A note on the stability of low-Prandtl-number Hadley circulations

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Abstract
This paper discusses the stability of the flow of a low-Prandtl-number liquid contained in a shallow slot with differentially heated vertical endwalls. The effect of thermally insulating boundaries at the top and bottom of the container on wavelength selection is emphasized. Stability calculations indicate that, for Prandtl number Pr in the range 0.015 < Pr < 0.27, the first perturbations to grow are overstable (oscillatory) longitudinal rolls with axes perpendicular to the endwalls, and with very large cross-stream wavelengths of about 9 to 15 layer depths. Previous studies using thermally conducting boundaries predict critical wavelengths of about three layer depths. The new results are in substantial agreement with an experiment using a differentially heated layer of mercury with aspect ratio (depth/length) 0.047 in both horizontal directions. The implications of the long-wavelength instability for the interpretation of thermal oscillations observed in other smaller-aspect-ratio configurations is discussed.