Yuri V. Lvov

Hamiltonian formalism and the Garrett-Munk spectrum of internal waves in the ocean.

Wave turbulence formalism for long internal waves in a stratified fluid is developed, based on a natural Hamiltonian description. A kinetic equation appropriate for the description of spectral energy transfer is derived, and its anisotropic self-similar stationary solution corresponding to a direct cascade of energy toward the short scales is found. This solution is very close to the high wave-number limit of the Garrett-Munk spectrum of long internal waves in the ocean. In fact, a small modification of the Garrett-Munk formalism includes a spectrum consistent with the one predicted by wave turbulence.

TOWARD REGIONAL CHARACTERIZATIONS OF THE OCEANIC INTERNAL WAVEFIELD

Many major oceanographic internal wave observational programs of the last 4 decades are reanalyzed in order to characterize variability of the deep ocean internal wavefield. The observations are discussed in the context of the universal spectral model proposed by Garrett and Munk. The Garrett and Munk model is a good description of wintertime conditions at Site-D on the continental rise north of the Gulf Stream. Elsewhere and at other times, significant deviations in terms of amplitude, separability of the 2-D vertical wavenumber - frequency spectrum, and departure from the models functional form are noted. Subtle geographic patterns are apparent in deviations from the high frequency and high vertical wavenumber power laws of the Garrett and Munk spectrum. Moreover, such deviations tend to co-vary: whiter frequency spectra are partnered with redder vertical wavenumber spectra. Attempts are made to interpret the variability in terms of the interplay between generation, propagation and nonlinearity using a statistical radiative balance equation. This process frames major questions for future research with the insight that such integrative studies could constrain both observationally and theoretically based interpretations.

Five-wave interaction on the surface of deep fluid

This article deals with the studying of the interaction of gravity waves propagating on the surface of an ideal fluid of infinite depth. The system of the corresponding equation is proven to be integrable up to the fourth order in power of steepness of the waves, but to be nonintegrable in the next, fifth, order. An exact formula for the five-wave scattering matrix element is obtained using diagram technique on the resonant surface. The stationary solutions of the five-wave kinetic equation are studied as well.

Noisy spectra, long correlations, and intermittency in wave turbulence.

We study the k-space fluctuations of the wave action about its mean spectrum in the turbulence of dispersive waves. We use a minimal model based on the random phase approximation (RPA) and derive evolution equations for the arbitrary-order one-point moments of the wave intensity in the wave-number space. The first equation in this series is the familiar kinetic equation for the mean wave-action spectrum, whereas the second and higher equations describe the fluctuations about this mean spectrum. The fluctuations exhibit a nontrivial dynamics if some long coordinate-space correlations are present in the system, as it is the case in typical numerical and laboratory experiments. Without such long-range correlations, the fluctuations are trivially fixed at their Gaussian values and cannot evolve even if the wave field itself is non-Gaussian in the coordinate space. Unlike the previous approaches based on smooth initial k-space cumulants, the RPA model works even for extreme cases where the k-space fluctuations are absent or very large and intermittent. We show that any initial non-Gaussianity at small amplitudes propagates without change toward the high amplitudes at each fixed wave number. At each fixed amplitude, however, the probability distribution function becomes Gaussian at large time.

Energy spectra of the Ocean's internal wave field: Theory and observations

The high-frequency limit of the Garrett and Munk spectrum of internal waves in the ocean and the observed deviations from it are shown to form a pattern consistent with the predictions of wave turbulence theory. In particular, the high-frequency limit of the Garrett and Munk spectrum constitutes an exact steady-state solution of the corresponding kinetic equation.

Joint statistics of amplitudes and phases in wave turbulence

Abstract Random Phase Approximation (RPA) provides a very convenient tool to study the ensembles of weakly interacting waves, commonly called wave turbulence. In its traditional formulation, RPA assumes that phases of interacting waves are random quantities but it usually ignores randomness of their amplitudes. Recently, RPA was generalised in a way that takes into account the amplitude randomness and it was applied to study of the higher momenta and probability densities of wave amplitudes. However, to have a meaningful description of wave turbulence, the RPA properties assumed for the initial fields must be proven to survive over the nonlinear evolution time, and such a proof is the main goal of the present paper. We derive an evolution equation for the full probability density function which contains the complete information about the joint statistics of all wave amplitudes and phases. We show that, for any initial statistics of the amplitudes, the phase factors remain statistically independent uniformly distributed variables. If in addition the initial amplitudes are also independent variables (but with arbitrary distributions) they will remain independent when considered in small sets which are much less than the total number of modes. However, if the size of a set is of order of the total number of modes then the joint probability density for this set is not factorisable into the product of one-mode probabilities. In the other words, the modes in such a set are involved in a “collective” (correlated) motion. We also study new type of correlators describing the phase statistics.

Wave turbulence in Bose-Einstein condensates

Abstract The kinetics of nonequilibrium Bose–Einstein condensates (BEC) are considered within the framework of the Gross–Pitaevskii (GP) equation. A systematic derivation is given for weak small-scale perturbations of a steady confined condensate state. This approach combines a wavepacket WKB description with the weak turbulence theory. The WKB theory derived in this paper describes the effect of the condensate on the short-wave excitations which appears to be different from a simple renormalization of the confining potential suggested in previous literature.

Probability densities and preservation of randomness in wave turbulence

Time evolution equation for the probability distribution function (PDF) is derived for system of weakly interacting w dominated by the four-wave process. It is shown that a steady state for such system may correspond to strong inte Numerical simulation performed on the surface gravity waves equations demonstrate an order of magnitude increase bilities of long large-amplitude waves with respect to Rayleigh distribution.  2005 Elsevier B.V. All rights reserved.

Resonant and near-resonant internal wave interactions

Abstract The spectral energy density of the internal waves in the open ocean is considered. The Garrett and Munk spectrum and the resonant kinetic equation are used as the main tools of the study. Evaluations of a resonant kinetic equation that suggest the slow time evolution of the Garrett and Munk spectrum is not in fact slow are reported. Instead, nonlinear transfers lead to evolution time scales that are smaller than one wave period at high vertical wavenumber. Such values of the transfer rates are inconsistent with the viewpoint expressed in papers by C. H. McComas and P. Muller, and by P. Muller et al., which regards the Garrett and Munk spectrum as an approximate stationary state of the resonant kinetic equation. It also puts the self-consistency of a resonant kinetic equation at a serious risk. The possible reasons for and resolutions of this paradox are explored. Inclusion of near-resonant interactions decreases the rate at which the spectrum evolves. Consequently, this inclusion shows a tendency...

A Hamiltonian Formulation for Long Internal Waves

A novel canonical Hamiltonian formalism is developed for long internal waves in a rotating environment. This includes the effects of background vorticity and shear on the waves. By restricting consideration to flows in hydrostatic balance, superimposed on a horizontally uniform background of vertical shear and vorticity, a particularly simple Hamiltonian structure arises, which can be thought of as describing a nonlinearly coupled infinite collection of shallow water systems. The kinetic equation describing the time evolution of the spectral energy of internal waves is subsequently derived, and a stationary Kolmogorov solution is found in the high frequency limit. This is surprisingly close to the Garrett–Munk spectrum of oceanic internal waves.