# American Institute of Mathematical Sciences

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 and 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, Durham University, work in preparation. [2] A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of 3d Euler and Navier-Stokes equations, Asymp. Anal., 15 (1997), 103-150. [3] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Math. Soc., 2002. [4] P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D, 30 (1988), 284-296. doi: 10.1016/0167-2789(88)90022-X. [5] C. R. Doering and J. D. Gibbon, "Applied Analysis of the Navier-Stokes Equations," Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511608803. [6] C. Foias, O. P. Manley, R. Temam and Y. M. Treve, Asymptotic analysis of the Navier-Stokes equations, Physica D, 9 (1983), 157-188. doi: 10.1016/0167-2789(83)90297-X. [7] I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves and convergence results, Mém. Soc. Math. France, 107 (2006), vi+116 pp. [8] P. B. Rhines, Waves and turbulence on a beta-plane, J. Fluid Mech., 69 (1975), 417-443. doi: 10.1017/S0022112075001504. [9] J. C. Robinson, "Infinite-Dimensional Dynamical Systems," Cambridge Univ. Press, 2001. [10] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Diff. Eq., 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157. [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), 289-302. doi: 10.1017/S0022112087002891. [12] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," SIAM, 2 ed., 1995. [13] --, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, 2 ed., 1997. [14] R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations, SIAM J. Math. Anal., 42 (2010), 427-458. arXiv:0808.2878. doi: 10.1137/080733358. [15] G. K. Vallis, "Atmospheric and Oceanic Fluid Dynamics," Cambridge Univ. Press, 2006. doi: 10.1017/CBO9780511790447. [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), 321-342.

show all references

##### References:
 [1] M. A. H. Al-Jaboori, Ph.D. thesis, Durham University, work in preparation. [2] A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of 3d Euler and Navier-Stokes equations, Asymp. Anal., 15 (1997), 103-150. [3] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Math. Soc., 2002. [4] P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D, 30 (1988), 284-296. doi: 10.1016/0167-2789(88)90022-X. [5] C. R. Doering and J. D. Gibbon, "Applied Analysis of the Navier-Stokes Equations," Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511608803. [6] C. Foias, O. P. Manley, R. Temam and Y. M. Treve, Asymptotic analysis of the Navier-Stokes equations, Physica D, 9 (1983), 157-188. doi: 10.1016/0167-2789(83)90297-X. [7] I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves and convergence results, Mém. Soc. Math. France, 107 (2006), vi+116 pp. [8] P. B. Rhines, Waves and turbulence on a beta-plane, J. Fluid Mech., 69 (1975), 417-443. doi: 10.1017/S0022112075001504. [9] J. C. Robinson, "Infinite-Dimensional Dynamical Systems," Cambridge Univ. Press, 2001. [10] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Diff. Eq., 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157. [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), 289-302. doi: 10.1017/S0022112087002891. [12] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," SIAM, 2 ed., 1995. [13] --, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, 2 ed., 1997. [14] R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations, SIAM J. Math. Anal., 42 (2010), 427-458. arXiv:0808.2878. doi: 10.1137/080733358. [15] G. K. Vallis, "Atmospheric and Oceanic Fluid Dynamics," Cambridge Univ. Press, 2006. doi: 10.1017/CBO9780511790447. [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), 321-342.
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