Nonlinear Boussinesq-type Model Research Articles

A high-order nonlinear Boussinesq-type model for internal waves over a mildly-sloping topography in a two-fluid system

With a rigid lid assumption on the upper boundary in a two-fluid system, a new high-order Boussinesq-type model for internal waves over a mildly-sloping topography is derived. The model is formulated in terms of computational velocity vectors defined at mid-water depth in each fluid. Stokes-type expansions are used to theoretically analyze the linear and nonlinear properties of the new equations. High accuracy in linear, nonlinear, shoaling and kinematic aspects is achieved and the model is applicable from deep to shallow water. For an equivalent water depth in upper and lower fluid, the model can be applied up to kh2 ≈ 10 (where k is the wave number and h2 is the water depth of the lower fluid) in linear dispersion, up to kh2 ≈ 7.0 in the second nonlinear property within 2% error, and up to 0 < kh2 < 10 in the linear shoaling property. Compared with most existing Boussinesq-type models, the accuracy of the horizontal and vertical velocity profiles along the water column is improved significantly and is applicable to up to kh2 ≈ 4.48.

Strongly non-linear Boussinesq-type model of the dynamics of internal solitary waves propagating in a multilayer stratified fluid

We propose a system of first-order balance laws that describe the propagation of internal solitary waves in a multilayer stratified shallow water with non-hydrostatic pressure in the upper and lower layers. The construction of this model is based on the use of additional variables, which make it possible to approximate the Green–Naghdi-type dispersive equations by a first-order system. In the Boussinesq approximation, the governing equations allow one to simulate the propagation of non-linear internal waves, taking into account fine density stratification, a weak velocity shear in the layers, and uneven topography. We obtain smooth steady-state soliton-like solutions of the proposed model in the form of symmetric and non-symmetric waves of mode-2 adjoining to a given multilayer constant flow. Numerical calculations of the generation and propagation of large-amplitude internal waves are carried out using both the proposed first-order system and Green–Naghdi-type equations. It is established that the solutions of these models practically coincide. The advantage of the first-order equations is the simplicity of numerical implementation and a significant reduction in the calculation time. We show that the results of numerical simulation are in good agreement with the experimental data on the evolution of mode-2 solitary waves in tanks of constant and variable height.

Anomalous wave statistics following sudden depth transitions: application of an alternative Boussinesq-type formulation

Recent studies of water waves propagating over sloping seabeds have shown that sudden transitions from deeper to shallower depths can produce significant increases in the skewness and kurtosis of the free surface elevation and hence in the probability of rogue wave occurrence. Gramstad et al. (Phys. Fluids 25 (12): 122103, 2013) have shown that the key physics underlying these increases can be captured by a weakly dispersive and weakly nonlinear Boussinesq-type model. In the present paper, a numerical model based on an alternative Boussinesq-type formulation is used to repeat these earlier simulations. Although qualitative agreement is achieved, the present model is found to be unable to reproduce accurately the findings of the earlier study. Model parameter tests are then used to demonstrate that the present Boussinesq-type formulation is not well-suited to modelling the propagation of waves over sudden depth transitions. The present study nonetheless provides useful insight into the complexity encountered when modelling this type of problem and outlines a number of promising avenues for further research.

Morphodynamic Modelling of Flash Rip Current Driven Coastal Sediment Transport

Park, S.; Kim, D.-H., Yoo, H., 2020. Morphodynamic modelling of flash rip current driven coastal sediment transport. In: Malvárez, G. and Navas, F. (eds.), Global Coastal Issues of 2020. Journal of Coastal Research, Special Issue No. 95, pp. 1229–1234. Coconut Creek (Florida), ISSN 0749-0208.The characteristics of the sediment transport driven by flash rip currents at nearshore areas was investigated using a morphodynamic model coupled with a fully nonlinear Boussinesq-type model and a sediment transport model. Wave transform processes in surfzone such as shoaling, refraction, breaking, runup, and rundown processes were included in the numerical model. The vorticity and turbulent eddy viscosity were considered to account for rotational and turbulent flow characteristics. In addition, the horizontal density variation and bottom evolution caused by sediment transport were integrated into the fully nonlinear Boussinesq-type model. A fourth-order finite volume method, based on the MUSCL scheme with an approximate Riemann solver, was used to solve the governing equations. Uniform and parallel wave fields were simulated on a plain beach with a uniform sandbar with very small disturbance on the bathymetry. Because the wave generator was parallel to the beach, no significant longshore current and longshore sediment transport were generated. From the tested case, transient offshore-directed strong rip currents (the wave-averaged flow velocity 0.5 m/s) appeared in various locations. Turbid suspended sediment currents were also randomly generated by the waves and currents. The modelling results and analysis showed that the offshore direction sediment transport was strongly associated with the flash rip current generated within the surfzone.

A 2DH fully dispersive and weakly nonlinear Boussinesq-type model based on a finite-volume and finite-difference TVD-type scheme

In this study, we developed a fully dispersive Boussinesq-type wave model in the modelling framework of the public-domain Boussinesq model, FUNWAVE-TVD. The model adopts the fully dispersive Boussinesq equations of Karambas and Memos (2009) and uses a finite-volume and finite-difference TVD-type scheme. A well-balanced conservative form of the governing equations is derived to facilitate the hybrid numerical scheme. Flux terms were computed by the MUSCL-TVD scheme up to the fourth-order accuracy within the Riemann solver. The third-order Strong Stability-Preserving (SSP) Runge–Kutta scheme was used for time stepping. The convolution integral terms were estimated by the numerical evaluation and the spatial derivative terms were computed by the finite difference scheme. Wave breaking is predicted by locally switching the Boussinesq equations to nonlinear shallow water equations with a Froude number criterion. The model is validated against the linear wave theory and various experiments to examine the capability of the model in simulating wave dispersion, shoaling, breaking, refraction, diffraction, and run-up.

UBathy: A New Approach for Bathymetric Inversion from Video Imagery

A new approach to infer the bathymetry from coastal video monitoring systems is presented. The methodology uses principal component analysis of the Hilbert transform of video images to obtain the components of the wave propagation field and their corresponding frequency and wavenumber. Incident and reflected constituents and subharmonics components are also found. Local water depth is then successfully estimated through wave dispersion relationship. The method is first applied to monochromatic and polychromatic synthetic wave trains propagated using linear wave theory over an alongshore uniform bathymetry in order to analyze the influence of different parameters on the results. To assess the ability of the approach to infer the bathymetry under more realistic conditions and to explore the influence of other parameters, nonlinear wave propagation is also performed using a fully nonlinear Boussinesq-type model over a complex bathymetry. In the synthetic cases, the relative root mean square error obtained in bathymetry recovery (for water depths 0.75 m ⩽ h ⩽ 8.0 m ) ranges from ∼1% to ∼3% for infinitesimal-amplitude wave cases (monochromatic or polychromatic) to ∼15% in the most complex case (nonlinear polychromatic waves). Finally, the new methodology is satisfactorily validated through a real field site video.

Discussion to “Boussinesq modeling of longshore currents in the SandyDuck experiment under directional random wave conditions” by J. Choi, J. T. Kirby and S. B. Yoon

The paper (Choi et al., 2015) shows the influence of the offshore wave field (frequency-directional spectra) on the nearshore circulation, particularly on the longshore current. In contrast to usual approaches, the numerical simulations are here carried out using a phase resolving model: the fully non-linear Boussinesq-type model FUNWAVE. Because the wave height plays a relevant role in the problem, I find that it could be of interest the following discussion, which has to do with the linear shoaling error of Boussinesq equations.

A space-time discontinuous Galerkin method for Boussinesq-type equations

In this work we propose a new high order accurate, fully implicit space-time discontinuous Galerkin (DG) method for advection-diffusion-dispersion equations, i.e. for so-called Korteweg-de-Vries-type equations. In particular, we focus on Boussinesq-type models for free surface flows, which are used for the modeling of water waves that travel in deep water, where the classical shallow water equations are not valid any more. Our method follows the ideas of the local DG method (LDG) for dispersion equations proposed by Yan and Shu (2002), who used an explicit Runge–Kutta method to integrate their scheme in time. However, such explicit time integrators applied to dispersive equations imply a very severe restriction on the time step, which has to be taken proportional to the cube of the mesh spacing, and which therefore can make even one-dimensional computations prohibitively expensive on fine grids. For the scalar case and with some simplifying assumptions, the scheme presented in this paper can be proven to be unconditionally stable in L2 norm. Furthermore, our method is based directly on a space-time finite element formulation, which also provides a natural way to discretize third order dispersive terms that contain a mixed space-time derivative. Such terms appear frequently in the context of Boussinesq-type models for free surface flows. We will show numerical convergence studies for linear scalar advection-diffusion-dispersion equations and furthermore, we will also study the convergence of our method using solitary wave solutions of the nonlinear Boussinesq-type model of Madsen, Murray and Sørensen (1991).

FULLY NONLINEAR BOUSSINESQ-TYPE MODELLING OF INFRAGRAVITY WAVE TRANSFORMATION OVER A LOW-SLOPING BEACH

Recent field studies over low sloping beaches have shown that infragravity waves could dissipate a significant part of their energy in the inner surf zone. This phenomenon and the associated short- and long-wave transformations are not well-understood. In this paper, we assess the ability of the fully nonlinear Boussinesq-type model introduced in Bonneton et al. (2011) to reproduce short and long wave transformation in a case involving a strong infragravity wave dissipation close to the shoreline. This validation study, based on van Dongeren et al. (2008)'s laboratory experiments, suggests that the model is able to predict infragravity wave breaking as well as the complex interactions between short and long waves in the surf zone.

Joint use of data and modeling in coastal wave transformation

In the framework of a research project entitled “BRISA—BReaking waves and Induced SAnd transport”, a methodology was devised to characterize the waves joining together in-situ measurements and numerical wave propagation models. With this goal in mind, a number of in-situ measurements were made, for selected positions in front of Praia de Faro (South Portugal), during four days (25th to 28th March, 2009) by using different types of equipments (e.g., resistive wave gauges, pressure sensors, currentmeters and a new prototype pore pressure sensor using optical fibre). Wave records were obtained simultaneously offshore (at a water depth of 11.7 m below mean sea level, MSL) and at the surf and swash zones. The data processing and analysis were made by applying classical time domain techniques. Numerical simulations of the wave propagation between offshore and inshore for the measurement period were performed with two numerical models, a 1D model based on linear theory and a nonlinear Boussinesq-type model, COULWAVE, both forced by the measured offshore wave conditions of 27th March 2009. Comparisons between numerical results and field data for the pressure sensors placed in the surf and swash zones were made and discussed. This approach enables to evaluate the performance of those models to simulate those specific conditions, but also to validate the models by gaining confidence on their use in other conditions.

Numerical modeling of surf beat generated by moving breakpoint

As an important hydrodynamic phenomenon in the nearshore zone, the cross-shore surf beat is numerically studied in this paper with a fully nonlinear Boussinesq-type model, which resolves the primary wave motion as well as the long waves. Compared with the classical Boussinesq equations, the equations adopted here allow for improved linear dispersion characteristics. Wave breaking and run-up in the swash zone are included in the numerical model. Mutual interactions between short waves and long waves are inherent in the model. The numerical study of long waves is based on bichromatic wave groups with a wide range of mean frequencies, group frequencies and modulation rates. The cross-shore variation in the amplitudes of short waves and long waves is investigated. The model results are compared with laboratory experiments from the literature and good agreement is found.

Derivation and application of new equations for radiation stress and volume flux

In this paper, new expressions of radiation stress and volume flux for long waves have been analytically derived by inclusion of higher-order surface elevations up to the sixth-order. To quantify these expressions, surface elevations along a beach are first simulated using the fully nonlinear Boussinesq-type model COULWAVE. Then, based on the large amount of numerical data, new equations for radiation stress and volume flux are statistically formulated. The research unveils the essential roles of the Ursell parameter, Irribarren number and wave steepness described by the local wave height, wave length and bottom slope. The study shows the importance of nonlinear wave properties in wave-induced currents and mean water levels (set-up/down). The higher-order formulations produce lower values for radiation stress and volume flux than calculated from the lower-order and linear waves. Case studies suggest that the new formulations produce an accurate estimation for mean water level. However, improvement on the computed current profiles is marginal for some cases. This implies that the accurate prediction of the current profile would require more than just the proposed improvement of the radiation stress and volume flux.