Effect of rotation and shear on the linear stability of double-diffusive convection

We study the instability of double-diffusive convection under the combined effects of rotation and Couette flow using a linear stability theory and energy analysis. Our results indicate that rotation inhibits instability in the absence of Couette flow. For Ekman-Couette double-diffusive convection, the inhibitory effect of rotation reduces at Reynolds number Re = 1, even causing the instability threshold to be lower than that in double-diffusive convection. In the presence of strong shear strength at Re = 500, four instability modes appear with the decrease in Ekman number Ek (note that smaller Ek denotes stronger rotation): a buoyancy-driven convective instability mode (type C), a shear-stabilized convective mode (type C_S), a type II mixed mode, and a thermohaline shear instability mode. With the increase in the rotation, the degree to which rotation weakens the inhibitory effect is not monotonic, and in the type II mixed mode, this weakening effect reaches its peak. Energy budget analysis shows that, unlike the other three modes where the main destabilizing factor is thermal buoyancy, for the type II mixed mode, the energy produced by base flow becomes the main destabilizing factor. Furthermore, in the type C mode, the energy absorbed by the salinity field instead of viscous dissipation becomes the most important stabilizing factor.