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Virtual states and exponential decay in small-scale dynamo

A. V. Kopyev, V. A. Sirota, A. S. Il'yn, K. P. Zybin

TL;DR

The paper extends the Kazantsev theory to small Prandtl-number regimes near the dynamo threshold, revealing that exponential decay observed just below threshold is a transient effect caused by a virtual energy level produced by the flattening of the velocity correlation at large scales. By analyzing the Schrödinger-type form of the Kazantsev equation with a VK-inspired velocity model, the authors identify a critical scale \(X_c\) (approximately 20.5 for \(s=1/3\)) and a corresponding critical magnetic Reynolds number \(Rm_c \approx 100\) that delineate growth and decay regimes. They derive a universal logarithmic dependence of the growth/decay rate on \(Rm/Rm_c\) both above and below threshold and estimate the lifetime of the virtual level, connecting these results to DNS data and showing consistency with observed linear-in-log behavior. The study provides practical, measurable parameters (\(r_d, X, \Lambda, L, v_{rms}, k_f\)) to compare across simulations, enabling better interpretation of SSD dynamics in low-Prandtl-number turbulence and offering a robust mechanism behind transient exponential damping near the threshold.

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.

Virtual states and exponential decay in small-scale dynamo

TL;DR

The paper extends the Kazantsev theory to small Prandtl-number regimes near the dynamo threshold, revealing that exponential decay observed just below threshold is a transient effect caused by a virtual energy level produced by the flattening of the velocity correlation at large scales. By analyzing the Schrödinger-type form of the Kazantsev equation with a VK-inspired velocity model, the authors identify a critical scale (approximately 20.5 for ) and a corresponding critical magnetic Reynolds number that delineate growth and decay regimes. They derive a universal logarithmic dependence of the growth/decay rate on both above and below threshold and estimate the lifetime of the virtual level, connecting these results to DNS data and showing consistency with observed linear-in-log behavior. The study provides practical, measurable parameters () to compare across simulations, enabling better interpretation of SSD dynamics in low-Prandtl-number turbulence and offering a robust mechanism behind transient exponential damping near the threshold.

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.

Paper Structure

This paper contains 8 sections, 66 equations, 1 figure.

Table of Contents

  1. Kazantsev equation
  2. Definition of parameters
  3. The shape of the potential
  4. Critical Reynolds number
  5. Below and over the critical Reynolds number
  6. Generation: $\gamma > 0$
  7. Damping: $\gamma <0$
  8. Conclusion

Figures (1)

  • Figure 1: The shape of the potential for $s=1/3$. The $\delta$-function is an artifact of the model, it does not affect the solutions essentially. The presence of a 'thick' maximum is of decisive importance.