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Prediction of frequencies in thermosolutal convection from mean flows

Motivated by studies of the cylinder wake, in which the vortex-shedding frequency can be obtained from the mean flow, we study thermosolutal convection driven by opposing thermal and solutal gradients. In the archetypal two-dimensional geometry with horizontally periodic and vertical no-slip boundary conditions, branches of traveling waves and standing waves are created simultaneously by a Hopf bifurcation. Consistent with similar analyses performed on the cylinder wake, we find that the traveling waves of thermosolutal convection have the RZIF property, meaning that linearization about the mean fields of the traveling waves yields an eigenvalue whose real part is almost zero and whose imaginary part corresponds very closely to the nonlinear frequency. In marked contrast, linearization about the mean field of the standing waves yields neither zero growth nor the nonlinear frequency. It is shown that this difference can be attributed to the fact that the temporal power spectrum for the traveling waves is peaked, while that of the standing waves is broad. We give a general demonstration that the frequency of any quasi-monochromatic oscillation can be predicted from its temporal mean.