On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

Walsh, Shane; Bustamante, Miguel D.

doi:10.1017/jfm.2019.949

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…It would also be of interest to consider clusters of GW–GW interactions with common triad members (cf. Walsh and Bustamante, 2020), and validate the approach against these. Finally, a challenge will be continuous GW spectra.…”

Section: Discussionmentioning

confidence: 97%

An application of WKBJ theory for triad interactions of internal gravity waves in varying background flows

Voelker

Akylas

Achatz

2021

Quart J Royal Meteoro Soc

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Motivated by the question of whether and how wave–wave interactions should be implemented into atmospheric gravity‐wave parametrizations, the modulation of triadic gravity‐wave interactions by a slowly varying and vertically sheared mean flow is considered for a non‐rotating Boussinesq fluid with constant stratification. An analysis using a multiple‐scale WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) expansion identifies two distinct scaling regimes, a linear off‐resonance regime, and a nonlinear near‐resonance regime. Simplifying the near‐resonance interaction equations allows for the construction of a parametrization for the triadic energy exchange which has been implemented into a one‐dimensional WKBJ ray‐tracing code. Theory and numerical implementation are validated for test cases where two wave trains generate a third wave train while spectrally passing through resonance. In various settings, of interacting vertical wavenumbers, mean‐flow shear, and initial wave amplitudes, the WKBJ simulations are generally in good agreement with wave‐resolving simulations. Both stronger mean‐flow shear and smaller wave amplitudes suppress the energy exchange among a resonantly interacting triad. Experiments with mean‐flow shear as strong as in the vicinity of atmospheric jets suggest that internal gravity‐wave dynamics are dominated in such regions by wave modulation. However, triadic gravity‐wave interactions are likely to be relevant in weakly sheared regions of the atmosphere.

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Section: Discussionmentioning

confidence: 97%

An application of WKBJ theory for triad interactions of internal gravity waves in varying background flows

Voelker

Akylas

Achatz

2021

Quart J Royal Meteoro Soc

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“…where β is an overall rescaling of the initial amplitudes and all other a k = 0 initially. Note that Equation (21) indicates that the phases of the waves are set initially to zero. Experiments, not shown, demonstrate that varying the phases has little qualitative effect on the results to follow, only adding some corrections, and maintaining the a K = a * −K symmetry leaves the results quantitatively identical.…”

Section: Numerical Simulations Of the Fifth-order Governing Equations In Natural Variablesmentioning

confidence: 99%

“…This approach assumes that the phases of the Fourier transforms do not play an important role in the dynamics of energy transfers. Interesting modern departures from these theories for water waves and other wave systems consider quasi-resonant scenarios [17,18] and finite-amplitude regimes in discrete wave turbulence for other systems (such as the barotropic vorticity equation and more recently in wave-mean flow models of solar cycle modulations), where the phases interact with the spectrum variables producing interesting effects, such as precession resonance [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

Lucas

Perlin

Liu

et al. 2021

Fluids

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In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance.

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Discrete resonant Rossby/drift wave triads: Explicit parameterisations and a fast direct numerical search algorithm

Hayat

Amanullah

Walsh

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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We report results on the explicit parameterisation of discrete Rossby-wave resonant triads of the Charney-Hasegawa-Mima equation in the small-scale limit (i.e. large Rossby deformation radius), following up from our previous solution in terms of elliptic curves (Bustamante and Hayat, 2013). We find an explicit parameterisation of the discrete resonant wavevectors in terms of two rational variables. We show that these new variables are restricted to a bounded region and find this region explicitly. We argue that this can be used to reduce the complexity of a direct numerical search for discrete triad resonances. Also, we introduce a new direct numerical method to search for discrete resonances. This numerical method has complexity O(N 3 ), where N is the largest wavenumber in the search. We apply this new method to find all discrete irreducible resonant triads in the wavevector box of size 5000, in a calculation that took about 10.5 days on a 16-core machine. Finally, based on our method of mapping to elliptic curves, we discuss some dynamical implications regarding the spread of quadratic invariants across scales via resonant triad interactions, in the form of sharp bounds on the size of the interacting wavevectors.

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On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

Cited by 5 publications

References 23 publications

An application of WKBJ theory for triad interactions of internal gravity waves in varying background flows

An application of WKBJ theory for triad interactions of internal gravity waves in varying background flows

Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

Discrete resonant Rossby/drift wave triads: Explicit parameterisations and a fast direct numerical search algorithm

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