Abstract
The quasigeostrophic model describes large scale and relatively slow fluid motion in geophysical flows. We investigate the quasigeostrophic model under random forcing and random boundary conditions. We first transform the model into a partial differential equation with random coefficients. Then we show that, under suitable conditions on the random forcing, random boundary conditions, viscosity, Ekman constant and Coriolis parameter, all quasigeostrophic motion approach a unique stationary state exponentially fast. This stationary state corresponds to a unique invariant Dirac measure.
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References
L. Arnold, Random Dynamical Systems. Springer-Verlag, Berlin, 1998.
A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes. J. Funct. Anal., 13 (1973), 195–222.
A. J. Bourgeois and J.T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean. SIAM J. Math. Anal., 25 (1994), 1023–1068.
J.R. Brannan, J. Duan and T. Wanner, Dissipative quasigeostrophic dynamics under random forcing. J. Math. Anal. Appl., 228 (1998), 221–233.
H. Crauel and F. Flandoli, Attractors for random dynamical systems. Prob. Th. Rel. Fields, 100 (1994), 365–393.
B. Desjardins and E. Grenier, Derivation of quasigeostrophic potential vorticity equations, to appear in Adv. Diff. Eqns., 1998.
P.F. Embid and A.J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Comm. PDEs, 21 (1996), 619–658.
R.Z. Hasńinskiĭ, Stochastic Stability of Differential Equations. Sijthoff & Nordhoff, Alphen aan den Rijn, 1980.
D.D. Holm, Hamiltonian formulation of the baroclinic quasigeostrophic fluid equations. Phys. Fluids, 29 (1986), 7–8.
P. Imkeller and B. Schmalfuss, The conjugacy of stochastic and random differential equations and the existence of global attractors. Submitted, 1999.
H. Keller and B. Schmalfuss, Attractors for stochastic hyperbolic equations via transformation into random equations. Institut für Dynamische Systeme, Universität Bremen, Report, 448 (1999).
P. Müller, Stochastic forcing of quasi-geostrophic eddies. In P. Müller R.J. Adler and B. Rozovskii, editors, Stochastic Modelling in Physical Oceanography, pages 381-396. Birkhäuser, Basel, 1996.
J. Pedlosky, Geophysical Fluid Dynamics. Springer-Verlag, Berlin, 1987.
J. Pedlosky, Ocean Circulation Theory. Springer-Verlag, Berlin, 1996.
G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge, 1996.
B. Schmalfuss, Qualitative properties of the stochastic Navier-Stokes equation. Nonlinear Analysis TMA, 28 (1997), 1545–1563.
B. Schmalfuss, A random fixed point theorem and the random graph transformation. J. Math. Anal Applns., 225(1) (1998), 91–113.
S.H. Schochet, Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation. J. Diff. Eqns., 68 (1987), 400–428.
T.G. Shepherd, T. Warn, O. Bokhove and G.K. Vallis, Rossby number expansions, slaving principle and balance dynamics. Quart. J. Roy. Met. Soc., 121 (1995), 723–739.
R. Temam, Navier-Stokes Equation-Theory and Numerical Analysis. North-Holland, Amsterdam, 1979.
G.K. Vallis, Potential vorticity inversion and balanced equations of motion for rotating and stratified flows. Quart. J. Roy. Met. Soc., 122 (1996), 291–322.
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Duan, J., Kloeden, P.E., Schmalfuss, B. (2001). Exponential stability of the quasigeostrophic equation under random perturbations. In: Imkeller, P., von Storch, JS. (eds) Stochastic Climate Models. Progress in Probability, vol 49. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8287-3_10
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DOI: https://doi.org/10.1007/978-3-0348-8287-3_10
Publisher Name: Birkhäuser, Basel
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