# American Institute of Mathematical Sciences

June  2015, 20(4): 1251-1259. doi: 10.3934/dcdsb.2015.20.1251

## Navier--Stokes equations on a rapidly rotating sphere

 1 Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom

Received  March 2014 Revised  October 2014 Published  February 2015

We extend our earlier $\beta$-plane results [al-Jaboori and Wirosoetisno, 2011, DCDS-B 16:687--701] to a rotating sphere. Specifically, we show that the solution of the Navier--Stokes equations on a sphere rotating with angular velocity $1/\epsilon$ becomes zonal in the long time limit, in the sense that the non-zonal component of the energy becomes bounded by $\epsilon M$. Central to our proof is controlling the behaviour of the nonlinear term near resonances. We also show that the global attractor reduces to a single stable steady state when the rotation is fast enough.
Citation: D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251
##### References:
 [1] M. A. H. Al-Jaboori and D. Wirosoetisno, Navier-Stokes equations on the $\beta$-plane,, Discr. Contin. Dyn. Sys. B, 16 (2011), 687. doi: 10.3934/dcdsb.2011.16.687. Google Scholar [2] C. Cao, M. A. Rammaha and E. S. Titi, The Navier-Stokes equations on the rotating 2-$d$ sphere: Gevrey regularity and asymptotic degrees of freedom,, ZAMP, 50 (1999), 341. doi: 10.1007/PL00001493. Google Scholar [3] B. Cheng and A. Mahalov, Euler equations on a fast rotating sphere - time-averages and zonal flows,, Eur. J. Mech. B/Fluids, 37 (2013), 48. doi: 10.1016/j.euromechflu.2012.06.001. Google Scholar [4] P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence,, Physica, 30 (1988), 284. doi: 10.1016/0167-2789(88)90022-X. Google Scholar [5] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [6] A. A. Il'in and A. N. Filatov, On unique solvability of the Navier-Stokes equations on the two-dimensional sphere,, Soviet Math. Dokl., 38 (1989), 9. Google Scholar [7] A. A. Ilyin, Navier-Stokes equations on the rotating sphere. A simple proof of the attractor dimension estimate,, Nonlinearity, 7 (1994), 31. doi: 10.1088/0951-7715/7/1/002. Google Scholar [8] A. A. Il'yin, Partly dissipative semigroups generated by the Navier-Stokes system on two-dimensional manifolds, and their attractors,, Russian Acad. Sci. Sb. Math., 78 (1994), 47. doi: 10.1070/SM1994v078n01ABEH003458. Google Scholar [9] M. N. Jones, Spherical Harmonics and Tensors for Classical Field Theory,, Research Studies Press, (1985). Google Scholar [10] National Institute for Standards and Technology, Digital library of mathematical functions, 2010,, URL , (). Google Scholar [11] P. B. Rhines, Jets,, Chaos, 4 (1994), 313. Google Scholar

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##### References:
 [1] M. A. H. Al-Jaboori and D. Wirosoetisno, Navier-Stokes equations on the $\beta$-plane,, Discr. Contin. Dyn. Sys. B, 16 (2011), 687. doi: 10.3934/dcdsb.2011.16.687. Google Scholar [2] C. Cao, M. A. Rammaha and E. S. Titi, The Navier-Stokes equations on the rotating 2-$d$ sphere: Gevrey regularity and asymptotic degrees of freedom,, ZAMP, 50 (1999), 341. doi: 10.1007/PL00001493. Google Scholar [3] B. Cheng and A. Mahalov, Euler equations on a fast rotating sphere - time-averages and zonal flows,, Eur. J. Mech. B/Fluids, 37 (2013), 48. doi: 10.1016/j.euromechflu.2012.06.001. Google Scholar [4] P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence,, Physica, 30 (1988), 284. doi: 10.1016/0167-2789(88)90022-X. Google Scholar [5] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [6] A. A. Il'in and A. N. Filatov, On unique solvability of the Navier-Stokes equations on the two-dimensional sphere,, Soviet Math. Dokl., 38 (1989), 9. Google Scholar [7] A. A. Ilyin, Navier-Stokes equations on the rotating sphere. A simple proof of the attractor dimension estimate,, Nonlinearity, 7 (1994), 31. doi: 10.1088/0951-7715/7/1/002. Google Scholar [8] A. A. Il'yin, Partly dissipative semigroups generated by the Navier-Stokes system on two-dimensional manifolds, and their attractors,, Russian Acad. Sci. Sb. Math., 78 (1994), 47. doi: 10.1070/SM1994v078n01ABEH003458. Google Scholar [9] M. N. Jones, Spherical Harmonics and Tensors for Classical Field Theory,, Research Studies Press, (1985). Google Scholar [10] National Institute for Standards and Technology, Digital library of mathematical functions, 2010,, URL , (). Google Scholar [11] P. B. Rhines, Jets,, Chaos, 4 (1994), 313. Google Scholar
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