On the role of thermal boundary conditions in dynamo scaling laws

Abstract

In dynamo power-based scaling laws, the power $P$ injected by buoyancy forces is measured by a so-called flux-based Rayleigh number, denoted as ${\rm Ra}_Q^*$ (see Christensen and Aubert, 2006). Whereas it is widely accepted that this parameter is measured (as opposite to controlled) in dynamos driven by differential heating, the literature is much less clear concerning its nature in the case of imposed heat flux. We clarify this issue by highlighting that in that case, the ${\rm Ra}_{Q}^*$ parameter becomes controlled only in the limit of large Nusselt numbers (${\rm Nu} \gg 1$). We then address the issue of the robustness of the original relation between $P$ and ${\rm Ra}_Q^*$ with the geometry and the thermal boundary conditions. We show that in the cartesian geometry, as in the spherical geometry with a central mass distribution, this relation is purely linear, in both differential and fixed-flux heating. However, we show that in the geometry commonly studied by geophysicists (spherical with uniform mass distribution), its validity places an upper-bound on the strength of the driving which can be envisaged in a fixed Ekman number simulation. An increase of the Rayleigh number indeed yields deviations (in terms of absolute correction) from the linear relation between $P$ and ${\rm Ra}_Q^*$. We conclude that in such configurations, the parameter range for which $P$ is controlled is limited.