Onset of wall-attached convection in a rotating fluid layer in the presence of a vertical magnetic field

A horizontal fluid layer heated from below and rotating about a vertical axis in the presence of a vertical magnetic field is considered. From earlier work it is known that the onset of convection in a rotating layer usually occurs in the form of travelling waves attached to the vertical sidewalls of the layer. It is found that this behaviour persists when a vertical magnetic field is applied. When the Elsasser number A is kept constant and the sidewall is thermally insulating the critical Rayleigh number R, increases in proportion to the rotation rate described by the square root of the Taylor number, T. This asymptotic relationship is found for an electrically highly conducting sidewall as well as for an electrically insulating one. At fixed rotation rate for Q >> tau, R, grows in proportion to Q when the sidewall is electrically highly conducting, and in proportion to Q(3/4) when the sidewall is electrically insulating. Here Q is the Chandrasekhar number which is a measure of the magnetic energy density, and a thermally insulating sidewall has been assumed. Of particular interest is the possibility that the magnetic field counteracts the stabilizing influence of rotation on the onset of sidewall convection in the case of thermally insulating sidewalls. When the sidewall is thermally highly conducting, R, for the sidewall mode grows in proportion to tau(4/3). This asymptotic behaviour is found for both cases of electrical boundary conditions, but it no longer precedes the onset of bulk convection for A greater than or similar to 1.