Magnetic energy transient growth in the subcritical Kazantsev model

We study average magnetic field growth in a mirror-symmetrical Kazantsev turbulent flow near the dissipative scales. Our main attention is directed to a subcritical regime, when an exponential decrease of magnetic energy is usually expected. We show that instead of damping, transient energy growth can be obtained, for example, in stochastic processes supported by the large-scale magnetic fields. We calculate the longitudinal correlation functions and demonstrate that they can tend to nonzero stationary solutions, whose localization width is inversely proportional to the square of the magnetic Reynolds numbers and with amplitude depending on the closeness of these numbers to the critical value. We present the local generation effect without any external support, predicted by Zeldovich in 1956. Numerically solving the initial-boundary Kazantsev problem on the nonuniform grids, we simulate this process by implicit schemes and discuss the possible consequences of subcritical growth for dynamo theory.

Figure 1

Second mode: (a) the evolution of the Kazantsev solution from nonzero initial perturbation (red line). Different snapshots are marked by $t_1, t_2$, and $t_3$. (b) The maximum of the correlation function as a function of time. Line 1 approximates the energy growth $[max|M|=5.6-6.1exp(-0.13t\left)\right]$; line 2 approximates the damping ($max\left|M\right|\sim t^{-2.1}$).

Figure 2

First mode: (a) The stationary solution (bold line) and the dynamical solutions for different time snapshots $t_1=1, t_2=2$ and $t_3=4$. (b) The dependences of the maximal deviation between stationary and dynamical solutions for $\text{Rm}=5$, 30, and 50. Red lines approximate initial intervals of rapid magnetic energy growth.

Figure 3

(a) The dependence of the amplitude of the stationary solution on $\text{Rm}$ (red line denotes the critical value ${\text{Rm}}_{cr}$). (b) The dependence of the width of the stationary solution on $\text{Rm}$ and its approximation $\mathrm{\Delta }r=-0.01+1.5\sqrt{2/\text{R}\text{m}}$, marked by the red line.