Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-13T23:40:30.103Z Has data issue: false hasContentIssue false
  • English
  • Français

The simplified quasigeostrophic equations: Existence and uniqueness of strong solutions

Published online by Cambridge University Press:  26 February 2010

A. F. Bennett  and
P. E. Kloeden
A. F. Bennett
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria, Australia, 3168.
P. E. Kloeden
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia, Australia, 6155.
Get access

Extract

The quasigeostrophic equations describe large scale motion in the atmosphere and oceans at middle latitudes. Being considerably simpler than the primitive equations, they have been widely used for modelling atmospheric and oceanic circulation, and for studies of stability, frontogenesis and turbulence. A number of assertions have been made about these equations; first, that finite element approximate solutions converge in an open flow region [7]; secondly, that solutions depend discontinuously or nondeterministically on initial data [13]; and thirdly, that the energy wave number spectra of the solutions asymptote to the statistical equilibrium spectra of the spectrally truncated equations [12].

Type
Research Article
Information
Mathematika , Volume 27 , Issue 2 , December 1980 , pp. 287 - 311
Copyright
Copyright © University College London 1980

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

1.Adams, R. A.. Sobolev spaces (Academic Press, New York, 1975).Google Scholar
2.Bennett, A. F. and Kloeden, P. E.. “The periodic quasigeostrophic equations: existence and uniqueness of strong solutions”. (Submitted for publication.)Google Scholar
3.Bennett, A. F. and Kloeden, P. E.. “The quasigeostrophic equations: approximation, predictability and equilibrium spectra of solutions”, Quart. J. R. Met. Soc, (to appear).Google Scholar
4.Charney, J. G.. “Geostrophic turbulence”, J. Atmos. Sci., 28 (1971), 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
5.Courant, R. and Hilbert, D.. Methods of mathematical physics, vo ll (Interscience, New York, 1962).Google Scholar
6.Dutton, J. A.. “The nonlinear quasigeostrophic equations: Existence and uniqueness of solutions on a bounded domain”, J. Atmos. Sci., 31 (1974), 422–33.2.0.CO;2>CrossRefGoogle Scholar
7.Fix, G. J.. “Finite element models for ocean circulation problems,” SI AM J. Appl. Math., 29 (1975), 371–387.CrossRefGoogle Scholar
8.Hartman, P.. Ordinary differential equations (Wiley, New York, 1964).Google Scholar
9.Judovich, V. I.. “Some bounds for the solutions of elliptic equations”, Matem. Sbornik, 59 (1962), (101) supplement, 229–244. (English translation in Tranl. Amer. Math. Soc., 57 (1966), 277–304.)Google Scholar
10.Judovich, V. I.. “A two-dimensional problem of steady flow of an ideal incompressible fluid across a given domain”, Matem. Sbornik, 64 (1964), (106), 562–588. (English translation in Tranl. Amer. Math. Soc., 57 (1966) 277–304.)Google Scholar
11.Kato, T.. “On classical solutions of the two-dimensional non-stationary Euler equation”, Arch. Rat. Mech. Anal., 25 (1967), 188–200.CrossRefGoogle Scholar
12.Kraichnan, R. H.. “Inertial ranges in two-dimensional turbulence”, Phys. Fluids, 10 (1967), 1417–1421.CrossRefGoogle Scholar
13.Lorenz, E. N.. “The predictability of a flow which possesses many scales of motion”, Tellus, 21 (1969), 289–307.CrossRefGoogle Scholar
14.McWilliams, J. C.. “A note on a consistent quasigeostrophic model in a multiply connected domain”, Dyn. Atmos. Oceans., 1 (1977), 427–441.CrossRefGoogle Scholar
15.Miranda, C.. “Sul problema misto per le equazioni lineari ellittiche”, Ann. Mat. pura. appl, 39 (1955), 279–303.CrossRefGoogle Scholar
16.Miranda, C.. Partial differential equation of elliptic type, 2nd revised edition (Springer-Verlag, Berlin, 1970).Google Scholar
17.Pedlosky, J.. “The stability of currents in the atmosphere and ocean: Part I”, J. Atmos. Sci., 21 (1964), 201–219.2.0.CO;2>CrossRefGoogle Scholar