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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.
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This content will become publicly available on January 1, 2026
A One-Dimensional Model for Investigating Scale-Separated Approaches to the Interaction of Oceanic Internal Waves
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.
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- PAR ID:
- 10613147
- Publisher / Repository:
- American Meteorological Society
- Date Published:
- Journal Name:
- Journal of Physical Oceanography
- Volume:
- 55
- Issue:
- 1
- ISSN:
- 0022-3670
- Page Range / eLocation ID:
- 3 to 28
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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