Numerical Analysis of Linear Traveling Wave in Rotating Rayleigh–Bénard Convection with an Adiabatic Sidewall

In rotating Rayleigh–Bénard problems, convection with traveling waves may occur near the sidewalls. The Rayleigh number, Taylor number and Prandtl number are involved in this phenomenon, and the convection mode is determined depending on their values. We focused on the onset of this convection with traveling waves under the assumption that centrifugal force is neglected. By conducting two-dimensional linear stability analyses assuming periodicity of the flow and temperature fields along the sidewall direction, we investigated the effect of the Taylor number and the Prandtl number on the critical Rayleigh number and also attempted to understand the phenomenon qualitatively through three-dimensional visualizations. It was exhibited that as the Taylor number increases, the wave number, the Rayleigh number and the phase speed are found to increase. On the other hand, as the Prandtl number decreases, the wavenumber and the Rayleigh number decrease, but the phase velocity increases. The present analyses suggest that convection modes localized near the sidewalls are unlikely to emerge for low Prandtl number cases, which are comparable to those of liquid metals.