Dynamos at extreme magnetic Prandtl numbers: Insights from shell models

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 $\mathrm{P}\mathrm{m}$; such computations are inaccessible to direct numerical simulations. For small $\mathrm{P}\mathrm{m}$, we observe that both kinetic and magnetic energy spectra scale as $k^{-5\mathrm{/}3}$ in the inertial range, but the dissipative magnetic energy scales as $k^{-11\mathrm{/}3}\exp (-k\mathrm{/}{k}_{\eta })$. 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-$\mathrm{P}\mathrm{m}$ dynamo has a similar behaviour except that the dissipative kinetic energy scales as $k^{-13\mathrm{/}3}$. 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.