Direct interaction approximation for non-Markovianized stochastic models in the turbulence problem

The purpose of this paper is to consider the application of the direct interaction approximation (DIA) [R. H. Kraichnan, J. Math. Phys. 2, 124 (1961); Phys. Fluids 8, 575 (1965)] to the non-Markovianized stochastic models in the turbulence problem. For illustration, a linear damped stochastic oscillator described by a stochastic equation is considered. This process is shown to lead to a functional equation, and construction of solutions of this equation is addressed within the framework of a continued fraction representation. The relation of the DIA solution to the perturbative solution is discussed in the Laplace transform framework, and qualitative differences between the time evolutions of the two solutions are pointed out. The perturbative solution is shown to exhibit an oscillatory behavior indefinitely, in contradiction to the fact that the linear stochastic oscillator in question is damped. The DIA solution, on the contrary, is shown to exhibit a decaying behavior in accord with the latter property. The DIA procedure is applied to the problem of wave propagation in a random medium, which is described by a stochastic differential equation, with the characteristics of the medium represented by stochastic coefficients. The results are compared with those given by the perturbative procedure.The purpose of this paper is to consider the application of the direct interaction approximation (DIA) [R. H. Kraichnan, J. Math. Phys. 2, 124 (1961); Phys. Fluids 8, 575 (1965)] to the non-Markovianized stochastic models in the turbulence problem. For illustration, a linear damped stochastic oscillator described by a stochastic equation is considered. This process is shown to lead to a functional equation, and construction of solutions of this equation is addressed within the framework of a continued fraction representation. The relation of the DIA solution to the perturbative solution is discussed in the Laplace transform framework, and qualitative differences between the time evolutions of the two solutions are pointed out. The perturbative solution is shown to exhibit an oscillatory behavior indefinitely, in contradiction to the fact that the linear stochastic oscillator in question is damped. The DIA solution, on the contrary, is shown to exhibit a decaying behavior in accord with the latter property...