Unexpected waves: Intermediate depth simulations and comparison with observations

In an earlier study, we defined an “unexpected wave” as, for example, a wave twice as high as any of the preceding 30 waves. Here we extend earlier deep water simulations to allow for the greater crest enhancement in water of finite depth and find that the predicted frequency of unexpected waves increases significantly. We also analyze data obtained by wave buoys off the east and west coasts of Canada. In both deep and intermediate depth water, the occurrence of unexpected waves is in reasonable accord with our simulations, supporting our assumption of random superposition of waves though with local crest enhancement by the non-resonant second harmonic.     

Regular waves onto a truncated circular column: A comparison of experiments and simulations

Accurate prediction of hydrodynamic forces on offshore structures is critical for safe and cost effective design of fixed and floating offshore structures exposed to a harsh environment. In the present paper, nonlinear interactions between regular waves and a single surface-piercing truncated circular column have been investigated using a frequency domain potential flow solver (DIFFRACT) and a full CFD solver in OpenFOAM for direct comparisons. Both the predicted free surface elevation around the column and the total force acting on the column have been analysed and compared with experimental data from MOERI. The degree of non-linearity and the contribution of each harmonic to the free surface run-up and wave forces have been examined, and evaluations of the accuracy and computational efficiency of the potential flow solver and the full CFD solver are provided and compared in the paper. Also of note are the local forms of the scattered waves around the column in numerical simulations, which are consistent with the Type-1 and Type-2 patterns identified in physical experiments at Imperial College.     

关于Boussinesq型水波方程理论和应用研究的综述

Boussinesq型方程是研究水波传播与演化问题的重要工具之一,本文就1967-2018年常用的Boussinesq型水波方程从理论推导和数值应用两个方面进行了回顾,以期推动该类方程在海岸(海洋)工程波浪水动力方向的深入研究和应用。此类方程推导主要从欧拉方程或Laplace方程出发。在一定的非线性和缓坡假设等条件下,国内外学者建立了多个Boussinesq型水波方程,并以Stokes波的相关理论为依据,考察了这些方程在相速度、群速度、线性变浅梯度、二阶非线性、三阶非线性、波幅离散、速度沿水深分布以及和(差)频等多方面性能的精度。将Boussinesq型水波方程分为水平二维和三维两大类,并对主要Boussinesq型水波方程的特性进行了评述。进而又对适合渗透地形和存在流体分层情况下的Boussinesq型水波方程进行了简述与评论。最后对这些方程的应用进行了总结与分析。     

Reynolds number dependence of mixing in a lock-exchange system from direct numerical and large eddy simulations

Turbulent mixing of water masses of different temperatures and salinities is an important process for both coastal and large-scale ocean circulation. It is, however, difficult to capture computationally. One of the reasons is that mixing in the ocean occurs at a wide range of complexity, with the Reynolds number reaching , or even higher.In this study, we continue to investigate whether large eddy simulation (LES) can be a reliable computational tool for stratified mixing in turbulent oceanic flows. LES is attractive because it can be times faster than a direct numerical simulation (DNS) of stratified mixing in turbulent flows. Before using the LES methodology to compute mixing in realistic oceanic flows, however, a careful assessment of the LES sensitivity with respect to Re needs to be performed first. The main objectives of this study are: (i) to investigate the performance of different LES models at high Re, such as those encountered in oceanic flows; and (ii) to study how mixing varies as a function of Re. To this end, as a benchmark we use the lock-exchange problem, which is described by unambigous and simple initial and boundary conditions. The background potential energy, which accurately quantifies irreversible mixing in an enclosed system, is used as the main criterion in a posteriori testing of LES.This study has two main achievements. The first is that we investigate the accuracy of six combinations of two different classes of LES models, namely eddy-viscosity and approximate deconvolution types, for 3×10³Re3×10⁴, for which DNS data is computed. We find that all LES models almost always provide significantly more accurate results than cases without LES models. Nevertheless, no single LES model that is persistently superior to others over this Re range could be identified. Then, an ensemble of the four best performing LES models is selected in order to estimate mixing taking place in this system at Re=10⁵ and 10⁶, for which DNS is presently not feasible. Thus the second achievement of this study is to quantify mixing taking place in this system over an Re range that changes by three orders of magnitude. We find that the background potential energy increases by about 67% when Re is increased from Re=10³ to Re=10⁶, within the computation period, with the most significant increase taking place from Re=3×10³ to Re=10⁵.     

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 new method for the generation of second-order random waves

In order to prevent the generation of spurious free sub- and superharmonics of random waves in a laboratory channel, the control signal for the wave board has to be derived according to a higher-order wave theory. An expression for this control signal has been derived with the perturbation method of multiple scales. It is much less complex and requires less computation time than the expressions obtained from the full second-order theory. The new method for second-order subharmonics was verified experimentally for waves with bichromatic and continuous first-order spectra. The data were analysed with the complex-harmonic principal-component analysis to reduce the influence of noise.     

A note on the theoretical comparison of wave staffs and wave rider buoys in steep gravity waves

The sea surface displacements measured by a wave staff, fixed in horizontal position, and a wave rider buoy, which moves with the water particles, are compared. Second-order theory shows that, to this approximation, the second harmonic in the surface displacement is not measured by the wave rider. Here a simple approximation to maximum amplitude gravity waves is considered; then a freely floating wave rider gives an approximately sinusoidal record at a lower frequency than the wave, while a tethered wave rider gives results depending on the mooring system. The horizontal motion of a tethered buoy still affects the measurements, particularly of the second and higher harmonics.     

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.     

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.     

A theory of semigeostrophic gravity waves and its application to the intrusion of a density current along a coast

Semigeostrophic gravity waves associated with a coastal boundary current, which has finite and uniform potential vorticity and is bounded away from the coastline by a density front on the ocean surface, are investigated. It is shown that the semigeostrophic coastal current has two waves which are named here the Semigeostrophic Coastal Wave (SCW) and the Semigeostrophic Frontal Wave (SFW). The SCW becomes an elementary Kelvin wave at some limit while the SFW is caused by the existence of the surface density front. The SCW appears mainly as variations in the upper layer depth at the coast and as alongshore velocity at the density front. On the other hand, the SFW appears mainly as variations in the width of the current. When the weak nonlinearity and ageostrophic effect are included, these semigeostrophic gravity waves satisfy the Kortweg- de Vries equation, which suggests that the local changes in the width and/or velocity of the semigeostrophic coastal current propagate as wave-like disturbances.     

Surface gravity waves formed in the process of motion of an ellipsoidal body submerged in the liquid

We study the problem of flow around an ellipsoidal body submerged in a liquid half space and obtain the solution for the free wave surface. The results of numerical analysis reveal the existence of a great variety of modes of motion characterized by substantially different shapes of the free surface. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

Improved explicit approximation of linear dispersion relationship for gravity waves: A discussion

Recently, an accurate explicit approximation to linear dispersion relationship is proposed based on Eckart's explicit relationship (Beji, 2013). The author has nicely improved Eckart's explicit dispersion relationship by introducing an empirical correction function. The resulting expression is valid for the entire range of relative water depths and accurate to within 0.044%.In this discussion, the proposed expression by the author is simplified and improved to an accuracy of 0.019%. Moreover, a near exact solution with 0.001% accuracy is also given.     

A numerical method for the two layer shallow water equations with dry states

A numerical method is proposed for solving the two layer shallow water equations with variable bathymetry in one dimension based on high-resolution f-wave-propagation finite volume methods. The method splits the jump in the fluxes and source terms into waves propagating away from each grid cell interface. It addresses the required determination of the system’s eigenstructure and a scheme for evaluating the flux and source terms. It also handles dry states in the system where the bottom layer depth becomes zero, utilizing existing methods for the single layer solution and handling single layer dry states that can exist independently. Sample results are shown illustrating the method and its handling of dry states including an idealized ocean setting.     

Source generation of nonlinear gravity waves with the boundary integral equation method

The source generation of nonlinear gravity waves in a boundary integral method is reported. Through a number of computational results it is demonstrated that virtually any type of two-dimensional wave field can be generated. It is furthermore shown that scattered waves can pass through the line of generation without significant reflection and that the process of generation is not affected by the scattered waves.     

欧拉模型模拟波浪的抑制数值耗散效应的一种简单方法

数值波浪水槽是研究波浪及波浪与结构相互作用的常用工具,可在真实尺度下产生波浪,并提供流场的详细数据。然而,大部分数值波浪水槽都存在数值耗散和数值色散问题,数值耗散使波能缓慢消散,数值弥散在波传播过程中使波频移。本文在有限差分法（FDM）求解欧拉方程的基础上,提出了一种抑制数值耗散效应的简单方法,考虑阻尼项的影响,对波的传播解进行了解析求解。该方法的主要思想是在动量方程中附加一个源项,其强度由数值阻尼效应的强度决定。本文通过对规则线性波、Stokes波和不规则波的数值模拟,验证了该方法的有效性。结果表明,本文方法可有效减小数值波浪水槽中存在的数值耗散现象。     

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.     

On the estimation of surface gravity waves from subsurface pressure records for estuarine basins

According to small-amplitude theory, the surface gravity-wave spectrum can be estimated from a subsurface pressure-fluctuation spectrum by applying a factor (K) that compensates for the attenuation of surface-wave amplitude as the depth below the water surface and the wave frequency increase.There are a number of factors, however, that cause K to be inaccurate over a large portion of the spectrum's frequency range. Numerous attempts have been made to derive an empirical correction factor (n) that could be applied to K to provide a better estimate of the surface-wave spectrum. This paper evaluates some of these empirical factors, specifically for use in an estuarine environment, and recommends Graces' (1978) equation for n as a function of the non-dimensional frequency parameter kh (where $\text{k =}\text{2π}\text{L}$ is the local wavenumber, h the local depth and L the wavelength).The paper also evaluates the maximum limit (K_max) on the magnitude of K suggested by Esteva & Harris (1970), where relative depth $\text{d}\text{h}$ (d is the pressure transducer height above the bottom) and k_oh (a parameter directly related for large values of kh to wave frequency by the dispersion relation) are the independent variables. The choice of K_max may be made unimportant if d is selected beforehand using an equation (Knowles, 1981a) for the minimum $\text{d}\text{h}$ limit affected by the choice of K_max.     

Filtering inertia-gravity waves from the initial conditions of the linear shallow water equations

A method for filtering inertia-gravity waves from elevation and depth-averaged velocity is described. This filtering scheme is derived from the linear shallow water equations for constant depth and constant Coriolis frequency. The filtered solution is obtained by retaining only the eigenvectors corresponding to the geostrophic equilibrium and by discarding explicitly the eigenvectors corresponding to the fast moving inertia-gravity waves. An alternative formulation is derived using a variational approach. Both filtering methods are tested numerically for a periodic domain with constant depth and the variational approach is implemented for a closed domain with large topographic variations. The filtering methods significantly reduce the amplitudes of the inertia-gravity waves while preserving the mean flow. The variational method is compared to the Incremental Analysis Update technique and the benefits of the variational filter are presented.     

Backscatter of microwaves from the sea surface and the modulation of spectra of short gravity waves

Backscatter cross-sections of microwaves from the sea surface are calculated by using the facet model and are compared withJonswap '75 experimental results. The principal features obtained are: (1) asymmetry of backscatter cross-sections between upwind and downwind directions is attributable to the modulation of the short gravity-capillary wave spectrum by a larger wave, and the non-Gaussian wave slope distribution has a tendency to cancel this effect, (2) angular spreading of the energy spectrum in the higher frequency range should have a narrower band than a simple cosine distribution, (3) the facet model itself should begin to break down at a larger incident angle than previously supposed.