JUNE 2004 NOTES AND CORRESPONDENCE 925
NOTES AND CORRESPONDENCE
A Method to Determine the Capping Inversion of the Convective Boundary Layer
G. RAMPANELLI AND D. ZARDI Dipartimento di Ingegneria Civile ed Ambientale, UniversitaÁ degli studi di Trento, Trento, Italy
11 July 2003 and 20 December 2003
ABSTRACT A simple mathematical algorithm is proposed to decipher the thermal structure of the convective boundary layer by means of best-®t analysis of soundings or airborne measurements with a smooth ideal pro®le. The latter includes a constant-potential-temperature mixed layer, a strongly strati®ed entrainment layer, and a constant- lapse-rate free atmosphere. The resulting pro®le depends on ®ve parameters amenable, through simple mathe- matical relationships, to physical variables de®ning the vertical structure of the layers. The method allows objective evaluation of parameters involved in the test pro®le and easy comparison of measurements with theoretically expected structure.
1. Introduction However, these theoretical results are not easily com- pared with either high-resolution model output (Khanna Various simpli®ed schemes have been proposed in and Brasseur 1997) or ®eld observations (Boers and the literature to model the vertical pro®le of potential Eloranta 1986; Cohn and Angevine 2000). A method temperature in the situation illustrated in Fig. 1. The for making such a comparison has been recently pro- latter reproduces the thermal structure of a convective posed using airborne lidar data (Davis et al. 2000). Like- boundary layer (CBL): the mixed layer (ML) below is wise four different criteria to identify the CBL structure connected to the stably strati®ed free atmosphere (FA) from large-eddy simulation models have been suggested above through the entrainment layer (EL) or ``interfacial layer'' (Deardorff 1979). Key parameters to describe by Sullivan et al. (1998). this structure are the potential temperature in the ML On the other hand, making a precise estimate of many quantities, such as boundary layer depth (Vogelezang m, inversion height h 0, entrainment-layer depth ⌬h, inversion strength ⌬, and lapse rate in the free at- and Holtslag 1996; Gryning et al. 1997), is a crucial mosphere ␥. step not only for applications (e.g., estimate of mixing Previous models (e.g., Deardorff 1979) capture the es- height for pollutant transport) but also for evaluating sential overall dynamics and diurnal evolution of the CBL theoretical similarity solutions as these quantities enter without explicit reference to turbulence at smaller scales. as scaling variables (Stull 1988; Johansson et al. 2000). In particular, the EL is modeled either as a layer dis- In order to describe the vertical structure of a CBL playing constant stability with rapid temperature variation from a sounding or model output in terms of the con- between the ML and the FA (``®rst-order jump;'' cf. Betts ceptual model sketched in Fig. 1, a best-®t analysis with 1974; Deardorff 1979), or as a sharp temperature dis- a simple test vertical pro®le (such as the curve in Fig. continuity (``zero-order jump;'' cf. Ball 1960; Tennekes 1) may seem like a reasonable procedure. However, the 1973; Betts 1973; Carson 1973; Carson and Smith 1974; usual least squares approach, using a piecewise constant Driedonks 1982). This approach allowed for a more com- test curve where both heights (hs, h 0, h 2) and thermal prehensive investigation of the various processes gov- structure parameters (⌬, ␥) are unknown quantities, erning the diurnal evolution of an inversion-capped CBL does not lead to an analytical solution, but rather it (Zilitinkevich 1975; Tennekes 1973; Deardorff 1979; Fe- requires an iterative search. It can be shown that in many dorovich and Mironov 1985). cases multiple combinations of the parameters may get close to a minimum scatter of data around the test pro- ®le, but the solution producing the absolute minimum Corresponding author address: Dino Zardi, UniversitaÁ degli studi may not provide the most reasonable result from a phys- di Trento, Dipartimento di Ingegneria Civile ed Ambientale, via Mes- iano 77, I-38050 Trento, Italy. ical viewpoint. On the contrary, a smooth test curve E-mail: [email protected] displaying regular merging of each layer into the ad-
᭧ 2004 American Meteorological Society
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FIG. 2. (a) The smoothed vertical pro®le of potential temperature and the symbols used in the present paper, associated with (b) the vertical structure of the sensible heat ¯ux.
pro®le displays a minimum (Fig. 2). Such a structure may be easily reproduced by means of a linear com- bination of functions in the form
FIG. 1. Sketch of a vertical pro®le of potential temperature in the convective boundary layer developing over ¯at uniform terrain (z) ϭ m ϩ af() ϩ bg(), (2)
[adapted from Garratt (1992); notation for h0 and h2 follows Deardorff (1979)]. where
z Ϫ l jacent ones, not only allows for a partly analytical so- ϭ (3) lution of the minimization problem, but is also more c⌬h likely to avoid ambiguity. Furthermore, as remarked by Deardorff (1979), a piecewise linear pro®le is a very is the vertical displacement from a reference height l rough approximation of the smooth transition from each (to be related later to the CBL height) scaled with the layer to those adjacent to it, because it leaves out various EL depth ⌬h and c is a constant (to be speci®ed later). ®nescale features. A similar strategy has been followed The structure of f and g is sketched in Fig. 3: the func- by Fitzjarrald and Garstang (1981) and more recently tion f provides the rapid transition from the constant by Steyn et al. (1999) for the analysis of lidar backscatter value m,as K 0, to a value of at the top of the pro®les. EL, matching the FA above, while the function g pro- In the present paper the smooth-test-curve approach vides the correct asymptotic behavior in accordance is adopted (section 2) and relationships between math- with (1) as k 0. ematical parameters de®ning the curve and the physical Accordingly, the following constraints have to be sat- variables de®ning the CBL are determined. Tests with is®ed: real data are provided (section 3), along with a discus- sion of the results and possible re®nements of the meth- od (section 4).
2. Outline of the method Following Deardorff (1979), the lowest part of a hor- izontally homogeneous CBL (apart from the surface lay- er region) can be represented as a layer displaying a constant potential temperature m up to a height h 0, then a rapidly increasing pro®le in the EL and asymptotically approaching, above a height h 2, the free atmosphere,
000(z) ϭ ϩ ␥z, (1) where 00 is a constant value of potential temperature at reference height (z ϭ 0) and ␥ is the constant vertical FIG. 3. Sketch of the basic functions f and g used in (2) to obtain gradient. The height h1 is where the vertical heat ¯ux the vertical pro®le.
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the real pro®le at the in¯ection point (Fig. 4b). A third criterion is based on evaluation of the heights at which the best-®t pro®le is close enough either to the ML
constant value m (h 0), or to the FA (h 2) (Fig. 4c). In the following, the latter criterion will be adopted.
The upper (h 2) and lower (h 0) limit of the EL can be calculated by imposing that the value of at that height must be very close to the asymptotic value, namely,
(h2002) ഠ ϩ ␥h and (11a) FIG. 4. Summary of criteria for the identi®cation of standard me- teorological parameters: (a) shaded areas are equal as a consequence (h0) ഠ m. (11b) of energy budget (Driedonks 1982), (b) linear potential temperature pro®le displaying the same lapse rate as the gradient of the real pro®le This can be obtained by setting at the in¯ection point, and (c) evaluation of heights at which the best-
®t pro®le is close enough either to the ML constant value m (h0)or h2 ϭ l ϩ c⌬h and (12a) to the FA (h2). h0 ϭ l Ϫ c⌬h, (12b) ⇒ where is a parameter related to the depth of the en- lim () ϭ m lim f (), g() ϭ 0, (4) →Ϫϱ →Ϫϱ trainment layer, setting the accuracy within which (11a) and (11b) are satis®ed. For large values of , the dif- lim f () ϭ 1, and (5) ferences between (h 2) and the asymptotic expression →ϩϱ in the mixed layer and in the free atmosphere are very dg small, but this is veri®ed only at heights very high in lim ϭ 1. (6) the FA or very close to the ground level. On the other →ϩϱ dz hand, low values of lead one to estimate a very thin As will be clear later, the limit value of 1 in (5) and (6) EL, with ⌬h vanishing as approaches 0. On the basis is chosen for convenience, but any constant value would of many applications of expression (2) to real data, a be acceptable. Asymptotic matching with the free at- value of ϭ 1.5 is recommended as a good compromise. mosphere requires, using (1), (5), and (6), This leads to a very strict ful®llment of (11a) and (11b). The actual depth of the EL can be calculated from (12a) d 1 df dg lim ϭ lim a ϩ b ϭ ␥, (7) and (12b) as →ϩϱ dz→ϩϱ c⌬hd d ⌬h ϵ h20Ϫ h ϭ 2c⌬h, (13) which implies which implies c ϭ 1/(2) ϭ 1/3. b ϭ c␥⌬h. (8) The relationships between the parameters a, b, and l in (2) and (3), and physical quantities resulting from the Among the many possible analytic functions satisfying above reasoning, are summarized in the following equa- the above requirements, consider the following expres- tions: sions: 2b tanh() ϩ 1 ␥ ϭ , (14) f () ϭ and (9a) ⌬h 2 h ϭ l Ϫ⌬h/2, and (15) ln[2 cosh()] ϩ 0 g() ϭ . (9b) 2 2b 00 ϭ a ϩ mmϪ ␥l ϭ a ϩ Ϫ l. (16) Equations (1), (2), (9a), and (9b), along with the re- ⌬h quired asymptotic behavior of (z) in the FA, imply The evaluation of the inversion strength requires some explanation, because it has been variously de®ned in ϭ ϩ a Ϫ ␥l. (10) 00 m the literature. The potential temperature jump (⌬)
To link the parameters a, b, m, l, and ⌬h to all of the across the EL is physical variables identifying the atmospheric-layer ⌬ ϭ ϩ ␥h Ϫ ϭ a ϩ b. (17) structure as shown in Fig. 1, some criteria have to be 00 2 m chosen to compare the smooth pro®le of (2) with the Betts (1974) suggests for the inversion strength the ex- usual conceptual schemes (Fig. 2). A ®rst criterion based pression on the energy budget (Driedonks 1982) requires that the ⌬Јϭ ϩ ␥h Ϫ , (18) shaded areas in Fig. 4a be equal. A second criterion is 00 1 m based on the averaged stability in the EL: accordingly, assuming h1 to be both the height where the mixed layer the best approximation is a linear potential temperature is upper bounded by the EL (h 0 in the present paper), pro®le displaying a lapse rate equal to the gradient in and the height where turbulent heat ¯ux displays a min-
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FIG. 6. An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the Adige valley, near the village of Besenello, south of the city of Trento (Italy) during the measurement ¯ights. (a) between 0930 and 0946 LST 1 Oct 1999 and (b) between 0935 and 0948 LST 26 May 1999.
N S ϭ [ Ϫ ( )]2 (21) ii 1 FIG. 5. Sketch of the case of pure encroachment and its interpretation with (17) and (19). to be a minimum; here, i is the value measured at height i ϭ (zi Ϫ l)/c⌬h, (i) is the value of potential tem- perature calculated from (2) at the same height, and N is the total number of data points. In particular, by min- imum. In fact, Deardorff (1979) shows that in general imizing S the optimal values of a, b, and m can be the two heights are different. Accordingly in the fol- related analytically to the values of l and ⌬h (see the lowing, h 0 will denote the height of the inversion base, appendix). whereas h1 will denote the heat ¯ux minimum height (Fig. 2). Straightforward application of (18) in the present 3. Application caseÐin which a smooth temperature pro®le is adopted, The method presented above has been applied to data using h 0 instead of h1 (as stated above)Ðleads to collected through airborne measurements in the atmo- spheric boudary layer, allowing for a very easy deter- ⌬Јϭ 00ϩ ␥h 0 Ϫ m ϭ a Ϫ b. (19) mination of the parameters identifying the atmospheric However, (19) would produce unphysical negative val- thermal structure at the sites of interest. Data have been collected in alpine valleys using an equipped motor glid- ues for ⌬Ј (see Fig. 5) in the limiting case of en- croachment (cf. Garratt 1992, p. 151; Stull 1988, p. er during various measurement ¯ights in the area sur- rounding the city of Trento in the Alps (northern Italy). 454). According to Deardorff (1979), h1 is roughly half- way between h and h . Thus, an optimal compromise Further details on the instruments and the measurements 0 2 can be found in de Franceschi et al. (2003). should be reached by estimating h1 ϭ (h 0 ϩ h 2)/2 ϭ l, and consequently The vertically spiraling path performed by the motor glider produced an essentially vertical sounding. A few major deviations from a standard vertical sounding are ⌬Јϭ 00ϩ ␥h 1 Ϫ m ϭ a. (20) related to horizontal displacements from a strictly ver-
Thus, the limiting case of encroachment (i.e., ⌬/␥h1 tical ascent and to the occurrence of signi®cant cross- ഠ 0; cf. Deardorff 1979) is recovered for vanishing a, valley temperature gradients, in connection with differ- when the only contribution of the curve g()in(2) ent sidewall exposure to incoming solar radiation and survives (see Figs. 3 and 5). The case of pure encroach- related thermally driven ¯ows (cf. Whiteman 1990). In ment will occur as a ϭ 0, as a limiting case that is rarely spite of these drawbacks the method allows for ef®cient met in real data. For this reason a value of a ϭ 0.2 may retrieval of a horizontally averaged basic vertical struc- be suggested as an upper limit for an encroachment ture and provides the ®rst step for subsequent analysis condition. of local cross-valley perturbations, as shown in Ram- The estimate of the parameters in (2) can be obtained panelli and Zardi (2000, 2002). by the usual least squares ®t to the sounding data, re- In Figs. 6±9 vertical pro®les of potential temperature quiring the functional data from diurnal survey ¯ights are shown. Figures 6,
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FIG. 7. An example of the analysis of airborne data using the FIG. 9. An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the Adige valley, method outlined in the paper. Data were collected during the mea- near the city of Trento (Italy), during the measurement ¯ights (a) surement ¯ight (a) between 1217 and 1229 LST 24 Oct 1998 near between 1023 and 1039 LST 2 Jul 1997 and (b) between 1039 and the town of Riva del Garda, in the lower Lakes Valley, about 30 km 1104 LST 22 Dec 2000. southwest of the city of Trento (Italy), and during the measurement ¯ight (b) between 0932 and 0958 LST 8 Jul 1999 in the Adige valley, near the city of Bolzano (Italy). 7, and 8 display examples of strong and deep entrain- ment layers capping a shallow mixed layer. These are consistent with what is usually found in deep mountain and h 2 is only about 100 m higher. This suggests that valleys. A detailed analysis of physical mechanisms the overall pro®le is characterized by a single layer, governing the diurnal evolution of thermal structure associated with a typical stable condition of a ground- within valleys can be found in Whiteman (1990) and based inversion. The values of the parameters h 0, h 2, Whiteman et al. (1996). Figures 9a,b show how the ⌬h, m, 00, ␥, ⌬, and ⌬Ј obtained from the analysis method performs under ``worst'' cases, such as (a) of these data are reported in Tables 1 and 2. In the same ``anomalous'' CBL development (the so-called en- tables the correlation coef®cient of the best ®t is also croachment) and (b) a ground-based inversion. Both shown. The general mean vertical structure provided by cases are relatively well captured using the proposed the dataset is well captured by the resulting continuous method. In fact, for Fig. 9a the algorithm produces a pro®le and the values of parameters de®ning the vertical very small inversion strength (⌬ЈϭϪ0.09 K), iden- structure appear to be in a reasonable range. tifying this case as an encroachment, and the transition Whenever vertical pro®les of other variables, such as zone between h 0 and h 2 is well localized and meaning- water vapor content q, are measured simultaneously ful. In Fig. 9b the ¯ight was performed in the early with the thermal pro®le, a similar approach can be fol- morning: the height h 0 turns out to be less than the lowed to design a shape function, such as (2), for q. height of the lowest measurement point of the pro®le, This function, speci®cally shaped for the water vapor content structure, can be used to ®t the q data and re- cover the strati®cation parameters that can be inferred from its pro®le. A separate ®t for each pro®le ( and q) is likely to produce different estimates of the same
parameters, such as h 0 and h 2. To overcome this problem and to strengthen the estimate of the parameters, a si- multaneous ®t is recommended, possibly by minimizing
TABLE 1. Parameters of the vertical pro®les obtained from data shown in Figs. 6±7. Variable Fig. 6a Fig. 6b Fig. 7a Fig. 7b ⌬h (m) 1070 644 272 1254
h0 (m) 585 517 628 525
h2 (m) 1555 1161 900 1780
m (K) 290.1 292.1 295.6 275.7 (K) 295.6 292.9 296.1 281.4 FIG. 8. An example of the analysis of airborne data using the 00 ␥ (K kmϪ1) 1.41 1.45 0.88 5.38 method outlined in the paper. Data were collected in the upper Lakes ⌬ (K) 7.63 2.53 1.21 15.36 Valley, near the village of Terlago, west of the city of Trento (Italy), ⌬Ј (K) 6.79 2.07 1.09 11.98 during the measurement ¯ight (a) between 1248 and 1304 LST 9 Sep R2 0.996 0.990 0.976 0.980 1998 and (b) between 1420 and 1452 LST 23 Sep 2001.
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TABLE 2. Parameters of the vertical pro®les obtained from data shown in Figs. 8±9. Variable Fig. 8a Fig. 8b Fig. 9a Fig. 9b ⌬h (m) 581 375 259 105 h0 (m) 961 1435 574 330 h2 (m) 1542 1811 833 436
m (K) 299.6 303.5 292.3 299.2
00 (K) 298.6 305.2 288.9 299.5 ␥ (K kmϪ1) 2.09 3.29 4.86 2.02 ⌬ (K) 2.19 2.23 0.71 1.16 ⌬Ј (K) 1.58 1.75 0.09 1.05 R2 0.978 0.980 0.989 0.976 the sum of the two functionals that calculate the distance of the theoretical pro®les to the data. Obviously, each FIG. 11. An example of the analysis of airborne data using the of these functions has to be normalized in order that method of the coupled ®t using the vertical pro®les of both and q. there be an equal effect of the information associated Vertical pro®les of q are shown: data (black dots), interpretation using with each pro®le (see the appendix). A comparison be- single ®t to the q data (thin line), and interpretation using ±q coupled tween the method of the coupled ®t and the method of ®t (thick line). Data were collected during the measurement ¯ight of the single ®t is shown in Figs. 10 and 11, and the results (a) Fig. 6a, and during the measurement ¯ight of (b) Fig. 8b. are given in detail in Tables 3 and 4. Notice that the differences between the results produced by the two pression. The adjustable parameters of this expression procedures are very small. However, the coupled meth- are amenable to the atmospheric variables generally od uses a larger number of data points to estimate the used for the description of the strati®cation (i.e., inver- inversion parameters l and ⌬h, and this obviously pro- sion height, entrainment depth, mixed layer potential duces a more stable and reliable result. For this reason, temperature, and free-atmosphere lapse rate). when a coupled series of measurements ( and q) are Application of the technique to real data produces available, the application of the coupled procedure is encouraging results. Furthermore, the conditions used recommended. for obtaining the suggested expression of the vertical pro®le are very general, and can be adopted to calculate 4. Summary other expressions for f() and g() as well. As a ®nal comment, we note that the suggested meth- A new technique to obtain the vertical structure of od assumes the database to be reasonably amenable to potential temperature from data collected within and the basic structure of a capping inversion because it is above a CBL using light airplanes or vertical soundings most commonly found. More complicated structures, has been introduced. The technique consists of a least such as multiple inversions, could be only poorly re- squares ®tting of data to a user-de®ned analytical ex- produced by the method. Possible application of the proposed pro®le in simpli®ed models for the diurnal evolution of the mixing height (Seibert et al. 2000) could improve the method introducing a more realistic capping inversion structure. An integration with similar methods for the identi®cation of the CBL upper structure from other kinds of measurements (lidar, wind pro®lers, so-
TABLE 3. Parameters of the vertical pro®les obtained from data shown in Fig. 10, and comparison between the single- and coupled- ®t procedure. Fig. 10a Fig. 10a Fig. 10b Fig. 10b Variable single ®t coupled ®t single ®t coupled ®t ⌬h (m) 1070 1083 375 310
h0 (m) 585 496 1435 1513
h2 (m) 1555 1579 1811 1823 (K) 290.1 289.8 303.5 303.5 FIG. 10. An example of the analysis of airborne data using the m (K) 295.6 294.9 305.2 305.4 method of the coupled ®t using the vertical pro®les of both and q. 00 ␥ (K kmϪ1) 1.41 1.70 3.29 2.49 Vertical pro®les of are shown: data (black dots), interpretation using ⌬ (K) 7.63 7.77 2.23 2.28 single ®t to the data (thin line), and interpretation using ±q coupled ⌬Ј (K) 6.79 6.93 1.75 1.90 ®t (thick line). Data were collected during the measurement ¯ight of R2 0.996 0.995 0.980 0.976 (a) Fig. 6a, and during the measurement ¯ight of (b) Fig. 8b.
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ץ TABLE 4. Parameters of the vertical pro®les obtained from data N shown in Fig. 11, and comparison between the single- and coupled- [ Ϫ ( )] ( ) ϭ 0, (A3a) ii i mץ t procedure. 1® ץ Fig. 10a Fig. 10a Fig. 10b Fig. 10b N [ Ϫ ( )] ( ) ϭ 0, (A3b) Variable single ®t coupled ®t single ®t coupled ®t iii aץ 1 ⌬h (m) 1321 1083 284 310 N ץ h0 (m) 274 496 1544 1513 h (m) 1595 1579 1828 1823 [iiiϪ ( )] ( ) ϭ 0, (A3c) 2 bץ Ϫ1 1 qm (g kg ) 7.5 7.2 9.7 9.7 q (g kgϪ1) 5.0 4.4 6.5 6.8 N ץ 00 q (g kgϪ1 kmϪ1) Ϫ0.60 Ϫ0.36 Ϫ0.19 Ϫ0.22 [iiϪ ( )( i) ϭ 0, and (A3d) Ϫ1 h⌬ץ q (g kg ) Ϫ3.3 Ϫ3.3 Ϫ3.54 Ϫ3.51 1⌬ ⌬qЈ (g kgϪ1) Ϫ3.1 Ϫ3.1 Ϫ3.27 Ϫ3.17 N ץ R2 0.994 0.990 0.929 0.928 [ Ϫ ( )] ( ) ϭ 0. (A3e) iii lץ 1 dar, etc.) is possible, in order to obtain a more accurate Note that the terms (A3d) and (A3e) depend on the knowledge of the physical variables. speci®c structure of the functions g and f, while by the de®nition provided in (2) we have Acknowledgments. Special thanks go to Massimiliano ץ de Franceschi, for providing the experimental setup of (i) ϭ 1, (A4a) mץ -the motor glider and technical support for the measure ments, and to the motor glider pilots, Andrea Ferrari ץ and Fabrizio Interlandi. This work has been partly sup- ( ) ϭ f ( ), and (A4b) a iiץ -ported by the Provincia Autonoma di Trento under Pro ץ -ject PAT-UNITN 2001, and by granting a leave of ab sence to G. Rampanelli for completing his Ph.D. pro- (ii) ϭ g( ), (A4c) bץ gram. This work has been also partly supported by the Italian National Institute for Scienti®c and Technolog- therefore, (A4a)±(A4c) become simply ical Research on the Mountain under the program INRM2000. N [ Ϫ ( )] ϭ 0, ii 1 APPENDIX N [ Ϫ ( )] f ( ) ϭ 0, and Minimizing the Functional S iii 1
For the estimation of the parameters a, b, m, l, and N ⌬h in (2), the usual least squares method requires that [ Ϫ ( )]g( ) ϭ 0, (A5) iii the function 1 N S ϭ [ Ϫ ( )]2 (A1) which upon substitution from (2) and rearrangements ii 1 gives
NNN be a minimum, where i is the value measured at height , ( ) is the value of the vertical pro®le at the same miiiN ϩ af( ) ϩ bg( ) ϭ , i i height, and N is the total number of data. The minimum 111 requirement can be imposed by setting NN N N f ( ) ϩ af2( ) ϩ bg( ) f ( ) mi i ii ץ [ Ϫ ( )]2 ϭ 0, 11 1 ii 1ץ m N N ϭ f ( ), and ii ץ [ Ϫ ( )]2 ϭ 0, 1 ii a 1ץ NN N N 2 (mig( ) ϩ af( ii)g( ) ϩ bg( i ץ [ Ϫ ( )]2 ϭ 0, 11 1 ii b 1ץ N N (ϭ g( ), (A6 ץ [ Ϫ ( )]2 ϭ 0, and ii ii 1 h 1⌬ץ N which is a linear symmetric algebraic system in the ץ [ Ϫ ( )]2 ϭ 0; (A2) variables a, b, and . Once the system is solved, a, b, ii m l 1ץ and m are directly related to l and ⌬, which are the then only free parameters left to be varied to minimize S.
Unauthenticated | Downloaded 10/01/21 06:31 AM UTC 932 JOURNAL OF APPLIED METEOROLOGY VOLUME 43 qץ In the case of a coupled ®t using vertical pro®les of N [q Ϫ q( )] ( ) ϭ 0, (A10d) potential temperature and water vapor content, the func- ii i qץ 1 tion to be minimized is m N qץ 22 NN [qiiϪ q( )] ( i) ϭ 0, (A10e) iiϪ ( ) q iiϪ q( ) aץ 1 S ϭϩ, (A7) q 11⌰ Q [][] N qץ [q Ϫ q( )] ( ) ϭ 0, (A10f) where qi is the q value measured at height i, q(i)is ii i bץ 1 the value of the vertical pro®le of water vapor content q at the same height, ⌰ and Q are two suitable scaling N ץ [( iiϪ (] factors (suggested values are the span of and of q in (i) 2 h⌬ץ⌰ 1 the vertical pro®le under analysis). According to Stull qץ [( and Garratt (1992), the shape function of the N [q Ϫ q( (1988) ϩ ii( ) ϭ 0, and (A10g) vertical pro®le of water vapor content can be assumed 2 i h⌬ץ Q 1 in the form N ץ [( iiϪ ()] q(z) ϭ qmqϩ af() ϩ bg q(), (A8) ( ) 2 i lץ⌰ 1 with the functions f and g identical to those used for N qץ [( the pro®le. Obviously, the linear behavior of (A8) in [qiiϪ q( ϩ ( ) ϭ 0. (A10h) the FA can produce negative values of water vapor when 2 i lץ Q 1 z is very large, but (A8) can be used as a linear ap- proximation of the vertical pro®le for the lower region The terms (A10g) and (A10h), as in the single ®t pro- of the FA. The minimum requirement of S can be im- cedure, depend on the speci®c structure of the functions posed by setting g and f, but even in this case we have N ץ qץ qץ 2 [iiϪ ( )] ϭ 0, (iii) ϭ 1, ( ) ϭ f ( ), and m 1ץ aץqmqץ N ץ qץ 2 [iiϪ ( )] ϭ 0, (ii) ϭ g( ); (A11) a 1ץ bqץ N ץ 2 [iiϪ ( )] ϭ 0, therefore, (A10a)±(A10f) after splitting and rearrange- b 1 ments become simplyץ N ץ [q Ϫ q( )]2 ϭ 0, NNN ii , q 1 N ϩ af( ) ϩ bg( ) ϭ ץ m miii 111 N ץ [q Ϫ q( )]2 ϭ 0, NN N ii ( aq 1 f ( ) ϩ af2( ) ϩ bg( ) f (ץ mi i ii 11 1 N ץ [q Ϫ q( )]2 ϭ 0, N ii ,( bq 1 ϭ f (ץ ii 1 NN22 [( iiϪ ( )] [q iiϪ q(] ץ ϩϭ0, and NN N 22 ( hΆ·11⌰ Q g( ) ϩ af( )g( ) ϩ bg2(⌬ץ mi ii i 11 1 [( NN[ Ϫ ( )]22 [q Ϫ q( ץ iiϩϭ ii0; (A9) N 22 (l 11⌰ Q ϭ g( ), (A12ץ Ά· ii 1 then NNN N , qN af( ) bg( ) qץ [ Ϫ ( )] ( ) ϭ 0, (A10a) mqϩ iqϩ iϭ i ii i 111 mץ 1
N NN N 2ץ [ Ϫ ( )] ( ) ϭ 0, (A10b) qfmiqiqii( ) ϩ af( ) ϩ bg( ) f ( ) iii a 11 1ץ 1 Nץ N [ Ϫ ( )] ( ) ϭ 0, (A10c) ϭ qf( ), and iii ii b 1ץ 1
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NN NDriedonks, A. G. M., 1982: Models and observations of the growth qg( ) ϩ af( )g( ) ϩ bg2( ) of the atmospheric boundary layer. Bound.-Layer Meteor., 23, miqiiqi 11 1 283±306. Fedorovich, E. E., and D. V. Mironov, 1995: A model for shear-free N convective boundary layer with parameterized capping inversion ϭ qg( ). (A13) ii structure. J. Atmos. Sci., 52, 83±95. 1 Fitzjarrald, D. R., and M. Garstang, 1981: Vertical structure of the These are two uncoupled linear symmetric algebraic tropical boundary layer. Mon. Wea. Rev., 109, 1512±1526. Garratt, J. R., 1992: The Atmospheric Boundary Layer. Cambridge systems with the variables a, b, m and aq, bq, qm, re- University Press, 316 pp. spectively. Gryning, S. E., F. Beyrich, and E. Batchvarova, Eds., 1997: The Once these two systems are solved, a, b, m and aq, determination of the mixing heightÐCurrent progress and prob- lems. EURASAP Workshop Proc., Roskilde, Denmark, Risù Na- bq, qm are directly related to l and ⌬h, as in the case of the single ®t to only one variable. This second procedure tional Laboratories, 157 pp. Johansson, C., A. S. Smedman, and U. HoÈgstroÈm, 2000: Critical test produces a single estimate for l and ⌬h for the vertical of the validity of Monin±Obukhov similarity during convective pro®les of both potential temperature and water vapor conditions. J. Atmos. Sci., 58, 1549±1566. content. An example is shown in Figs. 10 and 11 using Khanna, S., and J. G. Brasseur, 1997: Analysis of Monin±Obukhov the data displayed in Figs. 6a and 8b. The results are similarity from large-eddy simulation. J. Fluid Mech., 345, 251± reported in Tables 3 and 4, for a comparison with the 286. Rampanelli, G., and D. Zardi, 2000: Analysis of airborne data and single-®t procedure. identi®cation of thermal structures with geostatistical techniques. 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