Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. I. Incompressible isotropic turbulence and a singular integral equation

Abstract

Using Kraichnan's direct interaction approximation in the induction equations we set up the kinematic equations governing behavior of an ensemble average magnetic field under homogeneous, incompressible, mirror symmetric isotropic velocity turbulence. We demonstrate that the normal modes of the field depend on the solutions to a nonlinear single integral equation. For a simple form of the velocity turbulence we have investigated some of the properties of the integral equation. In particular, we have been able to construct a particular solution. We also point out what remains to be done if we are to obtain all the modes of the ensemble average magnetic field. We have done this calculation so that the behaviour and normal modes of the field can be investigated at arbitrary magnetic Reynolds' numbers. This is in contrast to customary approximations (like first‐order smoothing theory) which normally are valid only for very small Reynolds numbers, if at all, and which therefore omit large regimes of considerable physical interest.