Model III: Benard convection in the presence of horizontal magnetic field and rotation

Motivated by the need to understand better the roles of viscosity, electrical conductivity of the boundary and the interaction between all the possible wave motions in a magnetic rotating system, we study the linear stability of a simple system which can support five wave modes that can be excited in a rotating diffusive fluid under the influence of a magnetic field and gravity. This is a Benard layer rotating uniformly about a horizontal axis in the presence of a horizontal magnetic field inclined at an angle to the rotation vector, a situation previously studied succinctly under the name of model III. The stability is governed by seven dimensionless parameters: (i) the modified Rayleigh number, R, the Elsasser number, Λ, and the Ekman number, E, which, respectively, represent the ratios of the buoyancy, Lorentz and viscous forces to the Coriolis force, (ii) q and _{p m} which represent the ratios of thermal diffusivity and viscosity to magnetic diffusivity, (iii) r denoting the ratio of electrical conductivity of boundary to that of fluid, and (iv) f which measures the angle between field and rotation vector. A comprehensive investigation of the properties of these waves is carried out in the geophysically relevant case of small Ekman numbers, although some exact solutions are presented for other values of E. The preferred mode of convection is identified for each type of convection and the overall preferred mode is discussed and regime diagrams for the preference of the different modes are constructed in the parameter space. It is shown that viscosity plays a crucial role in the identification of the preferred mode of convection, although it may be very small. The assumption that E ≪ 1 naturally leads to the development of boundary layers, and a rich variety of them is found to exist in the system. The contribution of every boundary layer to the solution is found and the variables of the system strongly affected by the boundary layers are identified. The boundary layers are found to be affected by the dynamic and electrical properties of the boundary. Whereas free boundaries can sometimes allow exact solutions, rigid boundaries always develop boundary layers. The electrical conductivity of the boundary is found to have a quantitative effect on four modes while it leads to different mainstream solutions in the case of one mode, which can exist for a limited range of parameters and can take the form of a propagating wave only.