Benjamin Feir Research Articles

On the maximal temporal amplitude of down stream running nonlinear water waves

This paper concerns with the down-stream propagation of waves over initially still water. Such a study is relevant to generate waves of large amplitude in wave tanks of a hydrodynamic laboratory. Input in the form of a time signal is provided at the wave-maker located at one side of the wave tank; the resulting wave then propagates over initially still water towards the beach at the other side of the tank. Experiments show that nonlinear effects will deform the wave and may lead to large waves with wave heights larger than twice the original input; the deformations may show itself as peaking and splitting. It is of direct scientific interest to understand and quantify the nonlinear distortion; it is also of much practical interest to know at which location in the wave tank, the extreme position, the waves will achieve their maximum amplitude and to know the amplitude amplification factor. To investigate this, a previously introduced concept called Maximal Temporal Amplitude (MTA) is used: at each location the maximum over time of the wave elevation. An explicit expression of the MTA cannot be found in general from the governing equations and generating signal. In this paper we will use a Korteweg - de Vries (KdV) model and third order approximation theory to calculate the approximate extreme positions for two classes of waves. The classes are the wave-groups that originate from initially bi-chromatics and Benjamin-Feir (BF) type of waves, described by superposition of two or three monochromatic waves. We show that for initially bi-chromatics signals, the extreme position does not depend on the phases of the mono-chromatic components. For BF signals, however, the phases of the mono-chromatic components influence the extreme position essentially. The theoretical results are verified for the case of bi-chromatics with numerical as well as experimental results; for BF signals we use an analytical solution called the Soliton on Finite Background (SFB) for comparison.

Can swell increase the number of freak waves in a wind sea?

The effect of a swell on the statistical distribution of a directional short-wave field is investigated. Starting from Zakharov's spectral formulation, we derive a new modified nonlinear Schrödinger equation appropriate for the nonlinear evolution of a narrow-banded spectrum of short waves influenced by a swell. The swell-modified equation is solved analytically to yield an extended version of the result of Longuet-Higgins & Stewart (J. Fluid Mech., vol. 8, no. 4, 1960, pp. 565–583) for the modulation of a short wave riding on a longer wave. Numerical Monte Carlo simulations of the long-term evolution of a spectrum of short waves in the presence of a monochromatic swell are employed to extract statistical distributions of freak waves among the short waves. We find evidence that a realistic short-crested wind sea can on average experience a small increase in freak wave probability because of a swell provided the swell is not orthogonal to the wind waves. For orthogonal swell and wind waves we find evidence that there is almost no significant change in the probability of freak waves in the wind sea. If the short waves are unrealistically long crested, such that the Benjamin–Feir index serves as indicator for freak waves (Gramstad & Trulsen, J. Fluid Mech., vol. 582, 2007, pp. 463–472), it appears that the swell has much smaller relative influence on the probability of freak waves than in the short-crested case.

Transient spatio-temporal chaos in the complex Ginzburg–Landau equation on long domains

Numerical simulations of the complex Ginzburg–Landau equation in one spatial dimension on periodic domains with sufficiently large spatial period reveal persistent chaotic dynamics in large parts of parameter space that extend into the Benjamin–Feir stable regime. This situation changes when nonperiodic boundary conditions are imposed, and in the Benjamin–Feir stable regime chaos takes the form of a long-lived transient decaying to a spatially uniform oscillatory state. The lifetime of the transient has Poisson statistics and no domain length is found sufficient for persistent chaos.

Modelling of wave–current interactions at the mouths of the Danube

The target of the present study is the entrance to the Danube Delta in the Black Sea. The wave conditions in this coastal sector are usually significant from an energetic point of view and the relatively strong currents induced there by the outflow from the Danube lead to interactions between waves and currents. This process modifies considerably both the magnitude and direction waves, affecting also coastal navigation and sediment transport patterns. In order to assess the effects of the wave–current interactions, the simulating waves nearshore (SWAN) model was considered for developing a multilevel wave prediction system. Validations against measured data were carried out for each computational level. Five case studies corresponding to the most relevant patterns of the environmental matrix were analyzed. Finally, in order to assess the current effect for a longer timescale, an analysis concerning the variation of the main wave parameters was performed for a 3-month period considering some reference points. The results show that the currents produce considerable changes in the wave field, especially as regards the significant wave heights, mean wave directions and wavelengths. The Benjamin–Feir index was also estimated. The analysis of the variation induced by the current over this spectral shape parameter indicates that, in certain conditions, in the target area the wave heights cannot be considered Rayleigh distributed and freak waves may also occur.

Focusing of unidirectional wave groups on deep water: an approximate nonlinear Schrödinger equation-based model

This paper sets out an approximate analytical model describing the nonlinear evolution of a Gaussian wave group in deep water. The model is derived using the conserved quantities of the cubic nonlinear Schrödinger equation (NLSE). The key parameter for describing the evolution is the amplitude-to-wavenumber bandwidth ratio, a quantity analogous to the Benjamin–Feir index for random sea-states. For smaller values of this parameter, the group is wholly dispersive, whereas for more nonlinear cases, solitons are formed. Our model predicts the characteristics and the evolution of the groups in both regimes. These predictions are found to be in good agreement with numerical simulations using the NLSE and are in qualitative agreement with numerical results from a fully nonlinear potential flow solver and experimental results.

Nonlinear surface waves interacting with a linear shear current

To describe the evolution of fully nonlinear surface gravity waves in a linear shear current, a closed system of exact evolution equations for the free surface elevation and the free surface velocity potential is derived using a conformal mapping technique. Traveling wave solutions of the system are obtained numerically and it is found that the maximum wave amplitude for a positive shear current is much smaller than that in the absence of any shear while the opposite is true for a negative shear current. The new evolution equations are also solved numerically using a pseudo-spectral method to study the Benjamin–Feir instability of a modulated wave train in both positive and negative shear currents. With a fixed wave slope, compared with the irrotational case, the envelope of the modulated wave train grows faster in a positive shear current and slower in a negative shear current.

Secondary structures in a one-dimensional complex Ginzburg–Landau equation with homogeneous boundary conditions

Experiments in extended systems, such as the counter-rotating Couette–Taylor flow or the Taylor–Dean flow system, have shown that patterns with vanishing amplitude may exhibit periodic spatio-temporal defects for some range of control parameters. These observations could not be interpreted by the complex Ginzburg–Landau equation (CGLE) with periodic boundary conditions. We have investigated the one-dimensional CGLE with homogeneous boundary conditions. We found that, in the ‘Benjamin–Feir stable’ region, the basic wave train bifurcates to state with periodic spatio-temporal defects. The numerical results match the observations quite well. We have built a new state diagram in the parameter plane spanned by the criticality (or equivalently the linear group velocity) and the nonlinear frequency detuning.

Spectral data for travelling water waves: singularities and stability

In this paper we take up the question of the spectral stability of travelling water waves from a new point of view, namely that the spectral data of the water-wave operator linearized about fully nonlinear Stokes waves is analytic as a function of a height parameter. This observation was recently made rigorous by the author using a boundary perturbation approach which is amenable to approximation by a stable high-order numerical method. Using this algorithm, we investigate, for both super- and sub-harmonic disturbances, the evolution of the spectrum, in particular the ‘first collision’ of eigenvalues and the ‘smallest singularity’ in the perturbation expansion. The former is studied in response to MacKay & Saffman's (1986) work on the water-wave problem which demonstrated that instability can only arise after the collision of two eigenvalues of opposite Krein signature. However, we present results which show, quite explicitly, that eigenvalue collision (even of opposite Krein signature) is insufficient to conclude instability. With this in mind, we have identified a new criterion for the loss of spectral stability, namely the appearance of a singularity in the expansion of the spectral data (as a function of the height parameter mentioned above). We give some heuristic reasons why this should be so, and then provide complete numerical spectral stability results for four representative depths, two above (h = ∞, 2) and two below (h = 1, 1/2) Benjamin's (1967) critical value, hc ≈ 1.363, above which the Benjamin–Feir instability emerges. We find that the strongest (two-dimensional) instability appears to be among the long waves, but we notice that there is a sharp difference between ‘shallow-water’ and ‘deep-water’ waves in that first eigenvalue collision and smallest expansion singularity are synonymous for shallow water, while this is not so in deep water where ‘windows of stability’ beyond the first eigenvalue collision exist.

A numerical study of the transformation of water waves generated in a wave flume

This paper is concerned with the transformation of water waves generated in a numerical wave flume. To set up the numerical wave tank, the non-periodic spectral method is used to solve potential flow equations incorporated with fully nonlinear free-surface boundary conditions and the wavemaker boundary condition. The flume is employed to simulate the transformation of water waves with different steepness and wavelength. The calculation results of the free-surface elevation and the corresponding amplitude wave spectrum are compared with the linear wave solutions and the analytical solutions. The result of the investigations is that short waves agree well with the linear solutions, whereas the long waves with limited Ursell numbers are consistent with the analytical solutions. Physical experiments are conducted in a wave flume to verify the numerical scheme. Good agreement between the experimental and the numerical results is obtained. The time–frequency information of waves is also investigated based on the wavelet transform. The numerical results show that short waves with smaller Benjamin–Feir index (BFI) are more stable than long waves with larger BFI in finite wave depth.

Evolution of a Random Directional Wave and Freak Wave Occurrence

Abstract The evolution of a random directional wave in deep water was studied in a laboratory wave tank (50 m long, 10 m wide, 5 m deep) utilizing a directional wave generator. A number of experiments were conducted, changing the various spectral parameters (wave steepness 0.05 < ɛ < 0.11, with directional spreading up to 36° and frequency bandwidth 0.2 < δk/k < 0.6). The wave evolution was studied by an array of wave wires distributed down the tank. As the spectral parameters were altered, the wave height statistics change. Without any wave directionality, the occurrence of waves exceeding twice the significant wave height (the freak wave) increases as the frequency bandwidth narrows and steepness increases, due to quasi-resonant wave–wave interaction. However, the probability of an extreme wave rapidly reduces as the directional bandwidth broadens. The effective Benjamin–Feir index (BFIeff) is introduced, extending the BFI (the relative magnitude of nonlinearity and dispersion) to incorporate the effect of directionality, and successfully parameterizes the observed occurrence of freak waves in the tank. Analysis of the high-resolution hindcast wave field of the northwest Pacific reveals that such a directionally confined wind sea with high extreme wave probability is rare and corresponds mostly to a swell–wind sea mixed condition. Therefore, extreme wave occurrence in the sea as a result of quasi-resonant wave–wave interaction is a rare event that occurs only when the wind sea directionality is extremely narrow.

Bistability in the complex Ginzburg–Landau equation with drift

Properties of the complex Ginzburg–Landau equation with drift are studied focusing on the Benjamin–Feir stable regime. On a finite interval with Neumann boundary conditions the equation exhibits bistability between a spatially uniform time-periodic state and a variety of nonuniform states with complex time dependence. The origin of this behavior is identified and contrasted with the bistable behavior present with periodic boundary conditions and no drift.

Non-linearity and non-stationarity of the New Year abnormal wave

Generalized time–frequency spectrum, time–frequency bispectrum and time–frequency trispectrum are calculated from the in-situ data record to investigate the non-stationarity and non-linearity of the New Year abnormal wave. A Choi–Williams kernel is used to reduce the aliasing. It is shown that for a short series of 7 waves the second peak of the time–frequency spectrum exists probably due to Benjamin–Feir instability. There are five peaks of energy during the abnormal wave in the time–frequency spectrum calculated over a 4 min series. Frequencies of two pairs of peaks are related as the frequencies of five-wave resonant interactions. The time–frequency bispectrum calculated for a 4 min long record near the abnormal wave is used to describe how the bispectrum of wind waves is changing in time. The result is compared with the stationary bispectrum of the record containing the abnormal wave investigated in an earlier study using the complete time series of 20 min. It is shown that phase coupling takes place in the front side of the abnormal wave. The time–frequency trispectrum is calculated too. It is found that before the abnormal wave the time–frequency trispectrum shows very intensive non-linear interactions. This process is stronger before the abnormal wave than after it.

Can bottom friction suppress ‘freak wave’ formation?

The paper examines the effect of the bottom stress on the weakly nonlinear evolution of a narrow-band wave field, as a potential mechanism of suppression of ‘freak’ wave formation in water of moderate depth. Relying upon established experimental studies the bottom stress is modelled by the quadratic drag law with an amplitude/bottom roughness-dependent drag coefficient. The asymptotic analysis yields Davey–Stewartson-type equations with an added nonlinear complex friction term in the envelope equation. The friction leads to a power-law decay of the spatially uniform wave amplitude. It also affects the modulational (Benjamin–Feir) instability, e.g. alters the growth rates of sideband perturbations and the boundaries of the linearized stability domains in the modulation wavevector space. Moreover, the instability occurs only if the amplitude of the background wave exceeds a certain threshold. Since the friction is nonlinear and increases with wave amplitude, its effect on the formation of nonlinear patterns is more dramatic. Numerical experiments show that even when the friction is small compared to the nonlinear term, it hampers formation of the Akhmediev/Ma-type breathers (believed to be weakly nonlinear ‘prototypes’ of freak waves) at the nonlinear stage of instability. The specific predictions for a particular location depend on the bottom roughness ks in addition to the water depth and wave field characteristics.

Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus–Benjamin–Feir instability

A numerical study of the onset of thermal convection in a rotating circular cylinder of radius-to-depth ratio equal to four is considered in a regime dominated by the Coriolis force where the onset is to so-called wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and descending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dynamics are controlled by an Eckhaus–Benjamin–Feir instability, the most basic features of which had been captured by a complex Ginzburg–Landau equation model. Here, we analyse this instability in rotating convection using direct numerical simulations of the Navier–Stokes equations in the Boussinesq approximation. Several properties of the wall modes are computed, extending the results to far beyond the onset of convection. Extensive favourable comparisons between our numerical results and previous experimental observations and complex Ginzburg–Landau model results are made.

Spatial and temporal feedback control of traveling wave solutions of the two-dimensional complex Ginzburg–Landau equation

Previous work has shown that Benjamin–Feir unstable traveling waves of the complex Ginzburg–Landau equation (CGLE) in two spatial dimensions cannot be stabilized using a particular time-delayed feedback control mechanism known as ‘time-delay autosynchronization’. In this paper, we show that the addition of similar spatial feedback terms can be used to stabilize such waves. This type of feedback is a generalization of the time-delay method of Pyragas [K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170 (1992) 421–428] and has been previously used to stabilize waves in the one-dimensional CGLE by Montgomery and Silber [K. Montgomery, M. Silber, Feedback control of traveling wave solutions of the complex Ginzburg Landau equation, Nonlinearity 17 (6) (2004) 2225–2248]. We consider two cases in which the feedback contains either one or two spatial terms. We focus on how the spatial terms may be chosen to select the direction of travel of the plane waves. Numerical linear stability calculations demonstrate the results of our analysis.

Numerical simulations of binary fluid convection in large aspect ratio annular containers

The patterns arising in large aspect ratio annular containers heated from below are analyzed for water-ethanol mixtures with negative Soret coupling. The subcritical Hopf bifurcation at the onset of convection leads to a very rich dynamics. Using high resolution numerical tools based on spectral methods to solve the hydrodynamic equations we obtain and review the properties of the different regimes arising in early stages of convection: spatially extended stationary and travelling wave convective roll structures, dispersive chaotic states, and several types of localized convection. The dynamics triggered by the Eckhaus–Benjamin–Feir instability is presented for a mixture with a moderate negative value of the separation ratio.

Influence of crest and group length on the occurrence of freak waves

A large number of simulations have been performed to reveal how the occurrence of freak waves on deep water depends on the group and crest lengths for fixed steepness. It is found that there is a sharp qualitative transition between short- and long-crested sea, for a crest length of approximately ten wavelengths. For short crest lengths the statistics of freak waves deviates little from Gaussian and their occurrence is independent of group length (or Benjamin–Feir index, BFI). For long crest lengths the statistics of freak waves is strongly non-Gaussian and the group length (or BFI) is a good indicator of increased freak wave activity.

Dynamic instabilities in scalar neural field equations with space-dependent delays

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These nonlocal models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg–Landau equations describing a Turing–Hopf bifurcation with modulation group velocity of O ( 1 ) . Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but also those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin–Feir instabilities.

Governing equations of envelopes created by nearly bichromatic waves on deep water

In this paper, the author derives the modified Schrodinger equation that governs the envelope created by nearly bichromatic waves, which are defined by the waves whose energy is almost concentrated in two closely approached wavenumbers. The stability of the solution of the modified Schrodinger equation for nearly bichromatic waves on deep water is discussed and the fact that the Benjamin–Feir instability occurs in a condition is shown. Moreover, the solutions of the modified Schrodinger equation for nearly bichromatic waves on deep water are obtained and, in a special case, the solution becomes the standing wave solution is shown.

Stability of plane waves on deep water with dissipation

The Benjamin–Feir modulational instability effects the evolution of perturbed plane-wave solutions of the cubic nonlinear Schrödinger equation (NLS), the modified NLS, and the band-modified NLS. Recent work demonstrates that the Benjamin–Feir instability in NLS is “stabilized” when a linear term representing dissipation is added. In this paper, we add a linear term representing dissipation to the modified NLS and band-modified NLS equations and establish that the plane-wave solutions of these equations are linearly stable. Although the plane-wave solutions are stable, some perturbations grow for a finite period of time. We analytically bound this growth and present approximate time-dependent regions of wave-number space that correspond to perturbations that have increasing amplitudes.