Localized convection cells in the presence of a vertical magnetic field

Thermal convection in a horizontal fluid layer heated uniformly from below usually produces an array of convection cells of roughly equal amplitudes. In the presence of a vertical magnetic field, convection may instead occur in vigorous isolated cells separated by regions of strong magnetic field. An approximate model for two-dimensional solutions of this kind is constructed, using the limits of small magnetic diffusivity, large magnetic field strength and large thermal forcing. The approximate model captures the essential physics of these localized states, enables the determination of unstable localized solutions and indicates the approximate region of parameter space where such solutions exist. Comparisons with fully nonlinear numerical simulations are made and reveal a power-law scaling describing the location of the saddle-node bifurcation in which the localized states disappear.