Dynamos at extreme magnetic Prandtl numbers: Insights from shell models

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.

27 pages, 14 figures

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• Magnetic energy spectrum produced by turbulent dynamo: effect of time irreversibility
• No feedback is possible in small-scale turbulent magnetic field
• Partial invariants, large-scale dynamo action, and the inverse transfer of magnetic helicity
• Magnetohydrodynamic Turbulence: Chandrasekhar's Contributions \& Beyond