An Innovative Approach to Predicting Scour Depth Around Foundations Under Combined Waves and Currents in Large-Scale Tests Based on Small-Scale Tests

This study presents an innovative theoretical approach to predicting the scour depth around a foundation in large-scale model tests based on small-scale model tests under combined waves and currents.In the present approach,the hydrodynamic parameters were designed based on the Froude similitude criteria.To avoid the cohesive behavior,we scaled the sediment size based on the settling velocity similarity,i.e.,the suspended load similarity.Then,a series of different scale model tests was conducted to obtain the scour depth around the pile in combined waves and currents.The fitting formula of scour depth from the small-scale model tests was used to predict the results of large-scale tests.The accuracy of the present approach was validated by comparing the prediction values with experimental data of large-scale tests.Moreover,the correctness and accuracy of the present approach for foundations with complex shapes,e.g.,the tripod foundation,was further checked.The results indicated that the fitting line from small-scale model tests slightly overestimated the experimental data of large-scale model tests,and the errors can be accepted.In general,the present approach was applied to predict the maximum or equilibrium scour depth of the large-scale model tests around single piles and tripods.

Wave Force on the Crown Wall of Rubble Mound Breakwaters at Intermediate Depths

Rubble mound breakwaters with a crown wall are a common coastal engineering structure.The wave force on crown walls is an important parameter for the practice engineering design.Particularly,the wave force on crown walls under intermediate depths has been studied through physical model tests and numerical simulations.In this study,a three-dimensional numerical wave flume was developed to investigate monochromatic wave interactions in a rubble mound breakwater with a crown wall.Armor blocks were modeled in detail.The Navier-Stokes equations for two-phase incompressible flows,combined with shear stress transport k-ωturbulence model and volume of fluid method for tracking the free surface,were solved.A set of laboratory experiments were performed to validate the adopted model.Subsequently,a series of numerical simulations were implemented to examine the impacts of different hydrodynamic parameters(including wave height,incident wave period,and water depth)and the berm width on the wave force of the crown wall.Finally,a comparison of the experimental results and Martin method shows that the latter method is not suitable for this experimental scope.New empirical formulas are proposed to predict the wave force on crown walls under intermediate depth.The results can provide a basis for the design of crown wall of rubble mound breakwaters at intermediate depths.

Evaluation of Nonbreaking Wave-Induced Mixing Parameterization Schemes Based on a One-Dimensional Ocean Model

Surface waves have a considerable effect on vertical mixing in the upper ocean.In the past two decades,the vertical mixing induced through nonbreaking surface waves has been used in ocean and climate models to improve the simulation of the upper ocean.Thus far,several nonbreaking wave-induced mixing parameterization schemes have been proposed;however,no quantitative comparison has been performed among them.In this paper,a one-dimensional ocean model was used to compare the performances of five schemes,including those of Qiao et al.(Q),Hu and Wang(HW),Huang and Qiao(HQ),Pleskachevsky et al.(P),and Ghantous and Babanin(GB).Similar to previous studies,all of these schemes can decrease the simulated sea surface temperature(SST),increase the subsurface temperature,and deepen the mixed layer,thereby alleviating the common thermal deviation problem of the ocean model for upper ocean simulation.Among these schemes,the HQ scheme exhibited the weakest wave-induced mixing effect,and the HW scheme exhibited the strongest effect;the other three schemes exhibited roughly the same effect.In particular,the Q and P schemes exhibited nearly the same effect.In the simulation based on observations from the Ocean Weather Station Papa,the HQ scheme exhibited the best performance,followed by the Q scheme.In the experiment with the HQ scheme,the root-mean-square deviation of the simulated SST from the observations was 0.43℃,and the mixed layer depth(MLD)was 2.0 m.As a contrast,the deviations of the SST and MLD reached 1.25℃ and 8.4 m,respectively,in the experiment without wave-induced mixing.

Responses of the Southern Ocean mixed layer depth to the eastern and central Pacific El Niño events during austral winter

Based on the Ocean Reanalysis System version 5(ORAS5)and the fifth-generation reanalysis datasets derived from European Centre for Medium-Range Weather Forecasts(ERA5),we investigate the different impacts of the central Pacific(CP)El Niño and the eastern Pacific(EP)El Niño on the Southern Ocean(SO)mixed layer depth(MLD)during austral winter.The MLD response to the EP El Niño shows a dipole pattern in the South Pacific,namely the MLD dipole,which is the leading El Niño-induced MLD variability in the SO.The tropical Pacific warm sea surface temperature anomaly(SSTA)signal associated with the EP El Niño excites a Rossby wave train propagating southeastward and then enhances the Amundsen Sea low(ASL).This results in an anomalous cyclone over the Amundsen Sea.As a result,the anomalous southerly wind to the west of this anomalous cyclone advects colder and drier air into the southeast of New Zealand,leading to surface cooling through less total surface heat flux,especially surface sensible heat(SH)flux and latent heat(LH)flux,and thus contributing to the mix layer(ML)deepening.The east of the anomalous cyclone brings warmer and wetter air to the southwest of Chile,but the total heat flux anomaly shows no significant change.The warm air promotes the sea ice melting and maintains fresh water,which strengthens stratification.This results in a shallower MLD.During the CP El Niño,the response of MLD shows a separate negative MLD anomaly center in the central South Pacific.The Rossby wave train triggered by the warm SSTA in the central Pacific Ocean spreads to the Amundsen Sea,which weakens the ASL.Therefore,the anomalous anticyclone dominates the Amundsen Sea.Consequently,the anomalous northerly wind to the west of anomalous anticyclone advects warmer and wetter air into the central and southern Pacific,causing surface warming through increased SH,LH,and longwave radiation flux,and thus contributing to the ML shoaling.However,to the east of the anomalous anticyclone,there is no statistically significant impact on the MLD.

High-Order Models of Nonlinear and Dispersive Wave in Water of Varying Depth with Arbitrary Sloping Bottom

High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).

Oceanic pycnocline depth retrieval from SAR imagery in the existence of solitary internal waves

Oceanic pycnocline depth is usually inferred from in situ measurements. It is attempted to estimate the depth remotely. As solitary internal waves occur on oceanic pycnocline and propagate along it, it is possible to retrieve the depth indirectly in virtue of the solitary internal waves. A numerical model is presented for retrieving the pycnocline depth from synthetic aperture radar （SAR） images where the solitary internal waves are visible and when ocean waters are fully stratified. This numerical model is constructed by combining the solitary internal wave model and a two-layer ocean model. It is also assumed that the observed groups of solitary internal wave packets on the SAR imagery are generated by local semidiurnal tides. A case study in the East China Sea shows a good agreement with in situ CTD （conductivity-temperature-depth） data.

A study on the uncertainties of the centroid depth of the 2013 Lushan earthquake from teleseimic body wave data

Centroid depth of earthquakes is essential for seismic hazard mitigation. But, various studies provided different solutions for the centroid depth of the damaging 2013 Lushan earthquake, thus hindering further studies of the earthquake processes. To resolve its centroid depth and assess the uncertainties, we apply the teleseismic cut and paste method to invert for centroid depth with teleseismic body waves in the epicentral distance of 300-90~. We performed the inversion for P waves only as well the case of both P and SH waves and found that both cases lead to depth solutions with difference less than 0.5 km. We also investigated the effects on depth inversion from azimuth gap of seismic stations, source duration, and comer fre- quency of filter. These various tests show that even azi- muthal distribution of seismic stations is helpful for accurate depth inversion. It is also found that estimate of centroid depth is sensitive to source duration. Moreover, the depth is biased to larger values when corner frequency of low-pass filter is very low. The uncertainty in the velocity model can also generate some error in the depth estimation （- 1.0 km）.With all the above factors consid- ered, the centroid depth of Lushan earthquake is proposed to be around 12 km, with uncertainty about 2 km.

Modified Rayleigh Distribution of Wave Heights in Transitional Water Depths

This paper concerns the calculation of wave height exceedance probabilities for nonlinear irregular waves in transitional water depths, and a Transformed Rayleigh method is first proposed for carrying out the calculation. In the proposed Transformed Rayleigh method, the transformation model is chosen to be a monotonic exponential function, calibrated such that the first three moments of the transformed model match the moments of the true process. The proposed new method has been applied for calculating the wave height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform. It is demonstrated in this case that the proposed new method can offer better predictions than those by using the conventional Rayleigh wave height distribution model. The proposed new method has been further applied for calculating the total horizontal loads on a generic jacket, and its accuracy has once again been substantiated. The research findings gained from this study demonstrate that the proposed Transformed Rayleigh model can be utilized as a promising alternative to the well-established nonlinear wave height distribution models.

Numerical simulation and inversion of offshore area depth based on x-band microwave radar

A detection method of offshore area depth utilizing the x-band microwave radar is proposed. The method is based on the sea clutter imaging mechanism of microwave radar, and combined with dispersion equation of the liner wave theorem and least square method （LSM）, consequently get the inversion results of water depth in the detected region. The wave monitoring system OSMAR-X exploited by the Ocean State Laborato-ry, Wuhan University, based on a microwave radar has proven to be a powerful tool to monitor ocean waves in time and space. Numerical simulation and inversion of offshore area depth are carried out here; since JONSWAP model can give description of stormy waves in different growth phase, it is suitable for simulation. Besides, some results from measured data detected by OSMAR-X x-band radar located at Longhai of Fujian Province, China, validates this method. The tendency of the average water depths inferred from the radar images is in good agreement with the tide level detected by Xiamen tide station. These promising results suggest the possibility of using OSMAR-X to monitor operationally morphodynamics in coastal zones. This method can be applied to both shore-based and shipborne x-band microwave radar.

Depth decay rate for surface gravity wave pressure and velocity

Linear governing equations are formulated for the depth decay of the pressure and velocity variations associated with propagating surface gravity waves. These governing equations come from combining Bernoulli’s equation for steady frictionless flow along a streamline and the crossstream force balance involving gravity, the centrifugal force and a pressure gradient. Qualitative solutions show that the pressure decreases downward faster than the velocity does and at a rate that is probably not the normal exponential decrease, which does not agree with the classical result. The radius of curvature of the streamlines is a non-constant coefficient in these equations and it needs to be supplied, either from measurements or another theory, in order to complete the solution of the derived governing equations. There is no sensitivity of the solution to the exact path the radius of curvature takes between its minimum value at the surface of a crest and trough and infinity at great depth. In the future measurements, perhaps streak photographs, will be needed to distinguish between the new and old theories.

Numerical Calculation for Nonlinear Waves in Water of Arbitrarily Varying Depth with Boussinesq Equations

Based on the high order nonlinear and dispersive wave equation with a dissipative term, a numerical model for nonlinear waves is developed, It is suitable to calculate wave propagation in water areas with an arbitrarily varying bottom slope and a relative depth h/L(0)less than or equal to1. By the application of the completely implicit stagger grid and central difference algorithm, discrete governing equations are obtained. Although the central difference algorithm of second-order accuracy both in time and space domains is used to yield the difference equations, the order of truncation error in the difference equation is the same as that of the third-order derivatives of the Boussinesq equation. In this paper, the correction to the first-order derivative is made, and the accuracy of the difference equation is improved. The verifications of accuracy show that the results of the numerical model are in good agreement with those of analytical Solutions and physical models.

An Efficient Model for Transient Surface Waves in Both Finite and Infinite Water Depths

A numerical model is developed to simulate fully nonlinear extreme waves in finite and infinite water-depth wave tanks. A semi-mixed Enlerian-Lagrangian formulation is adopted and a higher-order boundary element method in conjunction with an image Green function is used for the fluid domain. The botmdary values on the free surface are updated at each time step by a fourth-order Runga-Kutta time-marching scheme at each time step. Input wave characteristics are specified at the upstream boundary by an appropriate wave theory. At the downstream boundary, an artificial damping zone is used to prevent wave reflection back into the computational domain. Using the image Green function in the whole fluid domain, the integrations on the two lateral walls and bottom are excluded. The simulation results on extreme wave elevations in finite and infinite water-depths are compared with experimental results and second-order analytical solutions respectively. The wave kinematics is also discussed in the present study.

Depth inversion in coastal water based on SAR image of waves

Wave-number spectrum technique is proposed to retrieve coastal water depths by means of Synthetic Aperture Radar （SAR） image of waves. Based on the general dispersion relation of ocean waves the wavelength changes of a surface wave over varying water depths can be derived from SAR. Approaching the analysis of SAR images of waves and using the general dispersion relation of ocean waves, this indirect technique of remote sensing bathymetry has been applied to a coastal region of Xiapu in Fujian Province, China. Results show that this technique is suitable for the coastal waters especially for the near-shore regions with variable water depths.

Empirical formula for wave length of ocean wave in finite depth water

In this paper, function characteristics of dispersion of ocean wave in finite depth water were analyzed systematically. The functional form of the fitting function is reasonably proposed, in which the parame- ters are optimally determined by the least square method (LSM). For infinitely deep and extremely shallow water, the fitting function fits strictly the dispersion to be fitted. A new technique is presented in application of LSM. An empirical formula with maximum error of less than 0.5% for computing wavelength in finite depth water is presented for practical applications.

Exciting Forces for a Wave Energy Device Consisting of a Pair of Coaxial Cylinders in Water of Finite Depth

Two coaxial vertical cylinders-one is a riding hollow cylinder and the other a solid cylinder of greater radius at some distance above an impermeable horizontal bottom,were considered.This problem of diffraction by these two cylinders,which were considered as idealization of a buoy and a circular plate,can be considered as a wave energy device.The wave energy that is created and transferred by this device can be appropriately used in many applications in lieu of conventional energy.Method of separation of variables was used to obtain the analytical expressions for the diffracted potentials in four clearly identified regions.By applying the appropriate matching conditions along the three virtual boundaries between the regions,a system of linear equations was obtained,which was solved for the unknown coefficients.The potentials allowed us to obtain the exciting forces acting on both cylinders.Sets of exciting forces were obtained for different radii of the cylinders and for different gaps between the cylinders.It was observed that changes in radius and the gap had significant effect on the forces.It was found that mostly the exciting forces were significant only at lower frequencies.The exciting forces almost vanished at higher frequencies.The problem was also investigated for the base case of no plate arrangement,i.e.,the case having only the floating cylinder tethered to the sea-bed.Comparison of forces for both arrangements was carried out.In order to take care of the radiation of the cylinders due to surge motion,the corresponding added mass and the damping coefficients for both cylinders were also computed.All the results were depicted graphically and compared with available results.

Wave-Induced Loads on Very Large FPSOs at Restricted Water Depth

The effects of water depth on the wave-induced vertical bending moment and shearing force on a very large FPSO are studied by experiments and computations for regular and irregular waves. The restricted water depth composite Green function is employed to develop a program for the computation of the hydrodynamic coefficients of the very large FPSO at shallow water. A three-segment model with 1∶100 scale is tested in the State Key Laboratory of Ocean Engineering at Shanghai Jiao Tong University for the verification of the numerical method. The experimental and computational results show that the water depth has a substantial effect on wave-induced loads. The wave-induced vertical loads increase with the decrease of water depth for shallow water. Especially, for ultra-shallow water these loads increase very evidently with the decrease of water depth. The long-term prediction values of wave-induced vertical loads increase with the decrease of the ratio of water depth to draught. The long-term prediction values of wave-induced vertical loads are about 8% larger than those for deep water when the ratio of water depth to draught is 3.0. However, water depth hardly affects the long-term prediction values of wave-induced loads when the ratio of water depth to draught is larger than 5.0.

EFFECT OF WATER DEPTH ON WIND-WAVE FREQUENCY SPECTRUM Ⅱ. VERIFICATION AND COMPARISON

The spectrum derived in Part 1 of the presert paper is here systematically verified with field data andcompared at some length with that obtained by multiplying the deep-water spectrum with theKitaigorodskii factor.

Water Wave Scattering by an Elastic Thin Vertical Plate Submerged in Finite Depth Water

The problem of water wave scattering by a thin vertical elastic plate submerged in uniform finite depth water is investigated here.The boundary condition on the elastic plate is derived from the Bernoulli-Euler equation of motion satisfied by the plate.Using the Green’s function technique,from this boundary condition,the normal velocity of the plate is expressed in terms of the difference between the velocity potentials(unknown)across the plate.The two ends of the plate are either clamped or free.The reflection and transmission coefficients are obtained in terms of the integrals’involving combinations of the unknown velocity potential on the two sides of the plate,which satisfy three simultaneous integral equations and are solved numerically.These coefficients are computed numerically for various values of different parameters and depicted graphically against the wave number in a number of figures.

LINEAR GRAVITY WAVES ON MAXWELL FLUIDS OF FINITE DEPTH

Linear surface gravity waves on Maxwell viscoelastic fluids with finite depth are studied in this paper.A dispersion equation describing the spatial decay of the gravity wave in finite depth is derived.A dimensionless memory(time)number θ is introduced.The dispersion equation for the pure viscous fluid will be a specific case of the dispersion equation for the viscoelastic fluid as θ=0.The complex dispersion equation is numerically solved to investigate the dispersion relation.The influences of θ and water depth on the dispersion characteristics and wave decay are discussed.It is found that the role of elasticity for the Maxwell fluid is to make the surface gravity wave on the Maxwell fluid behave more like the surface gravity wave on the inviscid fluid.

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 finite 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 determined by the water depth and the wave number 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.