Modelling small-scale dynamo by the Jacobi equation

M. E. Artyushkova - D. D. Sokoloff

Moscow State University, Russia

Abstract
The results of analytical investigations of the small-scale dynamo equation demonstrate an intermittent growth of the magnetic field. Many predictions of analytical theory have a very little connection with the current numerical results presumably because of limited possibilities of direct numerical simulations. We suggest to clarify this problem using a simple Jacobi equation with random coefficients as a model for the complicated small-scale dynamo equations. The Jacobi equation, being an ordinary differential equation, has features utilized in analytical theory to demonstrate the intermittent behaviour of the growing magnetic field. Extensive numerical studies undertaken for the Jacobi equation are reported below. The result is that a detailed reproduction of the intermittent behaviour of the growing solution requires about N=5·10⁵ independent realizations. This huge figure seems inaccessible for direct numerical simulations of available codes of 3D problems. On the other hand, the number of independent turbulent cells in, say, galactic MHD, is comparable with N. Tables 2, Figs 15, Refs 19.