Stability of convection with horizontal buoyancy gradients

Horizontal convection, in which variations in buoyancy exist on horizontal boundaries, are often contrasted to Rayleigh-Bénard convection as having no steady states. In fact there exist exact solutions to the equations of motion with constant horizontal buoyancy gradients and induced flows. The study of these flows goes back to Hart and Weber in the 1970s. We revisit this situation, with a focus on geophysically-relevant boundary conditions. Unlike the case of Rayleigh-Bénard and Orr-Sommerfeld stability, the marginal modes are no longer two-dimensional with zero frequency. We explore the stabilty boundary given by horizontal Rayleigh number in terms of vertical Rayleigh number. Fully three-dimensional modes are investigated. Finallly the energy stability problem is examined.