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Abstract High-frequency wave propagation in near-inertial wave shear has, for four decades, been considered fundamental in setting the spectral character of the oceanic internal wave continuum and for transporting energy to wave breaking. We compare idealized ray-tracing numerical results with metrics derived using a wave turbulence derivation for the kinetic equation and a path integral to study this specific process. Statistical metrics include the time-dependent ensemble mean vertical wavenumber, referred to as a mean drift; dispersion about the mean drift; time-lagged correlation estimates of wavenumber; and phase locking of the wave packets with the background. The path integral permits us to identify the mean drift as a resonant process and dispersion about that mean drift as nonresonant. At small inertial wave amplitudes, ray tracing, wave turbulence, and the path integral provide consistent descriptions for the mean drift of wave packets in the spectral domain and dispersion about the mean drift. Extrapolating these results to the background internal wavefield overpredicts downscale energy transports by an order of magnitude. At oceanic amplitudes, however, the numerics support diminished transport and dispersion that coincide with the mean drift time scale becoming similar to the lagged correlation time scale. We parse this as the transition to a non-Markovian process. Despite this decrease, numerical estimates of downscale energy transfer are still too large. We argue that residual differences result from an unwarranted discard of Bragg scattering resonances. Our results support replacing the long-standing interpretive paradigm of extreme scale-separated interactions with a more nuanced slate of “local” interactions in the kinetic equation. 

Wave turbulence provides a conceptual framework for weakly nonlinear interactions in dispersive media. Dating from five decades ago, applications of wave turbulence theory to oceanic internal waves assigned a leading-order role to interactions characterized by a near equivalence between the group velocity of high-frequency internal waves with the phase velocity of near-inertial waves. This scale-separated interaction leads to a Fokker–Planck (generalized diffusion) equation. More recently, starting four decades ago, this scale-separated paradigm has been investigated using ray tracing methods. These ray methods characterize spectral transport of energy by counting the amplitude and net velocity of wave packets in phase space past a high-wavenumber gate prior to ‘breaking’. This explicitly advective characterization is based on an intuitive assignment and lacks theoretical underpinning. When one takes an estimate of the net spectral drift from the wave turbulence derivation and makes the corresponding assessment, one obtains a prediction of spectral transport that is an order of magnitude larger than either observations or reported ray tracing estimates. Motivated by this contradiction, we report two parallel derivations for transport equations describing the refraction of high-frequency internal waves in a sea of random inertial waves. The first uses standard wave turbulence techniques and the second is an ensemble-averaged packet transport equation characterized by the dispersion of wave packets about a mean drift in the spectral domain. The ensemble-averaged transport equation for ray tracing differs in that it contains the intuitively motivated advective term. We conclude that the aforementioned contradiction between theory, numerics and observations needs to be taken at face value and present a pathway for resolving this contradiction. 

We use the Dyson–Wyld diagrammatic technique to analyse the infinite series for the correlation functions of the velocity in hydrodynamic turbulence. We demonstrate the fundamental role played by the triple correlator of the velocity in determining the entire statistics of the hydrodynamic turbulence. All higher-order correlation functions are expressed through the triple correlator. This is shown through the suggested triangular re-summation of the infinite diagrammatic series for multi-point correlation functions. The triangular re-summation is the next logical step after the Dyson–Wyld line re-summation for the Green's function and the double correlator. In particular, it allows us to explain why the inverse cascade of the two-dimensional hydrodynamic turbulence is close to Gaussian. Since the triple correlator dictates the flux of energy$$\varepsilon$$through the scales, we support the Kolmogorov-1941 idea that$$\varepsilon$$is one of the main characteristics of hydrodynamic turbulence. 

We calculate the net energy per unit time exchanged between two sets of modes in a generic system governed by a three-wave kinetic equation. Our calculation is based on the property of detailed energy conservation of the triadic resonant interactions. In a first application to isotropic systems, we re-derive the previously used formula for the energy flux as a particular case for adjacent sets. We then exploit the new formalism to quantify the level of locality of the energy transfers in the example of surface capillary waves. A second application to anisotropic wave systems expands the currently available set of tools to investigate magnitude and direction of the energy fluxes in these systems. We illustrate the use of the formalism by characterizing the energy pathways in the oceanic internal wavefield. Our proposed approach, unlike traditional approaches, is not limited to stationarity, scale invariance and strict locality. In addition, we define a number$$w$$that quantifies the scale separation necessary for two sets of modes to having negligible mutual energy exchange, with potential consequences in the interpretation of wave turbulence experiments. The methodology presented here provides a general, simple and systematic approach to energy fluxes in wave turbulence. 

Clebsch variables provide a canonical representation of ideal flows that is, in practice, difficult to handle: while the velocity field is a function of the Clebsch variables and their gradients, constructing the Clebsch variables from the velocity field is not trivial. We introduce an extended set of Clebsch variables that circumvents this problem. We apply this method to a compressible, chemically inhomogeneous, and rotating ideal fluid in a gravity field. A second difficulty, the secular growth of canonical variables even for stationary states of stratified fluids, makes expansions of the Hamiltonian in Clebsch variables problematic. We give a canonical transformation that associates a stationary state of the canonical variables with the stationary state of the fluid; the new set of variables permits canonical approximations of the dynamics. We apply this to a compressible stratified ideal fluid with the aim to facilitate forthcoming studies of wave turbulence of internal waves. 

We develop a theory of strong anisotropy of the energy spectra in the thermally driven turbulent counterflow of superfluid 4 He. The key ingredients of the theory are the three-dimensional differential closure for the vector of the energy flux and the anisotropy of the mutual friction force. We suggest an approximate analytic solution of the resulting energy-rate equation, which is fully supported by our numerical solution. The two-dimensional energy spectrum is strongly confined in the direction of the counterflow velocity. In agreement with the experiments, the energy spectra in the direction orthogonal to the counterflow exhibit two scaling ranges: a near-classical non-universal cascade dominated range and a universal critical regime at large wavenumbers. The theory predicts the dependence of various details of the spectra and the transition to the universal critical regime on the flow parameters. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’. 

We consider interactions between surface and interfacial waves in a two-layer system. Our approach is based on the Hamiltonian structure of the equations of motion, and includes the general procedure for diagonalization of the quadratic part of the Hamiltonian. Such diagonalization allows us to derive the interaction cross-section between surface and interfacial waves and to derive the coupled kinetic equations describing spectral energy transfers in this system. Our kinetic equation allows resonant and near-resonant interactions. We find that the energy transfers are dominated by the class III resonances of Alam ( J. Fluid Mech. , vol. 691, 2012, pp. 267–278). We apply our formalism to calculate the rate of growth for interfacial waves for different values of wind velocity. Using our kinetic equation, we also consider the energy transfer from wind-generated surface waves to interfacial waves for the case when the spectrum of the surface waves is given by the JONSWAP spectrum and interfacial waves are initially absent. We find that such energy transfer can occur along a time scale of hours; there is a range of wind speeds for the most effective energy transfer at approximately the wind speed corresponding to white capping of the sea. Furthermore, interfacial waves oblique to the direction of the wind are also generated.