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Nonlinear stability of multilayer quasi-geostrophic flow

Published online by Cambridge University Press:  26 April 2006

Mu Mu ,
Zeng Qingcun ,
Theodore G. Shepherd  and
Liu Yongming
Mu Mu
Affiliation:
LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100080, China
Zeng Qingcun
Affiliation:
LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100080, China
Theodore G. Shepherd
Affiliation:
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
Liu Yongming
Affiliation:
Institute of Mathematics, Anhui University, Heifei 230039, China
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Abstract

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 264 , 10 April 1994 , pp. 165 - 184
Copyright
© 1994 Cambridge University Press

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