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Wave Turbulence Theory for Understanding Oceanic Energy Cascade
Key words: wave-wave interaction, nonlinear physics, wave turbulence theory
Background
Background
Wave Turbulence Theory is a statistical framework that describes the evolution of wave action in a broad class of interacting wave systems, including water waves, waves in magnetized fluids (e.g., solar winds, interstellar media, and fusion plasmas), condensed matter (e.g., helium superfluids and Bose-Einstein condensates), nonlinear optics, and, more recently, gravitational waves in the early universe.
In the context of oceanography, the wave kinetic equation (WKE), a direct result of Wave Turbulence Theory, offers a distinctive approach to understanding energy transfer across scales due to interacting internal gravity waves. Originally envisioned by Nobel Prize winner Klaus Hasselmann over half a century ago, the theory's numerical implementation has only recently been made possible thanks to advances in high-performance computing. It has since been applied to global datasets of internal wave spectra and validated with microstructure observations, establishing itself as a powerful tool for estimating turbulent dissipation and improving parameterizations of ocean mixing in general circulation and climate models.
Instead of directly assessing small-scale dissipation processes, the WKE evaluates the energy flux down to a vertical scale of 10 meters, providing an estimate of the energy available for dissipation. In this project, we apply the Wave Turbulence framework to various oceanic scenarios to address theoretical gaps, resolve inconsistencies, and develop a physically grounded approach to understanding ocean mixing. The research involves (i) a comprehensive evaluation of turbulent dissipation across varying internal wave spectra, and (ii) an exploration of the time evolution of these spectra under realistic forcing conditions.
Historical Misconceptions and Research Gaps
The study of energy cascade in interacting internal gravity waves was pioneered by McComas et al. in a series of publications (McComas & Bretherton, 1977a; McComas & Müller, 1981a,b). A major argument made in these works is that the energy cascade is dominated by three non-local mechanisms (i.e., wave triads that are scale-separated in either vertical wavenumber or frequency, or both), namely parametric subharmonic instability (PSI), elastic scattering (ES), and induced diffusion (ID).
While the pioneering works by McComas et al. laid the cornerstone for finescale parameterization, the calculation of the downscale energy flux suffers from physical inconsistencies:
(a) The importance of local interactions:
A strong assumption put forward by McComas et al. and subsequent studies is that local interactions contribute minimally to the energy cascade—an assertion based on their secondary role in the interaction kernel. However, given the overwhelming abundance of local triads compared to non-local ones in a multiscale ocean, this argument remains preliminary and warrants skepticism. Dematteis et al. (2022) were the first to highlight the importance of local interactions. We emphasize the necessity of incorporating the cumulative effect of local interactions for more accurate energy flux estimates, as both their quality and quantity are crucial.
(b) The stationary solution:
Stationary solutions to the WKE have been theoretically predicted, including the convergent stationary solution (Lvov et al., 2010; Dematteis & Lvov, 2021) and the Pelinovsky-Raevsky stationary solution (Pelinovsky & Raevsky, 1977). However, these predictions have rarely been observed in the ocean, likely because they rely on the premise of a power-law relationship of the internal wave spectra, a condition only met in the high-frequency, high-wavenumber regime of the spectra. Furthermore, these theoretical predictions do not account for external forcing, which could play a deterministic role in shaping the spectra.
(c) The role of induced diffusion:
For decades, local interactions were overlooked, and their contributions were mistakenly attributed to ID. In fact, ID represents an equilibrium state for the Garrett-Munk spectrum, indicating no action diffusion resulted from the vanishing gradient. McComas et al. recognized this inconsistency and hence adopted a correction to artificially enforce an ID cascade, which apparently lacks physical justification. This paradox was partially resolved by Dematteis et al. (2022), confirming that ID does not contribute to the leading order of energy cascade in the Garrett-Munk spectrum. These updated insights prompt a comprehensive investigation of the induced diffusion mechanism across varying spectra.
(d) The role of elastic scattering:
Elastic scattering has been mistakenly believed to be an equilibrium state for vertically symmetric spectra, including the Garrett-Munk spectrum. The other two leading mechanisms, PSI and ID, inherently transfer energy towards low frequencies. This scenario is impossible for a balanced internal wave field unless there is an unrealistic energy injection at high frequencies or an unconfirmed mechanism that transfers energy towards high frequencies. A complete, physically consistent picture of energy cascade in the frequency domain is still awaited.
Numerical evaluation of the WKE has provided deeper insights into individual mechanism (i.e., PSI, ES, ID, and local interactions) and lead to a more comprehensive and physically grounded understanding.
Key Findings
Key Findings
The WKE serves as a prognostic tool for understanding the roles of individual mechanisms in oceanic energy cascade.
For the Garrett-Munk spectrum, we revised the underlying argument for finescale parameterization, namely, it should be PSI plus local interactions, rather than PSI plus ID, that constitute the downscale energy flux, thereby resolving Inconsistency (a).
For spectra other than Garrett-Munk, we highlighted the potential for the ID mechanism to reverse direction, which could facilitate either a forward cascade (towards smaller vertical scales) or a backward cascade (towards larger vertical scales), depending on the competing effects between the spectral slopes in frequency and vertical wavenumber of the internal wave spectra. This finding resolves Inconsistency (c).
For all spectra, we identified that ES is responsible for a forward frequency cascade (towards high frequencies), which resolves Inconsistency (d).
Publications
Publications
Wu, Y. and Pan, Y. (2025). Induced Diffusion of Internal Gravity Waves: Directionality and Role in Ocean Mixing. Journal of Fluid Mechanics, 1021, p. A48. Link
Wu, Y. and Pan, Y. (2023). Energy cascade in the Garrett–Munk spectrum of internal gravity waves. Journal of Fluid Mechanics, 975, p. A11. Link
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