Transient growth in diffusive convection with and without shear

We study the transient energy growth of linearly stable diffusive convection with and without a bounded Couette flow. In contrast to the one-component thermal (Rayleigh-Bénard) convection, diffusive convection can generate a nontrivial transient growth even in the absence of shear. This growth is characterized by (quasi)periodic oscillations, and the largest transient growth occurs at the peak of the first oscillation. Driven by the synchronous action of temperature and salinity fields, this growth results from resonance between the symmetric oscillatory modes, and also benefits from the catalytic enhancement by stationary modes of specific growth rates. In the presence of shear, the transient growth of longitudinal rolls is always the largest and three nonmodal instability regimes are identified with increasing density gradient. Unless the density gradient is very high, the lift-up mechanism is always the fundamental mechanism to drive the amplification of disturbances while double diffusion tends to enhance it by two means. Specifically, in regime I double diffusion can extend the duration of the lift-up mechanism and thus moderately enhance the nonmodal instability. In regime II where the transient growth of the mixed convection is the largest, double diffusion can effectively increase the working efficiency of the lift-up mechanism by feeding the perturbed vertical velocity. In regime III where the density gradient is sufficiently high, the high-frequency oscillation resulting from double diffusion can directly cause non-negligible transient growth but the duration of the lift-up mechanism is decreased.