Time-dependent rotating stratified shear flow: exact solution and stability analysis.

A solution of the Euler equations with Boussinesq approximation is derived by considering unbounded flows subjected to spatially uniform density stratification and shear rate that are time dependent [S(t)= partial differentialU3/partial differentialx2]. In addition to vertical stratification with constant strength N(v)2, this base flow includes an additional, horizontal, density gradient characterized by N(h)2(t). The stability of this flow is then analyzed: When the vertical stratification is stabilizing, there is a simple harmonic motion of the horizontal stratification N(h)2(t) and of the shear rate S(t), but this flow is unstable to certain disturbances, which are amplified by a Floquet mechanism. This analysis may involve an additional Coriolis effect with Coriolis parameter f, so that governing dimensionless parameters are a modified Richardson number, R=[S(0)2+N(h)4(0)/N(v)2]1/2, and f(v)=f/N(v), as well as the initial phase of the periodic shear rate. Parametric resonance between the inertia-gravity waves and the oscillating shear is demonstrated from the dispersion relation in the limit R-->0. The parametric instability has connection with both baroclinic and elliptical flow instabilities, but can develop from a very different base flow.