Slow Manifolds and Invariant Sets of the Primitive Equations

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. ABSTRACTThe 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.

Balance and Spontaneous Wave Generation in Geophysical Flows

“…This method has been applied for simple geophysical models by Cotter (2004), Cotter & Reich (2006) and Wirosoetisno (2004). More recently, Temam & Wirosoetisno (2007, 2011 considered the hydrostatic PEs including viscous terms and obtained a bound of the form exp(−α/ 1/3 ) using a method adapted to dissipative PDEs (Matthies 2001). The bound in this case relies on dissipation and increases rapidly with the Reynolds number; this limits its direct relevance to geophysical systems, where the Rossby number is much larger than the inverse Reynolds number.…”

Spontaneous inertia-gravity-wave generation by surface-intensified turbulence

“…By balanced ageostrophic, we mean the corrections to geostrophic balance that are obtained using higher-order balance relations which slave all the fields to a single one, say the potential vorticity. As mentioned, these relations can be obtained systematically using asymptotic expansions in powers of the Rossby number (Warn et al 1995); since they diverge, the expansions must be truncated; conceptually (if not practically), it is useful to think of the truncation as being an optimal one, near the smallest term (Bender & Orszag 1999), so that the remainder is dominated by unbalanced motion (Vanneste 2008b,a;Temam & Wirosoetisno 2011).…”

Approximation of Stochastic Invariant Manifolds

“…The theory of invariant manifolds for deterministic dynamical systems has been an active research field for a long time, and is now a very well-developed theory; see, e.g., [6,7,8,9,10,30,43,44,52,62,74,79,80,88,89,92,94,105,117,118,119,121,132,143,147,148,149,150,152,153,154]. Over the past two decades, several important results on random invariant manifolds for stochastically perturbed ordinary as well as partial differential equations (PDEs) have been obtained; these results often extend those found in the deterministic setting; see, e.g., [1,2,3,12,20,22,25,26,29,51,57,65,66,113,123,124,…”