Heat and momentum transfer for magnetoconvection in a vertical external magnetic field

The scaling theory of Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] for turbulent heat and momentum transfer is extended to the magnetoconvection case in the presence of a (strong) vertical magnetic field. A comparison with existing laboratory experiments and direct numerical simulations in the quasistatic limit allows us to restrict the parameter space to very low Prandtl and magnetic Prandtl numbers and thus to reduce the number of unknown parameters in the model. Also included is the Chandrasekhar limit, for which the outer magnetic induction field $\mathit{B}$ is large enough such that convective motion is suppressed and heat is transported by diffusion. Our theory identifies four distinct regimes of magnetoconvection that are distinguished by the strength of the outer magnetic field and the level of turbulence in the flow, respectively.

Figure 1

Magnetoconvection flow. The outer magnetic induction ${\mathit{B}}_0=B_0{\mathit{e}}_z$, the acceleration due to gravity $\mathit{g}=(0,0,-g)$, the temperature difference, and the characteristic large-scale velocity are indicated.

Figure 2

Phase diagrams of (a) Nu and (b) Re on the Ra-Ha plane for $\text{Pr}=0.025$. The symbols represent the data of our numerical simulations (squares) and the experiments by Cioni et al. [23] (circles) used for fitting the model parameters. The lines in the diagrams mark the position of the crossovers introduced in the model: Below ${\delta }_T={\delta }_v$, the scaling of the thermal BL dissipation changes, $\text{Re}={\text{Re}}^{*}$ marks the transition range from a weakly nonlinear to a fully turbulent bulk flow, and $\text{Ra}={\pi }^2{\text{Ha}}^2$ indicates the onset of convection. Regimes I–IV are marked as described in the text.

Figure 3

Dependence of the coefficients $c_1\text{–}{c}_5$ when fixing the sixth coefficient ${\text{Re}}^{*}$. Negative values of $c_2$ and $c_4$ are indicated by a dashed line.

Figure 4

Uncertainty of (a) Nusselt and (b) Reynolds number results in parameter space. The relative errors, $\mathrm{\Delta }\text{Nu}/\text{Nu}$ and $\mathrm{\Delta }\text{Re}/\text{Re}$, are shown in a logarithmic scale. The data are obtained by varying the fit coefficients $c_1,\cdots ,c_5$ and ${\text{Re}}^{*}$ independently of each other within their error bars. Regimes I–IV and their borders are marked as in Fig. 2, and the purely conductive regime has been masked.