On the stability of two-dimensional convection rolls in an infinite Prandtl number fluid with stress-free boundaries (Q1118483)

Numerical solutions for two-dimensional convection rolls in a fluid layer of infinite Prandtl number are obtained by the Galerkin method. Stress- free, isothermal boundaries are assumed at the horizontal boundaries of the fluid layer. The stability of the steady solutions with respect to three-dimensional disturbances is analyzed in the Rayleigh number-wave number space. It is found that even for Rayleigh numbers as high as several millions there appears to exist a region of the wavenumber \(\alpha\) where the convection rolls are stable. This result contrasts with the well known transition to three-dimensional bimodal convection in the presence of no-slip boundaries, but it agrees with simple arguments about the stability of the thermal boundary layers.