On Some Mathematical Aspects of the Direct-Interaction Approximation in Turbulence Theory*

The purpose of this paper is to clarify some mathematical issues associated with the direct-interaction approximation DIA.given by Kraichnan w Phys. Re¨. 109 1958., 1407; J. Fluid Mech. 5 1959., 497; Phys. Fluids 7 1964., 1723; 8 1965., 575; 9 1966., 1728xvia its application to Kraichnan’swJ. Math. Phys. 2 1961., 124x model equation. The latter process is shown to lead to a functional equation; the basic properties and construction of the solution of this equation are addressed. The relation of the DIA solution to the perturbative solution is discussed within the framework of a continued-fraction representation