Virtual states and exponential decay in small-scale dynamo

Abstract: We develop the Kazantsev theory of small-scale dynamo generation at small Prandtl numbers near the generation threshold and restore the concordance between the theory and numerical simulations: the theory predicted a power-law decay below the threshold, while simulations demonstrate exponential decay. We show that the exponential decay is temporary and owes its existence to the flattening of the velocity correlator at large scales. This effect corresponds to the existence of a long-living virtual level in the corresponding Schrodinger type equation. We also find the critical Reynolds number and the increment of growth/decay above and under the threshold; we express them in terms of the quantitative characteristic properties of the velocity correlator, which makes it possible to compare the results with the data of different simulations.