Heating a salinity gradient from a vertical sidewall: nonlinear theory

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Abstract
When a body of fluid with a vertical salinity gradient is heated from a vertical sidewall instabilities are sometimes observed. The linear stability of this basic state has been investigated by Kerr (1989). This linear theory predicts the onset of instability well when compared with experimental results; however, the form of the observed nonlinear instabilities does not coincide with the linear predictions (cf. Chen, Briggs & Wirtz 1971; Tsinober & Tanny 1986; Tanny & Tsinober 1988). In this paper we investigate some of the nonlinear aspects of the problem. A weakly nonlinear analysis reveals that the bifurcation into instability is subcritical, and that the initial trend along this branch of solutions is towards the co-rotating cells observed in experiments. The heating levels for which instabilities are absent are investigated by the use of energy stability analysis. This yields a weak result for arbitrary disturbances, showing that disturbances will decay for sufficiently low wall heating. This bound is greatly strengthened by imposing a vertical periodicity on the lengthscale proposed by Chen et al.

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