Convection rolls and their instabilities in the presence of a nearly insulating upper boundary

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Abstract
The problem of steady convection rolls and their instabilities in a fluid layer heated from below is studied numerically in the case of a highly conducting, rigid lower and a nearly insulating, stress-free upper boundary. A Galerkin method is used to obtain two-dimensional solutions in dependence on the Rayleigh number R and the wave number α for different values of the Prandtl number P. Their stability is analyzed through the superposition of general three-dimensional infinitesimal disturbances. Most instabilities correspond qualitatively to those found in the case of symmetric highly conducting, rigid boundaries. A new instability, the subharmonic varicose instability, is found, however, which restricts the region of stable rolls toward a higher Rayleigh number in the case of moderate Prandtl numbers.

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