Measurements of AMK-03 meteorological complex [11] are series of instantaneous values of meteorological parameters (vertical (w), southern (Vs), and eastern (Ve) wind speed components; air temperature (T), pressure, and relative humidity) grouped into files according to observation time intervals of a preset duration. Work [10] developed methods for post-detector (computer) processing of data from acoustic weather stations, which enabled calculating different statistical parameters of wind and turbulence based on time series of fluctuations in meteorological parameters. Energy spectra and structure functions are calculated in a specified averaging range; then, they are used for direct estimation of the parameters of a generalized power-law model (exponent, structural characteristic, and external scale) from samples from the time series of measured instantaneous values of meteorological parameters by means of power approximation.

A feature of the spectral processing algorithm is certain flexibility in setting key parameters, i.e., possibilities of varying the averaging interval and setting the boundaries of the spatial interval when assessing the power dependence of structure functions and the interval of spatial frequencies when assessing the slopes of energy spectra. The procedure for calculating numerical parameters of atmospheric turbulence based on the second moments of air temperature fluctuations and three wind speed components is also implemented for arbitrarily specified averaging interval. In particular, single-point moments for the three wind speed components and temperature are calculated, as well as the vertical turbulent momentum flux \(\tau = - \rho \left\langle ' w '} \right\rangle \) and heat flux \(H = _}\rho \left\langle ' w '} \right\rangle \), the speed scale (friction speed) \(u ^* = \sqrt \left\langle ' w '} \right\rangle } ,\) the temperature scale \(T ^* = \left\langle ' w '} \right\rangle } \mathord \left\langle ' w '} \right\rangle } ^*,}}} \right. \kern-0em} ^*,}}\) and the Monin–Obukhov scale \(L ^* = - \left\langle T \right\rangle ^*)}}^}} \mathord ^*)}}^}} ^*)}}} \right. \kern-0em} ^*)}}\) (u′, w′, and \(T '\) are the pulsations in the longitudinal and vertical air speed components and temperature; \(\left\langle T \right\rangle \) is the average absolute temperature; cp is the specific heat capacity of air at constant pressure; ρ is the air density; g is the acceleration of gravity).

In practice, the calculation of the friction velocity \(u ^* = \sqrt } \right. \kern-0em} \rho }} \) by the series of measured pulsations u′ and w′ in the atmosphere is complicated by the fact that τ becomes negative in some cases, where the similarity theory is violated [1], which implies impossibility of assessments of such characteristics as T* and L* within the classical definition. In order to exclude these situations, when assessing the speed scale with the use of, for example, software of AMK-03 meteorological complex, the module of \(\left\langle ' w '} \right\rangle \) is taken, i.e., its sign is ignored:

$$u_^ = \sqrt ' w '} \right\rangle } \right|} .$$

(1)

Then,

$$T_^ = \left\langle ' w '} \right\rangle } \mathord \left\langle ' w '} \right\rangle } ^}}} \right. \kern-0em} ^}}.$$

(2)

The occurrence of these situations apparently indicates the violation of horizontal isotropy conditions, when it is necessary to take into account other components of the turbulent momentum flux [12]. In [5, 13] a combination of two mixed moments is used to estimate the friction velocity \(u ^* = ' w '} \right\rangle }}^} + ' '} \right\rangle }}^}} \right)}^} \right. \kern-0em} 4}}}},\) where \( '\) is transverse speed pulsations. This excludes problems with the sign under the root. By analogy, we can introduce the total friction velocity

$$u_^ = ' w '} \right\rangle }}^} + ' '} \right\rangle }}^} + ' w '} \right\rangle }}^}} \right)}^} \right. \kern-0em} 4}}}}$$

(3)

and the total temperature scale

$$T_^ = \frac^}}\sqrt ' w '} \right\rangle }}^} + ' '} \right\rangle }}^} + ' u '} \right\rangle }}^}} ,$$

(4)

where all possible directions of turbulent transfer are taken into account. In the future, in practical assessments, we will use two calculation options: the classical MO scale \(L_^\) based on definitions (1) and (2), which describes the vertical stratification of SAL, and its analogue \(L_^\) based on Eqs. (3) and (4) taking into account all components of turbulent heat and momentum fluxes:

$$L_ 2}}^ = \frac 2}}^)}}^}}} 2}}^}}.$$

(5)

Note that the scale \(L_^\) does not characterize the vertical stratification of SAL, and the sign of temperature scale (4) does not change even in a horizontally isotropic medium, i.e., stable and unstable stratifications are not distinguished by this parameter.

Let us describe the temperature spectra using isotropic power-law model [14, 15], which, as shown in [10], well corresponds to measurement results. For the structure function in the power range of spatial separations ρ, we have

$$_}(\rho ) \sim \rho }^},$$

(6)

and in the corresponding range of spatial frequencies, the spectrum of temperature fluctuations also has a power-law form:

$$_}(\kappa ) \sim ^}}.$$

(7)

In the range 1 < ν < 3, the exponents in Eqs. (6) and (7) are related as μ = ν − 1. However, situations where ν < 1 can occur in the atmosphere [10]. In this case, the linear relationship between the exponents violates, and the exponent of the structure function weakly changes in the range 0 < μ < 0.2.

To estimate the exponents μ and ν from the temperature series and wind speed components we use the algorithm [10] based on the power approximation of the structure function calculated from experimental data as

$$_}(_}) = \frac = 0}^ - l - 1} ^}} - T_ + l}}^}})}}^}} }}},\,\,\,\,l = 0, \ldots ,N - 1,$$

(8)

and spectrum

$$\begin _}(_}) = \frac}\sum\limits_ = 0}^ - 1} = 0}^} \right. \kern-0em} s} - 1} + \fracN}}^}}}}^}}}}} } \right|}}^},} \\ m = 0, \ldots ,_} - 1. \\ \end $$

(9)

Here, f is the sampling rate of meteorological parameters; \(_} = \mathord }} \right. \kern-0em} }\) is the spatial separation; \(\left\langle V \right\rangle \) is average wind speed over the averaging interval Δt; N = fΔt is the number of samples in the averaging interval; s is the number of subintervals of the length Ns = N/s in the averaging interval; \(_} = \frac}_}\left\langle V \right\rangle }}\) is the spatial frequency.