Ptrl,- t6 P I=A:jf IN 1 - O.) .0 s PEC P0t- t7 ËP-q- 1 . oE-0;1 ìÉ tt * l+ * lt * ì' * s PEC POL 18 c * * i T t r * r * ¡l * ä s PEC POL 19 c I)ATA x I * l( I * X Ì t * t( l( * lÉ * * * *'( s PEC PO L 20 []* l+..{ lt * * * l( * s PEC PO L 2l C ê PEC 1 -1 J PEC P Ot_ .) t) s PEC S PEC POL c S PEC POL 7â t; S PEC POL ?7 C S PEC 3 S PEC A S PEC POL 30 C S PEC POL 31 S PEC POL z1 c S PEC 5 SPLUPÜL 3¿ t: S PEC 6 S PEC POL 3ó C S PEC 7 S PËC POL 38 t: S PEC I S PEC POL 40 c S PEC POL 41 t; s PEC POt- 42 t) S PEC POL 43 C S PEC POL 44 t: S PEC POL 45 CC S PEC I S PEC POL A7 t; S PEC POL 48 S PEC POL A9 S PEC POL 50 C S PEC POL 51 S PEC POL c S PEC POL 53 S PEC POL 5á S PEC POL S PEC POL tr6 46 S PEC POL ã7 S PEC POL 5B 60 Er. SPECPOL 59 PRl.llT S1' Ê-PS rE9.3.¡ SPEC)POL óo 8l F(]tiflAl r.1 x r ilTH I NT E,TJRATION ACCURÂCY IS = SPÊÚPOL ô! | 1. ) PRII.IT EI2' RADX :RÂDXII,I S PEC POL 61 BT F'IJRMAI'( 1 X r 1 1 l-ll X ÑADII == r?Fl1-4) a?.¡ s PEC POI- ó3 ÊìRlN't 83 r Pt]lx(. 1.] r PHJX S PEC POL 64 B3 FOF(HAT (: 1X r 1.'/l{TX D I POLE. ;ì ¡'l {:) t-[S = r!Ë11.1t) SPECPOL 65 Éi4 NAtrti, UÊ rJbt-l,Drl' ttr N PIiINT (l ', RX DIPOLE LENGTH IN HALF s PEC POL 66 El¿t FI}Rt,lAI C 1X' r ó8HN0 I: AERIÊI S laut.t-¡-HOlHrAt'lD ) s PËC POL ö/ $ uAt"/El-ÉltG 1 l-iS - I5r2F11-/¡ PEC POL óB PRrl,il tl5 r lll l X r HT ñ ECIATTËN (i 'r."ii'Åi,iìvE rHAtìE,HT uF RX AFovE TTIAGEtAND GRr¡ REF S PEC P OL -lo69 a5 FOFII'{Al .1X r ó9lltll Õ PEC P OL $l.t.t'1r0N c TJEI F r3F11.¿i.r s PEC F OL 7T PfilINl tìl¡r l{t¡L) LP LAYERTAND DEPTH OF LAYÊR = r2F1L'4tL\tl 5 PEC POL 7? a6 F ORHAl'(.1 X r 41H t-1 OF BASE OF PËC POL 73 $HÑMS J s PEC POL 7â f'tiIi'lT 87 r IPUL!ìTF rUIDH PULSE urDrtl = rAl0rF I 1'4) s PEC POL 75 a7 iijriiìÄrì:iiizitiÞLlLs¡--iVt'8,âr''lD 5 PEC P 0t_ 76 PR]-NT 88 77 t,1XI117IIt.OR EXP PULTìËIIJITITH,-IS UIDTH ÉETIJEEN-1/F: PTS OF POUË S PEC POL B€ FOF:i'-I11] ts rz8*UlliTH oF BAsE ) s PEC POt_ 7A $ri pRoFrLL C)F puLSr-::'rirË-i;'0é"Êtlr-sE'uliirH PEC POL 1Q PNINT B9 s PEC PO L 80 Élv f'I]RI,1AT(1XrIi111_.COSACIUAI.LYHEANSCOS*X4POUERPROFILEOFPULSE) s PEC PO L B1 l'ttINl 9l rËiE5¡lfix ô uRTO l DIPOLE SiUlll'lED FOI{ VERTICAL PROPO PEC PO L B2 9L trr.)Fì,lAT(: 1Xró2HtltJRFË-þlT TOTAL S PEC PO L 83 $GfiIION = tþ--11.4) S PÉ.C 10 DO 11 I=1r2O S PEC 11 (: tio=83;,- O+.F1...04T I - 1 ) S PEC POL Bó CV ri t( V"r iiË ili¡HüÂL-isli¡ rtADlAL vÉ:l'-0crrIÉs S PEC POL B7 T; N 0 0f: Lrl0P5 S PEC 12 Nl--11 S PEC POL B9 D(.1 I'l l. J= 1 r NL (:J-1 (Nt--1)+.02 S FEC 13 ü;- ö.:irioriT ., /FLoAr S PEC lô v:::\,*0.3 S PEC POL 9? (. Vti: J );V S PEC POL 93 C PllI l-fi"lIì:,^ S PEC POL 9^ Ctil.l'lciIS.rHË[ìlN['t}l-.].HEHAXTlrETAçoNSIDERED-_--THISMAXTHETA- Ëox-n S PEC POL 71.!' C D[:SIRED f'T'N cïI ON 1.t7 F'TJNCT(.J] S PEC POL =t-l S PEC POL l. 1B PRIi.IT 7B rVÊ iJ.'?FtJNCl(J) SPECPOL 119 ( 1 78 FuÌlhrrT Xrl [9).31 S PEC POL 1.2 0 21 C TIN'T I i.IU E POL 1.? r ctiL. t_. nIlr.Pt..l { u[L r l- tlNri] r -l'{l r I r 9H*RÉtD vELn r 7l-lx f,ol,JËR* ) SPEC 1?a I PHI.Ii-I- T PHASI: S PEC POr.- PRIN] 4.JóIRO' f,I1]IJIND SPEC POL I 2.1 4liô F 0tìllAl t.1X r I il lVl F:l t.lÉrl. t1l' t:? - 3 ¡ 1óHU I f.lD l)rtiEcTIoN = DIRECTI ;F'9.-!tBHPHôSìL" - ¡F9 l!,1.31-ltÂLL RêDIANs.''F9"3r30H ) SPEC POL tl4 $üN ûÊ TIL-ï '- S PEC POL 11q Al.{(ìi,11¡X=ASIiJ r. PHASE/ (.?. Or( P1.¡ ) * (.-1 -0) i ltlo.o/Pl pt{A$t1 (.'! x S PEC POL 126 VHAXi,l.= / " O P I ) r UÈIAXM s Pt¡c P0t- t'fil l.ll' 457 r A¡ltil'lÉrX I g PËC POL 138 r. OF 11É,X KESP0l'lsË. AL-ONG TILT DII{F:CTION - 4A7 F-OFii4AI IXr1rólJfiNGL.E S POL r29 t,F9,4t9H Dl:.iiFlt:Ë.S r111H ríll.lt) NOF{I'IALI S ED VELOCITY OF ÈlAX P0UER = PEC r'+tJl.i TOUÉrF:D AÉiFiAY ) ) 9 PEC POL 130 t,F?"4¡'.r19 S PEC POL 131 11 CONTINI.JE S PEI POL 132 Ei'¡D E3 (.PHl.) s PEEFO[- 133 F UNCT 1 ON PHINI S PEÈfÐL 13ô C OHHON-/ J ,/ þH IÙ l ND r v r PHÉrsË r T'llREË s PEC POL 135 f. X T E F(¡IAL CONV s PEC POL 13ó S PEC POL t3'l r TEN r RAI¡Xr'2) r PHIX t'3) C(ll'íMLrN;'Ul UAVt t' l r HTRE C r ltl I X DI PLËNIA'I S PEC POL 138 F-rr , (.99 ) CUhh(jN./ pHl Pt-t0 r'II.r r llA RAD S PEC POL 139 i;unmtl¡tzxz H'l, Dtrl-' r ur DH 'EPSrl'iOrPI S PEC POL 140 COMHON/DËLlDtJ.I- S PEC POL L47 P1l.?=l'l /?. SPECPOL 74') t: IN'IË.IJI{AND OF PHl If.IIF6RA TION SPEC POL L43 6 PEC FOL LAA AI]PIiOPfIIA TE TO UTPHI c FII{T) THE'IA j ( PHIUIND-PHI ) ) ) APEC POL r45 s1'tlA= t.2.O*vxV)/(. loiõosi:2. ox S PEC POL L46 'tl-lL.Tê (: ( -Ér5 I N S(ìR'l 5 THA-) ) S PEC POL L47 c Ë t. ND tìtrl.lvtlL,uT I ON S PEC POL L48 Ë[)UNDS uF col.lvÙ{-Lll Io S PEC POL r49 c ( -l t--us (: T HE TA) lìh I t.l=Hl /(.;OSì I l-ll A I * f.iMA!.=rHT+DE P.t/ S PEC POL 1.5 0 r [- P5-i: N r Pl'11.¡ cALL. Sl.11P:ìfl (cr.ìilu F( r F:l'f T N r FillAx rFI S PEC POL 151 ' l'- Aß RO c THIJSì f: I HOLDSì CL]I..IVOL- U T I ON UAI--uË ti(lF: S PEC POL 15! c Ft-!t..I. ]'NTE6RAND S PEC POL 153 Y= PÜLAR qTHF-l A r PllI l SPEC POL 15ó cc RFCALI- Pt)LTìFI FINI)5 AIIPLTTUDt- S PÉD POL 155 P[.1] N't =,F T xYrY*r,Hi; ( vl SPEC POL 15ó (: ) PllIi.l l.=PHLNI /f.ìfìHT I - 0-sl l.tA S PEC POL lí7 xx-0, Í:;* ( r .0+fl(J{i (? :0x(PHrulND-Pt{I,)) SPEC POL 158 PI'IIíI'I =,PHINT/XX S PEC POL 159 F(I.TUF(N gPEC POL 1ó0 ENI) S PEC POL 1ó1 f:rtNC f I 0N tì(ìr.lvr F,tr PHI.¡ SPEC POL 1ó2 c -"'ü¿1Mt4oÑ7izpllrurnDrvrPl{A$E'PHt([Ft;t)NvtJLul l.[JN Ë l.,l'l(.;'l ] 0N S PEC POL 1ó3 " S PEC POL I66 (:2 S PEC POL 1ó5 ÊN I ATTt-N I RADX t'2) t t')llI x ) col.fl'ltjN/u,/ tJAUt..t-l : H'l RF-[) r H'l 1 X r I)1 PL s PEC POt- 1ó¿ (et' ¡ COHHtJN/PHl i'IICl( 9I) I l'll\flFlI l(ÂD SPECPOI,- L6) õiiiìüóÑ7i{7 lti,ät p'uirit-t rEPSrRorPl S PËC POL tâB tìtìMfrr.lN/ I / L PUt Sìl P S PEC POL t69 slGilA=1.0 S PEC POL L70 ANIStITRqPY tli: scATTEti c "i'-ijijr-îl-Þultl's1.iìüÂ"ilt.gcniptls uqq S PEC POL L7L iìËrrE'¡ili-rjF-RX.F srrotrl'D-rit:ALLY'l Ë PEC POL L7? ð Ñ1 ( F I oF x PULSE) -'ð 'röuiii-ãn'1nÀÑsiljrri'i ""1É - S PËC POL L73 ì.ïPLli-s'iÞ. Nt: . slrEXP) Go I t) 77 B PEC POL t7â C "'-"ärih=lHo-n:iHERE PULSE, TYPT. ITì EXP(.uI S PEC POL L73 " ; (:l'io-R:r // DHxtJ IDt{) S PEÇ POL L76 (: (. --ARG ïì. ñr,.i; i cUa,/ Ri(ri: ) *É-x P ) S FEC POL L77 Giì Tt) 78 S PEC POL L78 ?7 T]ONTINUË S PEC POL 1.7 I tirìË FN 180 t: l-lE.F:ti.j:È'Þour_R r L-OS SPEC POL iÍ-' - ÞuL-i;r_ ¡s; ¿ssxx/r FN S PEC POL 181 - r:fia -t(.r,/ (:tJIDfl*8.,l ^, J POL 182 ctitjs-uOS( 1lI - 0 S PF:C iËì.Ãnãì:¡:o-¡ i . ör. ul' nsx¿' i ctìes=o' S PEC POL 183 s:ì x TEA i r Ñl'-c;oSsx r;lrsÍì*ctl (.Rxfi:)ç0ri5 S PEC PTL irXf ==Xfruf * (:5Itìl'f'A/ ) SPËC POL 185 7B t;t)l.iT l. NUE S PEC POL 186 coì{tr=.xrllT SPEC POL LA7 FL.TURN s PEC POt- 188 É:Nr) SPEC POL 189 FUNCl l.ON PT)LAI{( T HE IAtPHI) SPEC POL 190 CO},IHON/J/ PH I U,I. N D ,v r PHASÉ. t frHRËF S PEC POL 1.9 7 (:21 S PEC POL L?2 I EN I RADX (:2) r PHIX CIJH¡iÜN/U/ UAVF:I, T HTFIEC. ltT'l X r DI PLEN ATI 5 PËC POL L93 PHt).. I 9.r' r NAËli r RAD ( 99 I L?ó t;oMi'l0N/ Pll,/ rE.I,SrROr Pl S PEC POL C0Hi'1ON/H/ H1 rDti P ,uÍriíì- PHREFTHTRECTDTPLENT 5 FEC POL 195 Rt !ìatrf,RË!,j tlTllË.1  ! åili iii*vL,L rrFiAD, ÉxhíÉiìÃáÈixAERr S PEC POL 196 $ATT EN J O ¡ ATTEN) S PEC POL L?7 ( r UAVELT ¡ RADX I PtIIX I O' O I 2 r O' O r HTTX' I' F([fì=fiE5 xt{ f 'ftE:9 T l.lË I fr PtlI, S PEC P 0t- L9a PU[,åti-RE!:t S PEC POL L?9 ftEl utiN S PEC POL 200 E.NI) E4 sU trri orj'f'I s I¡.IPSN .'F ¡7cA tl r EPS r N I FI I PHI ) S PEC POL 201 |lE ' S 302 Ï TITE GF(A t-[.Sì F (.x)f.R OH. A_T[]H UITH AN ËRROR L.É:SS THAN ËPST PEC POL c S POL ?o3 IJTITJ NÏF-RV ALS PEC C tJl:iIN Cì N I POL U,TTH AN TJPPE Iì LIH IT OF 25OO INTEF{UALS S PEC ?0¿ c S PEC POL 205 (; TI.IE INTE,GRAL IS F I NOBt.. P?:19 S PEC POL 206 c ErUr-lL.? S PEC POL 1Q7 H'=O. ':ì')+ (. tt- A ) AJ=t.| xt.f:iArT tIETA'PllI) -lF{.Ë' I}iEfArPHIlr j S PEC POL 208 S PEC POL ?09 AIl=3-0*AJ S POL 210 N=1 PEC S PËC POL ?11 Etl=O - O 3 S PEC POL )12 0fJ 5li='1rN POL lF (. I ? Or ti- 1 - o ) {t.l r THETA r P}{I.r S PEC 313 :i lìB-BtJ A.l - s PEC POt_ ?IA +,1 - oxH.'6 fJLi ;lIO=A'f S PEC POL IFaAt.lSrAIOr -1. - oE-50 )ûo To 10 r1. LT. EPS ) GI] TO 1O S PEC POL :? 1ó 7 IFr:AttS(.rAl0-A /Ala). S PEC POL 2L7 _ (:AJ+A Irl AJ-O 25r S PEC POL 118 A I1.=A I u S PEC PDL a,lg N:?*N s PEC POf- F (:N . tìT . :.150Ô ) G0 TO 10 I S PEC POL :l? 1 H=.0 - -=;*H GOTOS S PÉC POL 7.22 1O FT=4I0./3-O S PEC POL 333 TtJIiN SPEC POL Rf S PEC POL END S PEC POL 226 SIJT{I{OLJTINE STHPSË (F TZTATFTEPS rNrFf.r I,JITH AN EKROR LESS THAN EPSt S PEC POL C INTEIJIIA1ES F(X)FRTIÈ1 A fIJE S PEC POL ?28 C IISING N SIJBINTERVÂI.-S SPEC POL 229 C UIl f{ AN LJI1PER LIHI.T OF 2iOO INTERVALS F]. S PEC POL 230 Tì ìIIE INTEIìTIAL I$ S PEO PT]L 231 r ? r Pll.J9 -la C N(llrt.E VÙL ' S PEC POL t H=0.5*(:F_A) a1a (.F (Al +F (:El ) S PEC POL AJ=ftx S PEC POL 23^ 41. I =.J. OìtAJ 5PEË POL N--- 1 S PEC POL 236 3 Ì111::0 - 0 S POL ?37 Dû 5 N=l rN PEC ( (::_ -1.. O)t+H) S PEC POL 238 :l ErE-Btr+F- A{ "O*l\ S PEC POL ?39 . Ol.H¡IJB S PEC POL 240 'à1.0=aìJ+4(.ñllsir'A o E GO TÜ IO If: It)r -Ll .1- -50.) S PEC POL ?AL ( ),/ A g).LT.E PS) GTJ TO 10 7 lF(ArrSì(: Ar.o-Af I I S PEC POL 24? fìJ=Ct. ?l;* {:AJ{AIO:' ?Á3 AI1.=Al.Lì SPEC POL N-ii tN S PEC POL 2t¿A TO 10 sPECP0r- ?lt3 l:t:cN"GT- ?500) GÛ S PEC POL 246 H'"0 ,5xH g PËC POL ?47 0r.r Tu:{ SPEC POL ?44 1O F-1-AIO./3'o S PEC POL 249 I(ET URN SPEC POL ?50 ËNT) SPECPOL 351 5 t.l tr ti lJ t.l't I NF: F:f ADE (. PttI rRADrNl'{tr) SPECPOL 35? (NNtJ ì Dl.M[.i.I ..iIoN tl-ll'lli,lt) ) ' F:AD SPECPOL !53 Pl-.1 . 1 4739 SPECPOL ?54 C RIiADS I N TEL I: S;i-;OPE D ¡.I'lENS I ON!ì SPECPOL 255 DO 11 l=1r 1:l SPECPOL 256 Jl.=', ( I -.1 .1 *6+ 1 SPECPTJL ?57 J:1.=.'l 1 SPECPOL ?3€J IF'I. Ë.Q.15;J2==J2 sPECPOL 2A9 riE Al.) 1O0,(nÁD(JK ilpxrr.¡xi ¡JK=J!,J2) SPECPTJL ?6O t oo Ft)fìHA T SPECPOL 2ôB f rRAD r PHI r FHASET NNOr PHREFTHTDIF'r SPECPOL 269 SPECPOL ?7O ARRAY IN DIRËCTI SPECPOL SE TIF I]IICIiLAND PARN SPECPOL "7L?7? SPECPOL ?73 ELENGTHS ÉE REAL I SPECPOL '!7 á n¡-lìÑ'üãrrEcrroH -cAN oNLY SPECPOL 173 S PEC POL ?7 6 SPECPOL ?77 INE SPECPOL. ?74 SPECPOL ?79 StJl'l=0. o SPECPOL :ì80 su¡'11==0"Õ SPECPOL 28I StJÈ'l!=0. o SPECPOL 282 C INC.LIilE DIPt]I-f;. RË:$PONSE SPECPOL ?83 FEIA= IEF:o ?84 ö aÊuarrr IIr * ( (ANGLÉ) **?' o ) :) SPECPOL s.t.NIr=ri0Rr ( I . o1 ì: öijs ! PH I O I xt(:ì - o) srN SPECPOL 285 t:srNu-LT -001 )t;o l'o-9, rË "0 ( SPECPOL 2A6 Cof-ì s=ST N ( AIJÛLE )'fCO5 PH I O:} 287 p ) r t'lrl SPECPOL RËÈí Ë =coä i ij I Þ[eH x Í,t 2 . o*cutìB /s SPECPOL 2BB GO l0 10 SPECPOL ?89 SPECPOL ?9O SPECPOL ?9L SPECPOL ??? I(I)] SPECPOL :I93 Y SPECPOL ?9á I ì.¿TRODUCED PHASE SPECPOL 293 HI(:l.t). SPECPOL 296 SPECPOL 297 iã[Fi5Ëfii'hll5, u"uou''-" SPECPOL ?'8 SPECPOI,- 299 SPECPOL 3OO coulD Mtrl.rIP SPECPOL 301 SPECPOL 302 ÊÄ?r,ohb FËt8oto'otlY-uE SPECPOL 3O3 N COruSTEruT FOR THAT CALL SPECPOL 30ó t. 1 DONTINUE SPECPOL 305 (. l¿ { H2 * * 2 o :) sÜiì = s Lì R I 5 U H 1 * 2 . O SU ' SPEDPOL 3Oó St.JH=SLlH I RES P SPECPOL 3O7 c Nou-ÏÑcõÈr'otrÃrr oF(ouND EFFEçr Ê . ^ ïs zERo SPECPOL 308 b i}iHe'58¡ir,îr,Fl¡turinui,rulËoÊîrÉE[jpriiÊii SPECPOL 309 :::"::l SPECPOL 31O suH=:ì(¡RT(: (:sìut.l{.srJH*fiTTENxCOScpHaÊljr-*n:.oi(:suM#ATTENTÊSIN(PH6RD)) SPECPOL 31 1 $ *x3.0) SPECPOL 312 ti PF( E !ì =S Ul'1 SPECPOL 313 RET UT{N SPECPOL 31¿ END APPENDIX F Program ScaÈPrf Program to Simulate ?artíaL Reflectíon Profiles For a Full Description of the Program, See Chapter VIII' Section 8.2.lb F1 2 (INPUTTOUTPUTTIAPEl¡lóOrTâPE2'1ó0) B8âTËãF 3 PROGRôH SC^IPRF SCATPRF Á c'r{pLEx 5 L2 ó 7 s I 10 i;l'îiîiiiifrãijilfi,î*î='"" ËEiiffi a)ll ruRN ,rRE REFLECTI0N coEFFs 88åiËftF 13 LA üHhlÄËt{nlÄ'ÉIËð"0* coEEF gEâ+lfif IJ Lê E,9[r'oouu' 3Ëni¡Xf 17 SCAT PRF 1B ( 19 6I (.10o) r PULsI? (:1oo) I FREOTJ 1oo) ?o 83ilËäF 2l a,) 2(1oo) 83âiË[F tl SCAT PRF SCTIT PRF 2A 88âi ËËF 26 27 uL 83âTF[F 28 SCAT PRÉ 29 30 DArA câRDs 38âlËÊF 31 PRF 32 SCAT a7 SI)AT PRF SCAT PRF 34 SCÊl PRF 35 3ó 33âi ËnF J7 SCAT PT(F JB 39 *No 0F FREnu trANDs trseD) 83Ê+ËñF 4a SCATPRF 6t UI,IES O VIT{TUALL r'HT TO BE 1ST A2 r Pr B3âTËËF 43 Aá TPr AND Pr usED FoR o vIRruAL B8åiËËF 43 46 â7 4B â9 50 5t 52 53 5á 55 56 57 5B SCAT PRF 59 gCAT PRF óo SCÂT PRF ât' SCÂ'T PRF 62 ó3 Ë3âT ËfrF 6¿t ó5 ó6 ËF'ilT'RIË'[To*'" tgl+fnf ó7 83â+ ËËF ó8 r SCA1 PRF 69 PRoFTL 70 BE usED FoR FrNAL ourPUr 33âiËãF 7l S- SCAT PRF 73 v LARGE IF uIsH ro Go ro END 33ÊiËËF 7tt SCAT PRF SCAT PRF 76 SCA T PRF 77 SCaìT PRF 7A SCAT PRF 79 80 83å+ ËnF e1 a2 BBâT Ë[F 83 84 Hr(AN cExP(rur) TIHE DEPENDENCE 83â+Ë[F 85 86 33å+ Ë[F a7 SCAT PRF 88 FALL SCAT PRF 89 so )*r2.orsc) B!âTFäF ?1 )x'?.ofsri) 92 83âiËËF 93 SCAT PRF 9á 9'¿ lt g8lltälForr(FREourpuLeEr-Nrr1rt7H*rllgc{rcRosEcs)rr11HÉAHPLTTUDE*) 96 ÊEâil[ESCAT PRF 97 REcEIvÊg"'Tu-r¿¡5¡cE ?8 c 79 FN sr.sIl[lnoår¡FTtpâbro$ãBro9uëul*Ëå'-vE FXEU ¡ËâiËiF 100 3 'lÊ[F^ï[ñ8"Ïi:;HTiã,-'',-.åíiiXi-ñqnårr õõäiÞn¡ 101 õ axp uoRK-.IouARg -Í'11ålì" ^És õóäipnr 10! E- îË-rõ-''i'6rçH'- .rrr quJ PUr õðäiþHþ 103 c,Ëcciur= SCATPRF 11ó (r) scâT PRi' tL7 ËCATFRF 118 t2 SCATPRF 1T9 SCAT PRF 120 C RES SCATPRF I21 ó41 I'HTaAHPLITUDET) SCATPRF L?2 rHE(Ì4fcRosEDS)rr SCIìT PRF L23 tiCdrT PRF 124 C BET SCAT PRF 125 SCAT PRF 12ó oAT(I-1) SCATPRF T27 OAT(I-NT-1) SCAT PRF T2B C Ft( sCrlf PRF 129 SCAT PRÍ' 130 131 c ALS T"X DËt.T SCATPRF cso SCAT PRF 1 32 SCAT PRF 1J3 SCAT PRF 134 13 SCATPRF 135 rNT) SCAT PRF I 3ó á78 ) SCATPRF T37 REO ( t{HZ ) * r SHèAt{pì SCATPRF 138 1 39 p-D-pp..D.-r' D D D DD DD DD DD D SCAT PRF c D D D D D D D D D D D D [' ÃNb sTURE IN xNTGRA- SCAT PRF IôO c cALCUT-rrrr rNr¿aiai iEñ¡-rörr-eÃc¡r rnuu SCAT PRF 14 1 DO ó1 I=1 ¡NT SCAT PRF 742 4l XNTGIIAI- (I)=(0.rO.) SCATPRF 14] 12=ii C H gCAfPRF 230 pHASpTH ( I ) -PHASPTH(I ) +NREFI *XHTSTP SCAT PRF 2f 1 c oerolf{3ãxREF1-NR ) SCATPRF 232 -- /c}rpLx(HpRES-FIZ'o.- SCA T PRF ?33 SCAT PRF ?34 c O" IEñI=IT åTå:SThIiiliÍ:ãEiåIäi:TÉ;âEEåil;;;;;;.;- SCÂTPRF 235 c SCAfPRF 236.¿37 ott SCAT PRF Ili+3Fât,r,=**,o*AL(I)llgTl:ircEXP(r'0'or-2'o)rcxPLX(2'oÉPIlcrFr SCATPRF 268 SCAf PRF 2ë,9 AGà{ETOIOITIC FQRXULAC scATPRf-- 270 c sCAÌ'PRF ?77 c 1?ÍT¿*3fifråoÛ8t*o APPË'x¡x^rIoÌ'rs SCATPRF 272 c LY 19á3) SCATPRF 273 c 0rPrGl rRrSr SCATPÑF 274 SCÂTPRF 271 SCÀTPRF 276 SCATPRF 277 UOrX¡ZñrYruL SCATPRF 278 HCH I UL SCATPRF 279 SCATPRF 2AO E-tï/ tãll/? ' IQS5€-3I/r SCATPRF 2B 1 SCATPRF 2B?. SCAT PRF 283 c r3 SCATPRF 28ó c c SCAT'PRF 281 c SCATPRF 2Bó c SCATPRF 2A7 E SCATPRF 284 SCATPRF 2â9 SCATPRF 29O co=CLIS ( Pl{I ) SCAT PflF SCATPRF "91297 õn=õiñ(pxirxsIN(PHr) SCATPRF 293 Ù=pJ*FñEßì+2.OÉ.O6 SÊATPFF 294 t 29= U[ì:E$ËHn"l,.oono.ÍEarEtl/EolEìr SCAIPRF 'J9 X=UOl (u*U) SCAT PRF 6 Zì1=vFt/U SCATPRF ?97 Y=UL,/U SCAT PRF 278 c órÊiNÈ coEFFrcIENrs SCAT PRF 299 GL=X/Ztí SCATPRF 30Ó SCAT PT(F 30 1 SCATPI(F 301 SCAT PRF 3O] SCAT PRF 3O 4 SCAT PRF 305 SCAT PRF 3Oó SCAT PRF AÔ7 SCAI PRF 30B SCAÍ PRF 30? c SCAIPRF 3IO c TENSOR COE FFICIENTS SCATPRF 3T1 EPSl=(: 1'O- A,_I14B scAT PttF 31 2 E PS2=O . 5* (: F-D)+0.5rI*( c-E) SCATPRF 3I3 E PS3= (:A-O. 5*(D+E))+IË( B-O.5È(t--+D)) SCAfPRF 314 ICIENTS SCATPRF J15 c I{ORE COEFF 314 c SCATPRF o I -OrEpS1i4(E PSl+EPS3) SCAT PTTF 3L7 P Ë PS3r (EPSl+ ÈPS3) +EP52rEP52 SCATPRF 318 o ? . O*EPSl rEP 32 SCATPRF 3I9 3'JO R 1 .OrEPSl SCTIT PRF c 1 SCAf PRF 32 I - o.rEPS3 REFRACTIVE INDEX N c c AL CULATE THE SOUARE OF THE SCATPRF 3?2 c SCAI PÃF 3]3 T=O+ SCAIPRF ]24 ARG= ËIElto'.o-o*e..corco scAT PRI-- 373 (êRG) sC¡1f PRF 3:lá ltÉ csoRT l.O.o)U--U êcet PnP 377 igqnrNno(ARG).L SCâTPRF 3?8 UÊR +S *SO SCATPRF J7? c OIìD I NARY REFT SCATPRF 3¡7 DOUBLE PRECIEION FUNCTION C5(X). SCATFRF 3JB ñËÃL-- itiiin i i a2 ¡o r t¡ 1 r 82 r 83 r B4 r X SCATPRF 339 DATA K/O/ ' SCATPRF 340 rF(K.OT.O )00 To 1 SCATPRF 3ó1 K-K+ 1 SCATPRF 342 ADEl.1ó3Oóól SCAT PRF 343 A1-16.?9o1OO2 SCATPRF 34á f\2-6-6945?39 SCAfPRF 345 EOsÁ.3óo573? SCATPRF 34é B1-ó4.0934óâ SCAT pRF J¡r7 B2-é4.92050S SCAT PRF 34S B3-3:t.3:JA25.7 SCATPRF 349 Èã-è. é31ââ?7 sCrlT PRF 350 1 õ;;ii;;3;Ãålx*x+arnx+âo)/(xrr5+84rxÍr4+B3rxrr3+82t.xrx+ SCAT PRF 35 1 I B1 IX+BO) SCATPRF 352 RET URN 353 END SCATPRF DOUBLE PRËCISION FIJNCTION C3(X) scA T PRF 6 REAL AOrAlrA2¡A3rBOrBlrB2rts3rB4rB5rX scA T PRF 3 5 SCA T PRF 3 J ó DAflt R/O/ a 1 scA T PRF 7 IF(K.GT.O )GO TO I K=K+ 1 scå I PRF 5 s A0=2.39834 7 6E*2 Ëcr\ T PRF 3 5 9 A1=1.12875 13E+1 SCA T FRF 3 6 o A2=1.13941 608+2 scA T PRF 3 1 A3=2.4â't3L l5E+1 3CA T PRF 3 ? IrO=1.80óó1 2AÊ.-2 scA ï PRF 3 3 ÊL=9.3A773 72 SC Êr T PRF _f â 4 B?=r .492L2 5àE+2 SCA I FÑF J B3=2 B?5BO B5E+2 5LH T PRF \ ,56 - PRF 84-1 .204?5 1 2E+2 5CA r -1 85=2. 4ó5óB 19E+1 5CA Y P F{F 3 óB 1 C3-(XÍlró{-A 3 EX f r3+42 x X)+X +A 1 |.X+lìO.),/ 5CA ï PRF J ó9 + SCA T PRF 70 1(.X*xó+85$ X * *5+ B4 * X * r{ ú {'[¡3 r X ** 3 + B? r X * X F I * X+ I 0 ] l SCA T PRF f 7\ RETURN 'r 72 END SCA PRF I SUBROUI'INE SIhPSN (F ¡Z¡l\tBrËPSTNTFI) scâT P RF 373 C INTEGRATES F (X) FROI'I A TÜB UITH AN ERR0R LESS THAI{ EPSr SCAT P r( tr 374 C USINTJ N SUHTNIËT{VALS SCAT P RF C UITH AN UPPER LIHIT OF ?5OO INTERUALS SCA T P F(F 376 C THE TNTËGRAL IS Ff SCAT P RF J77 C NOBLE r VOL. ?tP?39 scAt PRF 378 H-0,5r(E-A) SCAT PftF 379 AJ-H#(F(A)+F(Ê)) SCAT PRF 380 A I 1 =3.ol+AJ SCA T PRF 381 N=l SCfì T P RÉ- za2 J BE=O.O SCAT PRF 383 DO 5 K=lrN SCA'T P F{F 384 5 BB-BB.rF (r\+ 12. Oi{K-1 .0) äl.l'J SCAT PRF 3E5 AI0=rqJ{.4. OcHËBB SCAT PRI: 38é rF(AËS(ArO).LT" 1.OE-50)G0 lo 10 SCAÏ PRF 387 7 IF(Aús( (Ar0-Ar1) /A10).LT.EPS)60 TO lO SCAT PRF 3ËA AJ=o.25ré(AJ+AIu) SCAT 1 AI 1=AIO sc AT PF(F 389 N=2 rN sc AT PfIÉ 390 IF(H.GT-2s00) 60 TO 10 sc AT PRF 391 ll=0.5xH sc A] PRtr 39? GO TO3 sc f{T PRF 3?3 10 F1 =Al.-J/ 3.O 5L AT PÍ(f: 39d RE T URN sc ATP Rl- 39 F-N D AIP ftF 39ó FUN CT 10il coLL (.2¡L) 5L fìT PRF 39 7 c SUB PR OTìRAH IIJ CALCULATË COLLISION FREOUENCIES FOR 43S SC AT PTiÊ 3.1 I c I I - UINTER sc AT PRF 39 9 c ï 2 - Et¡UIN OX SCAT PRF 40 o c I 3 - SUT,f HE R SCA T P Rf; 40 t I F( z. GT.71.0) 60TOl SCAT P RF 4Q 2 I F( I -2s2t3t4 SCÊ Ï P RF qU 3 2C OL L =3.579E+ lo*exP<-z/7.Á7' SCAT P RF 40 4 R ETURN SCA T P RF 40 3C OLL=S. 2tZE+lOrEXp (-Z/7 .297, scAf P RF 40 6 R ETURN SCAT P RF 4ö 7 4C OLL=4. JSlE+lOíEXP <-Z/7 .55) SCAT P RF 40 a R ETURN 3CA I ? RF 40 9 1I F(7-?)Jt6t7 5CA T P RF lt oLL = 1 .4J4 Ê+ 1 1 rEXP (-Z/ 6.35) SCAT P RF 4 I R ET URH SCAT P ti F- 4 2 écoLL =2.?89E+1 IxEXP <-Z/ 6. L87) SCAI r, l{ F' 4 3 R ETURN scAf P RF 4 â 7C OLL =1 .392F.-+L?IEXP <-Z/'r - 45J SCAf P RF 4 J R ET URN SCA T P RF ¿t 6 E ND SCAT P RF 4 FUNCTION BFIELD(Z) SCAT PRF ó18 c FIELD AT 31S (UOOHER A) IN U/II'I2 SCAT PRF 4t9 (.L.O+Z/ 637O.Q ) **3 SCAÏ PRF 420 A-| . ö / SCAT PRF 42t BF I ELD=ó. OOOoE-05.4 SCAT PRF RETURN SCAT PRF a23 END r ¡ùl2rHJrX) !lc ATP RF 42^ COñPLEX FUNCT loN Lt(xl X2 rX3r111 êTF f(F 423 I'11 r rf2 t'13 r SUl'l sc COllPl-EX " 5C A'f P fiF 4?6 SUH=Ci"lPLX ( (X- iäl i ti-x¡>,/ (. cy,L-xzi r(x1-x3) ) r o - ) rt{1 ( (x2-xl ) * (x2-x¡i ) r )rH? sc ATP RF 4?7 SUH-SU¡'1+CH PLX c?i-i1i ii.x-xt> t Q. ATP RF /¡i]8 gg¡=gg¡+CH pLX ¿ ¿ i < (.x3-xL) i lx3-x2 ) ) r o - ) rl',t3 sc 1-1 i i t x-ii> / sc AfP RF 429 L2= SUI'1 5C ATP RF 430 F(ETURN ATP tiF 4lt EHD APPENDIX G Program Volscat See Chapter X for DescriPtion (equation 10.3.1. B) G1 T ? PRO G T(A H U0LSCAT(INPUT tOtlf PUT) voL scA E RN AL PHINT voL 5 cA T l EXÏ CA â H ON A VOLS ï coH /Í /THE.I T 5 HON/ P voLscA coH / PHI r:2) cA é cor.t HO¡l/ U IJAVEL T r HTt< EC I HT TX DI PLEN r ATTEN rRADX t.2) r PHIx voLs f / ( VOLS CA ì' 7 c0l'1 HON/ P H/ PHO< 99) rN AE R r RAD 99) coH HON,/H / liÏ, DE PIUID H ¡EPStROT PI voLscA T I coH HOr.t/D ELlDEL v0LscA T 9 v0Lsc A T 10 (;0 H t{oN/ I / I PULST P GRA C,ILCIJLA rrpecrrvE uoLUllE oF BACKScATIqE F¡orl A LAYER oF voLscA T t1 c PRO HTO ie VOLS CA T DEP TH DEPIL OUER ED cË Ãr ¡tÈlotli HTTFoR PoLAR 6PECIFIED BY PoLAR r2 c voLs CA T 13 xX=1.0 v0Ls CA T tô P I=3. I 413? ¡ E PS-1 .0E-02 rflltft T T T¡T rtIt}{l{irtt* UOLS CA T IJ CT VOLS CA T 1ó c DATA ã ¡a t I * i Í I Í af ì ì f lt t r I I t Ì r I v OL CA T t7 cr tit.r{t OL T COORDS v c t8 c TX OL I RADX (: 1 ) -12.5 I RADX(2)-12.5 PHIX(1 ) =Pr/2. t PIIIX(2)=-?I/2. OL SCtìT 20 L S r ANI) C00RDS VOLSCAT c RX AERIA VOLSCA T 2? NAER ? S CALL READE(PHOrRADt99) It =B UOLSCA T c UAUEI-ENG]' I'¡iÞo[T LTNCTX IN HALF UAVELENOTHS FOR RX ,) llAVEL r= 50.0 t DIPLEN-3.Ô UOL t HT oF Rx AERIALS ABovE Il'IAGES UOLSCAT c HT OF TX neniat-s-ÀbõÚÈ innoE UOL 1 l'lTT X*23 .o t HTREC=23.O UOLSCAT I ON COEFF OF GRD c REFLECT v0L A AT'iEll=t -o UOLSCAT 29 c 6ET HT ¡ DEP TH OF LAYER IN ÑIt HT*80. o ü DEP=?. O voL IS PULSE TYPE -EXP OR COS VOLS CAT 31 c IPULSTP PULqE VOLSD A T 32 c UIDH*- IF þúLsË iiÞE rs EXPTUTDH IS !1E HAI=¡ !IDTH oF ìa KH" cosrurDH=EôsE FULL uIDTll /8. IN Kh VOLSCAT c IN -:---ïr'ÈûisË-rYÞE'is S 34 USE snse'iJrbYir"7e. bô nÓrS Ñor gruEE qP-THETAL AND JHETôIN rN UOL CAT c HUSI "ïË-unñî-ÀiìGLÈé õF sI Is N0N zËR0 scArrER VOLSCAT 35 c FHII.IT lñÍÈcnÁrIg! IUERE voLscA 3ó cc AT START oÈ ÎñrEcnÀl t-ixlrs AND END 0F LIHrrs I I PUL!ìTP =3HEXP $ TJIDH=3.Ô VOL 6 c DEL= SHêLL ñö"Ci ÞuËðË"¡tÃt-È HIDTH AND LAYER DEPrH Éur Gr RoUNDoFF ERR VOLSCTìT 38 DEI-*1.08-1s VOLSCA T 39 UOLUIIE FOR RAN6E RO u0LsicAT 40 C FIHb_EFFECTIVE VOLSCAT 41 PI2*PL/2. VOLSCA T 42 C NORPIAI. IZATION FACTOR FOR I,ATER lllETA=0"o $ PHI=0.0 VIJL S CA T 43 C XX IS DUI.II{Y PAR/IÈ,IETER - NOT USÊD VOLSCA T 44 PRINT 4Á ' v0t-5cA T ó5 ( r PH I ) VOLSCAT 46 RESI''lAX- FOLAR THETA voL scAl 47 PRII-IT 4ó VOLSCAT 4B Á6 FORHAT(1Xr1H¡) A9 BO v0t.scAl PRINT VOL SCA T 50 BO FORHAT(1HT) VOL SCAT 51 ACY IS - rE9.3t VOLS I]AT ç, UOL SCAT 33 VOL 5CA T 5ô v0L sc T : ., VOL SCAT 5ó ::,,, VOL SCAT 57 ELEI{6THTANL RX DIPOLE LENGlll IN HALF UOLSCAI 58 VOLSCAT 59 Û- IJAVËLENIìTHS ,= I I5I2F1!.4) VOLSCA Ï 60 PRtNÍ BSTHTTXTHTI- 88 ( PHI ) v OL SCAT FUNCTION PHINT OL S CAT e9 EXTERNAL THETINT v OL gCAT 90 OL S CAT 9t coHHoH/u/UAVÊLTTHTRECIHJTXtqIFtENIATTENTRADX(2)TPHIX(2) v PHo(99) TNAERTRAD(99) OL SCAI 92 cni,i¡ioÑzFHz ,EPerRorPI-- voLscâT 93 õöüiìõñ2i{7'ni'.ueÞ'u¡ox voL SCAT 94 COITHON/ DEL/DEL voL SCAÌ 95 voL SCA T 96 c voL SCAT ?7 c voL SCAT 98 voL 5C^T 99 voL SCA T 00 voL SCA 01 VOL SCAT' 02 voL SCAT 03 PHINT=FI v0L 5CA T 0ô RETTJRN voL SCAT 05 END c2 (TllETA I PHI) VOL SCAT 10ó FUNCTION THETINT VOLSCA 1 LO7 EXTERNAL CONV VOL SCA T 109 t.2 ) VOL SCAI 109 c0Hl'{oN/u./ UAVELT I HTREC I I{TTX I D I PLEN I ATIEN T RADX I]2) I PHI X r NAERr RAD (99) VOLSCAT 110 c 0Èrl10N/ PH 'Ht'ógP,u¡Dtt/- ?HO<99 ) VTJLSC AT 111 c oHr{0N/H/ 'EPS'R0rPI VOLsCAT tt2 xX=1.O VO!- SCA T I l3 P f2=PI/2. 0 VOLSCAT 114 GRAT I OH OF CONVOL UTION TIHES POLAR DIAGS CI NIE ovÈn n vALUES IJITHIN LAYER UOL SCAT 115 CB DS OF R INT EGf{AL-ONL r'iñïecni{ÍÈ' VOLSCAT 11ó N=HT /C O5 ( THETA ) ¡ nrnX= (HT+DEP),/'COS(Ti{ETAl RI'{I TNTFITTHETATPHIT voL scA-t- tr) CAL L SI14P SN(CONU'R tR¡lÏti'RmnX'EPS VOLS C AT 118 PIJLAR DIAG FUNCTITJI.¡ c NOU H U LT I PLY BY VOLSCA T Lt9 P ( T HETAr PHI) OLAR ;SIN(THETA) UOLS CA T 120 TH E TINT=F IãYTY VOL S CAT 121 RE T URN VOL SCAT 122 EN D (.R¡ VOLS CAT 123 FUNC TION c0Nv THETA I PHI ) VOLSCAI L24 TIO N c coNvol,-u TION FUNC VOL 5CA T l?i (2 (.2 cot.,l¡loN/U/ UAuELT l HTREC t HTTX r p I PLEN l ATTEN r ÑADX ) t PH I x ) UOLSCAT PHo(99) TNAERTKAD(??) v0r_5cAT 727 eoHtt0N/PHl ,EPe'¡RorPr-- VOLSCAI 128 öij¡'ii,¡õñih7'Hî''nEpiu¡bH VOLSCAT l:9 COHIION/ ¡,/ I PULST P vot.scAT 130 UOLSC AT 131 c VOL sCAT 13? c VOLSCAT I Ì? c VOL SCAT 13ó VOL.SCAT r35 c vot_scAT 136 VOLSCAT 137 VOL SCAI 138 GO T0 78 UOL S CAÏ 139 77 CONT I NUE UOL SCA T 140 E HERE:USE CUS FN 141 FUI_SE IS COSrr4 FN v0LscôT c ïË..pOú¡n-- VOLSCA T 142 òosB;cos< ino-nl/(uIDHr8. )rf I) - UIIL-SCAT 143 . Gr -urDHrá, )coss=o' 0 iriÃss(Ro-h) *C0SS VOLSCAT 14A i t ¡lt-COSS¡COSSnC0SS VOL SCAT 145 XINT?XIl'lTxSIGl'tA VOI. SCAT t46 7B CO NT I NUE VOLS CAT 147 coNU=X I NT uot_5cA r 148 RETURN v0t_ 5 cA T 149 END PHI ) VIJLsCA T 150 FUNCTION POLAR(TI{ETAr VOI. S CA T 151 (: (:2) VOLS CA T 152 T }iTTX I TI I PLEN I ATTEN I RADX 2)' PH I X coHFl0N/u/ UAVELT, HTREC VOLS CA T 153 (99> r NAER I RAD (99) CO}1HON/ PH 'ur''õrPiulbtt/' PllO PI voL 3 cA T 154 colf )40N./H,/ voL 5C h T 1s5 (iirËih; Èü i ilinver-r' RnD' ÉÉb''EPS¡Ror o -o' NAER' o. 0 ¡ HrREC r D I PLEr'l r ArrEN t)REÉì.8 PRES v0L bL A T 1só (THETA I LJâVELT I RADX ¡ PHIX T O Ô I 2 ¡ O O ¡ HTTX 1 O ¡ ATTEN) vot_ A T t57 RES=RES * B PRES I PHI ' ' ' ' VDL- sc AT 158 P0L AR=RES VOLSC AT 159 RE TURN UOLSCAT 1ó0 EHD 1ó1 INE SI¡IPSN (F tZtArErE[¡SrNrF!¡IHETê1llI.l v0r-scAT 9UFROI.JT Ar'l ERRcR LESS THAN EPSr VOL S CAT t62 C I NTEGRATES FtïiFäoi'Á-ror,-uITll VOLSC A T tô3 USING N SU t{INTERUALS c pÈn INTEr VOLSCAT 18ó SUB ROU S IIlPSA ( LA7 TINE (x)FROtl 'll'ë¿Ë34',-rr.'no* rrr, VOL SCAÏ c I NT EGR AIES F 'Åtiâüt,illf, a VOLSCAT 188 c USING N SUTTI NT E RVAL tor VOLS CAT 189 c UITH A N UPPË R LII.IIT r=oo INTER,ALS UOLSCAT 190 c THE IN TË GftAL IS FI VOLSCAT L9r NOBLE r VOL.2 r P23 I c ') VOLSCAT I92 H=0.5r ( B-A VOL SCAT 193 AJ=H* ( F(A¡ PHI ) +F (Br PHI) I VOLSCAT 19ó AII=3. OTAJ UOLSCAT L9i N-- 1 VOLSCAT 196 3 BB=0.o UOLSCAÏ t?7 DO 5 K=lrN VOLSCAT 198 5 BB=BB+F (A+ (2, o* K-1 .0) rH I PHI ) VOLSCAT L99 AIo=AJrÁ.OrHf BF VOLSCAT 200 T. - oE-50)GO TO 10 IF(ABS(AIO).1 1o UOLgDAT 201 7 IF(ABS((âIO-A I1 7Ãroi.ur.EPs)co ro VOLSCAT 202 AJ=0.25r (AJ+A IO VOLSCAT 203 AII=AI0 VOLSCAT 20ô N=2tN VOLSCAT ?05 rF(N.GT.2500) c0 T0 10 VOLSCAT ?oó H=0. 5rH VOL SCA T 207 GOTO3 VOL SCAT 208 1O FI=AIO/3.O VOLSCAT :09 RET URN v0Lsc T 210 END 2l r SUTiROUTINE SIHPSB (F TZTATBTEPS rN'FI.l vot.scAT (X)FR AN ERROR LESS TiiAN EPSI VOL6CAT 212 c INTEGRATES F ott A TOÊ UITH voLscAT 2 I 3 c USING N SUB]NTERU ALS VOLSCAT 2IA c UITH AN UPPER LIT'I IT OF 25OO INTÊRVALS voLscAT ?l 5 F I c THE INTEGRIIL IS V0LSCAT 2l ¿1 c NOBLErVOL.2tP239 voLscAT 2t7 ]{æQ. gr ( 8-fi ) UOLSCâT 21R A,I-Hr(F(â)+F(B)) v0LscAT 2 I I AII=3.OÍAJ VOLSCAT 22C¡ N=1 VOLSCârT 227 3 EB= 0.0 uoLScAT 222 DO 5 K=1 rN voLscAÌ ?21 5 BE= BB+F(A+(2.OËl( -1. o) rH) voLECAl ?24 AIO =AJ+4. OTHiBB uoLscAT ?25 IF( AÉS(AIO).LT,1 . oE-50 )co TO l0 voLscAT 2?6 7 IF ( ABS((AIO-AI1) /àlo). LT.EFSJGO TO IO VoLSCAT 227 âJ o-25ñ(AJ+âI0) UOLSCAT T28 AI I =AIO ?29 N- a fN volscAT ( VOLSCAT 230 I F N. GT .2500) 60 T0 lo VOLSCAT 23 1 H 0 .5t H voLsCAT 232 G o TU3 voLScAl 233 t0 F I =AIO/3.0 VOLSCAT 2Jô R E I URN VOLSCAT 235 E ND VOLSCAÏ ?3ó SUBROUT I NE READE ( PH I I RAD I NN0 ) (NNo) (NNo) VOLSCAT 237 DixENs¡ox PHI r RAD VOLSCAT 23Ë PI=3.14159 VOLSCA T 239 C READS IN TELESCOPE DI}'IENSIONS D0 1l I-1r15 VOLSCAT 2AÕ J1.(l-1)¡ó+l VOLSCA T 2âr +5 VOL SCA T J2=J I voLscA T ?43 IF(I.E0.- l5)J2=J2-l CA (-RAD(Jx) PHr (Jr() r JK=Jl , J2) VOLS T !44 RËÀD 100; r UOI-SCA T 245 100 FoRHAT t PHI r PHASE I tlNO I PHREF r HTDI F ¡ v0L SCAT 353 r RAD v0L scAl uoL SCAT OF I{UCXLAND PARN AÑRAY IN DIRECTI VOLS CAT ?56 c UOL SCAT 237 c E NOTHS VOLSCAT c OÑ REFLECTION -CAN ONLY BE REAL I voLscAf 259 c VOLSCAT ?Á0 c voLScAl 2ór c E VOLSCAT c UOLS CA T VOLSCAT !ó4 SUI'l= 0 . O VOLSCAT SUll I =O.0 VOL SCAT 2óó SUH2=0.0 UOLSCAT 2-47 c INCÙÙDE DIPOLE RESPONSE VOL SCA T 2óB c BEUAITE IF IrEIA= ZERO VOLSCAT 269 "--êîÑs;boRl (cõs < PHI o :) xÉ2 - o ) t* (srN (ANGLE ) r*2' 0) ) iÏ - o: VOL SOAT !70 ircs¡x¡.LT.ô.ool)Go ro ? VOL S CAT óoSrl=s r x ( ArißLE ) xC0S t PH I Q l NB VOLSCAT ñË5Þ-õ0s (btPLEN*pLl2. o*cosÉ) /sI VOLSCAT 273 G0 TO 10 UOLSCAT 276 9 RES P=O. 0 U0LSCTIT l0 CONT I NUE UOLSCÂT 276 VOLS CAT ?77 I)) VOLSCAT 279 c TRODUCED PHASE VOLSC AT 279 c (r)l VOL S CAT ?80 FERENCE LINÊ VOL SCAT 281 c r PI +PHASEI( (DISI /UAVELT ) VOLSCâT 282 VOLSCAT 283 c vol_ scAT 2Aâ VOLS CAT 285 couLD r'luLTr P VOLSCAT 286 c Irohb EEtG0EaUALLY-uE VOL SCA T 2At c CONSTANT FOR THAT CALL VOL SCAT ?88 c VOL SCAT 389 l1 CONf I NUE VOL SCA Ï ?90 5u¡t:saltr ( sUH I * d 2 - o+SuÈ'l?r lf'J . ô l aa t GR0UND EFFEçT9 v0LSCA.l c NOU-'1ÑCORPORATE zERo-rlEANS sIN cPr Is zERo VOL SCAT 2?2 c iÄRE'ÞäÃ'dÊ-ôF,'nrÞtjrË"ãux hnovr-: Ae VOLSCAT 293 ' ""þxbäijlixîb I ËnbiiõiañGLE ) k:r. qÞ E I lÞôvELr+P I ) ¡'¡66,þ i i i*:. o+ ( suÌl¡Arl Er'l És I N ( PHoRD ) ) VOLS CAT 29^ bijË:õonÏ ¿ î silil;ËúùÏäiTËHröoÖ, VOLSCAT 7ri I #lf2.0) VOL SCAT 2?6 EPRES=SUÌ,I , VOLSCAT 297 RETURN VOLSCA T 2?A END Do 021?4.04 95325. OO 95135. OO i ï Ëi BB1ËO.00 tÊt : H i ii åå å, åitiEi.aliçi: ii 20 3á. 87 =e*i1t:l.12ZZ,1âti,â,gtä 25258. ó9 qi*i,at ei:iåîiË 32135. O0 :iËäËi'siãÉi.2ïi1l'ç^çtî9iilãåå¡,8oa ã5iÓt ¡r¿sz ' 0036s 7 ê?70 .OO äii .1pf iíô " i;äTii:cizi '91aÃÀ'-o'õé--¿'oo ' '2o?7o 90. O0 zol. eo-o01tr? 88 z?4. .po:)ö'õö-iiia.Àili¡'Qq cr'á4 tó28A.44 zzA,'t9^'rólõö¿si.ãd-ÞÕ'oo¿¿ã'ãszet'31377'02?trÁ'042Be 25 78.69 2030ó,87 :ËÍËiË,Es;El'cÉ:is'9iiËl:åã 7ó360 . o0 ZTe.äii "8?lóo.oo32e"ó? ql'ó?4 åË,îåiãã.li,ii.1i122 1ó341.5ó tBBl :äËtxB:ffiãÉá:iå,1!:tit9:i34.á¿:lãÍ?i:Eå'l?:31'n' APPENDIX H Coordinates of Some Important Middle-Atmosphere Observatories (Past and Present) OBSERVATORIES COORDINATES OF SOME IMPORTANT I"IIDDLE-ATMOSPHERE (PAST AND PRESENT) Geographic Geomagnetic L Shell Geo- Station Coordinates Coordinates [=1 /cos2 niagnetic (geomag. dip. Lat. Long.(E) LaÈ. Long. 0{) rat. ) l o o 209 L.75 -620 (Aust. ) 30045'S 136018 ' E -4ro o I^Ioomera 2l.3 2.0 -670 (Àust.) 34056's l3Bo30'E -4s o Adelaide o 2.0 -670 ßuckland Park (Aust.) 34038's 138o29'E -4s 2l_3 near Woodstock 2:rgo 1.3 -4Bo Townsvílle (Aust. ) 19040' s 146054'E -zgo (AusL.) Tantanool¿ 2.2 -6 7a near I'ft. Gambier 37050' s l4oo4 7'E -4Bo 2L6o Christchurch Meteor ,,' facitlty (N.2.) h3o27's L72024',E -4Bo z53o -7Lo Flat near o o o Birdlings 253 2.2 -7L Chrís rcl.rurc.h (N. Z . ) 440s t73on -48o o o E 214 1.8 -64 Broken Hill (Ausr. ) 31o54' s 141030 ' -¿+2 o o o 6L 309 4.35 78 Saslcatoon (Canada) 52ol¡ 10 7ow o o o 56 350 3.2 75 OtÈowa (Canada) 45ou 76otnl o o o 27 203 1.3 5 0 KyoÈo (Japan) 3405 1'N 136006'E o o 352 1.0 0 Jicamarca (Peru) rlo5 7' s 76052'w -loo o 49 3L7 2.3 6go Boulder (Co1, U. S .4. ) 4oo2'N lo5o16 'I^l (16 km West Sunset 2.3 6Bo of Boulder) 4oo2'N lo5o29'InI 4go 3170 Platterví11e o o o 49 3L70 2.3 6B u. s .4. ) 40 13 N 1o4o5o'I,I o o (col, o 4.0 7B (Alaska) 65 OB N L47o27>r'\l 60 3020 o Poker FlaÈ o o E 2260 1.9 -64 Sydney (Aust. ) 33 52 S 15 looo' -43 SOUSY (Harz MEs. o 53 g40 2.8 6go Gerrnany) 51042'N 10:30'E o 1170 6.7 77.50 Tromso (NorwaY) 69058'N Lgu22'E 67 Urbana (Illinois o u.s.A.) 4oooTtN 88o12't^l 510 336 2.5 7Lo (Georgia o o AÈlanta o 2.0 62 34ou 84oi'l 45 342o o u.s.A.) o 3.4s 6B Sheffield (U.K.) 5 3oN 2tl^l 57 B3 Ionospheric Research State College o Lab., 2.5 720 Pennsylvania U.S.A. 4oou 7low 51 35 10 Mawson base for o o o Research 67037's 62052'E 7I 103 9.4 -70o Australian o o 64 (France) 47oN 3oE 49 B4 2.3 o Garchy o o 52 Arecibo (Puerto Rico) l8o2o'N 66o45tt^I z9 2 1.3 References In the following list of references, the following abbreviations will be adherecl to as much as possible. 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"The interpretation of some observations of raclio VÍncent, J' Phys' \^Iaves scattered from the lower ionosphere"' Aust' ' 26,815, (1973). K'H" and Vincent, R.4., Stubbs, T.J', Pearson, P'H'O' ' Lloyd' Low, c.H. t'A comparison of partial reflection drifts roíth by rocket techníques"' J' Atmos' Terr' ' winds determinecl Phys ., 39, 813, (1977) ' R.4., and Ball, S.If. "Tides and gravíty waves in the Vincent, Phys. mesosphere at nid and low latitudesrt' J' Atmos' Terr' , 39, 965, (L977). t'A vincent, R.A., and stubbs, T.J. study of motíons in the l^Iinter Mesosphere using the partial reflection drift technique'r. Planet. SPace Sci., 25, l¡4t, (1917) ' Víncent, R.A., and Belrose, J'S' "TTre angular distribution of radio rr.r""-p"rtially reflec ed from the lower ionosphere"' J. Atmos. teir. PhYs.,40,35, (1978)' Vincent, R.4., and RäÈtger, J' "spaced anLenna VtlF radar observations of trãpospheric velocitíes and irregulariÈies"' Rad. Sci., 15, 319, (1980) ' t'Dífferentíal-phase Von Biel , H.4., Flood, VJ.A', and Camnitz, H'G' partíal-reflectíon technique for the determinatíon of D region ionization". J. Geophys' Res', 75, 4863' (1970)' VonBiel,H.A.''AmpliËudeDístributionsofD-RegionPartial Reflections". J. Geophys ' Res ' , 76, 8365' (1971) ' H.A. phase-switched correlatíon polarímeter - a von Biel, "The Atmos. ne\^I approach to tire partial reflection experiment". J. Terr.-PhYs,, 22, 769, (1977) ' t'The practical Von Bíel , H.A. use of arnplitude distributions in data assessment problems". J. Atmos. Terr. Phys,r 4I, I20I, (Le7e) . a 50-HHz tr{alcasugi, K., Kato, S., and Fukao, S' "Depolarizalion of radio wave back scattered frám the middle atmospherer'. Ract' Sci., 15, 439, (1980) ' lrleinstock,J.''onthetheoryofturbulenceínthebuoyancy subrange of stably stratifíed flows"' J' Atmos' Sci" ä' 634, (1978). Academic Press' Í,lhalen, A.D. "DeÈecËion of signals ín noiset" New York and Lond', (1971) t'The layer in the Itlhítehead, J.D. formation of the sporadic-B Þtty" 20' 49' (1961) temperate zones". J' Atmos' Terr' " ' gradíent instability ín trIhitehead, J.D. "The partially aligned the ionosphere". J. ceopirys' n""" 9L' 136I' (1976) ' the Upper S., and McElroY, M. "On vertical míxing ín çofsy, Res 78' 2619' Stratospheie arrd'Mesosphere"' J' Geophys' " (1e73). t'Radar and Guillen, A' observatíons of winds and !'loodman, R.F., J' Atmos' Èurbulence in the stratosphere and mesospheret" Sci., 31, 493, (I974)' stratospheric measurements !Íoodman, R.F."Hígh-altitude-resolution 4Il' (1980a)' with the Arecibo 430 \ft12 raclar"' Rad' sci.' 15' ttHigh-altítude-resolution stratospheric rueasurements l{oodman, R.F. 15'423' (1980b)' wíth the Arecíõo 23BO-Tß12 radar"' Rad' sci.' l,IoodsrJ.D."OrtRíchardsontsnumberasacriterionforlaminar- turbulenÈ-lamínartransitioníntheoceanandatmosphere''. Rad. Sci., 4, L289, (1969)' tr{ratt,D.S.,,Varíationsínelectrondensityinthemíddlelatitude D-regíonaboveUrbana,Illínois''.J.Atmos.Terr.Phys.,39, 607, (t977). radio drift !üright, J.W. "The, ínterpretatíon of ionospheric some results of comparisons with measurements. I. "tp"t1t"-otal 919' (1968) neutral t;;ã-profiles"' J' Atmos' Terr' Phys" 30' ' of Itrríght, J.W., and Píttevlay' M'L'V' 'rComputer simulation ionosphericradiodriftmeasurementsandtheíranalysísby correlation met,hods". Rad. sci', 13, 189, (1978)' tlunschrC."TemperaturemÍcrostructureonÈheBermudaslopewith(1972) application Lo the mean f l-ow"' Tellus' 2!' 350' ' anomal-y and Zimmerman, S.P., and Narcisi, R'S' "The winter related tránsport phenomenar'' J' Atmos' Terr' ?hys" 2Z' 1305, (1970). THBSIS ERRATA l. p.356 T lines and g lines down. "Fig 7.18" should read "Fig 7.19" ) Appendix H : Platteville NOT Plattorville and adtl a tt¡rrat end of 3. p. 389 delete one ")ttafterttequation B.2ltt; paragraph replace with "Rättgerrl 4. p. 23g: 10 lines down : dele.e "Rastogi" - 5. P. 97 : 4th Èo last line : ''t¿irte'' becomes ''v/rite'' "41/4T" 6. p. 9g : 2nd line , "1, = l/ka'ro= \f 4r", NoT Vradt 7. P. 35g : equaÈion 7.4.4.I2 : tr"dt' NOT 8. P. 3g5 : 6 lines up from bottom : "measurementstl 9. P. 398 z 2.BR on upleg ; 1-'6R on dov¡nleg tt ttcontroltt becomes ttcentraltt 10. p. 400 : Znd to bottom line ,'warranted'' ll. P. 401 : 8 lines down ''vlarranged'' becomes becomes t'lnleísskopft' L2. P. lr z '4 way down page : "I'ùeisokopftt + t' echoes appear" 13. P. 407 : (i) " echoes appear" p' I41) L4. 3rd page of t¿ible at end of Chapter 2 (after ;'l:-i;/x- ;' is wrong - should read 11 L - x'l4t¡ " 15. Fig 8.6 (i) "95.1 km" should be "95'9 km" (ii)The''800m''arro\4lshouldbestretchedto62mmlclrrg. Êd A-å uH' /L becomes ed - '11t lL' L6. p. 365 equation 7 -4.5.5 - ^i- 2 2 3 3 3 read tH 2 ct. (2r I 9"t) 17. EquaÈíon prior to 7'4'5'6b should l- ? z ptgt'J I In connection with 16 a.,.d, L7, also see "SPECIAL ERROR" ttTatarskítt lB. P. 366 : 9 lines from bottom : ) after F(8, Phi) 19. Page 8.4 : Program error : use F(A, Phi), and F(A + (2*K-1) ).:!;'fl, Phi) rr0rr is a function of R and Q í. e. de lete reference to the convolution only. 20. P. L74 : middle equaÈion (un-numb ered ) trsteradian is 2r¡ lh __t cl ur ds (Pt/h'? V o ) o eff' h L 2L. Equation10.3.1.6 A=exP {-+ NoT exp {- + s=o s=o 4o 4c R2 ( V Nor ( ) V 22. Equation 3.3.2.L9 hz- ) eff' ñ- eff 23. P. xii z para 4, line 4! ChaPter IV, NOT :II 24, P. xív Itne 10, after trdetermination of turbulence parametersil add "if beam broadening is sígniflcant" 25. P. 13 : llne 8, Dieminger ( le68) 26. P 30 para 2, line 9 : NOT "sodium-based vapour trails" but "Èri methyl aluminium vapour trailst' 27. P. 91 : last line < [ v (l) - V(5+¿¡ ]' >h tt 28. P. 93 line 7, "(ignoring small fluctuations) fj... insert ")" aftertrfluctuationstt] râ 29 . P. 1l_0 line 16! "viscous dissí-pation rate" should strictly be prefaced by "turbulent" 30. P. LL6 : line 12, add "(provided P, ís knowr)' af ter "ínto ít" do 31. P. LzO : bottom i g = f. + >rr NOT f. + 4r -:L. ff, Clr 32. P. I2L : Insert "inertialt' in front of "isotropic" on line 13. 33. P.141 5 lines from bottom : delete "by" which occurs in front of tt(ortt 34. P. L52 : line 11, Replace t'beaten down" with "heterodyned". 35. P. 32A : para 3, líne 1 : "cpapble" -> "capable" 36. P. 345: para 3, line 3, Delete reference to Elford and Roper, I96L. 37. P. 372 : 5 lines from boEtom : "sidebbe" + 'rsidelobe" SPECIAL ERRORg Thls item discusses the only serious errors found to date. ( ci, w, n) On P365 - 366, there are several errors. Flrstly, at the botÈom of P.365, I have given Èhe wrong normalízations. The bottom 5 lines of P. 365rand p. 366, should be replaced hlíth the followíng. (There was also a numerical error on P.366 r¿hich has now been corrected) (( Then k1 2 o. . .'/t k.-'/t dk Ei(ki)on, . Er(kí)dkÍ l_oa ]. ç r_ f I where kt = 2r/!.t I Recall that two normallzaÈions can be used; r-k (1) Ei(ki) dki + E. (k.)dk. = F and ) r a' l-' l- -æ k1 kr (íi) Ei(ki) dki + Ei(ki)dki = P/2. -ð krr where t¿t is the fluctuating component. I shall use case (i), as the formulae' derived willl involve longitudinal and tranaverse spectra, and the o,0, ot values given in Table 2.1, ChapÈer II, assumed this normalization ]. --2 lt 3 '/t Thus vH x2a td (2tt /9,)- , (note typographical errors in this l- 2 formula in original text), or t/, (7.4.s.6b) e ÈT -l ""' (assumj-ng (-T) - d 2D !,1 "nt), t/, where T = 0.827 (oi) 2D (tt can be seen that this differs from the equation Itz TZD o 11 .5 cr.' on P. 3662 this 1s partly due to the re-normalization and partly due to a numerical error on P. 366). Then by Tatarski (1961, EquaËíon 2.22), (1.4.5.6c) oí o 2.488 A. (after appropriate change of notation, and compensaEi-on for the facÈ that the normalizati-on E. (k,)at. = lrr/2 is used in Tatarski). t/, T 0.1026 A Hence 2D ^, L Thus, for Èhe longí.tudinal component, ff ,1¿ ] 2.0 then TZD*0.29, and for the transverse component in 2-D turbulence, usrng A. = 3'3' 'rzn N 0 .62. 4, It is lnstructive to compare 7.4.5-4 (í.e. td oAi ê tn> t/, vf with td = (.1026 Ai )-, /t, We then see ¡k o N 2.I3 v-- ?k rl- In equatign 7.4.5. 5 I assuned o È v- and we nohr see this is not valid. Yet both õ and tH rrlook like" RMS v'élocities. Whey are they different ? Consider the diagran below of the energy spectrum tß) l / .\ / \ / \ /\ :\ : : I -kl -k, o k In the derivation of t/ t (.1026 Ai ) -1 v /Lr, 'd H I assumed (defined) that thehaÈchedarea was equal to T. OnIy scales less than. ,01 contribute. But when the structure function bêtween two separated points *ith r"p.ration 9-t is formed, this in fact contains contribution from scales > .0r Ì".g. a scale of length 29"t can fit half an oscillation in this distance). ift" larger the sca-le, however, the less its contribution to the structure functioñ. Thus Õ2 can be regarded as the shaded area plus the dotted area. Hence it can be seen that there is a subtle difference between O and rq' (Note EhaÈ the equation ú = 2.13 v" is only valid if Èhe form.of E(k) ís t-7r over Èhe region (- - to - kz) and (kz to -) - otherwise the : result is different. If the spectral form changes, so will the constant. For example, if ,) the specLrurn fell abruptl-y to 0 at scales beLweetl - k¡ and kt, then O - vr) ' I shall not carry these corïectj-ons through the rest of the thesis. These results mean that ny e estimatesfollowing P.366 are too snall (although tlre calculations using Frg 7.22 will be accurate) If the corl-ect Tr.ì valucs are used, it will be seen th¿t the truet e estimates ur" ,n,-,ãñ-toåZ?ttg"-io ¡" clue to turbulence. Valuesl I Wk g- usual J. y ptoã.t."t e values of 0.1 Wkg-t . lesult, ancl turbulenc" -< I In connection with e, I -should point out a conceptual consideration. In 3-l] turbulence, e refers to the energy dissipation rate. Iror 2-D turbulence, tìrere is no energy clissipation. In two-dimensional atmospheric turbulence, the process of snaller-eddies ãcting coherently to produce larger ones is irnportant (e.g. tlreory clue to Kraic.hnanr see Phys. Fluids 10, 1417, (1967); J. Atnos. Scl", SS, IS2I, 19z6). It is not".g. correct to regard the energy as being dissipated as ñãát; - the srnallest scales involvecl are much larger than the Kolmogoroff microscale. So in this sense, it may be erroneous to interpretrrÊrr as determined from a spectïum of two-dimensional motions as an "energy dissipation raterr ] As a further point, notice that for transverse motion we ncx¿ have (1a) rd = 6.4 v3 lL This is more compatible \nrith, say, lrleirrstock, J. Atmos. Sci. 35, IO22 (f978) Equation 26, where he suggests t à1..5 'lt -2t or tì - v3 /9, (1b) 'f t: k' , a..4 Other aLlLtrdrs at tintes assume c=v'k-I, or 1(c) e È 6.3 v3 /!, e.g. Zimmerman ;lncl Mr.rrphy, "DynaniÍcal and Chemical Coupling between the neutral and ionized atmospheres" Proc. NATO Advanced SEudy Institute held at SpatirrJ., Norway, L977. P35-41. It will also be noted ín that reference that the formula ¿- = I/3 f. is derivecl (this formula is discussed on P. 37I of the thesis), dRMsD- "- and t*r, refers to the velocity associated with the transition wavenumber between the bouyant and inertlal subranges of turbulence (not the Kol-mogo-roff microscale, as I speculate on P. 37f) . In connectlon with the formula 3 V 'L I should mention another point. At times, Various authors (u.g. Cunnold, Ig: 5) have assumed r'- X/4t¡, \ being the radar wavelength. r made a similar assumption in Chapter II; e. g. pages 9I-99 . I have since realized that this is not valid. True, only eddies of scale - X/4tt will be seen by the radar. However, these wíll be carried by the motions of larger eddles; and if the radar volume ís much larger than the largest eddies, Lhe appropriate scale for 1, is the transítion scale between the inerlial and bouyancy ranges, since horizontal motions are suppresse{ at larger scales. If this is assumed, and we take the transition scale as L" - ei a.-3rz (..g. equation .- 2.2.4.2a, p. 108), we arrive at the f ormula t vro . Th.is is similar Lo the formula on P. 371 of the thesis. Perhaps one of the best derivations of the formula to date is that due to Wanstock, J. Atrnos. Sci., 38, 880-883, (1 9Bl) , who obtains 0.4 v r.,) e = R [.{S ' E¡*q2ô=o àz