Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom

We consider the effect of slowly varying depth on the values of skewness and kurtosis of weakly nonlinear irregular waves propagating from deeper to shallower water. It is known that the equilibrium value of kurtosis decreases with decreasing depth for waves propagating on constant depth. Waves propagating over a sloping bottom must continually adjust toward a new equilibrium state. We demonstrate that weakly nonlinear waves may need a considerable horizontal propagation distance in order to adjust to a new shallower environment, therefore the kurtosis can be notably different from the equilibrium value for each corresponding depth both on top of and beyond a bottom slope. A change of depth can provoke a wake-like spatially non-uniform distribution of kurtosis on the lee side of the slope. As an application, we anticipate that the probability of freak waves on or near the edge of the continental shelf may exhibit a rather complicated spatial structure for wave fields entering from deep sea

Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry

Experimental results describing random, uni-directional, long crested, water waves over non-uniform bathymetry confirm the formation of stable coherent wave packages traveling with almost uniform group velocity. The waves are generated with JONSWAP spectrum for various steepness, height and constant period. A set of statistical procedures were applied to the experimental data, including the space and time variation of kurtosis, skewness, BFI, Fourier and moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random field and traveling over shoals, valleys and slopes were compared with exact solutions of the NLS equation resulting in good matches and demonstrating that these packages are very similar to deep water breathers solutions, surviving over the non-uniform bathymetry. We also present events of formation of rogue waves over those regions where the BFI, kurtosis and skewness coefficients have maximal values.Comment: 41 pages, 21 figure

Freak Waves in Random Oceanic Sea States

Freak waves are very large, rare events in a random ocean wave train. Here we study the numerical generation of freak waves in a random sea state characterized by the JONSWAP power spectrum. We assume, to cubic order in nonlinearity, that the wave dynamics are governed by the nonlinear Schroedinger (NLS) equation. We identify two parameters in the power spectrum that control the nonlinear dynamics: the Phillips parameter $\alpha$ and the enhancement coefficient $\gamma$. We discuss how freak waves in a random sea state are more likely to occur for large values of $\alpha$ and $\gamma$. Our results are supported by extensive numerical simulations of the NLS equation with random initial conditions. Comparison with linear simulations are also reported.Comment: 7 pages, 6 figures, to be published in Phys. Rev. Let

Feeding of plankton in turbulent oceans and lakes

Analytical models for the statistical distribution of the gut content of fish larvae in a turbulent ocean environment are compared to data obtained in a field experiment. The proposed model allows the nutrition state and thereby the survival probability of plankton populations to be estimated for given conditions and parameters characterizing their environment, i.e., prey concentrations and turbulence levels. These parameters are all available in the field data. Other parameters such as the capture range and fields of view together with a characteristic time for digesting prey are assumed to be known. The analysis allows an estimate for the probability density of the gut content of plankton in terms of the number of nauplii in the gut. In particular, the analytical results give a basis for evaluating the average gut content of a given plankton population on the basis of basic information concerning the prey concentration and the turbulence intensity. Also analytical models for the prey capture rates are compared with results based on the field data. The analysis emphasizes the effects of turbulence.publishedVersio

Spectral up- and downshifting of Akhmediev breathers under wind forcing

We experimentally and numerically investigate the effect of wind forcing on the spectral dynamics of Akhmediev breathers, a wave-type known to model the modulation instability. We develop the wind model to the same order in steepness as the higher order modifcation of the nonlinear Schroedinger equation, also referred to as the Dysthe equation. This results in an asymmetric wind term in the higher order, in addition to the leading order wind forcing term. The derived model is in good agreement with laboratory experiments within the range of the facility's length. We show that the leading order forcing term amplifies all frequencies equally and therefore induces only a broadening of the spectrum while the asymmetric higher order term in the model enhances higher frequencies more than lower ones. Thus, the latter term induces a permanent upshift of the spectral mean. On the other hand, in contrast to the direct effect of wind forcing, wind can indirectly lead to frequency downshifts, due to dissipative effects such as wave breaking, or through amplification of the intrinsic spectral asymmetry of the Dysthe equation. Furthermore, the definitions of the up- and downshift in terms of peak- and mean frequencies, that are critical to relate our work to previous results, are highlighted and discussed.Comment: 30 pages, 11 figure

Low-frequency electrostatic waves in the ionospheric E-region: a comparison of rocket observations and numerical simulations

International audienceLow frequency electrostatic waves in the lower parts of the ionosphere are studied by a comparison of observations by instrumented rockets and of results from numerical simulations. Particular attention is given to the spectral properties of the waves. On the basis of a good agreement between the observations and the simulations, it can be argued that the most important nonlinear dynamics can be accounted for in a 2-D numerical model, referring to a plane perpendicular to a locally homogeneous magnetic field. It does not seem necessary to take into account turbulent fluctuations or motions in the neutral gas component. The numerical simulations explain the observed strongly intermittent nature of the fluctuations: secondary instabilities develop on the large scale gradients of the largest amplitude waves, and the small scale dynamics is strongly influenced by these secondary instabilities. We compare potential variations obtained at a single position in the numerical simulations with two point potential-difference signals, where the latter is the adequate representation for the data obtained by instrumented rockets. We can demonstrate a significant reduction in the amount of information concerning the plasma turbulence when the latter signal is used for analysis. In particular we show that the bicoherence estimate is strongly affected. The conclusions have implications for studies of low frequency ionospheric fluctuations in the E and F regions by instrumented rockets, and also for other methods relying on difference measurements, using two probes with large separation. The analysis also resolves a long standing controversy concerning the supersonic phase velocities of these cross-field instabilities being observed in laboratory experiments

Evolution of Rogue Waves in Interacting Wave Systems

Large amplitude water waves on deep water has long been known in the sea faring community, and the cause of great concern for, e.g., oil platform constructions. The concept of such freak waves is nowadays, thanks to satellite and radar measurements, well established within the scientific community. There are a number of important models and approaches for the theoretical description of such waves. By analyzing the scaling behavior of freak wave formation in a model of two interacting waves, described by two coupled nonlinear Schroedinger equations, we show that there are two different dynamical scaling behaviors above and below a critical angle theta_c of the direction of the interacting waves below theta_c all wave systems evolve and display statistics similar to a wave system of non-interacting waves. The results equally apply to other systems described by the nonlinear Schroedinger equations, and should be of interest when designing optical wave guides.Comment: 5 pages, 2 figures, to appear in Europhysics Letter

Horizon effects with surface waves on moving water

Surface waves on a stationary flow of water are considered, in a linear model that includes the surface tension of the fluid. The resulting gravity-capillary waves experience a rich array of horizon effects when propagating against the flow. In some cases three horizons (points where the group velocity of the wave reverses) exist for waves with a single laboratory frequency. Some of these effects are familiar in fluid mechanics under the name of wave blocking, but other aspects, in particular waves with negative co-moving frequency and the Hawking effect, were overlooked until surface waves were investigated as examples of analogue gravity [Sch\"utzhold R and Unruh W G 2002 Phys. Rev. D 66 044019]. A comprehensive presentation of the various horizon effects for gravity-capillary waves is given, with emphasis on the deep water/short wavelength case kh>>1 where many analytical results can be derived. A similarity of the state space of the waves to that of a thermodynamic system is pointed out.Comment: 30 pages, 15 figures. Minor change

Did the Draupner wave occur in a crossing sea?

The ‘New Year Wave’ was recorded at the Draupner platform in the North Sea and is a rare high quality measurement of a ‘freak’ or ‘rogue’ wave. The wave has been the subject of much interest and numerous studies. Despite this, the event has still not been satisfactorily explained. One piece of information which was not directly measured at the platform, but which is vital to understanding the nonlinear dynamics is the wave’s directional spreading. This paper investigates the directionality of the Draupner wave and concludes it might have resulted from two wave-groups crossing, whose mean wave directions were separated by about 90◦ or more. This result has been deduced from a set-up of the low frequency second order difference waves under the giant wave, which can be explained only if two wave systems are propagating at such an angle. To check whether second order theory is satisfactory for such a highly non-linear event, we have run numerical simulations using a fully non-linear potential flow solver, which confirm the conclusion deduced from the second order theory. This is backed up by a hindcast from ECMWF which shows swell waves propagating at ∼ 80◦ to the wind sea. Other evidence which supports our conclusion are the measured forces on the structure, the magnitude of the second order sum waves and some other instances of freak waves occurring in crossing sea states