Wave Interactions

Vanneste, Jacques

doi:10.1007/3-211-38025-6_3

Cited by 5 publications

(7 citation statements)

References 25 publications

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“…In general, maximum energy transfer between the three modes is ensured when the oscillations due to the detuning in the right-hand side of (3.1) are cancelled, that is, when Δω kpq → 0. If all three modes k, p and q are inertial waves, this leads to the well known mechanism of triadic resonance (Bretherton 1964;Vanneste 2005;Bordes et al 2012). When the mode k is imposed with an amplitude Ro and helicity s k , p and q grow exponentially with a rate proportional to kRo.…”

Section: The Low Rossby Number Limitmentioning

confidence: 99%

“…The studies of Le Reun et al (2019) and Brunet et al (2020) suggest that such a transfer occurs through an instability. Although wave-to-wave interactions are primarily governed by triadic resonance (Vanneste 2005;Bordes et al 2012), they cannot account for wave-to-geostrophic transfers (Greenspan 1969), at least in the asymptotic limit of vanishing velocity amplitude and dissipation. Several alternative mechanisms, outside the framework of Greenspan's theorem, have been proposed.…”

Section: Introductionmentioning

confidence: 99%

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Near-resonant instability of geostrophic modes: beyond Greenspan's theorem

et al. 2020

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Section: The Low Rossby Number Limitmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Near-resonant instability of geostrophic modes: beyond Greenspan's theorem

et al. 2020

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“…Following the argument in Vanneste (2005) (more details in Vanneste & Vial (1994), and Ripa (1981)) we define the nonlinear interaction coefficient in terms of the quadratic part of the energy and/or enstrophy. This is done by defining quadratic forms that, as well as giving the first terms of the conservation laws, form an orthogonality condition over the different modes.…”

Section: Resonant Triads and Conservation Lawsmentioning

confidence: 99%

“…where u is the velocity, u † is it's conjugate transpose and E is a Hermitian matrix. It can be proven (Vanneste 2005) that this must obey the following orthogonality relation:…”

Section: Resonant Triads and Conservation Lawsmentioning

confidence: 99%

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Fast and slow resonant triads in the two-layer rotating shallow water equations

Owen

Grimshaw

2018

J. Fluid Mech.

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In this paper we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect which cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multi-scale asymptotic expansion based on a small amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored with an emphasis on the parameter regime with small Rossby and Froude numbers.Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfmkeywords.pdf for the full list) † Email address for correspondence: ao306@exeter.ac.uk arXiv:1803.01736v1 [physics.flu-dyn]

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Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

2019

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In this paper, we present an experimental investigation of the turbulent saturation of the flow driven by parametric resonance of inertial waves in a rotating fluid. In our set-up, a half-meter wide ellipsoid filled with water is brought to solid body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the librationdriven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is injected into the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow mainly consists of a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly non-linear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wavedominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al. (2017) who introduced the possibility of these two turbulent regimes. The transition between these two regimes and their relevance to geophysical applications are finally discussed. †

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Wave Interactions

Cited by 5 publications

References 25 publications

Near-resonant instability of geostrophic modes: beyond Greenspan's theorem

Near-resonant instability of geostrophic modes: beyond Greenspan's theorem

Fast and slow resonant triads in the two-layer rotating shallow water equations

Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

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