The partitioning of attached and detached eddy motion in ... - CiteSeerX

The partitioning of attached and detached eddy motion in the atmospheric surface layer using Lorentz wavelet ltering Gabriel Katul1 Brani Vidakovic2

Duke University

Assistant Professor at School of Environment, Duke University, Durham, NC, 27708 [email protected]. 2Assistant Professor at ISDS, Duke University, PO Box 90251, Durham, NC, 27708-0251 [email protected]. 1

Abstract Townsend's (1976) attached eddy hypothesis states that the turbulent structure in the constant stress layer can be decomposed into attached and detached eddy motion. This paper proposes and tests methodology for separating the attached and detached eddy motion from time series measurements of velocity and temperature. The proposed methodology is based on the time-frequency localization and ltering capabilities of the orthonormal wavelet transforms. Using a relative entropy statistical measure, the optimal wavelet basis is identi ed rst. The turbulence time series measurements are then transformed into the wavelet domain where the contribution of speci c events in the time-frequency domain are identi ed. The ltering scheme utilizes a recently constructed Lorentz thresholding methodology that successfully eliminated all wavelet coecients associated with the detached eddy motion. While this ltering scheme lacks the compression eciency of the classical Donoho and Johnstone's universal thresholding model, it conserves the higher-order statistics and important turbulence interactions related to the Reynolds stresses. Following the ltering scheme, the attached eddy motion time series is re-constructed by an inverse wavelet transform of the non-zero wavelet coecients. The proposed partitioning methodology for attached and detached eddy motion is tested using 56Hz triaxial sonic anemometer velocity and temperature measurements above a uniform dry lake bed in Owens valley, California, for a wide range of atmospheric stability conditions. Validation that the wavelet ltered time series represented the attached eddy motion is also discussed in the context of conservation of turbulence energy and surface uxes.

2

Contents 1 Introduction

3

2 Wavelet Analysis

6

2.1 Wavelet Selection Criteria : : : : : : : : : : : : : : : : : : : : 2.2 Thresholding Wavelet Coecients : : : : : : : : : : : : : : : :

7 9

3 Experiment

12

4 Results and Discussions

14

4.1 Optimal Wavelet Selection : : : : : : : : : : : : : : : : : : : : 4.2 Wavelet Thresholding and Filtering : : : : : : : : : : : : : : : 4.3 Validation of the Organized/Less Organized Eddy Motion Partitioning : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

5 Conclusion

15 15

18

21

1 Introduction The structure of turbulent eddy motion in the atmospheric surface layer (ASL) plays a central role in the transport mechanisms of heat, mass, and momentum from the ground into the atmosphere. While the turbulent ow is continuous in time, laboratory and eld measurements demonstrated that the turbulent

uxes of heat, mass, and momentum are very intermittent with few events contributing a large percent to the mean surface ux value. Laboratory and eld experiments over the past three decades suggested that organized and coherent eddy motion are responsible for much of the heat and momentum transfer in boundary layer ows (see Raupach et al., 1991 for review). It also appears that superimposed on these coherent eddy motions are less organized smallscale uctuations that are well explained by Kolmogorov's (1941) (hereafter referred to as K41) theory (e.g., Katul et al., 1994 a,b). The quanti cation 3

of these organized large scale eddy motions from time series measurements of turbulent ow variables continuous to be an important research topic in land-atmosphere interaction and ASL studies. One of the diculties in separating the organized and less organized eddies from time series measurements is the locality and non-periodicity of the organized events. Traditional Fourier and eigenvector decompositions are less eective in separating these two dierent modes of turbulence properties. The footprint of such organized eddy motion from time series measurements is usually characterized by sharp edges (e.g., ramp-like structures in temperature measurements) that result in large local gradients. It was shown by Mahrt and Gamage (1987) that these sharp edges are somewhat ambiguous in Fourier space because these sharp edges contribute considerable spectral and co-spectral energy at scales much smaller than the structure itself. Localized transformations, such as wavelet transforms, utilize space-scale decomposition that isolate the scale contribution of speci c events in space. This desirable feature of capturing energy contributions of speci c events in the scale domain is well suited for identifying the coherent eddy motion observed in turbulence measurements (Collineau and Brunet, 1993; Gamage and Hagelberg, 1993; Gao and Li, 1993; Hagelberg and Gamage, 1994; Howel and Mahrt, 1994; Brunet and Collineau, 1994; Liandrat and Moret-Bailly, 1990). However, the key limitation to the routine applications of wavelet transforms is the need to specify, a priori, a wavelet basis function that is independent of the

ow conditions or properties. While Mahrt (1991) and Katul et al. (1994 a, b) used the Haar basis because of its excellent locality in physical space, other investigators used Daubechies wavelets (Meneveau, 1991; Katul and Parlange, 1994, 1995a), Battle-Meyer-Lemarie wavelets (Meneveau, 1991; Yamada and Ohkitani, 1990, 1991a,b), Mexican hat wavelets (Gao and Li, 1993; Everson et al., 1990; Barcy et al., 1991), and Morlet and rst derivative of Gaussian wavelets (see Farge, 1992 for review) for better frequency localization. The speci cation of such a basis function is not a trivial matter in turbu4

lence research. In fact, much of the interpretation of the localized mechanisms and events responsible for the turbulent activity in the wavelet domain might depend on the a priori and arbitrary choice of the wavelet function used. Recent advances in wavelet thresholding methodologies and wavelet ltering techniques (e.g., Donoho and Johnstone, 1992-1994, Tew k et al., 1992, Nason 1994, Vidakovic, 1994) oer promising tools for specifying wavelet functions that \optimally" identify the energy-containing eddy motion in the wavelet domain. It should be noted that such a conceptual partitioning of the eddy motion into organized and less-organized eddies is not recent in wall-bounded shear

ows. Townsend (1976 pp. 152-153) suggested that the mean- ow vorticity and the energy-containing turbulent motions, which include the Reynold shear stress motions, are caused by anisotropic coherent eddies attached to the wall (attached eddies). These coherent eddies are surrounded by a uid that contains ne-scale eddies (detached eddies) that are statistically isotropic and follow K41. Perry et al. (1986) and Perry and Li (1990) assumed that the ne-scale eddies contribute little to the turbulent intensities and make no contribution to the Reynold shear stresses and the turbulent uxes but are responsible for most of the energy transfer to the dissipative viscous scales. The detached eddies are the remainder of once attached eddies that have been stretched, distorted, and convected away from the near-wall region by other attached eddies. It is the recent advances in wavelet thresholding methods, the results of Perry et al. (1986) and Perry and Li (1990) regarding the dominant role of the attached eddies on the turbulence statistics that motivated this study. The objective of this study is to develop and test a methodology for partitioning the organized (attached) and less-organized (detached) eddy motion that is consistent with Townsend's \attached eddy" hypothesis using the ltering capabilities of the orthonormal wavelet transforms. The proposed wavelet ltering and partitioning methodology between organized and less organized events 5

is carried out in three stages. The rst stage identi es the \optimal" wavelet basis function for a particular time series and ow variable based on speci ed entropy measures. The second stage lters the time series measurements by eliminating wavelet coecients that contribute little to the total turbulent energy. Such elimination is carried out using a newly developed wavelet shrinkage model based on a Lorentz thresholding criteria (Goel and Vidakovic, 1995). Such a thresholding model proved to be better suited for turbulence analysis than the classical Donoho and Johnstone (1992, 1993, 1994) universal thresholding method. The third stage re-constructs the thresholded time series from the few energy-containing wavelet coecients. This reconstructed time series constitutes the wavelet- ltered time series representing the organized eddy motion. Validation schemes must be developed for testing the consistency with Townsend's (1976) attached eddy hypothesis. These validation schemes must demonstrate that: (i) the wavelet ltered time series explains the majority of the variance and turbulent uxes of heat and momentum, and thus represents the attached eddy structure. (ii) that the dierence between the original and wavelet ltered time series follows K41 scaling, and thus represents the detached eddy structure. The proposed methodology was tested using 56Hz triaxial sonic anemometer velocity and temperature measurements over a dry lake bed in Owens lake, California, for a wide range of atmospheric stability conditions. But before presenting the experimental setup, the theory section discusses the optimal wavelet selection and the wavelet shrinkage model.

2 Wavelet Analysis This Section is divided into two parts. In the rst part, the wavelet selection criteria are discussed in relation to Townsend's hypothesis. In the second part, 6

the wavelet ltering and shrinkage methodologies are presented. Wavelets, being well localized in time and scale, provide a useful tool in handling localized multiscaled phenomena. An important feature of wavelets is that they provide unconditional bases for not only L spaces, but also for a variety of smoothness spaces, such as Sobolev and Holder spaces. Faced with a multitude of dierent wavelet bases for performing a discrete wavelet transformation, researchers have been interested in an objective criterium for selecting \the best" basis. What constitutes the best basis is a question of application, but two common requirements are:  parsimonious representation of data (the basis compresses the data well), and  energy conservation (the loss of energy during compressing is small) A trade-o between these two con icting requirements usually gives a criterium for basis selection. 2

2.1 Wavelet Selection Criteria The optimal wavelet basis is not unique but depends on the measurements available and the process being investigated. In this study Townsend's hypothesis suggests that the ow variables can be decomposed into attached (energy containing) and detached (negligible energy). This decomposition clearly suggests an energy disbalance that can be well captured by some scalar measure. It is customary to use the Shannon measure of the entropy of the energy as a measure of this disbalance. For the purpose of this section we de ne the energy of a wavelet transformed signal d as the squared L -norm of its components. ~ Other necessary de nitions are considered next. The Shannon entropy measure for a discrete probability distribution p is ~ de ned as 2

H (p) = ?npn log pn ; (1) ~ where 0 log 0 def = 0: Let S be a normalized data set (i.e., jjSjj = 1 ) and let d ~ ~ ~ 7

be the sequence of corresponding wavelet coecients. The sequence d is also ~ normalized and can be viewed as a discrete probability measure corresponding to S , given the wavelet transformation Wf . ~ The standardized entropy of data S with respect to Wf is de ned as ~ (2) C (S~ jW ) = H (d~ )=H (( n1 ; : : :; n1 )): The entropy measures were rst used in wavelet-related problems by Coifman and Wickerhauser (1992) in the context of optimal construction of wavelet packets. Vidakovic and Katul (1994) applied the measure (2) to the problems of best wavelet selection in turbulence. The procedure for best basis selection is usually carried out as follows. (i) Let S represent the normalized time series measurements of the ow ~ variable under consideration (e.g., velocity, temperature, etc.). (ii) Choose a wavelet Wf from a library of wavelets under consideration. Such libraries may include: wavelets from Daubechies family, Coi ets, spline based wavelets, etc. (iii) The measurements S are transformed to the wavelet domain by the ~ wavelet Wf , d = Wf (S ): ~ ~ (iv) Compute the standardized entropy (2) for the vector d : ~ (v) Repeat (ii)-(iv) for all wavelets from the library and choose the wavelet that minimizes measure (2). We note that the probability measure in (2) is directly proportional to the magnitude of the squared wavelet coecients. If the wavelet coecients are small, i.e.,  > 0; then  log   ? and their contribution to the entropy is negligible. Thus, when a wavelet basis function produces a very small relative entropy measure, a few large wavelet coecients contribute to the entropy measure, and the rest of the wavelet coecients are negligible. This wavelet basis function is very desirable because our ability to partition the wavelet coecients into attached or detached is maximized. Conversely, if for some basis function the measure (2) is close to unity, then most of the wavelet coecients describing the ow variable are very dierent 2

2

8

from zero and comparable in magnitude. This wavelet basis function is not desirable because the partitioning between attached and detached eddies based on the wavelet coecients becomes dicult. Thus, minimizing the relative entropy measure maximizes our ability to discriminate between organized and less-organized eddies, and this de nes the \optimal" basis function.

2.2 Thresholding Wavelet Coecients The parsimony of wavelet transformations ensures that the attached eddy motion can be described by a relatively small number of wavelet coecients. Towards that end, elimination of wavelet coecients describing the detached motion is necessary. Such elimination can be achieved by wavelet thresholding. To threshold wavelet coecients, two decisions must be made: the choice of thresholding method and the choice of the threshold. The standard thresholding method is de ned by the threshold function , given by

(d; ) = d 1(jdj > );

(3)

where 1(A) is the indicator function of a condition A. Its value is unity if the condition A is satis ed and zero otherwise. (Donoho, 1993). Next we must choose a threshold : Donoho and Johnstone (1992, 1993, 1994) propose a threshold

q

U = 2 log n ^ ;

(4)

which they call universal. This threshold is one of the rst proposed and provides easy, fast, and automatic thresholding. The rationale of thresholding is to remove all wavelet coecients that are smaller than the expected maximum of the assumed uncorrelated Gaussian noise of a given size n, and ^ is an estimator of the standard deviation of the wavelet coecients (usually restricted to the nest scale). Dierent ways of specifying the threshold were proposed 9

by Nason (1994), Donoho and Jonstone (1992, 1993, 1994), and Vidakovic (1994), among others. Vidakovic and Katul (1994) have found that for ASL turbulence measurements, the universal threshold, based on sample standard deviation, does not conserve variances and covariances and thus is ill suited for this study. The possible causes for this variance-covariance loss are that large scales and small scales interact in ASL turbulence and the ne scales can not be modeled by white noise (Katul et al., 1995b). Therefore an alternative distribution-free method is required. Fortunately, we can exploit the fact that turbulence contains an energy disbalance in the wavelet domain. The most general measure of disbalance of energy is the Lorentz curve. The Lorentz curve is a convex curve describing the cumulative energy (y-axis, 0-100%) that is contained in p
100% smallest energy components (x-axis, 0-100%). In Figure 1, a sample Lorentz Curve is shown. Convexity of the curve, relative to the diagonal line (corresponding to balanced energy), is directly proportional to the energy disbalance. Goel and Vidakovic (1995) proposed a thresholding method based on the Lorentz curve for the energy in wavelet decomposition. This universal method can be simply described as: Replace the p  100% of the coecients with the smallest energy with 0. The proportion p is de ned as: (5) p = n1 i1(di  d ); where d is the mean of (d ; d ; : : : ; dn): 0

0

2

0

2 1

2

2 2

2

2

The point p is the proportion at which the gain (in parsimony) by thresholding an additional element will be smaller than the loss in the energy. Both losses are measured on a scale 0{1, and are equally weighted. Figure 1 shows the Lorentz Curve, the thresholding method, and the proportion p of the energy components removed in thresholding (p  63% and the energy loss is 28%, in this example). 0

0

0

10

1.0 0.8 0.6 energy loss 0.4 0.2 0.0

p0

0.0

0.2

0.4

0.6

0.8

1.0

percentage of coefficients rejected

Figure 1: Lorentz type wavelet thresholding. The diagonal corresponds to a balanced signal and the convex curve corresponds to turbulence measurements. Notice that the \optimal" proportion p is determined from the tangent parallel to the diagonal. 0

11

Figure 2 summarizes the wavelet ltering procedure described above that is used to extract the attached eddy motion from the original signal.

3 Experiment The three velocity components (U ; U ; U ) and air temperature (T ) measurements were collected during an experiment in June 1993 over a uniform dry lake bed (Owens lake) in Owens valley, California (elevation = 1,100 m). The lake bed is bounded by the Sierra Nevada range to the east and the White and Inyo Mountains to the west. The site's surface is a heaved sandy soil extending uniformly more than 11 km in the North-South direction. The momentum roughness height (z ) for this sandy surface is 0.13 mm (see Katul et al., 1994a,b; Katul, 1994). The three velocity components were measured at z = 2:0m ? 3:5m above the surface using a triaxial ultrasonic anemometer (Gill Instruments/1012R2). The sampling frequency fs was 56Hz and the sampling period Tp was 9.75 minutes resulting in 32,768 data points per velocity component. The short sampling period was necessary to achieve steady state mean meteorological conditions. The velocity components were rotated so that U is along the longitudinal direction, U and U are the lateral and vertical velocities, respectively. In this study, both meteorological and index notations (U = U ; V = U ; W = U ; x = x ; y = x ; z = x ) are interchangeably used. For the purpose of this study, we focus on four runs that represent a wide range of atmospheric stability conditions (z=L), where L is the Obukhov length given by (6) L = ? uH k g ( CpTa ) 1

2

3

0

1

2

3

1

2

3

1

2

3

3

q

and u (= ?huwi) is the friction velocity, H (=  Cp hwT i) is the sensible heat ux, Ta is the mean air temperature,  is the air density, k = 0:4 is Von 12

Figure 2: Methodology for separating the attached eddy motion from the measurements using wavelet ltering. 13

Table 1: A summary of the mean meteorological conditions for Runs #1 { #4. The DOY represents the day of year for each run. The time is in Paci c Daylights Savings Time. Run # z (m) DOY/Time u (m=s) H (W=m ) hU i (m=s) hT i (C ) L (m) 1 2.00 179/2134 0.25 -38 6.5 31.5 35.3 2 2.25 180/0702 0.18 34.6 4.1 26.3 -13.2 3 3.00 178/0853 0.14 94.9 1.9 34.8 -2.5 4 3.50 179/1710 0.11 158.4 6.2 43.7 -0.7 2

Karman's constant, g is the gravitational acceleration, Cp is the speci c heat capacity of dry air at constant pressure, u and w are the turbulent uctuations (huii = 0), and h i is time averaging assumed to converge to the ensemble averaging by the ergodic hypothesis as discussed in Monin and Yaglom (1971). The detailed structure of turbulence will be analyzed in the context of Townsend's attached and detached eddy hypothesis using the orthonormal wavelet transform and the procedure outlined in Figure 2. Table 1 summarizes the mean meteorological and turbulence conditions for these four runs.

4 Results and Discussions This section is divided into three parts. In the rst part, the \optimal" wavelet function is presented for all four runs and ow variables based on the entropy measures in Subsection 2.1. A wavelet decomposition is then carried out using this \optimal basis" function to transform the ow variables from time to wavelet domain. In the second part, the results of the thresholding methodology discussed in Subsection 2.2 are used to discard the wavelet coecients that are associated with the detached eddy motion. Recall that the detached 14

eddies contribute little to the turbulent energy content. In the third part, a validation that the attached eddies are the only eddies existing in the wavelet ltered time series is discussed. The validation is carried out by (i) evaluating how well the wavelet- ltered time series reproduces the variances (u; w ; and T ) and covariances (huwi; hwT i) of the original time series, and (ii) comparing the Fourier spectral density function for the wavelet ltered and original time series at low frequencies for all three ow variables.

4.1 Optimal Wavelet Selection As discussed in Subsection 2.1 the optimal wavelet basis was identi ed using the standardized entropy measure (2). This entropy measure was applied to all four runs described in the experimental setup. Table 2 summarizes the choice of the optimal wavelet basis function for all three ow variables and all four runs using the rst 20 Daubechies wavelets. We should note that for certain runs, the relative entropy measure did not vary signi cantly (< 5 %) with the variation in the wavelet basis function. For these cases, the choice of the wavelet basis function for partitioning the organized and less organized eddy motion becomes insigni cant and the Haar basis is used for its simplicity. The basis functions from Table 2 are used in the wavelet decomposition. Subsequent thresholding of the turbulent ow variables are discussed next.

4.2 Wavelet Thresholding and Filtering The optimal wavelet basis function presented in Table 2 was used to transform each ow variable into the wavelet domain. Using the resultant wavelet coecients, the thresholding methodology of Subsection 2.2 was applied. In essence, the thresholding methodology evaluates the gain in increasing the number of wavelet coecients versus the loss in energy of the turbulent ow variable. When the loss and gain are in balance, the thresholding criteria 15

Table 2: Summary of the optimal wavelet basis choice for runs #1{#4, and all ow variables. The relative entropy is also presented. The var is the variance in the relative entropy due to variation in the DAUB wavelet choice. Emax and Emin are the maximum and the minimum of calculated relative entropy for the library of Daubechies' wavelets. If Var is less than 5%, then the choice is the simplest wavelet (Haar or DAUB1). Otherwise, the wavelet corresponding to Emin is selected. Run # u w t 1 Var 0.01971 0.01827 0.02786 Emax 0.01378 0.62067 0.000341 Emin 0.01351 0.60954 0.000332 Choice Daub1 Daub1 Daub1 2 Var 0.03485 0.02826 0.02573 Emax 0.01602 0.66770 0.000293 Emin 0.01548 0.64935 0.000286 Choice Daub1 Daub1 Daub1 3 Var 0.08950 0.03691 0.02647 Emax 0.07457 0.60175 0.000568 Emin 0.06845 0.58329 0.000553 Choice Daub18 Daub1 Daub1 4 0.06472 0.02882 0.02030 Emax 0.03373 0.02882 0.000565 Emin 0.03168 0.62757 0.000554 Choice Daub16 Daub1 Daub1

16

Table 3: Summary of the proportion Lorentz thresholding. Run # u 1 93.7% 2 96.6 % 3 98.6 % 4 97.6 %

of rejected wavelet coecients due to

w 85.4% 89.9 % 94.3 % 91.0 %

t 94.7% 95.6% 96.5 % 95.1 %

is established and all wavelet coecients below this criteria are replaced by zero. The time series of the ow variable is then re-constructed by an inverse wavelet transform using the zero-wavelet coecients. The re-constructed time series is the wavelet- ltered time series of the ow variable and represents the contribution of the organized eddies in the time series. Table 3 summarizes the percent wavelet coecients set to zero based on the thresholding method of Subsection 2.2. This percent represents the percent of detached eddies in the ow variable that were ltered out by the proposed thresholding model. It is interesting to note from Table 3 that the organized eddy motion constitutes less than 7%, 15%, and 5% for the longitudinal velocity, vertical velocity, and temperature, respectively. We should note that Run 1 was under slightly stable conditions. For these stability conditions, the vertical velocity lacks any organized eddy structure because large scale events are destroyed by the density strati cation. This lack of self organization at the larger scales is responsible for the large number (15%) of wavelet coecients set to zero using the wavelet thresholding model. In order to display the eects of wavelet thresholding on the original time series, the re-constructed time series from the very limited wavelet coecients are presented in Figures 3 and 4 for U , W , and T , respectively, and for Runs 1 and 4. To better illustrate the comparison between these two time series, 17

only 30 s are displayed. Notice from Figures 3 and 4 that the larger scale eddy motion associated with large turbulent events are well captured by the proposed ltering methodology for unstable and stable atmospheric conditions. Superimposed on these large events are small-scale events representing the detached eddy motion.

4.3 Validation of the Organized/Less Organized Eddy Motion Partitioning Townsend's (1976) attached eddy hypothesis suggests that the velocity and temperature uctuations can be decomposed into

ui = uia + uid T = T a +T d ; ( )

( )

( )

(7)

( )

where the superscripts (a) and (d) represent the attached and detached eddy motion. In Townsend (1976, pp. 152-153), the energy-containing turbulent motions are caused by anisotropic coherent eddies attached to the wall (attached eddies) and scale by uia . The detached eddies are ne-scale statistically isotropic eddies surrounding the organized eddies and scale with uid . Townsend's (1976) hypothesis suggests that the attached eddies result in the following relations ( )

( )

hu a i = 0 hw a i = 0 u = h(u a ) i w = h(w a ) i T = h(T a ) i huwi = hu a w a i hwT i = hw a T a i ( )

( )

2

( ) 2

2

( ) 2

2

( ) 2

18

( )

( )

( )

( )

(8)

8 7 6

u (m/s)

5

0

5

10

15

20

25

30

20

25

30

20

25

30

0.0 -1.0

-0.5

w (m/s)

0.5

1.0

time (s)

0

5

10

15

t (C)

31.5

32.0

32.5

33.0

33.5

time (s)

0

5

10

15 time (s)

Figure 3: A comparison between measured (thin line) and Lorentz thresholded (thick line) time series for Run #1 (U is top, W is middle, and T is bottom panel). For comparison purposes, only the rst 30s are shown. 19

8 7 6

u (m/s)

5

0

5

10

15

20

25

30

20

25

30

20

25

30

w (m/s)

-1.0

-0.5

0.0

0.5

1.0

1.5

time (sec)

0

5

10

15

43

44

t (C)

45

46

time (sec)

0

5

10

15 time (sec)

Figure 4: Same as Figure 3 but for Run #4. 20

The determination of uia from ui is based on the wavelet- ltering approach discussed in Subsections 4.1 and 4.2 and shown in Figure 2. The validation of the ltering schemes are presented in Tables 4 (variance conservation) and 5 (covariance conservation). From Tables 4 and 5, it is evident that more than 85% of the turbulent variances and uxes are explained by the attached eddy motion that constitute less than 15% of the wavelet coecients. The detached eddies can be determined from the original time series and the attached eddies using uid = ui ? uia . According to Townsend's (1976) hypothesis, the detached eddies must follow K41 scaling with Euid (K )  K ? = ; where E(K ) is the Fourier power spectral density function for a ow variable ; and K is the wavenumber. The spectral density function was estimated by windowing 8,192 out of the 32,768 data points, cosine tapering 5% along each window edge, estimating the power spectral density function for each window, and averaging the four windows per ow variable per run. The spectral density functions for ui; uia , and uid as well as T; T a ; T d are presented in Figures 5 and 6 for Runs 1 and 4, respectively. In each gure, the Euid follows a ?5=3 power law consistent with K41. Notice in Figures 5 and 6 that the Fourier power spectra for the wavelet- ltered and original time series are in excellent agreement for frequencies less than 1Hz. That is, the wavelet ltered time series accurately reproduces the power spectral density function of the original time series at low frequencies that are typically associated with the large scale eddy motion. ( )

( )

( )

5 3

( )

( )

( )

( )

( )

( )

5 Conclusion This study evaluated the ltering capabilities of orthonormal wavelet transforms in partitioning the turbulent eddy motion into attached and detached in accordance with Townsend's attached eddy hypothesis. A methodology that combines the identi cation of the optimal wavelet basis with a wavelet 21

Table 4: Veri cation that the thresholded time series, related to the attached eddy motion, conserved the measured variances. For each run, the rst, second, and third rows represent the measured variance, the Lorentz thresholded variance, and percent retained of the variance in the thresholded time series, respectively. Run # u w t 0.494 0.124 0.244 1 0.459 0.108 0.230 93% 87 % 94 % 0.290 0.0499 0.145 2 0.280 0.045 0.139 96 % 90 % 96 % 0.512 0.069 0.504 3 0.504 0.066 0.488 98 % 94 % 97 % 1.37 0.168 0.763 4 1.33 0.153 0.728 97 % 91 % 95 %

22

Table 5: As Table 4 but for covariances. Run # huwi hwT i -0.06596 -0.02897 1 -0.05978 -0.02130 91 % 73 % -0.03540 0.03765 2 -0.03290 0.03450 98 % 92 % -0.01490 0.09389 3 -0.01460 0.09060 98 % 97 % -0.01630 0.17500 4 -0.00970 0.16400 59 % 94 %

23

residual

0.00001

0.00005

0.00050

0.00500

10^1 10^0 10^-2

power U (m*m/s)

10^-4 10^-6 0.001

0.010

0.100

1.000

10.000

0.001

0.010

0.100

1.000

10.000

1.000

10.000

1.000

10.000

frequency (Hz)

0.00050

residual

10^-3

0.00005

10^-4 10^-6

10^-5

power W (m*m/s)

10^-2

10^-1

frequency (Hz)

0.001

0.010

0.100

1.000

10.000

0.001

0.010

residual

0.00001

0.00010

0.00100

10^0 10^-2 10^-4 10^-6

power T (C*C*s)

0.100 frequency (Hz)

0.01000

frequency (Hz)

0.001

0.010

0.100 frequency (Hz)

1.000

10.000

0.001

0.010

0.100 frequency (Hz)

Figure 5: Comparison between measured and Lorentz thresholded Fourier power spectra for Run #1 (U is top, W is middle, and T is bottom panel). For comparison purposes, the spectra of the thresholded series are shifted down by one decade along the ordinate axis. The spectra of the residual time series (detached eddy motion) are shown on24 the side. The solid line is the ?5=3 power law.

10^-2

10^1

residual

10^-5

10^-4

10^-3

10^-1 10^-3

power U (m*m/s)

10^-5

10^-6

10^-7

10^-7

10^-9 0.001

0.010

0.100

1.000

10.000

0.001

0.010

0.100

1.000

10.000

1.000

10.000

1.000

10.000

frequency (Hz)

10^-6

10^-5

10^-4

residual

10^-2 10^-4 10^-6

power W (m*m/s)

10^-3

10^0

frequency (Hz)

0.001

0.010

0.100

1.000

10.000

0.001

0.010

0.100 frequency (Hz)

5*10^-5

5*10^-4

residual

10^-2 10^-4 0.001

0.010

0.100

1.000

5*10^-6

10^-6

power T (C*C*s)

10^0

5*10^-3

10^1

frequency (Hz)

10.000

frequency (Hz)

0.001

0.010

0.100 frequency (Hz)

Figure 6: Same as Figure 5 but for Run #4. 25

ltering model was proposed and tested. The identi cation was based on a relative entropy that measures the complexity of a given time series in the wavelet domain. The wavelet ltering model was based on a Lorentz type thresholding approach that balances the gain in information about localized events with the loss in the number of wavelet coecients. Applications of this proposed scheme to 56Hz triaxial sonic anemometer velocity and temperature measurements above a uniform dry lake bed were discussed. Our conclusions are summarized below: (i) Using the proposed wavelet ltering scheme, it is possible to partition the ASL turbulence activity, responsible for much of the heat and momentum surface uxes, by a two-mode eddy motion. These two modes are consistent with Townsend's attached eddy hypothesis, Kolmogorov (1941) theory, and the coherent and organized events well documented in ASL experiments. (ii) The choice of the orthonormal wavelet basis function was not critical for decomposing the turbulence structure into attached and detached eddy motion. This was tested using a wide range of Daubechies wavelets (DAUB1DAUB20) with varying degrees of locality and smoothness in both time and frequency domains. (iii) Based on the Lorentz wavelet ltering scheme, the attached eddy motion represented a small fraction of the turbulent ow variables time series and was well captured by 5% of the wavelet coecients for unstable and nearneutral stability conditions. For slightly stable conditions, for which the vertical velocity was simultaneously dampened by the ground and density strati cation, the attached eddies comprised about 15% of the wavelet coecients. These percent contributions agree qualitatively with many other ASL conditional sampling studies. (iv) Despite the limited number of wavelet coecients necessary to characterize the attached eddy motion, the variances and Reynold stresses were well conserved. The detached eddy motion, characterized by 95% of the wavelet coecients in the wavelet domain, contributed to less than 10% of the heat and 26

momentum uxes. These percent contributions agree with the assumptions of Townsend's hypothesis. (v) Using the Fourier power spectra, it was demonstrated that the attached eddy motion is associated with low frequencies that are of the order of 1Hz, while the detached eddy motion power spectrum follows K41 in agreement with Townsend's hypothesis.

Acknowledgments The authors would like to thank Marc Parlange, John Albertson, and Chia Ren Chu for their help in the data collection, and Scott Tylor for his support at Owens Lake. Guy Nason's wavethresh 2.2 software [26] provided an excellent computational environment for our study. This work was supported in part by National Science Foundation Award DMS-9404151 at Duke University and Environmental Protection Agency Grant 91-0074-94 at Duke University.

References [1] Barcy, E., Arneodo, A., Frisch, U., Gagne, Y., and Hopfinger, E. (1991). Wavelet analysis of fully developed turbulence data and measurements of scaling exponents, In: Turbulence and Coherent Structures, eds. O. Metais, and M. Lesieur, Kluwer Acadamic Press, 203-215. [2] Brunet, Y. and Collineau, S. (1994). Wavelet analysis of diurnal and nocturnal turbulence above a maize crop, 1994, In: Wavelets in Geophysics, eds. E. Foufoula, and P. Kumar, pp. 129-150. [3] Coifman, R. and Wickerhauser, V. (1992). Entropy based algorithms for best basis selection. IEEE Trans. Inform. Theory, 38 (2), 713718.

27

[4] Collineau, S. and Brunet, Y. (1993). Detection of turbulent coherent motions in a forest canopy: Part I: Wavelet analysis, Bound. Layer Meteorol., 65, 357-379. [5] Donoho, D. (1993). Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Data. Proceedings of Symposia in Applied Mathematics, American Mathematical Society. Biometrika. to appear. [6] Donoho, D., Johnstone, I., et al. (1992, 1993, 1994). Bank of wavelet-related Technical Reports at playfair.stanford.edu. [7] Everson, R., Sirovich, L., and Sreenivasan, K. (1990). Wavelet analysis of the turbulent jet, Physical Letters A, 145, 314-322. [8] Farge, M. (1992). Wavelet transforms and their application to turbulence, Ann. Rev. Fluid Mech., 24, 395-347. [9] Gamage, N. and Hagelberg, C. (1993). Detection and analysis of microfronts and associated coherent events using localized transforms, J. Atmos. Sci., 50, 750-756. [10] Gao, W. and Li, B. (1993). Wavelet analysis of coherent structures at the atmosphere-forest interface, J. Appl. Meteorol., 32, 1717- 1725. [11] Goel, P. and Vidakovic, B. (1995). Wavelet transformations as diversity enhancers, Discussion Paper 95-04, ISDS, Duke University. [12] Hagelberg, C. and Gamage, N. (1994). Applications of structures preserving wavelet decompositions to intermittent turbulence: A case study, In: Wavelets in Geophysics, eds. E. Foufoula, and P. Kumar, pp. 45-80. [13] Howell, J. and Mahrt, L. (1994). An adaptive decomposition: Application to turbulence, In: Wavelets in Geophysics, eds. E. Foufoula, and P. Kumar, pp. 107-128. 28

[14] Katul, G. (1994). A model for sensible heat ux probability density function for near-neutral and slightly stable atmospheric ows, Bound. Layer Meteorol., 71, 1-20. [15] Katul, G., Albertson, J., Chu, C., and Parlange, M. (1994a). Intermittency in atmospheric surface layer turbulence: The orthonormal wavelet representation. In: Wavelets in Geophysics, eds. E. Foufoula, and P. Kumar, pp. 82-105. [16] Katul, G. and Parlange, M. (1994). On the active role of temperature in surface layer turbulence, J. Atmos. Sci., 51, 2181- 2195. [17] Katul, G. and Parlange, M. (1995). The spatial structure of turbulence at production wavenumbers using the orthonormal wavelets, in Press, Bound. Layer Meteorol. [18] Katul, G., Parlange, M., and Chu, C. (1994b). Intermittency, local isotropy, and non-Gaussian statistics in atmospheric surface layer turbulence, Phys. Fluids, 6, 2480-2492. [19] Katul, G., Parlange, M., Albertson, J. and Chu, C. (1995). Local isotropy and anisotropy in the sheared and heated atmospheric surface layer, in Press, Bound. Layer Meteorol. [20] Kolmogorov, A.N. (1941). The local structure of turbulence in incompressible viscous uid for very large Reynolds numbers, Dokl. Akad. Nauk. SSSR, 30, 301-305. [21] Liandrat, J. and Moret-Bailly, F. (1990). The wavelet transform: some applications to uid dynamics and turbulence, Eur. J. Mech. B/ Fluids, 9, 1-9. [22] Mahrt, L. and Gamage, N. (1987). Observations of turbulence in strati ed ow, J. Atmos. Sci., 44, 1106-1121. 29

[23] Mahrt, L. (1991). Eddy asymmetry in the shear-heated boundary layer, J. Atmos. Sci., 48, 472-492. [24] Meneveau, C. (1991). Analysis of turbulence in the orthonormal wavelet representation, J. Fluid Mech., 232, 469-520. [25] Monin, A. S. and Yaglom, A. M. (1971). Statistical Fluid Mechanics, MIT Press, 769 pp. [26] Nason, G.P. and Silverman B. W. (1993). The discrete wavelet transform in S. Statistics Research Report 93:07, University of Bath, UK. [27] Perry, A., Henbest, S., and Chong, M. (1986). A theoretical and experimental study of wall turbulence, J. Fluid Mech., 165, 163- 199. [28] Perry, A. and Li, J. (1990). Experimental support for the attachededdy hypothesis in zero-pressure gradient turbulent boundary layers, J. Fluid Mech., 218, 405-438. [29] Raupach, M. Antonia, R., and Rajagopalan, S. (1991). Rough-wall turbulent boundary layers, Appl. Mech. Rev. 44, 1-25. [30] Tewfik, A., Sinha, D. and Jorgensen, P. (1992). On the optimal choice of a wavelet for signal representation. IEEE Transactions on Information Theory, 38, No 2, 747-765. [31] Townsend, A. (1976). The Structure of Turbulent Shear Flow. Cambridge University Press, 429 pp. [32] Vidakovic, B. (1994). Nonlinear wavelet shrinkage with Bayes rules and Bayes Factors. Discussion Paper 94-24, ISDS, Duke University, Submitted. [33] Vidakovic, B. and Katul, G. (1994). Wavelet shrinkage of turbulence signals. Discussion Paper 94-25, ISDS, Duke University. 30

[34] Yamada, M. and Ohkitani, K. (1990). Orthonormal wavelet expansion and its application to turbulence, Progress of Theoretical Physics, 83, 819823. [35] Yamada, M. and Ohkitani, K. (1991a). An identi cation of energy cascade in turbulence by orthonormal wavelet analysis. Progress of Theoretical Physics, 86, 799-815. [36] Yamada, M. and Ohkitani, K. (1991b). Orthonormal wavelet analysis of turbulence, Fluid Dynamics Research, 8, 101-115.

31