On the wave turbulence theory for stratified flows in the ocean

After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions.

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Numerical Investigation of Mechanisms Underlying Oceanic Internal Gravity Wave Power-Law Spectra

Journal of Physical Oceanography ◽

10.1175/jpo-d-20-0039.1 ◽

2020 ◽

Vol 50 (9) ◽

pp. 2713-2733

Author(s):

Yulin Pan ◽

Brian K. Arbic ◽

Arin D. Nelson ◽

Dimitris Menemenlis ◽

W. R. Peltier ◽

...

Keyword(s):

Power Law ◽

Gravity Waves ◽

High Frequency ◽

Internal Gravity Wave ◽

Numerical Data ◽

Field Measurements ◽

Internal Gravity Waves ◽

Collision Integral ◽

Turbulence Theory ◽

Nonlinear Interactions

AbstractWe consider the power-law spectra of internal gravity waves in a rotating and stratified ocean. Field measurements have shown considerable variability of spectral slopes compared to the high-wavenumber, high-frequency portion of the Garrett–Munk (GM) spectrum. Theoretical explanations have been developed through wave turbulence theory (WTT), where different power-law solutions of the kinetic equation can be found depending on the mechanisms underlying the nonlinear interactions. Mathematically, these are reflected by the convergence properties of the so-called collision integral (CL) at low- and high-frequency limits. In this work, we study the mechanisms in the formation of the power-law spectra of internal gravity waves, utilizing numerical data from the high-resolution modeling of internal waves (HRMIW) in a region northwest of Hawaii. The model captures the power-law spectra in broad ranges of space and time scales, with scalings ω−2.05±0.2 in frequency and m−2.58±0.4 in vertical wavenumber. The latter clearly deviates from the GM76 spectrum but is closer to a family of induced-diffusion-dominated solutions predicted by WTT. Our analysis of nonlinear interactions is performed directly on these model outputs, which is fundamentally different from previous work assuming a GM76 spectrum. By applying a bicoherence analysis and evaluations of modal energy transfer, we show that the CL is dominated by nonlocal interactions between modes in the power-law range and low-frequency inertial motions. We further identify induced diffusion and the near-resonances at its spectral vicinity as dominating the formation of power-law spectrum.

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Can laboratory experiments reach regimes relevant for the oceanic dynamics?

10.5194/egusphere-egu21-10904 ◽

2021 ◽

Author(s):

Costanza Rodda ◽

Clement Savaro ◽

Antoine Campagne ◽

Miguel Calpe Linares ◽

Pierre Augier ◽

...

Keyword(s):

Gravity Waves ◽

Scaling Laws ◽

Spatial Scales ◽

Scaling Law ◽

Internal Gravity Waves ◽

Energy Spectra ◽

Finite Size Effect ◽

Wave Turbulence ◽

Common Mechanism ◽

Buoyancy Frequency

Atmospheric and oceanic energy spectra are characterized by global scaling laws, suggesting a common mechanism driving the energy route to dissipation. Although several possible theories have been proposed, it is not clear yet what the phenomena contributing the most to the energy at the different spatial scales are. One possible scenario is that internal gravity waves, which can be ubiquitously found in the atmosphere and the ocean and play a fundamental role in the energy transfer, cause the observed spectral slopes at the mesoscales in the atmosphere and submesoscales in the oceans. In the context of this open field of investigation, we present an experimental study where internal gravity waves are forced at a given frequency by the oscillating walls of a large pentagonal-shaped domain filled with a stably stratified fluid. The setup is built inside the 13-meters-diameter tank at the Coriolis facility in Grenoble, where geophysical regimes (with high Reynolds number and low Froude) can be achieved and rotation can also be added. The purpose of our investigation is to determine whether it is possible to induce a wave turbulence cascade by forcing internal waves at the large scales. Following a previous study1, where instead of the pentagonal a square domain was utilized, we obtained the velocity field employing time-resolved particle image velocimetry and then calculated the energy spectra. The previous study inside a square domain showed some evidence of a cascade, but it was strongly affected by 2D modes that sharpened the spectrum. Therefore, we changed the domain shape to a pentagon to reduce this finite-size effect. When the waves are forced at frequency &#969;F=0.4 N, our data shows that the spectra follow the scaling law &#969;-2&#160;at frequencies larger than the forcing frequency and extending beyond N. The experimental spectra strikingly resemble the characteristic Garret-Munk spectrum measured in the ocean. As the interaction of weakly non-linear waves dominates the dynamics at frequencies smaller than the buoyancy frequency N, we can conclude that the experimental spectra are generated by weak internal wave turbulence driving the turbulent cascade at the high-frequency end of the spectrum.&#160;&#160;1 "Generation of weakly nonlinear turbulence of internal gravity waves in the Coriolis facility", C. Savaro, A. Campagne, M. Calpe Linares, P. Augier, J. Sommeria, T. Valran, S. Viboud, and N. Mordant, PRF 2020

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Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves

Boundary-Layer Meteorology ◽

10.1007/s10546-009-9424-0 ◽

2009 ◽

Vol 133 (2) ◽

pp. 139-164 ◽

Cited By ~ 30

Author(s):

S. S. Zilitinkevich ◽

T. Elperin ◽

N. Kleeorin ◽

V. L’vov ◽

I. Rogachevskii

Keyword(s):

Gravity Waves ◽

Internal Gravity Waves ◽

Stratified Flows ◽

Turbulence Closure ◽

Closure Model ◽

Turbulence Closure Model ◽

Stably Stratified Flows

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Amplitude Bounds for Nonlinear Internal Waves

Volume 3: Materials Technology; Ocean Engineering; Polar and Arctic Sciences and Technology; Workshops ◽

10.1115/omae2003-37458 ◽

2003 ◽

Author(s):

Nikolai I. Makarenko ◽

Janna L. Maltseva

Keyword(s):

Internal Waves ◽

Gravity Waves ◽

Density Profile ◽

Analytical Methods ◽

Internal Gravity Waves ◽

Stratified Flows ◽

Fine Scale ◽

Critical Amplitude ◽

The Mean

Amplitude bounds imposed by the conservation of mass, momentum and energy for internal gravity waves are considered. We discuss the theoretical schemes intended for a description of permanent waves just up to the broadening limit. Analytical methods which allow to determine the critical amplitude values for the current with a given density profile are considered. Attention is focused on the continuously stratified flows having multiple broadening limits. The role of the mean density profile and the influence of fine-scale stratification are analysed.

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Turbulence of capillary waves forced by steep gravity waves

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.467 ◽

2018 ◽

Vol 850 ◽

pp. 803-843 ◽

Cited By ~ 5

Author(s):

M. Berhanu ◽

E. Falcon ◽

L. Deike

Keyword(s):

Nonlinear Wave ◽

Power Law ◽

Gravity Waves ◽

Capillary Wave ◽

Capillary Waves ◽

Turbulence Theory ◽

Wave Turbulence ◽

Small Scale ◽

Turbulent Regime ◽

Strong Nonlinearity

We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

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Internal gravity waves in the energy and flux budget turbulence-closure theory for shear-free stably stratified flows

Physical Review E ◽

10.1103/physreve.99.063106 ◽

2019 ◽

Vol 99 (6) ◽

Cited By ~ 2

Author(s):

N. Kleeorin ◽

I. Rogachevskii ◽

I. A. Soustova ◽

Yu. I. Troitskaya ◽

O. S. Ermakova ◽

...

Keyword(s):

Gravity Waves ◽

Internal Gravity Waves ◽

Stratified Flows ◽

Turbulence Closure ◽

Stably Stratified Flows ◽

Closure Theory

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Experimental internal gravity wave turbulence

10.5194/egusphere-egu2020-18419 ◽

2020 ◽

Author(s):

Géraldine Davis ◽

Thierry Dauxois ◽

Sylvain Joubaud ◽

Timothée Jamin ◽

Nicolas Mordant ◽

...

Keyword(s):

Internal Wave ◽

Gravity Waves ◽

Closed Loop ◽

Internal Gravity Wave ◽

Internal Gravity Waves ◽

Wave Turbulence ◽

Large Set ◽

Weakly Nonlinear ◽

Resonant Interactions ◽

Set Up

Stratified fluids may develop simultaneously turbulence and internal wave turbulence, the latter describing a set of a large number of dispersive and weakly nonlinear interacting waves. The description and understanding of this regime for internal gravity waves (IGW) is really an open subject, in particular due to their very unusual dispersion relation. In this presentation, I will show experimental signatures of a large set of weakly interacting IGW obtained in a 2D trapezoidal tank.Due to the peculiar linear reflexion law of IGW on inclined slopes, this setup - for given excitation frequencies - focuses all the input energy on a closed loop called attractor. If the forcing is large enough, this attractor destabilizes and the system eventually achieves a nonlinear cascade in frequencies and wavevectors via triadic resonant interactions, which results at large forcing amplitudes in a k^-3 spatial energy spectrum. I will also show some results obtained in a much larger set-up -the Coriolis facility in Grenoble- with signature of 3D internal wave turbulence.

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Towards a Global Spectral Energy Budget for Internal Gravity Waves in the Ocean

Journal of Physical Oceanography ◽

10.1175/jpo-d-19-0022.1 ◽

2020 ◽

Vol 50 (4) ◽

pp. 935-944 ◽

Cited By ~ 1

Author(s):

Carsten Eden ◽

Friederike Pollmann ◽

Dirk Olbers

Keyword(s):

Kinetic Equation ◽

Gravity Waves ◽

Energy Budget ◽

Wave Energy ◽

Internal Gravity Waves ◽

Rate Of Change ◽

Spectral Energy ◽

Spectral Space ◽

Wave Interactions ◽

Energy Transfers

AbstractEnergy transfers by internal gravity wave–wave interactions in spectral space are diagnosed from numerical model simulations initialized with realizations of the Garrett–Munk spectrum in physical space and compared with the predictions of the so-called scattering integral or kinetic equation. Averaging the random phase of the initialization, the energy transfers by wave–wave interactions in the model agree well with the predictions of the kinetic equation for certain ranges of frequency and wavenumbers. This validation allows now, in principle, the use of the energy transfers predicted by the kinetic equation to design a global spectral energy budget for internal gravity waves in the ocean where divergences of energy transports in physical and spectral space balance forcing, dissipation, the energy transfers by the wave–wave interactions, or the rate of change of the spectral wave energy. First global estimates show indeed accumulation of the wave energy in a range of latitude ϕ consistent with tidal waves at frequency ωT propagating toward the latitudinal window where 2 < ωT/f(ϕ) < 3, as predicted by the kinetic equation.

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