Remote sensing of breaking wave phase speeds with application to non-linear depth inversions

A number of existing models for surface wave phase speeds (linear and non-linear, breaking and non-breaking waves) are reviewed and tested against phase speed data from a large-scale laboratory experiment. The results of these tests are utilized in the context of assessing the potential improvement gained by incorporating wave non-linearity in phase speed based depth inversions. The analysis is focused on the surf zone, where depth inversion accuracies are known to degrade significantly. The collected data includes very high-resolution remote sensing video and surface elevation records from fixed, in-situ wave gages. Wave phase speeds are extracted from the remote sensing data using a feature tracking technique, and local wave amplitudes are determined from the wave gage records and used for comparisons to non-linear phase speed models and for non-linear depth inversions. A series of five different regular wave conditions with a range of non-linearity and dispersion characteristics are analyzed and results show that a composite dispersion relation, which includes both non-linearity and dispersion effects, best matches the observed phase speeds across the domain and hence, improves surf zone depth estimation via depth inversions. Incorporating non-linearity into the phase speed model reduces errors to O(10%), which is a level previously found for depth inversions with small amplitude waves in intermediate water depths using linear dispersion. Considering the controlled conditions and extensive ground truth, this appears to be a practical limit for phase speed-based depth inversions. Finally, a phase speed sensitivity analysis is performed that indicates that typical nearshore sand bars should be resolvable using phase speed depth inversions. However, increasing wave steepness degrades the sensitivity of this inversion method.     

均匀剪切流场中的强非线性波及其色散关系

考虑了剪切流场中强非线性波,唯一的假设是水深与特征波长之比是小量,建立了基于任意水深处速度而不是通常所用的平均速度为速度变量的模型。不仅改进了色散关系,使模型的水深适用范围更大,而且由于整个推导过程对波的振幅没有做任何假设,因而所获得的模型可以用于任何振幅的波。     

Downward transfer of momentum by wind-driven waves

A model for the downward transfer of wind momentum is derived for growing waves. It is shown that waves, which grow due to an uneven pressure distribution on the water surface or a wave-coherent surface shear stress have horizontal velocities out of phase with the surface elevation. Further, if the waves grow in the x-direction, while the motion is perhaps time-periodic at any fixed point, the Reynolds stresses associated with the organized motion are positive. This is in agreement with several field and laboratory measurements which were previously unexplained, and the new theory successfully links measured wave growth rates and measured sub-surface Reynolds stresses. Wave coherent air pressure (and/or surface shear stress) is shown to change the speed of wave propagation as well as inducing growth or decay. From air pressure variations that are in phase with the surface elevation, the influence on the waves is simply a phase speed increase. For pressure variations out of phase with surface elevation, both growth (or decay) and phase speed changes occur. The theory is initially developed for long waves, after which the velocity potential and dispersion relation for linear waves in arbitrary depth are given. The model enables a sounder model for the transfer to storm surges or currents of momentum from breaking waves in that it does not rely entirely on ad-hoc turbulent diffusion. Future models of atmosphere-ocean exchanges should also acknowledge that momentum is transferred partly by the organized wave motion, while other species, like heat and gasses, may rely totally on turbulent diffusion. The fact that growing wind waves do in fact not generally obey the dispersion relation for free waves may need to be considered in future wind wave development models.     

Statistical Distribution of Depth-Integrated Local Horizontal Momentum for Second-Order Random Ocean Waves in Finite Water Depth

Based on the second-order random wave solutions of water wave equations in tinite water depth, statistical distributions of the depth-integrated local horizontal momentum components are derived by use of the characteristic function expansion method. The parameters involved in the distributions can be all detemained by the water depth and the wavenumber spectrum of ocean waves. As an illustrative example, a fully developed wind-generated sea is considered and the parameters are calculated for typical wind speeds and water depths by means of the Donelan and Pierson spectrum. The effects of nonlinearity and water depth on the distributions are also investigated.     

A hybrid model for numerical wave forecasting and its implementation——Ⅰ. The wind wave model

The authors make an endeavor to explain why a new hybrid wave model is here proposed when several such models have already been in operation and the so- called third generation wave modej is proving attractive. This part of the paper is devoted to the wind wave model. Both deep and shallow water models have been developed, the former being actually a special case of the latter when water depth is great. The deep water model is exceptionally simple in form. Significant wave height is the only prognostic variable. In comparison with the usual methods to compute the energy input and dissipations empirically or by "tuning", the proposed model has the merit that the effects of all source terms are combined into one term which is computed through empirical growth relations for significant waves, these relations being, relatively speaking, easier and more reliable to obtain than those for the source terms in the spectral energy balance equation. The discrete part of the model and the implementation of the mode     

波浪的非线性显式弥散关系

波浪的非线性弥散关系在应用于求解波浪的变形问题时很不方便，需要与含非线性效应的缓坡方程一起进行迭代运算，往往导致数值计算的计算量太大，计算过于复杂。采用显式形式表达非线性弥散关系，可以克服上述缺点，大为简化波浪变形数值计算的计算量。本文通过将现有的非线性弥散关系进行分析比较，给出了一个更为一般的非线性弥散关系及其显式表达式，经比较可知，该显式弥散关系与相对应非线性弥散关系吻合的很好。本文最后用该显式结合含弱非线性效应的缓坡方程，对复式浅滩地形上的波浪折射绕射进行了计算。结果表明，考虑弱非线性可以得出与实验数据更为相符的结果，而采用显式弥散关系可以有效提高计算效率,在波浪的非线性计算中不失为一种切实有效的方法。     

地形对陆架拦截波性质及机制的影响

陆架波的性质如频散关系、形成机制等受地形影响。研究地形对陆架波的性质影响具有重要意义。基于陆架拦截波理论,数值计算了分段线性地形下不同宽度陆架上陆架拦截波的频散关系、长波假设下波动的相速度、阻尼情况下的波动耗散率以及强迫波的外力影响因子。分析了陆架宽度及坡度对自由及强迫陆架拦截波性质的影响。陆架宽度影响陆架拦截波的频散关系。陆架变宽,使得长波频散曲线的斜率增大。陆架宽度的增加使第一模态陆架拦截波有明显的性质变化:相速度增大,波动受辐散影响的程度变大,摩擦衰减距离增大,且风应力旋度在波动的生成机制中起到的作用渐强。在宽的陆架上,研究陆架拦截波的生成及强迫波的振幅时,应充分考虑风应力旋度的作用。第二、三模态波动的相速度受陆架坡度的影响较大,但摩擦衰减距离基本都在200km左右,几乎不随陆架宽度改变,属于局地波。     

Hydroelastic interaction between obliquely incident waves and a semi-infinite elastic plate on a two-layer fluid

The hydroelastic response of a semi-infinite thin elastic plate floating on a two-layer fluid of finite depth due to obliquely incident waves is investigated. The upper and lower fluids with different densities separated by a sharp and stable interface are assumed to be inviscid and incompressible and the motion to be irrotational. Simply time-harmonic incident waves of the surface and interfacial wave modes with a given angular frequency are considered within the framework of linear potential flow theory. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, a closed system of simultaneous linear equations is derived for the reflection and transmission coefficients of the series solutions. Based on the dispersion relations for the gravity waves and the flexural–gravity waves in a two-layer fluid and Snell’s law for refraction, we obtain a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves. Graphical representations of the series solutions show the interaction between the water waves and the plate. The effects of several physical parameters, including the density and depth ratios of the fluid and the thickness of the plate, on the wave scattering and the hydroelastic response of the plate are studied. It is found that the variation of the thickness of the plate may change the wave numbers and the critical angles. The density ratio is the main factor to influence the wave numbers of the interfacial wave modes. Finally, the stress state is considered.     

Analysis of overtopping flow on sea dikes in oblique and short-crested waves

Dike resilience against wave overtopping has gained more and more attention in recent years due to the effect of expected future climate changes. The overtopping flow velocities and flow depths on dikes have recently been studied in 2D small-scale experiments. This has led to semi-empirical formulae for the estimation of flow depths and flow velocities across a dike. The results have been coupled to the actual erosion of the landward dike slope determined by full-scale 2D tests using the so-called “Overtopping Simulator”. This paper describes the results from 96 small-scale tests carried out in a shallow water basin at Aalborg University to cover the so far unknown 3D effects from oblique long-crested and short-crested waves. Based on results from the laboratory tests, expansions are proposed to the existing 2D formulae so as to cover oblique and short-crested waves. The wave obliquity is seen to significantly reduce the overtopping flow velocities and flow depths on especially the landward slope of a sea dike. Moreover, the tests showed that the average flow directions on the dike crest from oblique long-crested and short-crested waves correspond approximately to the incident wave direction. Flow depths and the squared flow velocities on the dike are concluded to be Rayleigh-distributed in case of both long-crested and short-crested waves for all considered incident wave obliquities. Findings in the present paper are needed to obtain more realistic estimates of dike erosion caused by wave overtopping.     

Depth of activation on a mixed sediment beach

The relationship between wave height and depth of sediment activation is evaluated on an estuarine beach to determine whether activation depth is less in pebbles than sand. Rods with washers were used to monitor three excavated beach plots filled with (1) pebbles with mean grain size of 11.5 mm; (2) sand and granules; and (3) sand, granules and pebbles. Plots were monitored for 26 events over 27 days. Significant wave heights ranged from 0.18 to 0.40 m and activation depths from 0.02 to 0.12 m. Activation depths in the pebble plot were less than the other two plots when waves reworked sediment not activated during previous tidal cycles. Proportionality coefficients for activation depth to wave height, when net change was < 0.02 m, were 0.24 in the pebble plot and 0.30–0.31 in the other plots when experimental fill sediments remained and 0.22 to 0.23 in all three plots over the entire monitoring period, which included activation of newly deposited native sediment. Results suggest that for similar wave heights, activation depths in pebbles is lower than in sand, granules and pebbles or sand and granules, but once waves have reworked the sediment there is little difference in activation depths.     

The eigenvalue equations of equatorial waves on a sphere

Zonally propagating wave solutions of the linearized shallow water equations (LSWE) in a zonal channel on the rotating spherical earth are constructed from numerical solutions of eigenvalue equations that yield the meridional variation of the waves' amplitudes and the phase speeds of these waves. An approximate Schrödinger equation, whose potential depends on one parameter only, is derived, and this equation yields analytic expressions for the dispersion relations and for the meridional structure of the waves' amplitudes in two asymptotic cases. These analytic solutions validate the accuracy of the numerical solutions of the exact eigenvalue equation. Our results show the existence of Kelvin, Poincaré and Rossby waves that are harmonic for large radius of deformation. For small radius of deformation, the latter two waves vary as Hermite functions. In addition, our results show that the mixed mode of the planar theory (a meridional wavenumber zero mode that behaves as a Rossby wave for large zonal wavenumbers and as a Poincaré wave for small ones) does not exist on a sphere; instead, the first Rossby mode and the first westward propagating Poincaré mode are separated by the anti-Kelvin mode for all values of the zonal wavenumber.     

一种计算有限水深波能功率的新方法

海洋波浪能平均功率的准确计算是波浪能开发和利用的基础。实践中,波浪能转换装置一般安装在有限水深区域。对于随机波,只有当详尽的波浪谱已知的时候,有限水深区的波能功率才能被准确计算出来。由于种种原因,实践中波浪的实测数据大多以散点图或有义波高和统计波周期的形式给出,而波浪谱信息有时则很难获得。基于这种情况,传统上人们利用无限水深条件下的相关公式来估算有限水深区域的波能功率,但这种做法会造成较大的误差。本研究显示,对于50 m水深的理论波谱JONSWAP谱来说该误差高达14.6%。为了提高波能功率计算的准确性,本文提出了一种基于能量频率的一阶和二阶近似算法,可以在未知波浪谱的情况下较为准确地计算不同水深时的波能功率。针对两种理论波浪谱的计算结果表明,本方法在计算有限带宽内的波能功率时计算误差低于2.8%。     

Finite-amplitude analysis of some Boussinesq-type equations

Recent progress in formulating Boussinesq-type equations includes improved features of linear dispersion and higher-order nonlinearity. Nonlinear characteristics of these equations have been previously analysed on the assumption of weak nonlinearity, being therefore limited to moderate wave height. The present work addresses this aspect for an important class of wave problems, namely, regular waves of permanent form on a constant depth. Using a numerical procedure which is valid up to the maximum wave height, permanent-form waves admitted by three sets of advanced Boussinesq-type equations are analysed. Further, the characteristics of each set of the Boussinesq-type equations are discussed in the light of those from the potential theory satisfying the exact free-surface conditions. Phase velocity, amplitude dispersion, harmonic amplitudes (namely, second and third) and skewness of surface profile are shown over a two-parameter space of frequency and wave height.     

Generation of internal waves of the near-inertial period by a periodic wind

Internal waves at the inertial periods and outside them generated by the tangential stress of a periodic wind are studied in a linear statement. At a constant Väisälä-Brunt frequency and the ocean's depth, formulae have been derived which permit the computation of parameters of forced internal waves close to and at the inertial periods. It is shown that when approaching the inertial period, one can observe not only a general increase of the amplitudes of waves and velocities, but also the emergence of separate spikes at the resonance periods. When the periods exceed the inertial one, the wave disturbances attenuate rapidly in the ocean's depths.Translated by Mikhail M. Trufanov.     

Experimental study on internal waves in a stratified shear flow

This paper describes experiments on interfacial phenomena in a stratified shear flow having a sharp velocity shear at a density interface. The interface was visualized in vertical cross-section using dye, and the flow pattern was traced using aluminum powder. Two kinds of internal waves with different phase velocities and wave profiles were observed. They are here named p(positive)-waves and n(negative)-waves, respectively. By means of a two-dimensional visualization technique, the following facts have been confirmed regarding these waves. (1) The two kinds of waves propagate in the opposite direction relative to a system moving with the mean velocity at the interface, and their dispersion relations approximately agree with the two solutions of interfacial waves in a two-layer system of a linear basic shear flow. (2) The p-wave has sharp crests and flat troughs, and the n-wave has the reverse of this. This difference in wave profile is due to the finite amplitude effect. (3) Phase velocity of each wave lies within the range of the mean velocity profile, so that a critical layer exists and each wave has a “cat's eye” flow pattern in the vicinity of the critical layer, when observed in a system moving with the phase velocity. Consequently, these two waves are symmetrical with respect to the interface. The mechanisms of generation of these waves, and the entrainment process are discussed. It is inferred that when the “cat's eye” flow pattern is distorted and a stagnation point approaches the interface, entrainment in the form of a stretched wisp from the lower to the upper layer occurs for the p-wave, and from the upper to the lower layer for the n-wave.     

Generation of internal waves by moving non-stationary disturbances in a real stratified ocean

The plane problem on the generation of linear internal waves by a moving area of time-harmonic surface pressures in a continuously-stratified ocean of constant depth is considered. An analytical relation has been derived for forced internal waves off the site of their generation in the form of an internal wave field superposition corresponding to individual vertical modes. The possible wave regimes are determined. For the Brunt-Väisälä frequency distribution in the North Atlantic, the generation conditions and amplitudes of diverse radiated waves are numerically determined.Translated by Vladimir A. Puchkin.     

Absorption of infrasonic waves in a cloudy medium

We study the absorption of infrasonic waves when they are propagating in cloud media. In this case, the main mechanism of absorption is the relaxation processes associated with vapor-drop phase transitions caused by infrasonic waves. For frequencies comparable to the inverse relaxation time, the acoustic waves in a cloud medium were shown to have a noticeable dispersion. We derive an expression for the frequency-dependent effective adiabatic index linking the oscillations of pressure and density in the wave, the dispersion equation, and the expression for the absorption coefficient. For typical values of cloud parameters, the absorption coefficient was estimated to be around 0.1 km^?1. We discuss observation data which indirectly confirm that clouds can considerably absorb infrasonic waves.     

Joint statistical distribution of two-point sea surface elevations in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a joint statistical distribution of two-point sea surface elevations is derived by using the characteristic function expansion method. It is found that the joint distribution depends on five parameters. These five parameters can all be determined by the water depth, the relative position of two points and the wave-number spectrum of ocean waves. As an illustrative example, for fully developed wind-generated sea, the parameters that appeared in the joint distribution are calculated for various wind speeds, water depths and relative positions of two points by using the Donelan and Pierson spectrum and the nonlinear effects of sea waves on the joint distribution are studied.     

Wave-power absorption by an array of attenuators oscillating with unconstrained amplitudes

An infinite array of evenly spaced groups of oscillating bodies is considered. All groups (or ‘attenuators’) are equal and they have the same directional orientation. The angle of wave incidence is arbitrary. Regular waves diffracted and radiated from the bodies interfere constructively into rays of plane waves propagating away from the array. The number of rays depends on the ratio between the wavelength and the interspacing between adjacent groups. To each ray there corresponds one term in the ‘array radiation resistance matrix’. The maximum wave power absorbed by the array is derived under the assumption of linear theory and of unconstrained amplitudes of the oscillating bodies. It is found that, apart from exceptional cases, all of the incident wave power may be absorbed by the array provided the total number of oscillating modes in each group is at least as large as the number of rays. It is then explicitly demonstrated that the condition for maximum power absorption is that all rays have a vanishing intensity. Further, some previously known general relations between scattered waves and radiated waves have been extended.