Turbulent Characteristics in Coherent Turbulence

Since coherent turbulence observed in the atmosphere differs from the Kolmogorov one, it becomes necessary to refine the domain of applicability of the Kolmogorov— Obukhov law. In particular, it is needed to refine the values of the Kolmogorov C and Obukhov Ce constants.

According to the Kolmogorov—Obukhov law, the structure function of fluctuations of the longitudinal velocity Drr(r) in the inertial range of scales r can be expressed through the structure characteristic CV of fluctuations of the longitudinal velocity: Drr (r) = CV ? r2/3 (see also Chapter 1). The structure function of temperature fluctuations Dr(r) can be expressed through the structure characteristic Ct of temperature fluctuations: Dr (r) = Cr ? r2 3. The structure characteristics Cy, Cr determine the intensity of velocity and temperature fluctuations and are important parameters of the turbulent motion of the medium. In their turn, CV and Cr depend on the average dissipation rate of kinetic energy e, temperature dissipation N, and Kolmogorov C and Obukhov Ce constants: CV = C ? e213, CV = Cg ? e-1/3 ? N. It is seen from here that at known e and N, velocity and temperature fluctuations are determined by the constants C and Ce. The constants C and Cg were measured in different media by different methods. The values C = 1.9 and Cg = 3.0 are recommended as the most probable estimates of C and Cg in References 1, 2. These estimates are average over data of different authors. The deviation from the average is rather large [1,2], for example, the values of 0.9, 1.6, and 2.8 were observed for C; and for Cg the observed values were 1.1, 1.4, 2.5, 2.7, 3.3, 3.5, 5.6, 5.8, 6.5, and 9.0. As was shown in References 1, 2, 4, the Kolmogorov constant C is connected with the asymmetry S of probability distribution of the longitudinal velocity difference as

The third moment of the longitudinal difference of wind velocities can be found from the temporal moment

and the condition of frozen turbulence [1,2,4]

Here, u'(r,t) = u(r,t) — (u(r,t)), where u(r,t) is the random value of the longitudinal velocity (usually, the projection of the random velocity vector onto the direction of the average velocity vector) at the point r at time t. The constant Cg is connected with the asymmetry S' of the probability distribution of temperature difference as [1,2,4]

The third spatial moment of the difference can also be found from the temporal moment and the condition of frozen turbulence

Here, T(r,t) = T(r,t) — (T(r,t)), u'(r,t) = u(r,t) — (u(r,t)) are random centered temperature and longitudinal velocity.

Figures 3.32 and 3.33 illustrate the process of determination of the Kolmogorov C and Obukhov Ce constants for the case of non-Kolmogorov turbulence in the atmosphere. The frequency spectrum depicted in Figure 3.32 has the long inertial range, in which WT ^ f—8/3.

The analysis of the data of more than 30 measurement points as revealed the following. If turbulence is close to Kolmogorov (smooth underlying surface, temporal spectra in the inertial range Wuf) ^ f—5/3, and others), then the values of the Kolmogorov and Obukhov constants can be taken C = 1.9 with an error of 1%—12% and Cg = 3.0 with an error not exceeding 30%. If turbulence deviates from the Kolmogorov one and is close to coherent (Wu(f) ^f—8/3 and others), then the Kolmogorov constant C falls within the range 0.9—3.7, while the Obukhov constant Cg is in the range 1.3—5.1. In this case, the error of C = 1.9 is 19%—93%, and the error of Cg = 3.° can achieve 70%.

As the experience shows, the measurements of turbulent characteristics of the atmosphere (usually conducted with the use of the Kolmogorov—Obukhov

148 ? Optical Waves and Laser Beams in the Irregular Atmosphere

Third moments of the longitudinal difference of velocity and temperatur

Figure 3.32 Third moments of the longitudinal difference of velocity and temperature Duuu(m3/s3), DuJT(deg2m/s). Smoothed spectrum (bottom left inset). Coherent turbulence, summer daytime measurements in mountains at a height of 680 m, July 2, 2007.

Kolmogorov C and Obukhov C constants

Figure 3.33 Kolmogorov C and Obukhov Ce constants. Coherent turbulence measured in mountains at a height of 680 m, July 2 of 2007. Vertical lines show the boundaries of the inertial range, and dashed lines are the average values of the constants C and Q.

law) are accompanied by significant errors, as a rule. The data of our measurements indicate the main cause for appearance of these errors. Variations of the Kolmogorov and Obukhov constants within 100% (depending on the observation point) lead to practically the same errors in determination of the characteristics Cr, CV, and Cl.

Another source of errors is the structure function of temperature fluctuations Dj(r) itself, if we determine the value of CT using the Kolmogorov—Obukhov law DT(r) = Ct ? r2/3. As known [4], the function Dj(r) can be represented in the form of

After the substitution of von Karman spectrum (3.30) written in the exponential form,

Spectrum (3.32) was obtained by us in References 82, 128 with allowance for the approximate relation between the outer scales for v = 5/6 and 1/3 (according to Table 3.1, they differ, on average, by the coefficient 2.3) and the relation at v = 1/3 between the used von Karman outer scale Lf and exponential L0 (usually, L0 = 0.54 L0, see References 175—177). Exponential spectrum (3.32) deviates from von Karman spectrum (3.30) only in the energy range, where ж2/*2 ^ 1. However, at v = 1/3, it gives practically the same results as (3.30) [175]. At the same time, it significantly simplifies the calculations. After calculation of the corresponding ranges, we obtain the asymptotic representations for the structure function Dj(r) in coherent turbulence (v = 5/6):

As can be seen, the function Dj(r) deviates from the Kolmogorov—Obukhov 2/3 law. At small r (r ^ /0), DT(r) ^ r, as in the Kolmogorov turbulence, but the coefficient in the asymptotic becomes dependent on the outer scale L0. In the inertial range (/0 ^ r ^ ), there is an extended initial part, in which Dj{f) ^ r5/3.

The parts with ^r2 and ^r5/3 intersect at r и 0.8 /0 (remind that the used inner scale /0 of coherent turbulence is an order of magnitude larger than the Kolmogorov one, see Table 3.1). The function f(x) has a maximum. Therefore, in the inertial range of coherent turbulence 3.0/0 < r < 0.4Lf, where this function can be considered approximately as a constant [f(x) и fmax = 2.17], the approximately 2/3

Kolmogorov part is observed, in which Dr (r) и CT ? fmax ? r2/3. It follows from here that in the measurements of Cr from the 2/3-asymptotic of Dr(r) we can expect that the values of CF will be more than twice overestimated.

 
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