October  2011, 16(3): 687-701. doi: 10.3934/dcdsb.2011.16.687

Navier--Stokes equations on the $\beta$-plane

1. 

Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom

Received  September 2010 Revised  March 2011 Published  June 2011

We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $\beta$-plane (i.e. with the Coriolis force varying as $f_0+\beta y$) will become nearly zonal: with the vorticity $\omega(x,y,t)=\bar\omega(y,t)+\tilde\omega(x,y,t),$ one has $|\tilde\omega|_{H^s}^2 \le \beta^{-1} M_s(\cdots)$ as $t\to\infty$. We use this show that, for sufficiently large $\beta$, the global attractor of this system reduces to a point.
Citation: Mustafa A. H. Al-Jaboori, D. Wirosoetisno. Navier--Stokes equations on the $\beta$-plane. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 687-701. doi: 10.3934/dcdsb.2011.16.687
References:
[1]

M. A. H. Al-Jaboori, Ph.D. thesis,, Ph.D. thesis, ().   Google Scholar

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V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", Attractors for Equations of Mathematical Physics, (2002).   Google Scholar

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I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves and convergence results,, Mém. Soc. Math. France, 107 (2006).   Google Scholar

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J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Infinite-Dimensional Dynamical Systems, (2001).   Google Scholar

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S. Schochet, Fast singular limits of hyperbolic PDEs,, J. Diff. Eq., 114 (1994), 476.  doi: 10.1006/jdeq.1994.1157.  Google Scholar

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T. G. Shepherd, Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere,, J. Fluid Mech., 184 (1987), 289.  doi: 10.1017/S0022112087002891.  Google Scholar

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R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Navier-Stokes Equations and Nonlinear Functional Analysis, (1995).   Google Scholar

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R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations,, SIAM J. Math. Anal., 42 (2010), 427.  doi: 10.1137/080733358.  Google Scholar

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G. K. Vallis, "Atmospheric and Oceanic Fluid Dynamics,", Atmospheric and Oceanic Fluid Dynamics, (2006).  doi: 10.1017/CBO9780511790447.  Google Scholar

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G. K. Vallis and M. E. Maltrud, Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence,, J. Fluid Mech., 228 (1991), 321.   Google Scholar

show all references

References:
[1]

M. A. H. Al-Jaboori, Ph.D. thesis,, Ph.D. thesis, ().   Google Scholar

[2]

A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of 3d Euler and Navier-Stokes equations,, Asymp. Anal., 15 (1997), 103.   Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", Attractors for Equations of Mathematical Physics, (2002).   Google Scholar

[4]

P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence,, Physica D, 30 (1988), 284.  doi: 10.1016/0167-2789(88)90022-X.  Google Scholar

[5]

C. R. Doering and J. D. Gibbon, "Applied Analysis of the Navier-Stokes Equations,", Applied Analysis of the Navier-Stokes Equations, (1995).  doi: 10.1017/CBO9780511608803.  Google Scholar

[6]

C. Foias, O. P. Manley, R. Temam and Y. M. Treve, Asymptotic analysis of the Navier-Stokes equations,, Physica D, 9 (1983), 157.  doi: 10.1016/0167-2789(83)90297-X.  Google Scholar

[7]

I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves and convergence results,, Mém. Soc. Math. France, 107 (2006).   Google Scholar

[8]

P. B. Rhines, Waves and turbulence on a beta-plane,, J. Fluid Mech., 69 (1975), 417.  doi: 10.1017/S0022112075001504.  Google Scholar

[9]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Infinite-Dimensional Dynamical Systems, (2001).   Google Scholar

[10]

S. Schochet, Fast singular limits of hyperbolic PDEs,, J. Diff. Eq., 114 (1994), 476.  doi: 10.1006/jdeq.1994.1157.  Google Scholar

[11]

T. G. Shepherd, Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere,, J. Fluid Mech., 184 (1987), 289.  doi: 10.1017/S0022112087002891.  Google Scholar

[12]

R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Navier-Stokes Equations and Nonlinear Functional Analysis, (1995).   Google Scholar

[13]

--, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Infinite-Dimensional Dynamical Systems in Mechanics and Physics, (1997).   Google Scholar

[14]

R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations,, SIAM J. Math. Anal., 42 (2010), 427.  doi: 10.1137/080733358.  Google Scholar

[15]

G. K. Vallis, "Atmospheric and Oceanic Fluid Dynamics,", Atmospheric and Oceanic Fluid Dynamics, (2006).  doi: 10.1017/CBO9780511790447.  Google Scholar

[16]

G. K. Vallis and M. E. Maltrud, Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence,, J. Fluid Mech., 228 (1991), 321.   Google Scholar

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