ISSN 1063-7729, Astronomy Reports, 2010, Vol. 54, No. 11, pp. 1032–1041. c Pleiades Publishing, Ltd., 2010. Original Russian Text c A.I. Efimov, T. Imamura, K.-I. Oyama, K. Noguchi,! L.N. Samoznaev, A.S. Nabatov, M.K. Bird, I.V. Chashei, 2010, published in Astronomicheski˘ı Zhurnal, 2010, Vol. 87,! No. 11, pp. 1120–1129.

Properties of Solar Wind Turbulence from Radio Occultation Experiments with the NOZOMI Spacecraft A. I. Efimov1,T.Imamura2,K.-I.Oyama3,K.Noguchi4, L. N. Samoznaev1,A.S.Nabatov5,M.K.Bird6,andI.V.Chashei7 1Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, Russia 2Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan 3Plasma and Space Science Center, National Cheng-Kung University, Tainan, Taiwan 4Department of Information and Computer Science, Nara Women’s University, Nara, Japan 5Institute of Radio Astronomy, National AcademyofSciencesofUkraine,Khar’kov,Ukraine 6Argelander Institute of Astronomy, University of Bonn, Bonn, Germany 7Pushchino Radio Astronomy Observatory, Astro Space Center, Lebedev Physical Insitute, Russian Academy of Sciences, Moscow, Russia Received March 31, 2010; in final form, June 8, 2010

Abstract—Radio-sounding experiments using signals from the Japanese NOZOMI spacecraft to probe the circum solar plasma were performed from December 2000 through January 2001. They can be used to obtain information about the properties of the solar wind plasma in the region where it is accelerated at heliocentric distances of 12.8 36.9Rs (where Rs is the radius of the Sun). Measurements of the intensity and frequency of the received− signals were carried out with high time resolution ( 0.05 sforthefrequency and 0.0064 sfortheintensity),makingitpossibletoinvestigatetheanisotropyofinhomogeneitiesand∼ the spatial∼ spectrum of the turbulence of the circum solar plasma. Analysis of these radio-sounding data has shown that the scintillation index and intensity of the frequency fluctuations decrease approximately according to a power law with increasing distance of the line of sight from the Sun. Measurements of the amplitude fluctuations and estimates of the solar wind velocity derived from spatially separated observations indicate the presence of small-scale inhomogeneities with sizes of the order of 50 km at heliocentric distances less than 25Rs,whichareelongatedintheradialdirectionwithanisotropycoefficients from 2.3 to 3.0. The inhomogeneities at heliocentric distances exceeding 30Rs become close to isotropic.

DOI: 10.1134/S1063772910110089

1. INTRODUCTION distances R<15Rs,thisindexdecreasesto3.0– Much information about the parameters of the 3.2 [1, 2]. According to certain as yet unconfirmed solar wind in its acceleration region, which is not data, the small-scale portion of the turbulence spec- accessible to direct measurements, can be inferred trum may be flatter [3, 4]. Analysis of radio-sounding from the analysis of radio signals emitted by space- data has also shown that the electron density in- craft and natural radio sources that pass through the homogeneities are anisotropic, and are elongated in solar wind plasma. Radio astronomy data and data the radial direction. The degree of anisotropy (the from radio physics experiments have been used to find axial ratio of the spatial scales of the plasma density the characteristics of inhomogeneities (turbulence) of fluctuations) depends on heliocentric distance, the the solar wind plasma, which give rise to fluctuations phase of the solar activity cycle, and the size of the of the amplitude, phase, and frequency of such radio inhomogeneities [5, 6]. waves. Radio sounding of the circum solar plasma Additional possibilities for investigating the small- has established that the index of the spatial spectrum scale structure of the solar wind are provided by high- of the plasma inhomogeneities p depends on the he- time-resolution amplitude and phase measurements liocentric distance. In the region where the flow has probing the circum solar plasma carried out with the formed (heliocentric distances exceeding 20Rs,where Japanese NOZOMI spacecraft [7], as well as the Rs is the radius of the Sun), the index p for the large- simultaneous reception of such signals at two ob- scale component (103 106 km in size)varies near serving points and the determination of the solar wind 3.67, close to a Kolmogorov− spectrum. At heliocentric velocity [8].

1032 SOLAR WIND TURBULENCE 1033 The current paper further develops the results of the plasma on the X-band signal propagating from of [7]. Our goal is to determine the characteristics the spacecraft to the reception point. of the turbulence of the solar wind plasma (the spatial An important property of the NOZOMI data is spectrum and level of the turbulence and the degree of their high time resolution, which makes it possible to anisotropy), and to obtain the radial dependences of carry out studies of the spatial spectrum of the tur- the scintillation index and intensity of the frequency bulence of small-scale inhomogeneities in the solar fluctuations based on radio sounding of the solar wind wind. The period of the frequency measurement cycle using signals from the NOZOMI spacecraft. is T =0.0512 s (corresponding to a sampling rate of F =19.531 Hz), while the period of the intensity measurement cycle was a factor of eight shorter (T = 2. CHARACTERISTICS 0.0064 s, F =156.25 Hz). The time resolution used OF THE EXPERIMENTS in the reduction of the NOZOMI data is much higher Radio sounding of the circum solar plasma was than in analogous experiments with the ULYSSES carried out using NOZOMI signals from Decem- (1991, 1995) and GALILEO (1995–2004) space- ber 6, 2000 through January 23, 2001 [7]. The craft, where the period T was 1 s [2]. observations (CR 1970–1972) took place during maximum solar activity: the Wolf number exceeded 3. SPECTRAL ANALYSIS 100, and sunspots encompassed heliolatitudes 40◦ (http://sidc.oma.be/html). The conditions on± the OF THE FREQUENCY FLUCTUATIONS Sun were fairly quiescent: only a few comparatively The spectral density of the fluctuation frequency weak (class M) flares occurred during this period. Gf (ν) is related to the characteristics of the sounded Therefore, we infer that these NOZOMI experiments turbulent plasma by the known expression [2] probed the region of circum solar plasma correspond- 2 2 ν 2 2 (2+α )/2 ing to the streamer belt. The sounding was carried G (ν) ν exp (ν + ν )− f , (1) f ∝ −ν2 0 out as the line of sight both approached the Sun ! m " (east limb) and receded from the Sun (west limb). where ν is the fluctuation frequency, ν = V/L , Seven sessions were conducted from December 6, to 0 0 ν = V/L are the fluctuation frequencies corre- December 28, 2000, as the distance R between the m m sponding to the outer L0 and inner Lm turbulence line of sight and the Sun decreased from 36.9Rs to scales, V is the velocity of the solar wind, and αf is the 12.8Rs.ThedistancesR in two sessions on January 22 and 23, 2001 as the line of sight receded from the index of the fluctuation frequency spectrum, which is the power-law index of the three-dimensional spatial Sun were 15.7Rs and 16.9Rs.Theprobedregions were located near the plane of the solar equator. spectrum of the turbulence p minus three [2]. The NOZOMI experiments were carried out in Using known data on Lm and L0 and assuming a coherent two-way mode. An S-band (wavelength V =300km/s, we can estimate the expected values λ =13.1 cm, carrier frequency f =2.1123 GHz) sig- for the frequencies ν0 and νm.Asfollowsfromradio nal radiated by ground-based systems passed near astronomy data [3] and radio physics experiments [9], the Sun and was then received by systems on the the inner scale, on average, increases with heliocen- tric distance R from 10 km at R =10R to 40 km at spacecraft. On board the spacecraft, the frequency ∼ s was multiplied by the coefficient q =880/221 to form R =30Rs;thefrequencyνm decreases accordingly, from 30 Hz (R =10R )to 7.5 Hz (R =30R ). an X-band (λ =3.6 cm, f =8.4109 GHz) response ∼ s ∼ s signal that was coherent with the original S-band The outer scale also increases in a regular fashion with distance [10], from 2R (R =10R ) to 6R signal; this signal was thentransmittedtowardanob- ∼ s s ∼ s serving point on Earth, where it was received. Fluc- (R =30Rs),whilethefrequencyν0 varies from 2 4 5 ∼ × tuations in the received X-band signal are due to two 10− to 7 10− Hz. The NOZOMI experiments × factors: fluctuations in the S-band signal arising in its have the minimum possible value ν ν0;asfollows % path from the original ground-based radiation point from (1), the spectral density can be represented in the to the spacecraft, multiplied by the coefficient q,and form 2 fluctuations in the X-band signal arising during its (p 3) ν G (ν) ν− − exp − . (2) propagation from the spacecraft to the ground-based f ∝ ν2 reception point. Since the intensity of the frequency ! m " fluctuations due to the presence of plasma is inversely Figure 1 shows four temporal spectra of the proportional to the frequency, the latter of these com- frequency fluctuations for the signals received at ponents is a factor of q2 smaller than the former in the ground-based reception points on December 28, power, and can be neglected. The fluctuations in the 2000 (R =12.8Rs), January 23, 2001 (R =16.8Rs), intensity (or amplitude) are due only to the influence December 18, 2000 (R =23.8Rs), and December 11,

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(a) (b) 100

10–1

10–2 /Hz 2 ), Hz

ν 0 (c) (d)

( 10 f G

10–1

10–2

10–2 10–1 100 101 10–2 10–1 100 101 ν, Hz

Fig. 1. Temporal spectra for the frequency fluctuations for (a) December 28, 2000 (R =12.8Rs, αf =0.34), (b) January 23, 2001 (R =16.8Rs, αf =0.56), (c) December 18, 2000 (R =23.8Rs, αf =0.69), and (d) December 11, 2000 (R =31.5Rs, αf =0.93).

2000 (R =31.5Rs). The presented spectra char- the propagation path. These results agree with data acterize the frequency fluctuations for the S-band obtained using other spacecraft [1, 2]. This is true for signal in the path from the ground-based radiation the growth in αf when perturbed flows of plasma pass point to the spacecraft multiplied by the coefficient through the propagation path (December 11, 2000; q.Thesewereobtainedbyaveraginganensembleof Fig. 1d) and the decrease (on average) of the spectral individual spectra derived by applying a Fast Fourier index of the frequency fluctuations that occurs when Transform (FFT) in successive time segments with the line of sight of the radio signal approaches the Sun durations of 8192 counts covering the entire mea- (December 28, 2000; Fig. 1a). surement session. The data of Fig. 1 show that the spectral density Gf (ν) for fluctuation frequencies An important property of the spectra depicted in 0.002 Hz ν 0.5 Hz can be approximated by a Fig. 1 is the excess spectral density of the frequency ≤ ≤ power law. The approximatingexponentialrelations fluctuations in the interval ν =0.5 3 Hz, compared are shown in Fig. 1 as linear functions on a log– to a power law. The detection of this− effect was possi- log scale. The power-law index (spectral index) for ble thanks to the high time resolution of the frequency these functions αf varies over an appreciable range, measurements. This observed enhancement in the from 0.34 (Fig. 1a) to 0.93 (Fig. 1d). In most cases, spectral density may be due to the influence of am- αf varies around 0.67, corresponding to Kolmogorov plitude fluctuations. At fairly low fluctuation frequen- turbulence. Higher values of αf could be associated cies, where amplitude fluctuations are insignificant, with the passage of coronal mass ejections through the spectral density of the frequency Gf (ν) and phase

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Gϕ(ν) fluctuations are related as [2] fluctuation, when the mean wave field becomes close to zero and the scintillation index equal to unity. G (ν) f Figure 2 shows four modified phase fluctuation Gϕ(ν)= 2 , (3) ν spectra derived for the same sessions, on December making it easy to find the phase spectra from the 28, 2000, January 23, 2001, December 18, 2000, and frequency spectra. However, this relation must be December 11, 2000. At frequencies below 0.1 Hz, the modified at frequencies comparable to the Fresnel fre- modified phase fluctuation spectra are close to power quency, since, in the standard definition of the phase laws and close to the phase fluctuation spectra. At and frequency, frequencies above 0.5 Hz, the spectra are broadened by the amplitude fluctuations. An interesting feature A(t) of the spectra presented on the log–log scale in Fig. 2 ϕ1(t)= i ln , (4) − A exp(iϕ(t)) is that they become nearly parallel at frequencies be- ! 0 " dϕ1 low 0.1 Hz and above 3 Hz. This property is due to 2πf1(t)= 2πf0, the fact that the power-law indices for the amplitude dt − and phase fluctuation spectra are equal at frequencies where ϕ1(t) and f1(t) are the modified phase and above the Fresnel frequency. Figure 3 shows the val- frequency, respectively, A(t) and A0 are the instan- ues of p determined from the frequency fluctuations taneous amplitude of the signal and its mean unper- at 0.002 Hz ν 0.1 Hz as a function of R.On ≤ ≤ turbed value, and f0 is the carrier frequency of the average, the power-law index p is close to 3.67, char- sounding radio signal. Relation (4) can be used to acteristic of Kolmogorov turbulence, with no obvious determine the correlation function of the fluctuations, dependence on the heliocentric distance. Only the value of p in the innermost point of the studied range Bf (τ)=Re f1(t)f1∗(t + τ) (5) ' ( of distances from the Sun ( 13RS)isslightlylower, d2 p =3.38. ∼ = (B (τ)+B (τ)) , dt2 ϕ χ The dependence of the rms frequency fluctuations on the distance of the line of sight from the Sun R where B (τ) and B (τ) are the correlation functions ϕ χ was obtained from our spectral analysis of the data. of the phase ϕ(t) and the level χ(t)=ln[A(t)/A0].In Figure 4 presents the values of σf (R) for the S-band contrast to (3), the temporal spectrum of the fluctua- signal on a log–log scale, obtained by integrating the tion frequency corresponding to (5) is spectral density from the minimum frequency ν to 2 Gf (ν)=ν [Gϕ(ν)+Gχ(ν)], (6) ν =3.0 Hz. The σf (R) can be approximated using a power law: and depends on both the spectra G (ν) and G (ν). ϕ χ b The form of the frequency-fluctuation spectra in σf (R) [Hz] = B(R/Rs)− , (7) Fig. 1 exactly corresponds to (6): at low frequencies, where B =35.80 and b =1.61.Thispower-lawfunc- where the signal-level fluctuations are low compared tion is shown by the straight line in Fig. 4. This value to the phase fluctuations, the spectra are close to of b is in agreement with data obtained using other power-law. The difference of the spectra from power spacecraft, which indicate power-law indices b in the laws at 0.5–3Hzisassociatedwiththeinfluence broad range 1.5–2.5 [12]. of the signal-level fluctuations, whose spectra (see Fig. 5) are close to flat at frequencies below the Fresnel frequency for the S band ( 0.5 Hz). The 4. SPECTRAL ANALYSIS OF THE possibility of a contribution of amplitude∼ fluctuations AMPLITUDE FLUCTUATIONS to the temporal spectra of the modified frequency The signal level is determined by the ratio of the is also supported by estimates of the scintillation amplitudes of the radio waves propagating in the index in the S band. When the X-band scintillation inhomogeneous medium A and in the unperturbed indices presented in Fig. 6 below are recalculated medium A0: to the S band taking into account the expected (p+2)/4 A wavelength dependence, σi λ [11], these χ =ln . (8) correspond to values 1.0 σ∝ 0.36 at heliocentric A0 ≥ i ≥ ! " distances (13 30)RS ;i.e.,theamplitudefluctuations For small fluctuations, the spectral density of the − in the S band are substantial. At the same time, signal-level fluctuation Gχ(ν) can be found from the measurements of the intrinsic frequency fluctuations temporal spectrum of the intensity fluctuations Gi(ν) with averaging times shorterthanthecharacteristic using the simple relation time scale for the fluctuation amplitude become difficult, especially in the case of saturated amplitude Gχ(ν)=0.25Gi(ν). (9)

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106 (‡) (b)

104

102

100

10–2 /Hz 2

), rad 6 (c) (d) ν

( 10 ϕ G

104

102

100

10–2

10–3 10–2 10–1 100 101 10–2 10–1 100 101 ν, Hz

Fig. 2. Temporal spectra of the phase fluctuations for (a) December 28, 2000 (αϕ =2.38), (b) January 23, 2001 (αϕ =2.64), (c) December 18, 2000 (αϕ =2.73), (d) December 11, 2000 (αϕ =2.95).

Figure 5 shows four temporal spectra for the in- temporal spectra for the amplitudes and intensities tensity fluctuations of the X-band signal calculated yields the characteristic frequency νc,givenby[13, from the radio-sounding measurements on Decem- 14] ber 28, 2000 (R =12.8Rs), December 23, 2000 (R = Vapp 18.3Rs), December 18, 2000 (R =23.8Rs), and De- νc = , (10) σprF cember 6, 2000 (R =36.9Rs). The spectra were ob- tained by averaging individual spectra in successive where σp is a weakly varying function of the index p sets of 8192 counts covering the entire session. The (it increases from 2.2 to 2.6 [6] as p varies from 3.0 data presented in Fig. 5 show that the scintillation to 4.0), ∼ is enhanced as the line of sight approaches the Sun, and the spectral density increases. The spectra L1L2 Gi r = λ (11) are informative up to some frequency ν ,abovewhich F L + L 1 # 1 2 they become noisy. Depending on R,thefrequencyν 1 is the radius of the first Fresnel zone, L is the dis- varies in the range 5–15 Hz. 1 tance between the scattering layer containing the Typical spectra of the scintillation amplitudes have inhomogeneities and the spacecraft, L2 is the dis- a flat section at low frequencies and fall off approxi- tance between the scattering layer and the ground- mately according to a power law at high frequencies. based station, and Vapp is the apparent velocity of The location of the intersection of the asymptotes the inhomogeneities through the line of sight (i.e., approximating the low-frequency and high-frequency the apparent velocity of the diffraction pattern). If

ASTRONOMYREPORTS Vol.54 No.11 2010 SOLAR WIND TURBULENCE 1037

p σf, Hz 1.0 0.9 3.9 0.8 0.7 3.8 0.6 σ [Hz] = 35.80R–1.61 0.5 f 3.7 0.4 0.3 3.6 0.2 3.5

3.4 0.10 0.09 3.3 0.08 0 15 20 25 30 35 R/Rs 3.2 15 20 25 30 35 R/Rs Fig. 4. Root-mean-square values σf (points) of the fre- quency fluctuations for the S-band signal for various he- Fig. 3. Index for the three-dimensional spatial spectrum liocentric distances R.Thestraightlineshowstheresult of the turbulence p as a function of heliocentric dis- of a least-squares fit. tance R.

during the entire measurement session. The FFT- the anisotropy parameter ζ =1 and the solar wind 1024 and FFT-128 algorithms were used to find the velocity V is much higher than the sound speed, as spectral density of the signal-level fluctuations and is true at fairly large distances from the Sun, then phase fluctuations, respectively. To increase the ac- Vapp = V [6]. In this case, relation (10) indicates curacy with which the spectral density of the signal- the possibility of determining the solar wind velocity level fluctuations is found from the derived spectra, from the characteristic frequency νc [14]. The ratio we subtracted the mean noise level in the interval of V/V represents an upper limit for ζ,namelyζ fluctuation frequencies ν>ν1. app ≤ V/Vapp [6]. AcomparisonofthecurvesinFig.7showsthat the behaviors of the phase and amplitude spectra are Figure 6 presents the rms values of the intensity indeed similar, and their values are numerically close, fluctuations (or scintillation index) of the X-band sig- although Gϕ >Gχ.Asimilarpictureisobservedfor nal σi as a function of the heliocentric distance R.The the other sessions. Only in one case (January 22, σi values were obtained by integrating the spectral 2001) was the difference between Gχ and Gϕ sub- density Gi from the lowest frequency (ν 0.02 Hz) stantial. This regularly observed difference in the ex- to ν .Thedependenceσ (R) (like σ (R))canalsobe≈ 1 i f perimental values of Gχ and Gϕ can be explained by approximated using a power law of the form (7), with the influence of the sphericity of the radio waves. As B =4.64 and b =1.305;thisfunctionisshownbythe follows from relations derived in [13, 14], the spheric- straight line on the log–log plot in Fig. 6. ity of the waves reduces the dispersion of the fluctu- 2 (2 p)/2 During the propagation of a plane radio wave in a ations σχ by an amount L1/(L1 + L2) − com- turbulent medium, the signal-level and phase spec- pared to the dispersion for a plane wave, which is a tra are identical at high frequencies (ν>νc)[15]. fairly strong dependence on p:thisquantityis1.56 Simultaneous measurements of the amplitude and when p =3.67 and 1.18 when p =2.6.Thesevalues frequency of the sounding signal with a small time are comparable to the ratio Gϕ/Gχ.Forexample, step can be used to verify this theoretical expectation. when ν =3 Hz, the ratio of the spectral densities Figure 7 shows the spectra of the phase fluctuations (Fig. 7) is 1.34. Gϕ (curve 1)andthesignal-levelfluctuations Gχ The presence of so-called Fresnel minima and (curve 2)forX-bandmeasurementscarriedouton maxima located at fluctuation frequencies ν>νc is January 23, 2001. The spectral density of the phase characteristic of the amplitude spectra [14]. In partic- fluctuations was determined from the spectrum of ular, the first Fresnel minimum of the spectral density the received signal by dividing it by q2.Bothspectra is at the frequency are the result of averaging anensembleofindividual Vapp νmin = , (12) spectra obtained in a succession of time segments rF

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–1 Gi(ν), Hz 1

10–2 2

3 –3 10 4

10–4 10–2 10–1 100 101 ν, Hz

Fig. 5. Temporal spectra of the intensity fluctuations of the X-band signal for (1)December28,2000(R =12.8Rs), (2)December23,2000(R =18.3Rs), (3)December18,2000(R =23.8Rs), and (4)December6,2000(R =36.9Rs).

with νmin/νc = δ.Thus,determiningthevalueνmin and durations of the data records. Figure 8 presents from the experimental amplitude spectrum Gχ(ν),we two amplitude spectra derived from measurements can find the velocity Vapp,thenusetheratioV/Vapp on December 28, 2000 (Fig. 8a) and December 23, to derive an upper limit on the degree of elongation of 2000 (Fig. 8b), in which the first Fresnel minimum the inhomogeneities in the radial direction. and maximum are visible. Both spectra were obtained using the FFT-2048 algorithm, via the averaging of The Fresnel minima and maxima are weakly ex- pressed, and are not observed in spectra averaged over eight data records. By means of illustration, curves alargenumberofdatarecords,ascanbeseenfrom approximating the low-frequency and high-frequency the spectra presented in Figs. 5 and 7. The Fresnel parts of the spectrum are shown in Fig. 8a. These extrema can be made to appear by varying the number

–1 Gχ, Gϕ, Hz σi 0.20 1 0.18 –3 0.16 10 2 0.14 0.12 –1.305 σi = 4.638R 0.10 10–4 0.08

0.06 10–5

0.04 10–6 1 2 3 4 5 6 7 8 9 15 20 25 30 35 R/Rs ν, Hz

Fig. 6. Root-mean-square values σi (points) of the in- Fig. 7. Comparison of the temporal spectrum of the phase tensity fluctuations for the X-band signal for various he- fluctuations (curve 1)andthespectrumofthesignal- liocentric distances R.Thestraightlineshowstheresult level fluctuations (curve 2)forX-bandmeasurementson of a least-squares fit. January 23, 2001.

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–1 Gχ(ν), Hz 10–2 (‡)

10–3

10–4

10–2 (b)

10–3

10–4

10–1 100 101 102 ν, Hz

Fig. 8. Spectra of the signal-level fluctuations for (a) December 28, 2000 (νmin =2.05 Hz) and (b) December 23, 2000 (νmin =2.23 Hz). intersect at ν = ν 0.84 Hz. The Fresnel minimum ν using (10). The value of σ was taken to be 2.2 (the c ≈ c p is observed at νmin 2.05 Hz, so that νmin/νc 2.4. theoretical value for p =3). Figure 9 presents spectra ≈ ≈ Using the fact that rF =56km together with formula derived from measurements made on December 11, (12), we can find the velocity of the diffraction pattern: 2000 (Fig. 9a) and December 6, 2000 (Fig. 9b). These Vapp =115km/s. spectra were obtained using the FFT-1024 algorithm To obtain an upper limit for the anisotropy pa- by averaging ensembles of temporal spectra corre- rameter ζ,weuseddataonthevelocityofthesolar sponding to a large number of data records (more wind obtained for six days in December 2000 (De- than 500) during the entire session and subtracting cember 6, 11, 18, 23, 27, and 28) via observations the spectral density for the noise fluctuations. This of the amplitude fluctuations made at two spatially method was adopted to obtain the maximum possi- separated points [8]. In particular, we find for De- ble accuracy in Gχ and the frequency νc.Weused the spectra shown in Fig. 9 to find the values of cember 28, 2000 the velocity V 270 km/s, and ν and V .ForthemeasurementsofDecember6, accordingly ζ =2.35.Valuesforζ ≈were found in the c app same way for December 18, 2000 (ζ =2.65), Decem- 2000, νc =2.8 Hz and Vapp =330km/s, while νc = ber 23, 2000 (ζ =2.64), and December 27, 2000 (ζ = 3.05 Hz and Vapp =375km/s for the measurements 2.96). Thus, the anisotropy parameter ζ varies in the of December 11, 2000. Both velocities Vapp are close range 2.35 ζ 2.96 in the interval of heliocentric to the corresponding solar wind velocities [8], indicat- distances (12≤.8 ≤23.8)R . ing that the anisotropy parameter ζ is close to unity. − s As the line of sight recedes from the Sun, the Fres- nel extrema become less clearly expressed, and they 5. CONCLUSION are essentially indistinguishable in the spectra derived for December 6, 2000 and December 11, 2000, when Our analyses of the temporal spectra for phase, R>30Rs.Inthesetwocases,theanisotropyparam- amplitude, and frequency fluctuations of radio- eter ζ is determined from the characteristic frequency sounding signals detectedinhigh-time-resolution

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–1 Gχ(ν), Hz (‡) 10–4

10–5

10–6

10–7

10–8 (b) 10–4

10–5

10–6

10–7

10–8 0.1 1 10 100 ν, Hz

Fig. 9. Spectra of the signal-level fluctuations for (a) December 11, 2000 (νс =3.05 Hz) and (b) December 6, 2000 (νс =2.80 Hz). measurements with the NOZOMI spacecraft have of the turbulent solar wind plasma is close to Kol- enabled us to study the turbulence of the small-scale mogorov at heliocentric distances from 15Rs to 30Rs, component of the solar wind plasma. possibly becoming flatter closer to the Sun. The degrees of elongation of small-scale plasma We have increased the accuracy with which we density inhomogeneities at heliocentric distances determine the characteristics of the turbulence of the of (13 25)R are 2.3–3. The inhomogeneities are sounding plasma using a method based on averag- s isotropic− at distances exceeding 30R ,inagreement ing an ensemble of spectra corresponding to a large s with the results of radio astronomy observations [16– (about 500) number of data records. We subtracted 18]. the spectral density of the noise fluctuations from the total spectral density of the amplitude fluctuations for We have obtained the radial dependences of the the sounding signals probing the solar wind. rms fluctuations of the frequency and the signal level at heliocentric distances (13 37)Rs.Boththesede- We also tested a method for finding the degree of pendences can be approximated− by power laws with anisotropy of the plasma density inhomogeneities and indices of 1.6 and 1.3,respectively. their elongation in the radial direction. This method − − is based on the relationship between the solar wind ACKNOWLEDGMENTS velocity derived from observations of the signal-level fluctuations at two spatially separated observation This work was supported by a collaboration be- points and the position on the frequency axis of the tween the Russian Foundation for Basic Research Fresnel minimum of the spectral density of the signal and the Deutsche Forschungsgemeinschaft (DFG) level. (project 09-02-91337) the Russian Foundation for Basic Research (project 10-02-00078a), and the Ba- The behavior of the temporal spectra of the fre- sic Research Program of the Division of Physical Sci- quency and phase fluctuations of the NOZOMI sig- ences of the Russian Academy of Sciences “Plasma nals testifies that the index of the spatial spectrum Processes in the Solar System”.Theauthorsare

ASTRONOMYREPORTS Vol.54 No.11 2010 SOLAR WIND TURBULENCE 1041 thankful to M.A. Livshits for drawing their attention 10. A. I. Efimov, I. V. Chashei, L. N. Samoznaev, et al., to the specificcharacterofconditionsontheSun Astron. Zh. 79,640(2002)[Astron.Rep.46,579 during the measurement period. (2002)]. 11. A. I. Efimov, N. A. Armand, L. A. Lukanina, et al., Radiotekh. i Elektron. 53,1257(2008).[J.Commu- REFERENCES nic. Technol. Electr. 53,1186(2008)]. 1. R. Woo and J. W. Armstrong, J. Geophys. Res. 84, 12. A. I. Efimov, L. N. Samoznaev, M. K. Bird, et al., Adv. 7288 (1979). Space Res. 42,117(2008). 2. N. A. Armand, A. I. Efimov, L. N. Samoznaev, et al., 13. N. A. Armand, A. I. Efimov, and O. I. Yakovlev, Prob- Radiotekh. i Elektron. 48,1058(2003).[J.Commu- nic. Technol. Electr. 48,970(2003)]. lems of Contemporary Radio Engineering and 3. W. A. Coles and J. K. Harmon, Astrophys. J. 337, Electronics (Nauka, Moscow, 1987) [in Russian]. 1023 (1989). 14. O. I. Yakovlev, Propagation of Radio Waves in 4. G. L. Tyler,J. F.Vesecky,M. A. Plume, H. T. Howard, Space (Nauka, Moscow, 1985) [in Russian]. and A. Barnes, Astrophys. J. 249,318(1981). 15. R. Woo, Astrophys. J. 201,238(1975). 5. J. W. Armstrong, W. A. Coles, S. R. Spangler, et al., 16. K. R. Grall, W.A. Coles, S. R. Spangler, et al., J. Geo- Astrophys. J. 358,685(1990). phys. Res. 102,263(1997). 6. A. I. Efimov, V. K. Rudash, M. K. Bird, et al., Adv. Space Res. 26,785(2000). 17. Y. Yamauchi, M. Tokumaru, M. Kojima, et al., J. Geo- 7. T. Imamura, K. Noguchi, A. Nabatov, et al., Astron. phys. Res. 103,6571(1998). Astrophys. 439,1165(2005). 18. I. V. Chashei, A. I. Efimov, V. K. Rudash, and 8. M. Tokumaru, K. Fujiki, H. Higashiyama, et al., un- M. K. Bird, Astron. Zh. 77,713(2000)[Astron.Rep. published manuscript (2004). 44,634(2000)]. 9. O. I. Yakovlev, A. I. Efimov, and S. N. Rubtsov, Astron. Zh. 65,1290(1988)[Sov.Astron.32,672 (1988)]. Translated by D. Gabuzda

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