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

Navier--Stokes equations on a rapidly rotating sphere

D. Wirosoetisno 1,

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.
Keywords: Navier--Stokes equations, global attractor., rotating sphere.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35R01, 76D0.
Citation: D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete and 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), 687-701, arXiv:1009.4538. doi: 10.3934/dcdsb.2011.16.687.

[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), 341-360. doi: 10.1007/PL00001493.

[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), 48-58, arXiv:1108.2536v1. doi: 10.1016/j.euromechflu.2012.06.001.

[4]

P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica, 30 (1988), 284-296. doi: 10.1016/0167-2789(88)90022-X.

[5]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3.

[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), 9-13.

[7]

A. A. Ilyin, Navier-Stokes equations on the rotating sphere. A simple proof of the attractor dimension estimate, Nonlinearity, 7 (1994), 31-39. doi: 10.1088/0951-7715/7/1/002.

[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), 47-76; Orig: Ross. Akad. Nauk Matem. Sbornik, 184 (1993), 55-88. doi: 10.1070/SM1994v078n01ABEH003458.

[9]

M. N. Jones, Spherical Harmonics and Tensors for Classical Field Theory, Research Studies Press, 1985.

[10]

National Institute for Standards and Technology, Digital library of mathematical functions, 2010, URL http://dlmf.nist.gov.

[11]

P. B. Rhines, Jets, Chaos, 4 (1994), 313-339.

show all references

References:
[1]

M. A. H. Al-Jaboori and D. Wirosoetisno, Navier-Stokes equations on the $\beta$-plane, Discr. Contin. Dyn. Sys. B, 16 (2011), 687-701, arXiv:1009.4538. doi: 10.3934/dcdsb.2011.16.687.

[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), 341-360. doi: 10.1007/PL00001493.

[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), 48-58, arXiv:1108.2536v1. doi: 10.1016/j.euromechflu.2012.06.001.

[4]

P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica, 30 (1988), 284-296. doi: 10.1016/0167-2789(88)90022-X.

[5]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3.

[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), 9-13.

[7]

A. A. Ilyin, Navier-Stokes equations on the rotating sphere. A simple proof of the attractor dimension estimate, Nonlinearity, 7 (1994), 31-39. doi: 10.1088/0951-7715/7/1/002.

[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), 47-76; Orig: Ross. Akad. Nauk Matem. Sbornik, 184 (1993), 55-88. doi: 10.1070/SM1994v078n01ABEH003458.

[9]

M. N. Jones, Spherical Harmonics and Tensors for Classical Field Theory, Research Studies Press, 1985.

[10]

National Institute for Standards and Technology, Digital library of mathematical functions, 2010, URL http://dlmf.nist.gov.

[11]

P. B. Rhines, Jets, Chaos, 4 (1994), 313-339.

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