Dynamo equations with random coefficients

E. A. Mikhailov¹ - I. I. Modyaev²

¹ Chair of Mathematics, Faculty of Physics, Moscow State University, Leninskie Gory GSP-1, Moscow 119991, Russia
² Chair of Probability Theory, Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory GSP-2, Moscow 119991, Russia

Abstract
Galactic dynamo is caused by two effects. One of them is caused by differential rotation and the other is determined by turbulent motions. In some galaxies there is a strong star formation or other processes which are connected with local regions of hot gas. Turbulent motions in such zones differ from the those in warm gas. It is useful to model such processes with dynamo equations that contain random coefficients. The coefficient of alpha-effect can take two different values. The first one is related to warm gas and it is the same as the coefficient for most of the galaxies studied before. The second one characterizes hot gas, which can be connected with the star formation or other processes with high velocities and large portions of energy. This coefficient is random and changes with a time, which is much less than typical times of galactic dynamo. The probability to obtain the second value of the coefficient is determined by the intensity of the star formation. We have obtained some critical values of probability, for which dynamo cannot support the magnetic field growth. Also we have calculated average velocities of the magnetic field growth and its dispersion. For calculations, we used both numerical and asymptotical methods. Tables 2, Figs 4, Refs 16.