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Yuri V. Lvov

Publications and source records attributed to Yuri V. Lvov.

32 records · Page 2Linked to original sources
arXiv2008

Nonlinear wave-wave interactions in stratified flows: Direct numerical simulations

To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynamical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we provide evidence that formation of the energy spectra in the inertial subrange is dominated by scale-separated interactions with the near-inertial waves. These findings support oceanographers' intuition that spectral energy density of internal waves is the result of predominantly wave-wave interactions.

physics.flu-dyn↗
arXiv2007

Interactions of renormalized waves in thermalized Fermi-Pasta-Ulam chains

The dispersive interacting waves in Fermi-Pasta-Ulam (FPU) chains of particles in \textit{thermal equilibrium} are studied from both statistical and wave resonance perspectives. It is shown that, even in a strongly nonlinear regime, the chain in thermal equilibrium can be effectively described by a system of weakly interacting \textit{renormalized} nonlinear waves that possess (i) the Rayleigh-Jeans distribution and (ii) zero correlations between waves, just as noninteracting free waves would. This renormalization is achieved through a set of canonical transformations. The renormalized linear dispersion of these renormalized waves is obtained and shown to be in excellent agreement with numerical experiments. Moreover, a dynamical interpretation of the renormalization of the dispersion relation is provided via a self-consistency, mean-field argument. It turns out that this renormalization arises mainly from the trivial resonant wave interactions, i.e., interactions with no momentum exchange. Furthermore, using a multiple time-scale, statistical averaging method, we show that the interactions of near-resonant waves give rise to the broadening of the resonance peaks in the frequency spectrum of renormalized modes. The theoretical prediction for the resonance width for the thermalized $β$-FPU chain is found to be in very good agreement with its numerically measured value.

nlin.CD↗
arXiv2006

Discreteness and its effect on the water-wave turbulence

We perform numerical simulations of the dynamical equations for free water surface in finite basin in presence of gravity. Wave Turbulence (WT) is a theory derived for describing statistics of weakly nonlinear waves in the infinite basin limit. Its formal applicability condition on the minimal size of the computational basin is impossible to satisfy in present numerical simulations, and the number of wave resonances is significantly depleted due to the wavenumber discreteness. The goal of this paper will be to examine which WT predictions survive in such discrete systems with depleted resonances and which properties arise specifically due to the discreteness effects. As in \cite{DKZ,onorato,naoto}, our results for the wave spectrum agree with the Zakharov-Filonenko spectrum predicted within WT. We also go beyond finding the spectra and compute probability density function (PDF) of the wave amplitudes and observe an anomalously large, with respect to Gaussian, probability of strong waves which is consistent with recent theory \cite{clnp,cln}. Using a simple model for quasi-resonances we predict an effect arising purely due to discreteness: existence of a threshold wave intensity above which turbulent cascade develops and proceeds to arbitrarily small scales. Numerically, we observe that the energy cascade is very ``bursty'' in time and is somewhat similar to sporadic sandpile avalanches. We explain this as a cycle: a cascade arrest due to discreteness leads to accumulation of energy near the forcing scale which, in turn, leads to widening of the nonlinear resonance and, therefore, triggering of the cascade draining the turbulence levels and returning the system to the beginning of the cycle. ~

math-ph↗
arXiv2005

Renormalized waves and discrete breathers in $β$-FPU chains

We demonstrate via numerical simulation that in the \textit{strongly} nonlinear limit, the $β$-FPU system in thermal equilibrium behaves surprisingly like weakly nonlinear waves in properly renormalized normal variables. This arises because the collective effect of strongly nonlinear interactions effectively renormalizes linear dispersion frequency and leads to effectively weak interaction among these renormalized waves. Furthermore, we show that the dynamical scenario for thermalized $β$-FPU chains is spatially highly localized discrete breathers riding chaotically on spatially extended, renormalized waves.

math-ph↗
arXiv2005

Wave Turbulence

In this paper we review recent developments in the statistical theory of weakly nonlinear dispersive waves, the subject known as Wave Turbulence (WT). We revise WT theory using a generalisation of the random phase approximation (RPA). This generalisation takes into account that not only the phases but also the amplitudes of the wave Fourier modes are random quantities and it is called the ``Random Phase and Amplitude'' approach. This approach allows to systematically derive the kinetic equation for the energy spectrum from the the Peierls-Brout-Prigogine (PBP) equation for the multi-mode probability density function (PDF). The PBP equation was originally derived for the three-wave systems and in the present paper we derive a similar equation for the four-wave case. Equation for the multi-mode PDF will be used to validate the statistical assumptions about the phase and the amplitude randomness used for WT closures. Further, the multi-mode PDF contains a detailed statistical information, beyond spectra, and it finally allows to study non-Gaussianity and intermittency in WT, as it will be described in the present paper. In particular, we will show that intermittency of stochastic nonlinear waves is related to a flux of probability in the space of wave amplitudes.

math-ph↗
arXiv2005

Joint statistics of amplitudes and phases in Wave Turbulence

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.

math-ph↗
arXiv2004

Probability densities and preservation of randomness in wave turbulence

Turbulence closure for the weakly nonlinear stochastic waves requires, besides weak nonlinearity, randomness in both the phases and the amplitudes of the Fourier modes. This randomness, once present initially, must remain over the nonlinear evolution time. Finding out to what extent is this true is the main goal of the present Letter. For this analysis we derive an evolution equation for the full probability density function (PDF) of the wave field. We will show that, for any statistics of the amplitudes, phases tend to stay random if they were random initially. If in addition the initial amplitudes are independent variables they will remain independent in a coarse-grained sense, i.e. when considered in small subsets which are much less than the total set of modes.

math-ph↗
arXiv2003

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 {\it exact} steady state solution of the corresponding kinetic equation.

math-ph↗
arXiv2003

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

We study the k-space fluctuations of the waveaction 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 wavenumber space. The first equation in this series is the familiar Kinetic Equation for the mean waveaction 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 wavefield 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 wavenumber. At each fixed amplitude, however, the PDF becomes Gaussian at large time.

math-ph↗
arXiv2003

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.

math-ph↗
arXiv2003

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 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 wavenumber 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.

nlin.CD↗
arXiv1999

Dispersion management in optical fiber links: Integrability in leading nonlinear order

We show that an integro-differential equation model for pulse propagation in optical transmission lines with dispersion management, is integrable at the {\it leading nonlinear order}. This equation can be transformed into the nonlinear Schroedinger equation by a near-identity canonical transformation for the case of weak dispersion. We also derive the next order (nonintegrable) correction.

chao-dyn↗
arXiv1997

Semiconductor Lasers and Kolmogorov Spectra

In this article, we make a prima facie case that there could be distinct advantages to exploiting a new class of finite flux equilibrium solutions of the Quantum Boltzmann equation in semiconductor lasers.

math-ph↗