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 ). 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 ) and a function involving the outer scale of the turbulence . It is shown that for a plane wave case, if the distance is within ( ), then the usual formula given by Tatarski is valid. But if the distance is between and where is the total transverse size of the turbulence, the variance of the wave is nearly constant, and if , the variance decays as . Similar conclusions are shown for a spherical wave case. Some examples are shown illustrating the effectiveness of this method.