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Slow Manifolds and Invariant Sets of the Primitive Equations

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  • 1 Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, Indiana
  • | 2 Mathematical Sciences, Durham University, Durham, United Kingdom
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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.

Corresponding author address: D. Wirosoetisno, Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom. Email: djoko.wirosoetisno@durham.ac.uk

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.

Corresponding author address: D. Wirosoetisno, Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom. Email: djoko.wirosoetisno@durham.ac.uk

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