Dynamos at extreme magnetic Prandtl numbers: Insights from shell models

TL;DR

This paper introduces an MHD shell model to study energy fluxes in turbulence at extreme magnetic Prandtl numbers, revealing distinct spectral behaviors and energy transfer mechanisms for small and large Pm, inaccessible to direct simulations.

Contribution

The paper develops a novel shell model enabling analysis of MHD turbulence at extreme magnetic Prandtl numbers, providing new insights into energy spectra and transfer processes.

Findings

Kinetic and magnetic spectra follow a $k^{-5/3}$ scaling at small Pm.

Dissipative magnetic energy scales as $k^{-11/3}$ with exponential cutoff.

Large Pm dynamo exhibits similar but distinct energy transfer characteristics.

Abstract

We present an MHD shell model suitable for computation of various energy fluxes of magnetohydrodynamic turbulence for very small and very large magnetic Prandtl numbers $\mathrm{Pm}$; such computations are inaccessible to direct numerical simulations. For small $\mathrm{Pm}$, we observe that both kinetic and magnetic energy spectra scale as $k^{-5/3}$ in the inertial range, but the dissipative magnetic energy scales as $k^{-11/3}\exp(-k/k_\eta)$. Here, the kinetic energy at large length scale feeds the large-scale magnetic field that cascades to small-scale magnetic field, which gets dissipated by Joule heating. The large-$\mathrm{Pm}$ dynamo has a similar behaviour except that the dissipative kinetic energy scales as $k^{-13/3}$. For this case, the large-scale velocity field transfers energy to the large-scale magnetic field, which gets transferred to small-scale velocity and magnetic fields; the energy of the small-scale magnetic field also gets transferred to the small-scale velocity field, and the energy thus accumulated is dissipated by the viscous force.