# Experimental Verification of Local Weak Anisotropy of the Mountain Boundary Layer via the Anisotropy Functions

According to the results of Section 3.2.2, for measurements in the mountain anisotropic boundary layer, the requirement of the stable regional meteorological situation is not principal. In this boundary layer, all turbulence characteristics become functions of the Monin—Obukhov number.

Figures 3.8 and 3.9 depict the experimental results for the scales of temperature T* and velocity *V** for all observation sessions as functions of the Monin— Obukhov number. In every session, the regional meteorological situation is described by its own set of the Monin—Obukhov numbers, varying in the entire range -581 < c < 0.3. As can be seen from Figures 3.8 and 3.9, the joining of all observation sessions in one does not leads to the significant spread of the data.

Figure 3.8 Turbulent scale of temperature field T* in the mountain boundary layer for all measurement sessions as a function of the Monin-Obukhov number *Z*

Figure 3.9 Turbulent scale of the velocity field V* in mountain boundary layer for all measurement sessions as a function of the Monin-Obukhov number Z.

All the data stably group around certain smoothed dependence curves shown in Figures 3.8 and 3.9. Some spread of the points observed in the zone of stable stratification and at (Z > +0.05) appears permanently in the most measurements of different turbulent characteristics in the atmosphere (taken from different authors). Thus, Monin and Yaglom [1,2] explained this fact by the intermittence of turbulence under stable conditions and, consequently, by the insufficient ordinary averaging time. As it follows from our measurements, in the mountain boundary layer, the experimental results, as functions of the universal parameter—the Monin— Obukhov number—can be joined regardless of the type of regional meteorological situation (at least, for meteorological situations observed during the period of measurements).

Figures 3.10 and 3.11 show the results of comparison of the semiempirical theory with the experiment for the functions

in the mountain boundary layer (see Equation 3.7, in which at the mean wind velocity along the axis *x _{1}* the horizontal transverse derivative

*dT/dx*can be neglected).

_{2}These figures also depict the experimental values of the derivatives *dT/dz* and *du/dz.* In the case of the isotropic boundary layer (when *Z* is fixed, there are no longitudinal derivatives and *D ^{T} = dT/dz, D^{V}* = du/dz), Equation 3.7 is fundamental in the semiempirical theory of turbulence (in the similarity theory) and are reliably confirmed by experiments [1,2,4].

In the anisotropic layer, all components of equalities (3.7) are functions of the number *Z* varying at an arbitrary shift of the observation point. As can be seen from

Figure 3.10 Comparison of experimental and theoretical values of the vertical derivative (with respect to z) of the mean temperature of air *dT/dz* in the mountain boundary layer. Open squares are for experimental values of *dT/dz* from measurements in the lower 5-m layer near LSVT (observation session 5). Open and closed circles are, respectively, for the semiempirical theory and experiment for the function *D ^{T}* in Equation 3.7. Straight lines are asymptotics of the theoretical function

*D*at |Z| —— 0 and

^{T}*Z*—— —oc.

Figure 3.11 Comparison of experimental and theoretical values of the vertical derivative (with respect to z) of the longitudinal component (u) of the mean wind velocity *du/dz* in the mountain boundary layer. Closed circles are experimental values of *du/dz* in all the measurement sessions. Open circles and open squares are, respectively, the semiempirical theory and experiment for the function *D ^{V }*[in Equation 3.7]. Straight lines are asymptotics of the theoretical function

*D*at

^{V}|Z| — 0 и z — —x.

*Atmospheric Turbulence in the Anisotropic Boundary Layer ?* 101

Figures 3.10 and 3.11, in the wide range of variation of the Monin—Obukhov number Z, the semiempirical theory is in satisfactory agreement with the experiment. The consideration of longitudinal derivatives improves this agreement.

It can be seen from Equations 3.7 and 3.3 that for the anisotropic layer the theoretical functions *D ^{T}* and

*D*can be represented in the form

^{V}

where the scales T*(Z) and V*(Z) are functions of Z (see Figures 3.8 and 3.9). The same representations are also valid for the isotropic layer, but the scales T* and V* in this case are constant.

As can be seen from the measurements (see Figures 3.10 and 3.11), inside the anisotropic layer there are local areas, in which we can neglect longitudinal derivatives in the functions *D*^{T}, *D ^{V}* in comparison with the vertical ones. This means that in these areas the conditions of the isotropic layer take place for the functions

*D*and

^{T}*D*The plane-parallel flow over an extended part of the surface can be considered as an extension of some small local area with the isotropic conditions. Consequently, in extended isotropic layers, the constant scales T* and V* are not arbitrary, but as can be seen from the theoretical reason for

^{V}.*D*and

^{T}*D*

^{V}, are determined by the particular value of

*Z*.

Figures 3.12 and 3.13 show the results of comparison of the semiempirical theory with the experiment for the energy and temperature anisotropy functions ^_{V}( Z) and <^t(Z). As can be seen from Figures 3.12 and 3.13, the theory is in the satisfactory agreement with the experiment. The theoretical values of these functions

Figure 3.12 Comparison of experimental and theoretical results for the temperature anisotropy function.

Figure 3.13 Comparison of experimental and theoretical results for the energy anisotropy function.

appear to be close to the experimental values in a wide range of variability of the Monin—Obukhov number (from stable to very unstable local temperature stratifications, —581 < *Z <* 0.3).

The anisotropy functions have maxima in different ranges of variability of the Monin—Obukhov number *(.* If *lj(Z)* is mostly concentrated in the range |Z| < 0.1, then *lv(Z)* concentrates in the range —1 > *Z* ^ -1000. Beyond these ranges, the both functions are close to zero, while at the maxima they achieve values close to 1000. Despite the low accuracy of measurement of the derivatives of hydrodynamic fields, the agreement is observed at the variation of the functions *lV(Z)* and *l(Z) *by more than three orders of magnitude. Therefore, it cannot be a consequence of experimental errors.

Since the functions *lV(Z)* and IjZZ) determine the dissipation rates e and N, it follows from Figures 3.12 and 3.13 that the anisotropic boundary layer influences the energy (e) and temperature (N) turbulence characteristic in the significantly nonsymmetrical way. The appearance of maxima of the anisotropy functions in different ranges of variation of the Monin—Obukhov number *Z* is associated with the corresponding behavior of the scales *T*(Z)* and V*(Z) in these areas. As can be seen from the smoothed empirical dependences for the scales *T*(Z)* and V*(Z) in Figures 3.8 and 3.9, at small values of |Z| (|Z| ^ 0) *T*(Z)* ^ 0, and at large negative values of *Z (Z ^* —ro) V*(Z) *^* 0. Therefore, if in these areas the derivatives *dT/dx** _{1}* (at |Z| ^ 0),

*dv*

_{1}*/dx*

*or*

_{1}*dv*

_{3}*/dx*

*(at*

_{3}*Z ^*—ro) are limited (and nonzero), then

*ljiZ) ^*ro at |Z| ^ 0 and

*lv(Z) ^*ro at

*Z ^*—ro (due to normalization to the scales

*T**and V*) according to Equations 3.16 and 3.12.

In the range of variation of the Monin—Obukhov number —0.1 > *Z* ^ — 1, in which the both anisotropy functions are simultaneously close to zero, Equations 3.8

and 3.10 for the anisotropic dissipation rates *e* and *N* coincide with the equations for the isotropic dissipation rates. Consequently, in this range of Z, the conditions of the isotopic layer take place in the anisotropic boundary layer.

Thus, it follows from the results of the measurements in the mountain boundary layer, the assumption on the local weak anisotropy of the arbitrary boundary layer is fulfilled with a good accuracy. Consequently, the arbitrary boundary layer can be considered as locally weakly anisotropic. This means that, by introducing the anisotropy functions *pO* and *p _{T}(Z),* we can extend the similarity theory for the isotropic dissipation rates to the arbitrary anisotropic boundary layer.