Averaging over Fast Gravity Waves for Geophysical Flows with Unbalanced Initial Data

Majda, Andrew J.; Embid, Pedro

doi:10.1007/s001620050086

Cited by 80 publications

(65 citation statements)

References 32 publications

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“…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.…”

Section: Slow Manifoldsmentioning

confidence: 99%

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

Vanneste

2008

Journal of the Atmospheric Sciences

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This paper discusses some of the mechanisms whereby fast inertia-gravity waves can be generated spontaneously by slow, balanced atmospheric and oceanic flows. In the small-Rossby-number regime relevant to mid-latitude dynamics, high-accuracy balanced models, which filter out inertia-gravity waves completely, can in principle describe the evolution of suitably initialised flows up to terms that are exponentially small in the Rossby number , i.e, of the form exp(−α/ ) for some α > 0.This suggests that the mechanisms of inertia-gravity-wave generation, which are not captured by these balanced models, are also exponentially weak. This has been confirmed by explicit analytical results obtained for a few highly-simplified models. We review these results and present some of the exponential-asymptotic techniques that have been used in their derivation. We examine both spontaneous-generation mechanisms which generate exponentially small waves from perfectly balanced initial conditions, and unbalanced instability mechanisms which amplify unbalanced initial perturbations of steady flows. The relevance of the results to realistic flows is discussed.2

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Section: Slow Manifoldsmentioning

confidence: 99%

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

Vanneste

2008

Journal of the Atmospheric Sciences

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“…In mid-latitudes, the fact that the rotational Coriolis terms are bounded away from zero leads to a strict temporal frequency scale separation between slow potential vorticity dynamics and fast gravity waves; theorems justifying the quasi-geostrophic mid-latitude dynamics have been proved even with general unbalanced initial data for both rapidly rotating shallow water equations and completely stratified flows [3,7,8,16,15,5]. However, the proofs require constant symmetric hyperbolic coefficients for the fast-wave dynamics in order to obtain higher derivative estimates on the solution.…”

Section: Introductionmentioning

confidence: 99%

Fast Wave Averaging for the Equatorial Shallow Water Equations

Dutrifoy

Majda

2007

Communications in Partial Differential Equations

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The equatorial shallow water equations in a suitable limit are shown to reduce to zonal jets as the Froude number tends to zero. This is a theorem of a singular limit with a fast variable coefficient due to the vanishing of the Coriolis force at the equator. Although it is not possible to get uniform estimates in classical Sobolev spaces (other than L 2 ) by differentiating the system, a new method exploiting the particular structure of the fast coefficient leads to uniform estimates in slightly different functional spaces.The computation of resonances shows that fast waves may interact with a strong external forcing, introduced to mimic the effects of moisture, to create zonal jets.

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“…e.g., Embid and Majda 1996;Chemin 1997;Gallagher 1998;Embid and Majda 1998;Majda and Embid 1998;Babin et al 2000Babin et al , 2001) based on the work of Schochet (1994), which is closely related to the method employed here. Mention should also be made of the work of Callet (1997), who analyzed four-wave resonances-these correspond to isolated interactions in our second-order renormalized system.…”

Section: ϫ3mentioning

confidence: 95%

Free Gravity Waves and Balanced Dynamics

Wirosoetisno¹,

Shepherd

Témam

2002

J. Atmos. Sci.

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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.

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Averaging over Fast Gravity Waves for Geophysical Flows with Unbalanced Initial Data

Cited by 80 publications

References 32 publications

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

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

Fast Wave Averaging for the Equatorial Shallow Water Equations

Free Gravity Waves and Balanced Dynamics

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