Scaling Laws in Rayleigh‐Bénard Convection

The heat transfer scaling theories for Rayleigh‐Bénard convection (RBC) are reviewed and discussed for configurations with and without rotation and magnetic fields. Scaling laws are a useful tool in studying and characterizing geophysical flows as they provide a basis for extrapolation to extreme parameter regimes that remain unobtainable by current computational and experimental efforts. Specifically, power law scalings that relate the efficiency of the heat transport, as measured by the nondimensional Nusselt number N u , to the thermal driving are pursued. Relations of the functional form N u ∝ ( R a / R a c ) α are considered. Given the strongly stabilizing influences of rotation and magnetic fields, thermal driving is considered in the context of the supercriticality of the system given by the ratio of the Rayleigh number R a , measuring the thermal forcing, to the critical R a c , above which convection occurs. Analytical predictions for the exponent α are presented for the regimes of convection, rotating convection, and magnetoconvection, and the scalings are benchmarked against available numerical and experimental results in the accessible regimes. The exponents indicate that the thermal bottleneck to heat transport occurs within the thermal boundary layers for nonrotating RBC and the turbulent interior for rotating RBC. For magnetoconvection, a single exponent of α =1 is obtained for all theories and no bottleneck is identified.