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

Vanneste, Jacques

doi:10.1175/2007jas2494.1

Cited by 31 publications

(25 citation statements)

References 55 publications

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“…Much of this has been motivated by theoretical issues related to the limitations of balanced models and the non-existence of exactly invariant slow manifolds (e.g., Vanneste 2008 and references therein). There are however practical implications, in particular for the parameterization of non-orographic GWs in general circulation models, since the (pseudo)momentum flux associated with the GWs generated spontaneously may well be significant.…”

Section: Introductionmentioning

confidence: 99%

“…The process we examine is also related to the work of Vanneste and Yavneh (2004) and Olafsdottir et al (2008) on GWs generated by PV anomalies in a horizontal shear with the difference that the wind shear is vertical in our case. A common feature of all these processes, one which is likely generic for spontaneous-generation phenomena (Vanneste 2008), is that the GW-activity is exponentially weak in the limit of small Rossby number, or equivalently large Richardson number. Our study, which is not restricted to this limit, confirms this conclusion; it nevertheless suggests that for reasonable values of the parameters, the GW amplitudes generated by PV anomalies in a vertical shear can be significant, comparable for instance with those observed in the low stratosphere by constant level balloons far from mountain ranges (Hertzog et al 2008).…”

Section: Introductionmentioning

confidence: 99%

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Gravity Waves Generated by Sheared Potential Vorticity Anomalies

Lott

Plougonven

Vanneste

2010

Journal of the Atmospheric Sciences

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The gravity waves generated by potential-vorticity anomalies in a rotating stratified shear flow are examined under the assumptions of constant vertical shear, twodimensionality and unbounded domain. Near a potential-vorticity anomaly, the associated perturbation is well modelled by quasi-geostrophic theory. This is not the case at large vertical distances, however, and in particular beyond the two inertial layers that appear above and below the anomaly; there, the perturbation is made of vertically propagating gravity waves. This structure is described analytically, using an expansion in the continuous spectrum of the singular modes that results from the presence of critical levels.Several explicit results are obtained. These include the form of the Eliassen-Palm flux as a function of the Richardson number N 2 /Λ 2 , where N is the Brunt-Väisälä frequency and Λ the vertical shear. Its non-dimensional value is shown to be approximately exp(−πN/Λ)/8 in the far-field, gravity-wave region, and approximately twice that between the two inertial layers. These results, which imply substantial wave-flow interactions in the inertial layers, are valid for Richardson numbers larger than 1, and for a large range of potential-vorticity distributions; In dimensional form they provide simple relationships between the Eliassen-Palm fluxes and the large-scale flow characteristics.As an illustration, we consider a potential-vorticity disturbance with an amplitude of 1 PVU and a depth of 1 km, and estimate that the associated Eliassen-Palm flux ranges between 0.1 mPa and 100 mPa for a Richardson number between 1 and 10. These values of the flux compare with those observed in the lower stratosphere, which suggests that the mechanism identified in this paper provides a substantial gravity-wave source, one that could be parameterized in GCMs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gravity Waves Generated by Sheared Potential Vorticity Anomalies

Lott

Plougonven

Vanneste

2010

Journal of the Atmospheric Sciences

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“…9 Theoretical arguments 10,11 indicate that this is a generic property, and the elliptical instability examined in this paper is no exception. In this case, the exponential smallness can be roughly understood by noting that in the manner typical of parametric instabilities, 12 the growth rates of the elliptical instability can be expected to be proportional to ⍀ n , where n is the order of the resonance.…”

Section: Introductionmentioning

confidence: 62%

Elliptical instability of a rapidly rotating, strongly stratified fluid

Aspden

Vanneste

2009

Physics of Fluids

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The elliptical instability of a rotating stratified fluid is examined in the regime of small Rossby number and order-one Burger number corresponding to rapid rotation and strong stratification.The Floquet problem describing the linear growth of disturbances to an unbounded, uniformvorticity elliptical flow is solved using exponential asymptotics. The results demonstrate that the flow is unstable for arbitrarily strong rotation and stratification; in particular, both cyclonic and anticyclonic flows are unstable. The instability is weak, however, with growth rates that are exponentially small in the Rossby number. The analytic expression obtained for the growth rate elucidates its dependence on the Burger number and on the eccentricity of the elliptical flow. It explains in particular the weakness of the instability of cyclonic flows, with growth rates that are only a small fraction of those obtained for the corresponding anticyclonic flows.The asymptotic results are confirmed by numerical solutions of Floquet problem.

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“…It has also been reported, the unsteady rotational flows with higher frequency tails always radiate spontaneous gravity waves irrespective of the low amplitude (Vanneste 2008). One possible way to investigate this spontaneous gravity wave radiation from higher frequency tails is through theoretical and numerical work such as aeroacoustic sound wave generation from a vortex pair (Mitchell et al 1995).…”

Section: Summary and Discussionmentioning

confidence: 99%

“…In contrast, in the process of spontaneous emission, gravity waves are radiated from unsteady rotational flows at any stage of time evolution. Although there are several theoretical (Vanneste and Yavneh 2004;Vanneste 2008) and experimental (Williams et al 2005;Afanasyev et al 2008) studies on spontaneous emission, the radiation processes of gravity waves are not fully understood.…”

Section: Introductionmentioning

confidence: 99%

Spontaneous Gravity Wave Radiation in a Shallow Water System on a Rotating Sphere

Sugimoto

Ishii

2012

Journal of the Meteorological Society of Japan

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The effect of the earth's rotation on spontaneous inertial gravity wave radiation from an unsteady rotational flow is investigated numerically using a shallow water system on a rotating sphere, changing three parameters, the Rossby number (1 ≤ Ro ≤ 30), the Froude number (0.1 ≤ Fr ≤ 0.7), and the latitudinal positions of the jet (11.25 • N ≤ θ 0 ≤ 78.75 • N). The spectral-like three-point combined compact difference (sp-CCD) scheme is used for numerical calculation to estimate the amplitude of gravity waves with high accuracy and resolution similar to the spherical harmonics model. The jet, which is initially balanced but evolves with time because of barotropic instability, is maintained by relaxation forcing. The time variations of nearly balanced rotational flows continuously radiate gravity waves. While the amount of gravity wave flux for large Ro (≥ 10) in all positions of the jet is almost constant, that for relatively small Ro (< 10) depends considerably on the positions of the jet and the observational latitude. First, there is almost no gravity wave radiation from the jet positioned at a high latitude. Second, even when gravity waves are radiated from the jet positioned at low latitude, they rarely propagate toward a higher latitude. To discuss the results, the source of gravity waves is introduced as an analogy to the aeroacoustic sound wave radiation theory (Lighthill theory) with an f -plane approximation at the position of the jet. Spectral analysis of the sources reasonably explains the waves. Because the effect of the earth's rotation varies with the latitude, gravity waves are radiated only from an unsteady source with higher frequency than the inertial cut-off frequency at the position of the jet. In a similar way, gravity waves that propagate toward high latitudes should have a higher frequency than the inertial cut-off frequency at that latitude.

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

Cited by 31 publications

References 55 publications

Gravity Waves Generated by Sheared Potential Vorticity Anomalies

Gravity Waves Generated by Sheared Potential Vorticity Anomalies

Elliptical instability of a rapidly rotating, strongly stratified fluid

Spontaneous Gravity Wave Radiation in a Shallow Water System on a Rotating Sphere

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