Magneto-convection in sunspot penumbrae

Abstract

Finite amplitude convection in the presence of a horizontal magnetic field has been investigated in a region where thermal diffusivity (κ) is less than magnetic diffusivity (η) and whenκ/η > 1,Q ≤Q _c, where

$$Q_c = \frac{{(1 + \sigma _1 )(\pi ^2 + q_c^2 )^2 }}{{q_c^2 (\sigma _2 - \sigma _1 )}}$$

,Q is the Chandrasekhar number,σ ₁ the Prandtl number,σ ₂ the magnetic Prandtl number, andq _c the critical wave number at the onset of stationary convection. We have derived a nonlinear time-dependent Landau—Ginzburg equation near the onset of supercritical stationary convection and a nonlinear, second-order equation at the Takens—Bogdanov bifurcation. We have obtained steady-state solutions of these equations, which describe the nonlinear behaviour near the onset of stationary convection.