Stabilities of Parallel Flow and Horizontal Convection

In the first part, the stability of two-dimensional parallel flow is discussed. A more restrictively general stability criterion for inviscid parallel flow is obtained analytically. First, a sufficient criterion for stability is found as either −μ1 < U ′′ U−Us < 0 or 0 < U ′′ U−Us in the flow, where Us is the velocity at the inflection point, and μ1 is the eigenvalue of Poincaré’s problem. Second, this criterion is generalized to barotropic geophysical flows in the β plane. Based on the stability criteria, the necessary condition for wave-mean flow interaction is also obtained. Then, the general stability criteria of two-dimensional inviscid rotating flow with angular velocity Ω(r) are obtained analytically. First, a necessary instability criterion for centrifugal flows is derived as ξ(Ω−Ωs) < 0 (or ξ/(Ω−Ωs) < 0 ) somewhere in the flow field, where ξ is the vortictiy of profile and Ωs is the angular velocity at the inflection point ξ ′ = 0 . Second, a criterion for stability is found as −(μ1 + 1/r2) < f(r) = ξ ′ Ω−Ωs < 0 , where μ1 is an eigenvalue. The new criteria are the analogues of the criteria for parallel flows, which are special cases of Arnol’d’s nonlinear criteria. Specifically, Pedley’s cirterion is proved to be an special case of Rayleigh’s criterion. Moreover, the criteria for parallel flows can also be derived from those for the rotating flows. The analogy between rotation and stratification in inviscid flow is also addressed. These results extend the previous theorems and would intrigue future research on the mechanism of hydrodynamic instability. Besides, the essence of shear instability is fully revealed within the linear context. The mechanism of shear instability is explored by combining the mechanisms of both the Kelvin-Helmholtz instability (K-H instability) and resonance of waves. The shear instability requires both a concentrated vortex (with speed of Us ) in the flow and resonant waves to interact with the concentrated vortex. Physically, the standing waves (with phase speed cr = Us ) can interact with the concentrated vortex, so they can trigger instability via K-H instability in the flows. While the travelling waves (with cr 6= Us ) have no interaction with the concentrated vortex, so that they can not trigger instability in the flows. The resonance of waves are totally within the linear context. In consequence, the above criteria would be helpful for understanding the wave-mean flow interaction, especially the Rossby wave-mean flow interaction in barotropic flows. According to the stable criteria, the necessary condition for wave-mean flow interaction can be obtained. And why the disturbed waves can’t take energy from the mean flow in the stable flow is revealed. If the flow is stable, there is no wave-mean flow interaction at all. This explains why the disturbed waves can’t take energy from the mean flow in the stable flow. In the second part, we report the numerical simulations of the partial-penetrating flow in horizontal convection within a squire cavity tank at high Rayleigh numbers 107 < Ra < 1010 . The partial-penetrating flow was first reported in the experiment by Wang and Huang (2005), which is though of an important material to understand ocean circulation energy budget. The fast established but slowly steadied flow is simulated, where a shallow and closed circulation cell is obtained numer3