# Outer Scale of Turbulence in the Anisotropic Boundary Layer

As is well known, the outer scale of turbulence *L** _{0}* can be determined in different ways. For example, Tatarskii [4] determines the vertical outer scale L from the condition of equality of the average squared difference of random temperature values at two points z

_{1}and z

_{2}to its systematic difference (scale of turbulent mixing). This condition gives

where, as before, *а* = Pr^{-1} и 1.17; *C _{e}* is the Obukhov constant. We can determine the outer scale

*L*from deviation of the structure function of temperature fluctuations from the 2/3 law. In the space of Fourier transforms, this scale corresponds to the scale L0 determined from deviations of the 1D spatial or temporal frequency spectra from the 5/3 law (see also information on turbulence spectral dependence in Chapter 1). There are also scales, which are parameters in different theoretical models of the energy range of the 3D spectrum of fluctuations (e.g., the von Karman outer scale

*L*). For practical needs, it is interesting to find the relations between these scales, to obtain the theoretical ideas about their applicability in the anisotropic boundary layer, and to compare the theoretical and experimental results in the unified manner.

For the von Karman model of the 3D spectrum of turbulence, the structure function D(r) and 1D spectral density *V(k)* are described by the equations [4]:

where r_{0} is some spatial scale (correlation radius); *aV„* is variance; and *K _{v}* is the MacDonald function. Considering, for example, temperature fluctuations, we should take

*v =*1/3 and account for r

_{0}

^{-1}=

*k*, where

_{0}= 2n/L*L*is the von Karman outer scale.

Expanding D_{v}(r) and *V _{v}(k)* at

*v*= 1/3 in power series of r/r

_{0}and k

_{0}/k, respectively, and at

*v*= 4/3 (assuming r

_{0}= r

_{1}, k

_{0}= k

_{1}) in power series of r/r

_{1}and k

_{1}/k, we finally get

Here, a_{0}, *a _{1}, *

*/*

*3*and Д are the positive constants dependent on йщ and r

_{0},_{0}; a

_{2 }and

*в*depend on aQ

_{2}_{/3}and r

_{1}. The parameters a

_{1}

^{2}/3, r

_{0}, and aQ

_{/3}, r

_{1}can be related to each other, if we impose the conditions a

_{0}= Cf

^{2}, a

_{2}= (dTdz)

^{2}on a

_{0}and a

_{2}. These conditions follow from definition (3.19). They allow us to find the relation between the Tatarskii scale LQ and other scales.

Determine the outer scale L, ( = 2n/L, *f* from the condition of intersection of V_{1/3}(k) and V4_{/3}(k) at the point k*, at which the relative deviation of V_{1/3} (k) from the dependence *@ _{0}* k

^{-5/3}(corresponding to the inertial range) is equal to the preset value S

_{v}Analogously, we can determine the outer scale

*L*|L

*= r*

**/(*

*aC*

*g*)

^{3/4}] from the condition of intersection of D

_{1/3}(r) and D

_{4/3}(r) at the point r*, at which the relative deviation of D

_{1/3}(r) from the inertial interval (dependence a

_{0}r

^{2/3}) is equal to S

_{d}. The deviations S

_{v}and S

_{d}turn out to be related. Thus, at |SV ^ 1 we have |S

_{d}| и 1.14 |4y|

^{3/4}.

Thus, we have four outer scales determined in different ways: L,, *L*, L0, and *L**)* . At small deviations *h** _{V}* and

*6*

*all these scales are related by linear dependences (with awkward expressions for the coefficients). For example, at*

_{D},*6*

*0.3 (Sd и 0.37), we obtain the following expressions of the scales through the Tatarskii scale:*

_{V}=

(or through the von Karman outer scale: LQ и 0.6Lf, и 0.06Lf, LQ и 0.08Lf).

With allowance for the known relation between the von Karman *L**,* and exponential outer scales *L _{0}* (usually, L = 0.54Lf or

*L*

*>*

*и*1.85L, [175—177]). Equations 3.21 establish the relations between five outer scales determined by five different methods: L

_{0}, L0,

*L*, LQ, and

*L*.

As follows from definitions (3.5), *Cf =* Qe^{-1/3} *N*. We can substitute Equations 3.8 and 3.10 for *e* and *N* in the anisotropic layer into this equation. The vertical derivative d77dz can be expressed from Equation 3.7, where *D*^{T} *=* —0.49 *dT/dx + dT/dz.* Upon the substitution of *Cf* and *dT7dz* into definition (3.19), we find the equation for the Tatarskii outer scale generalized for the case of an arbitrary anisotropic layer:

108 ? *Optical Waves and Laser Beams in the Irregular Atmosphere*

Assuming here *pj(Q* = 0, *pV(Q* = 0, *dT/dx =* 0, we obtain the well-known equation for the isotropic layer

In the isotropic layer, the simpler equation [4] *L = x ?* z/00) is also used. It differs insignificantly from Equation 3.23 in the limiting cases of very unstable and very stable stratification.

Compare the theory with the experiment. For this, we use different methods for obtaining of experimental values of the vertical outer scale.

One of such methods is the substitution of measured values of *CT* and *dT/dz *into definition (3.19) (conditionally, this method can be referred to as “by the Tatarskii definition”). As can be seen from Figure 3.10, the experimental values of *dT/dz* were determined from measurements in the lower 5-m layer (observation session 5, a total of six points for dT/dz), and they are relatively few. Therefore, for a more complete comparison, we present other independent methods allowing reconstruction of experimental values of the outer scale. These methods can be proposed from the results of measurement of temporal frequency spectra of temperature fluctuations.

Figure 3.16 shows sampled frequency spectra of temperature *W(f*) obtained in our measurements at different values of the Monin—Obukhov number Z As follows from Figure 3.16, all the spectra are characterized by the presence of the 5/3-inertial frequency range *f,* in which *W(f) ^* f^{-5/3}, and saturation in the range of low frequencies, which is well described by the von Karman model (see also Chapter 1).

Figure 3.16 Experimental nonnormalized spectra of temperature fluctuations. The upper curve in the low-frequency range corresponds to the very unstable stratification, while the lower curve corresponds to the stable stratification.

We apply the von Karman model of spectrum (3.20) for determination of the von Karman outer scale *L* from stable characteristics of the spectrum. These characteristics include the value of the spectrum at the lower boundary of the recorded frequency range (denote it as *W(0))* and the value of the coefficient w* of*f ^{-}*

*53*in the inertial range

*[W(f) = w**-f

^{53}]. We use the equation [4] V(k)

*= u*where

*u*is the absolute value of the average wind velocity vector (u = v). This equation relates the 1D spatial spectrum p

*V(k)*determined by Equation 3.20 with the temporal frequency spectrum Wt

_{p}(w) being the ordinary 1D Fourier transform of the correlation function (w

*= 2nf)*. Taking into account that

*W(f)*is the transformation by positive frequencies and

*W(f) = 4n - W*, we find two methods for determination of the von Karman scale

_{exp}(2nf)*L,*from characteristics of the spectra:

The second method for the frequencies of the inertial range simplifies and gives (2) *L и u -[W*(0)/w*]^{3/5}. We can conditionally refer to the first of these methods as “from spectra by saturation” and the second one as “from spectra by the 5/3 law.”

Figure 3.17 shows the results of comparison of experimental and theoretical results for the Tatarskii outer scale L in the mountain boundary layer. With the use of experimental values of the von Karman scale *L* obtained from spectra based

Figure 3.17 Comparison of experimental and theoretical results for the Tatarskii outer scale of turbulence *L _{0}* in the mountain anisotropic boundary layer: (1) experiment (from spectra by the 5/3 law); (2) experiment (from spectra by saturation); (3) experiment (by Tatarskii definition); (4) semiempirical theory for the anisotropic layer; (5) semiempirical theory for the isotropic layer.

on methods (3.24), the coefficient of the von Karman scale into the Tatarskii scale (3.21) was applied. This coefficient is applicable to any boundary layer.

The comparison of the scales Z_{0} measured by three methods (by Tatarskii definition, from spectra by saturation, and from spectra by the 5/3 law) shows that in the anisotropic boundary layer, the agreement between the experiment and the semiempirical theory is satisfactory (3.22).

For comparison, we used the data of all observation sessions (including session 5 of high-altitude observations). Therefore, because of the clear linear height dependence, the theoretical scales scale of both the isotropic [L = *xz/p(Z*)] and anisotropic layers (3.22) demonstrate jumps at some values of *Z* (where the height z differs from its permanent value of 2.7 m). As can be seen from Figure 3.17, for this *Z* the experimental data also show jumps. In addition, in the region with insignificant anisotropy (-0.1 > *Z* ^ -1), the theoretical values of the anisotropic and isotropic outer scales, as expected, turn out to be close (curves 4 and 5 have jumps at practically coinciding points Z).

In the region of very unstable local stratification, the anisotropic outer scale is smaller than the isotropic one. As follows from (3.22), this decrease is caused by the factor [^(Z) + ^y(Z) — Z]^{-1/4}, in which the values of *(pyiZ)* — Z) are large. The both scales (anisotropic and isotropic) decrease in the region of weakly stable stratification. The marked difference between them (anisotropic scale larger than isotropic one) is observed in the range of dynamic turbulence (neutral stratification). The increase of the anisotropic scale is connected with the growth of the function *PjiZ) *in this range. However, if we take into account the longitudinal derivative *dT/dx* in theoretical Equation 3.22, then the increase of the anisotropic outer scale becomes limited. As can be seen from Figure 3.17, this improves the agreement between the theory and the experiment.