On nonlinear symmetric stability and the nonlinear saturation of symmetric instability

Mu, M., Shepherd, T. G. ORCID: https://orcid.org/0000-0002-6631-9968 and Swanson, K. (1996) On nonlinear symmetric stability and the nonlinear saturation of symmetric instability. Journal of the Atmospheric Sciences, 53 (20). pp. 2918-2923. ISSN 0022-4928 doi: 10.1175/1520-0469(1996)053<2918:ONSSAT>2.0.CO;2

Abstract/Summary

A nonlinear symmetric stability theorem is derived in the context of the f-plane Boussinesq equations, recovering an earlier result of Xu within a more general framework. The theorem applies to symmetric disturbances to a baroclinic basic flow, the disturbances having arbitrary structure and magnitude. The criteria for nonlinear stability are virtually identical to those for linear stability. As in Xu, the nonlinear stability theorem can be used to obtain rigorous upper bounds on the saturation amplitude of symmetric instabilities. In a simple example, the bounds are found to compare favorably with heuristic parcel-based estimates in both the hydrostatic and non-hydrostatic limits.