Stability of the Slow Manifold in the Primitive Equations

Abstract

Abstract. We show that, under reasonably mild hypotheses, the solution of the forced-dissipative rotating primitive equations of the ocean loses most of its fast, inertia-gravity, component in the small Rossby number limit as t → ∞. At leading order, the solution approaches what is known as "geostrophic balance" even under ageostrophic, slowly time-dependent forcing. Higherorder results can be obtained if one further assumes that the forcing is timeindependent and sufficiently smooth. If the forcing lies in some Gevrey space, the solution will be exponentially close to a finite-dimensional "slow manifold" after some time.

Investigations of non-hydrostatic, stably stratified and rapidly rotating flows

“…The −5/3 (Smith & Waleffe 2002;Marino et al 2013) and −3 (Waite & Bartello 2006) spectral slopes are evident in simulations of triply periodic Boussinesq dynamics in the regime of low Rossby and Froude numbers, and both Waite & Bartello (2006) and Whitehead & Wingate (2014) observed energy accumulating in the vortical modes. These results underscore the importance of quasigeostrophic dynamics, and demonstrate that the theorem of Temam & Wirosoetisno (2010) applies qualitatively even in this stochastically-forced regime.…”

“…Time scale separation was exploited by Embid and Majda to rigorously prove the validity of the quasigeostrophic system even in the presence of wave modes with amplitudes comparable to the vortical modes, in contrast to the asymptotic derivation which assumes that any waves have low amplitude (Embid & Majda 1996. Temam & Wirosoetisno (2010 have also proven rigorously that, under mild assumptions, the small-Rossby, small-Froude dynamics eventually approaches a quasigeostrophic balance irrespective of the amplitude of wave modes in the initial condition. The quasigeostrophic system is thus a natural touchstone for geophysical turbulence, and the qualitative properties of turbulence in the quasigeostrophic system were presciently forecast by Charney (1971) based on an analogy with previous studies of two-dimensional turbulence.…”

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,…”

The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: variations from quasi-geostrophy

“…The current investigation is of particular interest in a geophysical context when considered in light of the result of [41] that indicates that QG dominates the solution for sufficiently long times for solutions of the hydrostatic primitive equations. This relies on the fact that for this system (as in the case of the Boussinesq system considered in [14,43]) and the corresponding asymptotic reduction, the influence of the fast waves on the slow dynamics vanishes to O (1).…”