Surface Gravity Waves In Deep Water Research Articles

Interaction of soliton gases in deep-water surface gravity waves.

Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at leading order. We report hydrodynamic experiments in which we investigate the interaction between two beams or jets of soliton gases having nearly identical amplitudes but opposite velocities of the same magnitude. The space-time evolution of the two interacting soliton gas jets is recorded in a 140-m-long water tank where the dynamics is described at leading order by the focusing one-dimensional nonlinear Schrödinger equation. Varying the relative initial velocity of the two species of soliton gas, we change their interaction strength and we measure the macroscopic soliton gas density and velocity changes due to the interaction. Our experimental results are found to be in good quantitative agreement with predictions of the spectral kinetic theory of soliton gas despite the presence of perturbative higher-order effects that break the integrability of the wave dynamics.

The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves

The role of the Lagrangian mean flow, or drift, in modulating the geometry, kinematics and dynamics of rotational and irrotational deep-water surface gravity waves is examined. A general theory for permanent progressive waves on an arbitrary vertically sheared steady Lagrangian mean flow is derived in the Lagrangian reference frame and mapped to the Eulerian frame. A Lagrangian viewpoint offers tremendous flexibility due to the particle labelling freedom and allows us to reveal how key physical wave behaviour arises from a kinematic constraint on the vorticity of the fluid, inter alia the nonlinear correction to the phase speed of irrotational finite amplitude waves, the free surface geometry and velocity in the Eulerian frame, and the connection between the Lagrangian drift and the Benjamin–Feir instability. To complement and illustrate our theory, a small laboratory experiment demonstrates how a specially tailored sheared mean flow can almost completely attenuate the Benjamin–Feir instability, in qualitative agreement with the theory. The application of these results to problems in remote sensing and ocean wave modelling is discussed. We provide an answer to a long-standing question: remote sensing techniques based on observing current-induced shifts in the wave dispersion will measure the Lagrangian, not the Eulerian, mean current.

Forward and inverse cascades by exact resonances in surface gravity waves.

We study the energy transfer by exact resonances for deep-water surface gravity waves in a finite periodic spatial domain. Based on a kinematic model simulating the generation of active wave modes in a finite discrete wave number space S_{R}, we examine the possibility of direct and inverse cascades. More specifically, we set an initially excited region which iteratively spreads energy to wave modes in S_{R} through exact resonances. At each iteration, we first activate new modes from scale resonances (which generate modes with new lengths), then consider two bounding situations for angle resonances (which transfer energy at the same length scale): the lower bound where no angle resonance is included and the upper bound where all modes with the same length as any active mode are excited. Such a strategy is essential to enable the computation for a large domain S_{R} with the maximum wave number R∼10^{3}. We show that for both direct and inverse cascades, the modal propagation to the boundaries of S_{R} can be established when the initially excited region is sufficiently large; otherwise a frozen turbulence state occurs, with a sharp transition between the two regimes especially for the direct cascade. Through a study on the structure of resonant quartets, the mechanism associated with the sharp transition and the role of angular energy transfer in the cascades are elucidated.

The role of bandwidth in setting the breaking slope threshold of deep-water focusing wave packets

We examine the role of wave packet bandwidth in modulating the breaking slope threshold for focusing deep-water surface gravity wave packets. Using a fully nonlinear potential flow solver and laboratory experiments, we show that the slope threshold may be strongly modulated by the wave packet bandwidth. We propose a new breaking threshold parameterization that shows that the slope threshold may be modulated by more than a factor of two by changes in the bandwidth. This has implications for parameterizations of the properties of the breaking induced flow (e.g., the energy dissipation) as they do not currently account for these effects.

On an eddy viscosity model for energetic deep-water surface gravity wave breaking

We present an investigation of the fundamental physical processes involved in deep-water gravity wave breaking. Our motivation is to identify the underlying reason causing the deficiency of the eddy viscosity breaking model (EVBM) in predicting surface elevation for strongly nonlinear waves. Owing to the limitation of experimental methods in the provision of high-resolution flow information, we propose a numerical methodology by developing an EVBM enclosed standalone fully nonlinear quasi-potential (FNP) flow model and a coupled FNP plus Navier–Stokes flow model. The numerical models were firstly verified with a wave train subject to modulational instability, then used to simulate a series of broad-banded focusing wave trains under non-, moderate- and strong-breaking conditions. A systematic analysis was carried out to investigate the discrepancies of numerical solutions produced by the two models in surface elevation and other important physical properties. It is found that EVBM predicts accurately the energy dissipated by breaking and the amplitude spectrum of free waves in terms of magnitude, but fails to capture accurately breaking induced phase shifting. The shift of phase grows with breaking intensity and is especially strong for high-wavenumber components. This is identified as a cause of the upshift of the wave dispersion relation, which increases the frequencies of large-wavenumber components. Such a variation drives large-wavenumber components to propagate at nearly the same speed, which is significantly higher than the linear dispersion levels. This suppresses the instant dispersive spreading of harmonics after the focal point, prolonging the lifespan of focused waves and expanding their propagation space.

Phase modulated domain walls and dark solitons for surface gravity waves

We report theoretical prediction of localized solutions for dynamics of surface gravity waves, at the critical point kh≈1.363, modelled by higher-order nonlinear Schrödinger equation. The model possesses domain walls (kink solitons) and dark solitons modulated through different phase profiles. The parametric domains are delineated for the existence of soliton solutions. The effects of wave parameters have been discussed on the amplitude of surface gravity waves. Our work is motivated by Tsitoura et al. [1], on experimental and analytical observation of phase domain walls for deep water surface gravity waves modelled by nonlinear Schrödinger equation.

Nonlinear Spectral Synthesis of Soliton Gas in Deep-Water Surface Gravity Waves.

Soliton gases represent large random soliton ensembles in physical systems that exhibit integrable dynamics at the leading order. Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media, their controlled experimental realization has been missing. We report a controlled synthesis of a dense soliton gas in deep-water surface gravity waves using the tools of nonlinear spectral theory [inverse scattering transform (IST)] for the one-dimensional focusing nonlinear Schrödinger equation. The soliton gas is experimentally generated in a one-dimensional water tank where we demonstrate that we can control and measure the density of states, i.e., the probability density function parametrizing the soliton gas in the IST spectral phase space. Nonlinear spectral analysis of the generated hydrodynamic soliton gas reveals that the density of states slowly changes under the influence of perturbative higher-order effects that break the integrability of the wave dynamics.

Theory of deep-water surface gravity waves derived from a Lagrangian

Theory of deep-water surface gravity waves derived from a Lagrangian

From modulational instability to focusing dam breaks in water waves

We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability but exhibits a specific, strongly nonlinear modulation, which propagates from the edges of the wavepacket towards the center with finite speed. Using numerical tools of nonlinear spectral analysis of experimental data we identify the observed envelope wave structures with focusing dispersive dam break flows, a peculiar type of dispersive shock waves recently described in the framework of the semi-classical limit of the 1D focusing nonlinear Schrodinger equation (1D-NLSE). Our experimental results are shown to be in a good quantitative agreement with the predictions of the semi-classical 1D-NLSE theory. This is the first observation of the persisting dispersive shock wave dynamics in a modulationally unstable water wave system.

Experimental study of particle trajectories below deep-water surface gravity wave groups

Owing to the interplay between the forward Stokes drift and the backward wave-induced Eulerian return flow, Lagrangian particles underneath surface gravity wave groups can follow different trajectories depending on their initial depth below the surface. The motion of particles near the free surface is dominated by the waves and their Stokes drift, whereas particles at large depths follow horseshoe-shaped trajectories dominated by the Eulerian return flow. For unidirectional wave groups, a small net displacement in the direction of travel of the group results near the surface, and is accompanied by a net particle displacement in the opposite direction at depth. For deep-water waves, we study these trajectories experimentally by means of particle tracking velocimetry in a two-dimensional flume. In doing so, we provide visual illustration of Lagrangian trajectories under groups, including the contributions of both the Stokes drift and the Eulerian return flow to both the horizontal and the vertical Lagrangian displacements. We compare our experimental results to leading-order solutions of the irrotational water wave equations, finding good agreement.

Focusing deep-water surface gravity wave packets: wave breaking criterion in a simplified model

Geometric, kinematic and dynamic properties of focusing deep-water surface gravity wave packets are examined in a simplified model with the intent of deriving a wave breaking threshold parameter. The model is based on the spatial modified nonlinear Schrödinger equation of Dysthe (Proc. R. Soc. Lond. A, vol. 369 (1736), 1979, pp. 105–114). The evolution of initially narrow-banded and weakly nonlinear chirped Gaussian wave packets are examined, by means of a trial function and a variational procedure, yielding analytic solutions describing the approximate evolution of the packet width, amplitude, asymmetry and phase during focusing. A model for the maximum free surface gradient, as a function of $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6E5}$, for $\unicode[STIX]{x1D716}$ the linear prediction of the maximum slope at focusing and $\unicode[STIX]{x1D6E5}$ the non-dimensional packet bandwidth, is proposed and numerically examined, indicating a quasi-self-similarity of these focusing events. The equations of motion for the fully nonlinear potential flow equations are then integrated to further investigate these predictions. It is found that a model of this form can characterize the bulk partitioning of $\unicode[STIX]{x1D716}-\unicode[STIX]{x1D6E5}$ phase space, between non-breaking and breaking waves, serving as a breaking criterion. Application of this result to better understanding air–sea interaction processes is discussed.

Numerical Simulations of Modulated Waves in a Higher-Order Dysthe Equation

The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrodinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.

On the inertial effects of density variation in stratified shear flows

In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) At≪O(1) or (ii) Frc2≪O(1), where At is the ratio of density variation to the mean density and Frc is the ratio of the phase speed to the long wave speed. The second clause implies that, in contrast to the commonly accepted notion, a flow with large density variations can also be Boussinesq. Indeed, we show that deep water surface gravity waves are Boussinesq, while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, the Boussinesq approximation implies the conventionally accepted At≪O(1). To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities and vorticities. Furthermore, three kinds of density interfaces—neutral, stable, and unstable—embedded in a background shear layer are investigated. Instabilities ensuing from these configurations, which include Kelvin-Helmholtz, Holmboe, Rayleigh-Taylor, and triangular-jet, are studied in terms of resonant wave interactions. The effects of density stratification and the shear on the stability of each of these flow configurations are explored. Some of the results, e.g., the destabilizing role of density stratification, stabilizing role of shear, etc., are apparently counter-intuitive, but physical explanations are possible if the instabilities are interpreted from wave interaction perspective.In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) At≪O(1) or (ii) Frc2≪O(1), where At is the ratio of density variation to the mean density and Frc is the ratio of the phase speed to the long wave speed. The second clause implies that, in contrast to the commonly accepted notion, a flow with large density variations can also be Boussinesq. Indeed, we show that deep water surface gravity waves are Boussinesq, while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, the Boussinesq approximation implies the conventionally accepted At≪O(1). To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities ...

Phase Domain Walls in Weakly Nonlinear Deep Water Surface Gravity Waves.

We report a theoretical derivation, an experimental observation and a numerical validation of nonlinear phase domain walls in weakly nonlinear deep water surface gravity waves. The domain walls presented are connecting homogeneous zones of weakly nonlinear plane Stokes waves of identical amplitude and wave vector but differences in phase. By exploiting symmetry transformations within the framework of the nonlinear Schrödinger equation we demonstrate the existence of exact analytical solutions representing such domain walls in the weakly nonlinear limit. The walls are in general oblique to the direction of the wave vector and stationary in moving reference frames. Experimental and numerical studies confirm and visualize the findings. Our present results demonstrate that nonlinear domain walls do exist in the weakly nonlinear regime of general systems exhibiting dispersive waves.

On the incomplete recurrence of modulationally unstable deep-water surface gravity waves

The issue of a recurrence of modulationally unstable water wave trains within the framework of the fully nonlinear potential Euler equations is addressed. It is examined, in particular, if a modulation, which appears from nowhere (i.e., is infinitesimal initially) and generates a rogue wave, can disappear with no trace. If so, this wave solution would be a breather solution of the primitive hydrodynamic equations. It is shown with the help of the fully nonlinear numerical simulation that when a rogue wave emerges from a uniform Stokes wave train, it excites other waves which have different lengths. This process prevents the complete recurrence and, eventually, results in a quasi-periodic breathing of the wave envelope. Meanwhile the discovered effects are rather small in magnitude, and the period of the modulation breathing may be thousands of the dominant wave periods. Thus, the obtained solution may be called a quasi-breather of the Euler equations.

Lagrangian transport for two-dimensional deep-water surface gravity wave groups

The Lagrangian trajectories of neutrally buoyant particles underneath surface gravity wave groups are dictated by two physical phenomena: the Stokes drift results in a net displacement of particles in the direction of propagation of the group, whereas the Eulerian return flow, as described by the multi-chromatic wave theory of Longuet-Higgins & Stewart (1962 J. Fluid Mech. 13 , 481–504. ( doi:10.1017/S0022112062000877 )), transports such particles in the opposite direction. By pursuing a separation of scales expansion, we develop simple closed-form expressions for the net Lagrangian displacement of particles. By comparing the results from the separation of scales expansion at different orders in bandwidth, we study the effect of frequency dispersion on the local Lagrangian transport, which we show can be ignored for realistic sea states.

Potential flow simulations of Peregrine‐type deep water surface gravity wave packets

AbstractApplying a higher order spectral method for potential flow we numerically determine the time evolution of non‐linear deep water wave packets originating from initial conditions derived from the Peregrine breather solution of the non‐linear Schrödinger equation (NLS). The spatio‐temporal evolution of the wave packets qualitatively agrees well with what would be expected from lowest order weakly nonlinear estimates (NLS). Some quantitative discrepancies do, however, exist: The maximum wave envelope amplification appears retarded with respect to the NLS predictions. The amplification factor slightly exceeds the factor known from the NLS solution. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Stability analysis from fourth order evolution equation for counter-propagating gravity wave packets in the presence of wind flowing over water

Fourth order nonlinear evolution equations are derived for two counter-propagating surface gravity wave packets in deep water in the presence of wind flowing over water. The resulting equations are asymptotically exact and nonlocal. Stability analysis is made for a uniform standing surface gravity wave train for longitudinal perturbation on the basis of these equations. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness for some different values of dimensionless wind velocity. Significant deviations are noticed between the results obtained from third order and fourth order nonlinear evolution equations. This paper has an application in rough waves. References T. Brooke Benjamin and J. E. Feir. The disintegration of wave trains on deep water part 1. theory. Journal of Fluid Mechanics , 27:417–430, 1967. doi:10.1017/S002211206700045X Sudebi Bhattacharyya and K. P. Das. Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline. The ANZIAM Journal , 39:214–229, 1997. doi:10.1017/S033427000000881X K. P. Das. On evolution equations for a three-dimensional surface gravity wave packet in a two-layer fluid. Wave Motion , 8(2):191–204, 1986. doi:10.1016/0165-2125(86)90024-7 A. Davey and K. Stewartson. On three-dimensional packets of surface waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 338(1613):101–110, 1974. doi:10.1098/rspa.1974.0076 Suma Debsarma and K. P. Das. Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water. The ANZIAM Journal , 43:513–524, 2002. doi:10.1017/S1446181100012116 A. K. Dhar and K. P. Das. A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water. Physics of Fluids A: Fluid Dynamics , 2(5), 1990. A. K. Dhar and K. P. Das. Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water. Physics of Fluids A: Fluid Dynamics , 3(12), 1991. A. K. Dhar and K. P. Das. Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear. The ANZIAM Journal , 35:348–365, 1994. doi:10.1017/S0334270000009346 V. D. Djordjevic and L. G. Redekopp. On two-dimensional packets of capillary-gravity waves. Journal of Fluid Mechanics , 79:703–714, 1977. doi:10.1017/S0022112077000408 J. C. Dungey and W. H. Hui. Nonlinear energy transfer in a narrow gravity-wave spectrum. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 368(1733):239–265, 1979. doi:10.1098/rspa.1979.0126 K. B. Dysthe. Note on a modification to the nonlinear schrodinger equation for application to deep water waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 369(1736):105–114, 1979. doi:10.1098/rspa.1979.0154 Tetsu Hara and Chiang C. Mei. Frequency downshift in narrow banded surface waves under the influence of wind. Journal of Fluid Mechanics , 230:429–477, 1991. doi:10.1017/S002211209100085X Tetsu Hara and Chiang C. Mei. Wind effects on the nonlinear evolution of slowly varying gravity-capillary waves. Journal of Fluid Mechanics , 267:221–250, 1994. doi:10.1017/S0022112094001163 S. J. Hogan. The fourth-order evolution equation for deep-water gravity-capillary waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 402(1823):359–372, 1985. doi:10.1098/rspa.1985.0122 Peter A. E. M. Janssen. On a fourth-order envelope equation for deep-water waves. Journal of Fluid Mechanics , 126:1–11, 1983. doi:10.1017/S0022112083000014 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water. I. superharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):471–488, 1978. doi:10.1098/rspa.1978.0080 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water II. subharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):489–505, 1978. doi:10.1098/rspa.1978.0081 R. D. Pierce and E. Knobloch. On the modulational stability of traveling and standing water waves. Physics of Fluids , 6(3), 1994. Michael Stiassnie. Note on the modified nonlinear Schrodinger equation for deep water waves. Wave Motion , 6(4):431–433, 1984. doi:10.1016/0165-2125(84)90043-X M. A. Weissman. Nonlinear wave packets in the Kelvin–Helmholtz instability. 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The mean flow and long waves induced by two-dimensional internal gravity wavepackets

Through theory supported by numerical simulations, we examine the induced local and long range response flows resulting from the momentum flux divergence associated with with a two-dimensional Boussinesq internal gravity wavepacket in a uniformly stratified ambient. Our theoretical approach performs a perturbation analysis that takes advantage of the separation of scales between waves and the amplitude envelope of a quasi-monochromatic wavepacket. We first illustrate our approach by applying it to the well-studied case of deep water surface gravity waves, showing that the induced flow, UDF, resulting from the divergence of the horizontal momentum flux is equal to the Stokes drift. For a localized surface wavepacket, UDF is itself a divergent flow and so there is the well-known non-local response manifest in the form of a deep return flow beneath the wavepacket. For horizontally periodic and vertically localized internal wavepackets, the divergent-flux induced flow, uDF, is found from consideration of the vertical gradient of the vertical flux of horizontal momentum associated with the waves. Because uDF is itself a non-divergent flow field, this accounts entirely for the wave-induced flow; there is no response flow. Our focus is upon internal wavepackets that are localized in the horizontal and vertical. We derive a formula for the divergent-flux induced flow that, as in this case of surface wavepackets, is itself a divergent flow. We show that the response is a horizontally long internal wave that translates vertically with the wavepacket at its group velocity. Scaling relationships are used to estimate the wavenumber, horizontal extent, and amplitude of this induced long wave. At higher order in perturbation theory we derive an explicit integral formula for the induced long wave. Thus, we provide validation of Bretherton's analysis of flows induced by two-dimensional internal wavepackets [F. P. Bretherton, “On the mean motion induced by gravity waves,” J. Fluid Mech. 36, 785–803 (1969)] and we provide further analyses that give intuitive insights into the physics governing the properties of the induced long wave. In particular, consistent with momentum conservation, we show that the horizontally-integrated horizontal flow associated with the long wave is given exactly by the horizontal integral of uDF. However, qualitatively different from horizontally periodic internal waves, the vertical profile of the induced horizontal flow across the horizontally localized wavepacket is positive at the leading edge and negative at the trailing edge. These results are validated by the results of numerical simulations of a Gaussian wavepacket initialized in an otherwise stationary ambient.

Vortex generation by deep-water breaking waves

AbstractThe connection between wave dissipation by breaking deep-water surface gravity waves and the resulting turbulence and mixing is crucial for an improved understanding of air–sea interaction processes. Starting with the ensemble-averaged Euler equations, governing the evolution of the mean flow, we model the forcing, associated with the breaking-induced Reynolds shear stresses, as a body force describing the bulk scale effects of a breaking deep-water surface gravity wave on the water column. From this, we derive an equation describing the generation of circulation, $\Gamma $, of the ensemble-average velocity field, due to the body force. By examining the relationship between a breaking wave and an impulsively forced fluid, we propose a functional form for the body force, allowing us to build upon the classical work on vortex ring phenomena to both quantify the circulation generated by a breaking wave and describe the vortex structure of the induced motion. Using scaling arguments, we show that $\Gamma = \alpha {(hk)}^{3/ 2} {c}^{3} / g$, where ($c, h, k$) represent a characteristic speed, height and wavenumber of the breaking wave, respectively, $g$ is the acceleration due to gravity and $\alpha $ is a constant. This then allows us to find a direct relationship between the circulation and the wave energy dissipation rate per unit crest length due to breaking, ${\epsilon }_{l} $. Finally, we compare our model and the available experimental data.