Slow manifolds and invariant sets of the primitive equations

Wirosoetisno, D.; Temam, R.

Abstract

The authors review, in a geophysical setting, several recent mathematical results on the forced–dissipative hydrostatic primitive equations with a linear equation of state in the limit of strong rotation and stratification, starting with existence and regularity (smoothness) results and describing their implications for the long-time behavior of the solution. These results are used to show how the solution of the primitive equations in a periodic box comes close to geostrophic balance as t → ∞. Then a review follows of how geostrophic balance could be extended to higher orders in the Rossby number, and it is shown that the solution of the primitive equations also satisfies a higher-order balance up to an exponentially small error. Finally, the connection between balance dynamics in the primitive equations and its global attractor, which is the only known invariant set (for a sufficiently general forcing), is discussed.

Citation

Wirosoetisno, D., & Temam, R. (2011). Slow manifolds and invariant sets of the primitive equations. Journal of the Atmospheric Sciences, 68(3), 675-682. https://doi.org/10.1175/2010jas3650.1

• Journal Article Type: Article
• Acceptance Date: Nov 2, 2010
• Online Publication Date: Mar 1, 2011
• Publication Date: Mar 1, 2011
• Deposit Date: Feb 17, 2011
• Publicly Available Date: Feb 17, 2017
• Journal: Journal of the Atmospheric Sciences
• Print ISSN: 0022-4928
• Electronic ISSN: 1520-0469
• Publisher: American Meteorological Society
• Peer Reviewed: Peer Reviewed
• Volume: 68
• Issue: 3
• Pages: 675-682
• DOI: https://doi.org/10.1175/2010jas3650.1
• Public URL: https://durham-repository.worktribe.com/output/1511071