Energy–Casimir and Energy–Lagrange Methods in the Study of Nonlinear Symmetric Stability Problems

Abstract

Abstract This paper is concerned with nonlinear symmetric stability problems. For the moist, adiabatic (saturated) system, the authors utilize the ECM (energy–Casimir method) to establish nonlinear stability criteria, which extends the previous work from the dry atmosphere to the moist case and demonstrates the complexity related to the moist symmetric instability problem. For the nonhydrostatic, Boussinesq equations on an f plane with the northward component of the earth rotation f = 2Ω cosϕ, which has been utilized to show the importance of f term in the mesoscale linear symmetric instability problem, both ECM and the ELM (energy–Lagrange method) are employed to study the “zonal” and“meridional” nonlinear symmetric stability problems. In both cases, the nonlinear stability of the basic states are obtained if the potential vorticity and the vertical component of absolute vorticity of the basic state are positive (for f > 0). In the zonal case, the potential vorticity depends upon f explicitly, and this s...