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

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 f̄ = 2Ω cosφ, which has been utilized to show the importance of f̄ 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 f̄ explicitly, and this shows the influence of the f̄ term to the nonlinear symmetric stability. In the meridional case the potential vorticity is independent of the f̄ term, which implies that the f̄ term plays no role in the nonlinear symmetric stability. The upper bounds on the disturbance field to the nonlinearly stable basic state are established, which consists of its initial value multiplied by an amplification factor independent of time. The ELM proposed by Xu is simplifed to be more concise and understandable. The applicable capacity of ECM and ELM is investigated. Both methods are applicable to the symmetric nonlinear stability problem of dry atmosphere, but only ECM has application to the moist problem.