When the full internal gravity wave spectrum propagates through a coherent mesoscale eddy, the waves are refracted, exchange energy with the eddy's shears, and break preferentially at the eddy rim near the surface — but existing simulations hold the eddy and the stratification stationary, treating the waves as a passive tracer of the background. The open problem, raised by Sebastia Saez, Eden, and Chouksey's Internal Wave Energy Model (IWEM) study of a Canary Current eddy, is the two-way coupled problem: how the wave–mean-flow energy exchange and wave-breaking-induced mixing feed back on the eddy's evolution and on the ambient stratification; what fraction of the wave energy removed across small-scale spectral thresholds produces secondary waves rather than turbulence (and what cutoff thresholds and wave-energy levels are physically justified); and whether the predicted eddy-rim-enhanced dissipation and diffusivities are confirmed by targeted in-situ measurements.

Background

Internal gravity waves (IGWs) are propagating oscillations of a stably stratified fluid, with frequencies between the local inertial (Coriolis) frequency ff and the buoyancy frequency N(z)N(z) that measures the strength of the density stratification. In the ocean they transport energy from their generation sites (wind forcing, tidal flow over topography) to distant dissipation sites, and their breaking drives the turbulent mixing that maintains the large-scale overturning circulation. Mesoscale eddies are coherent, roughly geostrophically balanced vortices of order 10–100 km that dominate the ocean's kinetic energy; when IGWs propagate through them, the eddy's horizontal and vertical shear refracts the waves and exchanges energy with them. The background wave field is conventionally described by the Garrett–Munk (GM) spectrum, an empirical model spectrum of wave energy versus frequency and wavenumber whose overall energy level is set by a parameter E0E_0. Sebastia Saez et al. (2024) developed the Internal Wave Energy Model (IWEM), which evolves the full IGW spectrum via a six-dimensional radiative transfer equation — three physical coordinates plus three wavenumber coordinates — derived under the WKBJ approximation (the assumption that the background currents and stratification vary slowly compared with a wavelength). In spin-down simulations initialized with the GM spectrum in an idealized mean flow and stratification modeled on a coherent cyclonic eddy of the Canary Current system, with the eddy and N(z)N(z) held stationary, they found: wave energy decreases at the eddy rim and increases toward the center; lateral (horizontal) shear transfers energy from the mean flow to the waves outside the rim via a developing horizontal anisotropy, while vertical shear extracts wave energy, most strongly at the rim and near the surface, where waves are refracted toward critical layers (levels where the mean flow matches the wave's phase speed, at which waves are absorbed and break); and wave breaking is diagnosed as the flux of wave energy across imposed vertical and horizontal wavenumber cutoffs (kzc=1 m−1k_z^c=1\,\mathrm{m^{-1}}, khc=0.5 m−1k_h^c=0.5\,\mathrm{m^{-1}}), converted to a vertical diffusivity via the Osborn (1980) closure κ=Γε/N2\kappa=\Gamma\varepsilon/N^2 with mixing efficiency Γ≈0.2\Gamma\approx0.2, giving κ∼10−7\kappa\sim10^{-7}–10−5 m2 s−110^{-5}\,\mathrm{m^2\,s^{-1}} concentrated at the rim near the surface. The study was deliberately one-way: the waves feel the eddy, but the eddy, the stratification, and the wave field's own stationary state never respond. The broader context includes Polzin's (2008, 2010) theory and Mid-Ocean Dynamics Experiment/PolyMode analyses of eddy–internal-wave coupling via wave capture (the exponential growth of a wave's wavenumber along a ray in a straining flow, which extracts eddy energy), the mooring observations of Cusack et al. (2020) in the Southern Ocean, which measured a net transfer of energy from the internal wave field to eddies of about 2.2 mW m−22.2\,\mathrm{mW\,m^{-2}} dominated by vertical shear, the realistically forced regional simulations of Barkan et al. (2021), in which turning on internal-wave forcing reduced the mesoscale kinetic energy of the flow — a two-way effect in the aggregate, though not for a single coherent eddy with evolving stratification — and the Lofoten Basin eddy dissipation measurements of Fer et al. (2018), which showed enhanced turbulent dissipation at both the rim and the center of a long-lived anticyclonic eddy, partly contradicting the rim-only enhancement found by IWEM. What none of these establish is the coupled evolution of a coherent eddy and its internal wave field.

Problem Statement

For a coherent mesoscale ocean eddy interacting with a realistic internal gravity wave field, determine the two-way coupled evolution and its observational signature. Specifically: (1) quantify how the exchange of energy between waves and the eddy's lateral and vertical shear, and the mixing induced by wave breaking, feed back on the eddy's currents and their evolution and on the ambient stratification N(z)N(z) — including whether the rim-concentrated, surface-intensified dissipation and the diagnosed tendency to accelerate the eddy inside and decelerate it outside its rim (with net deceleration overall) found under a stationary background survive once the mean flow and stratification are allowed to respond; (2) establish the fate of the wave energy removed at small scales, namely the partition between regeneration of waves at other wavelengths and irreversible conversion to turbulence, and the physically justified values (with sensitivities) of the vertical and horizontal wavenumber cutoffs and of the wave-energy level E0E_0 of the Garrett–Munk spectrum; and (3) test the predicted eddy-rim-enhanced dissipation rates and diffusivities (of order 10−710^{-7}–10−5 m2 s−110^{-5}\,\mathrm{m^2\,s^{-1}} in the source configuration) against targeted in-situ dissipation measurements resolving the rim versus the interior of coherent eddies. The method is unrestricted. Enabling developments the source explicitly defers — inclusion of tidal, wind-driven near-inertial, and topographic forcing; nonlinear wave–wave transfers or a scattering-integral (Wigner-transform kinetic) formulation with a consistent dissipation treatment; and adequate numerical resolution of small-scale spectral transfers — are components insofar as they affect these conclusions.

References

  1. Sebastia Saez, P., Eden, C., and Chouksey, M. (2024). Evolution of Internal Gravity Waves in a Mesoscale Eddy Simulated Using a Novel Model. Journal of Physical Oceanography 54(4), 985–1002. doi:10.1175/jpo-d-23-0095.1, https://doi.org/10.1175/jpo-d-23-0095.1
  2. Sebastia Saez, P., Eden, C., and Chouksey, M. (2023). Evolution of internal gravity waves in meso-scale eddies. arXiv:2311.11916, https://arxiv.org/abs/2311.11916
  3. Garrett, C., and Munk, W. (1975). Space-time scales of internal waves: A progress report. Journal of Geophysical Research 80(3), 291–297. doi:10.1029/JC080i003p00291, https://doi.org/10.1029/JC080i003p00291
  4. Osborn, T. R. (1980). Estimates of the Local Rate of Vertical Diffusion from Dissipation Measurements. Journal of Physical Oceanography 10(1), 83–89. doi:10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2, https://doi.org/10.1175/1520-0485(1980)010%3C0083:EOTLRO%3E2.0.CO;2
  5. Polzin, K. L. (2008). Mesoscale Eddy–Internal Wave Coupling. Part I: Symmetry, Wave Capture, and Results from the Mid-Ocean Dynamics Experiment. Journal of Physical Oceanography 38(11), 2556–2574. doi:10.1175/2008jpo3666.1, https://doi.org/10.1175/2008jpo3666.1
  6. Polzin, K. L. (2010). Mesoscale Eddy–Internal Wave Coupling. Part II: Energetics and Results from PolyMode. Journal of Physical Oceanography 40(4), 789–801. doi:10.1175/2009jpo4039.1, https://doi.org/10.1175/2009jpo4039.1
  7. Savva, M. A. C., Kafiabad, H. A., and Vanneste, J. (2021). Inertia-gravity-wave scattering by three-dimensional geostrophic turbulence. Journal of Fluid Mechanics 916, A6. doi:10.1017/jfm.2021.205, https://doi.org/10.1017/jfm.2021.205
  8. Fer, I., Bosse, A., Ferron, B., and Bouruet-Aubertot, P. (2018). The Dissipation of Kinetic Energy in the Lofoten Basin Eddy. Journal of Physical Oceanography 48(6), 1299–1316. doi:10.1175/jpo-d-17-0244.1, https://doi.org/10.1175/jpo-d-17-0244.1
  9. Cusack, J. M., Brearley, J. A., Naveira Garabato, A. C., Smeed, D. A., Polzin, K. L., Velzeboer, N., and Shakespeare, C. J. (2020). Observed Eddy–Internal Wave Interactions in the Southern Ocean. Journal of Physical Oceanography 50(10), 3043–3062. doi:10.1175/jpo-d-20-0001.1, https://doi.org/10.1175/jpo-d-20-0001.1
  10. Barkan, R., Srinivasan, K., Yang, L., McWilliams, J. C., Gula, J., and Vic, C. (2021). Oceanic Mesoscale Eddy Depletion Catalyzed by Internal Waves. Geophysical Research Letters 48(18), e2021GL094376. doi:10.1029/2021gl094376, https://doi.org/10.1029/2021gl094376
  11. Yang, L., Barkan, R., Srinivasan, K., McWilliams, J. C., Shakespeare, C. J., and Gibson, A. H. (2023). Oceanic eddies induce a rapid formation of an internal wave continuum. Communications Earth & Environment 4, 484. doi:10.1038/s43247-023-01137-1, https://doi.org/10.1038/s43247-023-01137-1
  12. Barkan, R., Srinivasan, K., and McWilliams, J. C. (2024). Eddy–Internal Wave Interactions: Stimulated Cascades in Cross-Scale Kinetic Energy and Enstrophy Fluxes. Journal of Physical Oceanography 54(6), 1309–1326. doi:10.1175/jpo-d-23-0191.1, https://doi.org/10.1175/jpo-d-23-0191.1
  13. Dematteis, G., Le Boyer, A., Pollmann, F., Polzin, K. L., Alford, M. H., Whalen, C. B., and Lvov, Y. V. (2024). Interacting internal waves explain global patterns of interior ocean mixing. Nature Communications 15, 7468. doi:10.1038/s41467-024-51503-6, https://doi.org/10.1038/s41467-024-51503-6
  14. Polzin, K. L., and Dematteis, G. (2025). Mesoscale Eddy–Internal Wave Coupling. III. The End of the Enstrophy Cascade and Maintenance of Gyre Scale Potential Vorticity Gradients. arXiv:2504.00486, https://arxiv.org/abs/2504.00486
  15. Chen, Y., Liu, C., Wang, F., Köhl, A., and Huang, R. X. (2025). Quantifying the Contribution of Mesoscale Eddies to Global Internal Wave‐Driven Turbulent Mixing. Journal of Geophysical Research: Oceans 130(8), e2025JC022416. doi:10.1029/2025jc022416, https://doi.org/10.1029/2025jc022416
  16. Uncu, J., and Grisouard, N. (2024). Wave scattering by an isolated cyclogeostrophic vortex. Journal of Fluid Mechanics 999, A100. doi:10.1017/jfm.2024.963, https://doi.org/10.1017/jfm.2024.963
  17. Wu, Y. C., and Pan, Y. (2025). Induced diffusion of internal gravity waves: directionality and role in ocean mixing. Journal of Fluid Mechanics 1021, A48. doi:10.1017/jfm.2025.10749, https://doi.org/10.1017/jfm.2025.10749
  18. Conn, S., and Callies, J. (2026). Global Near-Inertial Wave Spectra Shaped by Mesoscale Eddies. Journal of Physical Oceanography 56(3), 707–724. doi:10.1175/jpo-d-25-0163.1, https://doi.org/10.1175/jpo-d-25-0163.1