Equatorial symmetric stability.

摘要

Equatorial symmetric stability in the Earth's middle atmosphere, modelled by the inviscid adiabatic compressible Euler equations on a beta-plane, is investigated, taking into account the previously neglected Coriolis force terms due to the component of the planetary rotation vector tangent to the surface. Using an energy-Casimir method based on the underlying Hamiltonian structure of the governing equations, two conditions that, taken together, are sufficient for linear stability are derived: the well known condition that potential vorticity be positive in the northern hemisphere and negative in the southern; and the condition that entropy must increase (decrease) in the direction of the local planetary rotation vector in the northern (southern) hemisphere. Far from the equator, the latter reduces to the familiar condition for static stability. Explicit steady solutions are found that are stable when the horizontal Coriolis force terms are neglected but unstable when they are included.; The same problem is considered using an anelastic equations model, and conditions for stability under finite amplitude perturbations are derived. It is argued that only steady solutions that are even functions of latitude can be stable in the sense of Lyapunov. States that are demonstrably Lyapunov stable are used to estimate the growth of disturbances to unstable equilibria. The short time evolution of the system away from an unstable state with linear meridional shear in the zonal velocity is explicitly calculated, and the normal mode solution exhibits features commonly associated with symmetric instability.; Finally, the Rayleigh criterion for linear stability of inviscid Taylor-Couette flow between rotating cylinders, that the magnitude of angular momentum increase with distance from the axis of rotation, is shown to be valid for finite amplitude disturbances provided the radial gradient of angular momentum is not too high.