On the role of thermal boundary conditions in dynamo scaling laws

TL;DR

This paper investigates how thermal boundary conditions influence dynamo scaling laws, revealing that the relation between injected power and flux-based Rayleigh number varies with geometry and boundary conditions, affecting the control and robustness of dynamo models.

Contribution

It clarifies the conditions under which the flux-based Rayleigh number is controlled and demonstrates the geometry-dependent validity of the power-Ra_Q^* relation in dynamo simulations.

Findings

In Cartesian and certain spherical geometries, the power-Ra_Q^* relation is linear.

In spherical geometry with uniform mass distribution, the relation has an upper bound, limiting the parameter range.

Deviations from linearity occur at higher Rayleigh numbers, affecting control in simulations.

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.