Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere

Cho, H.-R., Shepherd, T. G. ORCID: https://orcid.org/0000-0002-6631-9968 and Vladimirov, V. A. (1993) Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere. Journal of the Atmospheric Sciences, 50 (6). pp. 822-836. ISSN 0022-4928 doi: 10.1175/1520-0469(1993)050<0822:AOTDLM>2.0.CO;2

Abstract/Summary

The problem of symmetric stability is examined within the context of the direct Liapunov method. The sufficient conditions for stability derived by Fjørtoft are shown to imply finite-amplitude, normed stability. This finite-amplitude stability theorem is then used to obtain rigorous upper bounds on the saturation amplitude of disturbances to symmetrically unstable flows.By employing a virial functional, the necessary conditions for instability implied by the stability theorem are shown to be in fact sufficient for instability. The results of Ooyama are improved upon insofar as a tight two-sided (upper and lower) estimate is obtained of the growth rate of (modal or nonmodal) symmetric instabilities.The case of moist adiabatic systems is also considered.