Asymptotic solutions for nonlinear magnetoconvection

Convection in a vertical magnetic field occurs in narrow cells in the physi cally relevant limit where the Chandrasekhar number Q becomes large, corres ponding to a strong field or small diffusion. This allows asymptotic soluti ons to be developed for fully nonlinear convection, requiring only the solu tion of a nonlinear boundary value problem. Solutions for steady and oscill atory magnetoconvection are obtained, with different scalings. In the stead y case, the heat flux and the fluid velocity are found at leading order in the asymptotic expansion and the vertical velocity scales as Q(1/6). In the oscillatory case, where it is necessary to continue to second order, the v ertical velocity is of order Q(1/3) and the frequency of the oscillations i s always greater than that predicted by linear theory. The heat flux does n ot depend on either the wavenumber or the planform.