The Onset of Thermal Convection Between Poorly Conducting Horizontal Boundaries in the Presence of a Shear Flow

We derive an evolution equation for three-dimensional thermal convection in an imposed shear flow between horizontal boundaries that conduct heat poorly. Through a long-wavelength asymptotic expansion the governing equations are reduced to a single partial differential equation for the depth-averaged temperature field. A linear stability analysis of this equation indicates that the first instability of the quiescent state is to convective rolls whose axes are aligned with the direction of the shear flow. This result complements previous numerical solutions of the linear stability problem for a fluid between perfectly conducting horizontal boundaries, where a similar preference for such rolls has been found.