Direct Interaction Approximation for Non-Markovianized Stochastic Models in the Turbulence Problem

TL;DR

This paper investigates the application of the direct interaction approximation (DIA) to non-Markovian stochastic models in turbulence, deriving a functional equation and comparing DIA solutions with perturbative methods.

Contribution

It introduces a functional equation framework for DIA in non-Markovian turbulence models and explores solution construction via continued fractions, linking DIA to perturbative solutions.

Findings

DIA leads to a specific functional equation in non-Markovian turbulence models.

Solutions of the DIA functional equation can be constructed using continued fractions.

DIA solutions are compared and related to perturbative solutions in wave propagation problems.

Abstract

The purpose of this paper is to explore mathematical aspects associated with the application of the direct interaction approximation (DIA) (Kraichnan [1],[2]) to the non-Markovianized stochastic models in the turbulence problem. 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. 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.