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 the 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.

Figure 1

Numerical and analytical approximation of ${\gamma }_2$ for small Strouhal numbers in the isotropic 3D flow model.

Figure 2

Numerical and analytical approximation of ${\gamma }_2$ for small Strouhal numbers in the isotropic 2D flow model.

Figure 3

Contour plot of the growth rate ${\gamma }_2$ distribution in the axisymmetric 3D flow model for fixed value $s=0.3$ and various anisotropy parameters $u$, $L$. Note the log scale of both axes. Black dot corresponds to the isotropic case $u=L=1$, and the green dot corresponds to the case $u=1.4$, $L=0.8$, both shown in Fig. 4. The gray area shows forbidden $u$ and $L$ values for the 3D model. The red line corresponds to the first inequality in (37), and the blue line corresponds to the last inequality in (37).

Figure 4

Growth rate for the isotropic 3D flow model $u=L=1$ (black line) in comparison to the anisotropy case $u=1.4$, $L=0.8$ (green line) for various Strouhal numbers $s$. Dashed lines show the corresponding analytical approximations. The vertical line shows the case $s=0.3$ given in Fig. 3.

Figure 5

Contour plot of the growth rate ${\gamma }_2$ distribution in the axisymmetric 2D flow model for fixed value $s=0.3$ and various anisotropy parameters $u$, $L$. Note the log scale of both axes. The black dot corresponds to the isotropic case $u=L=1$, and the green dot corresponds to the case $u=1.4$, $L=0.8$, both shown in Fig. 6.

Figure 6

Growth rate for the isotropic 2D flow model $u=L=1$ (black line) in comparison to the asymmetric case $u=1.4$, $L=0.8$ (green line) for various Strouhal numbers $s$. Dashed lines show the corresponding analytical approximations. The vertical line shows the case $s=0.3$ given in Fig. 5.