Experimental study of long wave generation on sloping bottoms

Low-frequency waves generated on steep (1:10) and mild (1:40) slopes by six series of bichromatic wave groups are studied experimentally. The shorelines for both slopes are replaced by horizontal reaches of small depth. This reduces the reflection of long waves near the shoreline significantly, which for the first time makes possible the explicit observation of outgoing breakpoint forced long waves. The breakpoint and released bound long wave mechanisms on the different slopes are compared. Generally, the breakpoint forced long waves dominate the low-frequency wave field on the steep slope, while the released bound long waves are found to be more significant on the mild slope. Two parameters indexing the effectiveness of the breakpoint mechanism are compared and the normalized slope tends to give more realistic results. Shoaling of bound long waves is analyzed and the shallow-water equilibrium limit ~ h^−5/2 exhibits a good prediction of the variation of the bound long waves on both slopes.     

Reflection of a long wave from an underwater slope

Reflection of long sea waves from an underwater slope described by a power law is studied within the shallow water theory. The slope is connected with the flat bottom. This model allows us to estimate the roles of a pointwise reflection from the inflection point of the bottom profile and distributed reflection at the underwater slope. The case of the underwater slope described by the so-called nonreflecting beach (h(x) ∼ x ^4/3, where h is the depth of the basin and x is the coordinate) when the wave is reflected only from the inflection point (pointwise reflection) is specially considered. The reflection and transmission coefficients over the bottom topography were calculated, and it was shown that the sum of the squared absolute values of these values differs from unity for all profiles except the nonreflecting one. This difference is related to the distributed re-reflections (resonances) over the underwater slope that lead to the deviations in the wave height from the known Green’s law.     

Infra-gravity wave generation by the shoaling wave groups over beaches

A physical parameter, μb, which was used to meet the forcing of primary short waves to be off-resonant before wave breaking, has been considered as an applicable parameter in the infra-gravity wave generation. Since a series of modulating wave groups for different wave conditions are performed to proceed with the resonant mechanism of infra-gravity waves prior to wave breaking, the amplitude growth of incident bound long wave is assumed to be simply controlled by the normalized bed slope, βb. The results appear a large dependence of the growth rate, α, of incident bound long wave, separated by the three-array method, on the normalized bed slope, βb. High spatial resolution of wave records enables identification of the cross-correlation between squared short-wave envelopes and infra-gravity waves. The cross-shore structure of infra-gravity waves over beaches presents the mechanics of incident bound- and outgoing free long waves with the formation of free standing long waves in the nearshore region. The wave run-up and amplification of infra-gravity waves in the swash zone appear that the additional long waves generated by the breaking process would modify the cross-shore structure of free standing long waves. Finally, this paper would further discuss the contribution of long wave breaking and bottom friction to the energy dissipation of infra-gravity waves based on different slope conditions.     

Some properties of surf-beats

Long ocean waves with periods of several minutes (surf-beats) were observed at a marine observation tower. We have analysed time series data of an envelope of incident swell, long period current velocity and surface elevation fluctuations. Current velocity was measued by an electromagnetic flow meter. Surf-beats amplitudeH ^(l) is shown to be proportional to 3/2 power of incident swell amplitudeH ^(s), and decreases with increase of depthh in proportional toh ^–1/2 such thatH ^(l) H ^(s) (H ^(s)/h)^1/2. Frequency energy density functionP _LL (f) of surface elevation had two dominant peaks whose frequencies were highly stable through the entire observational period. Cross-spectral analysis suggested that those peaks correspond to traveling edge waves caused by the excess momentum and mass flux in the surf zone. The forced long ocean waves predicted byLonguet-Higgins andStewart (1964) was ditected. Phase-shift and wave height of the wave with respect to those of incident swell envelope are shown to be in remarkable agreement with the predictions. However the forced long wave is only a minor component in the total energy of surf-beats. Current fields are shown to be largely composed of non-surface modes.     

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.     

江苏海岸中部近岸冬季潮汐和潮流特征

南黄海西侧的江苏海岸近岸区域,素以地形复杂、潮流强劲、悬沙输运剧烈著称,但是较长期的同步潮位和潮流观测数据仍然缺乏,尤其是在近岸(20 km)浅水(20 m)区域。2014年1月在大丰港附近开展了连续潮位和潮流观测,获得的数据揭示了一系列特征。此地潮汐潮流为正规半日潮,浅水分潮显著。平均潮差为3.05 m,最显著的两个分潮为M2和S2分潮,振幅分别为1.45 m和0.52 m。潮流最显著的半日分潮M2分潮和最显著的浅水分潮M4分潮在沿岸方向上振幅分别为0.84m/s和0.12m/s,在跨岸方向上振幅分别为0.24 m/s和0.01 m/s,沿岸方向占绝对优势。潮波的沿岸传播介于前进波和驻波之间,驻波的特征稍强。M2分潮潮流椭圆最大流(长轴)方向为南偏东7.4°。存在冬季沿岸向北的余流,垂向平均值的大小为2.2 cm/s。以上潮汐潮流特征为该区域海洋物质输运研究提供了基础资料。     

Wave set-up on coral reefs. 1. Set-up and wave-generated flow on an idealised two dimensional horizontal reef

Wave set-up may be significant in determining water levels on coral reefs particularly in microtidal environments and hence is an important factor for the design of reef-top structures and for the stability of reef-top islands. Laboratory experiments have been made on a two dimensional model of an idealised horizontal reef under two different conditions corresponding to a fringing reef (or closed lagoon) situation and a platform reef (or open lagoon) situation. Both wave set-up on the reef-top and the wave-generated flow across the reef were measured and related to wave and tide level conditions.All other factors being the same, wave set-up is greatest at low tide levels whereas wave-generated flow is greater at higher tide levels. The magnitude of the set-up on a platform reef with a wave-generated flow is less than on a fringing reef without any net flow by an amount equal to the velocity head of the flow across the reef. Dimensionless parameters and q/√gH_o³ are found to be functions of relative submergence parameters h_r/H_o or . For values of ( ) H_o > 1 waves break on the reef-top and radiation stress theory can be used to calculate set-up. For ( )H_o < 0.7 waves break on the reef-face and set-up is determined by broadcrested weir control at the reef-edge. (The symbols are defined as follows: g is gravitational acceleration; h_r is still water depth over horizontal reef-top; H_o is offreef wave height (equivalent deep water value); q is discharge per unit length of reef edge; T is wave period and is maximum wave set-up on reef-top.)     

Transformation of the Indonesian Throughflow Water by Vertical Mixing and Its Relation to Tidally Generated Internal Waves

Numerical experiments with two-dimensional nonhydrostatic model have been performed to investigate tidally generated internal waves at the Dewakang sill at the southern Makassar Strait where two large-amplitude “bumps” of relatively shallow water exist. We investigate the effect of these features on vertical mixing, with emphasis on the transformation of the Indonesian throughflow (ITF) water properties. The result shows that large-amplitude internal waves are generated at both bumps by the predominant M₂ tidal flow, even though the condition of the critical Froude number and the critical slope are not satisfied. The internal waves induce such vigorous vertical mixing in the sill region that the vertical diffusivity attains a maximum value of 6 × 10⁻³ m²s⁻¹ and the salinity maximum and minimum core layers characterizing the ITF thermocline water are considerably weakened. Close examination reveals that bottom-intensified currents produced mainly by the joint effect of barotropic M₂ flow and internal tides generated in the concave region surrounding both bumps can excite unsteady lee waves (Nakamura et al., 2000) on the inside slopes of the bumps, which tend to be trapped at the generation region and grow into large-amplitude waves. Such generation of unsteady lee waves does not occur in case of one bump alone. Trapping and amplification of the waves in the sill region induce large vertical displacements (∼60 m) of water parcels during one tidal period, leading to strong vertical mixing there. Since the K₁ tidal currents are relatively weak, large-amplitude internal waves causing intense vertical mixing are not generated. This revised version was published online in July 2006 with corrections to the Cover Date.     

A comparison of weakly and fully non-linear models of the shoaling of a solitary internal wave

This study investigates the behaviour of internal solitary waves crossing a continental slope in the presence of a seasonal thermocline. Comparisons are made between a fully non-linear computational fluid dynamics (CFD) model, and weakly non-linear theory. Previous observations suggested that the amplitudes of solitary waves are capped as they pass across the continental slope, which may be due to laminar dynamics, or due to the effect of turbulence. Across the continental slope, CFD and second order variable depth KdV (vEKdV) predictions agree well with observations of a limited change in solitary wave amplitude. First order variable depth KdV theory overpredicts the final amplitude significantly. In terms of the wave shape, the CFD modeled wave changes from a KdV shape in deep water towards an EkdV solution in shallow water, as observations suggest. The phase speed of the CFD and vEKdV waves are similar to that observed in waters of 400–500 m deep, but are slightly lower than observed in 140 m depth. CFD predictions using a standard k, turbulence model showed that turbulence had little effect on the amplitude. These preliminary results indicate that in this situation wave capping is due to laminar, large amplitude solitary wave dynamics and is independent of turbulent mixing.     

Topographically trapped waves over the continental shelf and slope

General characteristics of topographically trapped subinertia waves are discussed from the viewpoint of an eigenvalue problem and ray theory. Special attention is paid to the slope parameterS(x) (=(dh/dx)/h, wherex denotes the coordinate perpendicular to the shoreline, increasing seaward, andh(x) is the depth) which is a measure of the strength of the restoring force of the waves. Three cases for theS distribution are considered, in whichS is assumed to be positive at the coast and to tend to zero far from the coast. The first is whereS(x) decreases monotonically towards the open ocean. It is found in this case that waves are trapped near the coast. The second is whereS(x) does not decrease monotonically, but has a maximum. It is concluded that this case may contain two types of waves, i.e., those trapped near the coast and those trapped near the maximum, and the dispersion curves corresponding to different types may nearly intersect, namely, result in “kissing”. The third is whereS(x) has a negative region (corresponding to the presence of a trench). It is found in this case that an infinite sequence of waves is trapped in the negativeS region which propagate with the coast to their left (right) in the northern (southern) hemisphere besides the waves trapped near the coast. The topography in the second case corresponds to a typical continental shelf and a typical continental slope. It is shown by model calculation that trapped waves are present over the continental slope as well as over the continental shelf. Almost the same results are obtained for superinertia waves ifS is replaced by 1/h which is a measure of the restoring force of superinertia waves.     

Effects of longshore shelf variations on barotropic continental shelf waves,slope currents and ocean modes

Effects of continental shelf bends, converging depth contours and changing depth profiles are discussed. Some analysis is carried out for previously unstudied cases. Separate oceanic interior and shelf flow problems are formulated for a sufficiently narrow shelf. The ocean interior ‘sees’ only an integrated shelf effect, typically increasing shelf-edge amplitudes, retarding longshore Kelvin-wave propagation and increasing natural mode periods by 0 (10%). On the local shelf, the flow matches to the ocean interior and is nondivergent. Effects on shelf waves and slope currents depend subtly on the nature of the longshore variations. Curvature and contour convergence do not per se imply scaterring or generation of shelf waves. Indeed, any depth h(ξ) where ▽² ξ(x,y) = 0 (a condition approximating longshelf uniformity in the topography's convexity) supports essentially the same shelf waves as do straight depth contours (DAVIS, 1983), and slope currents follow depth contours. Scattering results rather from breaks in analyticity of the depth profile. Hence calculations for small isolated features (necessarily highly convex or concave) may overestimate scattering, and superposition for realistic topography may lead to much self-cancellation among scattered waves. Otherwise, examples show a strong preference for scattering into adjacent mode numbers and into any shelf wave mode near to its maximum frequency. A shelf sector, where the maximum shelf wave frequency maxω is less than the frequency ω of an incident shelf wave, causes substantial scattering unless maxω and ω are very close. Adjustment of slope currents to changed conditions takes place through (and over the decay distance of) scattered shelf waves.     

Explicit wave-averaged primitive equations using a generalized Lagrangian mean

The generalized Langrangian mean theory provides exact equations for general wave–turbulence–mean flow interactions in three dimensions. For practical applications, these equations must be closed by specifying the wave forcing terms. Here an approximate closure is obtained under the hypotheses of small surface slope, weak horizontal gradients of the water depth and mean current, and weak curvature of the mean current profile. These assumptions yield analytical expressions for the mean momentum and pressure forcing terms that can be expressed in terms of the wave spectrum. A vertical change of coordinate is then applied to obtain glm2z-RANS equations with non-divergent mass transport in cartesian coordinates. To lowest order, agreement is found with Eulerian mean theories, and the present approximation provides an explicit extension of known wave-averaged equations to short-scale variations of the wave field, and vertically varying currents only limited to weak or localized profile curvatures. Further, the underlying exact equations provide a natural framework for extensions to finite wave amplitudes and any realistic situation. The accuracy of the approximations is discussed using comparisons with exact numerical solutions for linear waves over arbitrary bottom slopes, for which the equations are still exact when properly accounting for partial standing waves. For finite amplitude waves it is found that the approximate solutions are probably accurate for ocean mixed layer modelling and shoaling waves, provided that an adequate turbulent closure is designed. However, for surf zone applications the approximations are expected to give only qualitative results due to the large influence of wave nonlinearity on the vertical profiles of wave forcing terms.     

Thermal structures in the Kuroshio extension measured with a towed thermistor chain

Two-dimensional temperature data observed by use of a 275 meter towed thermistor chain deployed from an oceanographic research vessel USS MARYSVILLE, which cruised with a speed of 6.2 knots in July 1966 across the Kuroshio Extension in the North Pacific, are investigated. Two-dimensional variations of the distribution of the isotherms along the ship's track are analyzed with special reference to their slope, wavelength and wave height. The results show that the slope and wave height of isotherms have a tendency to increase as the temperature decreases. Even if the contribution of wave heights smaller than 1.5 m is neglected, i.e., contribution of large scale slope with a horizontal scale of 5–30 km is subtracted, this tendency is still detected. In contrast to this, the wavelength evaluated by the crest to crest method has no dependency on the temperature. Power spectrum of the isotherm depth is proportional tok ^–1.87 for 13°C andk ^–2.13 for 27°C, wherek is the wave number. It is shown that the spectra of warmer isotherms are relatively well approximated by –2 power law (Garrett and Munk spectrum) for internal waves rather than the –5/3 power law (Kolmogorov spectrum) for three dimensional isotropic turbulence.     

Modeling wave�Ccurrent bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.     

Unusual features of the distribution of Radon-222 in the bottom waters of the continental slope of the Mid-Atlantic Bight: Hydrodynamic implications

²²²Rn was measured in the near-bottom waters of the continental slope of the Mid-Atlantic Bight. Separate measurements of the ²²²Rn supported by dissolved ²²⁶Ra allowed the excess ²²²Rn that is derived from the underlying sediments to be distinguished. Measurements of production of ²²²Rn by the sediments were used to calculate fluxes of ²²²Rn from sediments that would be expected as a result of molecular diffusion. On the upper slope and on the lower slope excess ²²²Rn standing crops were, respectively, greater than and consistent with fluxes of radon from sediments by molecular diffusion as are typical of most ocean environments. On the middle slope, however, observed excess ²²²Rn concentrations and standing crops were significantly lower than what would be expected from the calculated fluxes from the underlying sediments. This unusual feature of low radon concentrations on the middle slope is referred to as the low-radon zone (LRZ). This LRZ was always present over several years and seasons, but was variable in intensity (excess-radon concentration and standing crop) and in location on the slope. Low concentrations of suspended particulate matter and low current velocities observed by others in the same region are consistent with low mixing as a possible cause of the LRZ. Radon profile shapes and recent work by others on near bottom mixing due to interactions between topography and internal waves, however, suggest that high mixing due to internal waves is a more likely cause of the LRZ.     

Wave reflection characteristics of plane, dentated and serrated seawalls

The hydrodynamic performance of vertical and sloped plane, dentated and serrated seawalls were investigated using physical model studies. Regular and random waves of wide range of heights and periods were used. Tests were carried out for different inclinations of the seawall (i.e. θ=30, 40, 50, 60 and 90°) and for a constant water depth of 0.7 m. The wave reflection was measured to assess the dissipation character of the seawalls. It was observed that the serrated seawall was superior to the plane and dentated seawall in reducing the wave reflection. Even for the vertical case, the coefficient of reflection due to regular waves for dentated seawall ranged from 0.6–0.99 and for the vertical serrated seawall it was 0.45–0.98, whereas for the vertical plane wall, it was almost 1.0. It was found that the characteristic dimension of the seawall (i.e. L/W) and the relative water depth (i.e. d/L) were better influencing parameters compared to the conventionally used surf similarity parameter ‘ξ’ (ξ=tan θ/(H_i/L)^0.5) in predicting the reflection from the dentated and serrated seawall, where L is the local wave length, W the width of the dent along the length of the seawall slope, d the water depth at the toe and H_i is the incident wave height. A similar trend was observed for the random waves too. The reduction in the wave reflection due to random waves for the dentated seawall as compared to the plane seawall was about 18% and for the serrated seawall, it was 20%. It was observed that the reflection due to random waves was lesser for all the three different walls than the regular waves, due to the mutual interaction of random waves. Multiple regression analysis on the measured data points was carried out and predictive equations for the reflection coefficient were obtained for both regular and random waves. This study will be useful in the design of energy dissipating type vertical quay walls in ports and harbours, sloped seawalls for shore protection from erosion and sloped caisson as breakwaters. Comparison of predictive formulae with the experimental results revealed that the prediction methods were good enough for practical purposes.     

Short gravity waves on steady non-uniform currents obtained through up-/downwelling

Small amplitude water waves propagating in a medium with a steady non-uniform current are investigated. The non-uniform current is obtained by up- or downwelling through the horizontal bed. A new locally valid velocity potential correct to the second order is derived describing the combined wave–current motion. From this solution expressions for the local evolution of the wave amplitude and the wave number are extracted. These expressions are compared with the results found using the principle of wave action conservation and the linear dispersion relation, and good agreement is found at small distances compared to the wavelength. Unlike earlier works there is no restriction to deep water. The results valid for deep water are found as a special case of the general solution and agree with the solution found by Longuet-Higgins, M.S. and Stewart, R.W. (1961) The changes in amplitude of short gravity waves on steady non-uniform currents. Journal of Fluid Mechanics, 10(4), 529–549. Furthermore, it is shown that the principle of wave action conservation in fact holds for waves propagating in a medium with a steady non-uniform current maintained by up-/downwelling also on finite depth.     

The experimental study on infra-gravity wave

A series of experiments were conducted in a super-wave flume (300 m×5 m×5.2 m) to examine the low-frequency motion induced by waves with different incident steepness, sloping gradients and normalized frequency (sideband space). Two kinds of waves including initial uniform wave train and modulated wave train (one carrier with a pair of sidebands) are utilized for incident wave conditions. From the experimental results, it is found that for a given slope gradient the infra-gravity wave component decreases as wave nonlinearity increases and frequency downshift is a predominant factor. Furthermore, the magnitude of low-frequency component decreases with slope gradients for a given initial wave condition. In addition, the maximum value of low-frequency motion is found to be close to the normalized frequency, δ=1.0.     

Very large subaqueous sand dunes on the upper continental slope in the South China Sea generated by episodic,shoaling deep-water internal solitary waves

Very large subaqueous sand dunes were discovered on the upper continental slope of the northern South China Sea. The dunes were observed along a single 40 km long transect southeast of 21.93°N, 117.53°E on the upper continental slope in water depths of 160 m to 600 m. The sand dunes are composed of fine to medium sand, with amplitudes exceeding 16 m and crest-to-crest wavelengths exceeding 350 m. The dunes' apparent formation mechanism is the world's largest observed internal solitary waves which generate from tidal forcing on the Luzon Ridge on the east side of the South China Sea, propagate west across the deep basin with amplitudes regularly exceeding 100 m, and dissipate extremely large amounts of energy via turbulent interaction with the continental slope, suspending and redistributing the bottom sediment. While subaqueous dunes are found in many locations throughout the world's oceans and coastal zones, these particular dunes appear to be unique for two principal reasons: their location on the upper continental slope (away from the influence of shallow-water tidal forcing, deep basin bottom currents and topographically-amplified canyon flows), and their distinctive formation mechanism (approximately 60 episodic, extremely energetic, large amplitude events each lunar cycle).     

Origin of theAbiki phenomenon (a kind of seiche) in Nagasaki Bay

Large oscillations of water level in Nagasaki Bay are calledAbiki and are most frequently observed in winter. The largestAbiki recorded in the past 20 years at the tide station at Nagasaki occurred on March 31, 1979. Simultaneously, a distinct atmospheric pressure disturbance of solitary type with an amplitude of about 3 mb was recorded at several neighbouring stations in Kyûshû, which indicated the pressure disturbance probably travelled eastward with an average speed of about 110 km h^–1.The quantitative relation between this pressure disturbance and notable seiches observed in Nagasaki Bay is examined by means of numerical simulation, and it is confirmed that the exceptionally large range of oscillations in the bay, which reached 278 cm at the tide station, was indeed produced by this travelling pressure disturbance.The leading part of shallow water waves induced by the atmospheric pressure disturbance was amplified up to about 10 cm in amplitude, over the broad continental shelf region off China, because of near resonant coupling to the pressure disturbance. After leaving this continental shelf region, the amplified water wave converged into the shelf region (Gotô Nada) surrounded by the north-western coast of Kyûshû and the Gotô Islands and excited eigenoscillations on the shelf. A train of waves thus formed with a period of about 35 min entered Nagasaki Bay and was resonantly amplified at periods of 36 min and 23 min which are the eigen periods of the bay. Besides resonance, the combined effects of shoaling and reflection inside Nagasaki Bay also enhanced the amplification.