Transition from shear to sideways diffusive instability in a vertical slot

Stability of the steady motion of a fluid confined between two differentially heated rigid vertical plates is considered. When a stable, constant vertical salinity gradient is also present, the steady mean velocity in the vertical direction and the mean lateral salinity gradient are characterized by the solute Rayleigh number, R s . Experimental investigations (Elder 1965; Hart 1970) show that when R s = 0 the instability is induced by shear and occurs in the form of two-dimensional convection cells. However, at moderate values of R s , these shear instabilities are replaced by double-diffusive cellular convection (Thorpe, Hutt & Soulsby 1969; Paliwal & Chen 1980 a ). It is generally believed that the instability is stationary and cellular for all values of R s (Hart 1971; Paliwal & Chen 1980 b ). We have solved the general eigenvalue problem, and our results indicate that, during transition from the stationary shear-induced instability to stationary double-diffusive cellular convection, overstable motion occurs. Furthermore, in this transition region, over a range of moderately small values of R s , there is no preferred wavelength at the onset of instability.