Linear shoaling in Boussinesq-type wave propagation models

This work focuses on linear shoaling performance of low order Boussinesq-type equations. It is shown that the linear shoaling errors can be important in well known equations in the literature. New sets of coefficients are presented for three well known sets of equations. The sets are found so as to minimize a global linear error that includes celerity and shoaling errors. Finally, a new set of enhanced bilayer low order equations is presented, with much improved linear behavior (errors in wave celerity and wave amplitude below 1% up to kh = 20). For completeness, the equations are written in their fully nonlinear version, and the nonlinear coefficients are also given.     

Boussinesq–Green–Naghdi rotational water wave theory

Using Boussinesq scaling for water waves while imposing no constraints on rotationality, we derive and test model equations for nonlinear water wave transformation over varying depth. These use polynomial basis functions to create velocity profiles which are inserted into the basic equations of motion keeping terms up to the desired Boussinesq scaling order, and solved in a weighted residual sense. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Explicit coefficients are given at O(μ²) and O(μ⁴), while more generalized basis functions are given at higher order. Nonlinear performance is somewhat more limited as, for reasons of complexity, we only provide explicitly lower order nonlinear terms. Still, second order harmonics may remain good to kh ≈ 10 for O(μ⁴) equations. Numerical tests for wave transformation over a shoal show good agreement with experiments. Future work will harness the full rotational performance of these systems by incorporating turbulent and viscous stresses into the equations, making them into surf zone models.     

Comparison of time-dependent extended mild-slope equations for random waves

The extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Eng., 32, 91–117] and Lee et al. [Lee, C., Kim, G., Suh, K.D., 2003. Extended mild-slope equation for random waves. Coastal Eng., 48, 277–287] are compared analytically and numerically to determine their applicability to random wave transformation. The geometric optics approach is used to compare the two models analytically. In the model of Suh et al., the wave number of the component wave with a local angular frequency ω is approximated with an accuracy of O(ω − ω¯) at a constant water depth, where ω¯ is the carrier frequency of random waves. In the model of Suh et al., however, the diffraction effects and higher-order bottom effects are considered only for monochromatic waves, and the shoaling coefficient of random waves is not accurately approximated. This inaccuracy arises because the model of Suh et al. was derived for regular waves. In the model of Lee et al., all the parameters of random waves such as wave number, shoaling coefficient, diffraction effects, and higher-order bottom effects are approximated with an accuracy of O(ω − ω¯). This approximation is because the model of Lee et al. was developed using the Taylor series expansion technique for random waves. The result of dispersion relation analysis suggests the use of the peak and weighted-average frequencies as a carrier frequency for Suh et al. and Lee et al. models, respectively. All the analytical results are verified by numerical experiments of shoaling of random waves over a slightly inclined bed and diffraction of random waves through a breakwater gap on a flat bottom.     

Parameterization of nonlinear shallow water waves over sloping bottoms

Investigation of the bottom slope effects on the nonlinear transformation of irregular waves, which are generated based on JONSWAP spectra, is carried out in a physical wave flume with three slopes (β = 1/15, 1/30, 1/45). The slope effects on the estimation of representative wave height are examined first. To obtain a better estimation of wave height, the slope effect should be considered when slope is larger than 1/30. The nonlinear parameters (bicoherence, skewness and asymmetry) are estimated by using the wavelet-based bispectrum, and the empirical formulae regarding these nonlinear parameters as a function of the local Ursell number are derived based on the present data measured on each slope. The results indicate that the slopes have a negligible effect on the variations of the skewness. The fitted coefficients of the formulae for the other parameters on slope β = 1/15 are clearly different from the results on the slopes β = 1/30 and 1/45, indicating that slope influence on the parameterization cannot be ignored when β > 1/30. Hence, new formulae considering the slope effect are presented. Furthermore, the empirical formulae for the data in surf zone are recommended.     

A numerical study of nonlinear wave run-up on a vertical plate

A finite difference model based on a recently derived highly-accurate Boussinesq-type formulation is presented. Up to the third-order space derivatives in terms of the velocity variables are retained, and the horizontal velocity variables are re-formulated in terms of a velocity potential. This decreases the total number of unknowns in two horizontal dimensions from seven to five, simplifying the implementation, and leading to increased computational efficiency. Analysis of the embedded properties demonstrates that the resulting model has applications with errors of 2 to 3% for (wavenumber times depth) kh ≤ 10 in terms of dispersion and kh ≤ 4 in terms of internal kinematics. The stability and accuracy of the discrete linearised systems are also analysed for both potential and velocity formulations and the advantages and disadvantages of each are discussed. The velocity potential model is then used to study physically demanding problems involving highly nonlinear wave run-up on a bottom-mounted (surface-piercing) plate. New cases involving oblique incidence are considered. In all cases, comparisons with recent physical experiments demonstrate good quantitative accuracy, even in the most demanding cases, where the local wave steepness can exceed (waveheight divided by wavelength) H / L = 0.20. The velocity potential model is additionally shown to have numerical advantages when dealing with wave–structure interactions, requiring less smoothing around exterior structural corners.     

Wave attenuation in mangroves: A quantitative approach to field observations

Coastal mangroves, dwelling at the interface between land and sea, provide an important contribution to reducing risk from coastal hazards by attenuating incident waves and by trapping and stabilizing sediments. This paper focusses on relations between vegetation densities, wave attenuation rates, sediment characteristics and sedimentation rates in mangroves. These processes were studied along two cross-shore transects through mangroves fringing estuaries in the southern Andaman region of Thailand. Volumetric vegetation densities in these mangroves were ranging up to 32‰, depending on the water depth. Generalized total wave attenuation rates increased from 0.002 m^− 1 in the sparsely vegetated forest fringes with Avicennia and Sonneratia species, up to 0.012 m^− 1 in the dense Rhizophora vegetation in the back of the forests. The total wave attenuation rates integrate effects of shoaling and energy losses due to various bio-physical interactions within the mangrove ecosystem. Wave attenuation in the mangroves is presumably dominated by energy losses due to vegetation drag, since wave attenuation due to bottom friction and viscous dissipation on the bare mudflats is significantly lower than those inside the mangrove vegetation.Additionally, wave attenuation in the mangroves was found to facilitate enhanced net sediment deposition and a gradual fining of the bed material. These findings corroborate the coastal defence function of mangroves by quantifying their contribution to wave attenuation and sediment trapping. The explicit linking of these properties to vegetation composition and structure facilitates modelling studies investigating the mechanisms determining the coastal defence capacities of mangroves.     

Modeling wave-mud interaction on the central chenier-plain coast,western Louisiana Shelf,USA

The strong coupling between hydrodynamics and seafloors on shallow muddy shelves, and resulting bed reworking, have been extensively documented. On these shelves, spectral wave transformation is driven by a complex combination of forcing mechanisms that include nonlinear wave interactions and wave energy dissipation induced by fluid-mud at a range of frequencies. Wave-mud interaction is investigated herein by using a previously validated nonlinear spectral wave model and observations of waves and near-bed conditions on a mildly-sloping seafloor off the muddy central chenier-plain coast, western Louisiana Shelf, United States. Measurements were made along a cross-shelf transect spanning 1 km between 4 and 3 m water depths. The high-resolution observations of waves and near-bed conditions suggest presence of a fluid mud layer with thickness sometimes exceeding 10 cm under strong long wave action (1 meter wave height with 7 s peak period at 4 meter depth). Spectral wave transformation is modeled using the stochastic formulation of the nonlinear Mild Slope Equation, modified to account for wave-breaking and mud-induced dissipation. The model is used in an inverse manner in order to estimate the viscosity of the fluid mud layer, which is a key parameter controlling mud-induced wave dissipation but complicated to measure in the field during major wave events. Estimated kinematic viscosities vary between 10⁻⁴-10⁻³ m²/s. Combining these results of the wave model simulations with in-depth analysis of near-bed conditions and boundary layer modeling allows for a detailed investigation of the interaction of nonlinear wave propagation and mud characteristics. The results indicate that mud-induced dissipation is most efficient when the wave-induced resuspensions of concentrations  > 10 g/L settle due to relatively small bottom stresses to form a fluid mud layer that is not as thin and viscous as a consolidated seafloor in absence of wave action but also not as thick and soft as a near-bed high concentration layer that forms during strong wave action.     

Determining the weights of two types of artificial reefs required to resist wave action in different water depths and bottom slopes

The interaction between waves and artificial reefs (ARs; a hollow cube weighing 8.24 kN (0.84 t) and a water pipe weighing 1.27 kN (0.13 t)) in shallow waters was investigated with respect to variations in design weight, orientation (for cube; 45° and 90° angles, for pipe; 0°, 90°, and 180° angles to flow), depth (1–20 m), and bottom slope (10^?1, 30^?1, and 50^?1). Physics equations and FLUENT software were used to estimate resisting and mobilising forces, and drag coefficients. Drag coefficients for the hollow cube were 0.76 and 0.85 at 45° and 90° angles to the current, respectively, and 0.97, 0.38, and 1.42 for the water pipe at 0°, 90°, and 180° angles to the current, respectively. Deepwater offshore wave conditions at six stations were transformed into shallow nearshore waters representative of the artificial reef site. Waters deeper than 12 and 16 m are safe to deploy blocks with angles of 45° and 90°, respectively. However, water pipes constructed at angles of 90° and 180° to the current were estimated as being unstable for 365 out of 720 cases at all stations (only one station was stable for all cases). Water pipes angled at 0° were found to be stable in all 360 cases. Slope had a significant effect on weight and depth. Results from this study provide an important reference for engineers performing projects aiming to increase the performance and service life of ARs.     

Structure and mechanisms of the Arabian Sea variability during the winter monsoon

In the southern Arabian Sea (between the Equator and 10°N), the shoaling of isotherms at subsurface levels (20 °C isotherm depth is located at ∼90 m) leads to cooling at 100 m by 2–3 °C relative to surrounding waters during the winter monsoon. The annual and interannual variations of this upwelling zone, which we call the Arabian Sea dome (ASD), are studied using results from an eddy-permitting ocean general circulation model in conjunction with hydrography and TOPEX/ERS altimeter data. The ASD first appears in the southeastern Arabian Sea during September–October, maturing during November–December to extend across the entire southern Arabian Sea (along ∼5°N). It begins to weaken in January and dissipates by March in the southwestern Arabian Sea. From the analysis of heat-budget balance terms and a pair of model control experiments, it is shown that the local Ekman upwelling induced by the positive wind-stress curl of the winter monsoon generates the ASD in the southeastern Arabian Sea. The ASD decays due to the weakening of the cyclonic curl of the wind and the westward penetration of warm water from the east (Southern Arabian Sea High). The interannual variation of the ASD is governed by variations in the Ekman upwelling induced by the cyclonic wind-stress curl. Associated with the unusual winds during 1994–1995 and 1997–1998 Indian Ocean dipole (IOD) periods, the ASD failed to develop. In the absence of the ASD during the IOD events, the 20 °C isotherm depth was 20–30 m deeper than normal in the southern Arabian Sea resulting in a temperature increase at 97 m of 4–5 °C. An implication is that the SST evolution in the southern Arabian Sea during the winter monsoon is primarily controlled by advective cooling: the shoaling of isotherms associated with the ASD leads to SST cooling.     

Numerical study of turbulence and wave damping induced by vegetation canopies

Vegetation canopies control mean and turbulent flow structure as well as surface wave processes in coastal regions. A non-hydrostatic RANS model based on NHWAVE (Ma et al., 2012) is developed to study turbulent mixing, surface wave attenuation and nearshore circulation induced by vegetation. A nonlinear k − ϵ model accounting for vegetation-induced turbulence production is implemented to study turbulent flow within the vegetation field. The model is calibrated and validated using experimental data from vegetated open channel flow, as well as nonbreaking and breaking random wave propagation in vegetation fields. It is found that the drag-related coefficients in the k − ϵ model C_fk and C_fϵ can greatly affect turbulent flow structure, but seldom change the wave attenuation rate. The bulk drag coefficient C_D is the major parameter controlling surface wave damping by vegetation canopies. Using the empirical formula of Mendez and Losada (2004), the present model provides accurate predictions of vegetation-induced wave energy dissipation. Wave propagation through a finite patch of vegetation in the surf zone is investigated as well. It is found that the presence of a finite patch of vegetation may generate strong pressure-driven nearshore currents, with an onshore mean flow in the unvegetated zone and an offshore return flow in the vegetated zone.     

Waves in the nearshore waters of northern Arabian Sea during the summer monsoon

Waves at 15 m water depth in the northern Arabian Sea are measured during the summer monsoon for a period of 45 days and the characteristics are described. The significant wave height varied from 1.1 to 4.5 m with an average value of 2.5 m. 75% of the wave height at the measurement location is due to the swells arriving from the south-west and the remaining is due to the seas from south-west to north-west. Wave age of the measured data indicates that the waves in the nearshore waters of northern Arabian Sea during the summer monsoon are swells with young sea.     

Analysis of temporal and spatial characteristics of waves in the Indian Ocean based on ERA-40 wave reanalysis

Based on the 45-year (09/1957-08/2008) European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis (ERA-40) wave reanalysis dataset, this study analyzes interannual and interdecadal variabilities and intraseasonal oscillations of sea surface wind speed (WS), wind sea wave height (H^w), swell wave height (H^s) and significant wave height (H_s) in the Roaring Forties and tropical waters of the Indian Ocean, to determine swell propagation characteristics. The results show: (1) monthly variabilities of H^s in the Roaring Forties are in good agreement with those in tropical waters of the Indian Ocean; swell plays a dominant role in mixed waves throughout most of the Indian Ocean; and WS, H^w, H^s, and H_s exhibit a significant increasing trend over the 45-year study period. (2) H^s in the Roaring Forties and tropical waters of the Indian Ocean share a common period of 9.8–10.4 years on an interdecadal scale; and WS and H^s in the Roaring Forties and H^s in the tropical waters of the Indian Ocean share a common period of approximately 8 days (weekly oscillation) on an intraseasonal scale. (3) Swell of the Roaring Forties needs approximately 30 h to fully respond to the wind in this region. Approximately 84 h are required for H^s to propagate from the Roaring Forties to the tropical waters of the south Indian Ocean, while it takes approximately 132–138 h for H^s to propagate from the Roaring Forties to the tropical waters of the north Indian Ocean.     

Wave forces and moments on an intake well

A sea water intake well of size 20 m diameter and 15.5 m height in a water depth of 9.8 m is proposed north of the Visakhapatnam Port for a project to extract magnesia from sea water. A 1:25 scale model of the intake well was tested in the wave basin of the Ocean Engineering Centre, Indian Institute of Technology, Madras to measure the wave forces and moments on the intake well and the variation of water levels inside and outside the well. Accordingly, an intake well model of 0.8 m diameter and 0.62 m height was fabricated and fixed over a false bottom in a wave basin. The well model was subjected to the action of both regular waves for two test conditions, intake well inlet closed during installation and intake well inlet open. The experimental results on wave forces and moments were compared with the results of the Linear Diffraction Theory. The water level inside the well was measured to determine the submergence of suction pipes of pumps and location of the inlet opening of the intake well. The wave crest elevation in front of the well was also measured in order to fix the deck level of the well so as to avoid water overspill onto the deck. The salient results of the present study are presented and discussed in this paper.     

Community structure across a large-scale ocean productivity gradient: Marine bird assemblages of the Southern Indian Ocean

Our objective was to understand how marine birds respond to oceanographic variability across the Southern Indian Ocean using data collected during an 16-day cruise (4–21 January 2003). We quantified concurrent water mass distributions, ocean productivity patterns, and seabird distributions across a heterogeneous pelagic ecosystem from subtropical to sub-Antarctic waters. We surveyed 5155 km and sighted 15,606 birds from 51 species, and used these data to investigate how seabirds respond to spatial variability in the structure and productivity of the ocean. We addressed two spatial scales: the structure of seabird communities across macro-mega scale (1000 s km) biogeographic domains, and their coarse-scale (10 s km) aggregation at hydrographic and bathymetric gradients. Both seabird density and species composition changed with latitudinal and onshore–offshore gradients in depth, water temperature, and chlorophyll-a concentration. The average seabird density increased across the subtropical convergence (STC) from 2.4 birds km⁻² in subtropical waters to 23.8 birds km⁻² in sub-Antarctic waters. The composition of the avifauna also differed across biogeographic domains. Prions (Pachyptila spp.) accounted for 57% of all sub-Antarctic birds, wedge-tailed shearwaters (Puffinus pacificus) accounted for 46% of all subtropical birds, and Indian Ocean yellow-nosed albatross (Thallasarche carteri) accounted for 32% of all birds in the STC. While surface feeders were the most abundant foraging guild across the study area, divers were disproportionately more numerous in the sub-Antarctic domain, and plungers were disproportionately more abundant in subtropical waters. Seabird densities were also higher within shallow shelf-slope regions, especially in sub-Antarctic waters, where large numbers of breeding seabirds concentrated. However, we did not find elevated seabird densities along the STC, suggesting that this broad frontal region is not a site of enhanced aggregation.     

Hydrodynamic performance of double rows of piles suspending horizontal c shaped bars

The wave transmission, reflection, and energy dissipation of the double rows of vertical piles suspending horizontal steel C shaped bars are experimentally and theoretically studied under normal regular waves. Different wave and structural parameters are investigated e.g. the wave length, the C shaped bars draft and spacing, the supporting piles diameter and spacing, and the space between the double rows. Also, the theoretical model based on an eigenfunction expansion method is developed to study the hydrodynamic breakwater performance. In order to examine the validity of the theoretical model, the theoretical results are compared with the experimental and theoretical results obtained by different authors. Comparison between experiments and predictions showed that theoretical model provides a good estimate to the different hydrodynamic coefficients when the friction factors of the upper and the lower parts are f_U = 1.5 and f_L = 0.75. The present breakwater physical model gives efficiency near other similar systems of different shapes.     

Wave climate scatter performance of a cycloidal wave energy converter

A lift based cycloidal wave energy converter (WEC) was investigated using potential flow numerical simulations in combination with viscous loss estimates based on published hydrofoil data. This type of wave energy converter consists of a shaft with one or more hydrofoils attached eccentrically at a radius. The main shaft is aligned parallel to the wave crests and submerged at a fixed depth. The operation of the WEC as a wave-to-shaft energy converter interacting with straight crested waves was estimated for an actual ocean wave climate. The climate chosen was the climate recorded by a buoy off the north-east shore of Oahu/Hawaii, which was a typical moderate wave climate featuring an average annual wave power P_W = 17 kWh/m of wave crest. The impact of the design variables radius, chord, span and maximum generator power on the average annual shaft energy yield, capacity factor and power production time fraction were explored. In the selected wave climate, a radius R = 5 m, chord C = 5 m and span of S = 60 m along with a maximum generator power of P_G = 1.25 MW were found to be optimal in terms of annual shaft energy yield. At the design point, the CycWEC achieved a wave-to-shaft power efficiency of 70%. In the annual average, 40% of the incoming wave energy was converted to shaft energy, and a capacity factor of 42% was achieved. These numbers exceeded the typical performance of competing renewables like wind power, and demonstrated that the WEC was able to convert wave energy to shaft energy efficiently for a range of wave periods and wave heights as encountered in a typical wave climate.     

Surface wave predictions in weakly nonlinear directional seas

We have employed laboratory and numerical experiments in order to investigate propagation of waves in both long and short-crested wave fields in deep water. For long-crested waves with steepness, ϵ = k_ca_c = 0.1 (a fairly extreme case), reliable prediction can be performed with the modified nonlinear Schrödinger equation up to about 40 characteristic wavelengths. For short-crested waves the accuracy of prediction is strongly reduced with increasing directional spread.     

Focused wave evolution using linear and second order wavemaker theory

In this paper, the evolution of focused waves using different paddle displacements (piston type) under laboratory conditions is presented. It is well known that in intermediate water depths, linear paddle displacements will generate spurious, free, sub and super harmonics. Thus, a second order correction to suppress these spurious free sub and super harmonics was used to generate the focused waves. The focused waves were generated in the laboratory using a linear superimposition principle, in which the wave paddle displacement is derived based on the sum of a number of sinusoidal components at discrete frequencies, whose phases are accordingly set to focus at a particular location. For this method of generation, the second order wave maker theory proposed by Schäffer [24] can be easily adopted and was used in the present study. Two different centre frequencies (f_c = 0.68 Hz and 1.08 Hz) with three different bandwidth ratios (Δf/f_c = 0.5, 0.75 and 1.0) were tested in a constant water depth, to consider both narrow and broadband spectra. These test cases correspond to wave focusing packets propagating in intermediate and deep water regions. Further, for each wave packet, two different amplitudes were considered in order to analyze non-breaking and breaking cases. Thus, by systematically generating the wave packets using the linear and second order paddle displacements, the analysis was carried out for the spectral and temporal evolution of selected long waves. The temporal evolution of the selected harmonics was analyzed using the Inverse Fast Fourier Transform (IFFT), to show the propagation of the spurious, free, long waves. Further, the variations in energy for the lower, higher and primary frequency ranges are reported for different wave paddle displacements. The analysis revealed that for the broadband spectrum the differences are more pronounced when using linear paddle displacements. We have also noticed a shift in focusing/breaking location and time (i.e. premature) due to the increase in crest height using linear displacements. The experiment data used in this paper has been provided as a supplementary, which can be used to validate the numerical models.