Scaling behaviour of small-scale dynamos driven by Rayleigh–Bénard convection

A numerical investigation of convection-driven dynamos is carried out in the plane layer geometry. Dynamos with different magnetic Prandtl numbers Pm are simulated over a broad range of the Rayleigh number Ra. The heat transport, as characterized by the Nusselt number Nu, shows an initial departure from the heat transport scaling of non-magnetic Rayleigh–Bénard convection (RBC) as the magnetic field grows in magnitude; as Ra is increased further, the data suggest that Nu grows approximately as Ra2/7, but with a smaller prefactor in comparison with RBC. Viscous (ϵu) and ohmic (ϵB) dissipation contribute approximately equally to Nu at the highest Ra investigated; both ohmic and viscous dissipation approach a Reynolds-number-dependent scaling of the form Rea, where a≈2.8. The ratio of magnetic to kinetic energy approaches a Pm-dependent constant as Ra is increased, with the constant value increasing with Pm. The ohmic dissipation length scale depends on Ra in such a way that it is always smaller, and decreases more rapidly with increasing Ra, than the viscous dissipation length scale for all investigated values of Pm.