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 [Phys. Rev. 109 (1958), 1407; J. Fluid Mech. 5 (1959), 497; Phys. Fluids 7 (1964), 1723; 8 (1965), 575; 9 (1966), 1728] via its application to Kraichnan's [J . Math. Phys. 2 (1961), 124] model equation. The latter process is shown to lead to a functional equation; the basic properties and construction of th e solution of this equation are addressed. The relation of the DLA solution to the perturbative solution is discussed within the framework of a contin ued-fraction representation. (C) 1999 Academic Press.