Second Order In Wave Steepness Research Articles

On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves

This paper presents a novel theoretical framework in the Hamiltonian theory of nonlinear surface gravity waves. The envelope of surface elevation and the velocity potential on the free water surface are introduced in the framework, which are shown to be a new pair of canonical variables. Using the two envelopes as the main unknowns, coupled envelope evolution equations (CEEEs) are derived based on a perturbation expansion. Similar to the high-order spectral method, the CEEEs can be derived up to arbitrary order in wave steepness. In contrast, they have a temporal scale as slow as the rate of change of a wave spectrum and allow for the wave fields prescribed on a computational (spatial) domain with a much larger size and with spacing longer than the characteristic wavelength at no expense of accuracy and numerical efficiency. The energy balance equation is derived based on the CEEEs. The nonlinear terms in the CEEEs are in a form of the separation of wave harmonics, due to which an individual term is shown to have clear physical meanings in terms of whether or not it is able to force free waves that obey the dispersion relation. Both the nonlinear terms that can only lead to the forcing of bound waves and those that are capable of forcing free waves are demonstrated, in the case of the latter through the analysis of the quartet and quintet resonant interactions of linear waves. The relations between the CEEEs and two other existing theoretical frameworks are established, including the theory for a train of Stokes waves up to second order in wave steepness (Fenton, ASCE J. Waterway Port Coastal Ocean Engng, vol. 111, issue 2, 1985, pp. 216–234) and a semi-analytical framework for three-dimensional weakly nonlinear surface waves with arbitrary bandwidth and large directional spreading by Li & Li (Phys. Fluids, vol. 33, issue 7, 2021, 076609).

Statistics of weakly nonlinear waves on currents with strong vertical shear

We investigate the effect of a vertically sheared current on wave statistics, including the probability of rogue waves, and apply it to a real-world case using measured spectral and shear current data from the mouth of the Columbia River. A theory for weakly nonlinear waves valid to second order in wave steepness is derived and used to analyze statistical properties of surface waves. With the wave spectrum and velocity profile measured in the Columbia River estuary, our theory predicts that the probability of rogue waves is significantly reduced and enhanced during ebb and flood, respectively, supporting the need for shear currents to be accounted for in wave modeling and prediction.

Three-dimensional surface gravity waves of a broad bandwidth on deep water

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

Weakly nonlinear broadband and multi-directional surface waves on an arbitrary depth: A framework, Stokes drift, and particle trajectories

Surface gravity waves in coastal waters are broadband and multi-directional, whose quadratic properties are of considerable engineering and scientific interest. Based on a Stokes expansion and an envelope-type framework, a new semi-analytical approach is proposed in this paper for the description of weakly nonlinear broadband and multi-directional surface waves. This approach proposes solving for the second-order wave fields through the separation of harmonics, by using a Fast Fourier transform and a time integration method. Different from some other methods, e.g., the High-Order Spectral method, the approach introduces a spectral shift for the superharmonic waves, leading to computationally efficient and accurate spectral predictions. The approach has been validated through comparisons with the results based on Dalzell [“A note on finite depth second-order wave–wave interactions,” Appl. Ocean Res. 21, 105–111 (1999)]. An envelope-type framework for the fast prediction of particle trajectories and Stokes drifts up to the second order in wave steepness is also derived in this paper, based on the semi-analytical approach. This paper shows that the results based on a narrowband assumption lead to underestimates of Stokes drift velocities driven by broadband unidirectional focused wave groups. The cases, examined for particle trajectories below broadband unidirectional focused wave groups, show that a larger bandwidth and water depth can enhance the differences in the net mean horizontal displacement of particles at water surface relative to these at seabed.

Surface wavepackets subject to an abrupt depth change. Part 1. Second-order theory

Abstract

Evolution of high frequency waves in shoaling and breaking wave spectra

Mathematical derivation and numerical verification of a wave transformation model in the frequency domain are discussed. The model is a fully dispersive nonlinear wave model and is derived based on the boundary value problem. Transforming the problem in the frequency domain and using multiple scale analysis in space and perturbation theory, the model is expanded up to second order in wave steepness. This fully dispersive nonlinear wave model is a set of evolution equations which explicitly contains quadratic near-resonant interactions. The comparison between the presented model, the existing fully dispersive model, and a nearshore model with different sets of laboratory and field data shows that the presented model provides significant improvements particularly at higher frequency.

An improved Lagrangian model for the time evolution of nonlinear surface waves

Accurate real-time simulations and forecasting of phase-revolved ocean surface waves require nonlinear effects, both geometrical and kinematic, to be accurately represented. For this purpose, wave models based on a Lagrangian steepness expansion have proved particularly efficient, as compared to those based on Eulerian expansions, as they feature higher-order nonlinearities at a reduced numerical cost. However, while they can accurately model the instantaneous nonlinear wave shape, Lagrangian models developed to date cannot accurately predict the time evolution of even simple periodic waves. Here, we propose a novel and simple method to perform a Lagrangian expansion of surface waves to second order in wave steepness, based on the dynamical system relating particle locations and the Eulerian velocity field. We show that a simple redefinition of reference particles allows us to correct the time evolution of surface waves, through a modified nonlinear dispersion relationship. The resulting expressions of free surface particle locations can then be made numerically efficient by only retaining the most significant contributions to second-order terms, i.e. Stokes drift and mean vertical level. This results in a hybrid model, referred to as the ‘improved choppy wave model’ (ICWM) (with respect to Nouguier et al.’s J. Geophys. Res., vol. 114, 2009, p. C09012), whose performance is numerically assessed for long-crested waves, both periodic and irregular. To do so, ICWM results are compared to those of models based on a high-order spectral method and classical second-order Lagrangian expansions. For irregular waves, two generic types of narrow- and broad-banded wave spectra are considered, for which ICWM is shown to significantly improve wave forecast accuracy as compared to other Lagrangian models; hence, ICWM is well suited to providing accurate and efficient short-term ocean wave forecast (e.g. over a few peak periods). This aspect will be the object of future work.

A MODEL IN FREQUENCY DOMAIN FOR TRANSFORMATION OF FULLY DISPERSIVE NONLINEAR WAVES

In this study, mathematical derivation and numerical verification of a wave transformation model in frequency domain is discussed. This wave model is fully dispersive and nonlinear; and is derived based on the WKB assumptions. Transforming the problem into the frequency domain and using multiple scale analysis in space and perturbation theory, the model is expanded up to second order in wave steepness. This fully dispersive nonlinear wave model is a set of evolution equations which explicitly contains quadratic near-resonant interactions. The comparison between the presented model, the existing fully dispersive model and a nearshore model with different set of laboratory and field data shows that the presented model provides significant improvements particularly at higher frequencies.

Analysis of experimental data: The average shape of extreme wave forces on monopile foundations and the NewForce model

Experiments with a stiff pile subjected to extreme wave forces typical of offshore wind farm storm conditions are considered. The exceedance probability curves of the nondimensional force peaks and crest heights are analysed. The average force time history normalised with their peak values are compared across the sea states. It is found that the force shapes show a clear similarity when grouped after the values of the normalised peak force, F/(ρghR2), normalised depth h/(gT2p) and presented in a normalised time scale t/Ta. For the largest force events, slamming can be seen as a distinct ‘hat’ on top of the smoother underlying force curve. The force shapes are numerically reproduced using a design force model, NewForce, which is introduced here for the first time to both first and second order in wave steepness. For force shapes which are not asymmetric, the NewForce model compares well to the average shapes. For more nonlinear wave shapes, higher order terms has to be considered in order for the NewForce model to be able to predict the expected shapes.

Lagrangian flows within reflecting internal waves at a horizontal free-slip surface

In this paper sequel to Zhou and Diamessis [“Reflection of an internal gravity wave beam off a horizontal free-slip surface,” Phys. Fluids 25, 036601 (2013)], we consider Lagrangian flows within nonlinear internal waves (IWs) reflecting off a horizontal free-slip rigid lid, the latter being a model of the ocean surface. The problem is approached both analytically using small-amplitude approximations and numerically by tracking Lagrangian fluid particles in direct numerical simulation (DNS) datasets of the Eulerian flow. Inviscid small-amplitude analyses for both plane IWs and IW beams (IWBs) show that Eulerian mean flow due to wave-wave interaction and wave-induced Stokes drift cancels each other out completely at the second order in wave steepness A, i.e., O(A2), implying zero Lagrangian mean flow up to that order. However, high-accuracy particle tracking in finite-Reynolds-number fully nonlinear DNS datasets from the work of Zhou and Diamessis suggests that the Euler-Stokes cancelation on O(A2) is not c...

Eulerian Volume Transport Induced by Spatially Damped Internal Equatorial Kelvin Waves

AbstractThe Eulerian volume transport in internal equatorial Kelvin waves subject to viscous attenuation is investigated theoretically by integrating the horizontal momentum equations in the vertical. In terms of small perturbations, the time-averaged horizontal transports are determined to second order in wave steepness. The total Lagrangian volume transport in this problem consists of a Stokes transport plus an Eulerian transport. It is known that the Stokes transport, that is, the vertically integrated Stokes drift, in inviscid internal equatorial Kelvin waves vanishes identically in the rigid-lid approximation for arbitrary vertical variation of the Brunt–Väisälä frequency. The present study considers spatial wave damping due to viscosity. The Stokes transport still becomes zero, but now the radiation stresses due to decaying waves become source terms for the Eulerian mean transport. Calculations of the wave-induced Eulerian transport yield a jetlike symmetric mean flow along the equator from west to east for each baroclinic component, with compensating westward flows on both sides. The flow system scales as the internal equatorial Rossby radius in the north–south direction. The total eastward part of the Eulerian volume flux centered about the equator is estimated to about 0.2 Sv (1 Sv ≡ 106 m3 s−1) for the first baroclinic mode.

Mass Transport in the Stokes Edge Wave for Constant Arbitrary Bottom Slope in a Rotating Ocean

Abstract The Lagrangian mass transport in the Stokes surface edge wave is obtained from the vertically integrated equations of momentum and mass in a viscous rotating ocean, correct to the second order in wave steepness. The analysis is valid for bottom slope angles β in the interval 0 < β ≤ π/2. Vertically averaged drift currents are obtained by dividing the fluxes by the local depth. The Lagrangian mean current is composed of a Stokes drift (inherent in the waves) plus a mean Eulerian drift current. The latter arises as a balance between the radiation stresses, the Coriolis force, and bottom friction. Analytical solutions for the mean Eulerian current are obtained in the form of exponential integrals. The relative importance of the Stokes drift to the Eulerian current in their contribution to the Lagrangian drift velocity is investigated in detail. For the given wavelength, the Eulerian current dominates for medium and large values of β, while for moderate and small β, the Stokes drift yields the main contribution to the Lagrangian drift. Because most natural beaches are characterized by moderate or small slopes, one may only calculate the Stokes drift in order to assess the mean drift of pollution and suspended material in the Stokes edge wave. The main future application of the results for large β appears to be for comparison with laboratory experiments in rotating tanks.

Mass transport in internal coastal Kelvin waves

We investigate theoretically the mass transport in internal coastal Kelvin waves by integrating the horizontal momentum equations in the vertical. Applying a perturbation method, the time-averaged Lagrangian horizontal fluxes are determined to second order in wave steepness. The linear wave field is expanded in the vertical using orthogonal functions. Due to the orthogonality property of these functions, formulae for the non-linear Stokes drift and the mean vertically-averaged Eulerian transport driven by the radiation stress can be derived for arbitrary vertical variation of the Brunt–Väisälä frequency N. For values of N typical of the thermocline in the Caspian Sea, the calculation of the non-linear transports yields a jet-like mean flow along the coast, limited in the off-shore direction by the internal Rossby radius. It is suggested that this wave-induced mean drift may contribute to the mean circulation in the Caspian Sea.

Mean drift velocity in the Stokes interfacial edge wave

[1] The Stokes interfacial edge wave in a viscous rotating two-layer system is studied theoretically. The mean wave-induced Lagrangian drift velocity is obtained from the vertically integrated Eulerian equations of momentum and mass, correct to second order in wave steepness. The analysis is valid for shallow-water waves in the case when the upper layer is much thicker than the lower layer. In the lower layer the effect of viscosity is confined to a frictional boundary layer at the bottom. The waves are trapped by the bottom slope and can propagate in either direction along the bottom contours (in the y direction). Assuming that the waves attenuate in space as they propagate, this yields a Stokes drift velocity and a mean energy density E that decay exponentially in y. In this problem −∂E/∂y is the relevant radiation stress forcing in the wave propagation direction. It is explained why this differs from the radiation-stress forcing of −∂E/∂y for plane waves in an unbounded nonrotating shallow ocean. The bottom stress acting on the mean Eulerian wave-induced flow is modeled by a turbulent friction coefficient. The results show that the maximum mean Eulerian drift current is considerably larger than the maximum Stokes drift velocity. Since the Eulerian current becomes negative at larger seaward distances, the total mean Lagrangian drift current is confined to a rather narrow wedge in the lower layer.

Mass transport induced by internal Kelvin waves beneath shore‐fast ice

A one‐layer reduced‐gravity model is used to investigate the wave‐induced mass flux in internal Kelvin waves along a straight coast beneath shore‐fast ice. The waves are generated by barotropic tidal pumping at narrow sounds, and the ice lid introduces a no‐slip condition for the horizontal wave motion. The mean Lagrangian fluxes to second order in wave steepness are obtained by integrating the equations of momentum and mass between the material interface and the surface. The mean flow is forced by the conventional radiation stress for internal wave motion, the mean pressure gradient due to the sloping surface, and the frictional drag at the boundaries. The equations that govern the mean fluxes are expressed in terms of mean Eulerian variables, while the wave forcing terms are given by the horizontal divergence of the Stokes flux. Analytical results show that the effect of friction induces a mean Eulerian flux along the coast that is comparable to the Stokes flux. In addition, the horizontal divergence of the total mean flux along the coast induces a small mass flux in the cross‐shore direction. This flux changes the mean thickness of the upper layer outside the trapping region and may facilitate geostrophically balanced boundary currents in enclosed basins. This is indeed demonstrated by numerical solutions of the flux equations for confined areas larger than the trapping region. Application of the theory to Arctic waters is discussed, with emphasis on the transport of biological material and pollutants in nearshore regions.

Mass transport in the Stokes edge wave

The Lagrangian mass transport in the Stokes progressive edge wave is obtained from the vertically integrated equations of momentum and mass, correct to second order in wave steepness. The cross-shore momentum balance is between the mean pressure at the sloping bottom, the radiation stress, and the pressure gradient due to the mean surface slope. In the alongshore direction, the effect of viscosity leads to a wave attenuation, and hence a radiation stress component. The frictional effect on the mean Eulerian motion is modeled through a turbulent bottom drag. The alongshore momentum balance is between the mean pressure gradient due to the surface slope, the radiation stress, and the turbulent drag on the mean Eulerian flow. It is shown that E/ y, where E is the total mean energy density for waves along the y-axis, is the wave-forcing term for the total mean Lagrangian momentum in the trapping region. This result is independent of the bottom slope angle. Vertically-averaged drift velocity components are obtained from the fluxes, divided by the local depth. Utilizing physical parameters relevant for field conditions, it appears the traditional Stokes drift in the Stokes edge wave is negligible compared to the mean Eulerian velocity component. The importance of this drift for the near-shore transport of effluents and suspended light sediments is discussed.

Low-frequency spectra in a harbour excited by short and random incident waves

We present a spectral theory of the low-frequency response of a harbour to short and random incident waves. Assuming the incident sea to be stationary and Gaussian, nonlinear extensions are made for the response spectrum. Advantage is taken of the typical wind-wave spectrum which is dominated by high-frequency components. After showing that nonlinearity is needed only up to the second order in wave steepness, we extend the mild-slope approximation for constructing the transfer functions. Numerical examples are presented for a square harbour and constant depth. Discounting friction losses, the effects of different entrances are compared.

Second-order refraction and diffraction of surface water waves

The mild-slope equation is an effective approximation for treating the combined effects of refraction and diffraction of infinitesimal water waves, for it reduces the spatial dimension of the linear boundary-value problem from three to two. We extend this approximation to nonlinear waves up to the second order in wave steepness, in order to simplify the inherently three-dimensional task. Assuming that the geometrical complexity is restricted to a finite, though large, horizontal domain, the hybrid-element method designed earlier for linearized problems is modified for the two-dimensional elliptic boundary-value problems at the second order. In the special case of a semi-circular peninsula (or a vertical cylinder on a cliff) in a sea of constant depth, the solution is analytical. Effects of the angle of incidence are examined for the free-surface height along the cylinder. For a cylinder standing on a shoal of radially varying depth, numerical results are discussed.

Third order wave force on axisymmetric bodies

The third order triple-frequency wave load on fixed axisymmetric bodies by monochromatic waves is considered within the frame of potential theory. Waves are assumed to be weak non-linearity and a perturbation method is used to expand velocity potentials and wave loadings into series according to a wave steepness of kA. Integral equation method is used to compute velocity potentials up to second order in wave steepness. The third order triple-frequency wave loads are computed by an indirect method and an efficient method is applied to form the third order forcing term on the free surface quickly. The method can be used to compute third order triple-frequency surge force, heave force and pitch moment on any revolution bodies with vertical axes. The comparison with Malenica and Molin's results is made on surge force on a uniform cylinder, and comparison with experimental results is made on third order surge force, heave force and pitch moment on a truncated cylinder. More numerical computations are carried out for third order forces and moments on a uniform cylinder, truncated cylinders and a hemisphere.

Deterministic wave model for short-crested ocean waves: Part I. Theory and numerical scheme

A directional hybrid wave model (DHWM) has been developed for deterministic prediction of short-crested irregular ocean waves. In using the DHWM, a measured wave field is first decomposed into its free-wave components based on as few as three point measurements. Then the wave properties are predicted in the vicinity of the measurements based on the decomposed free-wave components. Effects of nonlinear interactions among the free-wave components up to second order in wave steepness are considered in both decomposition and prediction. While the prediction scheme is straightforward, the decomposition scheme is innovative and accomplished through an iterative process involving three major steps. The extended maximum likelihood method is employed to determine the directional wave spreading; the initial phases of directional free-wave components are determined using a least-square fitting scheme; and nonlinear effects are computed using both conventional and phase modulation methods to achieve fast convergence. The free-wave components are obtained after the nonlinear effects being decoupled from the measurements. Variety of numerical tests have been conducted, indicating that the DHWM is convergent and reliable.