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Astron. Astrophys. 351, 347-358 (1999)

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Appendix A: derivation of the Eq. 38

The diffusion coefficient as functions of the variables in Fourier space is:

[EQUATION]

The Fourier transform is given by the following expression:

[EQUATION]

(with [FORMULA]).

The velocity can be expressed with the inverse Fourier transform:

[EQUATION]

as well as the partial derivative of the velocity:

[EQUATION]

where the monochromatic component [FORMULA] (using Eqs. 21 and 30) is written as:

[EQUATION]

[FORMULA] is the vertical amplitude at the boundary. [FORMULA], the horizontal amplitude, is taken equal to the turbulent mean square velocity in the convective zone generating the waves.

[EQUATION]

We therefore have the expression of the vertical diffusion coefficient:

[EQUATION]

where

[EQUATION]

and

[EQUATION]

From now on [FORMULA], the turbulent motion at the boundary. In order to calculate the ensemble average, we follow the rule given by Knobloch (1977), which consists in replacing the ensemble average of four terms by the products of two ensemble averages of two terms.

This implies the presence of three products, but the integral taken over two of them does vanish. Therefore, the ensemble average of the four terms in Eq. (37) can be obtained as the product of two ensemble averages:

[EQUATION]

The first ensamble average is given by the following equation:

[EQUATION]

with

[EQUATION]

the term between brackets tends to [FORMULA] when [FORMULA] is large and when [FORMULA]:

[EQUATION]

So,

[EQUATION]

and

[EQUATION]

where [FORMULA] is the spectrum of turbulence. If we can factorize the auto-correlation function as:

[EQUATION]

and

[EQUATION]

[EQUATION]

from Lesieur (V-5-4):

[EQUATION]

and the mean square velocity for 3D and 2D turbulence is given by:

[EQUATION]

Furthermore, for the dependence on frequency of the turbulence spectrum, we chose an exponentiel one:

[EQUATION]

and,

[EQUATION]

So, the expression [FORMULA] takes the following aspect:

[EQUATION]

from Lessieur (p.111), for 3D case, [FORMULA], with [FORMULA] given by a Kolmogorov law [FORMULA]. Then the expression for [FORMULA] is:

[EQUATION]

which leads to a new expression of [FORMULA]:

[EQUATION]

From Eq. (A8) and Eq. (A10) we hve a new expression of the diffusion coefficient:

[EQUATION]

[EQUATION]

Introducing a new variable [FORMULA], its value at [FORMULA], [FORMULA]; and using the function [FORMULA]:

[EQUATION]

with [FORMULA] being here the incomplete Gamma Function, the diffusion coefficient takes the following expression:

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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