A new approach to the problem of wave fluctuations in localized smoothly varying turbulence

In the past, smoothly varying turbulence has been studied by changing the structure constant to the function
C_{n}^{2}(\bar{r}
). The purpose of this paper is to show that this approach is insufficient, and that a random process developed by Silverman can be used to describe the wave fluctuations in localized smoothly varying turbulence. The localized turbulence is characterized by a correlation function which is a product of a function of the average coordinate and a function of the difference coordinate. The corresponding spectrum is also given by a product of a function of the difference wavenumber and a function of the average wavenumber. They are related to each other through two Fourier transform pairs. Making use of the preceding representations, the fluctuations of a wave propagating through such a turbulence can be given either by the integrals with respect to the two wavenumbers or by a convolution integral of the structure constant
C_{n}^{2}(\bar{r}
) and a function involving the outer scale of the turbulence
L_{0}
. It is shown that for a plane wave case, if the distance
L
is within (
L_{0}^{2}/\lambda
), then the usual formula given by Tatarski is valid. But if the distance is between
L_{0}^{2}/\lambda
and
(bL_{0})/\lambda
where
b
is the total transverse size of the turbulence, the variance of the wave is nearly constant, and if
L \gg (bL_{0})/\lambda
, the variance decays as
L^{-2}
. Similar conclusions are shown for a spherical wave case. Some examples are shown illustrating the effectiveness of this method.