Unifying heat transport model for the transition between buoyancy-dominated and Lorentz-force-dominated regimes in quasistatic magnetoconvection

Abstract

In magnetoconvection, the flow of electromagnetically conductive fluid is driven by a combination of buoyancy forces, which create the fluid motion due to thermal expansion and contraction, and Lorentz forces, which distort the convective flow structure in the presence of a magnetic field. The differences in the global flow structures in the buoyancy-dominated and Lorentz-force-dominated regimes lead to different heat transport properties in these regimes, reflected in distinct dimensionless scaling relations of the global heat flux (Nusselt number $\text{Nu}$) versus the strength of buoyancy (Rayleigh number $\text{Ra}$) and electromagnetic forces (Hartmann number $\text{Ha}$). Here, we propose a theoretical model for the transition between these two regimes for the case of a quasistatic vertical magnetic field applied to a convective fluid layer confined between two isothermal, a lower warmer and an upper colder, horizontal surfaces. The model suggests that the scaling exponents $\gamma$ in the buoyancy-dominated regime, $\text{Nu}\sim {\text{Ra}}^{\gamma }$, and $\xi$ in the Lorentz-force-dominated regime, $\text{Nu}\sim ({\text{Ha}}^{-2}\text{Ra}{)}^{\xi }$, are related as $\xi =\gamma /(1-2\gamma )$, and the onset of the transition scales with ${\text{Ha}}^{-1/\gamma }\text{Ra}$. These theoretical results are supported by our Direct Numerical Simulations for $10\le \text{Ha}\le 2000$, Prandtl number $\text{Pr}=0.025$ and $\text{Ra}$ up to $10^9$ and data from the literature.