Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. II. Incompressible, helical velocity turbulence and a pair of coupled, singular, nonlinear integral equations

Abstract

Using Kraichnan's direct interaction approximation, we set up the equations governing the normal modes of the ensemble average magnetic field under incompressible, nonmirror symmetric velocity turbulence. We show that (i) the Green's stress tensor enjoys equipartition of its symmetric and antisymmetric parts at the normal mode frequencies of the ensemble average field, (ii) for static velocity turbulence, including helicity, there are no growing modes, (iii) the commonly used first order smoothing theory approximation is invalid when compared to the Kraichnan equations, for the Kraichnan equations do not satisfy Hammerstein's theorem while first order smoothing theory requires the satisfaction of Hammerstein's theorem, (iv) if there is to be any growth of the ensemble average magnetic field it must come from time dependent velocity turbulence, and when the velocity turbulence is time dependent we have so far been unable to solve the Kraichnan equations. We have done these calculations for two reasons. First to illustrate, by exact solution, the manner in which the normal modes of the ensemble average magnetic field depend on the helicity and Reynolds number of the turbulent velocity field. Second to show that approximate treatments of the hydromagnetic equation (like first order smoothing theory), rather than exact solution, are liable to give rise to substantial error in view of the fact that the Kraichnan equations do not satisfy Hammerstein's theorem.