Long-wavelength thermal convection in a weak shear flow

The problem of thermal convection in an imposed shear flow isexamined for a horizontal layer of fluid between poorly conductingboundaries. The horizontal scale H of the convective motionnear its onset is much greater than the depth h of the fluidlayer, with h/H being proportional to the one-fourth power ofa Biot number appearing in the condition applied to the temperatureat the horizontal boundaries. It is known that an asymptoticexpansion in powers of h/H yields a nonlinear long-wavelengthevolution equation for the depth-averaged temperature fieldthat is spatially isotropic in the absence of an imposed shearflow, but is strongly anisotropic for ‘strong’ shear.We derive in this paper a nonlocal long-wavelength equationthat bridges these two cases, and that contains each case inthe zero-shear and large-shear limits. Using this evolutionequation, we show how the shear flow stabilizes the longitudinalrolls to the zigzag instability, and how a preference for asquare planfonn on a periodic square lattice gives way to apreference for longitudinal rolls near onset. The longitudinalrolls may then become unstable as the Rayleigh number is increased.The analytical work is illustrated by some numerical simulationsof the full three-dimensional Boussinesq Navier-Stokes equations.The problem of pattern selection on a hexagonal lattice is alsodiscussed, and some new results are presented.