An approximate method for determining the stability of two-scale flow in the mantle

Summary An approximate method of calculating the stability of two-dimensional convection rolls to cross-roll disturbances is evaluated by comparison with the results from exact analyses and good agreement is obtained. In particular, investigation of the second mode of the disturbance provides a qualitative estimate of the terms excluded in making the approximation. We conclude that this approximate method, sensibly used, gives a good indication of the stability of rolls to cross-roll disturbances.Application of this method is made to convection rolls heated partially or wholly from within, and to rolls in the presence of a longitudinal shear flow. The approximation indicates that, at high Rayleigh Numbers, growth rates are underestimated and so the amount of shear calculated to stabilise longitudinal rolls is a lower bound. Our results suggest that two-dimensional rolls are unlikely to be a stable flow pattern beneath even the fastest moving plates.