Linear stability analysis of Richter rolls

[1] The planform of sublithospheric convection is studied by a 3-D linear stability analysis of longitudinal rolls in the presence of vertical shear. Rayleigh numbers up to 10 are considered. The transition Rayleigh number over which longitudinal rolls are unstable is derived as a function of surface velocity. Effects of nonlinear vertical shear are shown to be insignificant unless nonlinearity becomes extreme. Our results are in good agreement with previous studies, except at high Rayleigh numbers where the meaning of roll stability becomes ambiguous owning to inherently time-dependent convection. When applied to small-scale convection in the upper mantle, our results suggest that the roll stability is very sensitive to plate velocity if the upper mantle viscosity is relatively high ( 10 Pa s); Richter rolls may be expected only beneath fast-moving plates such as the Pacific plate.