## Re Recrit

patch dissipated by viscosity

Turbulence inner scale ~ mm

Fig. 6. Graphical representation of the vortex cascades. The turbulent flow generates a vortex of linear dimension L (outer scale).The Reynolds number Re of this vortex is greater than the critical one (Re)crit. and the vortex is split into another vortex of a smaller linear dimension. If this vortex still has Re >> (Re)crit., the process is iterated, generating the vortex cascade. The cascade stops when the the vortex energy is dissipated by viscosity. The linear dimension of the smallest vortex is called inner scale

These vortexes will have some velocity fluctuation and will have a Reynolds number greater than the critical one. As a result of this condition, they vortexes will split in to other vortexes having a smaller scale and other velocity fluctuations. This process will stop when the small vortexes have a Reynolds number lower then the critical one. At this point, they will be dissipated by viscous friction in the fluid and will not split anymore. The spatial scale where this happens is called the turbulence inner scale and is of the order of a few mm. Using dimensional consideration and the above given description of the turbulence regime, Kolmogorov was able to express the so-called structure function of the velocity fluctuations that we report below where r is the distance between the two considered points, and C is a constant that accounts for the strength of the velocity fluctuations.3 Tatarsky applied this expression in the computation of the phase perturbation experienced by an electromagnetic wave propagating in an inhomogeneous medium (a medium where the refraction index is not constant in space), introducing the concept of a physical quantity that is a conserved passive additive [27]. The relevant example is the atmospheric temperature. This quantity does not change in any way the fluid dynamics, and its values are not modified by atmospheric turbulence. For example, a colored dye added to a turbulent flux would be a passive conserved additive and would be useful for tracing the velocity fluctuations. Tatarsky demonstrates that the fluctuation distributions of any such quantity in a turbulent flow are described by the same structure function of the velocity fluctuations. Using this result, he described the temperature fluctuation in the turbulent atmosphere by the following structure function where r is the distance of the two considered points, and C^ is a constant accounting for the strength of the temperature fluctuations. The next step in Tatarsky's work is to consider the expression of the atmospheric refraction index as a function of pressure P and temperature T. This common expression is reported below for easy reading.

From this equation, it is easy to find out the expression of refraction index fluctuation as a function of temperature. In the following, we will neglect the fluctuations due to pressure changes because pressure fluctuations are usually

3 We report here the mathematical definition of the structure function Dx (p) of a random function xDx(p) = {(x(r + p) — x(r))2). The quantities r and p can be one, two, or three dimensional quantities. The last case is the case of the velocity fluctuation structure function.

much less in percentage than the temperature fluctuations. With this in mind we find that

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