Multiple equilibria and stable oscillations in thermosolutal convection at small aspect ratio

Abstract

For thermosolutal convection in an enclosure of arbitrary vertical aspect ratio, mixed boundary conditions – with salt flux and temperature prescribed on a horizontal boundary – can lead to symmetry breaking via pitchfork bifurcation. In the present paper we consider an enclosure of very small height-to-length aspect ratio δ, as encountered in the world's oceans. In this case, if the ratios of the vertical to horizontal components of viscosity, [Pscr ], and of diffusivity, [Lscr ], are of order unity, advective transport cannot set in even at very high Rayleigh numbers. The ratios [Pscr ] and [Lscr ] must be substantially less than unity in order for convection to dominate the heat and solute transport.We have investigated numerically the effects of monotonic and non-monotonic temperature and salinity boundary conditions in a two-dimensional domain at constant δ = 0.01 and constant [Pscr ] = [Lscr ] = 0.01. This ratio of eddy-mixing coefficients reflects the different scales of motion – vertical and horizontal – in the ocean, rather than a physically realizable laboratory fluid. It is found that when the salt-flux strength, γ, is sufficiently large, the system undergoes a second bifurcation for both types of boundary conditions. It is a Hopf bifurcation, leading from the asymmetric steady states produced by the first one to oscillatory solutions. These periodic solutions are stable and very robust. An approximate Hopf bifurcation diagram has been produced. We conclude that non-monotonic salt-flux conditions are neither necessary nor sufficient to induce the oscillations, while the strength of the salt flux is crucial.