Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-09T16:30:40.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows

Published online by Cambridge University Press:  21 April 2006

David G. Dritschel
David G. Dritschel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Rigorous bounds are obtained on the mean normal displacement of vorticity or potential vorticity contours from their undisturbed parallel (or concentric) positions for incompressible planar flow, flow on the surface of a sphere, and three-dimensional quasi-geostrophic flow. It is required that the basic flows have monotonic distributions of vorticity, and it is this requirement that turns a particular linear combination of conserved quantities, a combination involving the linear or angular impulse and the areas enclosed by vorticity contours, into a norm when viewed in a certain hybrid Eulerian–Lagrangian set of coordinates. Liapunov stability theorems constraining the growth of finite-amplitude disturbances then follow merely from conservation of this norm. As a corollary, it is proved that arbitrarily steep, one-signed vorticity gradients are stable, including the limiting case of a circular patch of uniform vorticity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 191 , June 1988 , pp. 575 - 581
Copyright
© 1988 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abarbanel, H. D. I., Holm, D. D., Marsden, J. E. & Ratiu, T. 1986 Nonlinear stability analysis of stratified fluid equilibria. Phil. Trans. R. Soc. Lond. A 318, 349–409.Google Scholar
Arnol'd, V. I. 1966 On an a priori estimate in the theory of hydrodynamical instability. Izv. Vyssh. Uchebn. Zaved. Matematika 54, 3–5. (English transl. Am. Math. Soc. Trans., Series 2 79, 267–269 (1969).)Google Scholar
Holliday, D. & McIntyre, M. E. 1981 On potential energy density in an incompressible, stratified fluid. J. Fluid Mech. 107, 221–225.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116.Google Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential-vorticity maps. Q. J. R. Met. Soc. 111, 877–946. Also 113, 402–404.Google Scholar
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7, 157–167.Google Scholar
McIntyre, M. E. & Shepherd, T. G. 1987 An exact local conservation theorem for finiteamplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid Mech. 181, 527–565.Google Scholar
Shepherd, T. G. 1987 Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere. J. Fluid Mech. 184, 289–302.Google Scholar
Shepherd, T. G. 1988 Nonlinear equilibration of baroclinic instability. J. Atmos. Sci. (in the Press).Google Scholar
Wan, Y. H. & Pulvirenti, M. 1985 Nonlinear stability of circular vortex patches. Commun. Math. Phys. 99, 435–450.Google Scholar