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, Rs. Experimental investigations (Elder 1965; Hart 1970) show that when Rs = 0 the instability is induced by shear and occurs in the form of two-dimensional convection cells. However, at moderate values of Rs, 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 Rs (Hart 1971; Paliwal & Chen 19806). 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 Rs, there is no preferred wavelength at the onset of instability.