A nonlinear model for nonbreaking shoaling random waves

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Irrotational,progressive surface gravity waves near the limiting height

Numerical solutions of irrotational, progressive surface gravity waves in water of a constant depth are obtained by means of an iterative method. Our results suggest that waves with the surface slope angle greater than/6 may exist. The calculated phase velocity of deep water waves near the wave steepness 0.14 is significantly smaller than the value given by the Stokes' fourth approximation.In order to check our method, we apply it to the problem proposed byDavies (1951), which is hypothetical but similar to the present problem, and for which the exact solution is known. In this case our results show good agreement with the exact solution.     

Trapped bottom topographic waves over the inclined bottom

In the Boussinesq approximation, we study baroclinic topographic waves trapped by the flat meridional slope. The existence of these waves is explained by stratification, inclined bottom, and Earth's rotation. We deduce the evolutionary equation for the square of the envelope of a narrow-band wave packet of trapped waves. In the second order of smallness relative to the wave amplitude, we find the mean fields of velocity and density induced by the packet. It is shown that, in the limiting case of weakly nonlinear plane waves, the induced current is zonal. In the Northern hemisphere, depending on the slope of the bottom γ₁, the sign of the phase velocity σ/k (k is the zonal wave number) is either always positive (for γ₁>γ_1cr) or always negative (for γ₁<γ_1cr). If we neglect the vertical component of the Coriolis acceleration, then γ_1cr=0. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

On the second approximation to mass transport in the bottom boundary layer

The second approximation is obtained for the mass transport velocity within the oscillatory bottom boundary layer beneath sinusoidal progressive and standing waves of finite amplitude. This approximation includes a simple new term, which essentially ensures continuity of the vertical gradient of mass transport at the edge of the layer and is of third-order in the perturbation (or wave-slope) parameter. For long progressive waves in conditions of zero net mass flow, the term represents a moderate reduction in mass transport at the edge of the layer, compared with the first approximation of Longuet-Higgins. For standing waves of arbitrary length, the mass transport is reduced (increased) far from (near) the bottom, except near nodal locations where an increase (a reduction) is predicted. The proposed correction to the first approximation yields clearly improved results when compared with appropriate experimental evidence. Deficiencies in the higher-order theories of Sleath and Isaacson for propagating waves are disclosed.     

The effect of the vertical density structure on internal waves in the neighbourhood of the intertial period

With the latitudinal variability of the Coriolis parameter considered, the influence of the vertical density structure on the parameters of free internal waves is stuided. Numerical calculations are performed for Brunt-Väisälä frequency profiles characteristic of the four seasons of the year in the deep section of the Black Sea. Results of the analysis of the dependences of the wavenumber and group velocity on the frequency are presented, and the domain of application of thef-plane approximation is defined.Translated by V. Puchkin.     

The influence of quasi resonant internal waves on the radar imaging mechanism of shallow sea bottom topography

During previous field experiments in the North Sea it was often assumed that the water column in such shallow coastal tidal waters is vertically well mixed and stratification was neglected when discussing the Normalized Radar Cross Section modulation caused by the sea floor. In this paper the influence of quasi resonant internal waves with the sea bed on the radar imaging mechanism of submarine sand waves itself is investigated. In situ data of the tidal current velocity and several water quality parameters such as sea surface temperature, fluorescence, and beam transmittance were measured in the Southern Bight of the North Sea in April 1991. Simulations of the total NRCS modulation caused by sand waves and internal waves as a function of the current gradient or strain rate induced by the internal wave current field at the sea surface have been carried out using the quasi-steady approximation and linear internal wave theory. As a first approximation the strain rate depending on stratification was calculated using the two-layer model. These simulations demonstrate that at least a density difference between the two layers of the order of Δρ ≈ 1 kg m^–3 is necessary for a sinusoidal thermocline to effect the total NRCS modulation considerably. The NRCS modulation as a function of wind friction velocity has been calculated independently and is discussed with regard to the strain rate of the surface current field caused by the superimposed imaging mechanisms of sand waves and internal waves. It turned out that the existence of a surface roughness-wind stress feedback mechanism cannot be excluded.     

An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

Nonlinear wave effect on the slow drift motion of a floating body

The slow drift motion of a floating body in a two-dimensional wave field has been investigated using a time-domain, fully nonlinear numerical model with non-reflective open boundaries. Preliminary computations were conducted for incident bichromatic waves, in which wave theories with different orders were applied in generating the waves required. The results show that the use of low-order theories generates undesirable free waves, and that fourth-order terms contribute markedly to low-frequency input. The motion of a rectangular floating body in response to nonlinear bichromatic waves was computed. The numerical results for small-amplitude incident waves agree reasonably well with the second-order approximation for both the steady and difference-frequency (Δσ) components in the body's motion. For relatively large waves, however, the 2Δσ component becomes predominant compared with the Δσ component. The motion of the body in irregular waves with different wave parameters has also been presented in order to discuss the validity range of a second-order approximation.     

Simulation of wave propagation over a submerged bar using the VOF method with a two-equation k– turbulence modeling

A two-equation k– turbulence model is used in this paper to simulate the propagation of cnoidal waves over a submerged bar, where the free surface is handled by the volume-of-fluid (VOF) method. Using a VOF partial-cell variable and a donor–acceptor method, the model is capable of treating irregular boundaries, including arbitrary bottom topography and internal obstacles, where the no-slip condition is satisfied. The model also allows the viscous sublayer to be modeled by a wall function approximation implemented in the grid nodes that are immediately adjacent to a wall boundary. The numerical model applied to the propagation of cnoidal waves over a submerged bar can produce results that are in general agreement with some laboratory measurements. Some remarks arising from the comparison between the computational and experimental results are presented.     

An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

Generation of long internal waves in the ocean by a moving pressure zone

The generation of plane internal waves in a continuously stratified ocean by a steadily moving anomaly of the atmospheric pressure is studied in a linear statement using a quasi-static approximation. Quantitative estimates for the parameters of internal waves in the wake are obtained for the average distribution of the Väisälä-Brunt frequency in the Caribbean Sea.Translated by Mikhail M. Trufanov.     

Numerical calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-section

Most off-shore oil platforms are supported by vertical cylinders extending to the ocean floor. An important problem in off-shore engineering is the calculation of the wave loading exerted on these vertical cylinders. Analytical solutions have been found for the case of plane incident waves incident on a circular cylinder by MacCamy and Fuchs [(1954), Wave forces on piles: a diffraction theory. U.S. Army Corps of Engineering, Beach Erosion Board, Technical Memorandum No. 69] and also for short-crested waves incident on a circular cylinder by Zhu [(1993), Diffraction of short-crested waves around a circular cylinder. Ocean Engng 20, 389–407]. However, for a cylinder of arbitrary cross-section, no analytic solutions currently exist. Au and Brebbia [(1983), Diffraction of water waves for vertical cylinders using boundary elements. Appl. Math. Modelling 7, 106–114] proposed an efficient numerical approach to calculate the wave loads induced by plane waves on vertical cylinders by using the boundary element method. However, wind-generated waves are better modelled by short-crested waves. Whether or not these short-crested waves can induce larger wave forces on a structure is of great concern to ocean engineers. In this paper wave loads, induced by short-crested incident waves, on a vertical cylinder of arbitrary cross-section are discussed. For a cylinder of certain cross-section, the wave loads induced by short-crested waves can be larger than those induced by plane waves with the same total wave number.     

Wave groupiness variations in the nearshore

This paper proposes a new definition of the groupiness factor, GF, based on the envelope of the incident-wave time series. It is shown that an envelope-based GF has several important advantages over the SIWEH-based groupiness factor, including objective criteria for determining the accuracy of the envelope function and well-defined numerical limits.Using this new GF, the variability of incident wave groupiness in the field is examined both temporally, in unbroken waves at a fixed location, and spatially, in a cross-shore array through the surf zone. Contrary to previous studies using the SIWEH-based GF, results suggest that incident wave groupiness may not be an independent parameter in unbroken waves; through a wide range of spectral shapes, from swell to storm waves, the groupiness did not vary significantly. As expected, the groupiness decreases rapidly as waves break through the surf zone, although significant wave height variability persists even through a saturated surf zone. The source of this inner surf zone groupiness is not identified; however, this observation implies that models of long wave generation must account for nonsteady radiation stress gradients landward of some narrow zone near the mean breakpoint.     

Design weight of armour stone considering the effect of extreme waves

In this study, waves with the heights higher than H_1/3 in an irregular wave train are called as extreme waves and defined with the help of extreme wave parameter, α_extreme. In order to see the effect of extreme waves on the design weight of armour stone, stability analysis is carried out based on the hydraulic model test results. The test results of high α_extreme cases (HE) and low α_extreme cases (LE) are compared with currently used van der Meer's formulae with permeability factor P=0.4 and 0.45 and Hudson formula by using H_1/3 and H_1/10 in terms of the design weight of armour stone. As a result of the comparison, it is found that Hudson formula by using H_1/3 underestimates the necessary armour weight. Usage of H_1/10 instead of H_1/3 in Hudson formula doubles the weight which seems overestimated when Irribaren number is away from the transition zone in which both wave run-up and run-down forces become effective. However, it seems underestimated near the transition zone where experiment case HE gives higher armour weights. When the design weight of armour stone is calculated by van der Meer's formulae with P=0.4, it may be necessary to increase the weight up to 30% in the case of high extreme waves. On the other hand, van der Meer's formulae may overestimate the weight 14% when the extreme waves are low.     

Dispersal of droplets over breaking wind waves under the direct action of wind

This paper describes the dispersal of droplets over breaking wind waves under the direct action of wind, based on a comparison between the actual distribution of droplet velocity and the wind field measured in a wind-wave tank (reference wind speed 16 m sec^–1). The velocity distribution of droplets with a diameterd>0.81 mm over breaking wind waves was measured by Koga (1981). In this paper the wind field over breaking wind waves is measured by a flow visualization technique using styrofoam flakes as a tracer. The comparison allows a clear interpretation of droplet movement over the wave profile, and shows that the horizontal movement of the droplets ofd>0.81 mm is approximately determined by acceleration by the wind while their vertical movement is determined by acceleration due to gravity. These observations offer some support for the dispersion model proposed by Koga and Toba (1981).     

A note on boundary layers and wakes in rotating fluids

The flow induced by the two-dimensional line vortex moving in a rotating fluid is discussed. The governing vorticity equation is linearized adopting the Oseen approximation.First, the problem is considered on a constantf-plane. The solution shows that the Stewartson E^1/4 layer is transformed into the Oseen wake as the role of the advection becomes important.Second, the problem is considered on a-plane. When the line vortex moves westward, the solution shows a pattern of Rossby lee waves decaying downstream of the vortex and alternating flows far upstream. When the line vortex moves eastward, the inviscid solution shows definite alternating jets downstream. In a viscous case, however, the jets become less definite and identical with the above mentioned alternating flows in the far field. Far upstream, there are no disturbances because of the special propagation characteristics of Rossby waves.     

Experimental study of the transformation of bound long waves over a mild slope with ambient currents

The effect of currents on the variation of cross-shore bound long waves forced by bichromatic waves over a plane slope was investigated in the laboratory. In still water the growth rate of the shoaling bound long waves over the slope is proportional to h^– 5/2 (h is still-water depth). It was found that the opposing current makes the amplitudes of the bound long waves greater than those of still water for all cases. However, the amplitudes of bound long waves in a following current are reduced in the weakly modulated cases but are enhanced in the fully modulated case.     

An approximated method for the solution of elliptic problems in thin domains: Application to nonlinear internal waves

Realistic numerical simulations of nonlinear internal waves (NLIWs) have been hampered by the need to use computationally expensive nonhydrostatic models. In this paper, we show that the solution to the elliptic problem arising from the incompressibility condition can be successfully approximated by a few terms (three at most) of an expansion in powers of the ratio (horizontal grid spacing)/(total depth). For an n dimensional problem, each term in the expansion is the sum of a function that satisfies a one-dimensional second-order ODE in the vertical direction plus, depending on the surface boundary condition, the solution to an n-1 dimension elliptic problem, an evident saving over having to solve the original n-dimensional elliptic problem. This approximation provides the physically correct amount of dispersion necessary to counteract the nonlinear steepening tendency of NLIWs. Experiments with different types of NLIWs validate the approach. Unlike other methods, no ad hoc artificial dispersion needs to be introduced.     

Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport

A large number of studies have been done dealing with sinusoidal wave boundary layers in the past. However, ocean waves often have a strong asymmetric shape especially in shallow water, and net of sediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviors influenced by the effect of asymmetry are different from those in sinusoidal waves. Characteristics of the turbulent boundary layer under breaking waves (saw-tooth) are investigated and described through both laboratory and numerical experiments. A new calculation method for bottom shear stress based on velocity and acceleration terms, theoretical phase difference, φ and the acceleration coefficient, a_c expressing the wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined empirically from both experimental and baseline k–ω model results. The new calculation has shown better agreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by other existing methods. It was further applied into sediment transport rate calculation induced by skew waves. Sediment transport rate was formulated by using the existing sheet flow sediment transport rate data under skew waves by Watanabe and Sato [Watanabe, A. and Sato, S., 2004. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. Proc. of 29th ICCE, ASCE, pp. 1703–1714.]. Moreover, the characteristics of the net sediment transport were also examined and a good agreement between the proposed method and experimental data has been found.     

Doppler-shift approximations of encountered wave statistics

In this paper the joint density of waveheight and half-wavelength is considered for waves observed at a fixed time point and encountering waves that are overtaking a ship from behind. The densities for these two cases are related by a Doppler shift, expressed in terms of the relative velocity of the waves and the ship. Based on this observation, an approximation of the encountered density is proposed. This approximation is then investigated for a Gaussian sea having a Pierson–Moskowitz spectrum.