j - umass amherst...oct 14, 2009 · the classical theory of homogeneous turbulence (batchelor...

J. Other

J.1. Signed Waiver of Access form

Has been submitted to the ECE Department O�ce.

J.2. Four (4) important publications

Publication #1:Muschinski, A., 1996: A similarity theory of locally homogeneous and isotropic turbulence gener-ated by a Smagorinsky-type LES. J. Fluid Mech., 325, 239-260.

Publication #2:Muschinski, A., R. G. Frehlich, and B. B. Balsley, 2004: Small-scale and large-scale intermittencyin the nocturnal boundary layer and residual layer. J. Fluid Mech., 515, 319-351.

Publication #3:Muschinski, A., 2004: Local and global statistics of clear-air Doppler radar signals. Radio Sci., 39,doi:10.1029/2003RS002908.

Publication #4:1Cheon, Y., V. Hohreiter, M. Behn, and A. Muschinski, 2007: Angle-of-arrival anemometry by meansof a large-aperture Schmidt-Cassegrain telescope equipped with a CCD camera. J. Opt. Soc. Am.A, 24, 3478-3492.

1The lead author and all coauthors were members of A.M.'s research team when the paper was prepared, submitted,and published.

1

d. Ffuid Mech. ( 1 996), vol. 325, p p . 239-260 Copyright @ 1996 Cambridge University Press

239

A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type LES

By A N D R E A S M U S C H I N S K I Institut fur Meteorologie und Klimatologie der Universitat Hannover,

30419 Hannover, Germany

(Received 4 August 1995 and in revised form 13 May 1996)

A Kolmogorov-type similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type large-eddy simulation (LES) at very large LES Reynolds numbers is developed and discussed. The underlying concept is that the LES equations may be considered equations of motion of specific hypothetical fully turbulent non-Newtonian fluids, called ‘LES fluids’. It is shown that the length scale ls = csd, which scales the magnitude of the variable viscosity in a Smagorinsky- type LES, is the ‘Smagorinsky-fluid’ counterpart of Kolmogorov’s dissipation length rj = v 3 / 4 ~ - 1 / 4 for a Newtonian fluid where v is the kinematic viscosity and E is the energy dissipation rate. While in a Newtonian fluid the viscosity is a material parameter and the length q depends on E , in a Smagorinsky fluid the length I S is a material parameter and the viscosity depends on E. The Smagorinsky coefficient cs may be considered the reciprocal of a ‘microstructure Knudsen number’ of a Smagorinsky fluid. A combination of Lilly’s (1967) cut-off model with two well- known spectral models for dissipation-range turbulence (Heisenberg 1948; Pao 1965) leads to models for the LES-generated Kolmogorov coefficient RLES as a function of cs. Both models predict an intrinsic overestimation of CXLES for finite values of cs. For cs = 0.2 Heisenberg’s and Pao’s models provide RLES = 1.74 (16% overestimation) and c~LES = 2.14 (43% overestimation), respectively, if limcs-tJ=(ctLES) = 1.5 is ad hoc assumed. The predicted overestimation becomes negligible beyond about cs = 0.5. The requirement cs > 0.5 is equivalent to d < 21s. A similar requirement, L < 2q where L is the wire length of hot-wire anemometers, has been recommended by experimentalists. The value of limcs+sc(xLES) for a Smagorinsky-type LES at very large LES Reynolds numbers is not predicted by the models and remains unknown. Two critical values of cs are identified. The first critical cs is Lilly’s (1967) value, which indicates the cs below which finite-difference-approximation errors become important; the second critical cs is the value beyond which the Reynolds number similarity is violated.

1. Introduction The classical theory of homogeneous turbulence (Batchelor 1953) may be considered

the backbone of the physics of fully developed turbulence. It is the present-day view (see, for example, Hunt, Phillips & Williams 1991; Yaglom 1981, 1994) that the decisive breakthrough towards modern turbulence physics is to be attributed

240 A. Muschinski

to Kolmogorov’s (1941u,b) two classical papers. It is important to note, however, that Kolmogorov’s similarity theory does not rely on the Navier-Stokes equations. Kolmogorov (1941~) did not even mention them. On the other hand, the Navier- Stokes equations are widely believed to be the first principles for the physics of fluid turbulence, also at high Reynolds numbers, and much work has been done to gain insight into Kolmogorov’s turbulence laws from the viewpoint of the Navier-Stokes equations, see, for example, McComb (1990).

Since the advance of supercomputers, it has been possible to leave any turbulence theory completely aside and, instead, to straightforward numerically integrate the Navier-Stokes equations under the constraint of the external forcing. This approach is known as direct numerical simulation (DNS). See, for example, McComb (1990), Reynolds (1990) or Chen et al. (1993). DNS is free from any assumption on the statistical nature of developed turbulence and, to a certain extent, may be considered the numerical counterpart of a laboratory experiment. The DNS technique, however, is limited to moderate Reynolds numbers Re since at very high Re as encountered in the atmosphere or in the ocean the number of hydrodynamical degrees of freedom is far beyond present-day computer capacities.

A very efficient technique to reduce the numerical expense by many orders of magnitude is large-eddy simulation, abbreviated as LES (Smagorinsky 1963; Lilly 1967; Deardorff 1970). The technique’s history and development have been compiled in a monograph edited by Galperin & Orszag (1993). A critical review of the technique was given by Mason (1994).

LES relies on both the Navier-Stokes equations and on a reasonable model for the small-scale turbulence. In LES the Navier-Stokes equations and the other diagnostic and prognostic equations are used in a spatially filtered form. Lilly (1967) used a three-dimensional top-hat filter. Leonard (1974) generalized the filtering concept. Moeng & Wyngaard (1988) compared ‘empirical’ spectra of the turbulent kinetic energy (TKE) generated in an LES with ‘theoretical’ TKE spectra that they obtained by applying the specific filter associated with their specific LES equations to Kolmogorov’s inertial-range TKE spectrum. But the theoretical spectra did not compare well with the empirical LES spectra. They presumed that the discrepancy might be to be attributed to the fact that the theoretical spectra were obtained by explicitly filtering while the empirical LES TKE spectra were the result of a filtering operation that is “to some extent implicit” (Moeng & Wyngaard 1988,

It is the purpose of this paper to give an elementary physical picture of this ‘implicit’ filtering. To a certain extent, we adopt the philosophy which was described by Mason (1994).

The paper is organized as follows. In $2, the difference between Navier-Stokes equations and LES equations is discussed. It is shown that LES equations may be considered equations of motion of specific hypothetical non-Newtonian turbulent fluids, called ‘LES fluids’. Section 3 contains the essence of this paper. Somewhat in analogy to Kolmogorov’s (19414 similarity theory, a similarity theory of locally homogeneous and isotropic turbulence generated by a finite-difference Smagorinsky- type LES is put forward. It is shown that the LES-generated Kolmogorov coefficient C X L E ~ is sensitive to the Smagorinsky coefficient cs if cs is smaller than Lilly’s (1967) value for cs; on the other hand, LxLES is asymptotically universal for cs larger than Lilly’s cs. The similarity theory is generalized to account for turbulence generated by an anisotropic-grid LES. In 54, some implications of the similarity theory are discussed. A summary and conclusions are given in $5.

p. 3577).

Similarity theory of LES-generated turbulence 24 1

2. The nature of the LES equations and the concept of LES fluids In this Section a physical interpretation of the nature of the LES equations will be

given in some detail. It will be shown that the LES equations may be considered the equations of motion of specific hypothetical non-Newtonian turbulent fluids. These fluids will be called ‘LES fluids’. The concept of LES fluids is useful to get a physical picture of the similarity theory of LES-generated turbulence that will be put forward in this paper.

2.1. Navier-Stokes equations and LES equations: the concept of LES j u i d s DNS is a numerical integration of the Navier-Stokes equations. If a turbulent flow at a high Reynolds number Re is to be modelled using DNS it must be guaranteed that the mesh width A of the numerical grid is smaller than the smallest curvature radii of the iso-surfaces of any physical quantity to be predicted by the DNS. It is generally accepted that this requirement is fulfilled if d is a few times smaller than Kolmogorov’s ( 1 9 4 1 ~ ) dissipation length

114

v = ( ’ > ,

where v is the molecular kinematic viscosity and E the mean energy dissipation rate. It is known that the ratio of the outer scale of the turbulence L and the inner scale

q depends on the turbulent Reynolds number R e :

4 N (2.2) v Thus, DNS of a fully turbulent flow requires a number of grid points N that increases dramatically as a function of Re:

3

N - ( 4 ) - Re9I4, (2.3)

see, e.g., Corrsin (1961). Large-eddy simulation (LES) is a technique that allows numerical simulation of

turbulent flows for arbitrarily large Re. In contrast to DNS, in LES not all turbulent structures are resolved but only the ‘large’ eddies, i.e. those having length scales larger than a certain length lf. The ‘inner inertial range’, i.e. eddies having length scales between q and lf is parameterized. The length 1, is defined by the spatial filter that is applied to the Navier-Stokes equations in order to get a specific set of LES equations. Note that, like the Navier-Stokes equations, the LES equations are a priori partial differential equations. A numerical integration of the LES equations is called a LES.

The LES equations contain a term zij that is physically interpreted as a Reynolds stress tensor that is variable in space and time. This tensor is to be parameterized in terms of the local instantaneous velocity field generated by the LES, and this is usually done on the basis of an eddy-viscosity hypothesis (Lilly 1967, p. 203; Leonard 1974, p. 240):

Here, vLES is a kinematic viscosity which is not constant but varies temporally and spatially and depends on the local and instantaneous ( a priori spatially filtered) velocity field ui generated by the LES. Now, in order to close the LES equations an

242 A. Muschinski

additional equation is needed, namely the equation that allows the parameterization of VLES itself in terms of ui.

A fluid in which the viscosity depends on the shear is a non-Newtonian fluid; therefore, the LES equations may be considered the equations of motion for a specific hypothetical non-Newtonian fluid. In the following, we will call such a hypothetical non-Newtonian fluid an ‘LES fluid’, which is physically specified by the equation for VLES or, more generally, by the parameterization of zi, in terms of the ui.

The spatial filter that is inherent in the LES equations smears out the fluctuations that are considerably smaller than I f . A physical interpretation is that the ‘small’ eddies are damped out by the eddy viscosity, which depends on (or defines) l f and which is usually several orders of magnitude larger than the molecular viscosity v of the (Newtonian) fluid to be modelled. In other words l f is a property of the specific LES fluid while the molecular viscosity is a property of a Newtonian fluid. While the smallest eddies in a fully turbulent Newtonian fluid are on the order of the dissipation length y, the size of the smallest eddies in a fully turbulent (non- Newtonian) LES fluid is on the order of the filter length I f . One might presume that the role that y plays in Navier-Stokes turbulence is similar to that of l f in LES turbulence. In the next Sections we will show in more detail that, in full analogy to the dissipation length y in a Newtonian fluid, l f is physically a dissipation length in an LES fluid.

The Navier-Stokes equations describe the energy dissipation of the turbulent kinetic energy due to the random motion of the molecules. The molecular motion itself, however, is not explicitly described but is parameterized. In other words the Navier-Stokes equations do not know anything about the existence of individual molecules; in a Navier-Stokes fluid (a fluid that is described by the Navier-Stokes equations), there are a priori no structures at length scales considerably smaller than y. Even if it were possible to carry out a DNS with a grid spacing A smaller than the size of the molecules one could not expect that the DNS would provide insight into the existence of individual molecules. Correspondingly, the LES equations describe the energy dissipation of the large eddies due to the effects of the eddies smaller than l f . The motion of the small eddies itself, however, is not explicitly described but is parameterized. In other words the LES equations do not know anything about the existence of individual eddies smaller than l f , i.e. of eddies within the inner inertial range; in an LES fluid, there are a priovi no structures at length scales considerably smaller than l f . Even if a LES were carried out with A smaller than y one could not expect that the LES would resolve the eddies smaller than l f if l f is several orders of magnitude larger than y, which is usually the case.

It is important to note that up to now we have considered partial differential equations (or finite-difference equations with an arbitrarily small A ) : on the one hand the Navier-Stokes equations and on the other hand the LES equations. As stated above, in DNS d must be chosen equal to or smaller than a fraction of y since the size of the smallest nonlinear structures in a turbulent Newtonian fluid is on the order of q . Correspondingly, in LES d must be chosen smaller than a critical length that is defined by If (Mason 1994, p. 5) .

The LES technique was developed to minimize the computational expense of the simulation of turbulent flows with very high Reynolds numbers. Thus, there is generally a need to chose A as large as possible, and there is a need to know the critical A as precisely as possible. Note, however, that the need to maximize A is simply a consequence of limited computer resources. Thus, the grid spacing is maximized for technical reasons but not for physical reasons.

Similarity theory of LES-generated turbulence 243

Obviously, the value of A must have an influence on the LES results if A is close to or even larger than the critical A , which depends on 1, since 1, defines the length scales of the smallest structures in the LES-generated velocity field. On the other hand, however, it is to be expected that for a given 1, the LES results will be insensitive to A if it is chosen considerably smaller than its critical value.

In summary, we point out that it is very important to carefully distinguish between physically resolved length scales and numerically resolved length scales, and it is to be realized that the spatial filter defined by the model for zij has a priori nothing to do with the numerical grid that is used for the numerical integration of the LES equations.

2.2. A specijic LES fluid: the Smagorinsky ,fluid Following Smagorinsky (1963), Lilly (1967) suggested an eddy viscosity VLES that is proportional to the deformation tensor amplitude :

where

compare, for example, Leonard (1974), Schmidt & Schumann (1989), and Mason & Brown (1994). We call an LES fluid defined by (2.5) and (2.6) a ‘Smagorinsky fluid’ and the length 1s the ‘Smagorinsky length’. Usually, but not necessarily, I s is stated in units of a grid spacing A ,

1s = cs A ,

where the numerical coefficient cs is known as the ‘Smagorinsky coefficient’ (e.g. Schmidt & Schumann 1989, p. 556) or the ‘Smagorinsky constant’ (e.g. Germano et al. 1991). It is to be emphasized, however, that there is no need to introduce a finite grid spacing A at this point. The coefficient cs has no physical relevance as long as the LES equations, which are a priori partial differential equations, are not replaced with their finite-difference counterparts.

Obviously, ls defines the magnitude of vLES and simultaneously the magnitude of z I j . Since, in turn, the parameterization of zi, in terms of ui defines the effective spatial filter associated with the LES equations the filter length l f is defined by l s . In the next Section, we will show that the Smagorinsky length ls is the Smagorinsky-fluid counterpart of the Navier-Stokes-fluid dissipation length q.

3. Spectral analysis of homogeneous and isotropic LES-generated turbulence

3.1. The eflective spatial Jilter There are two different interpretations of the nature of the spatial filter inherent in the LES equations. The traditional approach (Lilly 1967; Leonard 1974) is as follows. The Navier-Stokes equations are spatially filtered. The filter is explicitly defined and a priori known. The result are equations of motion, the form of which is “precisely that of the Navier-Stokes equations with ziJ replacing the viscous term” (Mason 1994, p. 3 ) . Thus, the Reynolds stress tensor T~~ contains the full information concerning the

244 A. Muschinski

spatial filter. If z i j is properly parameterized the LES will provide the TKE spectrum

F(k ) = ~ i E ~ ’ ~ k - ~ ’ ~ A ~ ( k ) , (3.1)

where A ( k ) is the transfer function which is the Fourier transform of the spatial filter function (Leonard 1974, equations 4.23 and 4.24; see also Moeng & Wyngaard 1988, pp. 3574ff.). Since the filter is a priori known the transfer function and the LES-generated TKE spectrum are also a priori known. So far this is the traditional philosophy.

An alternative interpretation has been given by Mason and his co-workers (Mason & Callen 1986; Mason & Brown 1994; Mason 1994). The pivotal issue is that in an LES z i j is to be parameterized in terms of the filtered field variables, but the relation between z i j and the filtered variables is not exactly known. The Smagorinsky-type parameterization, for example, is reasonable but nevertheless ad hoc. Mason (1994, p. 4): “Since the Smagorinsky model only involves a single scalar variable 10 [the Smagorinsky-length ts in the present paper] it is immediately apparent that, whatever the approach, only the characteristic scale of a filter can be represented. A key, but as yet unanswered, question is: what particular shape of filter operation does the Smagorinsky model correspond to?”

We will not give a conclusive answer but we will see that the physical nature of this effective ‘Smagorinsky-filter’ is similar to the ‘diffuse cutoff’ (Moeng & Wyngaard 1988, p. 3578) of the inertial-range TKE spectrum of Navier-Stokes turbulence at wavenumbers on the order of q-’. In the next subsection we interpret turbulence generated by a Smagorinsky-type LES as turbulence in a hypothetical Smagorinsky fluid and get some insight into the LES-generated (‘resolved-scale’) TKE spectrum by making use of a Kolmogorov-type dimensional analysis.

This is the central question that the present paper deals with.

3.2. Dimensional analysis of homogeneous and isotropic turbulence

Assuming L+lS we postulate three similarity hypotheses for statistically isotropic and homogeneous turbulence generated by a cubic-grid LES that relies on a Smagorinsky- type parameterization of the Reynolds stress tensor:

in a Smagorinsky jluid

First similarity hypothesis: F (k ) is determined by E, 1s and A. Second similarity hypothesis : At wavenumbers k considerably smaller than l;’, F ( k )

is determined by E and A but does not depend on 1s. Third similarity hypothesis: F(k) is determined by E and I s but does not depend on

A if A is considerably smaller than ls. The first two hypotheses are similar to Kolmogorov’s (1941~) two hypotheses. The

third hypothesis, however, is postulated because the LES equations are a priori partial differential equations and because it is to be expected that the numerical solution of these equations becomes asymptotically independent of A if A is chosen considerably smaller than a certain critical value (Mason 1994, pp. 7ff.).

Defining the two dimensionless parameters

and


and making use of the n-theorem (see, e.g., Gortler 1975 and the references cited by him) we find that in the general case the LES-generated TKE spectrum may be written

F ( k ) = E2.’3k-5’3G(L’,, nl), (3.4)

where G is a dimensionless function of two dimensionless variables. The inertial- range limit of G(nl,I12) is G(0,U2) and may be considered the LES counterpart of the Kolmogorov constant a :

~ L E S ( C S ) = G(0, nz). (3.5)

Note that in the general case the LES-generated Kolmogorov coefficient uLES is a function of cs. Introducing

provides

where

for any value of cs.

3.3. Lilly’s ( 1 967) assumptions considered from the viewpoint of the similarity theory Lilly’s (1967) well-known relationship between the Smagorinsky coefficient and the Kolmogorov constant cc can be rederived from the general LES TKE spectrum, (3.7), as follows. Lilly (1967) assumed

and

~ L E S ( C S ) = u, (3.10)

where u is the quasi-universal Kolmogorov constant which is empirically known from real-world turbulence. Muschinski & Roth (1993) suggested a local effective cut- off wavenumber n / ( 2 z ) (where z is the distance from the surface) for surface-layer turbulence. Schumann (1994) pointed out that this cut-off wavenumber is closely related to Lilly’s (1967) cut-off wavenumber n/A for LES-generated turbulence.

Making use of approximations for the ensemble averages r: and VLES in homogeneous LES turbulence,

and

(3.11)

(3.12)

246 A. Muschinski

one obtains Lilly’s (1967) result

c s = - ( l ) 1 3a 3/4 . 71

(3.13)

3.4. The nature of the Smagorinsky length The Smagorinsky length ls defines the magnitude of the variable kinematic viscosity of a fully turbulent Smagorinsky fluid, see (2.5). In analogy to Kolmogorov’s (1941~) dissipation length, (2.11, we define a dissipation length for a LES fluid:

(3.14)

Using (3.11) and (3.12) we obtain

~ L E S = CS A = 1s. (3.15)

Thus, we have shown that the Smagorinsky-length is the dissipation length of the Smagorinsky-fluid, and we can rewrite (3.7):

(3.16)

3.5. Comparison with Kolmogorov’s (1 941 ,) similarity theory According to Kolmogorov’s (1941~) similarity theory, the three-dimensional TKE spectrum in locally homogeneous and isotropic fully developed turbulence is uniquely determined by the mean energy dissipation rate E and the molecular kinematic viscosity v :

F(k) = aE2/3k-5/3 f (kv), (3.17)

where 1 i 4

q = (;) and

limf(x) = 1. X 4

(3.18)

(3.19)

In the early 1940s, it was pointed out by Landau that the spatial distribution of E

should be taken into account in a reliable similarity theory for turbulence at high Reynolds numbers. This was done by Kolmogorov (19621, but the influence of the small-scale intermittency on the shape of the TKE spectrum has proved to be small. It is negligible in many applications.

According to the third similarity hypothesis postulated above we expect for large Smagorinsky coefficients :

It is tempting to presume

lim ~ L E S ( C S ) a Cs+m

(3.21)

and

(3.22)


where a is the quasi-universal Kolmogorov constant and f(x) the quasi-universal ‘damping function’ for Navier-Stokes turbulence. In other words : one might presume that Kolmogorov’s (1941a) similarity theory not only holds for Navier-Stokes turbulence but also for turbulence generated by a Smagorinsky-type LES if cs is sufficiently large.

This presumption appears to be justified if the details of the spatial and temporal distribution of vLES in a fully turbulent Smagorinsky fluid may be ignored and if its mean value is the important parameter that characterizes the TKE spectrum. Such a ‘vLEs-homogeneity assumption’ is encouraged by the success of the &-homogeneity assumption that Kolmogorov’s (19414 theory relies on. Moreover, the numerical experiments carried out by Bardina, Ferziger & Reynolds (1983) and by Mason & Brown (1994) may also be seen as an empirical verification of the unimportance of the details of the distribution of VLES in the case of homogeneous turbulence: “In the flow interior, it seems that only the mean value of the eddy-viscosity is at issue” (Mason & Brown 1994, p. 134).

3.6. LES dissipation spectra The normalized dissipation spectrum is given by

g L E S ( X , C S ) = X ’ / 3 f L E S ( X , C S ) , (3.23)

where

x = ~ L E S (3.24)

is the dimensionless wavenumber. Figure 1 shows three models for gLEs(x,cs) . Two of the three curves correspond with the semi-empirical damping functions after Heisenberg (1948),

(3.25)

and Pa0 (1965),

(3.26)

respectively. We have assumed the standard value a = 1.5. Although the discussion about the definitive asymptotic form of f(x) for large x has not been settled (see, for example, Schumann 1994 and Saddoughi & Veerevalli 1994), the models by Heisen- berg (1948) and Pa0 (1965) are used here since they fulfil the accuracy requirements that are appropriate in the present context.

The third curve in figure 1 is the dissipation spectrum that follows from Lilly’s (1967) model, see (3.13) and (3.9):

i a 413 f P ( X ) = exp (-yx ) ,

where 314

xs = (i) (3.27)

(3.28)

is the dimensionless cut-off wavenumber according to Lilly (1967). In figure 1, CI = 1.5 has been assumed for Lilly’s dissipation spectrum also.

248 A . Muschinski

1.2

0.9

- . - - . - . - - . - - - I , ‘ I , , I

-

3.7. The LES-generated Kolmogorov coeficient as a function of

Now we derive the LES-generated Kolmogorov coefficient as a function of the Smagorinsky coefficient. Combining (3.1 1) and (3.12) yields

the Smagorinsky coeficient

E = ( c s A ) ~ (2 I“ F(k)k2dk) 3 i 2 .

Inserting (3.7) and resolving for RLES leads to

where

x = kls = kcsA.

(3.29)

(3.30)

(3.31)

A simple model for ~ L E S ( X , cs) is the large-cs limit, limes+oo ~ L E S ( X , CS), cut-off at

X, = kscS A = ZCS. (3.32)

Such a cut-off has already been suggested by Lilly (1967) but simultaneously he made use of the crude assumption f(x) = 1. Assuming Kolmogorov’s (1941~) universal damping function, f(x), as the large-cs limit of ~ L E S ( X , CS), see (3.22), we obtain

(3.33)

It is easy to verify Lilly’s a-cs-relationship, (3.13), for f (x) = 1. As two more precise models for f(x), which take the decoupling of aLES from cs for

large cs into account (which Lilly’s model does not), we suggest the above-mentioned


models by Heisenberg (1948) and Pao (1965). Inserting Heisenberg’s model, (3.25), into (3.33) reveals a model for aLEs(cS) in closed form. After solving the integral and after some elementary manipulations we obtain

(3.34)

This result and the alternative model for c ( L E S ( C S ) which relies on Pao’s (1965) model will be discussed in $4.

3.8. Dimensional analysis of homogeneous and isotropic turbulence generated

Consider a finite-difference Smagorinsky-type LES with an anisotropic grid. Let A 1, A 2 and A 3 be the three grid spacings, where

A I ,< A2 ,< A3. (3.35)

We modify the similarity theory described above in order to account for anisotropic grids. Assuming again L+ls we postulate:

with an anisotropic-grid LES

First similarity hypothesis: F ( k ) is determined by E, I S , A , , A 2 , and A3. Second similarity hypothesis: At wavenumbers k considerably smaller than l;’, F ( k )

Third similarity hypothesis: F ( k ) is determined by E and I s but does not depend on is determined by 8, A , , A 2 , and 43 but does not depend on 1s.

A , , A 2 , and A , if A l , A 2 , and A 3 are considerably smaller than 1s. We define four dimensionless variables :

I71 = k l s , (3.36)

and

(3.37)

(3.38)

(3.39)

where

AD = ( A , A2A3)’ /3 (3.40)

is a measure for the grid spacing as used by Deardorff (19701, and where a1 and a2 are ‘aspect ratios’ that quantify the anisotropy of the grid (Scotti, Meneveau & Lilly 1993). The Il-theorem leads to

F ( k ) = :2/3k-5/3G,(ni,n2,n3,114) (3.41)

where G, is a dimensionless function of four dimensionless variables. The suffix a stands for anisotropic.

We normalize G,(nl, n2, n3, n4) with the inertial-range limit, G,(O, II2 , n3, n4), and get

(3.42)

250

where

A. Muschinski

(3.43)

G,(0,f72,n,,n4) is the LES-generated Kolmogorov coefficient in the case of an anisotropic grid, and it is appropriate to express it in terms of the LES-generated Kolmogorov coefficient in the case of an isotropic grid and an ‘anisotropy factor’ Pa which depends in the general case on CS, a1 and a2:

~LES(CS)P~(CS, ai, a21 = GJO, n2, n3, n4). (3.44)

B a ( C s , 41) = 1. (3.45)

In the case of an isotropic grid we have

Scotti et al. (1993) generalized Lilly’s (1967) analysis for an anisotropic grid, assuming a three-dimensional anisotropic cut-off filter defined by three different cut-off wavenumbers, corresponding with the three different grid spacings. Moreover, they assumed

~LES(CS)P~(CS, ai, a21 = a (3.46)

and obtained cs as a function of the empirical Kolmogorov constant a and of the grid aspect ratios a1 and a2.

Finally we consider the case that A l , A 2 , and A3 are considerably smaller than E s . Then, according to the third similarity hypothesis, aL~s(cS) is equal to the isotropic large-cs limit, and Pa(cs,a1,a2) = 1 even for an anisotropic-grid LES.

It seems possible that dimensional analysis can lead to further insight into the asymptotic behaviour of the anisotropy function Pa(cs, al, a2) for various limits of CS,

al, and a2. I believe in this case, instead of making use of cs, a l , and a2, it would be more appropriate to introduce three different Smagorinsky coefficients cs1 = Zs/d 1,

cs2 = ls/A2, and cs3 = l s / A 3 . As one of the reviewers pointed out, it would be of interest to consider the case that one (two) of the three Smagorinsky coefficients is (are) small and two (one) are (is) large. A detailed discussion of this problem, however, is beyond the scope of this paper.

4. Discussion 4.1. The relevance of the Smagorinsky coeficient for the LES-generated TKE

Figure 1 shows clearly the difference between LES with a small cs and a LES with a large cS. The dissipation spectra g H ( x ) and g p ( x ) have maxima at wavenumbers x about half the cut-off wavenumber x, suggested by Lilly (1967), indicating insensitivity of the TKE budget of the LES-generated turbulence if x, is chosen considerably larger than Lilly’s x,. The maximum of Lilly’s crude cut-off dissipation spectrum, gL(x), however, is at xs, implying a priori a coupling between the TKE budget and both the magnitude of the cut-off wavenumber and of the grid spacing A .

Figure 2 shows several models for C I L E S ( C S ) . The solid lines represent C I L E S ( C S )

obtained from inserting f L ( x ) , f H ( x ) , and f p ( x ) , respectively, into Eq. (3.33). The dashed line marks the value of limcs+co(aLES), which has ad hoc been assumed to be 1.5. Since Lilly’s (1967) model assumes a cut-off of the spectrum at the largest wavenumber resolvable by the grid, it is a priori clear that it fails for larger C S , i.e. in the case of a decoupling of the LES turbulence from the numerics. Note that

spectrum and the LES-generated Kolmogorov coeficient

Similarity theory of LES-generated turbulence 25 1

Heisenberg (1948)

0.1 0.2 0.3 0.4 0.5

CS

FIGURE 2. Models for the Kolmogorov coefficient C ( ~ E S generated by a Smagorinsky-type LES as a function of the Smagorinsky coefficient cs. The three solid curves show CILES(CS) according to Lilly (1967) and two models for aLEs(cS) that rely on the spectral models by Heisenberg (1948) and Pao (1965), respectively. For the asymptotic value, lim,.s,x zLES (cs) 1.5 has been assumed (dashed line).

the Lilly asymptote and the (constant) large-cs asymptote intersect exactly at Lilly’s (1967) value for cs. Thus, Lilly’s (1967) cs may be considered a critical cs.

Both Pao’s (1965) and Heisenberg’s (1948) models also imply a systematic overestimation of aLES for cs larger than 0.2 if the energy at the smallest grid scales is not artificially removed. Pao’s (1965) model leads to a larger predicted overestimation than Heisenberg’s (1948) because the ‘diffuse cut-off’ of the TKE spectrum according to Pao’s model is more diffuse than that according to Heisenberg’s model, see figure 1. For lim,,,,(aLES) = 1.5 and cs = 0.2 we obtain XLES = 1.74 (16% overestimation) from Heisenberg’s model and XLES = 2.14 (43% overestimation) from Pao’s model. A possible intrinsic overestimation of LES-generated Kolmogorov coefficients has been reported by Chasnov (1991): “ ... it appears that the value of K O [CILES in the present paper] obtained from numerical simulations is approximately 30% higher than that obtained in high Reynolds number atmospheric experiments. The origin of this discrepancy remains to be understood.” Chasnov (1991) tested a possible cut-off effect by comparing a 643 run with a 12S3 run and obtained an even slightly larger ~ L E S for the 12S3 run than for the 643 run, which seems to exclude that the overestimation is to be attributed to a truncation effect. On the other hand, the physics of Chasnov’s spectral wavenumber-space LES, which is based on a spectral subfilter closure, is different from the physics of a basic finite-difference Smagorinsky-type LES, which is carried out in physical space. Hence, the reliability of a direct comparison between Chasnov’s results and results expected for the basic Smagorinsky-type LES, might be doubtful.

In most LES applications, values for cs between 0.1 and 0.2 have been used. These values are fairly close to Lilly’s (1967) value, which marks the intermediate range between the small-cs asymptote and the large-cs asymptote. Using a cs in the intermediate range has three practical advantages : first, the LES-generated Kolmogorov coefficient is not very sensitive to cs ; second, the LES-generated Kolmogorov coeffi-

252 A. Muschinski

cient is expected to be not too far from the quasi-universal Kolmogorov coefficient a; third, the numerical expense of the LES is minimized.

For large values of cs the length scale defined by the model for zij (and not the grid spacing) determines the width of the effective spatial filter. For small values of cs, however, the effective spatial filter is determined by the numerical grid. In a smal lq LES, the grid spacing A plays the role of the transport length that determines the magnitude of the effective subfilter viscosity. In the intermediate range, i.e. at Smagorinsky coefficients close to Lilly’s (1967) value, obviously both the numerical grid and the zij closure determine the effective spatial filter. Thus, in an intermediate- cs LES the nature of the turbulent motions at the smallest scales is ambiguous. The physical properties are defined at distinct points, namely at the grid points. One might speculate that for small and intermediate values of cs the dynamics of the velocity fluctuations at scales comparable with the grid spacing is somewhat similar to the microphysical dynamics of a solid at the smallest scales.

The grid spacing may be considered the LES-fluid counterpart of the mean free path in a gaseous Navier-Stokes fluid. Tennekes & Lumley (1972, p. 23) called the ratio between the mean free path and the dissipation length,

A

r K n , = -,

a “microstructure Knudsen number”. It is generally expected that turbulence can be properly described within the framework of continuum mechanics as long as K n, is smaller than 1. The LES-fluid counterpart of K n , is

Thus, the Smagorinsky coefficient is physically the reciprocal of a microstructure Knudsen number (or a ‘grid Knudsen number’). Mason (1994, p. 8), however, considers cs the reciprocal of the square root of a “mesh Reynolds number”.

Figure 2 indicates that the overestimation becomes negligible for cs larger than about 0.5. Experimentalists (see Kutznetsov, Praskovsky & Sabelnikov 1992, p. 602) have recommended using a wire length L (which defines the spatial resolution in hot-wire anemometry) smaller than 217 for investigations of fully developed turbulence. Since ls is the Smagorinsky- fluid counterpart of 7 the requirement cs <0.5 is the Smagorinsky-fluid counterpart of the requirement L<2y. This analogy may be seen as a further indication for the reasonableness of the presumption that there is a similarity between the ‘diffuse cut-offs’ in Smagorinsky-fluid turbulence and in real-world turbulence.

This is equivalent to the requirement A<21~.

4.2. Subgrid scales, subfilter scales, and resolved scales It is worthwhile to look more closely at the different terminologies used by the different authors. Most researchers in the LES community (see, e.g., Deardorff 1980; Moeng & Wyngaard 1988; Schmidt & Schumann 1989; Kaltenbach, Gerz & Schumann 1994) use the term ‘subgrid scales’. Mason and his co-workers (see, e.g., Mason 1994; Mason & Brown 1994), however, prefer the term ‘subfilter scales’. A third notion is ‘resolved scales’. What is the difference between these terminologies? Here I suggest a definition.


Subgrid scales are length scales that cannot be resolved by the numerical grid, i.e. length scales smaller than

A, = 24, (4.3) where d is the grid spacing. The factor 2 is due to the sampling theorem.

A precise definition for the subfilter scales is more problematic. Consider a large-cs LES. There is a length scale Af at which the LES-generated three-dimensional TKE spectrum drops off significantly in comparison with the inertial-range k-5/3-spectrum. If we define a decrease by a factor e = 2.7181 ... as significant and if we assume Pao’s (1965) spectrum as a reasonable approximation for the postulated universal TKE spectrum of turbulence generated by a large-cs LES, A f is approximately given by

exp (-px;/3) 3 = - 1 e’

where

(4.4)

We have shown that in the case of the Smagorinsky-type eddy-viscosity parameterization ~ L E S is given by

V L E S = c s 4 (4.6) and it follows that

314

Af = 271 cs ( :a ) A .

Inserting Lilly’s value of the Smagorinsky coefficient,

see (3.13), leads to

(4.7)

Thus, subfilter scales may be considered length scales smaller than lf. Obviously, l f equals A, if Lilly’s value for the Smagorinsky coefficient is used. In this case the terms subfilter and subgrid may be considered synomous. Using a ‘non-optimal’, i.e. a larger value of cs, however, leads to Af>R,. In this case a length scale A larger than A, but smaller than Af can be numerically resolved but not physically. This is the case in a large-cs LES. Thus, it is necessary to distinguish carefully between ‘physically resolved scales’ and ‘numerically resolved scales’.

4.3. Non-homogeneous LES turbulence The main purpose of LES is to model turbulence under real conditions. Real turbulence is often inhomogeneous owing to rigid boundaries or to stable stratification. Thus, it is very important to establish the reliability of the LES technique for non- homogeneous conditions. Recently, encouraging results of LES of a neutrally stratified boundary layer (Andren et al. 1994) and even of stably stratified flows (Kaltenbach et al. 1994; Schumann & Gerz 1995) have been published.

A ‘dynamic’ subgrid model has been developed (German0 et al. 1991) that allows

254 A . Muschinski

more of the information about the local and instantaneous state of the flow at the smallest resolved scales to be used in order to account for anisotropy at these scales. This procedure results in a locally and temporally changing Smagorinsky coefficient cs. The utility of Germano et d ’ s (1991) model has been called in question by Mason (1994). On the other hand, Kaltenbach et a!. (1994, pp. 26 ff.) are in agreement with Germano et al.’s procedure: “Our simulations lack an inertial subrange which can hardly be obtained unless the resolution is much finer than we were able to provide. A common practice in LES of a wide variety of flows is to account for the lack of an inertial subrange by adjusting CSGS [CS in the terminology of the present paper] in such a manner that energy is dissipated at a sufficient rate.” ‘Dynamic’ models might be the only way to guarantee the applicability of the LES technique in the case of stably stratified turbulence if the turbulence is anisotropic at all resolved length scales. But such procedures are not quite satisfactory. Making use of them seems to be a step backwards from the original concept of LES since a major advantage of the LES technique lies in its conceptual simplicity.

Even worse than in the presence of stably stratified regions is the situation in the vicinity of rigid boundaries: “It is disappointing to find that the boundary regions in large-eddy simulations contain serious errors. This cannot, however, be considered too surprising, as close to the surface the potential rationality of the large-eddy simulation vanishes as the dominant eddy-scales become comparable with, and smaller than, the filter-scale’’ (Mason 1994, p. 17).

Generally a Smagorinsky-type LES is expected to be successful if the flow is fully turbulent, if the length 1s is always significantly smaller than the local outer scale and larger than the local inner scale of the turbulence to be modelled, and if the turbulence to be modelled is statistically isotropic at length scales comparable with ls. Since the outer scales are drastically reduced in regions with high static stability, realistic LES of flows under these conditions require a considerably smaller 1s than, for example, an LES of a convective boundary layer.

Recently, subfilter models have been developed which take into account backscatter of subfilter energy into the resolved-scale regime and reduce the intrinsic determinism of the basic Smagorinsky model by introducing a stochastic force at the smallest resolvable scales (see, e.g., Mason & Thomson 1992; Mason 1994; Schumann 1995). A further subfilter model is the structure function model put forward by Mitais & Lesieur (1992).

However, an interpretation of the relationships between the concepts developed in the present paper, which rely on classical models of homogeneous turbulence, and the more sophisticated subfilter closures mentioned above is not given here.

4.4. Intermittency and Reynolds-number similarity Another problem of the LES technique is the effective Reynolds number of the LES- generated turbulence. In LES the dissipation length q of the real-world turbulence is replaced with ~ L E S = csd. Thus,

(4.10)

may be defined as the turbulent Reynolds number of LES-generated turbulence, where Re is the Reynolds number of the real-world turbulence to be simulated by the LES, see $2. In the atmospheric boundary layer, q is on the order of 1 cm but ~ L E S is usually chosen on the order of 10 m. In this case ReLEs is about lo4 times smaller than the Reynolds number Re of the turbulence to be modelled. One must be careful that


ReLEs remains sufficiently large. Otherwise the Reynolds number similarity might not be fulfilled, and the LES results could not be considered representative for a large-Re flow (see, e.g., Moeng & Wyngaard 1988).

It is known, however, that some statistical properties of fully developed turbulence are not independent of the Reynolds number even at very large Re, i.e. even if the Reynolds number similarity is to be expected to be guaranteed. Wyngaard & Tennekes (1970) observed that the skewness and kurtosis of the first derivative of the streamwise velocity component depend on Re. The 4/3-ratio between longitudinal and transversal velocity correlations that is expected from the classical theory (equation 24 in Kolmogorov 1941~) has been observed. See, e.g., Hauf (1984). Mestayer (1982), however, reported experiments that did not give support to the theoretical 4/3-ratio. Also the LES results by Moeng & Wyngaard (1988) did not reproduce the theoretical value, compare also Schmidt & Schumann (1989, p. 528). Mestayer suggested that the 4/3-ratio is to be expected mainly at the small-scale regime of the inertial subrange. This implies that the existence of a 4/3-ratio depends on the Reynolds number. Another Reynolds-number dependency has been suggested by Oncley et al. (1990). They found empirically a relation between von Karmin’s constant K and the roughness Reynolds number, leading to a smaller IC for smooth terrain than for rough terrain.

Moeng & Wyngaard (1988) considered the influence of the spatial fluctuations of LES-generated energy dissipation E on the TKE budget and pointed out that the LES-generated Kolmogorov coefficient depends on the width of the E distribution. It seems that the small-scale intermittency of LES-generated turbulence is an important effect at least for simulations of the convective boundary layer.

Recently, Schumann (1995) investigated LES-generated turbulence for three different values of cs: 0.083, 0.165 and 0.330. While cs = 0.165 (which corresponds to Lilly’s cs for 2 = 1.6) led to a realistic inertial-range spectrum with a ~ ~ s c 1 . 6 (see Erratum to Schumann 1995) the doubled and halved cs provided unrealistic spectra (Schumann 1995, p. 311). This appears to be incompatible with the presumption that the LES results are asymptotically independent of cs if c g is chosen larger than Lilly’s cs-value (see figure 2). Here I suggest a possible explanation of Schumann’s (1995) finding. Let N be the number of the grid points in one dimension. For simplicity we identify the side length L of a cubic modelled volume with the outer scale of the LES-generated turbulence. Then we have

and finally

(4.1 1)

(4.12)

Thus, ReLEs is reduced if cs is enlarged and vice versa, provided that N is kept constant. The number of grid points in Schumann’s (1995) LES was N = 64, corresponding to ReLEs = 7069 (2823, 1122) for cs = 0.083 (0.165,0.330). These values for ReLEs are not very large, and a violation of the Reynolds-number similarity seems possible. It is to be expected that there is a critical ReLEs and that the Smagorinsky fluid is not fully turbulent if ReLEs is smaller than this critical value. In this case, LES cannot generate an inertial-range spectrum. Possibly, ReLEs = 1122 is below this critical value. It would be of great interest to test the cs-sensitivity of a basic

256 A. Muschinski

4

3

1

0 0.1 0.2 0.3 0.4 0.5

CS

FIGURE 3. Qualitative picture of the expected behaviour of CLLES(CS) for finite LES Reynolds numbers ReLEs, i.e. for finite numbers of grid points N , where N1 < N2 < N3 < .... Note the expected plateau for large grids and the similarity to Lilly’s (1967) CLLES(CS) model for small grids.

Smagorinsky-type LES without changing simultaneously the value of R e L E s . This can be done by varying N proportionally to cs ; in other words A has to be varied while L and 1s have to be kept constant.

Figure 3 shows qualitatively CILES(CS) which is to be expected for different values of N where N1 < N2 < N3 < .... For a given N it is to be expected that C I L ~ S ( C ~ )

drops off beyond a critical c$ which is related to the critical LES Reynolds number Re:,, (which is expected to be not universal but to depend on the type of the flow) as follows:

cg = N . (4.13)

It is to be expected that for small N there is no plateau between Lilly’s cs (the critical cs for discretization errors) and c$ (the critical cs associated with the critical Reynolds number of the LES fluid). Such a plateau, however, is to be expected for larger N . In present-day LES we have N = 100, and it is possible that the expected plateau of C I ~ ~ ~ ( C S ) cannot be observed with such small values of N . Note that C I ~ ~ ~ ( C ~ ) for a small N (see figure 3) exhibits some similarity with Lilly’s (1967) model for CILES(CS),

see figure 2. This similarity, however, appears to be just a misleading coincidence.

5. Summary and conclusions Mason and his co-workers (Mason & Callen 1986; Mason 1994; Mason & Brown

1994) have pointed out that the effective spatial filter of an LES is not defined by the ‘conceptual’ filter that is applied to the Navier-Stokes equations in order to get the (continuous) LES equations. Neither is the filter determined by the numerical grid, provided the grid spacing is sufficiently small in comparison with the width of the ‘conceptual’ filter. Rather, the effective spatial filter is defined by the closure for the subfilter turbulence. The filter shape associated with the widely used Smagorinsky-type subfilter model, however, is unknown (Mason 1994). Mason’s philosophy contrasts somewhat with the traditional philosophy (Lilly 1967; Leonard 1974), which assumes


that the subfilter closure is a priori determined by the conceptual filter and that the width of the conceptual filter is simultaneously the grid spacing.

In the present paper, the LES equations have been considered equations of motion of specific hypothetical turbulent non-Newtonian fluids, called LES fluids. The Smagorinsky-fluid is by definition the LES fluid that is specified by the Smagorinsky- type subfilter closure. A Kolmogorov-type similarity theory has been suggested which leads for very large LES Reynolds numbers to the TKE spectrum

F ( k 1 = ~ L E S ( C S )C f L E S ( x , CS), (5.1) -213k-513

where fLES(x, cs) is a dimensionless function of the dimensionless wavenumber x = kls. Here, cs is the Smagorinsky coefficient and 1s = csd is the Smagorinsky length which scales the magnitude of the variable viscosity vLES. d is the mesh width (or in the case of anisotropic grid a measure of the mesh width).

The effective spatial filter is specified by the dimensionless function fLES(X, cs). From the Leonard (1974) point of view fLEs(kEs,cs) is A2(k) where A ( k ) is the Fourier transform of the conceptual filter. The key hypothesis of the present paper, however, is that the asymptote of fLES(k&,(.S) for large cs is physically the Smagorinsky-fluid counterpart of Kolmogorov’s ( 1941a) damping-function f ( x ) which describes the drop-off of the TKE spectrum at wavenumbers in the vicinity of q-’. This hypothesis is encouraged by the physical equivalence of ls and Kolmogorov’s dissipation length q , which is shown by using Lilly’s (1967) approximations for the ensemble averages E and E. One of the anonymous reviewers remarked that the pivotal issue in this paper is the assumption that for homogeneous LES-generated turbulence the eddy viscosity can be replaced with a constant viscosity. It is clear that this assumption is closely related to the hypothesis of the equivalence of limcs+s(fLES(kls, CS)) and f ( k q ) . But the ‘equivalence hypothesis’ appears to be more general since it implies that Kolmogorov’s (19414 similarity does not hold only for turbulent Navier-Stokes fluids but also for a certain (but up to now not specified) class of turbulent non- Newtonian fluids. In this respect, the concept of LES fluids seems reasonable, and it might be useful for defining alternative subfilter closures.

If Mason is right in saying that for homogeneous turbulence LES provides similar results to DNS with a constant (properly defined) viscosity, what is the advantage of LES compared with DNS? Both LES and DNS provide spatial and temporal distributions of energy dissipation rates, variances, and fluxes. However, there is a decisive difference: while in a Navier-Stokes fluid the viscosity is the material parameter and the dissipation length is variable, in a Smagorinsky fluid the dissipation length is the material parameter and the viscosity is variable. Thus, replacing a turbulent Navier-Stokes fluid with an equivalent turbulent LES fluid allows simulation of turbulence at arbitrarily large Reynolds numbers by making optimal use of the limited wavenumber-space regime that can be represented by a finite numerical grid.

We have combined Lilly’s (1967) cut-off approximation with two classical dissipation -range models (Heisenberg 1948; Pa0 1965). The combined models predict an intrinsic overestimation of the LES-generated Kolmogorov coefficient for finite values of cs. For cs = 0.2 we obtain 16% overestimation from Heisenberg’s (1948) model and 43% from Pao’s (1965) model. The predicted overestimation becomes negligible for values of cs beyond about 0.5. In other words both models predict non-negligible finite-difference-approximation errors for cs smaller than about 0.5. It has been shown that the requirement cs>0.5 is equivalent to the requirement L<2y ( L is the wire length of a hot-wire anemometer) which has been recommended by experimentalists.

258 A. Muschinski

It has been pointed out that for a constant number of grid points N an enlargement of cS implies a reduction of the LES Reynolds number ReLEs, and it is to be expected that for a finite N there is a critical value for cs associated with a critical ReLES. Thus, for any finite N there are two critical Smagorinsky coefficients: first Lilly’s (1967) C S ,

indicating for which values of cs finite-difference-approximation errors are important, and second a critical cs beyond which the Reynolds similarity is violated. Reasonable LES results are to be expected if cs is chosen not smaller than the first and not larger than the second critical Smagorinsky coefficient.

There appears to be a lack of a systematic study of homogeneous turbulence generated by a basic Smagorinsky-type finite-difference LES for different values of cs (between, say, 0.2 and 1.0) without changing simultaneously the LES Reynolds number. Such numerical experiments could empirically provide the function ~ L E S ( X , cs) and the LES-generated Kolmogorov-coefficient CLLES(CS). While it is to be expected that f L ~ s ( x , c S ) and aLES(cS) are not universal for small and intermediate cs but rather depend on the specific finite-difference-approximation scheme, it would be of great interest to get insight into the asymptotic shape of the Smagorinsky filter, limcs+03 fLEs(x, cs), and to determine the asymptotic value limcs+m CLLES(CS), i.e. the Kolmogorov coefficient for continuous Smagorinsky turbulence.

I thank P. J. Mason and U. Schumann for their valuable comments on earlier versions of this paper. Thanks also to R. Blender, P. Chilson, S. Raasch, V. Schilling, and Z. Sorbjan for helpful discussions and comments. I appreciate the valuable comments by three anonymous reviewers, and I would like to give thanks to F. Herbert and F. Fiedler who invited me to seminar talks in Frankfurt and Karlsruhe, respectively, on the topic of this paper, giving me the occasion to discuss my results.

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319

Small-scale and large-scale intermittencyin the nocturnal boundary layer

and the residual layer

By ANDREAS MUSCHINSKI1†, ROD G. FREHLICH2

AND BEN B. BALSLEY2

1CIRES, University of Colorado and NOAA Environmental Technology Laboratory,325 Broadway, R/ET2, Boulder, Colorado 80305-3328, USA

2CIRES, University of Colorado, Campus Box 216, Boulder, Colorado 80309, USA

(Received 11 September 2003 and in revised form 6 May 2004)

In high Reynolds-number turbulence, local scalar turbulence structure parameters,(C2

θ )r , local scalar variance dissipation rates, χr , and local energy dissipation rates,εr , vary randomly in time and space. This variability, commonly referred to asintermittency, is known to increase with decreasing r , where r is the linear dimensionof the local averaging volume. Statistical relationships between χr , εr , and (C2

θ )rare of practical interest, for example, in optical and radar remote sensing. Some ofthese relationships are studied here, both theoretically and on the basis of recentobservations. Two models for the conditionally averaged local temperature structureparameter, 〈(C2

θ )r |εr〉, are derived. The first model assumes that the joint probabilitydensity function (j.p.d.f.) of χr and εr is bivariate lognormal and that the Obukhov–Corrsin relationship, (C2

θ )r = γ ε−1/3r χr , where γ = 1.6, is locally valid. In the second

model, small-scale intermittency is ignored and C2θ and ε are treated traditionally,

that is, as averages over many outer scale lengths, such that C2θ and ε change only as

a result of large-scale intermittency. Both models lead to power-law relationships ofthe form 〈(C2

θ )r |εr〉 = c εδr , where c is a constant. Both models make predictions for the

value of the power-law exponent δ. The first model leads to δ = ρxyσy/σx −1/3, whereσx and σy are the standard deviations of the logarithms of εr and χr , respectively,and ρxy is the correlation coefficient of the logarithms of χr and εr . This model leadsto δ = 1/3 if ρxy = 2/3 and if σx = σy . The second model predicts δ = 2/3, regardlessof whether (i) static stability and shear are statistically independent, or (ii) theyare connected through a Richardson-number criterion. These theoretical predictionsare compared to fine-wire measurements that were taken during the night of 20/21October 1999, at altitudes of up to 500 m in the nocturnal boundary layer and theoverlying residual layer above Kansas. The fine-wire sensors were moved up anddown with the University of Colorado’s Tethered Lifting System (TLS). The datawere obtained during the Cooperative Atmosphere-Surface Exchange Study 1999(CASES-99). An interesting side result is that the observed frequency spectra of thelogarithms of εr and χr are described well by an f −1 law. A simple theoreticalexplanation is offered.

† Present address: Electrical and Computer Engineering Department, University ofMassachusetts, Knowles Building, 151 Holdsworth Way, Amherst, MA 01003-9284, USA

320 A. Muschinski, R. G. Frehlich and B. B. Balsley

1. IntroductionAtmospheric turbulence occurs at time and length scales that range over many

orders of magnitude. Obukhov (1962) suggested distinguishing between small-scaleturbulence and large-scale turbulence, where small scales are scales at which “thehypothesis of three-dimensional isotropy is valid in a certain rough approximation”,and large scales are those at which the fluctuations are necessarily anisotropic becausebeyond a certain scale, atmospheric flow is always quasi-two-dimensional.

Now, let L be the diameter of the largest turbulent eddies that are still part ofthe small-scale turbulence. That length scale is known variously as the ‘outer scale’,‘large-eddy scale’, or ‘overturning length’. In the vicinity of the ground or a wall, L

is of the order of the distance between the observation point and the boundary. Athigher levels in the atmosphere, the small-scale turbulence does not ‘feel’ the presenceof the ground, and L is determined by the local mean vertical gradient of the potentialtemperature and the local mean vertical shear of horizontal velocity.

Large-scale turbulence (mesoscale variability, gravity waves, or synoptic-scaledisturbances) is known to have drastic effects on the statistical parameters of thesmall-scale turbulence. Randomness in small-scale turbulence statistics is generallyreferred to as intermittency. It is important, however, to distinguish between theintermittency that is an inherent part of small-scale turbulence (Kolmogorov 1962;Obukhov 1962) and the intermittency that is associated with, and caused by, large-scale variability of the flow. The problem of distinguishing between the two doesnot arise in wind-tunnel experiments because large-scale turbulence simply does notexist there. As pointed out by Obukhov (1962, p. 78): “In the study of turbulencein wind tunnels one usually takes averages over a certain ensemble corresponding toa time ensemble, and the averaging period is larger than the life-time of the largesteddies. In the case of atmospheric turbulence there arise specific difficulties; so thatthe life-time of the largest eddies in the atmosphere greatly exceeds, as a rule, thetime of measuring turbulent characteristics.”

In the turbulence literature, the term ‘intermittency’ has been used more broadlythan outlined above, and with different meanings that vary across authors anddisciplines. Intermittency has been defined or characterized simply as “another namefor nonstationarity” (Trevino & Andreas 2000) or as “an unexpected high probabilityof large velocity fluctuations” (Boettcher et al. 2003). Sreenivasan (1999, p. S389)defines turbulence intermittency as the non-Gaussianity of small-scale statistics:“Roughly speaking, intermittency means that extreme events are far more probablethan can be expected from Gaussian statistics and that the probability densityfunctions of increasingly smaller scales are increasingly non-Gaussian . . . ”

Sreenivasan & Antonia (1997, p. 441) divide small-scale intermittency intodissipation-scale intermittency and inertial-range intermittency: “Batchelor &Townsend (1949) showed that the non-Gaussian behavior in the pdf of dissipationquantities increased with decreasing scale. In a complementary sense, dissipationquantities become increasingly non-Gaussian as the Reynolds number increases. Theseare the two hallmarks of dissipation-scale intermittency. For the inertial range, sincethe Reynolds number variation should be irrelevant, intermittency requires that thepdfs of wavenumber bands show increasingly flared-out tails for increasing midbandwavenumber, or that the flatness of velocity increments increases with decreasingscale.” Mahrt (1989) distinguishes between “small scale or microscale intermittency”and “global intermittency”. This is exactly what we will refer to here as small-scaleintermittency and large-scale intermittency, respectively.

Intermittency in the nocturnal boundary layer and residual layer 321

For many years, clear-air Doppler radars have been used to remotely sense energydissipation rates and refractive-index structure parameters in the troposphere andthe stratosphere (e.g. VanZandt et al. 1978; Gage 1990; Nastrom & Eaton 1997;Doviak & Zrnic 1993; Muschinski & Lenschow 2001). Clear-air radar data areusually processed with averaging times ranging from minutes to hours and withaveraging volumes with diameters ranging from tens of metres to a few kilometres.Recently, Muschinski (2004) generalized the existing theory of clear-air radio-wavebackscatter and analysed the effects of spatial and temporal variability of localturbulence statistics within the radar’s resolution volume and during the radar’s dwelltime. There is a similar interest in small-scale intermittency in the area of opticalremote sensing (e.g. Frehlich 1992; Wheelon 2001, 2003).

In this study, we investigate both small-scale and large-scale intermittency in thenocturnal boundary layer (NBL) and in the lower part of the residual layer (RL). Weanalyse measurements collected with vertical arrays of fine-wire anemometers andthermometers lifted to altitudes as much as 500 m above ground level. The data weretaken on 21 October 1999 during Intensive Observational Period 9 (IOP 9) of themonth-long field campaign of the Cooperative Atmosphere-Surface Exchange Study1999 (CASES-99) conducted near Leon, Kansas (Poulos et al. 2002).

2. Small-scale turbulence and small-scale intermittency2.1. Small-scale turbulence

One of the main goals of understanding small-scale turbulence is to establish lawsfor the statistical properties of two-point velocity increments,

�u ≡ u(x ′′, t) − u(x ′, t), (2.1)

where u is the local and instantaneous velocity vector, and x ′ and x ′′ are two fixedlocations.

An important working assumption is that in a space–time domain D whosespatial dimensions are sufficiently small, the statistical properties of �u are nearlyhomogeneous, isotropic, and stationary. In other words, the statistics of �u areassumed to be independent of time t , independent of the mid-point locationx ≡ (x ′+x ′′)/2, and independent of the orientation of the separation vector s ≡ x ′′−x ′,such that the statistics are functions only of the magnitude s ≡ |s| of the separationvector. Mid-point and separation coordinates had already been used by Tatarskii(1961). The advantage of using mid-point and separation coordinates instead of usingthe ‘end-point coordinates’ x ′ and x ′′ were recently discussed by Hill & Wilczak (2001).

According to classical turbulence theory (Kolmogorov 1941), any �u statistic for agiven D should depend only on the molecular kinematic viscosity ν and on the averagevalue of the energy dissipation rate, ε, over that specific domain D (Kolmogorov’sfirst similarity hypothesis). Furthermore, in the inertial range, i.e. for r large comparedto the dissipation length, or Kolmogorov microscale,

η =

(ν3

ε

)1/4

, (2.2)

but still small compared to the outer length scale L, any �u statistic should beindependent of ν and depend solely on ε (Kolmogorov’s second similarity hypothesis).

In a broader sense, turbulence is also randomness in the field of a scalar variable,θ , like temperature, humidity, refractive index, etc. We refer to such turbulence as‘scalar turbulence’ as opposed to ‘velocity turbulence’. The functional form of the


second-order structure function of scalar increments

�θ ≡ θ(x ′′, t) − θ(x ′, t) (2.3)

in the inertial subrange was predicted by Obukhov (1949) and Corrsin (1951).

2.2. Small-scale intermittency

In their refined similarity theory, Kolmogorov (1962) and Obukhov (1962) introducedlocal averages εr over (small) spheres with radius r � L. They assumed that εr islognormally distributed and that the variance σ 2

x of the logarithm of εr increases withdecreasing r like

σ 2x = A + µ ln

L

r, (2.4)

where µ is a universal constant and the coefficient A “depends on the macrostructureof the flow” (Kolmogorov 1962, p. 83). Equation (2.4), together with the lognormalityassumption, is sometimes referred to as Kolmogorov’s third similarity hypothesis(e.g. Sreenivasan, Antonia & Danh 1977).

Van Atta (1971) hypothesized, firstly, that local averages χr of scalar variancedissipation rates should be lognormally distributed and that the variance σ 2

y of thelogarithm of χr increases with decreasing r:

σ 2y = Aθ + µθ ln

Lθ

r, (2.5)

where Lθ and µθ are the scalar counterparts of L and µ, respectively. Secondly, VanAtta (1971) hypothesized that the joint probability of εr and χr should be bivariatelognormal. It is natural to assume that L and Lθ are of comparable magnitude.Relationships resulting from the joint lognormality hypothesis are summarized inAntonia & Van Atta (1975). A review on intermittency in passive-scalar turbulencewas given by Warhaft (2000).

3. Fine-wire measurements of local dissipation rates and structure parameters:concepts and procedures

Local dissipation rates can be retrieved from fine-wire measurements by meansof two different techniques: the direct dissipation technique and the inertial-rangetechnique. Both retrieval techniques rely on a number of assumptions. An outline isgiven in the following.

For incompressible, Newtonian fluids at high Reynolds numbers, εr and χr aregiven by

εr =

⟨ν

2

3∑i=1

3∑j=1

(∂ui

∂xj

+∂uj

∂xi

)2⟩

r

(3.1)

(e.g. Taylor 1935, equation 41 on p. 436) and

χr =

⟨2κ

3∑i=1

(∂θ

∂xi

)2⟩

r

(3.2)

(e.g. Sreenivasan 1996), where ui and uj are the ith and j th, respectively, componentsof the fluctuating part of the velocity vector, θ is the fluctuating part of thetemperature, and κ is the molecular heat conductivity. The spatial averaging, denotedby 〈·〉r , is performed over a domain of linear size r . Kolmogorov (1962) specified theaveraging volume as a sphere with radius r .


With a single fine wire, however, only fluctuations in the streamwise direction canbe measured, such that (3.1) and (3.2) cannot be directly applied for two reasons:first, a fine wire measures only fluctuations along a line, which makes averaging overvolumes impossible; second, (3.1) and (3.2) also involve gradients in the transversedirections, which cannot be measured with a single fine wire.

It is commonly assumed that the shape of the averaging volume is of minorimportance (Monin & Yaglom 1975, p. 591) and that it is only the linear dimension r

of the averaging domain that matters. Under these assumptions, volume averaging canbe replaced by line averaging. Moreover, it is commonly assumed that the fluctuationsare statistically isotropic at scales comparable to the Kolmogorov length, which makesdirect measurement of the transverse gradients unnessary because in that case (3.1)and (3.2) are equivalent to

εr = 15ν

⟨(∂u1

∂x1

)2⟩

r

(3.3)

(first derived by Taylor 1935, equation 45 on p. 437) and

χr = 6κ

⟨(∂θ

∂x1

)2⟩

r

, (3.4)

respectively, where ‘1’ stands for the streamwise direction.Because fine wires measure time series and not spatial series, a local version of

Taylor’s frozen-turbulence hypothesis has to be used to retrieve streamwise gradientsfrom time derivatives:

εr = 15ν

U 2r

⟨(∂u1

∂t

)2⟩

τ

(3.5)

and

χr = 6κ

U 2r

⟨(∂θ

∂t

)2⟩

τ

. (3.6)

Here, Ur is the true air speed (i.e. the magnitude of the sensor’s velocity vectorrelative to the air) averaged over the time period τ . For a given τ , we have r = Urτ .If the fluctuations of the true air velocity vector during τ are significant comparedto Ur , then Taylor’s hypothesis in its original form (Taylor 1938) may lead to seriousoverestimations of 〈(∂u1/∂x1)

2〉 and 〈(∂θ/∂x1)2〉 and, therefore, of energy and scalar

variance dissipation rates. Equations to quantify and correct biases resulting from theoriginal, i.e. ‘global’, Taylor hypothesis (Taylor 1938) have been worked out, e.g., byHeskestad (1965), Lumley (1965) and Wyngaard & Clifford (1977). Wyngaard &Clifford (1977) found that global energy dissipation rates are overestimated by15〈u2

1〉/U 2 and global temperature variance dissipation rates by 9〈u21〉/U 2, where

U is the global mean true air speed and 〈u21〉 is the global variance of the streamwise

velocity. These results have been confirmed by Hill (1996).Correspondingly, the local version of Taylor’s hypothesis, which is also known

as the ‘random Taylor hypothesis’ (e.g. Tennekes 1975) or ‘sweeping decorrelationhypothesis’ (Praskovsky et al. 1993), leads to overestimations by 15〈u2

1〉r/U 2r in εr

and by 9〈u21〉r/U 2

r in χr , where 〈u21〉r is the local variance of the streamwise velocity.

Because 〈u21〉r/〈u2

1〉 is of order (r/L)2/3, the biases resulting from the local Taylorhypothesis are only of order 15〈u2

1〉(r/L)2/3/U 2 and 9〈u21〉(r/L)2/3/U 2, respectively.


In summary, (3.5) and (3.6) rely on the following three assumptions: (i) negligibilityof the shape of the local averaging volume; (ii) statistical isotropy at length scalescomparable to the Kolmogorov length; and (iii) the validity of the local Taylorhypothesis. Although errors resulting from violations of one or more of these threeassumptions may lead to significant biases, measurements of εr and χr by meansof (3.5) and (3.6) are usually, and somewhat misleadingly, called ‘direct’ dissipationmeasurements.

Direct dissipation measurements in high-Reynolds-number flows require extremelyhigh spatial and temporal resolution. According to Kuznetsov, Praskovsky &Sabelnikov (1992, p. 602), the spatial resolution has to be 2η or better, which translatesinto the following requirement for the time resolution:

�t �2

Ur

(ν3

εr

)1/4

. (3.7)

(Muschinski (1996) pointed out that there is a relationship between this ‘2η criterion’and the requirement that the Smagorinsky coefficient in large-eddy simulations mustexceed a certain critical value.) In the NBL, a typical wind speed is 10 m s−1 and onehas to be prepared for local values of the energy dissipation rate as large as 0.1 m2 s−3.With ν = 1.5 × 10−5 m2 s−1 this leads to 2η = 0.86 mm and �t � 86 µs. That is, thesampling rate has to be 11.7 kHz or higher. Although this is feasible – Kuznetsovet al. (1992), for example, sampled wind-tunnel turbulence at 32 kHz – the difficultiesassociated with direct dissipation measurements in the open atmosphere, particularlyonboard airborne platforms, usually do not justify the expenses.

As an alternative to the direct dissipation technique, one can retrieve dissipationrates with the inertial-range technique. As described in the following, local structureparameters are extracted from the inertial subrange of the measured spectra, anddissipation rates are then computed from the structure parameters.

Local structure functions for streamwise velocity and for temperature are definedby

D(r)11 (s) = 〈[u(x + s/2) − u(x − s/2)]2〉r (3.8)

and

D(r)θθ (s) = 〈[θ(x + s/2) − θ(x − s/2)]2〉r , (3.9)

respectively. Here, s is the separation in the streamwise direction, and 〈·〉r means thatthe structure functions are estimated from spatial series of length r . In the inertialrange, the structure functions follow the two-thirds laws

D(r)11 (s) =

(C2

u

)rs2/3 (3.10)

and

D(r)θθ (s) =

(C2

θ

)rs2/3, (3.11)

where (C2u)r is the local structure parameter of streamwise velocity and (C2

θ )r isthe local temperature structure parameter. The corresponding local, one-sided, one-dimensional, streamwise wavenumber spectra are

F(r)11 (k1) =

2

3�(1/3)

(C2

u

)rk

−5/31 (3.12)

and

F(r)θθ (k1) =

2

3�(1/3)

(C2

θ

)rk

−5/31 , (3.13)


where 2/3�(1/3) = 0.2489. Inertial-range theory for velocity turbulence (Kolmogorov1941; Obukhov 1941a , b) and for scalar turbulence (Obukhov 1949; Corrsin 1951)leads to

F(r)11 (k1) = α1ε

2/3r k

−5/31 (3.14)

and

F(r)θθ (k1) = γ ε−1/3

r χrk−5/31 . (3.15)

The Kolmogorov coefficient α1 is quasi-universal and has a value close to 0.5 (Yaglom1981; Sreenivasan 1995; Gotoh, Fukayama & Nakano 2002). The Obukhov–Corrsincoefficient γ is also quasi-universal and has a value close to 0.4 (Sreenivasan 1996).This leads to (

C2u

)r= 2.0 ε2/3

r (3.16)

and (C2

θ

)r= 1.6 ε−1/3

r χr , (3.17)

respectively.Usually, the local wavenumber spectra F

(r)11 (k1) and F

(r)θθ (k1) are not directly measured

but are, by means of the local Taylor hypothesis, obtained from the one-sided, localfrequency spectra S

(r)11 (f ) and S

(r)θθ (f ). The relationships are

F(r)11 (k1) =

Ur

2πS

(r)11 (f ) (3.18)

and

F(r)θθ (k1) =

Ur

2πS

(r)θθ (f ), (3.19)

respectively, where

f =Ur

2πk1 (3.20)

is frequency.The inertial-range technique relies on the following assumptions: (i) negligibility of

the shape of the local averaging volume; (ii) statistical isotropy in the wavenumberrange within which the model spectrum is fitted to the measured spectrum; (iii) theexistence of inertial subranges in the velocity and temperature fields; (iv) the assump-tions that the inertial-subrange theories are locally valid and that the Kolmogorovand Obukhov–Corrsin coefficients are universal and known; and (v) the validity ofthe local Taylor hypothesis.

The εr and χr data that will be analysed in the following were obtained withthe inertial-range technique. The measured spectra were fitted to refined models forF

(r)11 (k1) and F

(r)θθ (k1), respectively. These models account for the ‘Hill bump’ (Hill 1978)

in the scalar spectrum, the drop-off of spectral density at wavenumbers where dis-sipation is no longer negligible, and white noise in the sensors. A detailed descriptionof the models and the estimation algorithms was given by Frehlich et al. (2003).

Because the measurements were made in the stably stratified atmosphere, theturbulence intensities

√〈u2

1〉r/Ur were very small compared to 1, such that errorscaused by the local Taylor hypothesis were negligible and no correction was made.

4. Two theoretical models of conditionally averaged local turbulence statisticsOne purpose of this study is to analyse and theoretically explain observations

of 〈(C2θ )r |εr〉, that is, of conditional averages of (C2

θ )r for fixed values of εr . We


will see that the observed values of 〈(C2θ )r |εr〉 are proportional to εδ

r , where δ isa numerical exponent. The main two questions that we will address are: First,is the power-law behaviour of 〈(C2

θ )r |εr〉 a consequence of the lognormality ofsmall-scale intermittency statistics, or can it be explained within the frameworkof classical concepts of turbulence? Second, what is the physics behind the power-lawexponent δ?

We will present two different theoretical models of 〈(C2θ )r |εr〉. The first model builds

on the joint lognormality assumption for small-scale intermittency. The second modelignores small-scale intermittency entirely and, instead, explains 〈C2

θ |ε〉 on the basis ofsimple mixing-length arguments. (Note that in the second scenario we have suppressedthe suffix r because, in that case, the size of the averaging volume plays no role inthe theory. It is only assumed that the averaging volume is large compared to L.)

Although these two theoretical concepts have very little in common, both lead topower laws for 〈(C2

θ )r |εr〉 and 〈C2θ |ε〉, respectively.

4.1. The small-scale intermittency model

Let us assume that the joint lognormality hypothesis (Van Atta 1971; Antonia & VanAtta 1975) is valid, such that the joint probability density function (j.p.d.f.) of χr andεr is

p (εr, χr ) =1

2πεrχrσxσy

√1 − ρ2

xy

exp

−

x2

σ 2x

− 2ρxy

x

σx

y

σy

+y2

σ 2y

2(1 − ρ2

xy

)

, (4.1)

where

x = lnεr

εr

(4.2)

is the (natural) logarithm of the normalized energy dissipation rate εr (εr is chosensuch that 〈ln(εr/εr )〉 = 0), with σx as the standard deviation of x,

y = lnχr

χr

(4.3)

is the logarithm of the normalized temperature variance dissipation rate, χr (χr ischosen such that 〈ln(χr/χr )〉 = 0), with σy as the standard deviation of y, and

ρxy =〈xy〉σxσy

(4.4)

is the correlation coefficient of x and y.Let us follow Peltier & Wyngaard (1995, p. 3642) and assume that the Corrsin

(1951) relationship,

C2θ = γ ε−1/3χ, (4.5)

is locally valid, such that (C2

θ

)r= γ ε−1/3

r χr , (4.6)

where the Obukhov–Corrsin coefficient γ is assumed to have a universal value closeto 1.6, and that γ does not vary with r; see also (3.17).


The conditional average of (C2θ )r for a fixed value of εr is

⟨(C2

θ

)r

∣∣εr

⟩=

∫p(χr, εr )γ ε−1/3

r χr dχr∫p(χr, εr ) dχr

, (4.7)

which gives (see Appendix A)

⟨(C2

θ

)r

∣∣εr

⟩= γχrε

−1/3r exp

[1 − ρ2

xy

2σ 2

y

] (εr

εr

)ρxyσy/σx

. (4.8)

That is, we have the power law ⟨(C2

θ

)r

∣∣εr

⟩∝ εδ

r (4.9)

with the exponent, or ‘logarithmic slope’,

δ = ρxy

σy

σx

− 1

3. (4.10)

As described in § 3, the primary observables obtained with the inertial-range techniqueare εr and (C2

θ )r . Therefore, it is more natural to express 〈(C2θ )r |εr〉 in terms of (C2

θ )rand εr statistics, rather than in terms of χr and εr statistics. In logarithmic notation,the Obukhov–Corrsin relation is z = −x/3 + y, where

z = ln

(C2

θ

)r(

C2θ

)r

(4.11)

is the logarithm of the normalized temperature structure parameter, where (C2θ )r is

chosen such that 〈z〉 = 0, and where σz is the standard deviation of z. It is knownthat x and z = ax + by (with real numbers a and b) are jointly normal if x andy are jointly normal (e.g. Davenport & Root 1958, p. 151). Therefore, (C2

θ )r and εr

are jointly lognormal if χr and εr are jointly lognormal. Based on the same datasetthat we will analyse in the following, Frehlich et al. (2004) have shown that jointlognormality for (C2

θ )r and εr is a good approximation in the shear region of thenocturnal low-level jet.

With (A 14), we obtain

⟨(C2

θ

)r

∣∣εr

⟩=

(C2

θ

)rexp

[1 − ρ2

xz

2σ 2

z

] (εr

εr

)ρxzσz/σx

, (4.12)

where ρxz is the correlation coefficient of the logarithms of (C2θ )r and εr . Therefore,

the slope in terms of inertial-range measurables is

δ = ρxz

σz

σx

. (4.13)

From (2.4) and (2.5) we find that

σy

σx

=

√Aθ + µθ ln(Lθ/r)

A + µ ln(L/r). (4.14)

For r � L and r � Lθ , the ratio σy/σx approaches the asymptotic value√

µθ/µ.Antonia & Van Atta (1975) hypothesized that µθ ≈ µ. In that case, σy/σx ≈ 1, and


because the correlation coefficient ρxy cannot exceed 1, δ cannot exceed the value2/3. This argument, however, is valid only if r is ‘deep in the inertial range’, suchthat r/L � 1. Also, one has to keep in mind that the ratios Lθ/r and L/r enter onlylogarithmically in the expression for σy/σx , such that unrealistically large values forLθ/r and L/r may have to be assumed to infer that σy/σx is close to one. Therefore,the result that δ cannot exceed the value 2/3 has to be taken with caution, particularlysince little is known about the magnitude of the coefficients A and Aθ .

Antonia & Van Atta (1975, p. 280) argued that in order to be consistent with ther2/3 dependence of the second-order scalar structure function (i.e. of 〈(�θ)2〉), ρxy isrequired to be equal to 2/3. With the additional assumption σx = σy , this leads toa power-law exponent δ = 1/3. We will return to these issues when we discuss theresults from the fine-wire measurements.

4.2. The large-scale intermittency model

In micrometeorology, it is a traditional working assumption that estimates of low-order turbulence statistics like variances and structure parameters are statisticallystable if the length of the time series from which the statistics are estimated is atleast 10 min or so. The underlying assumption is that there is a sufficiently clearscale separation between small-scale turbulence and large-scale turbulence. This scaleseparation is also known as the ‘mesoscale gap’ or ‘spectral gap’ (Van der Hoeven1957); see also Lumley & Panofsky (1964, pp. 42ff.), Fiedler & Panofsky (1970), andMahrt, Moore & Vickers (2001). Recently, Vickers & Mahrt (2003) have describeda ‘co-spectral gap,’ which they found in spectra of turbulent fluxes measured duringCASES-99.

Let us denote with σ 2u and σ 2

θ the variances of streamwise velocity and temperature,respectively, associated with three-dimensional turbulence. Let us assume that thestructure functions D11(s) and Dθθ (s) follow the inertial-range laws (3.8) and (3.9)for separations s up to some outer scales Lu and Lθ , respectively, and that forlarger separations the structure functions approach the constant values 2σ 2

u and2σ 2

θ , respectively. It is natural to define Lu and Lθ as the separations at which theinertial-range asymptotes and the constant asymptotes intersect, such that

C2uL

2/3u = 2σ 2

u (4.15)

and

C2θ L

2/3θ = 2σ 2

θ . (4.16)

Now, we follow Prandtl (1925) and define a velocity mischungsweglange, or velocitymixing length, lu and a scalar mixing length lθ such that the magnitude of thesystematic change across the mixing length is equal to the standard deviationassociated with the turbulent fluctuations:∣∣∣∣∂U

∂z

∣∣∣∣ lu = σu (4.17)

and ∣∣∣∣∂Θ

∂z

∣∣∣∣ lθ = σθ . (4.18)

Here, ∂U/∂z and ∂Θ/∂z are the vertical gradients of mean velocity and the meanpotential temperature, respectively.


After combining these last four equations and eliminating σu and σθ , we obtain arelationship between C2

u and C2θ :

C2θ =

L2/3u l2θ

L2/3θ l2u

(∂Θ/∂z)2

(∂U/∂z)2C2

u, (4.19)

or

C2θ = CKab ε2/3, (4.20)

where

a ≡ L2/3u l2θ

L2/3θ l2u

(4.21)

and

b ≡(

∂Θ∂z

)2

(∂U∂z

)2(4.22)

are parameters that are allowed to vary randomly as a result of large-scaleintermittency, and where CK = 2.0 is the coefficient in (3.16).

Here, we consider two special cases: first, the parameter

q ≡ CKab =L2/3

u l2θ

L2/3θ l2u

(∂Θ/∂z)2

(∂U/∂z)2(4.23)

is statistically independent of ε; second, shear and stratification are coupled throughthe Richardson criterion.

If q is statistically independent of ε, we have pqε(q, ε) = pq(q)pε(ε), where pqε(q, ε)is the j.p.d.f. of q and ε, pq(q) is the p.d.f. of q , and pε(ε) is the p.d.f. of ε. Therefore,

⟨C2

θ

∣∣ε⟩ ≡

∫pqε(q, ε)C2

θ (q, ε) dq∫pqε(q, ε) dq

=

∫pq(q)pε(ε)qε2/3 dq∫

pq(q)pε(ε) dq

, (4.24)

and we obtain immediately ⟨C2

θ

∣∣ε⟩ = 〈q〉ε2/3. (4.25)

That is, the slope is δ = 2/3 if q is statistically independent of ε.In the second scenario, we assume that the turbulence is in ‘Richardson equilibrium’

everywhere in the space–time window under consideration. That is, the statisticalensemble is assumed to contain only combinations of mean shear ∂U/∂z and meanstratification ∂Θ/∂z that satisfy the Richardson criterion

Ric =g

Θ

∂Θ/∂z

(∂U/∂z)2. (4.26)

Here, Ric is a critical Richardson number, which we assume to have a universal valueclose to its traditional value 1/4, g is acceleration due to gravity, and Θ is the meanpotential temperature at the altitude of interest.

Solving (4.26) for (∂U/∂z)2 and inserting into (4.19) leads to

C2θ = aRic

Θ

g

∂Θ

∂zC2

u (4.27)


or

C2θ = aRicCK

Θ2

g2N2ε2/3, (4.28)

where

N =

√g

Θ

∂Θ

∂z(4.29)

is the Brunt–Vaisala frequency.The coefficients CK and Ric and the parameter g/Θ may be considered as constants.

Furthermore, let us follow conventional wisdom and assume that all four length scalesare of order L, such that a may also be treated as a quasi-universal coefficient. Thenthe coefficient aRicCK is also a constant of order unity. If N is statistically independentof ε, which is the case if fluctuations in ε result from changes in shear, rather thanfrom changes in thermal stability, we find

⟨C2

θ

∣∣ε⟩ = aRicCK

Θ2

g2〈N2〉ε2/3. (4.30)

That is, the large-scale intermittency model predicts an ε2/3 law for 〈C2θ |ε〉, regardless

of whether (i) shear and stratification are statistically independent, or (ii) shear andstratification ‘track’ each other because they are deterministically connected throughthe Richardson criterion.

5. Experimental setup and meteorological situation5.1. CIRES tethered lifting system (TLS) and turbulence sensors

The observations reported here were obtained from high-resolution in-situ turbulenceobservations in the lowest few hundred metres of the night-time atmosphereapproximately 40 km east of Wichita, Kansas, during the Cooperative Atmosphere-Surface Exchange Study (CASES-99). CASES-99 was designed to study the structureand dynamics of the night-time stable boundary layer (Poulos et al. 2002). In situdata were recorded using the Cooperative Institute for Research in EnvironmentalSciences (CIRES) Tethered Lifting System (TLS), which employs either a kite oran aerodynamic balloon (kites for moderate-to-strong wind conditions; balloons forlow wind conditions) to carry a suite of lightweight instruments from the ground upthrough the first few kilometres of the atmosphere (Balsley, Jensen & Frehlich 1998;Muschinski et al. 2001; Balsley et al. 2003; Frehlich et al. 2003).

For CASES-99, the TLS instrumentation included up to five ‘turbulence payloads’(TPs) separated vertically by pre-selected spacings. The TPs could be attached aboveor below the basic meteorological payload (BMP). The measurements collected withthe TPs were calibrated and converted into accurate turbulence data only after thefield campaign. The BMP consisted of a Vaisala RS-80 radiosonde and a Tmax-boardinterface, and its main purpose was to provide real-time data of pressure, wind speedand direction, and temperature, thereby enabling scientists and technicians on theground to respond to the ever-changing structure and dynamics of the NBL and tomove the TPs up and down as required. None of the data collected with the BMPwere used for the calibration of the temperature and velocity data measured withthe TPs. After the field campaign, the turbulence data obtained from each of thefive TPs were calibrated independently from the BMP and independently from eachother. The pressure data from the BMP were used to reconstruct the time series ofthe altitudes of each TP.


Winddirection

Kite

Tether

D

A

B

C

BMP

Figure 1. Sketch of the CIRES Tethered Lifting System (TLS), carrying a vertical array offour turbulence sensors (A, B, C, and D) and the basic meteorological payload (BMP) up toaltitudes of 500m above ground level (AGL).

Each TP archived data measured with both cold-wire and hot-wire sensors. Thecold- and hot-wire data were sampled at 200 Hz. Each TP also carried conventionallow-frequency sensors (e.g. a Pitot tube, a solid-state temperature sensor, and a piezo-electric pressure sensor) for sampling wind speed, temperature, and pressure, alongwith a 3-axis tilt sensor and a magnetic compass. Separate archiving for each TP wasaccomplished using onboard digital flash-memory storage, with data downloads todisk occurring when the packages were brought down to the ground at the end ofeach night of operation. Figure 1 is a sketch of the TLS and the sensor-array in theconfiguration of the CASES-99 deployment during the night of 20/21 October 1999.During that night, the TLS was operated with a kite, and the sensor array consistedof the four TPs A, B, C, and D (from bottom to top), and the BMP. The verticalspacing between the turbulence payloads was 6m.

The last step of the calibration procedure was to ‘merge’ the low- and high-frequencydata measured with each TP into 200 Hz time series of temperature and streamwisevelocity. This procedure and the specifications of the individual sensors are describedin detail in Frehlich et al. (2003). The merged time series inherit the high long-termstability of the low-frequency sensors and the excellent sensitivity and short responsetimes of the fine-wire sensors. The absolute accuracy of the merged temperature


and velocity data is better than 0.5 K and 1 m s−1, respectively. The cold and hotwires were operated with low-pass filters with 3 dB cutoff frequencies of 500 Hz and2 kHz, respectively. Although no anti-aliasing filter was used, the uncorrelated noisestandard deviations in the temperature and velocity samples were as small as 1 mKand 1.7 mm s−1, respectively. These estimates were obtained from the spectra shownin figures 8 and 9 in Frehlich et al. (2003).

Consecutive one-second periods of temperature and streamwise velocity fluctuationssampled at 200 Hz were spectrally analysed to produce estimates of (C2

θ )r andεr , respectively, by means of the inertial-range technique described in § 3. Thesampling uncertainty of these 1 s estimates is better than 15%, which implies a 15%accuracy for (C2

θ )r and 22.5% accuracy for εr ; see Frehlich et al. (2003) for furtherdetails.

As described in § 3, the local Taylor hypothesis was used. Because the averagingtime τ is related to the averaging length r via r = Urτ , a fixed averaging time (τ = 1 s)leads to a changing r if the wind speed changes. Because in the data set Ur rangedbetween about 5 m s−1 and 13 m s−1, r varied between 5 m and 13 m. The variancesσ 2

x and σ 2z are expected to increase with decreasing r , such that the sections with low

wind speeds should reveal larger values of σ 2x and σ 2

z than the sections with higherwind speeds. However, since σ 2

x and σ 2y are expected to vary only with the logarithm

of r , we neglect this effect.

5.2. Meterological situation

Figure 2 shows three vertical profiles of wind speed and temperature, respectively,observed with the TLS during three ascents. The first ascent (20:58 LST to 21:12LST) reached an altitude of 340 m, the second (22:40 LST to 23:04 LST) reached330 m, and the third ascent (01:10 LST to 01:31 LST) reached 450 m. All data pointsshown here are averages over 1 min. That is, a typical ascent rate was 20 mmin−1, orabout 0.3 m s−1.

The dynamics in the lowest 200 m AGL were dominated by a low-level jet witha wind speed maximum between 12 m s−1 and 14 m s−1 at altitudes around 150 m(figure 2a). After midnight, the intensity of the jet as well as the magnitudes of theshear both below and above the the jet decreased significantly, which led to a drasticreduction of turbulence production in the upper part of the NBL. As a result, a verysharp inversion at about 190 m was formed. This inversion marked the top of theNBL. As documented by Balsley et al. (2003), the thickness of this inversion was assmall as 5 cm. The temperature change across these 5 cm was about 2 K.

As can be seen in figure 2(b), the thermal stratification in the NBL was rather strong,with ∂Θ/∂z ≈ 0.05 Km−1, which is five times as much as in an isothermal atmosphere.During the second ascent, the shear in the lowest 100 m was 0.10 s−1 and the potentialtemperature gradient was 0.04 Km−1, resulting in a gradient Richardson number of0.13 below the jet, about half the traditional value of the critical Richardson number,0.25. That is, the NBL below the jet was dynamically unstable and turbulent, alsolater in the night, which is evident from the fairly intense turbulence, as will bedocumented in the following. During the night, the NBL cooled down with typicalrates of 1Kh−1.

In the residual layer above the NBL top, thermal stratification and wind shear weremuch less than in the NBL. Gradient Richardson numbers (estimated from altitudeintervals of 20 m or so) in the residual layer were typically much larger than 0.25.


500

450

400

350

300

250

200

150

100

50

0

500

450

400

350

300

250

200

150

100

50

05 10 15

20:58–21:12 (ascent)22:40–23:04 (ascent)01:10–01:31 (ascent)

286 288 290 292 294 296 298

Wind speed (m s–1) Potential temperature (K)

h (m

AG

L)

(a) (b)

Figure 2. Vertical profiles of 1 min averages of (a) the wind speed and (b) the potentialtemperature. Data were taken with sensor C during three different ascents. The wind speedmaximum associated with the low-level jet was at about 150 m AGL. The top of the nocturnalboundary layer (NBL) was between 150 and 200 m AGL. Thermal stratification (b) wasstrongly stable in the NBL and weakly stable in the overlying residual layer.

6. Observations6.1. Time series

Figure 3 gives an overview of the measurements taken with sensor C during thenight of 20/21 October 1999. Ten-hour time series of 1 s estimates of the followingquantities are shown: (a) the sensor altitude h above ground level; (b) the windspeed u; (c) the air temperature T ; (d ) the local energy dissipation rate εr ; and(e) the local temperature variance dissipation rate χr . In figure 3(a), three episodesare marked: NBL, for ‘nocturnal boundary layer’, between 03:00 and 05:00 LST ataltitudes between 52 m and 74 m AGL; RL, for ‘residual layer’, between 21:20 and22:20 LST at altitudes between 167 m and 224 m AGL; and AD, for ‘ascent/descent’,between 01:10 and 01:50 LST at altitudes between 32 m and 452 m AGL. The NBLand RL episodes were chosen because the sensors were ‘parked’ at roughly constantheight during those episodes, and the observed fluctuations were nearly statisticallystationary. The AD episode consists of two vertical soundings of the nocturnalboundary layer and the lower residual layer.

Time series measured during the episodes NBL, RL, and AD are shown in moredetail in figures 4, 5, and 6, respectively.


600

400

200

021:00 00:00 03:00 06:00

20

15

10

5

20

15

10

5

021:00 00:00 03:00 06:00

21:00 00:00 03:00 06:00

21:00 00:00 03:00 06:00

21:00 00:00 03:00 06:00

10–1

10–3

10–5

10–7

10–1

10–3

10–5

10–7

(a)

(b)

(c)

(d)

(e)

RLAD

NBLh (m

AG

L)

u (m

s–1

)T

(°C

)ε

(m2 s

–3)

C2 θ (K

2 m–2

/3)

Time LST (hh:mm)

Figure 3. Overview of the turbulence measurements collected with sensor package C duringthe night of 20/21 October 1999. (a) Sensor altitude above ground level (AGL); (b) wind speed(1 s averages); (c) air temperature (1 s averages); (d) local energy dissipation rate, εr , estimatedfrom 1 s time series of wind speed fluctuations sampled at 200 Hz; (e) local temperaturestructure parameter, (C2

θ )r , estimated from 1 s time series of temperature fluctuations sampledat 200Hz. Detailed time series measured during the episodes annotated in (a) as NBL (for‘nocturnal boundary layer’, from 03:00 LST to 05:00 LST), as RL (for ‘residual layer’, from21:20 LST to 22:20 LST), and as AD (for ‘ascent/descent’, from 01:10 LST to 01:50 LST) areshown in figures 4(a–e), 5(a–e), and 6(a–e), respectively.


80

70

60

5003:00 03:30 04:00 04:30 05:00

10

8

6

4

12

11

10

9

8

10–1

10–3

10–5

10–7

10–1

10–3

10–5

10–7

(a)

(b)

(c)

(d)

(e)

h (m

AG

L)

u (m

s–1

)T

(°C

)ε

(m2 s

–3)

C2 θ (K

2 m

–2/3

)

03:00 03:30 04:00 04:30 05:00

03:00 03:30 04:00 04:30 05:00

03:00 03:30 04:00 04:30 05:00

03:00 03:30 04:00 04:30 05:00

Time LST (hh:mm)

Figure 4. Same as figure 3(a–e) but for the NBL (nocturnal boundary layer) episodebetween 03:00 LST and 05:00 LST.

6.2. Velocity and temperature spectra

Frequency spectra Suu(f ) and Sθθ (f ) of wind speed and temperature fluctuationsmeasured during the NBL episode are shown in figure 7(a). Both Suu(f ) and Sθθ (f )show two regimes in which the spectral densities vary with f −5/3. Both in Suu(f ) and


240

220

200

180

021:20 21:30 21:40 21:50 22:00 22:2022:10

14

12

10

8

19

18.5

18

17.5

17

10–1

10–3

10–5

10–7

10–1

10–3

10–5

10–7

(a)

(b)

(c)

(d)

(e)

h (m

AG

L)

u (m

s–1

)T

(°C

)ε

(m2 s

–3)

C2 θ (K

2 m

–2/3

)

Time LST (hh:mm)

21:20 21:30 21:40 21:50 22:00 22:2022:10

21:20 21:30 21:40 21:50 22:00 22:2022:10

21:20 21:30 21:40 21:50 22:00 22:2022:10

21:20 21:30 21:40 21:50 22:00 22:2022:10

Figure 5. Same as figure 3 (a–e) but for the RL (residual layer) episode between21:20 LST and 22:20 LST.

Sθθ (f ), the two f −5/3 regimes are separated by a plateau. Figure 7(b) shows the samedata as in figure 7(a) but in ‘area-preserving’ representation. That is, figure 7(b) showsf Suu(f ) and f Sθθ (f ) in a semilogarithmic diagram.


600

400

200

001:10 01:20 01:30 01:5001:40

01:10 01:20 01:30 01:5001:40

01:10 01:20 01:30 01:5001:40

01:10 01:20 01:30 01:5001:40

01:10 01:20 01:30 01:5001:40

15

10

5

20

15

10

10–1

10–3

10–5

10–7

10–1

10–3

10–5

10–7

(a)

(b)

(c)

(d)

(e)

h (m

AG

L)

u (m

s–1

)T

(°C

)ε

(m2 s

–3)

C2 θ (K

2 m

–2/3

)

Time LST (hh:mm)

Figure 6. Same as figure 3 (a–e) but for the AD (ascent/descent) episode between01:10 LST and 01:50 LST.

In both Suu(f ) and Sθθ (f ), the plateau ranges from 0.003 Hz (5.5 min) to 0.03 Hz(0.5 min), corresponding to wavelengths between 230 m and 2.3 km. Note thatthe frequencies of the minima and maxima in the area-preserving spectra shown


104

103

102

101

100

10–1

10–2

10–3

10–4 10–3 10–2 10–1 100

f (Hz)10–4 10–3 10–2 10–1 100

f (Hz)

S uu(

f)

(m2 s

–2 H

z–1 )

, Sθθ(

f)

(K–2

Hz–

1 )

fSuu

(f

) (m

2 s–

2 ),

fSθθ(

f)

(K2 )

Suu( f )

f –5/3

f –5/3

Sθθ( f )

fSuu( f )

f Sθθ( f )

0.10

0.08

0.06

0.04

0.02

0

(a) (b)

Figure 7. (a) Frequency spectra of wind speed and temperature fluctuations indouble-logarithmic representation. The data were collected during the NBL episode (from03:00 LST to 05:00 LST). Two regimes showing a −5/3 power law are separated by a plateaubetween 0.005Hz and 0.02 Hz, corresponding to time scales ranging from 50 s to 3.3 minand (at a mean wind speed of U = 7 m s−1) to length scales between 350 m and 1400m.(b) Same data as in (a) but in ‘area-preserving’ representation. The high-frequency andlow-frequency boundaries of the plateau in (a) coincide with the spectral peak and the spectralgap, respectively, in (b).

in figure 7(b) coincide with the frequencies at the lower and upper, respectively,boundaries of the plateaux in figure 7(a), as expected.

The wavelength corresponding to the high-frequency end of the plateau marks thelarge-scale boundary of the inertial subrange of the three-dimensional, Kolmogorov-type turbulence. Muschinski & Roth (1993) argued that at an altitude h the largestfeatures that could be statistically isotropic in three dimensions cannot have a radiuslarger than h, or a diameter larger than 2h, or a wavelength larger than

λm = 4h. (6.1)

The data presented in figure 7 were taken at altitudes around 65 m AGL. This leads toλm = 260 m and agrees well with the observed wavelength of 230 m. More discussionon this issue is given in § 7.1.

6.3. Scatter diagrams of εr and (C2θ )r ; conditional averages of (C2

θ )r for specifiedvalues of εr

The local energy dissipation rates, εr , and local temperature structure parameters,(C2

θ )r , which were shown as time series in figures 3–6, are presented in the form ofscatter diagrams in figures 8(a), 9(a), 10(a), and 11(a), respectively. Figures 8(b), 9(b),10(b), and 11(b) show, for the respective ensemble (the entire night, NBL, RL, or AD),conditional averages 〈(C2

θ )r |εr〉 as functions of εr . The dots are linear averages of (C2θ )r

computed for specific εr bins, each of which has a width of 1/20 decade. The lines are


10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1

C2 θ (K

2 m

–2/3

)10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1

�C

2 θ|ε� (

K2

m–2

/3)

(a) (b)

ε (m2 s–3) ε (m2 s–3)

Figure 8. (a) Scatter diagram of all (C2θ )r and εr values measured with sensor C during the

night of 20/21 October 1999. The plot contains 33 660 data points. Statistical parameters:σx = 1.86, σz = 1.82, ρxz = 0.70. (b) Conditional averages of (C2

θ )r for specified valuesof εr . The width of the εr bins is 1/20 decade. The solid line is the model resulting from(4.12), which assumes joint lognormality of (C2

θ )r and εr . The slope predicted from (4.12) isδ = ρxzσz/σx = 0.69.

10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1

C2 θ (K

2 m

–2/3

)

10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1

�C

2 θ|ε� (

K2

m–2

/3)

(a) (b)

ε (m2 s–3) ε (m2 s–3)

Figure 9. Same as figure 8(a, b) but for the NBL (nocturnal boundary layer) episode between03:00 LST and 05:00 LST. The scatter diagram (a) contains 6859 data points. Statisticalparameters: σx = 0.93, σz = 1.09, ρxz = 0.32. The slope predicted from (4.12) is δ = 0.38.


10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1

C2 θ (K

2 m

–2/3

)10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1�

C2 θ|ε� (

K2

m–2

/3)

(a) (b)

ε (m2 s–3) ε (m2 s–3)

Figure 10. Same as figure 8(a, b) but for the RL (residual layer) episode between 21:20 LSTand 22:20 LST. The scatter diagram (a) contains 3430 data points. Statistical parameters:σx = 0.83, σz = 1.01, ρxz = 0.38. The slope predicted from (4.12) is δ = 0.46.

10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1

C2 θ (K

2 m

–2/3

)

10–1

10–2

10–3

10–4

10–5

10–6

10–7 10–6 10–5 10–4 10–3 10–2 10–1

�C

2 θ|ε� (

K2

m–2

/3)

(a) (b)

ε (m2 s–3) ε (m2 s–3)

Figure 11. Same as figure 8(a, b) but for the AD (ascent/descent) episode between 01:10LST and 01:50 LST. The scatter diagram (a) contains 2284 data points. Statistical parameters:σx = 2.51, σz = 2.29, ρxz = 0.80. The slope predicted from (4.12) is δ = 0.72.

the theoretical power laws obtained from (4.12) with the respective values of (C2θ )r , εr ,

σx , σz, and ρxz. Table 1 gives an overview of the statistical parameters characterizingthe four ensembles. Relationships between these statistical parameters are derived inAppendix B.


(C2θ )r

[K2 m−2/3

]εr [m2 s−3] σx σz ρxz σy ρxy σy/σx δ

all 4.3 × 10−4 4.3 × 10−4 1.86 1.82 0.70 2.30 0.83 1.24 0.69NBL 8.3 × 10−4 9.7 × 10−4 0.93 1.09 0.32 1.23 0.54 1.32 0.38RL 1.9 × 10−4 4.0 × 10−4 0.83 1.01 0.38 1.14 0.58 1.37 0.46AD 2.1 × 10−4 1.8 × 10−4 2.51 2.29 0.80 3.00 0.89 1.19 0.72

Table 1. Statistical parameters characterizing the four data ensembles ‘all’ (20:12 LST to 06:01LST), NBL (03:00 to 05:00 LST), RL (21:20 to 22:20 LST), and AD (01:10 to 01:50 LST).

103

102

101

100

10–1

10–2

10–4 10–3 10–2 10–1 100

f (Hz)10–4 10–3 10–2 10–1 100

f (Hz)

S xx(

f)

(Hz–

1 ), S

yy(

f)

(Hz–

1 ), S

zz(

f)

(Hz–

1 )

fSxx

(f

), f

S yy(

f),

fS z

z(f

)

Sxx( f )

Syy( f )

Szz( f )

f Sxx( f )

f Syy( f )

f Szz( f )

f –1

0.4

0.3

0.2

0.1

0(a) (b)

Figure 12. Same as figure 7 but showing spectra of x (the logarithm of εr ), y (the logarithm ofχr , calculated from εr and (C2

θ )r through the Obukhov–Corrsin relation), and z (the logarithmof (C2

θ )r ). As in figure 7, data were collected during the NBL episode. At high frequencies,the spectra of x, y, and z show a f −1 power law, in agreement with Kolmogorov’s (1962)lognormality hypothesis.

6.4. Spectra of the logarithms of εr , χr , and (C2θ )r

Figures 12 and 13 show frequency spectra of x, y, and z (i.e. of the centralizedlogarithms of εr , χr , and (C2

θ )r ) observed during the NBL and RL episodes,respectively. At frequencies higher than about 0.03 Hz (wavelengths shorter than230 m), all spectra follow (approximately) an f −1 power law.

7. Discussion7.1. Three-dimensional and quasi-two-dimensional regimes in stratified shear flow

The velocity and temperature spectra in figure 7 show f −5/3 regimes at frequencieslower than 0.003 Hz (wavelengths longer than 2.3 km) and at frequencies higher than0.03 Hz (wavelengths shorter than 230 m). These two −5/3 regimes are separated bya plateau.


103

102

101

100

10–1

10–2

10–4 10–3 10–2 10–1 100

f (Hz)10–4 10–3 10–2 10–1 100

f (Hz)

S xx(

f)

(Hz–

1 ), S

yy(

f)

(Hz–

1 ), S

zz(

f)

(Hz–

1 )

fSxx

(f

), f

S yy(

f),

fS z

z(f

)

Sxx( f )

Syy( f )

Szz( f )

f Sxx( f )

f Syy( f )

f Szz( f )

f –1

0.4

0.3

0.2

0.1

0(a) (b)

Figure 13. Same as figure 12 but for the RL episode.

The −5/3 power-law behaviour at small scales is expected from classical inertial-range theories for three-dimensionally isotropic velocity turbulence (Kolmogorov1941; Obukhov 1941a ,b) and three-dimensionally isotropic temperature turbulence(Obukhov 1949; Corrsin 1951). The observed wavelength of the largest eddies in thatregime agrees well with the ‘isotropic cutoff’ wavelength λm = 4h (h being the altitudeabove ground level) predicted by the simple Muschinski & Roth (1993) model. Ofcourse, λm can be only a rough approximation of the wavelength that separates theisotropic from the anisotropic regime. In the real atmosphere, the transition fromthree-dimensional to two-dimensional flow occurs gradually. Although it may wellbe that the features with the horizontal wavelength λm were strongly squeezed in thevertical direction and therefore strongly anisotropic, we cannot quantify the degreeof anisotropy based on our data. Regardless of the possible anisotropy, however,λm = 4h appears to be a good approximation of the horizontal wavelength at whichthe streamwise spectra show the transition from the plateau to the −5/3 power lawcharacterizing the small-scale turbulence.

The physics of the −5/3 power law at wavelengths longer than 2.3 km is less clear.The low-frequency parts of the spectra represent quasi-two-dimensional velocity andtemperature fluctuations advected past the sensors by the horizontal wind. Spectrafollowing a −5/3 power law in the quasi-two-dimensional wavenumber regime havepreviously been observed in the ocean (Ozmidov 1965) and in the atmosphere (e.g.Gage 1979; Nastrom & Gage 1985) but their physical nature is still a matter ofdebate.

Ozmidov (1965) suggested that in general there is a forward energy cascade,transferring energy from the largest scales to the smallest scales. According to hisconceptual model, the spectral density varies like k

−5/31 (k1 being the wavenumber

in the streamwise direction) except within wavenumber bands of intensified energysupply, where spectral ‘bumps’ and plateaux occur. Ozmidov (1965) identified threesources of energy supply for oceanic motion: major atmospheric disturbances, inertial


and tidal oscillations, and wind waves. In our case, gravity-wave breaking and Kelvin–Helmholtz (KH) instability are two mechanisms through which kinetic energy fromthe horizontal flow could be converted into three-dimensional turbulent kinetic energy.The wavelength of the fastest growing KH billows is about 7.5 times the thickness D

of the shear layer (e.g. Fritts & Rastogi 1985). If D is of order 100 m, then the lengthof the dominating KH billows is of order 750 m. The horizontal wavelength of thedominating gravity waves is 2πU/N , where 2π/N is the Brunt–Vaisala period, whichin the NBL is about 1 min, which leads to a wavelength of 600 m for a wind speedof 10 m s−1. Both wavelengths are between 230 m and 2.3 km, i.e. within the spectralplateau.

The idea that the energy generally cascades all the way down from the very largescales to the dissipation scales is inconsistent with two-dimensional turbulence theory(Kraichnan 1967), which requires an inverse cascade. A forward cascade, however,is consistent with the theory of gravity-wave breaking. Lilly (1983) investigated theratio of the forward and inverse energy transfer rates and concluded that “the upscaleescape of only a few percent of the total energy released by small-scale turbulenceis apparently sufficient to explain the observed mesoscale energy spectrum of thetroposphere”. Recently, Cho & Lindborg (2001) and Lindborg & Cho (2001) analysedaircraft measurements collected in the upper troposphere and lower stratosphere andfound evidence that in the quasi-two-dimensional, mesoscale regime between 10 kmand 100 km the energy is cascaded to smaller scales, not to larger scales as predictedby two-dimensional turbulence theory. Also the recent quasi-geostrophic two-levelmodel simulations by Tung & Orlando (2003) support the notion of an energy sourceat very large scales but do not support the existence of an inverse cascade that wouldhave to be fed by an energy source at small scales. The Tung & Orlando (2003)analysis, however, has to some extent been challenged by Smith (2004). See also thereply by Tung (2004).

Although we present no positive evidence for the forward cascade here, in our casea forward cascade appears to be a more natural explanation than the idea that theentire mesoscale spectrum would be generated and maintained by energy supply atscales of order 1 km.

7.2. Classical spectra and intermittency spectra

As shown in § 6, our NBL observations of the ‘classical’ turbulence spectra Suu(f )and Sθθ (f ) and of the ‘intermittency spectra’ Sxx(f ), Syy(f ), and Szz(f ) are allcharacterized by plateaux at intermediate frequencies and by power-law roll-offs atfrequencies higher than 0.03 Hz, i.e. at wavelengths shorter than 230 m.

While the plateaux in the classical spectra separate small-scale turbulence fromlarge-scale turbulence, the plateaux in the intermittency spectra may be seenas separating small-scale intermittency from large-scale intermittency. Small-scaleintermittency is the random variability of the ‘turbulence parameters’ εr , χr , and(C2

θ )r in the three-dimensional regime, while large-scale intermittency is the randomvariability of εr , χr , and (C2

θ )r in the quasi-two-dimensional regime.That the outer scale L is relevant for the small-scale intermittency statistics was

hypothesized with great intuition by Kolmogorov (1962) and Obukhov (1962), and ourobservations support their view. However, Kolmogorov (1962) and Obukhov (1962)say nothing about large-scale intermittency. Figures 12 and 13 show clearly thaton the low-frequency side of the plateaux, the spectral densities in the intermittencyspectra increase with decreasing frequency, or with increasing length scales. Therefore,intermittency statistics like σx , σy , σz, ρxy , and ρxz vary with the sample size, implying


that any small-scale intermittency model is necessarily an incomplete description ofatmospheric intermittency. The larger the sample, i.e. the wider the p.d.f.s of εr , χr ,and (C2

θ )r , the closer to 1 the correlation coefficients ρxy and ρxz become.

7.3. The f −1 law and the intermittency exponents

While, as expected from classical theory, Suu(f ) and Sθθ (f ) drop off like f −5/3 at highfrequencies, the intermittency spectra Sxx(f ), Syy(f ), and Szz(f ) decrease like f −1. Inthe following, we offer a simple explanation for the f −1 law.

As described in § 2.2, Kolmogorov (1962) and Obukhov (1962) hypothesized thatthe variance of the logarithm of the locally averaged energy dissipation rate, εr (wherer is the linear dimension of the averaging volume) increases with decreasing r as

σ 2x = A + µ ln

L

r. (7.1)

Therefore, the increment dσ 2x is given by

dσ 2x = −µ

dr

r. (7.2)

Since σ 2x may be written as an integral of the intermittency spectrum Sxx(f ), we have

dσ 2x = Sxx(f ) df, (7.3)

such that

Sxx(f ) = −µ

r

dr

df. (7.4)

With Taylor’s hypothesis, U = rf , we obtain

Sxx(f ) = µf −1 (7.5)

and, correspondingly, for the scalar intermittency spectrum,

Syy(f ) = µθf−1. (7.6)

That is, we have a theoretical explanation for the f −1 power law observed inthe intermittency spectra, and the constants of proportionality turn out to be theintermittency exponents µ and µθ themselves. Therefore, in the frequency regimewhere the f −1 law is valid, µ is equal to f Sxx(f ) and can be directly obtained fromfigures 12(b) and 13(b). We find µ ≈ 0.15 and µθ ≈ 0.3 in the NBL as well in theRL episodes. These results agree within a factor of two with values reported earlierand as reviewed by Sreenivasan & Kailasnath (1993). Note that in their review,Sreenivasan & Kailasnath (1993) discuss spectra and autocovariance functions of εr

but they do not consider spectra or autocovariance functions of the logarithm of εr .Using one of the evaluation methods described by Sreenivasan & Kailasnath

(1993), Frehlich et al. (2004) find µ ≈ 0.5 and µθ ≈ 0.6 from the data of the NBLepisode. Frehlich et al.’s µ is twice as large as the value 0.25 that is considered quasi-universal by Sreenivasan & Kailasnath (1993). It has to be kept in mind, however,that practically all previously reported intermittency exponents have been measuredeither in the laboratory or in the atmospheric surface layer. In other words, very littleis known about the intermittency exponents in the atmosphere in the stably stratifiedsurface layer or above the surface layer.


7.4. The power laws of the conditionally averaged temperature structure parameters

In § 3.1, we have shown that if (C2θ )r and εr are jointly lognormally distributed, then

the conditional average 〈(C2θ )r |εr〉 for a specified value of εr is proportional to εδ

r ,where the exponent is δ = ρxyσy/σx −1/3 or, alternatively, δ = ρxzσz/σx; see equations(4.8) and (4.12), respectively. Figures 8–11 show that the theoretical power-law modelsfor 〈(C2

θ )r |εr〉 agree quite well with the empirical conditional averages drawn from thefour ensembles ‘all’, NBL, RL, and AD, respectively. At the tails of the εr distributions,the empirical conditional averages are typically smaller than the respective theoreticalmodel. The reason is that mean values of a lognormally distributed population arenegatively biased, and the bias increases with decreasing population size. Therefore,the 〈(C2

θ )r |εr〉 bias has the largest magnitude in the tails of the εr population.Figures 8–11 and table 1 show that δ is smaller (closer to 1/3) for the constant-

altitude episodes NBL and RL and larger (closer to 2/3) for the episodes ‘all’ andAD, which contain samples from a wide variety of altitudes. This is consistent withthe observation that the larger the ensemble size, the more closely the correlationcoefficients ρxy and ρxz approach the value 1.

8. Summary and conclusionsTen hours of high-resolution, kite-borne turbulence measurements in the lowest

500 m of the night-time troposphere over land have been analysed. Frequency spectraSuu(f ) and Sθθ (f ) of the wind speed and temperature fluctuations and frequencyspectra Sxx(f ), Syy(f ), Szz(f ) of the logarithms of local dissipation rates and localtemperature structure parameters have been presented. In addition, p.d.f.s and jointp.d.f.s of local dissipation rates, εr, and local temperature structure parameters, (C2

θ )r ,and conditionally averaged (C2

θ )r for specified values of εr have been evaluated. Themain results are as follows:

(1) The ‘classical’ spectra Suu(f ) and Sθθ (f ) show two −5/3 regimes which areseparated by a platea. The high-frequency −5/3 regimes characterize three-dimen-sional, Kolmogorov-type small-scale turbulence; the low-frequency −5/3 regimesrepresent quasi-two-dimensional mesoscale motion, probably gravity waves. In ‘area-preserving’ representation, a deep spectral gap separates the two −2/3 regimes. Thespectral plateau observed in the NBL comprises streamwise wavelengths between230 m and 2.3 km.

(2) The ‘intermittency spectra’ Sxx(f ), Syy(f ), Szz(f ) also show plateaux separatingtwo regimes in which the spectral densities decrease with increasing f . The plateauxin the intermittency spectra appear at the same frequencies as in the classicalspectra. In this paper, the small-scale (high-frequency) variability in the intermittencyspectra is referred to as small-scale intermittency, while the large-scale (low-frequency)variability in the intermittency spectra is called large-scale intermittency.

(3) At high frequencies, Sxx(f ), Syy(f ), Szz(f ) decrease proportionally to f −1. It isanalytically shown that Sxx(f ) = µf −1 follows from Kolmogorov’s (1962) lognormalityhypothesis, σ 2

x = A + µ ln(L/r). That is, the (small-scale) intermittency exponents µ

and µθ can be empirically determined from plots of f Sxx(f ) and f Syy(f ), respectively.(4) Conditional averages 〈(C2

θ )r |εr〉 as functions of εr tend to be proportional toεδ

r , where the ‘logarithmic slope’ δ varies with the statistical ensemble. From fourdifferent ensembles, slopes between 0.38 (close to 1/3) and 0.72 (close to 2/3) areobtained. The two ensembles in which data were taken from a single altitude providedthe smallest slopes, while the two ensembles that contained data from all altitudesprovided the largest slopes. Two theoretical models, both of which provide power laws


for 〈(C2θ )r |εr〉, have been developed. The first model, referred to as the ‘small-scale

intermittency model’, assumes that (C2θ )r and εr are jointly lognormally distributed

and leads to δ = ρxyσy/σx − 1/3, where σx and σy are the standard deviations of thelogarithms of εr and χr , and ρxy is the correlation coefficient between the logarithmsof εr and χr . With the assumptions ρxy ≈ 2/3 and σy/σx ≈ 1 (Antonia & Van Atta1975), the small-scale intermittency model leads to δ = 1/3. An alternative model,referred to as the ‘large-scale intermittency model’, has been constructed on thebasis of heuristic mixing-length arguments. With some not too restrictive additionalassumptions, it leads to δ = 2/3.

This study was supported by the National Science Foundation through grant ATM-0128089. In addition, A.M.’s contribution was supported by the US Army ResearchOffice through grant DAAD 19-00-1-0527. Michael L. Jensen and Yannick Meillierwere instrumental in collecting and archiving the measurements. The authors aregrateful to Drs Robert Banta and Reginald Hill and to the anonymous reviewers fortheir comments and suggestions.

Appendix A. Conditional averages of jointly normally distributed variablesConsider the joint probability density function (p.d.f.) of two jointly normally

distributed, zero-mean, unity-variance variables X and Y :

pXY (X, Y ) =1

2π√

1 − ρ2exp

{−X2 − 2ρXY + Y 2

2(1 − ρ2)

}(A 1)

(Parzen 1960, p. 357), where ρ is the correlation coefficient of X and Y .Now, let X be the (natural) logarithm of another random variable, x, such that

X =ln(x/x0)

σX

, (A 2)

where x0 is chosen such that 〈X〉 = 0, as was assumed. Correspondingly,

Y =ln(yy0)

σY

. (A 3)

Here, σX and σY are the standard deviations of ln(x/x0) and ln(y/y0), respectively.If the joint p.d.f. pXY (X, Y ) of X and Y is jointly normal, then, by definition,

pxy(x, y) (the joint p.d.f. of x and y) is jointly lognormal. The two joint p.d.f.s areconnected through

pXY (X, Y ) dX dY = pxy(x, y) dx dy. (A 4)

Therefore,

pxy (x, y) =1

x0σXeσXX

1

y0σY eσY YpXY (X, Y ) . (A 5)

Now, consider the conditional average of xayb for a specified value of x, where a

and b are real numbers:

〈xayb|x〉 ≡

∫ ∞

0

pxy(x, y) xayb dy∫ ∞

0

pxy(x, y) dy

= xa〈yb|y〉. (A 6)


The integrations have to be performed between 0 and ∞ because lognormality of x

and y implies that neither x nor y can assume negative values. The integrations canbe conveniently carried out in logarithmic coordinates. With

yb = ebσY Y yb0 (A 7)

and

dy = y0σY eσY Y dY, (A 8)

we find ∫ ∞

x=0

pxy(x, y)yb dy =yb

0

x0σX

e−σXX

∫ ∞

Y=−∞ebσY Y pXY (X, Y ) dY. (A 9)

The remaining integral is of the form∫ ∞

−∞exp(−αY 2 + βY ) dY =

√π

αexp

(β2

4α

), (A 10)

where

α =1

2(1 − ρ2)(A 11)

and

β = bσY +ρ

1 − ρ2X. (A 12)

Therefore,

∫ ∞

x=0

pxy(x, y)yb dy =√

2π(1 − ρ2)yb

0

x0σX

e−σXX exp

[1 − ρ2

2

(bσY +

ρ

1 − ρ2X

)2]

.

(A 13)

This leads to ⟨yb|x

⟩= yb

0 exp

[1 − ρ2

2b2σ 2

Y

](x

x0

)ρbσY /σX

. (A 14)

With 〈xayb|x〉 = xa〈yb|x〉, we find

〈xayb|x〉 = yb0 exp

[1 − ρ2

2b2σ 2

Y

]xa

(x

x0

)ρbσY /σX

. (A 15)

That is, if x and y are jointly lognormal, then 〈xayb|x〉 as a function of x is a powerlaw,

〈xayb|x〉 = cxδ, (A 16)

with the coefficient

c = x−ρbσY /σX

0 yb0 exp

[1 − ρ2

2b2σ 2

Y

](A 17)

and the exponent, or ‘logarithmic slope’,

δ = a + ρbσY

σX

. (A 18)


Appendix B. Relationships between variances and correlation coefficients of thelogarithms of εr , χr , and (C2

θ )r

Consider the random variable

z = ax + by, (B 1)

where a and b are constant real numbers and x and y are zero-mean random variableswith a correlation coefficient

ρxy =〈xy〉σxσy

, (B 2)

where σ 2x ≡ 〈x2〉 and σ 2

y ≡ 〈y2〉 are the variances of x and y, respectively.By definition, the correlation coefficient of x and z is

ρxz =〈xz〉√

〈x2〉〈z2〉=

a⟨x2

⟩+ b 〈xy〉√

〈x2〉(a2 〈x2〉 + 2ab 〈xy〉 + b2 〈y2〉

) , (B 3)

which gives

ρxz =a + bρxyσy/σx√

a2 + 2abρxyσy/σx + b2σ 2y /σ 2

x

. (B 4)

In addition, we can express the variance σ 2z ≡ 〈z2〉 in terms of σx and σy:

σ 2z = a2σ 2

x + 2abρxzσxσy + b2σ 2y . (B 5)

Now, consider the case where

x = lnεr

εr

, (B 6)

y = lnχr

χr

(B 7)

and

z = ln

(C2

θ

)r(

C2θ

)r

. (B 8)

Here, εr , χr , and (C2θ )r are the ‘geometric mean values’ (Obukhov 1962, p. 79) of

εr , χr , and (C2θ )r , respectively, such that x, y, and z are indeed zero-mean variables.

Through the Obukhov–Corrsin relation,(C2

θ

)r= γ ε−1/3

r χr , (B 9)

we find

a = − 13

(B 10)

and

b = 1. (B 11)

This leads to

ρxz =− 1

3+ ρxyσy/σx√

19

− 23ρxyσy/σx + σ 2

y /σ 2x

(B 12)

and

σ 2z = 1

9σ 2

x − 23ρxyσxσy + σ 2

y . (B 13)


These two equations, (B 12) and (B 13), relate five quantities with each other: thethree standard deviations σx , σy , and σz, and the two correlation coefficients ρxy andρxz. Three of these five, namely σx , σz, and ρxz, can be measured with fine wiresoperating in the inertial range. The other two, σy and ρxy , can be directly measuredonly if the dissipation scales are resolved. But σy and ρxy can still be retrieved frominertial-range measurements of σx , σz, and ρxz. From y = (z − ax)/b we find

σ 2y =

a2

b2σ 2

x − 2a

b2ρxzσxσz +

1

b2σ 2

z (B 14)

and

ρxy =−(a/b)σ 2

x + (1/b)ρxzσxσz

σx

√(a2/b2)σ 2

x − 2(a/b2)ρxzσxσz + (1/b2)σ 2z

. (B 15)

With a = −1/3 and b = 1, this leads to

σ 2y = 1

9σ 2

x + 23ρxzσxσz + σ 2

z (B 16)

and

ρxy =σx + 3ρxzσz√

σ 2x + 6ρxzσxσz + 9σ 2

z

. (B 17)

Note that in this Appendix, no assumptions on the p.d.f.s or joint p.d.f.s of x, y,and z, or of εr , χr , and (C2

θ )r , have been made.

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Sreenivasan, K. R. 1996 The passive scalar spectrum and the Obukhov-Corrsin constant. Phys.Fluids 8, 189–196.

Sreenivasan, K. R. 1999 Fluid turbulence. Rev. Mod. Phys. 71, S383–S395.

Sreenivasan, K. R. & Antonia, R. A. 1997 The phenomenology of small-scale turbulence. Annu.Rev. Fluid Mech. 29, 435–472.

Sreenivasan, K. R., Antonia, R. A. & Danh, H. Q. 1977 Temperature dissipation fluctuations ina turbulent boundary layer. Phys. Fluids 20, 1238–1249.

Sreenivasan, K. R. & Kailasnath, P. 1993 An update on the intermittency exponent in turbulence.Phys. Fluids A 5, 512–514.

Tatarskii, V. I. 1961 Wave Propagation in a Turbulent Medium . McGraw-Hill.

Taylor, G. I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. A 151, 421–478.

Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476–490.

Tennekes, H. 1975 Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech.67, 561–567.

Trevino, G. & Andreas, E. L. 2000 Averaging intervals for spectral analysis of nonstationaryturbulence. Boundary-Layer Met. 95, 231–247.

Tung, K. K. 2004 Reply. J. Atmos. Sci. 61, 943–948.

Tung, K. K. & Orlando, W. W. 2003 The k−3 and k−5/3 energy spectrum of atmospheric turbulence:Quasigeostrophic two-level model simulation. J. Atmos. Sci. 60, 824–835.

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Local and global statistics of clear-air Doppler radar

signals

Andreas Muschinski

CIRES, University of Colorado, and NOAA Environmental Technology Laboratory, Boulder, Colorado, USA

Received 2 May 2003; revised 6 October 2003; accepted 30 October 2003; published 27 January 2004.

[1] A refined theoretical analysis of the clear-air Doppler radar (CDR) measurementprocess is presented. The refined theory builds on the Fresnel-approximated (as opposedto Fraunhofer-approximated) radio wave propagation theory, and turbulence statistics likelocally averaged velocities, local velocity variances, local dissipation rates, and localstructure parameters are allowed to vary randomly within the radar’s sampling volume andduring the dwell time. A local version of the moments theorem and the random Taylorhypothesis are used to derive first-principle formulations of all higher moments of theDoppler cross-spectrum. The mth moment is written as a convolution integral of a spectralsampling function and a generalized, mth-order refractive-index spectrum or, alternatively,as a convolution integral of a lag-space sampling function and a spatial cross-covariancefunction of the local refractive-index fluctuations and their local mth-order timederivatives. Closed-form expressions of the first three moments (i.e., m = 0, 1, 2) of theDoppler spectrum for the monostatic, single-signal case are derived. This refined theory,or ‘‘local sampling theory,’’ enables one to correctly interpret CDR observations that arecollected under conditions where the applicability of the traditional ‘‘global samplingtheory’’ is questionable. The commonly used global sampling assumptions (Bragg-isotropy, homogeneity, and stationarity of all turbulence statistics within the samplingvolume and during the dwell time) may be invalid for small-scale intermittency in themixed layer, for refractive-index sheets corrugated by gravity waves or instabilities, andfor layered turbulence in the stably stratified atmosphere. INDEX TERMS: 0669

Electromagnetics: Scattering and diffraction; 3360 Meteorology and Atmospheric Dynamics: Remote sensing;

3379 Meteorology and Atmospheric Dynamics: Turbulence; 6964 Radio Science: Radio wave propagation;

KEYWORDS: clear-air radars, turbulence, scattering theory

Citation: Muschinski, A. (2004), Local and global statistics of clear-air Doppler radar signals, Radio Sci., 39, RS1008,

doi:10.1029/2003RS002908.

1. Introduction

[2] Ground-based, phase-coherent clear-air radars, alsoknown as clear-air Doppler radars (CDRs), UHF/VHFradars, or radar wind profilers, have been established asstandard instruments for remote sensing of winds, waves,and turbulence in the atmospheric boundary layer (ABL),the free troposphere, and the lower stratosphere for bothresearch and operational purposes [e.g., Woodman andGuillen, 1974; Gage and Balsley, 1978; Balsley andGage, 1982; Gossard and Strauch, 1983; Weber et al.,1990; Mead et al., 1998; Steinhagen et al., 1998]. Thestate-of-the-art of CDR science and technology untilaround 1990 was reviewed by Gage [1990], Gossard

[1990], Rottger and Larsen [1990], and Doviak andZrnic [1993]. More recent developments were summa-rized by Luce et al. [2001a], Gage and Gossard [2003]and Fritts and Alexander [2003].[3] CDRs transmit electromagnetic pulses into the

atmosphere and detect their echoes, which are caused byscatter or reflection from small-scale spatial perturbationsof the instantaneous refractive-index field [Tatarskii,1961; Doviak and Zrnic, 1993]. In contrast to lidars andsodars, CDRs are phase-coherent, such that the phaseinformation is not lost between subsequent pulses. Also incontrast to lidars and sodars, the CDR Doppler shifts arenot obtained by spectral analysis of a single echo but byspectral analysis of a sequence of echoes, where the delaytime (between transmitting and receiving of a pulse) iskept fixed. A typical pulse repetition period of a CDR is100 ms, and a typical ‘‘dwell time’’ (the length of a signal

RADIO SCIENCE, VOL. 39, RS1008, doi:10.1029/2003RS002908, 2004

Copyright 2004 by the American Geophysical Union.

0048-6604/04/2003RS002908$11.00

RS1008 1 of 23

time series fromwhich aDoppler shift is computed) is 10 s.That is, a single Doppler shift is typically estimated from asequence containing on the order of 105 echoes, whereasweather radars can make these estimates using far shorterdwell times (e.g., shorter than 0.1 s) and with far fewersamples (e.g., 50). There are two main reasons for this.First, clear-air reflectivities are usually much smaller thanreflectivities from hydrometeors. Second, weather radarsare horizontally scanning radars operating typically atwavelengths of 10 cm, while CDRs have usually near-vertical beam-pointing directions and operate at longerwavelengths. As a result, compared to weather radars,CDRs are typically operated at much smaller signal-to-noise ratios (SNRs) and with much longer signal correla-tion times. This requires a much larger number of samplesand therefore much longer dwell times for CDRs than forweather radars.May and Strauch [1989] show that mean-ingful Doppler velocities can be retrieved even if the CDRis operated at an SNR as small as �35 dB.[4] Through temporal changes of the echoes’ ampli-

tudes and phases, CDRs can ‘‘see’’ mean and turbulentmotion in the optically clear atmosphere, like an observerlooking down from a bridge across a river can recognizespeed, direction, and turbulence intensity of the waterflow by evaluating the spatiotemporal patterns of lightreflected and scattered from the water surface. In contrastto the observer on the bridge, however, CDRs can detectechoes not only from a discrete surface but from refrac-tive-index irregularities that quasi-continuously populatethe atmospheric boundary layer and also the overlyingfree atmosphere. This offers the possibility to use CDRsto monitor the optically clear atmosphere quasi-continu-ously in time and height. The first three moments of thesignal spectrum can be used to track the refractive-indexvariability, the mean air motion, and the turbulenceintensity [Woodman and Guillen, 1974].[5] Since Woodman and Guillen’s [1974] pioneering

experiments at the Jicamarca VHF radar, CDR researchand technology has benefited from major advancementsin several directions: (1) CDR sensitivity has beenimproved through the development of better antennas,better receivers, and more powerful transmitters; (2) UHFCDRs have been developed, which have become knownas boundary layer profilers and have been used to probethe atmospheric boundary layer (ABL) at altitudes downto less than 200 m AGL [Ecklund et al., 1988; Wilczak etal., 1996]; (3) interferometric and imaging techniqueshave been developed that take advantage of multiplereceivers [e.g., Briggs et al., 1950; Rottger, 1981; Doviaket al., 1996; Chau and Balsley, 1998; Palmer et al.,1998; Mead et al., 1998], multiple carrier frequencies[e.g., Kudeki and Stitt, 1987; Chilson et al., 1997;Muschinski et al., 1999a; Palmer et al., 1999; Luce etal., 2001b; Chilson et al., 2003], or both [e.g., Yu andPalmer, 2001] to overcome resolution limitations set by

a finite pulse length and a finite angular beam width;(4) advanced signal processing allows the atmosphericsignal to be separated from ground and sea clutter,intermittent echoes from birds and aircraft, radio interfer-ence, and external and internal noise [e.g., Lehmann andTeschke, 2001, and references therein]; (5) the Fresnelapproximation [Doviak and Zrnic, 1984], which retainsthe quadratic phase term as function of the transversespatial coordinates and is therefore more accurate than thetraditional Fraunhofer approximation, serves as a unifyingtheoretical framework for Bragg scatter, Fresnel scatter,and Fresnel reflection; (6) advancements in computationalfluid dynamics [e.g., Werne and Fritts, 1999; Smyth andMoum, 2000a, 2000b] and in airborne in situ sensor andplatform technology [e.g., Dalaudier et al., 1994; Balsleyet al., 1998; Muschinski and Wode, 1998; Luce etal., 2001a; Muschinski et al., 2001; Muschinski andLenschow, 2001; Siebert et al., 2003; Frehlich et al.,2003; Balsley et al., 2003] have contributed to morerealistic simulations of and to a better observationalaccessibility to the fine-structure of atmospheric velocityand scalar fields in the atmosphere far from the ground;(7) realistic numerical simulation of CDR signals in theatmospheric boundary layer has become possible bycombining the large-eddy simulation (LES) techniquewith first-principle radio-wave propagation physics forforward scatter [Gilbert et al., 1999] and backscatter[Muschinski et al., 1999b].[6] There are a number of problems, however, that have

been puzzling the CDR community for many years. Whatare the reasons of the biases that are consistently found inVHF and UHF vertical-velocity observations and thatcannot be attributed to instrumental deficiencies [Nastromand VanZandt, 1994;Muschinski, 1996b; Angevine, 1997;Worthington et al., 2001; Lothon et al., 2002]?What is themeaning of spectral moments estimated from signal timeseries as short as 1 s [e.g., Pollard et al., 2000], i.e., shortcompared to the ‘‘renewal time’’ (the time needed for anair parcel to be advected across the CDR’s resolutionvolume)? Down to what length and timescales, and inwhat sense, are signal contributions from different loca-tions within the radar’s resolution volume and differentinstants during the dwell time localizable? What is thecorrect interpretation of spectral moments estimated fromlong signal time series measured in an intermittent orstatistically nonstationary atmosphere?[7] One reason why there is no consensus of how to

address these problems is the lack of a unifying theorythat allows the effects of variations of the local andinstantaneous turbulence characteristics within the reso-lution volume and during the dwell time to be examinedboth generally and specifically. Such a theory is pre-sented in the following. The theoretical developmentbuilds on Fresnel-approximated radio-wave scatteringtheory [Doviak and Zrnic, 1984, 1993], and it allows

RS1008 MUSCHINSKI: STATISTICS OF CLEAR-AIR DOPPLER RADAR SIGNALS

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turbulence statistics like local velocity variances, localenergy dissipation rates, and local structure parameters,which in classical turbulence theory are deterministicvariables, to vary randomly within the CDR’s resolutionvolume and during the dwell time. In the fluid mechanicscommunity, the treatment of local turbulence statistics asrandom flow variables has long been common [e.g.,Oboukhov, 1962; Kolmogorov, 1962; Kuznetsov etal., 1992; Praskovsky et al., 1993; Peltier and Wyngaard,1995; Sreenivasan and Antonia, 1997; Wang et al., 1996,1999; Wyngaard et al., 2001].[8] The paper is organized as follows. In Section 2,

equations for instantaneous covariances eC12(m)(t) �

hI*1 (t)I2(m) (t)i are developed, where I1(t) and I2(t) are

two phase-coherently measured, complex CDR signals,where I2

(m) (t) � @mI2(t)/@tm is the mth time derivative of

I2(t), and where t is time. (Here, I1 and I2 could be twophase-coherent signals measured with two differenttransmitting and/or receiving antennas, at two differentcarrier frequencies, at two different delay times, or‘‘range gates,’’ with two different transmitted pulsedurations, or with two different receiver bandwidths.)Then it is shown that eC12

(m)(t) can be written as aconvolution product over six-dimensional x-r space (xis the location vector, r is the spatial lag vector):

eC mð Þ12 tð Þ ¼

ZZZ ZZZG12 x; rð ÞeR mð Þ

nn x; r; tð Þd3xd3r; ð1Þ

where the local cross-covariance function

eR mð Þnn x; r; tð Þ � en x� r

2; t

� �@menD

xþ r

2; t

� �=@tm

Eð2Þ

is a random function of its arguments, G12(x, r) is adeterministic instrument function, and en is the refractive-index fluctuation with respect to the sampling-volumeaverage of the refractive index [Doviak and Zrnic, 1993,p. 427].[9] Throughout the paper, a tilde over a symbol stands

for a variable that is allowed to vary randomly within theCDR’s resolution volume and during the dwell time, andwhose local mean value is not necessarily zero.[10] In section 3, the Fresnel approximation is used to

derive a model for the single-signal (I1(t) � I2(t))sampling function G11(x, r) for a monostatic radar. It isshown that eC11

(m)(t) can be written as a weighted x-spaceintegral over contributions eFnn

(m)(x, �bkB(x), t), wherebkB(x) is the local Bragg wave vector, and eFnn(m)(x,

k, t) is the local spectrum associated with eRnn(m)(x, r, t).

[11] In section 4, explicit models for eFnn(1)(x, k, t) andeFnn

(2)(x, k, t) are derived based on the assumptions thatvelocity and generalized potential refractive index areconserved quantities and that the viscous terms may beneglected.[12] In section 5, equations for the zeroth, first, and

second moments of the instantaneous Doppler spectrum

in the monostatic, single-signal case are derived. Adiscussion follows in section 6, and a summary andconclusions are given in section 7.[13] Throughout the paper, only the statistics of atmo-

spheric signals are analyzed. That is, any difficultiesassociated with the estimation and removal of any non-atmospheric components in the CDR measurements (i.e.,receiver noise, cosmic noise, ground clutter, radar inter-ference, aircraft echoes, etc.) are not discussed.

2. Two-Signal Statistics: Basic Theory

2.1. Instantaneous Covariances of Signals andSignal Time Derivatives

[14] Consider the scattering integrals for the two sig-nals I1(t) and I2(t) measured at the same time t:

I1 tð Þ ¼ZZZ

G1 x0ð Þena x0; tð Þd3x0 ð3Þ

and

I2 tð Þ ¼ZZZ

G2 x00ð Þena x00; tð Þd3x00 ð4Þ

[Doviak and Zrnic, 1993, equation (11.115) on p. 456],where ena is the fluctuating (with respect to the resolution-volume average) part of the actual refractive index. Notethat no averaging is involved at this point, neither timeaveraging nor spatial averaging nor ensemble averaging.The sampling functions G1(x

0) and G2(x00) are determi-

nistic functions of location. Here we assume that the twosignals are measured at fixed delay times, such that G1

and G2 do not depend on the time t. If ena is a randomvariable somewhere in the resolution volume, orsampling volume (the volume within which the respec-tive sampling function has nonnegligible weight), thenthe signals I1 and I2 are random variables. However, ifena is a deterministic variable everywhere in the samplingvolume, then the signals are deterministic variables.[15] The mth time derivative of I2(t) is

Imð Þ

2 tð Þ � @mI2 tð Þ@tm

¼ZZZ

G2 x00ð Þen mð Þa x00; tð Þd3x00; ð5Þ

where ena(m)(x00, t) � @mena(x00, t)/@tm is the mth local timederivative of ena at the location x00. In the remainder of thepaper, we take advantage of the fact that the generalizedpotential refractive index, for (the fluctuations of) whichwe use the symbol en, is a conserved quantity, in contrastto the actual refractive index, which is not conserved[Ottersten, 1969]. By definition, en = ena at an altitude thatis of specific interest (here the altitude of the center of theresolution volume). In an altitude region around thisreference level, relative pressure changes are muchsmaller than 1 as long as that altitude region (here the


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vertical extent of the resolution volume) is small, whichwe assume. Therefore, in the scattering integrals for I1and I2, and in the expressions for their time derivatives,ena may be replaced with en [see also Doviak and Zrnic,1993, p. 466]. A very similar argument was made byLumley and Panofsky [1964, p. 62] for the relationshipbetween temperature fluctuations and potential tempera-ture fluctuations.[16] Now, we introduce the instantaneous covariance

of I1(t) and I2(m) (t) as

eC mð Þ12 tð Þ � I1* tð ÞI mð Þ

2 tð ÞD E

: ð6Þ

In practice, the instantaneous ensemble average (over alarge number of realizations at the same time t) has to bereplaced by a time average over the temporal evolutionof a single realization of the pair {I1(t), I2(t)}. CDRsignals have usually very short integral timescales (or‘‘correlation times’’ or ‘‘fading times’’), such thatstatistically meaningful signal covariances can bemeasured (quasi-) instantaneously. An excellent discus-sion on the relationships between spatial averages, timeaverages, and ensemble averages of turbulent variablescan be found in Monin and Yaglom [1975, pp. 205 ff].[17] Using the relationships given above, we find


ZZZ ZZZG1* x0ð ÞG2 x00ð Þ

en x0; tð Þen mð Þ x00; tð ÞD E

d3x0d3x00; ð7Þ

where

en mð Þ x00; tð Þ � @m

@tmen x00; tð Þ: ð8Þ

[18] Now, we follow Tatarskii [1961, p. 52ff.] andintroduce sum and difference coordinates. We call

x ¼ x0þx00

2ð9Þ

the location and

r ¼ x00 � x0 ð10Þ

the spatial lag. Furthermore, we define the ‘‘signalcovariance sampling function’’

G12 x; rð Þ ¼ G1* x0ð ÞG2 x00ð Þ ð11Þ

and the local and instantaneous, spatial cross-covariancefunction of the (generalized potential) refractive indexand its mth local time derivative:

eR mð Þnn x; r; tð Þ ¼ en x0; tð Þen mð Þ x00; tð Þ

D E: ð12Þ

We obtain the very general result


ZZZ ZZZG12 x; rð ÞeR mð Þ

nn x; r; tð Þd3xd3r: ð13Þ

Note that the covariances eC12(m)(t) are defined as

instantaneous covariances, such that the eC12(m)(t) are

allowed to be a random function of time.[19] Equation (13) generalizes equation (11.121a) in

Doviak and Zrnic [1993, p. 458] in several respects: first,(13) describes both the two-signal case and the single-signal case; second, eRnn

(m) is allowed to vary randomlywithin the resolution volume and during the dwell time,while Doviak and Zrnic [1993] assume eRnn

(m) to beindependent of x and t); third, (13) describes also thehigher-order (m > 0) Doppler moments, whileDoviak andZrnic [1993] consider only the zeroth moment (m = 0);and fourth, the instrument function G12(x, t), which wehave shown to be the same for all m, is left unspecifiedin (13). Doviak and Zrnic [1993, p. 458] mention thatone ‘‘could separate R(r, r0)[eRnn

(0) (x0, x00, t) in thenotation in this paper] into a product of two functions[of position and of spatial lag, respectively]. . . Thiswould allow the variance of Dn to be spatially depen-dent, a feature that would be especially useful if scat-tering irregularities did not fill the resolution volume.’’The assumption of separability appears to be an unnec-essarily strong assumption, and Doviak and Zrnic [1993]do not justify it. Therefore, in the following we make noattempt to separate eRnn

(m)(x, r, t) into products. Instead,we allow eRnn

(m)(x, r, t) to vary with all of its argumentssimultaneously.[20] Often it is more convenient to work in wave-

number space rather than in lag space. Applying thecorrelation theorem (see Appendix B) to (13) leads to


ZZZ ZZZH12 x; kð Þ eF mð Þ

nn x;�k; tð Þd3xd3k;

ð14Þ

where

H12 x; kð Þ ¼ZZZ

exp �ik � rð ÞG12 x; rð Þd3r ð15Þ

and

eF mð Þnn x; k; tð Þ¼ 1

2pð Þ3ZZZ

exp �ik � rð Þ eR mð Þnn x; r; tð Þd3r:

ð16Þ

[21] In the remainder of this paper, we will considerthe instantaneous covariances eC12

(m)(t) as the primaryCDR observables. In the following subsection, we willsee that, via the moments theorem, the eC12

(m)(t) may be


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interpreted as, and estimated by, the moments of instan-taneous Doppler (cross-) spectra.

2.2. Instantaneous Doppler Spectra andInstantaneous Spectral Moments

[22] Now, consider two signals, I1(t) and I2(t0), mea-

sured at two different times t and t0, where t0 = t + t andwhere t is the time lag. The instantaneous cross-covari-ance function iseC12 t; tð Þ ¼ I1* tð ÞI2 t þ tð Þh i: ð17Þ

In general, CDR signals are not statistically stationary,but often the nonstationarity can be ignored withinnarrow time windows. We refer to such processes asquasi-stationary.[23] In the case of quasi-stationarity we may define the

instantaneous (frequency) cross-spectrum ef12(t, w) as(2p)�1 times the Fourier transform of eC12(t, t) withrespect to t:

ef12 t;wð Þ � 1

2p

Z 1

�1eC12 t; tð Þ exp �iwtð Þdt: ð18Þ

We call ef12(t, w) the instantaneous Doppler cross-spectrum. In practice, estimates of ef12(t, w) are mean-ingful if the dwell time (the length of the signal timeseries from which a spectrum is estimated) contains atleast a few signal correlation times. For UHF clear-airradars operating in the mixed layer, the correlation timesare typically on the order of a tenth of a second, such thatdwell times as short as 1 s usually lead to meaningfulestimates of the instantaneous Doppler spectrum [e.g.,Muschinski et al., 1999b; Pollard et al., 2000], providedthe signal-to-noise ratio is sufficiently high. A quantita-tive discussion of the dependencies of the correlationtimes on the various parameters characterizing the CDRand the micrometeorological conditions in the samplingvolume is beyond the scope of this paper.[24] The integral of ef12(t, w)wm over all frequencies is

the mth moment of the instantaneous Doppler spectrum,or the instantaneous mth spectral moment:

eM mð Þ12 tð Þ �

Z ef12 t;wð Þwmdw: ð19Þ

(Here and in the following, omission of the integrationlimits means that the integration is to be performed from�1 to +1.)[25] Now, consider the mth-order t-derivative ofeC12(t, t) at zero time lag:

eC mð Þ12 t; 0ð Þ � @m

@tmeC12 t; tð Þ

��t¼0

¼ @m

@tmI1* tð ÞI2 t þ tð Þh i

��t¼0

;

ð20Þ

which leads to

eC mð Þ12 t; 0ð Þ ¼ I1* tð ÞI mð Þ

2 tð ÞD E

: ð21Þ

[26] If the signals are statistically stationary or quasi-stationary, then eC12

(m)(t, 0) is directly connected witheM12(m)(t) through the moments theorem, which is derived

in Appendix A:

eC mð Þ12 t; 0ð Þ ¼ im eM mð Þ

12 tð Þ: ð22Þ

That is, the instantaneous covariances eC12(m)(t, 0) are

(apart from the factor im) identical to the instantaneousspectral moments eM12

(m)(t) in the case of quasi-station-arity. In the remainder of this paper, we consider onlythe case t = 0. Therefore, in the following we omitthe argument ‘‘0’’ in eC12

(m)(t, 0), as in the previoussubsection.[27] An interesting alternative approach to the analysis

of CDR signals has recently been put forward byPraskovsky and Praskovskaya [2003]. Instead of analyz-ing auto- and cross-covariance functions at zero time lagand the moments of Doppler spectra and Doppler cross-spectra, as described in this paper, they have investigatedproperties of the auto- and cross-structure functions ofCDR signals. It is not clear at this point to what extentthe theory presented here and the one developed byPraskovsky and Praskovskaya [2003] are consistent witheach other, and what their relative advantages and dis-advantages are.

3. Single-Signal Sampling Functions for a

Monostatic CDR

3.1. General Theory

[28] If the angle between the axes of the transmittingand receiving beams is small, the two-way samplingfunction G(x) for a signal I is given by

G xð Þ ¼ FT xð Þ exp ijT xð Þ½ � FR xð Þ exp ijR xð Þ½ �; ð23Þ

where FT(x) and FR(x) are the one-way electrical fieldweighting patterns of the transmitting antenna and thereceiving antenna, respectively, and jT (x) is the phasechange that the transmitted pulse undergoes along thestraight path from the center of the transmitting antennato the location x within the radar’s resolution volume.Correspondingly, jR(x) is the phase change that thebackscattered pulse undergoes along the straight pathfrom the location x to the center of the receiving antenna.[29] Let us consider here a single signal measured with

a monostatic radar, where the same antenna is used for


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transmitting and receiving, such that FT (x) = FR(x) �F(x). In this case,

G xð Þ ¼ A xð ÞP xð Þ; ð24Þ

where A(x) � F 2(x) is the two-way amplitude weightingfunction and

P xð Þ ¼ exp �ikBj � r0 þ xj½ � ð25Þ

is the two-way phase factor. Here,

kB ¼ 4pl

ð26Þ

is the Bragg wavenumber, and r0 is the range vector,which points from the origin of the x coordinate system

to the antenna center; see Figure 1. We choose the xcoordinate system such that its origin coincides with thecenter of the sampling volume, and that the z-axis pointsin the beam direction (opposite to the direction of r0).Expanding the length j�r0 + xj, which is the distancebetween the antenna center and the scattering point x, ina Taylor series up to second order yields

P xð Þ ¼ exp �ikB r0 þ zþ x2 þ y2

2r0

� �: ð27Þ

Doviak and Zrnic [1984, p. 328] point out that the first-order theory [Tatarskii, 1961], which neglects quadraticand higher-order terms in x and y (Fraunhoferapproximation), is invalid for some relevant applica-tions. We follow Doviak and Zrnic [1984] and retain thequadratic terms in (27). That is, we use the Fresnelapproximation.

Figure 1. Schematic sketch of the sampling geometry of a clear-air Doppler radar (CDR). Theorigin of the coordinate system coincides with the center of the CDR’s sampling volume. A localaveraging volume within the CDR’s sampling volume is depicted. Note that the sketch is not onscale; the beam width is greatly exaggerated.


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[30] According toDoviak and Zrnic [1984, equation (6)]in the case of narrow beams and small pulse widths,A(x) may be written as

A xð Þ ¼ g

lr20

Pt

2R

1=2

f 2Qx

r0;y

r0

Wr zð Þ; ð28Þ

where g is the antenna gain, Pt is the transmitted power,fQ2 (Qx, Qy) is the angular two-way electrical field patternof the antenna, Wr(z) is the combined weighting of thetransmitted pulse and the receiver bandwidth, and R isthe receiver resistance. The beam pattern is normalizedsuch that fQ

2 (0, 0) = 1. For a narrow-beam antenna, thegain g is given by

g ¼ 4pZ Zf 2Q Qx;Qy

�dQxdQy

: ð29Þ

Inserting (27) and (28) into (24) yields

G xð Þ ¼ g

lr20

Pt

2R

1=2

f 2Qx

r0;y

r0

Wr zð Þ

exp �ikB r0 þ zþ x2 þ y2

2r0

� �: ð30Þ

The value offfiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pfor which the phase of G(x)

changes by p (compared to x = y = 0 at the same z) iscalled the radius f of the first half-period Fresnel zone[e.g., Jenkins and White, 1976, p. 380] or, shorter, theradius of the first Fresnel zone [e.g., Doviak and Zrnic,1993, p. 459]. From (30) we find

f ¼ffiffiffiffiffiffiffir0l2

r; ð31Þ

in agreement with Jenkins and White [1976, p. 381,equation (18b)] and Doviak and Zrnic [1993, p. 459].Some authors [Jenkins and White, 1937, p. 175] definethe Fresnel zones based on the one-way (as opposed totwo-way) phase differences. This leads to a Fresnel-zoneradius of

ffiffiffiffiffiffiffir0l

p, [Jenkins and White, 1937, p. 182], which

is byffiffiffi2

psmaller than f as it is defined here.

[31] Now, let us write the covariance sampling func-tion in (13) for the single-signal case (i.e., G12(x, r) �G11(x, r) � G*(x � r/2) G(x + r/2)) as a product of anamplitude factor A11(x, r) and a phase factor P11(x, r):

G11 x; rð Þ ¼ A11 x; rð ÞP11 x; rð Þ; ð32Þ

where

A11 x; rð Þ ¼ A x� r

2

� �A xþ r

2

� �ð33Þ

and

P11 x; rð Þ ¼ P* x� r

2

� �P xþ r

2

� �: ð34Þ

The second-order expansion of P(x) given in (27) yields

P11 x; rð Þ ¼ exp �ikB rz þx

r0rx þ

y

r0ry

� �; ð35Þ

where rx, ry, and rz are the components of the lag vector r.[32] From (28) we obtain

A11 x; rð Þ ¼ g2

l2r40

Pt

2R f 2Q

x� rx

2r0

;y� ry

2r0

0@ 1A f 2Q

xþ rx

2r0

;yþ ry

2r0

0@ 1AWr z� ry

2

� �Wr zþ ry

2

� �: ð36Þ

Therefore,

G11 x; rð Þ ¼ g2

l2r40

Pt

2R f 2Q

x� rx

2r0

;y� ry

2r0

0@ 1A f 2Q

xþ rx

2r0

;yþ ry

2r0

0@ 1AWr z� ry

2

� �Wr zþ ry

2

� � exp �ikB rz þ

x

r0rx þ

y

r0ry

� �: ð37Þ

[33] This equation is very general. No specific assump-tions about the antenna pattern fQ

2(Qx, Qy) and the rangeweighting function Wr(z) have been made so far. Inparticular, we have not assumed that fQ

2(Qx, Qy) andWr(z) are Gaussian, which was assumed by Doviak andZrnic [1984] from the beginning. The Fourier transformof G11(x, r) is

H11 x; rð Þ ¼ZZZ

exp �ik � rð ÞG11 x; rð Þd3r; ð38Þ

and from (14) we obtain a very general equation for theinstantaneous single-signal (I1 = I2) covariances:eC mð Þ

11 tð Þ �ZZZ ZZZ

H11 x; kð ÞeF mð Þnn x;�k; tð Þd3xd3k:

ð39Þ

[34] There are two paths along which one can proceedfrom here. The first path is to assume homogeneity of


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eFnn(m)(x, k, t) within the sampling volume. Then eFnn

(m)(x,k, t) does not depend on x and may be taken out ofthe integral over x:


ZZZh11 kð ÞeF mð Þ

nn �k; tð Þd3k; ð40Þ

where

h11 kð Þ �ZZZ

H11 x; kð Þd3x: ð41Þ

This is the traditional path, which was explored byDoviak and Zrnic [1984], but only for m = 0.[35] The second path takes advantage of the fact that

H11(x, k) may be interpreted as a local (with respect to x)spectral sampling function, such that for a specificlocation x within the sampling volume the integral overk can be evaluated. Then eC11

(m)(t) may be written as aspatial integral over local contributions ec11(m) (x, t):


ZZZ ec mð Þ11 x; tð Þd3x; ð42Þ

where

ec mð Þ11 x; tð Þ �

ZZZH11 x; kð ÞeF mð Þ

nn x;�k; tð Þd3k: ð43Þ

This second path does not require homogeneity, such thateFnn(m) (x, k, t) is allowed to vary with x within the

sampling volume. In practically all CDR applications,H11(x, k) samples only a small region within k-space, aregion that we will refer to as the ‘‘local Bragg window,’’such that H11(x, k) may be approximated by a deltafunction in k-space. This delta function has its peak atthe local Bragg wave vector. In the following, we willrefer to the first path as global sampling and to thesecond path as local sampling.

3.2. Local Sampling

[36] In order to derive a specific model for H11(x, k)and h11(k), we now follow Doviak and Zrnic [1984] andassume that the antenna pattern and range weightingfunctions are Gaussian:

f 2Qx

r0;y

r0

¼ exp � x2 þ y2

2Q20r

20

!ð44Þ

and

Wr zð Þ ¼ exp � z2

4s2r

: ð45Þ

We will see that for Gaussian beam patterns the angle

Q0 ¼Q1ffiffiffiffiffiffiffiffiffiffiffi8 ln 2

p ¼ 0:425Q1 ð46Þ

is a natural measure for the beam width. In the following,we will refer to Q0 as the ‘‘characteristic beam width,’’ asan alternative to the commonly used half-power beamwidth Q1.[37] For a Gaussian beam pattern the gain is

g ¼ 2

Q20

: ð47Þ

The effective antenna area Ae is defined by

Ae ¼gl2

4pð48Þ

[Doviak and Zrnic, 1993 p. 45]. For a rotationallysymmetric antenna, an effective antenna diameter De canbe defined via Ae = p(De/2)

2, which yields

De ¼ffiffiffig

p

pl: ð49Þ

With (47), we obtain for the effective antenna diameterof a Gaussian antenna:

De ¼ffiffiffi2

p

plQ0

: ð50Þ

[38] From (37) we obtain

G11 x; rð Þ ¼ g2

l2r40

Pt

2R exp � x2 þ y2

Q20r

20

� z2

2s2r

!

exp �r2x þ r2y

4Q20r

20

� r2z8s2r

!

exp �ikB rz þxrx

r0þ yry

r0

� �: ð51Þ

After taking the Fourier transform with respect to lagcoordinates and after elementary rearrangements weobtain

H11 x; kð Þ ¼ 25=2p3=2 g2

l2r20

Pt

RQ

20sr

exp � x2 þ y2

Q20r

20

� z2

2s2r

!

exp �Q20r

20 kx þ

x

r0kB

2" #

exp �Q20r

20 ky þ

y

r0kB

2" #

exp �2s2r kz þ kBð Þ2h i

: ð52Þ


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That is, for a given location x, the sampling functionH11(x, k) gives maximum weight to the wave vector

bkB xð Þ ¼ �kB

x

r0y

r0

1

0BBBB@1CCCCA ¼ �kB

Qx

Qy

1

0BBBB@1CCCCA; ð53Þ

where Qx � x/r0 and Qy � x/r0 are angular coordinates.We call bkB(x) the ‘‘local Bragg wave vector’’ and thek-space region around bkB(x) within which the magnitudeof H11(x, k) for a fixed x is nonnegligible the ‘‘localBragg window.’’ The local Bragg wave vector pointsfrom the scattering point x to the antenna center.[39] If we write the Gaussian terms in standard form,

i.e., like exp�� kx þ x

r0kBÞ2=2k2lt�, we obtain the trans-

verse and radial widths of the local Bragg window,

klt ¼1ffiffiffi

2p

r0Q0

ð54Þ

and

klr ¼1

2sr: ð55Þ

For practically all CDR applications, the relative trans-verse width, klt/kB = 1/

ffiffiffi2

pr0Q0kB = De/8r0, is very small

compared to 1. Also the relative radial width, klr/kB =1/2srkB = l/8psr, is very small compared to 1. Therefore,within the local Bragg window, changes of eFnn

(m)(x, k, t)with respect to k can usually be neglected, and we mayapproximate the local spectral sampling functionH11(x, k)by a delta function that peaks at �bkB(x):H11 x; kð Þ ¼ 4p3 g2

l2r40

Pt

R exp � x2 þ y2ð Þ

Q20r

20

� z2

2s2r

! d k þ bkB xð Þ

h i: ð56Þ

Therefore,

eC mð Þ11 tð Þ ¼ 4p3 g2

l2r40

Pt

RZZZ

exp � x2 þ y2ð ÞQ

20r

20

� z2

2s2r

! eF mð Þ

B x; tð Þd3x; ð57Þ

where we have introduced

eF mð ÞB x; tð Þ � eF mð Þ

nn x;�bkB xð Þ; th i

ð58Þ

as the local Bragg component of eFnn(m)(x, k, t).

[40] Ultrawideband radars [e.g., Dvorak et al., 1997]operate with pulse lengths so short that the bandwidthand the carrier frequency are of the same magnitude.

Although in such cases the spectral sampling functioncannot be longer approximated as a delta function, localsampling theory is still applicable.

3.3. Global Sampling

[41] Now, let us assume homogeneity across the sam-pling volume, such that eFB

(m)(x, t) does not depend on x.In this case, we may integrate (52) over x-space andobtain the global spectral sampling function in theFresnel approximation:

h11 kð Þ ¼ p2g2

Pt

R

s2rqr20

exp � 1

qQ20k

2B

k2x þ k2y

� �" #

exp � 2s2rq

þ 2s2rq

r2cr20

kB þ kzð Þ2

� �: ð59Þ

Here we have introduced the near-field parameter

q ¼ 1þ 1

Q40k

2Br

20

¼ 1þ rc

r0

2

; ð60Þ

where

rc ¼1

Q20kB

¼ p8

D2e

l: ð61Þ

By definition, the sampling volume lies in the far field ofthe antenna if r0 is large compared to rc. If that is thecase, then q = 1 is a good approximation, such that in(59) the term 2sr

2/(qr02/rc

2) may be neglected in the farfield. Hence, the global spectral sampling function in thefar field is

h11 kð Þ ¼ p2g2

Pt

R

s2rr20

exp � 1

Q20k

2B

k2x þ k2y

� �" # exp �2s2r kB þ kzð Þ2

h i: ð62Þ

Obviously, the global Bragg wave vector is

kB ¼ �0

0

kB

0@ 1A ð63Þ

and is simultaneously the sampling-volume average ofthe local Bragg wave vector.

4. Generalized Spatial Refractive-Index

Spectra

[42] In order to express the instantaneous covarianceeC11(m)(t), as given in (57), explicitly in terms of local

turbulence statistics, specific models for the ‘‘generalizedspatial spectra’’ eFnn

(m)(x, k, t) are required. In the follow-ing, we derive models for eFnn

(1)(x, k, t) and eFnn(2)(x, k, t).


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We use the compact notation suggested by Lumleyand Panofsky [1964, p. 5f.]: @ui/@xj � ui,j, @ui/@t �ui,t, @n/@xi � n,i, etc.

4.1. Spatial Cross-Spectrum of Refractive-IndexFluctuations and Their First Local Time Derivatives

[43] In order to derive models for eFnn(1)(k), we consider

the spatial cross-covariance function

eR 1ð Þnn x0; x00ð Þ ¼ en x0ð Þ @

@ten x00ð Þ

� ; ð64Þ

where en is the generalized potential refractive indexfluctuation with respect to the resolution-volume average[Doviak and Zrnic, 1993, p. 427], which we assume tobe a conserved quantity. Neglecting molecular dissipa-tion and compressibility effects, we haveen;t ¼ �euien;i; ð65Þ

where eui ¼ Ui þ ui ð66Þ

is the velocity written as the sum of a local average Ui

and a randomly fluctuating part ui. Correspondingly,en ¼ N þ n: ð67Þ

Note that N is several orders of magnitude smaller thanthe refractive-index itself, which is close to 1 in theoptically clear atmosphere.[44] Using themore compact notationN(x)�N,N(x0 )�

N0, Ui (x00) � U00

i, @Ui(x00)/@x00i � U00

i, j, etc., we findeR 1ð Þnn x0; x00ð Þ ¼ en0 �eu00i en00;i� �D E

¼ �U 00i en0en00;iD E

� en0u00i en00;iD E¼ �U 00

i N0N 00

;i � U 00i n0n00;i

D E� N 0 u00i n

00;i

D E� N 00

;i n0u00i! "

� n0u00i n00;i

D E: ð68Þ

By definition, the local averagesUi,N, andN,i are constantwithin the local averaging volume centered at x � (x0 +x00)/2, such that Ui(x

0) = Ui(x00) = Ui(x) etc. Therefore,eR 1ð Þ

nn x0; x00ð Þ ¼ � UiNN;i � Ui n0n00;i

D E� N u00i n

00;i

D E� N;i n

0u00i! "

� n0u00i n00;i

D E: ð69Þ

Now, we evaluate the second term, �Uihn0n00,i i, which, aswe will see in section 5.2, leads to the commonly assumedproportionality between Doppler shift and radial windvelocity. (The fourth term,�N,ihn0u00i i, was first identified

byMuschinski [1998]. It was discussed in detail and calledthe ‘‘correlation velocity’’ term by Tatarskii andMuschinski [2001].)[45] The covariance hn0n00,i i can be evaluated by using

the Fourier-Stieltjes representation of n:

n00 ¼ZZZ

exp ik00 � x00ð ÞdZn k00ð Þ: ð70Þ

Therefore,

n00;i ¼ZZZ

ik 00i exp ik00 � x00ð ÞdZn k00ð Þ ð71Þ

and (see Appendix B)

n0n00;i

D En0*n00;i

D E¼

ZZZexp ik0 � x0ð ÞdZn k0ð Þ

*

�

ZZZik 00i exp ik00 � x00ð ÞdZn k00ð Þ

¼ZZZ ZZZ

ik 00i exp i k00 � x00 � k0 � x0ð Þ½ �

dZn* k0ð ÞdZn k00ð Þh i

¼ZZZ ZZZ

ik 00i exp i k00 � k0ð Þ � x0½ �

� exp ik00 � r½ �

eFnn k0ð Þd k00 � k0ð Þd3k 0d3k 00

¼ZZZ

iki exp ik � rð ÞeFnn kð Þd3k: ð72Þ

(Herewehave takenadvantageof the fact that the refractiveindex is a real quantity, such that n0 � n0*.) We obtaineF 1ð Þ

B x; tð Þ ¼ �i �kB xð Þ � U x; tð Þð ÞeF 0ð ÞB x; tð Þ

¼ �ikBVr xð ÞeF 0ð ÞB x; tð Þ;

ð73Þ

where eFB(m)(x, t) is defined in (58) and where

Vr x; tð Þ ¼ �kB xð Þ � U x; tð ÞkB

ð74Þ

is the local and instantaneous radial wind velocity. Apositive Vr means that the air moves away from the radar.Equation (73) means that the ratio eFB

(1)(x, t)/eFB(0)(x, t)

is proportional to the local radial wind velocity.[46] We have derived (73) from (69) by neglecting all

terms in (69) except the second term. Therefore, (73) isin general invalid if any of the other four terms in (69) isnonnegligible.

4.2. Spatial Cross-Spectrum of Refractive-IndexFluctuations and Their Second Local TimeDerivatives

[47] Consider the spatial cross-covariance function ofen at x0 and its second local time derivative, en(2), at x00:


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RS1008

eR 2ð Þnn x0; x00ð Þ ¼ en x0ð Þ @

2

@t2en x00ð Þ

� : ð75Þ

The second local time derivative of en is

@2

@t2en � en;tt¼ �euien;i �

;t

¼ �eui;ten;i � euien;it¼ � �eujeuj;i �en;i � eui �eujen;ij �¼ eujeuj;ien;i þ euieujen;ij: ð76Þ

Here, we have assumedeui;t ¼ �eujeuj;i; ð77Þ

that is, compared to equation (2.8) in Lumley andPanofsky [1964, p. 61] we have neglected the pressureterm, the buoyancy term, and the viscosity term. Withthis simplification, we obtain

eR 2ð Þnn x0; x00ð Þ ¼ eu00j eu00j;ien0en00;iD E

þ eu00i eu00j en0en00;ijD E: ð78Þ

Each of the two terms contains four factors, each ofwhich is the sum of a local average and a randomfluctuation. Therefore, expanding eRnn

(2)(x0, x00) leads toan expression that contains 32 terms. Out of these32 terms, eight are of the form N0U 00

jU00j,ihn00,ii; they are

zero by definition. That is, there are 24 terms that are notnecessarily zero. It is beyond the scope of this paper tosystematically analyze the relative importance of these24 terms.[48] Let us consider the simplified case that the refrac-

tive-index fluctuations are statistically independent of thevelocity fluctuations, such thateu00j eu00j;ien0en00;iD E

¼ eu00j eu00j;iD E en0en00;iD E¼ UjUj;i þ u00j u

00j;i

D E� � NN;i þ en0en00;iD E� �

:

ð79Þ

Neglecting NN,i leads to

eu00j eu00j;ien0en00;iD E¼ UjUj;i þ u00j u

00j;i

D E� � en0en00;iD E: ð80Þ

Note that hu00j u00j,ii � hujuj,ii is a single-point covariance,while hen0en;00i i is a two-point covariance. This distinctionis important because two-point statistics depend on bothx and the lag vector r � x00 � x0, while single-pointstatistics do not depend on r. Then, after neglecting NN,ij

in heu00i eu00j en0en00;iji, we obtaineR 2ð Þnn x0; x00ð Þ ¼ UjUj;i þ ujuj;i

! " �n0n00;i

D Eþ UiUj þ uiuj

! " �n0n00;ij

D E: ð81Þ

[49] In the previous subsection, we have shown

n0n00;i

D E¼ZZZ

iki exp ik � rð ÞeFnn kð Þd3k: ð82Þ

Differentiation with respect to xj00 leads to

n0n00;ij

D E¼ZZZ

�kikj exp ik � rð ÞeFnn kð Þd3k: ð83Þ

Therefore,

eR 2ð Þnn x0; x00ð Þ ¼ �

ZZZi UjUj;i þ ujuj;i

! " � ki exp ik � rð ÞeFnn kð Þd3k

�ZZZ

UiUj þ uiuj! " �

kikj exp ik � rð ÞeFnn kð Þd3k: ð84Þ

This leads to

eF 2ð Þnn kð Þ ¼ � kikj UiUj þ uiuj

! " �eFnn kð Þ

� iki UjUj;i þ ujuj;i! " �eFnn kð Þ: ð85Þ

[50] The second spectral moment of a single-signal(auto-) spectrum is real; therefore only the real partof eFnn

(2)(k) can contribute, and we may ignore theimaginary part. (Note that the imaginary part must notbe ignored in the two-signal case.) Let us further assumethat huiuji is negligible compared to UiUj. Then we have

eF 2ð ÞB x; tð Þ ¼ � �kB xð Þ � U x; tð Þ½ �2eF 0ð Þ

B x; tð Þ

¼ �k2BV2r x; tð ÞeF 0ð Þ

B x; tð Þ; ð86Þ

where Vr(x, t) is the radial wind velocity defined in (74).

5. Single-Signal Statistics for a Monostatic

CDR

[51] In this section, we apply the general theorypresented above to the first three spectral moments of asingle signal measured with a monostatic CDR.

5.1. Zeroth Spectral Moment: Radar Equationsand VHF Aspect Sensitivity

[52] The instantaneous power ePr(t) received at theantenna is proportional to the instantaneous signal var-iance eC11

(0)(t):

ePr tð Þ ¼ R

2eC 0ð Þ11 tð Þ ð87Þ


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[e.g., Doviak and Zrnic, 1984, p. 327]. From localsampling, (57), we obtain the following radar equation:ePr tð ÞPt

¼2p3 g2

l2r40

ZZZexp � x2 þ y2ð Þ

Q20r

20

� z2

2s2r

! eF 0ð Þ

B x; tð Þd3x: ð88Þ

From global sampling, (62), we find an alternative radarequation:ePr tð ÞPt

¼ p4g2

s2rr20

ZZZ exp � 1

Q20k

2B

k2x þ k2y

� �"�2s2r kB þ kzð Þ2

# eFnn �k; tð Þd3k: ð89Þ

Note that global sampling assumes homogeneity acrossthe radar’s sampling volume, which is why we haveomitted the argument x in the integrand in (89).5.1.1. Isotropic Turbulence[53] In the case of Bragg-isotropic and homogeneous

turbulence, both radar equations lead toePr tð ÞPt

¼ 21=2p9=2 gsrf 4eF 0ð ÞB tð Þ: ð90Þ

[54] Probably (90) is the most concise formulation forthe radar equation that is possible. An important propertyof (90) is that also in the Fresnel approximation theintegral length scale of the refractive-index perturbationsdoes not enter in the radar equation in the case of Bragg-isotropy, in agreement with traditional theory [Tatarskii,1961], which relies on the Fraunhofer approximation.[55] Since the radar equation (90) looks very different

from the clear-air radar equations found elsewhere in theliterature, a consistency check is in order. Doviak andZrnic [1993, p. 75] present the radar equation in thefollowing form:

Pr

Pt

¼ g2l2hctppQ21

4pð Þ3r20lr16 ln 2; ð91Þ

where

h ¼ p2

2k4BF

0ð ÞB ð92Þ

[e.g., Doviak and Zrnic, 1993, equation (11.93) onp. 450] is the volume reflectivity, and

lr ¼ctp=2Z 1

�1W 2

r zð Þdz¼ ctp=2Z 1

�1exp �z2=2s2r �

dz

¼ ctp=2ffiffiffiffiffiffi2p

psr

ð93Þ

is the loss due to finite receiver bandwidth (where tp isthe length of the transmitted pulse and c is the speed oflight). Inserting the inertial-range spectral density at theBragg wavenumber, FB

(0)= 0.033Cn

2kB�11/3 [Tatarskii,

1961, equation (3.25), p. 48], provides the well-knownequation h = 0.379Cn

2l�1/3. The relationship (47)between gain and beam width, g = 2/Q0

2 = 16 ln 2/Q12,

is confirmed by Doviak and Zrnic [1993, p. 478]. Afterelementary rearrangements one finds that (91) isconsistent with (90).[56] That in the case of a Bragg-isotropic refractive-

index field the Fraunhofer approximation and the Fresnelapproximation lead to the same radar equation seems tobe at odds with by Tatarskii [2003], who concludes thatthe Fraunhofer approximation ‘‘can never be used in thetheory of scattering by distributed scatterer (such asturbulence) in the far zone. . .,’’ a statement that is inopposition to his earlier view [Tatarskii, 1961].5.1.2. Bragg-Anisotropy and VHFAspect Sensitivity[57] Consider the simple Gaussian model for the spa-

tial autocovariance function of the refractive-index, assuggested by Doviak and Zrnic [1984]:

Rnn rð Þ ¼ s2n exp �r2x þ r2y

2L2

!d rzð Þ: ð94Þ

Here, L is a correlation length, and sn2 is the variance of

the refractive index. That is, the refractive-indexperturbations are assumed to be delta-correlated in thez-direction. The Wiener-Khintchine relation,

Fnn kð Þ ¼ 1

2pð Þ3ZZZ

exp �ik � rð ÞRnn rð Þd3r; ð95Þ

provides

Fnn kð Þ ¼ 1

4p2s2nL

2 exp � L2

2k2x þ k2y

� �� : ð96Þ

Now, we tilt the plane of the laminae by replacing ky with(kycos � � kzsin �), where � is the angle between theplane of the refractive-index laminae and the x-y plane(i.e., the plane perpendicular to the beam axis). We obtain

Fnn kð Þ ¼ 1

4p2s2nL

2 exp � L2

2k2x

exp � L2

2ky cos�� kz sin� �2� �

: ð97Þ

In the case of horizontal laminae,� is the off-zenith angleof the beam direction. Here, we use the small-angleapproximations cos � = 1 and sin � = � and obtain

Fnn kð Þ ¼ 1

4p2s2nL

2 exp � L2

2k2x þ ky � kz�

�2� �� :

ð98Þ


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Inserting into (88) leads, after elementary rearrange-ments, to the radar equation

Pr �ð ÞPt

¼ 23=2p5=2s2ngsrf 4

1

k2BQ2s

exp ��2

Q2s

!; ð99Þ

where

Q2s ¼

2

L2k2BþQ

20: ð100Þ

For L large compared to the critical length

Lc ¼ffiffiffi2

p

Q0kBð101Þ

we obtain Qs = Q0. A convincing empirical verificationof this asymptotic behavior of Qs can be found inFigure 4 in Hocking et al. [1986], where an observedvertical profile of Qs is shown. The smallest values ofQs were reached in the very stably stratified lowerstratosphere, i.e., at altitudes between about 12 km and20 km AGL. In that region, Qs = 2.3� was found. TheSOUSY radar, which was used for these observations, hadQ1 = 5�, corresponding toQ0 = 1/

ffiffiffiffiffiffiffiffiffiffiffi8 ln 2

p 5� = 2.1�. That

is, the observedminimum ofQs is close toQ0, as predictedby (100).[58] Using (50), it can be shown that the critical length

Lc amounts to one quarter of the effective antennadiameter De:

Lc ¼ffiffiffi2

p

4plQ0

¼ De

4: ð102Þ

That is, Lc is not directly related to the Fresnel length f.This result is in agreement with Gurvich and Kon [1992]but is contrary to common belief, as will be documentedin section 6.

5.2. First Spectral Moment: Doppler Velocity

[59] We define the instantaneous Doppler shift as theratio of the first and the zeroth instantaneous spectralmoments:

ewD tð Þ �eM 1ð Þ11 tð ÞeM 0ð Þ11 tð Þ

¼ 1

i

eC 1ð Þ11 tð ÞeC 0ð Þ11 tð Þ

: ð103Þ

From (52) follows

ewD tð Þ ¼ 1

i

Z Z ZA2 xð ÞeF 1ð Þ

B x; tð Þd3xZ Z ZA2 xð ÞeF 0ð Þ

B x; tð Þd3x; ð104Þ

where A(x) is the amplitude weighting function definedin (28). The common definition of the Doppler velocity

as vD = �wD/kB leads naturally to the definition of theinstantaneous Doppler velocity:

evD tð Þ � � ewD tð ÞkB

; ð105Þ

such that

evD tð Þ ¼ i

kB

Z Z ZA2 xð ÞeF 1ð Þ

B x; tð Þd3xZ Z ZA2 xð ÞeF 0ð Þ

B x; tð Þd3x: ð106Þ

Note that this formulation is very general.[60] Inserting (73), which neglects the correlation

velocity term in eFB(1)(x, t), leads to

evD tð Þ ¼ZZZ evP x; tð Þd3x; ð107Þ

where

evP x; tð Þ ¼ A2 xð ÞeF 0ð ÞB x; tð ÞZ Z Z

A2 xð ÞeF 0ð ÞB x; tð Þd3x

Vr x; tð Þ ð108Þ

is the instantaneous power-weighted radial velocity, i.e.,the local and instantaneous, radial velocity weighted withthe normalized product of A2(x) and eFB

(0)(x, t).[61] An equation similar to (107) has been derived

earlier [e.g., Doviak and Zrnic, 1993, p. 110] for asampling volume populated with point scatterers, and ithas been used heuristically also for scatter from contin-uous refractive fields [e.g., Frisch and Clifford, 1974;Hocking et al., 1986; Muschinski et al., 1999b; Gorsdorfand Lehmann, 2000; Johnston et al., 2001].

5.3. Second Spectral Moment: Spectral Width

[62] Of particular interest for turbulence measurementsis the instantaneous second central moment, normalizedby the instantaneous zeroth moment:

es2w ¼

Z ef11 t;wð Þ w� ewD tð Þð Þ2dwZ ef11 t;wð Þdw

¼eM 2ð Þ11 tð ÞeM 0ð Þ11 tð Þ

� ew2D tð Þ: ð109Þ

The moments theorem (Appendix A) leads to

es2w tð Þ ¼ �eC 2ð Þ11 tð ÞeC 0ð Þ11 tð Þ

� ew2D tð Þ: ð110Þ


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Defining the instantaneous second central moment invelocity units,

es2v tð Þ ¼ es2w tð Þk2B

; ð111Þ

leads to the very general formulation

es2v tð Þ ¼ � 1

k2B

ZZZA2 xð ÞeF 2ð Þ

B x; tð Þd3xZZZA2 xð ÞeF 0ð Þ

B x; tð Þd3x

� 1

k2B

ZZZA2 xð ÞeF 1ð Þ

B x; tð Þd3xZZZA2 xð ÞeF 0ð Þ

B x; tð Þd3x

0BB@1CCA

2

: ð112Þ

[63] The simplified models for eFB(1)(x, t) and eFB

(2)(x, t)given in (73) and (86), respectively, yield

es2v tð Þ ¼ZZZ ev2P x; tð Þd3x�

ZZZ evP x; tð Þd3x 2

:

ð113Þ

This simplified relationship has been derived earlier forpoint scatterers [e.g., Doviak and Zrnic, 1993, p. 110].Frisch and Clifford [1974] anticipated that this relation-ship is valid also for clear-air backscatter and used it toretrieve energy dissipation rates from esv2 measurementsin the atmospheric boundary layer. A discussion of theFrisch and Clifford [1974] approach can be found inDoviak and Zrnic [1993, p. 408f.].

6. Discussion

[64] The three common assumptions in the literatureon CDR theory are that refractive-index and velocityperturbations are (1) statistically homogeneous withinthe radar’s sampling volume, (2) statistically isotropic atthe Bragg wavenumber, and (3) statistically stationaryduring the dwell time. There is a wide consensus,however, that one or more of these assumptions areusually not fulfilled, neither in the free atmosphere[e.g., VanZandt et al., 1978; Gage et al., 1981; Woodmanand Chu, 1989; Gage, 1990; Fairall et al., 1991;Dalaudier et al., 1994; Chilson et al., 1997; Muschinski,1997; Muschinski and Wode, 1998; Luce et al., 2001a]nor in the atmospheric boundary layer [e.g., Gossard etal., 1984; Gossard, 1990; Eaton et al., 1995; Muschinskiet al., 1999b; Pollard et al., 2000;Wyngaard et al., 2001;Balsley et al., 2003; Chilson et al., 2003].[65] The theory developed in this paper allows the

robustness of CDR retrieval techniques against violations

of these three assumptions to be analyzed both specifi-cally and generally. In the following discussion, a num-ber of implications of the refined theory are addressed inan illustrative, rather than in a systematic manner.

6.1. Why Local Sampling?

[66] There are basically two different mechanisms forCDR backscatter. First, scatter from locally Bragg-iso-tropic, fully developed refractive-index turbulence[Tatarskii, 1961]; second, scatter from refractive-indexdiscontinuities (‘‘sheets’’) that are thin compared to theradar wavelength but whose extent in the directionstransverse to the radial direction is much larger thanthe radar wavelength [Metcalf and Atlas, 1973; Gageand Green, 1978; Rottger and Liu, 1978; Doviak andZrnic, 1984; Woodman and Chu, 1989; Tsuda et al.,1997a, 1997b; Worthington et al., 1999; Luce et al.,2001a].[67] The first mechanism is usually referred to as

‘‘isotropic scatter,’’ ‘‘Bragg scatter,’’ or ‘‘turbulencescatter.’’ The second mechanism has been termed ‘‘par-tial reflection,’’ ‘‘quasi-specular reflection,’’ ‘‘Fresnelscatter,’’ or ‘‘Fresnel reflection.’’ It is commonly acceptedterminology that Fresnel scatter is scatter from a popula-tion of parallel sheets with random relative spacing, whileFresnel reflection is scatter (or reflection) from a singlesheet. That is, in contrast to Fresnel reflection, Fresnelscatter requires a statistical approach [Doviak and Zrnic,1984].[68] As Gossard et al. [1984, p. 1532] pointed out,

‘‘under conditions when partial reflection can be impor-tant, the return is from only a tiny fraction of theresolution cell of the radar.’’ The case study by Gageet al. [1981] corroborates this statement. Using a narrow-beam, vertically pointing 50-MHz CDR, they observedspecular reflection from sheets whose tilt angles variedbecause of gravity-wave motion. When the ‘‘specularpoint’’ moved out of the CDR’s angular field of view, theecho intensity dropped dramatically. Within minutes,echo intensity variations of up to 40 dB were observed.[69] More recently, Chau and Balsley [1998] and

Palmer et al. [1998] used interferometric and imagingtechniques to track the angular position of the specularpoint as a function height and time. Hocking et al. [1986]pointed out that the position of the specular point relativeto the beam axis leads to a bias in the retrieved radialwind velocity and suggested a correction formula.Muschinski [1996b] suggested a simple statistical modelof this bias in the case of Fresnel scatter from laminae atthe edges of a train of Kelvin-Helmholtz billows, and heobtained biases as large as a few tens of centimeters persecond for moderate beam widths and for wind speedstypical for upper-tropospheric jet streams. The modelpredicts a decrease of the bias magnitude with decreasingbeam width. In a recent case study on VHF CDR


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observations in the upper troposphere over Indonesia,Yamamoto et al. [2003] report quasi-persistent KHI andgood quantitative agreement with the Muschinski[1996b] model, both with respect to the sign and mag-nitude of the vertical-velocity bias and the sign of thestreamwise echo-intensity imbalance.[70] Woodman and Chu [1989] discussed a conceptual

model for backscatter from a turbulent shear layerembedded in a stably stratified atmosphere. Figure 2 isa modification of Figure 4 in Woodman and Chu [1989].They presumed that the upper and lower edges of theturbulent layers are usually marked by steep verticaltemperature gradients that persist longer than the ran-domly oriented gradients in the interior of the turbulentlayer. If these edges, or interfaces, are thin compared tol/2 and not too rough (also relative to l/2), then the echointensity from the edges may well exceed the echointensity from the more or less isotropic turbulencewithin the layer. Recent high-resolution in situ observa-tions of turbulent layers in the lower free troposphere[e.g., Muschinski and Wode, 1998; Muschinski et al.,2001; Balsley et al., 2003] and advanced direct numer-ical simulations [e.g., Werne and Fritts, 1999; Gibson-Wilde et al., 2000] have provided supporting evidencefor the existence and persistence of these sharp edges.

Figure 2, in contrast to Figure 4 in Woodman and Chu[1989], emphasizes the roughness of the layer edges.[71] The scattering theory presented in this paper can

be used to analyze, in a quantitative fashion, the effectsof various characteristics of turbulent layers on meanvalues and fluctuations of the moments of the CDRspectra. Refined models of turbulent layers could includeparameters like the shear layer thickness, the timeelapsed since the onset of the turbulence, the ‘‘back-ground’’ vertical gradients of horizontal velocity, poten-tial temperature, and specific humidity, the edge regions’roughness statistics, and the larger-scale layer tilt statis-tics. In a further step, the single-layer scenario could begeneralized by allowing the CDR sampling volume to befilled by a population of such turbulent layers, as out-lined by VanZandt et al. [1978] and Fairall et al. [1991].[72] Spatial variability of CDR reflectivity and radial

wind velocity, particularly in the atmospheric boundarylayer, can be directly observed with imaging CDRs likethe Turbulent Eddy Profiler of the University of Massa-chusetts [Mead et al., 1998; Pollard et al., 2000].

6.2. Why Instantaneous Sampling?

[73] For a statistically meaningful measurement of theDoppler shift, the CDR signal time series must contain a

Figure 2. Sketch of a turbulent layer extending across the CDR’s sampling volume, adapted fromFigure 4 in Woodman and Chu [1989]. The thickness of the layer may or may not be larger than theBragg wavelength. The layer ‘‘edges’’ may or may not be smooth with respect to the Braggwavelength. The small diagram on the right-hand side of the main sketch shows schematically aninstantaneous and local vertical profile (thin line) of the refractive-index fluctuations across thelayer. Note the steep gradients in the mean vertical profile (bold line, schematic) at the upper andlower edges of the layer.


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sufficient number of independent samples. In otherwords, the dwell time must be long compared to thecorrelation time tc. If the sampling volume is filled withfully developed turbulence, tc is approximately

tc ¼1

kBst; ð114Þ

where st is the standard deviation of the radial velocityfluctuations within the resolution volume. For a 915-MHzCDR (l = 0.33 cm), tc is as short as 0.25 s even ifst is only 0.1 m s�1. Therefore, a dwell time of 1 s isusually long enough to provide statistically meaningfulspectral moments, which is in agreement, e.g., with thesimulations by Muschinski et al. [1999b]. One second isshort compared to typical ‘‘renewal time’’ scales, i.e., thetime needed for an air parcel to be advected across theradar’s sampling volume.[74] Frisch and Clifford [1974] introduced a technique

to retrieve the energy dissipation rate e from the spectralwidth. This method requires that the observed spectralwidth represents the instantaneous spatial variability ofVr within the sampling volume and that the effect of thetemporal variability of Vr during the dwell time may beneglected. Therefore, the Frisch and Clifford [1974]method requires the dwell time to be short compared tothe renewal time. The effects of ‘‘temporal spectralbroadening’’ associated with dwell times comparablewith or longer than the renewal timescale were quanti-tatively discussed by White et al. [1999].

6.3. Taylor Hypothesis and Random TaylorHypothesis

[75] Heuristic CDR theory often assumes that withinthe CDR’s sampling volume and during the dwell time,refractive-index perturbations are statistically stationaryand homogeneous and are advected with the mean windvector, in agreement with Taylor’s frozen turbulencehypothesis [Taylor, 1938]. This assumption neglectsany correlations between velocity fluctuations andrefractive-index fluctuations from the outset. Frischand Clifford [1974] made one step further by allowingthe velocity to randomly vary within the CDR’s samplingvolume but they still assumed statistical homogeneityand statistical stationarity.[76] The CDR theory developed in this paper allows

mth-order moments of instantaneous Doppler spectra tobe expressed in terms of generalized spectra, eFnn

(m)(x, k, t),which were defined in (16). Models for the generalizedspectra’s Bragg components, eFB

(m)(x, t) � eFnn(m)[x,

�bkB(x), t], for m = 1 and m = 2 were developed insection 4. These models rely on (65), i.e., on the assump-tion that for the local rates of change of the refractiveindex the local and instantaneous advection dominatesand that compressibility effects and molecular dissipation

may be neglected. This assumption is also known asthe ‘‘random Taylor hypothesis’’ [e.g., Tennekes, 1975;Tsinober et al., 2001] or the ‘‘sweeping decorrelationhypothesis’’ [e.g., Praskovsky et al., 1993].

6.4. Velocity Biases Resulting From IncoherentAveraging of Doppler Spectra

[77] Often, dwell times much longer than the renewaltime have to be used in order to separate the atmosphericpart of the Doppler spectrum from system or cosmicnoise. For example, data of the NOAA Profiler Networkare routinely processed with a dwell time of about 1 min[Barth et al., 1994].[78] Now, consider the biases that may result from

using time-averaged spectral moments, i.e., momentscomputed from ‘‘incoherently averaged’’ Doppler spec-tra. The Doppler shift estimated from time-averagedmoments is

VD �eM 1ð Þ11eM 0ð Þ11

: ð115Þ

Obviously, VD is in general different from the timeaverage ewD of the instantaneous Doppler shifts:

ewD ¼eM 1ð Þ11eM 0ð Þ11

!: ð116Þ

Only if Vr(x, t) and eFB(0)(x, t) are uncorrelated, then VD =ewD. On the other hand, if the joint statistics of Vr(x, t)

and eFB(0)(x, t) are known, then specific expressions for

the difference between VD and ewD can be derived, whichmay be used to remove Doppler-velocity biases thatresult from using incoherently averaged Doppler spectra.[79] Two different mechanisms that lead to a correla-

tion between Vr(x, t) and eFB(0)(x, t) in the free atmo-

sphere have been suggested. According to Nastrom andVanZandt [1994], vertically propagating gravity wavesare associated with a correlation between Vr(x, t) andeCn

2(x, t). Muschinski [1996b] pointed out that inshear regions populated with Kelvin-Helmholtz billows,instantaneous Doppler velocities and eFB

(0)(x, t) arecorrelated because the local tilting angles of quasi-specular refractive-index interfaces have a skewedprobability density function.[80] Vertical-velocity biases have been found not only

in VHF CDR observations of the free atmosphere butalso in UHF CDR observations in the daytime ABL.Angevine [1997], Beyrich et al. [1998], and Lothon et al.[2002] report downward biases in the daytime ABL of upto 30 cm s�1. Because the daytime ABL is unstable,gravity waves and Kelvin-Helmholtz instability donot occur there. Therefore, the scenarios suggested byNastrom and VanZandt [1994] and Muschinski [1996b]do not apply for the daytime ABL. In order to understand


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the biases reported by Angevine [1997] and Lothon etal. [2002], further research on the joint statistics Vr(x, t)and eFB

(0)(x, t) in the ABL is needed.

6.5. Aspect Sensitivity, Fresnel Scatter,and Asymptotic Specularity

[81] In section 5, we have shown that the antennadiameter De, not the Fresnel length f, is the relevantlength scale for aspect sensitivity. This is in agreementwith Doviak and Zrnic’s [1984] analytical result that theshape of the global sampling function h11(k) is indepen-dent of range. If f were relevant, then h11(k) would haveto depend on range because f increases proportional tothe square root of the range.[82] In section 5.1.2, we have shown that for a refrac-

tive-index spectrum that is Gaussian in the transversedirections and flat, or white, in the radial direction, thewidth Qs of the angular pattern of backscattered powerdepends on the Bragg wavenumber kB, the transverserefractive-index correlation length L, and the character-istic beam width Q0 in a way that Qs approaches Q0 if Lis large compared to a critical value Lc. We have shownthat Lc amounts to one quarter of the effective diameterand is therefore independent of range. In particular, Lc isnot related to f.[83] The result that a minimum of Qs is set by the

beam width Q0 is in contrast to Hocking and Rottger[2001, p. 941]: ‘‘In principle, specular reflectors can haveany value of Qs;. . .’’ Rottger and Larsen [1990, p. 242]write: ‘‘The terms Fresnel scatter and Fresnel reflectionhave been introduced because the horizontal correlationdistance of the discontinuities is longer than the radarwavelength but of the order of the Fresnel zone. . .’’ Asimilar view is documented in Gage [1990, p. 551]. Thisdisagrees with our result that the critical length scale Lc isequal to De/4 and not equal to f. In the transition regionbetween the near field and the far field, f and De/4 are ofcomparable magnitude, which may be the reason of themisinterpretation. In the ‘‘outer far field,’’ however,where f is large compared to De/4, ‘‘asymptotic specu-larity’’ is reached at much smaller L than previouslythought. (With asymptotic specularity we mean indepen-dence of the degree of aspect sensitivity as function ofL if L is larger than Lc.) For VHF CDR observations inthe mesosphere, f is of order 1 km and much larger thanDe/4, which for a VHF radar is of order 20 m.

7. Summary and Conclusions

[84] In this paper, the existing Fresnel-approximatedtheory of radio-wave backscatter observed with CDRs[Doviak and Zrnic, 1984, 1993] has been extended in twomain directions. First, local sampling has been introduced,such that the local velocity and refractive-index statisticsare allowed to vary randomly within the radar’s sampling

volume and during the dwell time. Second, a first-princi-ple formulation of all moments of the instantaneousDoppler spectrum has been derived from a local versionof the moments theorem in combination with the randomTaylor hypothesis. Although it is beyond the scope of thispaper to give a comprehensive discussion of the implica-tions, a number of conclusions can be drawn:[85] 1. The Fraunhofer approximation, which neglects

effects of the curvature of the wave fronts within thesampling volume [Tatarskii, 1961], and the Fresnelapproximation, which takes the curvature of the wavefronts within the sampling volume into account [Doviakand Zrnic, 1984, 1993], lead to the same radar equationif the refractive-index perturbations are statistically iso-tropic at the Bragg wavenumber.[86] 2. If the local refractive-index spectrum is strongly

Bragg- anisotropic, and if the laminae’s planes are nearlyperpendicular to the beam axis, then the echo intensitytends to become aspect sensitive, as is commonly observedin the lower VHF regime at near-zenith beam directions. Ifthe echo intensity is not aspect sensitive, however, then therefractive-index spectrum is either Bragg-isotropic at leastsomewhere within the sampling volume, or the variabilityof Bragg-anisotropic laminae orientations is large com-pared to the radar’s beam width, or both. Therefore,absence of aspect sensitivity at VHF and UHF frequenciesdoes not necessarily imply Bragg-isotropy, in contrast towhat appears to be widely believed.[87] 3. For asymptotic specularity to be reached, the

refractive-index correlation length perpendicular to thebeam axis must be larger than one quarter of the effectiveantenna diameter but not necessarily larger than theFresnel-zone radius. (Here we have assumed a Gaussianbeam and a refractive-index spectrum that is Gaussian inthe transverse directions and white in the radial direction.)This result is in agreement with Gurvich and Kon [1992]but in contrast to what appears to be widely believed.[88] 4. The refined theory gives a unifying description

of three mechanisms that cause systematic differencesbetween Doppler velocities and radial wind velocities:(1) a correlation between Bragg-isotropic radar reflectiv-ity and radial wind velocity [Nastrom and VanZandt,1994]; (2) a correlation between the orientation of Bragg-anisotropic scatterers and the radial wind velocity[Muschinski, 1996b; Yamamoto et al., 2003]; and (3) anonzero Bragg wave vector component of the spatialquadrature spectrum of refractive-index and radial windvelocity [Tatarskii and Muschinski, 2001]. Whether therefined theory is useful to explain the puzzling down-ward bias of a few tens of centimeters per second, whichis often observed with UHF CDRs in the atmosphericboundary layer [e.g., Angevine, 1997; Beyrich et al.,1998; Lothon et al., 2002], remains to be seen.[89] 5. If the correlation velocity [Tatarskii and

Muschinski, 2001] is negligible and the refractive-index


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perturbations Bragg-isotropic, then the instantaneousDoppler spectra may be interpreted as histograms ofpower- and reflectivity-weighted radial wind velocities.This is not necessarily true for incoherently averaged(i.e., time-averaged) Doppler spectra.[90] 6. The local sampling theory presented here

allows CDR signals to be expressed in terms of localturbulence statistics that may vary randomly in space andtime. In this regard, there is a conceptual similaritybetween the refined CDR theory and the large-eddysimulation (LES), which has become the key techniqueto computationally simulate high-Reynolds number tur-bulent flows [e.g., Lilly, 1967; Leonard, 1974; Schmidtand Schumann, 1989; Muschinski, 1996a; Lesieur andMetais, 1996; Stevens and Lenschow, 2001; Muschinskiand Lenschow, 2001; Wyngaard et al., 2001].[91] 7. Perhaps the most important feature of this

theoretical development, which in its general form hasbeen presented in sections 2 and 4, is that it offers arigorous and systematic access to the zeroth as well asthe higher-order moments of the Doppler spectrum (inthe single-signal case) and of the Doppler cross-spec-trum (in the two-signal case) of observed and simulatedCDR signals under a wide variety of meteorologicalconditions in the atmospheric boundary layer and thefree atmosphere. This has been achieved by combiningthe random Taylor hypothesis, the Fresnel approxima-tion, and a local version of the moments theorem. Alllocal turbulence statistics are allowed to vary randomlyin time and space, without the need to assume isotropy,homogeneity, or stationarity of the refractive-index andvelocity turbulence within the sampling volume andduring the dwell time. Moreover, the general theorymakes no assumptions on the specific form of theinstrument functions. The theory can be used to analyt-ically explore and optimize the design of multifrequencyand multireceiver CDRs as well as the setup of computersimulations based on modern computational fluid dy-namics techniques.

Appendix A: Moments Theorem

[92] The moments theorem allows the mth moment ofa cross-spectrum f12(w) to be expressed in terms of themth derivative of the corresponding cross-covariancefunction C12

(m)(t) at zero lag:

Mmð Þ

12 ¼ 1

imC

mð Þ12 0ð Þ: ðA1Þ

A proof is given in the following.[93] By definition,

Mmð Þ

12 ¼Z

f12 wð Þwmdw: ðA2Þ

Consider the Fourier expansion of C12(t),

C12 tð Þ ¼Z

exp iwtð Þf12 wð Þdw: ðA3Þ

Expanding the exponential into a Taylor series,

exp iwtð Þ ¼X1k¼0

itð Þk

k!wk ; ðA4Þ

and taking the mth derivative at t = 0 leads to

Cmð Þ12 0ð Þ � @

@tmC12 tð Þ

��t¼0

¼ @m

@tm

Z X1k¼0

itð Þk

k!wk

!f12 wð Þdw

��t¼0

: ðA5Þ

With

@m

@tmX1k¼0

itð Þk

k!wk

��t¼0

¼ imwm ðA6Þ

we obtain

Cmð Þ12 0ð Þ ¼ im

Zf12 wð Þwmdw ¼ imM

mð Þ12 : ðA7Þ

Therefore,

Mmð Þ

12 ¼ 1

imC

mð Þ12 0ð Þ; ðA8Þ

q.e.d.

Appendix B: Fourier-Stieltjes

Representations

B1. Statistical Orthogonality

[94] Consider the Fourier-Stieltjes representations oftwo complex, random quantities, p(t) and q(t):

p tð Þ ¼Z

exp iwtð ÞdZp wð Þ ðB1Þ

and

q tð Þ ¼Z

exp iwtð ÞdZq wð Þ: ðB2Þ

Now, assuming statistical stationarity, we define a cross-covariance function:

Cpq tð Þ ¼ p* tð Þq t þ tð Þh i: ðB3Þ


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Taking the conjugate complex of p(t + t) instead ofp(t + t) is an arbitrary but commonly acceptedconvention. Taking

Cpq tð Þ �Z

exp iwtð Þfpq wð Þdw ðB4Þ

as the defining equation for the cross-spectrum fpq(w),we find

Cpq tð Þ �ZZ

exp iwtð Þ exp i w0 � wð Þt½ �

fpq wð Þd w0 � wð Þdwdw0: ðB5Þ

(This looks artificially complicated. But we will see ina moment why we write Cpq(t) in this peculiar way.)[95] On the other hand, we find from the Fourier-

Stieltjes representations:

Cpq tð Þ ¼Z

exp iwtð ÞdZp wð Þ �

*

Z

exp iw0 t þ tð Þ½ �dZq w0ð Þ

¼ZZ

exp iwtð Þ exp i w0 � wð Þt½ �

dZp* wð ÞdZq w0ð Þ! "

: ðB6Þ

Comparing with (B5) leads to the statistical orthogon-ality relationship

dZp* wð ÞdZq w0ð Þ! "

¼ fpq wð Þd w0 � wð Þdwdw0: ðB7Þ

B2. Correlation Theorem

[96] Consider the expression

C ¼Z

G rð ÞR rð Þdr; ðB8Þ

where G(r) is a window function and R(r) is a covariancefunction. Write G(r) and R(r) in terms of their Fouriertransforms, H(k) and F(k), where H(k) is a spectraltransfer function and F(k) is a spectrum:

G rð Þ ¼ 1

2p

Zexp ikrð ÞH kð Þdk ðB9Þ

and (according to the Wiener-Khinthine theorem)

R rð Þ ¼Z

exp ik 0rð ÞF k 0ð Þdk 0: ðB10Þ

Therefore

C ¼Z

1

2p

Zexp ikrð ÞH kð Þdk

Zexp ik 0rð ÞF k 0ð Þdk 0

dr

¼ZZZ

1

2pexp i k þ k 0ð Þr½ �H kð ÞF k 0ð Þdkdk 0dr

¼ZZ

1

2p

Zexp i k þ k 0ð Þr½ �dr

H kð ÞF k 0ð Þdkdk 0

¼ZZ

1

2p2pd k þ k 0ð Þð ÞH kð ÞF k 0ð Þdkdk 0

¼Z

H kð ÞF �kð Þdk; ðB11Þ

where we have used the identityZexp ikr½ �dr ¼ 2pd kð Þ: ðB12Þ

Comparing the two expressions for C leads to thecorrelation theorem:Z

G rð ÞR rð Þdr ¼Z

H kð ÞF �kð Þdk: ðB13Þ

Note the minus sign in F(�k).

[97] Acknowledgments. This work would have been im-possible without innumerable discussions with colleagues inthe various disciplines that are connected by an interest inCDR. I am grateful to Steve Frasier and Paco Lopez-Dekker(both Univ. Mass.) for many interesting discussions and forproviding me access to their beautiful and inspiring TEP andFMCW radar measurements; to Rod Frehlich (CIRES), PeterSullivan (NCAR), Dave Fritts, Joe Werne (both at ColoradoResearch Associates), and John Wyngaard (Pennsylvania StateUniv.) for many discussions about boundary-layer turbulence,LES, DNS, and intermittency; to my colleagues at CIRES andthe NOAA Environmental Technology Laboratory: Bob Banta,Alan Brewer, Phil Chilson, Chris Fairall, Graham Feingold,Reg Hill, Rich Lataitis, Dan Law, Ken Moran, VladimirOstashev, Barry Rye, Valerian Tatarskii, Bob Weber, and AlanWhite for countless conversations on turbulence and atmo-spheric remote sensing; to Phil Chilson, Bob Palmer (Univ.Nebraska), and Tian-You Yu (Univ. Oklahoma) for a decade ofjoint efforts to get the operational community excited about thepotential of frequency-domain interferometry (FDI) and rangeimaging (RIM); and to Peter Czechowsky, Jurgen Klostermeyer,Rudiger Ruster, and Gerhard Schmidt from the SOUSY group,who introduced me to the world of radar remote sensing in theearly 1990s. Thanks are due to Koki Chau (Jicamarca Observa-tory, Peru), Rod Frehlich (Univ. Colorado), Ken Gage (NOAAAeronomy Lab.), Reg Hill, Volker Lehmann (German WeatherService, Lindenberg, Germany), Hubert Luce (Univ. de Toulon


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et du Var, France), Alex Praskovsky and Eleanor Praskovskaya(both NCAR), Valerian Tatarskii, Ulrich Schumann (DLROberpfaffenhofen, Germany), Gerd Teschke (Univ. Bremen,Germany), and John Wyngaard for many helpful comments onearlier versions of the manuscript. A particularly thorough andvaluable review was given by Dick Doviak (NOAA NationalSevere Storms Lab.), who served as one of the Radio Sciencereviewers. I am particularly indebted to Ben Balsley (CIRES),Steve Clifford (CIRES), Dick Doviak, Earl Gossard (CIRES,now retired), Mike Hardesty (NOAA-ETL), Don Lenschow(NCAR), M. J. Post (CIRES), and Dick Strauch (formerlyNOAA-ETL, now NOAA Forecast Systems Lab.) for manyyears of support, encouragement, and collaboration. This studywas supported by the U.S. Army Research Office under grant40136-EV (program management Dr. Walter Bach).

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1Wtflaiooprttwat

tltTistqtfioquttmm

3478 J. Opt. Soc. Am. A/Vol. 24, No. 11 /November 2007 Cheon et al.

Angle-of-arrival anemometry by means of alarge-aperture Schmidt–Cassegrain telescope

equipped with a CCD camera

Yonghun Cheon,* Vincent Hohreiter, Mario Behn, and Andreas Muschinski

Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, 151 Holdsworth Way,Amherst, Massachusetts 01003-9284, USA

*Corresponding author: [email protected]

Received March 21, 2007; revised July 10, 2007; accepted August 10, 2007;posted August 30, 2007 (Doc. ID 81236); published October 12, 2007

The frequency spectrum of angle-of-arrival (AOA) fluctuations of optical waves propagating through atmo-spheric turbulence carries information of wind speed transverse to the propagation path. We present the re-trievals of the transverse wind speed, vb, from the AOA spectra measured with a Schmidt–Cassegrain telescopeequipped with a CCD camera by estimating the “knee frequency,” the intersection of two power laws of theAOA spectrum. The rms difference between 30 s estimates of vb retrieved from the measured AOA spectra and30 s averages of the transverse horizontal wind speed measured with an ultrasonic anemometer was 11 cm s−1

for a 1 h period, during which the transverse horizontal wind speed varied between 0 and 80 cm s−1. Potentialand limitations of angle-of-arrival anemometry are discussed © 2007 Optical Society of America

OCIS codes: 010.1300, 010.1330, 010.7350.

2attt

otTeoflcitmqq

mtgmoTtttsfva

m

. INTRODUCTIONhen an optical wave propagates through atmospheric

urbulence, the wave is affected by the refractive-indexuctuations along the optical path. In particular, thengles-of-arrival (AOAs) of a wave incident upon a receiv-ng aperture, or upon the receiver of a two-point interfer-meter, are functions of the mean values and fluctuationsf the refractive-index gradient along the propagationath. Since temporal changes of the local (optical)efractive-index gradient are dominatingly determined byhe local wind field and the local temperature field, theemporal characteristics of the AOA fluctuations observedith a telescope or an interferometer carry informationbout both the wind field and the temperature field alonghe propagation path.

Assuming that AOAs are measured with a two-point in-erferometer with a baseline b, and assuming that Tay-or’s frozen-turbulence hypothesis �Eq. (7) in [1]� and Ry-ov’s approximation (see pp. 124 ff. in [2]) are valid,atarskii �Eqs. (33)–(36) on p. 268 and Eq. (40) on p. 269

n [3]� developed a theoretical model of the frequencypectrum of AOA fluctuations of a plane wave propagatinghrough homogeneous and isotropic turbulence. The fre-uency spectrum is predicted to be proportional to f−2/3 forhe low-frequency portion �f� fk� and to f−8/3 for the high-requency portion �f� fk�, where f is the frequency and fks the “knee-frequency,” defined as the intersection pointf the two asymptotes. Gurvich et al. [4] measured the fre-uency spectrum of AOA fluctuations of a plane wave bysing a receiver with an aperture of 8 cm and a propaga-ion path length of 650 m. They reported a time-averagedransverse wind speed of �2 m s−1. They compared theeasured frequency spectrum with Tatarskii’s theoreticalodel and also retrieved a transverse wind speed of

1084-7529/07/113478-15/$15.00 © 2

m s−1, indicating some initial agreement between theorynd experiment. They did not mention in detail, however,he manner in which the transverse wind speed was ob-ained from the frequency spectrum. Nor did they quan-ify estimation errors or other errors.

Clifford [5] later investigated the frequency spectrumf AOA fluctuations of a spherical wave observed with awo-point interferometer and obtained results similar toatarskii’s, but with a different weighting function. Raot al. [6] studied numerically the high-frequency behaviorf the frequency spectra of log-amplitude, phase, and AOAuctuations for spherical waves propagating through lo-ally isotropic turbulence and evaluated the effect of thenner scale of turbulence on the high-frequency portion ofhe frequency spectra. They showed that for turbulenceodels that include the inner scale of turbulence, the fre-

uency spectrum drops off steeper than −8/3 at high fre-uencies, where viscous smoothing becomes important.Recently, Lüdi and Magun [7] developed a theoreticalodel and conducted experiments for the frequency spec-

ra of two-dimensional AOAs for a millimeter wave propa-ating through anisotropic turbulence. They fitted theirodel to the measured spectra and retrieved outer scales

f turbulence for the horizontal and vertical directions.he measured spectrum for the horizontal direction for

he low frequencies corresponding to the outer scale ofurbulence was smaller than that for the vertical direc-ion. By fitting the theoretical model to the measuredpectrum, they found that the outer scale of turbulenceor the horizontal direction was larger than that for theertical direction in the case of a stable and stratifiedtmosphere.The overwhelming majority of theoretical and experi-ental work in the area of optical wave propagation

007 Optical Society of America

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Cheon et al. Vol. 24, No. 11 /November 2007 /J. Opt. Soc. Am. A 3479

hrough the turbulent atmosphere has focused on ampli-ude fluctuations (scintillation). The few studies on AOAs“quivering” and “image motion”) were aimed mostly aterifying theoretical predictions of asymptotic poweraws. The usefulness of AOA spectra for the retrieval ofind velocities, however, has not been studied in detailet.

In this paper, we present AOA anemometry as aethod for retrieving the path-averaged transverse wind

peed from the frequency spectra of AOA fluctuations.ield measurements were conducted at the Boulder At-ospheric Observatory (BAO) near Erie, Colorado. The

ptical equipment consisted of a commercial Schmidt–assegrain telescope with an aperture diameter of 36 cm,CCD camera, and krypton flashlight bulbs. An ultra-

onic anemometer was used to measure the transverseind speed independently.This paper is organized as follows. In Section 2, the

heory for the spectrum of AOA fluctuations for thepherical wave is summarized, and the knee frequency isefined. In Section 3, each instrument is described, asell as how the instruments were set up. In Section 4, the

alculation of AOAs and the retrieval of the path-veraged transverse wind speed from the frequency spec-rum of AOA fluctuations are discussed. In Section 5, thexperimental results are presented, and path-averagedransverse wind speeds retrieved from the frequencypectra of AOA fluctuations are compared with the trans-erse wind speeds measured by the ultrasonic anemom-ter. A summary and conclusions are given in Section 6.

. THEORYet us assume that an optical wave propagates throughomogeneous and isotropic turbulence and is observed bytelescope at a distance L from a light source. Let us de-ne the x axis as the propagation direction from theource to the telescope, and the y and z axes such that aight handed coordinate system is formed with the xylane parallel to the ground and the z axis pointing up-ard.To interpret our measured AOA spectra, one needs a

heory that describes the experimental setup with suffi-ient accuracy. According to Tatarskii [3], Clifford [5], andawrence and Strohbehn [8], the essential physics of AOApectra observed with a filled aperture is captured with aigorous theory of AOA spectra observed with a two-pointnterferometer. Lawrence and Strohbehn (p. 1542 in [8])rgue, “…if one is using an interferometer, then the anglef arrival is the phase difference measured at two pointseparated by some distance, call it b. Obviously, the pa-ameter b affects the measurements. If a telescope issed, the angle of arrival is closely related to the averageilt of the wave across the aperture of the telescope. Inhis case spatial variations in phase tend to average out ifhey are smaller than the diameter of the aperture.” Fromhis statement, one might expect that high-frequencypectral densities of filled-aperture AOAs are lower thanhose of interferometric AOAs, which may lead to ateeper dropoff for filled-aperture as opposed to interfero-etric AOA spectra. However, Clifford (p. 1290 in [5]),ithout offering a proof, claims that the main result of his

nterferometric theory “also applies to about the same or-er of approximation for a telescope of diameter b.” It isot clear whether his statement is meant to imply thathe power law of telescopic and interferometric AOA spec-ra should be the same.

In this paper, we use Clifford’s heuristic argument as aorking hypothesis and interpret our telescopic AOA

pectra on the basis of his theory [5] of spherical-waveOA spectra observed with a two-point interferometer. Inubsection 2.A, we discuss Clifford’s theory in some detailnd show that his main result can be obtained by neglect-ng diffraction effects. In Subsection 2.B, we come back tohe question of the differences between filled-aperturend interferometric AOA spectra.

. Reevaluation of Clifford’s (1971) Theory of thenterferometric AOA Frequency Spectrum forSpherical Wavee follow Tatarskii �Eq. (25) on p. 185 in [3]� and denoteith � the AOA retrieved from the instantaneous phaseifference measured with a two-point interferometer withbaseline b. In our experiments we use small krypton

ashlight lamps as sources, which have a diameter ofbout Ds=2 mm. For visible light ��5�10−7 m�, thiseads to a far-field range of 2Ds

2 /��16 m. The propaga-ion path length in the experiment described here was80 m; thus we assume that the unperturbed wave is apherical wave, not a plane one. (Plane waves are usuallyssumed for astronomical observations.)Clifford [5] developed a theoretical model for the fre-

uency spectrum S��f� of AOA fluctuations of a sphericalave propagating through locally homogeneous and iso-

ropic turbulence, assuming that � is measured with awo-point interferometer, where the two point receiversre separated by a baseline of length b. Clifford assumedhat the transverse component of the wind vector is par-llel to the baseline, and he neglected effects of the longi-udinal wind. (Effects of longitudinal wind componentsre known to be small compared with those caused byransverse wind.) The central equation in Clifford’s papers his Eq. (19):

W�S�f� = 32�2k2�0

L

dx sin2��bfx

vbL

��2�f/vb

�

dK Kn�K�

1 + cosK2x�L − x�

kL ��Kvb�2 − �2�f�2

, �1�

here W�S�f� is the interferometric phase-difference fre-uency spectrum, f is frequency, k=2� /� is the opticalave number, L is the propagation path length, K is theagnitude of the (turbulence) wave vector, and n�K� is

he three-dimensional wave number spectrum of theefractive-index fluctuations (assuming isotropy at theave number K). Further assumptions and approxima-

ions that Clifford has made for the derivation of Eq. (1)re that Rytov’s approximation (the approximation ofmooth perturbations) is valid, Taylor’s frozen-turbulenceypothesis is valid for the refractive-index fluctuations,nd the wind velocity is a deterministic variable.

adevf

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w

W

�

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T

w

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In his further development, Clifford assumed that vbnd n�K� do not vary along the propagation path, but heid not simplify the mathematics by ignoring diffractionffects. In the following, we do the opposite: we will allowb and n�K� to vary with x, but we will assume that dif-raction effects can be neglected.

With S��f�=W�S�f� / �kb�2, vb=vb�x�, and n�K�n�K ,x�, Eq. (1) leads to

S��f� = 32�2b−2�0

L

dx sin2 �bfx

vb�x�L��

2�f/vb�x�

�

dK Kn�K,x�

1 + cosK2x�L − x�

kL �� Kvb�x��2 − �2�f�2

.

�2�

. Geometrical-Optics Approximationhe geometrical-optics approximation is valid if diffrac-ion effects can be neglected. This is the case if in Eq. (2)e can approximate

cosK2x�L − x�

kL � � 1. �3�

his approximation is valid if K2x�L−x� / �kL��1, whichs fulfilled for all x��0,L� if K2�L /2��L−L /2� / �kL��1, or

K � 2�k/L. �4�

hat is, Eq. (3) is valid if refractive-index fluctuationsith wavenumber K larger than the Fresnel wavenumberk /L contribute only a negligible amount to the K inte-ral in Eq. (2). Because Kn�K� /��Kvb�2− �2�f�2 decreasesapidly with increasing K [note that n�K� varies like−11/3 in the inertial subrange], this can be safely as-

umed if the wavenumber 2�f /vb (the lower integrationimit in the K integral) meets criterion (4), such that

2�f/vb � 2�k/L. �5�

his translates into a criterion for the frequencies f forhich the geometrical-optics approximation, Eq. (3), isalid:

f ��2

�

vb

��L. �6�

hat is, for frequencies small compared to the Fresnel fre-uency,

fF = vb/��L, �7�

e have

S��f� = 64�2b−2�0

L

dx sin2 �bfx

vb�x�L��

2�f/vb�x�

�

dKKn�K,x�

� Kvb�x��2 − �2�f�2. �8�

e insert Tatarskii’s inertial-range spectrum

n�K,x� = cTCn2�x�K−11/3 �9�

Eq. (3.24) on p. 48 in [2]�, where

cT =�3�8/3�

8�2 = 0.03305, �10�

s Tatarskii’s coefficient, and we introduce the dimension-ess integration variable �,

� = Kvb�x�

2�f �2

− 1. �11�

hen we can evaluate the K integral exactly and find

S��f� =27/3

9�7/6�5/6�b−2f−8/3�

0

L

Cn2�x�vb

5/3�x�sin2 �bfx

vb�x�L�dx,

�12�

here we have used Eq. (10) and

�0

�

�−1/2�1 + ��−11/6d� =4

15�3/2

�3

�2/3��5/6�. �13�

The numerical coefficient in Eq. (12) has the value.1305.]Let us consider the simplest case where Cn

2 and vb doot vary with x. Then Eq. (12) gives

S��f� =24/3

9�7/6�5/6�Cn

2vb5/3Lb−2f−8/31 −

sin�2�bf/vb�

2�bf/vb� ,

�14�

here the numerical coefficient has the value 0.06524.quation (14) is identical to Clifford’s main result, his Eq.

28). [His slightly different coefficient, 0.066, is probablyue to rounding errors, and his factor k2 instead of b−2 isue to an erroneous conversion from W�S�f� to S��f�.] It isuite remarkable that Eq. (14) can be easily obtained viahe geometrical-optics approximation, Eq. (3), without theumbersome integrations that Clifford had to carry outecause he did not approximate the cosine term as aonstant.

For frequencies large compared with the Fresnel fre-uency vb /��L, the cosine term in Eq. (1) oscillates rap-dly between −1 and 1, such that the contribution to the Kntegral becomes negligible. As a result, for f�vb /��L,he magnitude of S��f� is, compared with Eq. (14), reducedy a factor of 2, in agreement with Eq. (27) in [5]. Thisactor of 2 is also consistent with Tatarskii’s result �Eqs.20) and (21) on p. 289 in [3]� that the AOA variance ob-erved with a large-aperture telescope (aperture diameter��L) is twice the AOA variance observed with a

mall-aperture telescope �D��L�. In their recent analy-is based on the Rytov approximation, Cheon anduschinski [9] showed that the large-aperture asymptote

f the circular-aperture AOA variance is reached within% for plane waves if D�1.65��L and for spherical wavesf D�1.84��L.

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. Knee Frequencyet us evaluate Eq. (14) further, with the goal of using its the model to extract vb from measured S��f�. For�bf /vb�1, or

f �1

2�

vb

b, �15�

he second term in the square brackets of Eq. (14) can beeglected, and we have

S��f� = 0.06524Cn2vb

5/3Lb−2f−8/3. �16�

or 2�bf /vb�1, however, we can expand the sine func-ion up to third order, sin ��−�3 /6, and we obtain

S��f� = 0.065242

3�2Cn

2vb−1/3Lf−2/3. �17�

he high-frequency f−8/3 asymptote and the low-frequency−2/3 asymptote intersect at the knee frequency,

fk =�6

2�

vb

b, �18�

r fk=0.39vb /b. If the baseline length b [or an effectiveaseline, since we retrieve S��f� from filled-aperture cen-roids] is known, then vb can be retrieved from a measure-ent of fk. On the other hand, if both fk and vb are mea-

ured, then Eq. (18) allows us to determine the effectiveaseline length empirically. This baseline calibration isecessary if S��f� is measured with an instrument that isot a two-point interferometer (as in our case) or if theath-transverse wind component is not parallel to the in-erferometer baseline.

. Frequency Dependence of Path-Weighting Functionsccording to Eq. (12), the path-weighting function for��f� in the case of a spherical wave is given by

Wp�f,x� = Cn2�x�vb

5/3�x�sin2 �bfx

vb�x�L� . �19�

f wind speed and turbulence are homogeneous, then theath-weighting is purely geometric:

Wg�f,x� = sin2��bfx

vbL . �20�

t is important to note that Wg�f ,x� is frequency depen-ent. For �� /4, the small-angle approximation sin �� is accurate within 11%, and sin2��2 is accurateithin 23%, such that Wg�f ,x� varies approximately like

2 within �0,L� for �bfx / �vbL�� /4, or

f �1

4

vb

b. �21�

hat is, for frequencies smaller than the knee frequency,he centroid of W �f ,x�,
p
xc�f� =

�0

L

Wg�f,x�xdx

�0

L

Wg�f,x�dx

, �22�

s independent of f and approaches the value xc= �3/4�L,hat is, about L /4 away from the receiver. For frequenciesarge compared with fk, Wp�f ,x� oscillates rapidly between

and 1, such that the path weighting is practically uni-orm. The first null in Wp�f ,x� within the propagationath, i.e., for x� �0,L�, appears at the frequency

f =1

�

vb

b= 0.81fk. �23�

. Effects of Variability of vb and Cn2 on AOA

pectrahe path-weighting becomes more complicated when vbnd Cn

2 vary along the path. The atmospheric weightingunction

Wa�x� = Cn2�x�vb

5/3�x�, �24�

eads to higher contributions to S��f� from locations xhere Cn

2 and vb are large. While Cn2 enters linearly into

a�x�, the baseline wind speed vb enters with the power of/3. The effect of wind-speed variability on the magnitudef S��f�, however, is almost compensated because vb en-ers as vb

−2 through the geometric weighting (as long as fs not too large), such that the effect of inhomogeneousind speed on the AOA spectral density is expected to be

mall compared with effects of inhomogeneous Cn2. The

ain effect of a varying wind speed, regardless of whetherithin the path or during the observation time duringhich an estimate of S��f� is taken, is that there is a dis-

ribution of knee frequencies, such that the sudden dropf S��f� is smeared over a certain range of frequencies.

One could approach the problem in a probabilistic fash-on: if one allows Cn

2 and vb to vary randomly, such that�Cn

2 ,vb� is their joint probability density function, thenhe expected AOA spectrum is

�S��f�� =�� p�Cn2,vb�S��f;Cn

2,vb�dCn2dvb, �25�

here S��f ;Cn2 ,vb� is given, say, by Eq. (14). An investiga-

ion along these lines, however, is beyond the scope of thisaper.

. Interferometric versus Filled-Aperture AOAse come back to the question of whether Clifford’s [5] in-

erferometric model for spherical-wave AOAs is adequateor interpreting AOAs retrieved from circular-apertureentroids.

Hogge and Butts [10] carried Clifford’s analysis furthernd derived a theoretical model for the frequency spec-rum of wavefront tilts over a filled, circular aperture.hey expanded the wavefront aberration into Zernikeolynomials and found −2/3 and −11/3 asymptotes forrequencies smaller and larger than v /D, respectively,here v is the cross-wind speed and D is the aperture di-

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meter. Along the lines of the remarks by Lawrence andtrohbehn [8] quoted at the beginning of this Section,ogge and Butts (p. 148 in [10]) offer an interpretation of

he steeper asymptote (as compared with the f−8/3 lawserived by Tatarskii [3] for plane waves and by Cliffordor spherical waves): “The different dependence on fre-uency we believe can be explained in the following way.or the phase difference spectrum, one measures the dif-

erence between the phase fluctuations at precisely twooints in space. The phase fluctuations at each point havepower spectrum that varies for large f like f−8/3. Hence,

he phase difference power spectrum for the higher fre-uency components (i.e., for those phase fluctuations withcale sizes smaller than the separation of the two measur-ng points) will also vary as f−8/3. On the other hand, thengle-of-arrival power spectra obtained from a leastquares fit to the phase distortions is obtained by per-orming a spatial averaging over the fluctuations in theollecting aperture. Thus, even though distinct points inhe field of the collecting aperture may have phase fluc-uations that behave as f−8/3, the averaged wavefront tiltill not contain as much power at the higher frequencies,nd hence, varies for large f as f−11/3.” However, whileogge and Butts’s explanation helps to understand why

entroid-derived AOAs (observed with an aperture of di-meter D) have a somewhat smaller variance than AOAsbtained from a two-point interferometer with baseline=D, it leaves it unclear why aperture averaging should

ead to a spectral density with an f−11/3 law, as opposed todifferent power law or an unchanged (f−8/3, say) power

aw with just a smaller constant of proportionality.Here we offer a simple physical explanation of the f−11/3

aw. Consider the interferometric AOA variance S�I �f�qf

ontained in some frequency band around f with a rela-ive width q (1/2, say). Then, according to Taylor’s frozen-urbulence hypothesis, l=vb / f is the transverse scale sizef the refractive-index perturbations associated with theOA fluctuations at frequency f, where vb is the baselineind speed. Therefore, the number of independent events

eddies) of size l along a line across the aperture is ND / l�Df /vb. According to the law of large numbers, av-

raging over N independent events reduces the variancey the factor N, and we obtain S�

FA�f��S�I �f� /N

�v /D�S�I �f�f−1, where S�

FA�f� is the AOA spectrum ob-ained with a filled aperture. This, with Eq. (16), leads to

S�FA�f� � D−3LCn

2vb8/3f−11/3. �26�

This result is consistent with Tyler’s result �Eq. (48) in11]; note that v=vb /D in Tyler’s notation�, who workedut the circular-aperture AOA spectrum for plane waves.The same result, but with a different numerical coeffi-ient, should follow from Hogge and Butts’s spherical-ave theory [10].)Interestingly, there has been very little observational

upport for the f−11/3 law in the published literature. Oneeason is that most AOA measurements have been madeith interferometric configurations, and those measure-ents are usually consistent with an f−8/3 law (see, e.g.,ig. 64 on p. 305 in [3]; Fig. 4 in [5]; Fig. 3 in [7]), in agree-ent with theory. The two reports on observations of

entroid-derived AOA spectra that we are aware of are

omewhat contradictory: Acton et al. [12] observed wave-ront tilt power spectra from the image motion of solarores. They write, “Between 5 and 150 Hz, the slopes arepproximately −2.6. This is not consistent with either ofhe theoretical predictions. It is interesting to note, how-ver, that both the measurements of −0.5 and −2.6 areonsistent with the slope predictions of −2/3 (low) and8/3 (high) for a phase-difference power spectrum. Tatar-kii also derived the phase-difference PSD slopes andhen stated without proof that the same slopes should bepplicable to a wave-front tilt PSD.” On the other hand,ore recent observations by McGaughey and Aitken [13]

o support an f−11/3 power law (Fig. 3 in [13]). McGaugheynd Aitken point out that “the expected −11/3 index isvident only when the estimated noise level is subtractedrom the curves.”

. EXPERIMENTield experiments were conducted after sunset on Sep-

ember 27 and 28, 2006, at BAO near Erie, Colorado. TheAO is a research facility operated by NOAA (the Na-

ional Oceanic and Atmospheric Administration) with theurpose of facilitating investigations of the atmosphericoundary layer. The primary feature of the BAO facility is300 m tower erected over wide open and reasonably flat

errain [14,15]. All instruments were set up at the south-ast side of the BAO tower, as shown in Fig. 1. Theround cover surrounding the experiment site was uni-orm in all directions and consisted of dry brush with aeight of 30 to 90 cm as shown in Fig. 2.

. Experimental Setuplarge-aperture telescope (Model LX200GPS, Schmidt–

assegrain type, manufactured by Meade Instrumentsorporation of Irvine, California) was set up approxi-ately 100 m from the BAO tower, and the legs of the

elescope frame adjusted such that the center of the aper-ure was 1.5 m above ground level (AGL). Four kryptonashlight bulbs, each with an aperture diameter of2 mm, were arranged on a sturdy metal tripod that was

ositioned 180 m southeast of the telescope (bearing of40.8 deg relative to North). The individual bulbs wererranged at the corners of a rectangle with a spacing of0 cm in the horizontal and 9 cm in the vertical. The cen-er of the array of bulbs was at 0.94 m AGL.

After the telescope was focused, a black-and-white CCDamera (Model 075M, manufactured by Lumenera Corpo-ation of Ottawa, Ontario, Canada) was mounted at theear-side observation port of the telescope. A laptop com-uter with an external hard disk drive was connected tohe CCD camera and ran software that controlled the pa-ameters associated with image capture (i.e., frame ratend exposure time; see Subsection 3.B for more detail).An ultrasonic anemometer–thermometer (Model

1000, manufactured by R. M. Young Company ofraverse City, Michigan), from this point onward referredo as the “sonic,” was set up at 64 m from the telescopelong the line of the optical propagation path at a heightf 1.5 m AGL and within 1 m of the propagation path hori-

zcv

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arasot�=ate

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ontally. The sonic measured three components of the lo-al, instantaneous wind velocity, which we call vx, vy, andz (see Subsection 5.B).

Fifteen self-logging, thermistor-based thermometersModel HOBO H8 Pro, manufactured by Onset Computerorporation of Pocasset, Massachusetts) were deployed in

hree vertical arrays (of five sensors each) that were posi-ioned 31, 56, and 94 m from the telescope, along the op-ical propagation path. Each array of five sensors wasounted securely to a stable stand (photographic light

tand with tripod legs). The sensors were spaced verti-ally along each stand at heights 0.3, 0.5, 1.0, 1.5, and.0 m AGL.

. Technical Details and Operation of Instrumentshe telescope was of the Schmidt–Cassegrain type. That

ig. 2. Experimental setup with the 14-in. telescope, ultrasonicnemometer, and self-logging thermometers at BAO.

ig. 1. Diagram of position of the instruments and the propagaado, generated from DeLorme Topo USA 3.0. BAO, BAO tower;onic anemometer tower, respectively; LG, light.

s, its physical aperture area was annular such that light r

rriving at the aperture entered the telescope through aegion concentric about a circular, center obstruction. Theperture had a 36 cm outer diameter, and the center ob-truction had a diameter of 12.7 cm. The focal length, F,f the telescope was 3556 mm. The angular resolution ofhe telescope was, according to the Rayleigh criterionEq. (1) in [16]�, 1.22� /D=1.71 rad=0.35 arcsec for �

500 nm. The telescope was focused on the bulb arraynd the CCD camera attached as discussed above. Theelescope adjustment remained fixed for the duration ofach experiment.

The CCD camera had 640�480 pixels, and the spacingetween each pixel was 7.4 m in both directions, trans-ating to an AOA increment of 2.08 rad per pixel. Theeak sensitivity of the CCD was to wavelengths of00 nm. The camera was controlled by a laptop computerhich ran software (Stream Pix, Version 3.21.0) that

pecified the number of pixels, the exposure time, therame rate, and the bit depth. The number of pixels was40�480 pixels. The exposure time of the CCD cameraas set to 35 s (from an available range of 1 s to 400 s),hich was determined as the optimal value to maximize

he signal-to-noise ratio and yet avoid saturation. Therame rate was set to 30 frames/s, which was found to behe maximum frame rate for which all frames could be re-iably saved. The bit depth was set to 12 bits.

The image data, saved in an external hard disk drive,ere given individual time stamps that were extracted

rom the system clock of the computer. The system clockas manually synchronized with that of the data-logging

ystem (developed by the authors M. Behn and A.uschinski) for the sonic, to ensure consistency of time-

eries data between the two instruments with an accu-

ath at Boulder Atmospheric Observatory (BAO) near Erie, Colo-lescope; HT and ST, self-logging thermometer towers and ultra-

tion pTS, te

acy of about 1 s.

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The sonic, at the setting used for the present experi-ents, had a measurable wind speed range from 0 to

0 m s−1 with a resolution of 0.01 m s−1 and a measurableemperature range from −50 to +50°C with a resolutionf 0.01°C. The sampling rate was adjustable within theange 0–32 Hz, of which only the maximum 32 Hz wassed. The data were saved in a local storage of the data-

ogging system. A time stamp was provided for each dataoint by means of a GPS (Global Positioning System) re-eiver, which automated the synchronization of the on-oard system clock of the data-logging system with a GPSlock.

The self-logging thermometers were rated to measureemperature in the range −30–50°C with an accuracy of0.2°C at 25°C. The sampling period of 5 s was used

from an available range of 0.5 s to 9 h), and all data weretored in the on-board memory. As with the CCD camera,he internal clock of each temperature sensor was syn-hronized with that of the data-logging unit for the sonic,o maintain consistency in time between all data sources.

Utilizing the instruments as described above, CCD im-ges of the bulb array along with supporting wind andemperature information from the sonic and self-logginghermometers were collected in two approximately 1 h in-ervals, the first between 21:00 and 22:00 on September7, 2006, and the second between 18:00 and 19:00 on Sep-ember 28, 2006. Processing of the data and experimentalesults of the first data set are presented in the sectionshat follow.

. DATA PROCESSINGn this section, we describe a procedure to estimate windpeeds from digital images of a point source. We proceedn three steps: first, time series of vertical and horizontalOAs are estimated from the sequences of digital images;econd, we estimate three parameters (Cn

2, the modelaseline wind speed vb, and a noise floor) in Clifford’sodel for the AOA frequency spectrum by fitting theodel spectrum to the observed AOA spectra; third, we

etrieve the calibrated baseline wind speed vaoac , assuming

linear relationship between vb and vaoac .

. Estimating Image Centroids and AOAshe digital images are discrete-time sequences of the in-ensity field measured in the focal plane. Assuming thathe telescope is focused on a point source and neglectingiffraction effects due to the finite size of the aperture16], a spherical wave emitted from the source would bebserved as a point image, whose pixel coordinates i and jead to the AOAs

��y

�z =

1

F��yi

�zj = ��yi

��zj , �27�

here F is the focal length, and �y and �z are the physicalixel spacings in the horizontal and vertical direction, re-pectively. In our case, �y=�z=7.4 m and F=3.556 m,hich gives the angular pixel resolutions ��y=��z2.08 rad.However, turbulent refractive-index perturbations

long the propagation path lead to phase-front distor-

ions, causing local, instantaneous AOAs to vary acrosshe aperture and resulting in a blurred image. Accordingo Gurvich and Kallistratova [17], the instantaneous AOAentroids �y and �z,

��y

�z =

�i,j��yi

��zjIij

�i,j

Iij

, �28�

ay be used as aperture-averaged AOAs.For calibration of CCD images (p. 157 in [18]), a master

ark image (average of dark images) can be used to re-ove a dark current caused by thermal noise. Both before

nd after recording light images, dark images were re-orded by covering the aperture of the telescope. After re-oving the dark current by subtracting the master dark

mage, we divided images into four sections, one for eachf the four light sources, and each section was centeredith the maximum intensity of each light. The diametersf the (blurred) images were about 30 pixels, that is,bout 60 rad. Howell (p. 181 in [19]) claimed that in or-er to determine the mean of the background intensity,he size of the image should be about three times the sizef the spot. Thus, we chose a section size of 101�101 pix-ls. We subtracted the mean background intensity forach section, which was calculated after the bright spotas excluded. Then, we calculated the aperture-averagedOAs by using Eq. (28).

. Least-Squares Optimizationhe frequency spectrum of the AOA fluctuations, ashown in Eq. (14), is a function of the refractive-indextructure constant, Cn

2, and the transverse wind speed, vb,or the given propagation distance, L, and baseline, b, forhe interferometer or aperture size, D, for the telescope.he frequency spectrum that is distinct enough to easilyefine the knee frequency might not always be obtainedecause of intermittency of turbulence or the presence ofhigh noise floor. Thus, it is necessary to have a system-

tic way for retrieving the knee frequency or the trans-erse wind speed. We used Eq. (14) to fit a function to thebserved frequency spectrum of AOA fluctuations by aeast-squares optimization.

Let S�,im �f ,Cn

2 , vb� be the observed frequency spectrumnd S�,i

t �f ,Cn2 , vb� be the Clifford model. It is necessary to

nd Cn2 and vb (L and D are given) for minimizing a cost

unction, C, where

C = �i=1

N

log10 S�,it �f,Cn

2, vb� − log10 S�,im �f,Cn

2, vb��2, �29�

S�,it �f,Cn

2, vb� = 0.065Cn2Lvb

5/3D−21 −sin�2�Df/vb�

2�Df/vb�f−8/3

+ Snoise, �30�

nd f= �f1 , f2 , ¯ , fN�. Note that we used the aperture di-meter D as (a first guess of) the effective baseline beff. Weill determine beff in terms of D a posteriori.The logarithm of the spectra is taken in the cost func-

ion to avoid underevaluation for the high-frequency por-

tfoofTmifqsartwSq

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5A(ssvm

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ion of the spectra. The noise, Snoise was added to the Clif-ord model to take into account the noise floor of thebserved frequency spectra. Thus, the variables for theptimization are Cn

2, vb, and Snoise. A MATLAB function,minsearch, was used for the least-squares optimization.o define the frequency range for the optimization, weultiplied f+5/3 times the spectrum so that the spectrum

s proportional to f+1 and f−1 for the low- and high-requency portions, respectively, and obtained fm, the fre-uency for the maximum of the f+1/ f−1 representation. Wepecified the beginning frequency, f1 for the optimizations fm /M, and M=9 was chosen to obtain the minimumms difference between the transverse wind speed re-rieved from the frequency spectra and the transverseind speed measured by the sonic (discussed further inection 5). The end frequency fN was specified as the Ny-uist frequency.

. Calibration of Transverse Wind Speedet us designate fk=0.39vb /b as the knee frequency re-rieved from the spectrum of AOA fluctuations and vb ashe time-averaged transverse wind speed measured byhe sonic. To calibrate the transverse wind speed, we com-ared fk with vb by plotting a scatter plot and obtained aalibration line, fk=p1vb+p0, by interpolating the dataet. The y intercept, p0 (zero-mean wind offset, ideally0=0) was added to account for the fluctuations of theransverse mean wind speed. To eliminate the effect ofutliers on the calibration, we calculated the standard de-iation, �d of the distance between each data point andhe calibration line and removed the data outside theound of ±2�d. Then, we calculated a new calibration line,nd the transverse wind speed can be calibrated as

vaoac =

bfk/0.39 − p0

p1. �31�

Fig. 3. Image of four lights measu

. RESULTS AND DISCUSSIONs mentioned in Section 3, four krypton flashlight bulbs

two bottom lights and two top lights) were used asources. In this paper, only the AOAs and the frequencypectra of AOA fluctuations from the bottom lights (asiewed from the telescope) were analyzed. Analysis forultiple lights will be the subject of future work.

. Angle-of-Arrivaligure 3 shows an image of the four lights measured at1:00:10 LT (local time), September 27, 2006. The inten-ity of light in the image is linear in scale, and the maxi-um value of the intensity is 1529 ADU (analog-to-digitalnit) at a pixel location (horizontal, vertical) of (177, 374),hich corresponds to the bottom left-hand light. Theean of the intensity over the entire image is 3.3 ADU,

nd the intensity of the background is in the range of 0 to5 ADU. The ratio of the maximum intensity of the lightith the maximum intensity of the background is 46 dB.igure 4 shows 20 images of the bottom left-hand light,ach with a size of 51�51 in pixel number. The intensityuctuations and blurring of the light images are evident.The centroids for the horizontal and vertical directions

in pixel number) were obtained, and the horizontal andertical AOAs, �y and �z, of the bottom left-hand lightere calculated as discussed previously. Figure 5 shows

he AOAs for (a) the horizontal direction and (b) the ver-ical direction for the bottom left-hand light measured oneptember 27, 2006. The means are subtracted for bothirections. The standard deviations for �y and �z are 5.44nd 27.49 rad, respectively.At night, it can be assumed that the atmosphere is

tratified parallel to the ground and that the vertical tem-erature gradient is dominant compared with the hori-ontal. Furthermore, the temperature near the groundecreases more rapidly than that at a certain height, and

21:00:10 LT, September 27, 2006.
red at

Fb

FT


ig. 5. AOA time series for (a) the horizontal direction and (b) the vertical direction of the bottom left-hand light measured on Septem-er 27, 2006.

ig. 4. Sequence of subimages (51�51 pixels) of the bottom left-hand light, measured at 21:00:10.9–21:00:11.6 LT, September 27, 2006.he first row, first column is the first image, the first row, fourth column is the fourth image.

trwpshtbplmgwcdbf

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CTbflts

qma2ow

of�ottqsof

tssoqsllttAtthtrpt

Fm


he mean vertical temperature gradient is positive. Thus,ays of the optical wave in the current setup bend down-ard at the aperture of the telescope, and the apparentosition of the light is therefore higher than the real po-ition of the light. The higher temperature gradient weave, the higher �z we have. Figure 6 shows �z of the bot-om left-hand light and the vertical temperature gradientetween 1 and 2 m (recall that the optical propagationath is 1.5 m AGL) measured by the middle array of self-ogging thermometers (56 m from the telescope). The

ean and standard deviation of the vertical temperatureradient were 0.89 K m−1 and 0.27 K m−1, respectively. Itas observed that if the vertical temperature gradient in-

reases, �z increases, and if the vertical temperature gra-ient decreases, �z decreases. The correlation coefficientetween �z and the vertical temperature gradient wasound to be 0.8.

. Wind Speede defined vx as the wind speed along the optical propa-

ation path and vy and vz as the wind speeds perpendicu-ar to the propagation path. Wind blowing from the lightso the telescope is denoted +vx, wind blowing from right toeft viewed from the light (as shown in Fig. 1) is denotedvy, and wind blowing upward from ground is denotedvz. The transverse horizontal wind speed, vb= �vy�, with0 s averaging time, is shown in Fig. 10 below (solid curven top panel), and the mean and standard deviations of vb,ere 0.37 and 0.18 m s−1, respectively.

. Frequency Spectrum of AOA Fluctuationshe frequency spectra of AOA fluctuations were obtainedy taking the Fourier transform of the time series of AOAuctuations. Figure 7 shows the averaged frequency spec-ra of 12 frequency spectra of AOA fluctuations (eachpectrum was obtained for 10 s of time duration) and fre-

ig. 6. Vertical AOA, �z (thin curve) of the bottom left-hand lighteasured by the self-logging thermometers at 56 m from the tele

uency spectra averaged over intervals of equal logarith-ic width for 2 min of time duration for the horizontal

nd vertical directions. The observation time was1:20:10 to 21:22:10 LT, September 27, 2006. During thebservation time, the mean of the transverse horizontalind speed, vb, was 0.74 m s−1.From Fig. 7, there appears to be three distinct regimes

f the spectra for the horizontal AOA fluctuations: a low-requency band, f�0.3 Hz; a mid-range band, 0.3� f

2 Hz; and a high-frequency band, f�2 Hz. The previ-usly defined knee frequency is the intersection betweenhe mid-range and high-frequency bands. For the horizon-al direction, the knee frequency is �2 Hz, and the fre-uency spectrum is proportional to f−2/3 for frequenciesmaller than the knee frequency, but is above the low cut-ff at 0.3 Hz and f−8/3 for frequencies larger than the kneerequency.

Below 0.3 Hz, the frequency spectrum for the horizon-al direction is divergent from the −2/3 power law andmaller than that for the vertical direction, which corre-ponds to the outer scale of turbulence that is of the orderf meters. Lüdi and Magun [7] observed that the fre-uency spectrum of the horizontal AOA fluctuations ismaller than that of the vertical AOA fluctuations in theow-frequency band with a larger outer scale of turbu-ence in the horizontal direction than in the vertical direc-ion for a stable and stratified atmosphere; this is similaro our result. This result is understandable given that theOAs are the aperture-averaged phase differences over

he aperture and that inhomogeneities cross the beam ofhe wave. When the outer scale of the turbulence in theorizontal direction is larger than in the vertical direc-ion, the spatial correlation length for the horizontal di-ection is larger than that for the vertical direction. Thehase difference in the horizontal direction is smallerhan that in the vertical direction.

e vertical temperature gradient (thick curve) between 1 and 2 mwith 5 s averaging time measured on September 27, 2006.

and thscope

rctFm

btqpar

Fbs

Ftar


For f�5 Hz, the frequency spectrum for the vertical di-ection deviated from the −8/3 power law because of noiseontamination, which may be due to mechanical vibra-ions caused by wind passing over the telescope housing.igure 2 shows the telescope mounting, which is clearlyore susceptible to vibration in the vertical (due to the

ig. 7. Averaged frequency spectra of AOA fluctuations for the hhe bottom left-hand light measured on September 27, 2006. Eacre the frequency spectra averaged over intervals of equal logariation, respectively. The observation time was 21:20:10–21:22:10

earing on which it pivots) than in the horizontal direc-ion. The telescope vibrations can cause a spike in the fre-uency spectrum [20], which is not distinctive in our ex-eriment. The standard deviations of the noise were �0.7nd �2 rad for the horizontal and vertical directions,espectively.

al direction (solid line) and the vertical direction (dashed line) ofrum is for 10 s of time duration. The black dots and open circleswidth for horizontal and vertical directions of 2 min of time du-

ig. 8. Averaged over logarithmically equidistant steps, observed and fitted frequency spectra of horizontal AOA fluctuations of theottom left-hand light measured on September 27, 2006. The observation time was 21:06:10–21:06:40 LT. The dots are for the observedpectrum, and the solid curve is for the fitted spectrum.

orizonth spectthmicLT.

Ftdc

FTchrl


ig. 9. Scatter plot of the knee frequency, fk, from the frequency spectra of horizontal AOA fluctuations (bottom left-hand light) versushe time-averaged transverse horizontal wind speed measured by the ultrasonic anemometer, vb, measured on September 27, 2006. Theots are the data inside the bound �±2�d�, the open circles are the data out of the bound, and the solid line is the calibration line. The

−1
alibration line is fk=2.0 m �vb+0.19 Hz. The averaging time is 30 s.
ig. 10. Comparison of vaoac with vb (top panel) and rms difference between vaoa

c and vb (bottom panel) measured on September 27, 2006.he solid curve is for the 30 s averaged transverse horizontal wind speed measured by the ultrasonic anemometer. The filled and openircles (top panel) are for the calibrated path-averaged transverse horizontal wind speeds retrieved from the frequency spectra of theorizontal AOAs fluctuations for 30 s of time duration for the bottom left-hand and bottom right-hand lights as viewed from the telescope,espectively. The filled and open circles (bottom panel) show the rms difference between vaoa

c and vb for the bottom left- and right-handights for 5 min of averaging time, respectively.

DOtbtpautt2thslsw

fwtbdattrtpct0t6iptt

hmcsfl(wrtsliwbvoamr(9sv

swttd

EOrmnwiwtmmkcrpf

1O21tmATafstb

2AbhtcitHp

3Fmitsbfgttc


. Comparison of fk with vbnce the frequency spectrum of the horizontal AOA fluc-

uations was obtained, the frequency spectrum was fittedy using the Clifford model by least-squares optimizationo obtain the refractive-index structure constant, Cn

2, theath-averaged transverse horizontal wind speed, vaoa,nd the level of noise floor, Snoise with 30 s intervals. Fig-re 8 shows the observed (bottom left-hand light) and fit-ed frequency spectra of AOA fluctuations for the horizon-al direction for the observation time of 21:06:10 to1:06:40 LT, September 27, 2006. During the observationime, the mean and standard deviation of the transverseorizontal wind speed, vs, were 0.51 and 0.06 m s−1, re-pectively. The fitted frequency spectrum obtained byeast-squares optimization follows the observed frequencypectrum very well, and the knee frequency fk=0.91 Hzas obtained.A calibration line was obtained from the comparison of

k with vb. For 30 s averaging time, the calibration lineas fk=1.99 m−1�vb+0.25 Hz. Then, the standard devia-

ion of the distance between each data point and the cali-ration line, �d, was found to be 0.32 Hz. As previouslyiscussed, the data out of the bound ±2�d were removed,nd the calibration line was recalculated. Figure 9 showshe scatter plot of fk (bottom left-hand light) versus vb af-er the out-of-bound data (open circle in the figure) wereemoved for 30 s averaging time. The calibration line washen fk=2.0 m−1�vb+0.19 Hz. For comparison, the samerocess was performed for 60 s averaging time, and thealibration line was fk=2.0 m−1�vb+0.07 Hz. The correla-ion coefficients between fk and the calibration lines were.78 and 0.85 for 30 and 60 s averaging times, respec-ively. We observe in the calibration lines, for both 30 and0 s of averaging times, two deviations from ideal behav-or; a slope �1 and an intercept �0. Both indicate theresence of features in the physical measurement of AOAhat are not accounted for in the Clifford model. Some ofhese are discussed in the following section.

Figure 10 (top) shows the time-averaged transverseorizontal wind speed vb, with 30 s of averaging timeeasured by the sonic at 64 m from the telescope, and the

alibrated path-averaged transverse horizontal windpeed vaoa

c , from the frequency spectra of horizontal AOAuctuations from the bottom left- and right-hand lightsas viewed from the telescope) using Eq. (31); vaoa

c agreesith vb reasonably well. Figure 10 (bottom) shows the

ms difference between vaoac and vb for 5 min of averaging

ime. The rms differences for both lights are increasedignificantly after 21:50 LT, which might be caused by aow signal-to-noise ratio. The ratio between the maximumntensity of light and the maximum background intensityas �2/3 of that at the beginning of experiment. For theottom left-hand light, the mean rms differences between

aoac and vb are 11 cm s−1 for 30 s averaging, which is 29%f the mean transverse horizontal wind speed �37 cm s−1�,nd 8.8 cm s−1 for 60 s averaging, which is 23% of theean transverse horizontal wind speed. For the bottom

ight-hand light, the mean rms differences are 11 cm s−1

29% of the mean transverse horizontal wind speed) and.6 cm s−1 (26% of the mean transverse horizontal windpeed) for 30 and 60 s averaging times, respectively. Sincec is the path-averaged transverse horizontal wind
aoa
peed and vb is the time-averaged transverse horizontalind speed measured at one position, vaoa

c fluctuates morehan vs; the standard deviations of vaoa

c are 0.22 m s−1 forhe bottom left- and right-hand lights, and the standardeviation of vs is 0.17 m s−1.

. Limitations of Applicability of Theoretical Modelsn the one hand, we have shown that there is a high cor-

elation between the retrievals of fk and the independenteasurements of vb. On the other hand, there are quite a

umber of reasons, some of which we list in the following,hy it would be unrealistic to expect our data to follow an

dealized theoretical model precisely, regardless ofhether we would have chosen the (perhaps more realis-

ic) Hogge and Butts model [10] instead of Clifford’sodel [5], or some other model. The point that we want toake is that our method relies on the detectability of the

nee, rather than on a precise fitting of our data to a spe-ific model. Theoretical models can be refined in severalespects, as indicated in the following. Such model im-rovements, however, may turn out to be of limited valueor the improvement of wind retrieval techniques.

. Extended Sourceur light source was a bulb with a diameter of aboutmm, corresponding to an angular diameter of about0 rad at a range of L=180 m. Therefore, the unper-urbed wave was not a spherical one, contrary to ourodel assumption. Kallistratova and Kon [21] studiedOA fluctuations of light waves from an extended source.hey assumed that each point of the extended source actss an independent source and then averaged over the dif-erent contributions. They showed that if the size of theource is increased by a factor of 30, the knee frequency ofhe frequency spectrum of AOA fluctuations is decreasedy a factor of 10.

. Anisotropy of Turbulencell the standard models [3,5,10,11] assume that the tur-ulence is statistically isotropic. Atmospheric turbulence,owever, is rarely isotropic (p. 96 in [22]), particularly inhe atmospheric surface layer under clear skies, as in ourase. Lee and Harp [23] suggested an ad hoc model for an-sotropic turbulence, which has been used by various au-hors mostly because of its mathematical convenience.owever, the Lee and Harp anisotropy model is not sup-orted by turbulence theory.

. Outer-Scale and Inner-Scale Effectsor the sake of mathematical convenience, all standardodels assume that the inertial-range turbulence model

s valid for all wave numbers over which the K-space in-egration has to be performed. For the technique de-cribed in this paper, it is important that the effectiveaseline is well within the inertial subrange. This wasulfilled in our measurements, since the height aboveround level (which in the atmospheric surface layer setshe outer scale of turbulence) was significantly largerhan the effective baseline of 20 cm, and the inner scalelose to the ground is typically smaller than 1 cm.

4WI(sptirtdiqS

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Aftlfumtuoclav

tsrf

ht

0wa

twcm(

hficta1pbctbmoeericbycf

ff(swwsa

sdttstwtofsa

AWs


. Inhomogeneity and Nonstationarity of the Transverseind

n all standard models, including the Clifford model, Eq.14), the transverse horizontal mean wind speed, vb, is as-umed to be homogeneous along the optical propagationath, and fluctuations of the mean wind speed are notaken into account. This assumption is often unrealistic,n particular in complex terrain and in situations whereadiative forcing is important, such as in daytime convec-ion and in the nocturnal boundary layer. The probabilityensity function of the cross-wind speed could be takennto account for a more sophisticated model of the fre-uency spectrum of AOA fluctuations, as mentioned inubsection 2.A (see also p. 266 in [24]).

. SUMMARY AND CONCLUSIONSe have presented measurements of optical angle-of-

rrival (AOA) fluctuations along a 180 m long, horizontalropagation path at the Boulder Atmospheric Observa-ory (BAO) site near Erie, Colorado. The propagation pathas 1.5 m above ground level (AGL), and the measure-ents were taken during a 1 h long period after sunset onclear, calm, late-September night with strong radiative

ooling near the ground. Bright, small krypton flashlightamps served as light sources. Time series of vertical andorizontal AOAs were retrieved from centroids calculated

rom digital images (30 images/s) taken with a CCD cam-ra connected to a commercial Schmidt–Cassegrain tele-cope with an aperture diameter of 36 cm. During the 1 hong observational period, 1 min averages of the cross-ath wind (as measured with the sonic) varied betweeness than 10 cm s−1 and �1 m s−1.

We analyzed frequency spectra S��f� of the horizontalOA fluctuations, with the main goal of exploring the use-

ulness of cross-path wind-speed retrievals from observa-ions of the knee frequency fk [the frequency at which theow-frequency f−2/3 asymptote of S��f� intersects the high-requency f−8/3 asymptote]. High time resolutionltrasonic-anemometer–thermometer (“sonic”) measure-ents of the wind vector were made at a location along

he propagation path at the same height (1.5 m AGL). Wesed Clifford’s [5] theoretical model for the AOA spectrumbserved with a two-point interferometer (two point re-eivers separated by a baseline of length b) to derive a re-ationship between the interferometer baseline length bnd the baseline wind (cross-path wind speed in our case)b. From Clifford’s theory, we find fk=0.39vb /b.

Our main empirical results are the following:

1. Frequency spectra S��f� of horizontal AOA fluctua-ions estimated from periods ranging from a few tens ofeconds to a few minutes show fairly consistently a f−2/3

egime at lower frequencies and a f−8/3 regime at higherrequencies, with fk of the order of 1 Hz.

2. 30 s estimates of fk and 30 s averages of vb areighly correlated, and a linear regression gives the rela-ionship

f = 2.0 m−1 � v + 0.19 Hz. �32�
k b
3. Interpreting the slope of the regression line as.39/beff gives an effective baseline length of beff=20 cm,hich is about half the physical diameter of the telescopeperture �D=36 cm�.4. Interpreting the zero-wind offset fk0=0.19 Hz as due

o an rms value �v of a variable cross wind along the path,e obtain �v= fk0 / �2.0 m−1�=9 cm s−1, an amount that is

onsistent with the rms difference between the in situeasured and optically retrieved cross-path wind speeds

see Fig. 10).

On the basis of data collected in a field experiment, weave demonstrated that the knee frequency obtainedrom horizontal AOA frequency spectra estimated frommages taken with a CCD camera mounted on a commer-ial, large-aperture telescope are useful for the quantita-ive retrieval of cross-path wind speeds. In our case, theccuracy of the retrieved wind speeds was about0 cm s−1. This technique relies on the detectability of theredicted rapid dropoff of S��f� at fk=0.39vb /beff, whereeff is the effective baseline of the optical sensor, in ourase a certain fraction of the aperture diameter. In ordero retrieve wind speeds quantitatively, beff has to be cali-rated in an experiment where the actual wind speed iseasured independently, as described in this paper. For

ur Schmidt–Cassegrain telescope (36 cm aperture diam-ter, with a circular, central obstruction of 13 cm diam-ter), we found empirically that beff=20 cm. Although theatio beff /D=0.55 is in the range of what one would expectntuitively, it would be interesting to compare this empiri-al result against a first-principle theory prediction ofeff /D. To the best of our knowledge, however, the two as-mptotes of S��f� for spherical waves observed with a cir-ular aperture have not yet been worked out in explicitorm.

We used Clifford’s model, Eq. (14), which predicts an−8/3 asymptote for f� fk, and we expect that a high-requency asymptote with a dropoff steeper than f−8/3

such as f−11/3, as predicted by Hogge and Butts [10] for apherical wave observed with a circular, filled aperture)ould provide only slightly different results. In otherords, we expect the method to be quite robust with re-

pect to changes in the actual high-frequency power law,s long as S��f� drops steeply at high frequencies.Unfortunately, the described retrieval technique is in-

ensitive to the sign of the cross-path wind. That is, it isifficult to separate a zero-mean, transverse wind speedhat fluctuates during the observation period and/or alonghe propagation path with an rms value of �v from a con-tant transverse wind speed vb comparable with �v. Fu-ure work could take advantage of spatial filtering [25,26]ith the goal of manipulating the path-weighting func-

ion in order to obtain range-resolved wind retrievals, aspposed to path averages. This could include analyzingrequency cross spectra of AOAs from pairs of pointources observed simultaneously, either with the sameperture or with spaced apertures.

CKNOWLEDGMENTSe thank Dan Wolfe and Chris Fairall for helpful discus-

ions and providing access to the BAO site. Thanks are

aLmAEULmAOF

R

1

1

1

1

1

1

1

1

1

1

2

2

2

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2

2

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lso owed to Steve Clifford, Rod Frehlich, Reg Hill, Richataitis, and Doug Looze for their interest and encourage-ent, and to two reviewers for their valuable comments.. Muschinski is the first holder of the Jerome M. Parosndowed Professorship in Measurement Sciences at theniversity of Massachusetts at Amherst, and he thanksinda and Jerome Paros for their generous support. Thisaterial is based on work supported in part by the U.S.rmy Research Laboratory and the U.S. Army Researchffice under grant 49393-EV and by the National Scienceoundation under grant ATM-0444688.

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