Deep Water Limit Research Articles

Deep-water and shallow-water limits of the intermediate long wave equation

In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs -Sobolev space for s>12 , to those of BO in the deep-water limit (as δ→∞ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s>32 in the deep-water limit and s⩾2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity ∂x(uk) for k⩾2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s⩾34 . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.

Mean flow modeling in high-order nonlinear Schrödinger equations

The evaluation and consideration of the mean flow in wave evolution equations are necessary for the accurate prediction of fluid particle trajectories under wave groups, with relevant implications in several domains, from the transport of pollutants in the ocean to the estimation of energy and momentum exchanges between the waves at small scales and the ocean circulation at large scale. We derive an expression of the mean flow at a finite water depth, which, in contrast to other approximations in the literature, accurately accords with the deep-water limit at third order in steepness and is equivalent to second-order formulations in intermediate water. We also provide envelope evolution equations at fourth order in steepness for the propagation of unidirectional wave groups either in time or space that include the respective mean flow term. The latter, in particular, is required for accurately modeling experiments in water wave flumes in arbitrary depths.

Perturbation analysis of nonlinear partial reflected wave on a sloping bottom

In this paper, we use the perturbation method to develop a new mathematical derivation to describe nonlinear partial standing wave over uniformly sloping bottoms. In the Lagrangian coordinate system, the particle trajectories are obtained as a function of the nonlinear order parameter ɛ and the bottom slope α to the second order of the perturbation. The setups and mean return flow for an arbitrary bottom in the Lagrangian framework are also found as part of the solutions. The direct influences of reflection coefficient, wave steepness and bottom slope on the surface profiles of partially reflected waves are also derived. This nonlinear analytical solution is verified by reduction to the classical Stokes solution of progressive waves in both the deep water and constant depth limit. This solution allows to describe the successive deformation of partial reflected wave profiles and water particle trajectories prior to wave breaking to be described. The dynamic properties including mass transport, and Lagrangian mean level and radiation stress for nonlinear partially reflected waves on various sloping bottoms are investigated.

Depth and temperature profiles reflect individual differences in the daytime diving behaviours of pelagic thresher sharks

We used acoustic telemetry to investigate the roles of depth and temperature in the daytime foraging behaviours of 13 tagged pelagic thresher sharks by monitoring their fine scale vertical movements in the Philippines. Cumulatively, pelagic thresher shark dives traversed the entire water column where they encountered temperatures that ranged from 33°C at the surface to 12°C at 250 m depths throughout the day, but the movements of individuals varied in the extent of both their deep and shallow water limits. Dives were not synchronized to diurnal cycles, and periodicity reflected cycles of similar dives, the dives themselves, deviations, cruising, and individuality. Pelagic thresher shark movements between the warm surface layer and cooler waters below the thermocline (155 – 175 m) may reflect a common Alopiid strategy that balances maintaining tolerable ambient water temperatures with opportunities to search for and forage on spatially patchy distributions of prey.

On Stokes wave solutions

Using a perturbation method with the aid of MATLAB®Symbolic Toolbox, a new set of Stokes wave solutions, up to the fifth order, are derived. The new solutions are expressed in terms of free surface profile, velocity potential and wave celerity (or frequency dispersion relation). The solutions in the deep water limit are also deduced and discussed. The new solutions are compared with the existing solutions up to the fifth order. Differences appear among the existing and the present solutions, starting at the third order. The causes for the differences are identified. The characteristic differences were highlighted in the figures for high order solutions of Stokes waves. Numerical examples are provided to quantify the differences between the new fifth-order solutions and Fenton’s, and Skjelbreia and Hendrickson’s solutions.

Radiation-induced instability of a finite-chord Nemtsov membrane

We consider a problem of stability of a membrane of an infinite span and a finite chord length, submerged in a uniform flow of finite depth with free surface. In the shallow water approximation, Nemtsov [“Flutter effect and emission in the region of anomalous and normal Doppler effects,” Radiophys. Quantum Electron. 28(12), 1076–1079 (1985)] has shown that an infinite-chord membrane is susceptible to flutter instability due to the excitation of long gravity waves on the free surface if the velocity of the flow exceeds the phase velocity of the waves and related this phenomenon with the anomalous Doppler effect. In the present work, we derive a full nonlinear eigenvalue problem for an integrodifferential equation for the finite-chord Nemtsov membrane in the finite-depth flow. In the shallow- and deep-water limits, we develop a perturbation theory in the small added mass ratio parameter acting as an effective dissipation to find explicit analytical expressions for the frequencies and the growth rates of the membrane modes coupled to the surface waves. We find an intricate pattern of instability pockets in the parameter space and describe it analytically. The case of an arbitrary depth flow with free surface requires numerical solution of a new non-polynomial nonlinear eigenvalue problem. We propose an original approach combining methods of complex analysis and residue calculus, Galerkin discretization, Newton method, and parallelization techniques implemented in MATLAB to produce high-accuracy stability diagrams within an unprecedented wide range of a system's parameters. We believe that the Nemtsov membrane plays the same paradigmatic role for understanding radiation-induced instabilities as the Lamb oscillator coupled to a string has played for understanding radiation damping.

Bäcklund transformation, infinite number of conservation laws and fission properties of an integro-differential model for ocean internal solitary waves

This paper presents an analytical investigation of the propagation of internal solitary waves in the ocean of finite depth. Using the multi-scale analysis and reduced perturbation methods, the integro-differential equation is derived, which is called the intermediate long wave (ILW) equation and can describe the amplitude of internal solitary waves. It can reduce to the Benjamin–Ono equation in the deep-water limit, and to the KdV equation in the shallow-water limit. Little attention has been paid to the features of integro-differential equations, especially for their conservation laws. Here, based on Hirota bilinear method, Bäcklund transformations in bilinear form of ILW equation are derived and infinite number of conservation laws are given. Finally, we analyze the fission phenomenon of internal solitary waves theoretically and verify it through numerical simulation. All of these have potential value for the further research on ocean internal solitary waves.

An investigation of the properties of flexural-gravity wave propagation in a coupled submerged and floating plate system

The propagation of flexural-gravity waves in a coupled submerged and floating plate system is studied, and the complex flexural gravity wave propagation properties are investigated in the frequency and time domain, The system exhibits two modes of propagation analogous to the surface and internal waves in a two-layer fluid. The effect of the plate properties on the phase velocities of both the faster (floating plate in general) and the slower (submerged plate in general) modes are analysed in detail. Both modes exhibit complex properties, including waves with negative kinetic energy and points of blocking at which no energy can propagate. They also show complex coupling and energy transfer in the time-domain. Furthermore, the dependency of the blocking frequency and the critical opposing current is analysed in both the shallow and deep water limits. The occurrence of blocking in the internal mode under a specific constraint is also demonstrated. As a special case, the influence of the submerged plate’s position and compressive force on the dynamics of the free surface flexural-gravity wave blocking is studied. The characteristics of the phase speed in both modes are studied, and the occurrences/non-occurrences of optima are mathematically proved. Some of the analytic results are illustrated by time-domain simulations which illustrate the complex wave propagation phenomena of this system.

Analysis of the linear version of a highly dispersive potential water wave model using a spectral approach in the vertical

The properties and accuracy of the linearized version of the fully dispersive and nonlinear wave model developed in Yates and Benoit (2015) and Raoult et al. (2016) are analyzed for both flat and variable bottom bathymetries. This model considers only a single layer of fluid and uses a basis of orthogonal Chebyshev polynomials to project the vertical structure of the potential. This approach results in an exponential convergence rate with the maximum degree of the Chebyshev polynomial, denoted NT, while only first- and second-order derivatives in space need to be evaluated.For the constant water depth case, the linear dispersion relation of the model is derived analytically, and expressions are established for NT ranging from 2 to 15. The analysis shows a rapid increase in accuracy in the deep water range with increasing NT. For instance, the relative error in the calculated wave celerity (in comparison with Stokes’ analytical solution) remains smaller than 2.5% for deep water cases with kh up to 100 using NT≥9 (k and h are the representative wavenumber and water depth, respectively). The wave kinematics, vertical profiles of the horizontal and vertical orbital velocities, converge to the Stokes profiles for kh up to 60 when using a sufficiently high value of NT. The vertically-averaged relative errors of the horizontal and vertical velocities remain below 6% and 3%, respectively, for kh up to 60 when using NT≥11. The presented model shows better dispersive properties in deep water than several high-order Boussinesq-type models.For variable bottom bathymetries, the shoaling properties of the model are studied numerically, exhibiting good agreement with results from Stokes linear theory in the case of mild bottom slopes, using a sufficiently high value of NT with respect to the offshore relative water depth. For an offshore water depth of kh=10 (i.e. more than 3 times the deep water limit), accurate wave heights in shallow water (kh=0.25) are obtained with NT=6 (or higher). Finally, the linear version of the model is validated with comparisons to analytical solutions of the reflection and transmission coefficients of regular waves over Roseau-type bathymetric profiles. Two bottom profiles are considered, including one with a steep slope, whose maximum value reaches about 1:0.7 (i.e. an angle of about 54.9 deg.). Using NT=7, small differences (<0.4%) with the analytical solution are observed for the four considered cases, confirming the ability of the linear model to represent accurately the effects of steep bottom gradients on wave propagation dynamics.

Wind-wave amplification mechanisms: possible models for steep wave events in finite depth

Abstract. We extend the Miles mechanism of wind-wave generation to finite depth. A β-Miles linear growth rate depending on the depth and wind velocity is derived and allows the study of linear growth rates of surface waves from weak to moderate winds in finite depth h. The evolution of β is plotted, for several values of the dispersion parameter kh with k the wave number. For constant depths we find that no matter what the values of wind velocities are, at small enough wave age the β-Miles linear growth rates are in the known deep-water limit. However winds of moderate intensities prevent the waves from growing beyond a critical wave age, which is also constrained by the water depth and is less than the wave age limit of deep water. Depending on wave age and wind velocity, the Jeffreys and Miles mechanisms are compared to determine which of them dominates. A wind-forced nonlinear Schrödinger equation is derived and the Akhmediev, Peregrine and Kuznetsov–Ma breather solutions for weak wind inputs in finite depth h are obtained.

Estimating the potential of ocean wave power resources

The realistic assessment of an ocean wave energy resource that can be converted to an electrical power at various offshore sites depends upon many factors, and these include estimating the resource recognizing the random nature of the site-specific wave field, and optimizing the power conversion from particular wave energy conversion devices. In order to better account for the uncertainty in wave power resource estimates, conditional probability distribution functions of wave power in a given sea-state are derived. Theoretical expressions for the deep and shallow water limits are derived and the role of spectral width is studied. The theoretical model estimates were compared with the statistics obtained from the wave-by-wave analysis of JONSWAP based ocean wave time-series. It was shown that the narrow-band approximation is appropriate when the variability due to wave period is negligible. The application of the short-term models in evaluating the long-term wave power resource at a site was illustrated using wave data measured off the California coast. The final example illustrates the procedure for incorporating the local wave data and the sea-state model together with a wave energy device to obtain an estimate of the potential wave energy that could be converted into a usable energy resource.

A fourth‐order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq‐type equations. Part II: boundary conditions and validation

AbstractThis paper supplements the validation of the fourth‐order compact finite volume Boussinesq‐type model presented by Cienfuegos et al. (Int. J. Numer. Meth. Fluids 2006, in press). We discuss several issues related to the application of the model for realistic wave propagation problems where boundary conditions and uneven bathymetries must be considered. We implement a moving shoreline boundary condition following the lines given by Lynett et al. (Coastal Eng. 2002; 46:89–107), while an absorbing‐generating seaward boundary and an impermeable vertical wall boundary are approximated using a characteristic decomposition of the Serre equations. Using several benchmark tests, both numerical and experimental, we show that the new finite volume model is able to correctly describe nonlinear wave processes from shallow waters and up to wavelengths which correspond to the theoretical deep water limit. The results compare favourably with those reported using former fully nonlinear and weakly dispersive Boussinesq‐type solvers even when time integration is conducted with Courant numbers greater than 1.0. Furthermore, excellent nonlinear performance is observed when numerical computations are compared with several experimental tests on solitary waves shoaling over planar beaches up to breaking. A preliminary test including the wave‐breaking parameterization described by Cienfuegos (Fifth International Symposium on Ocean Wave Measurement Analysis, Madrid, Spain, 2005) shows that the Boussinesq model can be extended to deal with surf zone waves. Finally, practical aspects related to the application of a high‐order implicit filter as given by Gaitonde et al. (Int. J. Numer. Methods Engng 1999; 45:1849–1869) to damp out unphysical wavelengths, and the numerical robustness of the finite volume scheme are also discussed. Copyright © 2006 John Wiley &amp; Sons, Ltd.

A Comparison of Nonlinear Water Wave Models

We compare the numerical evolution of one-dimensional gravity waves in response to a traveling surface pressure pulse using a highly accurate boundary integral method and two relatively efficient approximate models (West et al. and Benney–Luke). In both water of finite-depth and in the deep-water limit, the steady state effect of the decaying pressure ramp is to create a profile which approximates a Stokes wave. Moreover, the transient surface profile appears to evolve through a series of Stokes waves of time-varying amplitude. Results show all three models yield similar predictions for lower amplitude waves, while the West et al. and boundary integral predictions differ from the Benney–Luke model at higher amplitudes.

A numerical study of submarine–landslide–generated waves and run–up

A mathematical model is derived to describe the generation and propagation of water waves by a submarine landslide. The model consists of a depth–integrated continuity equation and momentum equations, in which the ground movement is the forcing function. These equations include full nonlinear, but weak frequency–dispersion, effects. The model is capable of describing wave propagation from relatively deep water to shallow water. Simplified models for waves generated by small seafloor displacement or creeping ground movement are also presented. A numerical algorithm is developed for the general fully nonlinear model. Comparisons are made with a boundary integral equation method model, and a deep–water limit for the depth–integrated model is determined in terms of a characteristic side length of the submarine mass. The importance of nonlinearity and frequency dispersion in the wave–generation region and on the shoreline movement is discussed.

Dark soliton generation for the intermediate nonlinear Schrödinger equation

The generation of dark solitons due to a small change of the initial data is studied within the framework of the intermediate nonlinear Schrödinger (INLS) equation. In particular, we analyze the spectral problem associated with the INLS equation when the potential consists of a small perturbation imposed on a constant background. We derive a criterion for the perturbation to generate a pair of new discrete eigenvalues as well as their explicit expressions in terms of the perturbation. In addition, we demonstrate that the eigenvalues appear without a threshold on the magnitude of the perturbation. We also consider both the shallow- and deep-water limits of various results obtained for the INLS spectral problem. In the former case, the limiting procedure can be performed smoothly whereas in the latter case, the spectral equation exhibits a new feature of bound states, showing that the eigenvalue is exponentially small compared with the magnitude of the perturbation.

N-soliton formulae for theintermediate nonlinear Schrödinger equation

A direct proof is given to the N-soliton solution of theintermediate nonlinear Schrödinger (INLS) equation describingenvelope waves. The proof relies only on an elementary theory ofdeterminants and knowledge of the inverse scatteringtransform method is not required. In particular, when theN-soliton solution is substituted into the system of bilinearequations for the INLS equation, the system is found to reduceto Jacobi's formula for determinants. A special class ofN-soliton solutions is also presented which is expressed interms of exponential functions. In the deep-water limit, thesolution reduces to the algebraic N-soliton solution of anonlocal NLS equation with the Hilbert kernel whereas in theshallow-water limit, the solution reduces to the N-solitonsolution of the defocusing NLS equation.

Mechanism for the faraday instability in viscous liquids

This report presents a discussion of the wave number selection mechanism in the Faraday instability, which arises when standing waves form on the free surface of a liquid subject to vertical sinusoidal oscillation. The focus is on the case where the viscous effects are strong, i.e., when the wavelength is of the same order of magnitude as the boundary layer thickness at the free surface. We investigate the relationship between the Faraday instability and the Rayleigh-Taylor instability by performing linear stability calculations. In the deep water limit (wavelength << liquid depth), our results indicate that the preferred wave number in the Faraday instability is primarily determined through a Rayleigh-Taylor instability. In the case of shallow water (wavelength approximately liquid depth), the agreement between the Rayleigh-Taylor and Faraday wave numbers does not appear to be as good, probably due to the interaction between the oscillatory motion of the standing waves and the bottom boundary.

Wave‐wave interactions in finite depth water

In this study we present a new formulation for the nonlinear wave‐wave interaction source function in finite water depth. The formulation, denoted the reduced integration approximation (RIA), is shown to compare well with published formulations, both for shallow water wave‐wave interactions [Hertench and Hasselmann, 1980; Polnikov, 1997; Hashimoto et al., 1998; A. Masuda and K. Komatsu, manuscript in preparation, 1998] and also for the asymptotic deep water limit: (1) the Hamiltonian formulation proposed by Lin and Perrie [1997], by (2) Hasselmann and Hasselmann [1981], and (3) the line integral transformation of Webb [1978] and Resio and Ferne [1991]. Of these deep water formulations, that of Lin‐Perrie generalizing the Hamiltonian representation of Zakharov [1968] to finite depth water, is notable for its simplicity, efficiency and its ability to apply to very shallow water (kh ≈ 0.3), and highly nonlinear (ε≤0.3) interactions. RIA is based on an analysis of the main resonance domain, which reduces the six‐dimensional integration to a quasi‐line integral to minimize computational time. In terms of computational time, RIA is a thousand times faster than the EXACT‐NL version formulated by Hasselmann and Hasselmann [1981], with similar accuracy. Thus RIA can be considered a candidate for operational forecasting in finite depth water, in the sense that the discrete interaction approximation was presented as a candidate for operational deep water wave forecasting by Hasselmann et al. [1988].

A time–dependent nonlinear mild slope equation for water waves

A weakly nonlinear and dispersive water wave equation, which in linearized form yields a new version of the time–dependent mild–slope equation of Smith and Sprinks (1975), is derived. The applicable spectral width of the new wave equation for random waves is found to be more satisfactory than that of Smith and Sprinks (1975). For very shallow depths the equation reduces to the combined form of Airy9s nonlinear non–dispersive wave equations; if the lowest–order dispersion is retained it produces the combined form of Boussinesq9s equations. In the deep–water limit the equation admits the second–order Stokes waves as analytical solutions. Furthermore, by introducing a right–moving coordinate transformation, the equation is recast into a unidirectional form, rendering the KdV equation in one limit while reproducing the second–order Stokes waves in the other.

The bifurcation of steady gravity water waves in (R, S) parameter space

The bifurcation of steady periodic waves from irrotational inviscid streamflows is considered. Normalizing the flux Q to unity leaves two other natural quantities R (pressure head) and S (flowforce) to parameterize the wavetrain. In a well-known paper, Benjamin &amp; Lighthill (1954) presented calculations within a cnoidal-wave theory which suggested that the corresponding values of R and S lie inside the cusped locus traced by the sub- and supercritical streamflows. This rule has been applied since to many other flow scenarios. In this paper, regular expansions for the streamfunction and profile are constructed for a wave forming on a subcritical stream and thence values for R and S are calculated. These describe, locally, how wave brances in (R, S) parameter space point inside the streamflow cusp. Accurate numerics using a boundry-integral solver show how these constant-period branches extend globally and map out parameter space. The main result is to show that the large-amplitude branches for all steady Stokes’ waves lie surprisingly close to the subcritical stream branch, This has important consequences for the feasibility of undular bores (as opposed to hydraulic jumps) in obstructed flow. Moreover, the transition from the ‘long-wave region’ towards the ‘deep-water limit’ is char-acterized by an extreme geometry, bith of the wave branches and how they sit inside each other. It is also shown that a single (Q, R, S) trriple may represent more than one wave since the global branches can overlap in (R, S) parameter space. This non-uniqueness is not that associated with the known premature maxima of wave propertties as functions of wave amplitude near waves of greatest height.