Predictive scaling laws for spherical rotating dynamos

State of the art numerical models of the Geodynamo are still performed in a parameter regime extremely remote from the values relevant to the physics of the Earth's core. In order to establish a connection between dynamo modelling and the geophysical motivation, it is necessary to use scaling laws. Such scaling laws establish the dependence of essential quantities (such as the magnetic field strength) on measured or controlled quantities. They allow for a direct confrontation of advanced models with geophysical constraints. We combine a numerical approach, based on a multiple linear regression method in the form of power laws, applied to a database of 102 direct numerical simulations (courtesy of U. Christensen), and a physical approach, based on energetics and forces balances. We show that previous empirical scaling laws for the magnetic field strength essentially reflect the statistical balance between energy production and dissipation for saturated dynamos. Such power based scaling laws are thus necessarily valid for any dynamo in statistical equilibrium and applicable to any numerical model, irrespectively of the dynamo mechanism. We show that direct numerical fits can provide contradictory results owing to biases in the parameters space covered in the numerics and to the role of a priori hypothesis on the fraction of ohmic dissipation. We introduce predictive scaling laws, that is relations involving input parameters of the governing equations only. We guide our reasoning on physical considerations. We show that our predictive scaling laws can properly describe the numerical database and reflect the dominant forces balance at work in these numerical simulations. We highlight the dependence of the magnetic field strength on the rotation rate. Finally, our results stress that available numerical models operate in a viscous dynamical regime, which is not relevant to the Earth's core.