Free Gravity Waves and Balanced Dynamics

Abstract

It is shown how a renormalization technique, which is a variant of classical Krylov-Bogolyubov-Mitropol'skii averaging, can be used to obtain slow evolution equations for the vortical and inertia-gravity wave components of the dynamics in a rotating flow. The evolution equations for each component are obtained to second order in the Rossby number, and the nature of the coupling between the two is analyzed carefully. It is also shown how classical balance models such as quasigeostrophic dynamics and its second-order extension appear naturally as a special case of this renormalized system, thereby providing a rigorous basis for the slaving approach where only the fast variables are expanded. It is well known that these balance models correspond to a hypothetical slow manifold of the parent system; the method herein allows the determination of the dynamics in the neighborhood of such solutions. As a concrete illustration, a simple weak-wave model is used, although the method readily applies to more complex rotating fluid models such as the shallow-water, Boussinesq, primitive, and 3D Euler equations.

Exponential Smallness of Inertia–Gravity Wave Generation at Small Rossby Number

“…The most standard of these is averaging, whereby (2.2)-(2.3) are averaged over the fast O( −1 ) time scale to leave only slow equations for s and for slowly-varying amplitudes of the fast variables f. Averaging has been applied only recently in the geophysical context, however (e.g. Majda & Embid 1998, Babin et al 2000, Wirosoetisno et al 2002, and with rather theoretical motivations. Instead, what has been extensively employed, is the idea of balance, which seeks to filer out fast inertia-gravity waves completely, mainly on the ground that these are weak in most of the atmosphere and oceans as well as poorly constrained by observations.…”

Inertia–Gravity Waves Emitted from Balanced Flow: Observations, Properties, and Consequences

“…A reasonably defined slow manifold may persist even when inertia-gravity waves are energized (Warn et al 1995;Vallis 1996;Wirosoetisno et al 2002). Furthermore, the waves may be sufficiently weak that they merely perturb the slow manifold into a fuzzy manifold (Warn and Menard 1986), which retains many of its useful properties.…”

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

“…We consider the formalism of multiple scale asymptotics in detail (see [13,14,43]) and briefly discuss a similar derivation via the renormalization group method (see [44,31]). The focus on the formalism of multiple scales is due to the simplicity of the calculation that follows.…”

“…The previous two Sections considered the effect of two rapid, 'wave-generating' linear operators using the formalism of multiple time scale asymptotics. In the current Section we derive the same limiting system (with less attention paid to detail), but here using the methods of cancellation of fast oscillation (see [3,23,34,19]) and renormalization ( [44,39,31,40]). Although the same result is achieved for each of these approaches, the outline of each is included here to demonstrate the insight gained via each method.…”