Closed-form linear stability conditions for magneto-convection

Abstract

Chandrasekhar (1961) extensively investigated the linear dynamics of Rayleigh–Bénard convection in an electrically conducting fluid exposed to a uniform vertical magnetic field and enclosed by rigid, stress-free, upper and lower boundaries. He determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection as a function of the Chandrasekhar number $Q$ or Hartmann number squared. No closed-form formulae appeared to exist and the results were tabulated numerically. We have discovered simple expressions that concisely describe the stability properties of the system. When the Prandtl number $Pr$ is greater than or equal to the magnetic Prandtl number $Pm$ the marginal stability boundary is described by the curve $Q = \pi^{-2}[R -R_c^{1/3}R^{2/3}]$ where $R$ is the Rayleigh number and $R_c = (27/4)\pi^4$ is Rayleigh's famous critical value for the onset of stationary convection in the absence of a magnetic field ($Q = 0$). When $Pm > Pr$ the marginal stability boundary is determined by this curve until intersected by the curve[Q = \frac{1}{\pi^2}\left[\frac{\Pm^2(1+\Pr)}{\Pr^2(1+\Pm)}R \right. -\left. \left(\frac{(1+\Pr)(\Pr+\Pm)}{\Pr^2}\right)^{1/3}\left( \frac{\Pm^2(1+\Pr)}{\Pr^2(1+\Pm)} \right)^{2/3} R_c^{1/3}R^{2/3}\right].]An expression for the intersection point is derived and also for the critical horizontal wavenumbers for which instability sets in along the marginal stability boundary either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Also we show that in the limit of vanishing magnetic diffusivity, or infinite electrical conductivity, the system is unstable for sufficiently large $R$. Instability in this limit always sets in via overstability.