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Viscous motion in an oceanic circulation model

Published online by Cambridge University Press:  17 April 2009

A. F. Bennett  and
P. E. Kloeden
A. F. Bennett
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
P. E. Kloeden
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia 6150, Australia.
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Abstract

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The barotropic motion of a viscous fluid in a laboratory simulation of ocean circulation may be modelled by Beards ley's vorticity equations. It is established here that these equations have unique smooth solutions which depend continuously on initial conditions. To avoid a boundary condition which involves an integral operator, the vorticity equations are replaced by an equivalent system of momentum equations. The system resembles the two-dimensional incompressible Navier-Stokes equations in a rotating reference frame. The existence of unique generalized solutions of the system in a square domain is established by modifying arguments used by Ladyzhenskaya for the Navier-Stokes equations. Smoothness of the solutions is then established by modifying Golovkin's arguments, again originally for the Navier- Stokes equations. A numerical procedure for solving the vorticity equations is discussed, as are the effects of reentrant corners in the domain modelling islands and peninsulae.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 3 , June 1981 , pp. 443 - 460
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Adams, Robert A., Sobolev spaces (Pure and Applied Mathematics, 65. Academic Press [Harcourt Brace Jovanovich], New York, San Francisco, London, 1975).Google Scholar
[2]Arakawa, Akio, “Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I”, J. Comput. Phys. 1 (1966–1967), 119–143.CrossRefGoogle Scholar
[3]Beardsley, R.C., “A numerical model of the wind-driven ocean circulation in a circular basin”, Geophys. Fluid Dyn. 4 (1972–1973), 211–241.CrossRefGoogle Scholar
[4]Beardsley, Robert C., “The ‘sliced-cylinder’ laboratory model of the wind-driven ocean circulation. Part 2: Oscillatory forcing and Rossby wave resonance”, J. Fluid Mech. 69 (1975), 41–64.CrossRefGoogle Scholar
[5]Beardsley, R.C. and Robbins, K., “The ‘sliced-cylinder’ laboratory model of the wind-driven ocean circulation. Part 1: Steady forcing and topographic Rossby wave instability”, J. Fluid Mech. 69 (1975), 27–4O.CrossRefGoogle Scholar
[6] К.К. Головкин [Golovkin, K.K.] “О плоском движении вязкой несжимаемой жидкости” [The plane motion of a viscous incompressible fluid], Trudy Mat. Inst. Steklov 59 (1960), 37–86. English Transl: Amer. Math. Soc. Transl. (2) 35 (1964), 297–350.Google Scholar
[7]Hardy, G.H., Littlewood, J.E., Pólya, G., Inequalities (Cambridge University Press, Cambridge, 1934).Google Scholar
[8]Holland, William R., “The role of mesoscale eddies in the general circulation of the ocean - numerical experiments using a wind-driven quasi-geostrophic model”, J. Phys. Oceanogr. 8 (1978), 363–392.2.0.CO;2>CrossRefGoogle Scholar
[9]Holland, William R. and Bennett, A.F., “Numerical simulation of the long-term motion in a laboratory oceanic circulation experiment”, in preparation.Google Scholar
[10] А.М. Ильин, А.С. Калашников, О.А. Олейник [II'in, A.M., Kalashnikov, A.S., Oleinik, O.A.], “Линейные уравнения второго порядка параболического типа” [Lineaŕ equations of the second order of parabolic type], Uspehi Mat. Nauk 17 (1962), no. 3, 3–146; English Transl: Russian Math. Surveys 17 (1962), no. 3, 1–143.Google Scholar
[11]Ladyzhenskaya, O.A., The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged (Translated by Silverman, Richard A. and Chu, John. Mathematics and its Applications, 2. Gordon and Breach, New York, London, Paris, 1969).Google Scholar
[12]Leray, Jean, “Essai sur les mouvements plans d'un liquide visqueux que limitent des parois”, J. Math. Pures Appl. (9) 13 (1934), 331–418.Google Scholar
[13]Pedlosky, Joseph, Geophysical fluid dynamics (Springer-Verlag, New York, Heidelberg, Berlin, 1979).CrossRefGoogle Scholar
[14]Yosida, Kôsaku, Functional analysis 3rd edition (Die Grundlehren der mathematischen Wissenschaften, 123. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar