Finite memory time and anisotropy effects for initial magnetic energy growth in random flow of conducting media

The dynamo mechanism is a process of magnetic field self-excitation in a moving electrically conducting fluid. One of the most interesting applications of this mechanism related to the astrophysical systems is the case of a random motion of plasma. For the very first stage of the process, the governing dynamo equation can be reduced to a system of first order ordinary differential equations. For this case we suggest a regular method to calculate the growth rate of magnetic energy. Based on this method we calculate the growth rate for random flow with finite memory time and anisotropic statistical distribution of the stretching matrix and compare the results with corresponding ones for isotropic case and for short-correlated approximation. We find that for moderate Strouhal numbers and moderate anisotropy the analytical results reproduce the numerically estimated growth rates reasonably well, while for larger governing parameters the quantitative difference becomes substantial. In particular, analytical approximation is applicable for the Strouhal numbers s<0.6 and we find some numerical models and observational examples for which this region might be relevant. Rather unexpectedly, we find that the mirror asymmetry does not contribute to the growth rates obtained, although the mirror asymmetry effects are known to be crucial for later stages of dynamo action.