Internal wave spectrum in shallow water: measurement and comparisonwith the Garrett-Munk model

The covariance matrix of sound-speed variations is determined from yo-yo CTD data collected during the SWARM 95 experiment at a fixed station. The data covered approximately 2 h and were collected during a period when nonlinear solitary internal waves were absent or negligible. The method of empirical orthogonal functions (EOF) is applied to the sound-speed covariance matrix assuming that the internal wave modes are uncorrelated. The first five eigenvectors are found to agree well with the theoretically modeled eigenfunctions based on the measured buoyancy frequency and the internal wave eigenmode equation. The mode amplitudes for the first five modes are estimated from the corresponding eigenvalues. They agree with the Garrett-Munk model if j*=1 is used instead of j*=3. A second method is used to deduce the mode amplitudes and mode frequency spectra by projecting the sound-speed variation (as a function of time) onto the theoretical mode depth functions. The mode amplitudes estimated with this method are in agreement with the EOF results. A modified Garrett-Munk model is proposed to fit the frequency spectrum of linear internal waves in shallow water     

Internal tide and nonlinear internal wave behavior at the continental slope in the northern south China Sea

A field program to measure acoustic propagation characteristics and physical oceanography was undertaken in April and May 2001 in the northern South China Sea. Fluctuating ocean properties were measured with 21 moorings in water of 350- to 71-m depth near the continental slope. The sea floor at the site is gradually sloped at depths less than 90 m, but the deeper area is steppy, having gradual slopes over large areas that are near critical for diurnal internal waves and steep steps between those areas that account for much of the depth change. Large-amplitude nonlinear internal gravity waves incident on the site from the east were observed to change amplitude, horizontal length scale, and energy when shoaling. Beginning as relatively narrow solitary waves of depression, these waves continued onto the shelf much broadened in horizontal scale, where they were trailed by numerous waves of elevation (alternatively described as oscillations) that first appeared in the continental slope region. Internal gravity waves of both diurnal and semidiurnal tidal frequencies (internal tides) were also observed to propagate into shallow water from deeper water, with the diurnal waves dominating. The internal tides were at times sufficiently nonlinear to break down into bores and groups of high-frequency nonlinear internal waves.     

Wash Waves Generated by High Speed Displacement Ships

It is difficult to compute far-field waves in a relative large area by using one wave generation model when a large calculation domain is needed because of large dimensions of the waterway and long distance of the required computing points. Variation of waterway bathymetry and nonlinearity in the far field cannot be included in a ship fixed process either. A coupled method combining a wave generation model and wave propagation model is then used in this paper to simulate the wash waves generated by the passing ship. A NURBS-based higher order panel method is adopted as the stationary wave generation model; a wave spectrum method and Boussinesq-type equation wave model are used as the wave propagation model for the constant water depth condition and variable water depth condition, respectively. The waves calculated by the NURBS-based higher order panel method in the near field are used as the input for the wave spectrum method and the Boussinesq-type equation wave model to obtain the far-field waves. With this approach it is possible to simulate the ship wash waves including the effects of water depth and waterway bathymetry. Parts of the calculated results are validated experimentally, and the agreement is demonstrated. The effects of ship wash waves on the moored ship are discussed by using a diffraction theory method. The results indicate that the prediction of the ship induced waves by coupling models is feasible.     

On the possible generation of low-frequency microseisms by internal waves in the ocean

The conditions under which the interaction between internal waves and the rough bottom topography may be the reason for synchronous fluctuations of pressure at the bottom have been studied. Disturbances of the depth are assumed to be small compared with the mean depth of the ocean, and the Väisälä-Brunt frequency is constant. It is shown that synchronous fluctuations of pressure exist with a frequency equal to that of internal waves. The amplitude of pulsations can be approximately an order lower than the amplitude of fluctuations generated by standing surface waves. However, local maxima can exist in the low-frequency minimum of the spectrum of microseisms (in the range of 20–1000 s).Translated by Mikhail M. Trufanov.     

Transformation of internal waves over an uneven bottom: Analytical results

The transformation of internal waves over the oceanic shelf of variable depth is studied analytically within a linear theory of a two-layer flow. It is shown that, at a specific character of depth variation, the internal wave propagates without reflection from the slope even if it is sufficiently steep. The properties of such progressive waves are studied—their form and the current structure in the upper and lower layers. The transformation of the wave propagating from the open ocean, where the depth is assumed to be constant, is considered. It is shown that the wave is transformed at the shelf edge and does not change its form in the course of time during its further propagation over the shelf. The height and form of the internal wave are calculated at the interface of the transition of the two-layer flow into the one-layer flow. Applications of the developed analytical theory to the estimation of internal wave transformation over a real shelf are discussed.     

Numerical Analysis of the Electromagnetic Fields Induced by Wave Motions in the North Part of the Aegean Sea

The results of investigations of internal waves carried out in three test ranges located in the north part of the Aegean Sea and characterized by different dynamics of wave motions and water masses are used to study the electromagnetic waves induced by wave motions. We analyze their dependence on the period of waves and wavelength, reveal specific features of the behavior of the electric and magnetic fields and their phases as functions of depth, and compare the intensities of electromagnetic fields obtained under different hydrological conditions.     

两相流模型在南海内孤立波与沙波相互作用中的应用

为研究内孤立波与沙波的相互作用,本文对基于OpenFOAM的SedWaveFoam求解器进行改进,建立了内孤立波-泥沙运动欧拉两相流模型。在利用试验资料对模型进行验证的基础上,在南海北部典型代表性条件下,模拟分析了500 m水深位置沙波床面上内孤立波作用下的水动力变化和泥沙运动。结果表明,内孤立波逐渐离开沙波时,海底沙波背流面处出现与内孤立波背景流速反向的流速,在内孤立波导致的流场作用下,沙波床面上的泥沙悬起并运动到床面以上的水体中。振幅100 m的内孤立波可以导致床面以上14 m高的位置处出现约0.07 kg/m³的悬沙浓度。     

A new approximate solution to the surface wave problem

Recent measurements of wave kinematics, showing that the horizontal velocity under the trough is in absolute value greater than the velocity under the crest, can not be explained by the Stokes theories up to and including the fifth order theory which are normally used by the offshore industry in the design process. This has led to a reconsideration of the surface wave problem. By applying a Green function technique a solution is obtained which reflects these recent experimental results on wave kinematics. This solution and the Stokes higher order solutions represent different approximate solutions to the surface wave problem. It turns out, however, that this new approximate solution gives a better fit to the measurements than do the approximate solutions obtained from the Stokes higher order theories. In the present paper deep water waves are discussed, but the method of solution can just as well be applied to the finite water depth case.     

Generation of spatial internal waves in a rotating ocean by moving non-stationary disturbances

In a general linear statement, the kinematic structure of spatial internal waves generated by a uniformly moving area of oscillating surface pressures in a continuously stratified ocean of constant depth is studied. The earth's rotation effects are considered. Possible types of ocean wave fields with a constant Brunt-Väisälä frequency are examined. The wave regimes for individual modes of internal and gyroscopic waves are classified on the basis of estimating the integrals asymptotically.Translated by V. Puchkin.     

Mass transport in a two-layer wave boundary layer

The transport of a chemical species under the pure action of surface progressive waves in the benthic boundary layer which is loaded with dense suspended sediments is studied theoretically. The flow structure of the boundary layer is approximated by that of a two-layer Stokes boundary layer with a sharp interface between clear water and a heavy fluid. The simplest model of constant eddy diffusivities is adopted and the exchange of matter with the bed is ignored. For a thin layer of heavy fluid, whose thickness is comparable to the surface wave amplitude and the Stokes boundary layer thickness, effective transport equations are deduced using an averaging technique based on the method of homogenization. The effective advection velocity is found to be equal to the depth-averaged mass transport velocity, while the dispersion coefficient can be shown to be positive definite. Explicit expressions for the transport coefficients are obtained as functions of fluid properties and flow kinematics. Physical discussions on their relations are also presented.     

Second-Oorder Analytic Solutions of Nonlinear Interactions of Edge Waves on A Plane Sloping Bottom

Based on the full water-wave equation, a second-order analytic solution for nonlinear interaction of short edge waves on a constant plane sloping bottom is presented in this paper. For special case of slope angle b=p/2, this solution can be reduced to the same order solution of deep water gravity surface waves traveling along parallel coastline. Interactions between two edge waves including progressive, standing and partially reflected standing waves were also discussed. The unified analytic expressions with transfer functions for kinematic-dynamic elements of edge waves were also discussed. The random model of the unified wave motion processes for linear and nonlinear irregular edge waves is formulated, and the corresponding theoretical autocorrelation and spectral density functions of the first and second orders are derived. The boundary conditions for the determining determination of the parameters of short edge wave are suggested, that may be seen as one special simple edge wave excitation mechanism and an extension to the sea wave refraction theory. Finally some computation results are demonstrated.     

Penetration depth of the dynamic response of seabed induced by internal solitary waves

We use flume experiments and numerical modeling to examine the penetration depth of internal solitary waves (ISWs) on partially saturated porous sandy silt and clayey silt seabed. The results of the experiment and model showed that the instantaneous excess pore water pressure in both the sandy silt and clayey silt seabed followed the same trend of decreasing with the seabed depth. In general, the excess pore water pressure generated by the sandy silt was bigger than that by clayey silt at the same depth. The ISW-induced excess pore water pressure greatly influenced the surface seabed and showed a linear relationship. The penetration depth was approximately one order of magnitude smaller than the half-wavelength of the ISWs, which might be larger than the penetration depth induced by surface waves. Our study results are helpful for understanding the damage that ISWs inflict upon the seabed and for informing future field experiments designed to directly measure the interaction between ISWs and seabed sediments.     

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.     

Wave interaction with ‘’-type breakwaters

The wave transmission, reflection and energy dissipation characteristics of ‘’-type breakwaters were studied using physical models. Regular and random waves in a wide range of wave heights and periods and a constant water depth were used. Five different depths of immersion (two emerged, one surface flushing and two submerged conditions) of this breakwater were selected. The coefficient of transmission, K_t, and coefficient of reflection, K_r, were obtained from the measurements, and the coefficient of energy loss, K_l was calculated using the law of balance of energy. It was found that the wave transmission is significantly reduced with increased relative water depth, d/L, whether the vertical barrier of the breakwater is surface piercing or submerged, where ‘d’ is the water depth and ‘L’ is the wave length. The wave reflection decreases and energy loss increases with increased wave steepness, especially when the top tip of the vertical barrier of this breakwater is kept at still water level (SWL). For any incident wave climate (moderate or storm waves), the wave transmission consistently decreases and the reflection increases with increased relative depth of immersion, Δ/d from −0.142 to 0.142. K_t values less than 0.3 can be easily obtained for the case of Δ/d=+0.071 and 0.142, where Δ is the height of exposure (+ve) or depth of immersion (−ve) of the top tip of the vertical barrier. This breakwater is capable of dissipating wave energy to an extent of 50–80%. The overall performance of this breakwater was found to be better in the random wave fields than in the regular waves. A comparison of the hydrodynamic performance of ‘’-type and ‘T’-type shows that ‘T’-type breakwater is better than ‘’-type by about 20–30% under identical conditions.     

Internal waves on the White Sea shelf according to observations

Experimental data obtained in the summer of 2010 are used for the analysis of the intense internal waves and their relation with the barotropic tides. It is shown that these waves are characterized by significant nonlinearity with their amplitudes amounting to half values of the water depth and periods of tens of minutes. The contribution of intense internal waves to the vertical exchange is assessed.     

A numerical study of the generation and propagation of internal solitary waves in the Luzon Strait

A new composite model, which consists of a generation model of the internal tides and a regularized long wave propagation model, is presented to study the generation and evolution of internal solitary waves in the sill strait. Internal bores in the sill strait are first simulated by the generation model, and then the internal tidal field outside of the sill region is given as input for the propagation model. Numerical experiments are carried out to study the imposing tide, depth profile, channel width and shoaling effect, etc., on the generation and evolution of internal solitary waves. It is shown that only when the amplitude of internal tide at the forcing boundary of the propagation model is large enough that a train of internal solitary waves would be induced. The amplitude of the imposing tide in the generation model, shoaling effect, asymmetry of the depth profile and channel width have some effects on the amplitude of the induced internal solitary wave. The imposing tidal flow superimposed on a constant mean background flow has a great damping effect on the induced internal waves, especially on those propagate against the background flow direction. The generation and propagation of internal solitary waves in three possible straits among the Luzon Strait are simulated, and the reasons for the asymmetry of their propagation are also explained.     

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.     

Estimation of infragravity waves at intermediate water depth

The accuracy of nearshore infragravity wave height model predictions has been investigated using a combination of the spectral short wave evolution model SWAN and a linear 1D SurfBeat model (IDSB). Data recorded by a wave rider located approximately 3.5 km from the coast at 18 m water depth have been used to construct the short wave frequency-directional spectra that are subsequently translated to approximately 8 m water depth with the third generation short wave model SWAN. Next the SWAN-computed frequency-directional spectra are used as input for IDSB to compute the infragravity response in the 0.01 Hz–0.05 Hz frequency range, generated by the transformation of the grouped short waves through the surf zone including bound long waves, leaky waves and edge waves at this depth. Comparison of the computed and measured infragravity waves in 8 m water depth shows an average skill of approximately 80%. Using data from a directional buoy located approximately 70 km offshore as input for the SWAN model results in an average infragravity prediction skill of 47%. This difference in skill is in a large part related to the under prediction of the short wave directional spreading by SWAN. Accounting for the spreading mismatch increases the skill to 70%. Directional analyses of the infragravity waves shows that outgoing infragravity wave heights at 8 m depth are generally over predicted during storm conditions suggesting that dissipation mechanisms in addition to bottom friction such as non-linear energy transfer and long wave breaking may be important. Provided that the infragravity wave reflection at the beach is close to unity and tidal water level modulations are modest, a relatively small computational effort allows for the generation of long-term infragravity data sets at intermediate water depths. These data can subsequently be analyzed to establish infragravity wave height design criteria for engineering facilities exposed to the open ocean, such as nearshore tanker offloading terminals at coastal locations.     

Standing wave pressures on walls

The pressure variations exerted on a vertical wall in a constant water depth are computed from the calculation of a short-crested wave system using the Fourier series approximation method. The numerical results have been compared with experimental results from literature. The comparison with the linear solution of the present theory gives a good estimate for the variation of dynamic pressures. The double peak formation in the experimental results for pressure curves at still water level and at the bottom for the steeper waves was not observed for the linear case. In the case of surface elevation, deviations with the third order perturbation results are observed for higher waves. In addition, deviations are also observed within the elevation results for higher waves.     

Internal inlet for wave generation and absorption treatment

A new method of implementing, in two-dimensional (2-D) Navier–Stokes equations, a numerical internal wave generation in the finite volume formulation is developed. To our knowledge, the originality of this model is on the specification of an internal inlet velocity defined as a source line for the generation of linear and non-linear waves. The use of a single cell to represent the source line and its transformation to an internal boundary condition proved to be an interesting alternative to the common procedure of adding a mass source term to the continuity equation within a multi-cell rectangular region. Given the reduction of the source domain to a one-dimensional region, this simple new type of source introduced less perturbation than the 2-D source type. This model was successfully implemented in the PHOENICS code (Parabolic Hyperbolic Or Elliptic Numerical Integration Code Series). In addition, the volume of fluid (VOF) fraction was used to describe the free surface displacements. A friction force term was added to the momentum transport equation in the vertical direction, in order to enhance wave damping, within relatively limited number of cells representing the sponge layers at the open boundaries. For monochromatic wave, propagating on constant water depth, numerical and analytical results showed good agreements for free surface profiles and vertical distribution of velocity components. For solitary wave simulation, the wave shape and velocity were preserved; while, small discrepancy in the tailing edge of the free surface profiles was observed. The suitability of this new numerical wave generation model for a two source lines extension was investigated and proven to be innovative. The comparisons between numerical, analytical and experimental results showed that the height of the merging waves was correctly reproduced and that the reflected waves do not interact with the source lines.