Two remarks on Heisenberg’s theory of isotropic turbulence
Equation (3) states that the spatial rate of change of the distribution function 2V(r, s) at the position r, in the direction of the unit vector s, is due to three effects which appear on the right side of the equation. They are, in order of their appearance, first, radiation lost through absorption, and scattering out of the direction s; secondly radiation which is scattered from all 4ir solid angle into the direction s. p(0) is the angular differential cross section for the specific process involved. Thirdly there is the source contribution <S. It is assumed in the above equation that a scatter is not accompanied by a change in wave length, n is the total narrow beam absorption coefficient. When the solution of (3) is through expansion processes, Eq. (2) becomes a useful integral formula for reducing the integral of (3) to its equivalent summation. For instance, consider the problem which permits the unit vector s to be replaced by the ordinate angle 6 (e.g., spherical symmetry and infinite plane source). In this case the distribution function may be expanded as