Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-09T00:27:40.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave energy-momentum and pseudoenergy-momentum conservation for the layered quasi-geostrophic instability problem

Published online by Cambridge University Press:  26 April 2006

P. Ripa
P. Ripa
Affiliation:
Centro de Investigación Cientifica y de Educación Superior de Ensenada, 22800 Ensenada, Baja California, México
Get access Rights & Permissions [Opens in a new window]

Abstract

Evolution equations and conservation laws are derived for a quite general layered quasi-geostrophic model: with arbitrary thickness and stratification structure and with either a free or a rigid (including the possibility of topography) boundary condition, at the top and bottom. The system is shown to be Hamiltonian, and Arnol'd stability conditions are derived, in the sense of both the first and second theorem, i.e. for pseudowestward and pseudoeastward basic flows, respectively, and for arbitrary perturbations of potential vorticity and Kelvin circulations.

Two examples of parallel basic flow in a channel are analysed: the sine profile in the so-called equivalent barotropic model (one layer with a free boundary) and Phillips’ problem (uniform flow in each of two layers with rigid boundaries). Using the second theorem with the optimum combination of pseudoenergy and pseudomomentum it is shown that, in both cases, the basic state is nonlinearly stable if the channel width L is small enough, namely, ΛL < π and $2(f_0L/\pi)^2 < g^{\prime}(H_1H_2)^{\frac{1}{2}} $, respectively. (In the first problem, Λ is the wavenumber of the sine profile; in the second one, g′ is the reduced gravity, H1 and H2 are the layer thicknesses, and f0 is the Coriolis parameter). The stability condition of either problem is found to be also a necessary one: as soon as it is violated a grave mode becomes unstable. It is shown explicitly that the second variation of the pseudoenergy and pseudomomentum of a growing (decaying) normal mode is identically zero, defining the direction of the unstable (stable) manifold.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 235 , February 1992 , pp. 379 - 398
Copyright
© 1992 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

Andrews, D. G. 1984 On the existence of nonzonal flows satisfying sufficient conditions for stability. Geophys. Astrophys. Fluid Dyn. 28, 243–256.Google Scholar
Arnol'd, V. I.1965 Condition for nonlinear stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSR 162, 975–978; (English transl. Sov. Maths 6, 773–777, 1965).Google Scholar
Arnol'd, V. I.1966 On an apriori estimate in the theory of hydrodynamical stability. Izv. Vyssh. Uchebn. Zaved. Matematika 54, 3–5; (English transl. Am. Math. Soc. Transl. Ser. 2, 79, 267–269, 1969.Google Scholar
Benzi, R., Pierini, S., Vulpiani, A. & Salusti, E. 1982 On nonlinear hydrodynamic stability of planetary vortices. Geophys. Astrophys. Fluid Dyn. 20, 293–306.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Gill, A. E., Green, J. S. A. & Simmons, A. J. 1974 Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep Sea Res. 21, 499–528.Google Scholar
Held, I. M. 1985 Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci. 42, 2280–2288.Google Scholar
Henrotay, P. 1983 Nonlinear baroclinic instability: An approach based on Serrin's energy method. J. Atmos. Sci. 40, 762–768.Google Scholar
Holm, D. D. 1986 Hamiltonian formulation of the baroclinic quasigeostrophic fluid equations. Phys. Fluids 29, 7–8.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinsten, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116.Google Scholar
Lipps, F. B. 1963 Stability of jets in a divergent barotropic fluid. J. Atmos. Sci. 20, 120–129.Google Scholar
McIntyre, M. E. & Shepherd, T. G. 1987 An exact local conservation theorem for finite-amplitude 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
Pedlosky, J. 1964 The stability of currents in the Atmosphere and the Ocean: Part I. J. Atmos. Sci 21, 201–219.Google Scholar
Pedlosky, J. 1979 Geophysical fluid dynamics. Springer. 624 pp.
Phillips, N. A. 1951 A simpel three-dimensional model for the study of large-scale extratropical flow patterns. J. Atmos. Sci. 8, 381–394.Google Scholar
Phillips, N. A. 1954 Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6, 273–286.Google Scholar
Ripa, P. 1981a On the theory of nonlinear wave-wave interactions among geophysical waves. J. Fluid Mech. 103, 87–115.Google Scholar
Ripa, P. 1981b Symmetries and conservation laws for internal gravity waves. AIP Proc. 76, 281–306.Google Scholar
Ripa, P. 1986 Evaluation of vertical structure functions for the analysis of oceanic data. J. Phys. Ocean. 16, 223–232.Google Scholar
Ripa, P. 1990 Positive, negative and zero wave energy and the flow stability problem, in the Eulerian and Lagrangian-Eulerian descriptions. Pure Appl. Geophys. 133, 713–732.Google Scholar
Ripa, P. 1991a General stability conditions for a multi-layer model. J. Fluid Mech. 222, 119–137.Google Scholar
Ripa, P. 1991b A tale of three theorems. Rev. Mex. FiAs. submitted.Google Scholar
Sakai, S. 1989 Rossby-Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176.Google Scholar
Shepherd, T. G. 1989 Nonlinear saturation of baroclinic instability. Part II: continuously stratified fluid. J. Atmos. Sci. 46, 888–907.Google Scholar
Shepherd, T. G. 1990 Symmetries, conservation laws, and hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32,Google Scholar
Swaters, G. E. 1986 A nonlinear stability theorem for baroclinic quasigeostrophic flow. Phys. Fluids 29, 5–6.Google Scholar
White, A. A. 1982 Zonal translation properties of two quasi-geostrophic systems of equations. J. Atmos. Sci. 39, 2107–2118.Google Scholar