Wave Action Spectrum Research Articles

Non-local gravity wave turbulence in presence of condensate

The $k^{-23/6}$ wave action spectrum with an inverse cascade is one of the fundamental Kolmogorov–Zakharov solutions for gravity wave turbulence, which is part of the citation for the Dirac Medal in 2003. Instead of confirming this solution, however, several existing simulations and experiments suggest a spectrum of $k^{-3}$ in set-ups corresponding to the inverse cascade. We provide a theoretical explanation for the latter, considering the condensate that naturally forms in finite domains of experiments/simulations. Our new theory hinges on: (1) derivation of a spectral diffusion equation when non-local interactions with the condensate become dominant, for the first time systematically formulated for quartet-interaction systems; and (2) careful analysis of the asymptotics of interaction coefficient with a remarkable cancellation of all leading-order terms.

Testing wave turbulence theory for the Gross-Pitaevskii system.

We test the predictions of the theory of weak wave turbulence by performing numerical simulations of the Gross-Pitaevskii equation(GPE) and the associated wave-kinetic equation(WKE). We consider an initial state localized in Fourier space, and we confront the solutions of the WKE obtained numerically with GPE data for both the wave-action spectrum and the probability density functions (PDFs) of the Fourier mode intensities. We find that the temporal evolution of the GPE data is accurately predicted by the WKE, with no adjustable parameters, for about two nonlinear kinetic times. Qualitative agreement between the GPE and the WKE persists also for longer times with some quantitative deviations that may be attributed to the combination of a breakdown of the theoretical assumptions underlying the WKE as well as numerical issues. Furthermore, we study how the wave statistics evolves toward Gaussianity in a timescale of the order of the kinetic time. The excellent agreement between direct numerical simulations of the GPE and the WKE provides a solid foundation to the theory of weak wave turbulence.

Nonequilibrium Bose-Einstein condensation

We investigate formation of Bose-Einstein condensates under non-equilibrium conditions using numerical simulations of the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and the forcing at high wave numbers, and study propagation of the turbulent spectrum toward the low wave numbers. Our primary goal is to compare the results for the evolving spectrum with the previous results obtained for the kinetic equation of weak wave turbulence. We demonstrate existence of a regime for which good agreement with the wave turbulence results is found in terms of the main features of the previously discussed self-similar solution. In particular, we find a reasonable agreement with the low-frequency and the high-frequency power-law asymptotics of the evolving solution, including the anomalous power-law exponent $x^* \approx 1.24$ for the three-dimensional waveaction spectrum. We also study the regimes of very weak turbulence, when the evolution is affected by the discreteness of the Fourier space, and the strong turbulence regime when emerging condensate modifies the wave dynamics and leads to formation of strongly nonlinear filamentary vortices.

Thermalization of an Oscillating Bose Condensate in a Disordered Trap

Previously, we numerically showed that thermalization can occur in an oscillating Bose–Einstein condensate with a disordered harmonic trap when the healing length $$\xi $$ of the condensate is shorter than the correlation length $$\sigma _{D}$$ of the Gaussian disorder [see, for example, the experiment reported in Dries et al. (Phys Rev A 82:033603, 2010)]. In this work, we investigate and show that in the $$\xi >\sigma _{D}$$ (Anderson localization) regime, the system can also exhibit a relaxation process from nonequilibrium to equilibrium. In such an isolated quantum system, energy and particle number are conserved and the irreversible evolution toward thermodynamic equilibrium is induced by the disorder. The thermodynamic equilibrium is evidenced by the maximized entropy $$S\left[ n_{k}\right] $$ in which the waveaction spectrum $$n_{k}$$ follows the Rayleigh–Jeans distribution. Besides, unlike a monotonic irreversible process of thermalization to equilibrium, the Fermi–Pasta–Ulam–Tsingou recurrence arises in this system, manifested by the oscillation of the nonequilibrium entropy.

Strongly interacting soliton gas and formation of rogue waves

We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional Nonlinear Schr{\"o}dinger (NLS) equation. In order to model such gas we use N-soliton solutions (N-SS) with $N\sim 100$, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum and the probability density function (PDF) of wave intensity. We show that in the case of small soliton density the kinetic and potential energies, as well as the kurtosis, are very well described by the analytical relations derived without taking into account soliton interactions. With increasing soliton density and velocities, soliton interactions enhance, and we observe increasing deviations from these relations leading to increased absolute values for all of these three characteristics. The wave-action spectrum is smooth, decays close to exponentially at large wavenumbers and widens with increasing soliton density and velocities. The PDF of wave intensity deviates from the exponential (Rayleigh) PDF drastically for rarefied soliton gas, transforming much closer to it at densities corresponding to essential interaction between the solitons. Rogue waves emerging in soliton gas are multi-soliton collisions, and yet some of them have spatial profiles very similar to those of the Peregrine solutions of different orders. We present example of three-soliton collision, for which even the temporal behavior of the maximal amplitude is very well approximated by the Peregrine solution of the second order.

Reflected wave solution of Alfvén wave turbulence

It was found numerically in Galtier et al (2000 J. Plasma Phys. 63 447–88) that the evolution towards the Kolmogorov–Zakharov (KZ) wave action spectrum of weak magneto–hydrodynamic turbulence (here is the wave number component perpendicular to the external magnetic field) consists of two stages. Stage one is characterised by a wave action spectrum with a front propagating to the maximum wave number kmax in a finite time t*, arguably with as , and with a power law which is steeper than KZ, . This stage was recently studied in Bell et al (2017 J. Phys. A: Math. Theor. 50 435501) by identifying it with a self-similar solution of the second kind. In particular, a corrected value of and the structure of the spectral front were found. In the present paper, we study the second (post-t*) stage which is characterised by formation of the KZ spectrum as an inversely propagating wave, from kmax toward lower wave numbers. We identify this stage with a self-similar solution of the third kind, whose similarity coefficients are fixed by the previous (pre-t*) self-similar stage rather than the energy conservation (as in the first-kind self-similarity) or by a nonlinear eigenvalue problem (as in the second-kind self-similarity).

Nonstationary distributions of wave intensities in wave turbulence

We obtain a general solution for the probability density function (PDF) of wave intensities in non-stationary wave turbulence. The solution is expressed in terms of the initial PDF and the wave action spectrum satisfying the wave-kinetic equation. We establish that, in the absence of wave breaking, the wave statistics converge to a Gaussian distribution in forced-dissipated wave systems while approaching a steady state. Also, we find that in non-stationary systems, if the statistic is Gaussian initially, it will remain Gaussian for all time. Generally, if the statistic is not initially Gaussian, it will remain non-Gaussian over the characteristic nonlinear evolution time of the wave spectrum. In freely decaying wave turbulence, substantial deviations from Gaussianity may persist infinitely long.

Integrable turbulence and formation of rogue waves

In the framework of the focusing nonlinear Schrödinger equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. The development of the MI leads to the formation of ‘integrable turbulence’ (Zakharov 2009 Stud. Appl. Math. 122 219–34). We study the time evolution of its major characteristics averaged across realizations of initial data—the condensate solution seeded by small random noise with fixed statistical properties.We observe that the system asymptotically approaches to the stationary integrable turbulence, however this is a long process. During this process momenta, as well as kinetic and potential energies, oscillate around their asymptotic values. The amplitudes of these oscillations decay with time t as t−3/2, the phases contain the nonlinear phase shift that decays as t−1/2, and the frequency of the oscillations is equal to the double maximum growth rate of the MI. The evolution of wave-action spectrum is also oscillatory, and characterized by formation of the power-law region ∼|k|−α in the small vicinity of the zeroth harmonic k = 0 with exponent α close to 2/3. The corresponding modes form ‘quasi-condensate’, that acquires very significant wave action and macroscopic potential energy.The probability density function of wave amplitudes asymptotically approaches the Rayleigh distribution in an oscillatory way. Nevertheless, in the beginning of the nonlinear stage the MI slightly increases the occurrence of rogue waves. This takes place at the moments of potential energy modulus minima, where the PDF acquires ‘fat tales’ and the probability of rogue waves occurrence is by about two times larger than in the asymptotic stationary state.Presented facts need a theoretical explanation.

Scattering of surface gravity waves by bottom topography with a current

A theory is presented that describes the scattering of random surface gravity waves by small-amplitude topography, with horizontal scales of the order of the wavelength, in the presence of an irrotational and almost uniform current. A perturbation expansion of the wave action to order η2 yields an evolution equation for the wave action spectrum, where η = max(h)/H is the small-scale bottom amplitude normalized by the mean water depth. Spectral wave evolution is proportional to the bottom elevation variance at the resonant wavenumbers, representing a Bragg scattering approximation. With a current, scattering results from a direct effect of the bottom topography, and an indirect effect of the bottom through the modulations of the surface current and mean surface elevation. For Froude numbers of the order of 0.6 or less, the bottom topography effects dominate. For all Froude numbers, the reflection coefficients for the wave amplitudes that are inferred from the wave action source term are asymptotically identical, as η goes to zero, to previous theoretical results for monochromatic waves propagating in one dimension over sinusoidal bars. In particular, the frequency of the most reflected wave components is shifted by the current, and wave action conservation results in amplified reflected wave energies for following currents. Application of the theory to waves over current-generated sandwaves suggests that forward scattering can be significant, resulting in a broadening of the directional wave spectrum, while back-scattering should be generally weaker.

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.

Statistics of gravity waves obtained by direct numerical simulation

We perform direct numerical simulations of dynamic equations of decaying gravity waves on infinite-depth water. Power-law behaviour of the wave action spectrum and structure functions of the surface elevation is obtained. These power laws agree with the prediction of the weak turbulence theory. The probability density function (p.d.f.) of the surface elevation is close to the Gaussian distribution around the mean value which seems to be consistent with the random phase approximation. However, the p.d.f. deviates weakly from the Gaussian in the tail region. This deviation is significant and can be amplified by taking the Laplacian. In addition, intermittency and breakdown of the weak turbulence theory are discussed.

A comparison of two cartographic exposure methods using Fucus vesiculosus as an indicator

Morphological variation and vertical distribution of Fucus vesiculosus were quantified at several sites in the Finnish archipelago (Baltic Sea). F. vesiculosus samples were obtained from skerries at geographical distances of 1 km or more (large scale) and at intervals of ca 100 m around a single island (small scale). The results were examined in relation to wave exposure, calculated by Baardseth and effective fetch cartographic methods. Despite the fact that the exposure indices were calculated differently they correlated strongly. Vegetative morphological characteristics of F. vesiculosus illustrate the morphological differences both within and between exposure gradients. The tallest and widest F. vesiculosus plants were found at the sheltered end of the large-scale exposure gradient. Those from equally sheltered sites of the island were smaller in all respects. Thus, the trend from small narrow plants to large wide sheltered plants was expressed differently over the different geographical scales. Consequently localities with similar exposure indices may have morphologically different F. vesiculosus populations. Shores with similar cartographic exposure indices can be different in nature. Underwater topography and shore locations, either close to the mainland or at the outermost sites of the archipelago, affect the exposure. Although a sheltered shore is indicated, the sublittoral zone may be quite exposed to the movements of water. In contrast, in an open shore environment underwater rocks, boulders and shallow water areas can provide sheltered habitats. The depth range of the F. vesiculosus belt exhibited two distinctive patterns. At sheltered sites, around islands in the outermost reaches of the archipelago F. vesiculosus can grow to a maximum depth of 5 m. In exposed habitats the belt becomes narrower, reaching a maximum depth of 3 m. Closer to the mainland F. vesiculosus is found at exposed sites to a maximum depth of 5 m; the depth range at sheltered sites is narrower, only reaching depths of 2 m or less. In conclusion, the changes in plant morphology and in the vertical belt distribution are similar to each other along both gradients at the exposed ends of the wave action spectrum; however, the two gradients diverge at the sheltered ends of the spectrum.

A Eulerian–Lagrangian method for the simulation of wave propagation

A Eulerian–Lagrangian method (ELM) is employed for the simulation of wave propagation in the present research. The wave action conservation equation, instead of the wave energy balance equation, is used. The wave action is conservative and the action flux remains constant along the wave rays. The ELM correctly accounts for this physical characteristic of wave propagation and integrates the wave action spectrum along the wave rays. Thus, the total derivative for wave action spectrum may be introduced into the numerical scheme and the complicated partial differential wave action balance equation is simplified into an ordinary differential equation. A number of test cases on wave propagation are carried out and show that the present method is stable, accurate and efficient. The results are compared with analytical solutions and/or other computed results. It is shown that the ELM is superior to the first-order upwind method in accuracy, stability and efficiency and may better reflect the complicated dynamics due to the complicated bathymetry features in shallow water areas.

Numerical investigation of depth and current refraction of waves

A numerical investigation of depth and current refraction of surface waves for several simple test cases is presented. The refraction scheme is incorporated into a third‐generation wave prediction model. It models the effect upon wave propagation of the temporally and spatially varying sea surface elevations and mean currents associated with tides and storm surges. The scheme solves the transport equation for the wave action spectrum. The magnitude of bathymetric depth refraction is large for the lower frequency swell waves. The accuracy to which the proposed scheme can predict refraction is dependent upon the directional resolution of the spectrum. A directional resolution of at least 15° is required; the use of a higher directional resolution may be limited by computational resource restrictions. Current refraction effects are very significant for wave propagation across a simple current eddy. Refraction effects are not limited to just a turning of the waves, but also involve significant changes in the spectral shape. Some of the test results exhibit excessive numerical dispersion associated with the use of upwind differencing. The magnitude of this dispersion is much reduced by employing a simple hybrid differencing scheme in wave frequency‐direction space. The high degree of numerical dispersion present in the model results is also a result of the nature of the simplified test cases. The magnitude of the numerical dispersion produced by the upwind differencing technique may not be significant when the model is used for storm simulation.