A note on wave energy dissipation over steep beaches

This paper revisits the derivation of the parametric surf zone model proposed by Baldock et al. [Baldock, T. E., Holmes, P., Bunker, S. & Van Weert, P. 1998 Cross-shore hydrodynamics within an unsaturated surf zone. Coast. Eng. 34, 173–196.]. We show that a consistent use of the proposed Rayleigh distribution for surf zone wave heights results in modification of the expressions for the bulk dissipation rate and enhanced dissipation levels on steep beaches and over-saturated surf zone conditions. As a consequence, the modification proposed herein renders the model robust even on steep beaches where it could otherwise develop a shoreline singularity.     

Shallow water gravity waves: A note on the particle orbits

When surface gravity waves of small amplitude progress in shallow water of constant mean depth, the fluid particle orbits are observed to be oval, where the longer axis of the oval is parallel to the flat bottom, and at the bottom the orbits are straight lines. Potential flow, upon which the standard wave theory is based, predicts that the oval orbits are ellipses, but by a rather lengthy mathematical procedure that is founded on the questionable assumption of irrotationality. Using a more elementary and physical method, that does not employ the irrotational assumption, the elliptical orbits can be understood much more easily. The elementary method features a balance of two oppositely directed forces on each fluid particle: the outward centrifugal force and the inward pressure force.     

A note on significant wave height

In this note conservative bounds for significant crest height and amplitude obtained from the crossing intensity of a sea are presented. For Gaussian models of a sea level, the Rayleigh approximation for the distributions of amplitude and crest height is proved to provide conservative values for the expected significant wave characteristics. The results are illustrated by examples in which both Gaussian and non-Gaussian models for a sea are considered.     

Wave prediction using wave rider position measurements and NARX network in wave energy conversion

Several control methods of wave energy converters (WECs) need prediction in the future of wave surface elevation. Prediction of wave surface elevation can be performed using measurements of surface elevation at a location ahead of the controlled WEC in the upcoming wave. Artificial neural network (ANN) is a robust data-learning tool, and is proposed in this study to predict the surface elevation at the WEC location using measurements of wave elevation at ahead located sensor (a wave rider buoy). The nonlinear autoregressive with exogenous input network (NARX NN) is utilized in this study as the prediction method. Simulations show promising results for predicting the wave surface elevation. Challenges of using real measurements data are also discussed in this paper.     

A note on the horizontal momentum exchange in combined waves and current

The horizontal exchange of momentum due to the organized motion in combined waves and current has been analyzed. The combination of the vertical orbital wave motion and the mean current gives a periodic variation in the horizontal velocity in addition to the wave orbital motion. This periodic variation, combined with the wave orbital motion, gives a significant contribution to the momentum exchange. Two examples are considered, the interaction of a pure wave motion and a current normal to the direction of wave propagation, and a wave driven longshore current with an undertow velocity profile. It is demonstrated that the new contribution changes the resulting momentum exchange considerably.     

A mathematical representation of random gravity waves in the ocean

We give a mathematical representation of random ocean surface waves in the gravity-wave regime. The so-called random gravity waves are treated as an asymptotic phenomenon when the wind pressure acting on the surface and the dissipation become negligible. We adopt a phenomenological model for the wind pressure such that it excites a surface consisting of wind-driven sea and swell. Starting from the Navier-Stokes equations, we derive a general system of the first-order perturbation equations governing the surface waves, and solve them with this wind pressure as the excitation. The resulting solution is decomposed into a part which is asymptotically dominant and another which is asymptotically negligible. The former consists of two groups: one which is a sum of superpositions of uncorrelated plane waves having approximate dispersion relations and the other a sum of random plane waves with their wavenumbers and frequencies approximately satisfying the dispersion relation. They correspond to the dominant parts of the wind-driven sea and the swell, respectively. Finally, we derive a limiting form of the directional-frequency spectrum in the gravity-wave regime.     

Surface gravity waves from direct numerical simulations of the Euler equations: A comparison with second-order theory

When the wave spectrum is sufficiently narrow-banded and the wave steepness is sufficiently high, the modulational instability can take place and waves can be higher than expected from second-order wave theory. In order to investigate these effects on the statistical distribution of long-crested, deep water waves, direct numerical simulations of the Euler equations have been performed. Results show that, for a typical design spectral shape, both the upper and lower tails of the probability density function for the surface elevation significantly deviate from the commonly used second-order wave theory. In this respect, the crest elevation is observed to increase up to 18% at low probability levels. It would furthermore be expected that wave troughs become shallower due to nonlinear effects. Nonetheless, the numerical simulations show that the trough depressions tend to be deeper than in second-order theory.     

A note on wave action conservation in a dissipative current wave motion

We consider steady, slowly varying water waves propagating on a steady current over a gently sloping bed, so-called current depth refraction. All expressions are correct to second order in wave amplitude. Formulating the energy equation for the fluctuating motion in terms of wave action (wave energy divided by intrinsic angular frequency) results in an expression, where the dissipative term is strikingly similar to wave action itself. It is simply the ‘extra’ dissipation (per unit area) caused by the fluctuating motion (i.e. total dissipation minus the effect of current acting on total mean bed shear stress) divided by the intrinsic angular frequency. We call it ‘wave action dissipation’. An inconsistency in Phillips' (1977) book is pointed out. A new formula for the calculation of wave amplitudes along rays is set forth.     

A note on the direct injection of turbulence by breaking waves

We investigate the turbulence induced by wave-breaking at the ocean surface. Two recent models use a mechanism of direct depth injection of turbulent kinetic energy (TKE) by breaking waves. Those models aim to reproduce the near-surface mean and turbulent properties, in particular the TKE dissipation rates. Of critical importance are the injection depth of each breaking wave and the size distribution of those breaking waves. The models by Sullivan et al. (2007) and by Kudryavtsev et al. (2008) have very different parameterizations, and those differences are reviewed here and compared to available observations. Using realistic parameterizations in these models leads to TKE injections too shallow to compare to observations, in particular for developed seas. The near-surface turbulence is thus still not well understood to the zeroth order. For instance, whether developed seas produce deeper or shallower mixing than young seas is neither well understood nor well modelled. Additional dedicated measurements as well as investigations of breaking non-breaking wave interactions are needed.     

A note on nearshore wave features: Implications for wave generation

This paper analyses 10 years of wave data from the Mediterranean Spanish (Catalan) coast considering the mean wave climate and storm events from the standpoint of wind-wave momentum transfer and wave prediction. The data, registered by a buoy at about 12 km from the coastline, revealed two main groups of wave storms, with NW and E directions. NW storms correspond to a fetch-limited situation since the intense wind blows from land. Low-pressure centres located over the Mediterranean Sea produce easterly storms. Near the coast the eastern winds from the sea are replaced by NW winds coming from meteorological patterns over northern Spain and south-western France. Wave storms are classified and studied to obtain their main features (including spectral width, wave length, wave age and bimodality) and discussed in terms of wind-wave momentum transfer for operational wave predictions. Observations show a complex coastal wave climate. Fetch-limited storms presented smaller spectral widths while varying wind situations presented larger widths due to the presence of bimodal spectra. These wave features are highly relevant for wind–ocean momentum transfer and, thus, for current and wave predictions. The spectral width proved to be a good indicator of sea complexity and is thus applicable for improved wind drag estimations. A new drag coefficient formulation is proposed, based on existing wind dependent drag expressions, but including also spectral wave properties (a spectral width parameter) that highlights the characteristics of wind-wave generation under pre-existing swell. Such a formulation, once properly validated with field observations, is expected to improve wind-wave predictions.     

A note on wave-induced set-up by obliquely incident waves

Water level variations due to obliquely incident, shoaling and breaking waves on a plane sloping beach were discussed recently by Hsu et al. (Coastal Engineering, 53, 865–877, 2006). An inconsistency in this work with respect to the set-down, and its implications to circulation offshore of the breakpoint, was pointed out by Shi and Kirby (Coastal Engineering, 55, 1246 – 1249, 2008). Here we extend that discussion to include the surfzone momentum balance and wave-induced set-up. We discuss some remaining inconsistencies in the approximation of the surfzone momentum balance, derive and present a consistent approximation, and validate the new approximation through numerical comparison to a more exact model.     

On local and convective accelerations in steep wave events

The local and convective accelerations of steep irregular wave events have been investigated. These properties are measured by a two-camera PIV technique. Furthermore, the experiments are compared with two different theories including a fully nonlinear and a simplified analytical model. An important result is that the convective term is of the same order of magnitude and of opposite sign as the local acceleration. The convective acceleration term can therefore not be neglected in acceleration and force estimates.     

A note on the derivation of potential energy for two-dimensional water waves

The potential energy available in a two-dimensional progressive water wave can be calculated in numerous ways. One derivation of this energy based on the first law of thermodynamics and on the linearized velocity potential for waves, is presented in this paper. The energy densities and total energy expressions are given for deep and finite depth water waves. It is also shown that the travelling component of the energy for deep water waves is the potential energy component.     

A fast convolution approach to the transformation of surface gravity waves: Linear waves in 1DH

The transformation of irrotational surface gravity waves in an inviscid fluid can be studied by time stepping the kinematic and dynamic surface boundary conditions. This requires a closure providing the normal surface particle velocity in terms of the surface velocity potential or its tangential derivative. A convolution integral giving this closure as an explicit expression is derived for linear 1D waves over a mildly sloping bottom. The model has exact linear dispersion and shoaling properties. A discrete numerical model is developed for a spatially staggered uniform grid. The model involves a spatial derivative which is discretized by an arbitrary-order finite-difference scheme. Error control is attained by solving the discrete dispersion relation a priori and model results make a perfect match to this prediction. A procedure is developed by which the computational effort is minimized for a specific physical problem while adapting the numerical parameters under the constraint of a predefined tolerance of damping and dispersion error. Two computational examples show that accurate irregular-wave transformation on the kilometre scale can be computed in seconds. Thus, the method makes up a highly efficient basis for a forthcoming extension that includes nonlinearity at arbitrary order. The relation to Boussinesq equations, mild-slope wave equations, boundary integral equations and spectral methods is briefly discussed.     

基于MEMS加速度计空投浮标的波浪测量方法试验研究

针对基于MEMS加速度传感器的空投波浪浮标存在采样频率与测波精度低的问题,根据频域衰减积分算法,提出一种相应的波浪测量算法,为了验证该算法测波的准确性,开展了多功能水槽试验研究。该算法旨在将MEMS加速度传感器输出的加速度与姿态角转化为浮标运动的波形,首先将加速度与姿态角信号进行竖向处理获得竖直方向的加速度,再利用离散傅里叶变换将竖向加速度转化为频域内的加速度复数序列,然后引入控制函数减弱低频噪声,经过频域积分、离散傅里叶逆变换、时域积分获得竖直方向的位移,最后通过后处理得到最终的波形。多功能水槽试验采取10中不同波高和周期的工况,对比空投波浪测量浮标与波高仪的测量结果,试验结果表明,浮标的测量误差在10%以内,达到测波标准。     

A note on the derivation of wave action balance equation in frequency space

In this paper the wave action balance equation in terms of frequency-direction spectrum is derived.A theoretical formulation is presented to generate an invariant frequency space to replace the varying wavenumber space through a Jacobian transformation in the wave action balance equation.The physical properties of the Jacobian incorporating the effects of water depths are discussed.The results provide a theoretical basis of wave action balance equations and ensure that the wave balance equations used in the SWAN or other numerical models are correct.It should be noted that the Jacobian is omitted in the wave action balance equations which are identical to a conventional action balance equation.     

Effects of rotation on self-resonant internal gravity waves in the ocean

The Resonant Triad Model (RTM) developed in (Ibragimov, 2007), is used to study the Thorpe’s problem (Thorpe, 1997) on the existence of self-resonant internal waves, i.e., the waves for which a resonant interaction occurs at second order between the incident and reflected internal waves off slopes. The RTM represents the extension of the McComas and Bretherton’s three wave hydrostatic model (McComas and Bretherton, 1977) which ignores the effects of the earth’s rotation to the case of the non-hydrostatic analytical model involving arbitrarily large number of rotating internal waves with frequencies spanning the range of possible frequencies, i.e., between the maximum of the buoyancy frequency (vertical motion) and a minimum of the inertial frequency (horizontal motion). The present analysis is based on classification of resonant interactions into the sum, middle and difference interaction classes. It is shown in this paper that there exists a certain value of latitude, which is classified as the singular latitude, at which the coalescence of the middle and difference interaction classes occurs. Such coalescence, which apparently had passed unnoticed before, can be used to study the Thorpe’s problem on the existence of self-resonant waves. In particular, it is shown that the value of the bottom slope at which the second-order frequency and wavenumber components of the incident and reflected waves satisfy the internal wave dispersion relation can be approximated by two latitude-dependent parameters in the limiting case when latitude approaches its singular value. Since the existence of a such singular latitude is generic for resonant triad interactions, a question on application of the RTM to the modeling of enhanced mixing in the vicinity of ridges in the ocean arises.