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In magnetoconvection, the flow of electromagnetically conductive fluid isdriven by a combination of buoyancy forces, which create the fluid motion dueto thermal expansion and contraction, and Lorentz forces, which distort theconvective flow structure in the presence of a magnetic field. The differencesin the global flow structures in the buoyancy-dominated andLorentz-force-dominated regimes lead to different heat transport properties inthese regimes, reflected in distinct dimensionless scaling relations of theglobal heat flux (Nusselt number $\textrm{Nu}$) versus the strength of buoyancy(Rayleigh number $\textrm{Ra}$) and electromagnetic forces (Hartmann number$\textrm{Ha}$). Here, we propose a theoretical model for the transition betweenthese two regimes for the case of a quasistatic vertical magnetic field appliedto a convective fluid layer confined between two isothermal, a lower warmer andan upper colder, horizontal surfaces. The model suggests that the scalingexponents $\gamma$ in the buoyancy-dominated regime,$\textrm{Nu}\sim\textrm{Ra}^\gamma$, and $\xi$ in the Lorentz-force-dominatedregime, $\textrm{Nu}\sim(\textrm{Ha}^{-2}\textrm{Ra})^\xi$, are related as$\xi=\gamma/(1-2\gamma)$, and the onset of the transition scales with$\textrm{Ha}^{-1/\gamma}\textrm{Ra}$. These theoretical results are supportedby our Direct Numerical Simulations for $10\leq \textrm{Ha}\leq2000$, Prandtlnumber $\textrm{Pr}=0.025$ and $\textrm{Ra}$ up to $10^9$ and data from theliterature.

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