High-order Boussinesq-type Equations Research Articles

Strong earthquakes and tsunami potential in the Hellenic Subduction Zone

The tsunamigenic potential of an earthquake depends on its size, source depth and focal mechanism. The Hellenic Subduction Zone (HSZ) has been selected in the paper to study this important issue. The HSZ was ruptured by 11 strong (Mw6.0) earthquakes in the time period 2009–2023. One earthquake ruptured onshore but only three out of ten offshore earthquakes produced tsunamis: 1 July 2009 (Mw6.4), 25 October 2018 (Mw6.8), 5 May 2020 (Mw6.6). For each one of the two more recent earthquakes of 5 May 2020 (tsunamigenic, thrust faulting) and 12 October 2021 (non-tsunamigenic, strike-slip faulting) we developed heterogeneous fault models from the inversion of teleseismic P-waveforms, and homogeneous fault models from published moment-tensor solutions. For each fault model tsunami generation and propagation was numerically simulated based on an advanced phase-resolving wave model with the use of higher-order Boussinesq-type equations. The modelled tsunami mareograms are consistent with tide records of the small tsunami (height ∼30 cm) produced by the 2020 earthquake. For the 2021 earthquake the modelled mareograms showed tsunami-like disturbance with amplitude not exceeding the noise level. The tsunamigenic earthquakes of 2009, 2018 and 2020 shared magnitude Mw≥ 6.4, shallow depth (h<20 km), moderate-to-high dip-angle and thrust faulting or oblique slip with significant thrust component. In the remaining seven non-tsunamigenic earthquakes, including the 2021 one, at least one of these features is missing. The results obtained help to better understand the seismic mechanisms of tsunami generation in the HSZ. Further investigation is needed for the central HSZ segment to the south of Crete Island, which historically has not been ruptured by large (Mw>7.0) tsunamigenic earthquakes. In contrast, the western and eastern HSZ segments ruptured by the large 365 AD and 1303 AD tsunamigenic earthquakes.

High-Order Boussinesq Equations for Water Wave Propagation in Porous Media

To accurately capture wave dynamics in porous media, the higher-order Boussinesq-type equations for wave propagation in deep water are derived in this paper. Starting with the Laplace equations combined with the linear and nonlinear resistance force of the dynamic conditions on the free surface, the governing equations were formulated using various independent velocity variables, such as the depth-averaged velocity and the velocity at the still water level and at an arbitrary vertical position in the water column. The derived equations were then improved, and theoretical analyses were carried out to investigate the linear performances with respect to phase celerity and damping rate. It is shown that Boussinesq-type models with Padé [4, 4] dispersion can be applied in deep water. A numerical implementation for one-dimensional equations expressed with free surface elevation and depth-averaged velocity is presented. Solitary wave propagation in porous media was simulated, and the computed results were found to be generally in good agreement with the measurements.

Modelling the Impact of Climate Change on Coastal Flooding: Implications for Coastal Structures Design

In the present work, the impact of climate change on coastal flooding is investigated through a set of interoperable models developed by the authors, following a modular modelling approach and adapting the modelling sequence to two separate objectives with respect to inundation over large-scale areas and coastal protection structures’ design. The modelling toolbox used includes a large-scale wave propagation model, a storm-induced circulation model, and an advanced nearshore wave propagation model based on the higher order Boussinesq-type equations, all of which are presented in detail. Model capabilities are validated and applications are made for projected scenarios of climate change-induced wave and storm surge events, simulating coastal flooding over the low-lying areas of a semi-enclosed bay and testing the effects of different structures on a typical sandy beach (both in northern Greece). This work is among the few in relevant literature that incorporate a fully non-linear wave model to a modelling system aimed at representing coastal flooding. Results highlight the capabilities of the presented modelling approach and set the basis for a comprehensive evaluation of the use of advanced modelling tools for the design of coastal protection and adaptation measures against future climatic pressures.

Numerical simulation of ship-borne waves using a 2DH post-Boussinesq model

This work presents results on the simulation of the generation and propagation of ship-borne waves, using an advanced nonlinear dispersive wave model based on the higher order Boussinesq-type equations. The model includes a single frequency dispersion term, expressed through convolution integrals; it is adapted to represent ship-borne waves by adopting the approach for wave generation by a moving local pressure disturbance, achieved through adding a respective pressure term in its governing equations. The model is tested against an analytical solution for the calculation of the angles of ship wakes and laboratory experiments of waves produced by a high-speed ship in a channel. Results compare well to data from both the aforementioned studies, confirming the model's capabilities and highlighting its accuracy in the representation of ship-borne waves; the model's applicability to practical coastal engineering applications is also investigated, through a set of numerical experiments of waves generated at sea and in a harbour.

Water wave solutions of Zufiria’s higher-order Boussinesq type equations and its stability

The weakly nonlinear Hamiltonian theory for two-dimensional irrotational shallow water waves is formulated. Zufiria’s higher-order Boussinesq model for waves on water of finite depth is obtained. The formulation Zufiria’s higher-order Boussinesq type equations are studied by transforming them into solvable ordinary differential equations. By implementing different types of extended direct algebraic method, we show that in fact this model has new exact solutions. An extended direct algebraic methods are applied to construct solitary wave solutions.

Determination of fractional energy loss of waves in nearshore waters using an improved high-order Boussinesq-type model

Fractional energy losses of waves due to wave breaking when passing over a submerged bar are studied systematically using a modified numerical code that is based on the high-order Boussinesq-type equations. The model is first tested by the additional experimental data, and the model's capability of simulating the wave transformation over both gentle slope and steep slope is demonstrated. Then, the model's breaking index is replaced and tested. The new breaking index, which is optimized from the several breaking indices, is not sensitive to the spatial grid length and includes the bottom slopes. Numerical tests show that the modified model with the new breaking index is more stable and efficient for the shallow-water wave breaking. Finally, the modified model is used to study the fractional energy losses for the regular waves propagating and breaking over a submerged bar. Our results have revealed that how the nonlinearity and the dispersion of the incident waves as well as the dimensionless bar height (normalized by water depth) dominate the fractional energy losses. It is also found that the bar slope (limited to gentle slopes that less than 1:10) and the dimensionless bar length (normalized by incident wave length) have negligible effects on the fractional energy losses.

Boussinesq-type equations for nonlinear evolution of wave trains

Nonlinear evolution of wave trains involves amplitude dispersion and four-wave resonant interaction and hence is difficult to describe using a simple wave equation such as the cubic Schrödinger equation or conventional Boussinesq equations. The present study develops a set of improved higher-order Boussinesq equations with a wide accuracy range of third-order nonlinear characteristics, including amplitude dispersion, and with superior performance for simulations of the nonlinear evolution of wave trains. The equations are obtained by enhancing the higher-order Boussinesq-type equations developed by Zou [Z.L. Zou, A new form of high-order Boussinesq equations, Ocean Eng. 27 (2000) 557–575] through introducing two nonlinear terms into the expression for the computation velocity. The new terms can improve the nonlinear property at higher order by adjusting their free parameters to match the theoretical solutions for amplitude dispersion and the third-order transfer function. Super- and sub-harmonics of bichromatic waves are also improved. The new equations are applied to simulate the nonlinear evolution of wave groups along a 1D wave tank with flat bottom, and nonlinear refraction and diffraction of regular wave trains over a cylindrical ramp, good effectiveness is found.

Travelling wave solutions to Zufiria’s higher-order Boussinesq type equations

Zufiria’s higher-order Boussinesq type equations are studied by transforming them into solvable ordinary differential equations. Various families of their travelling wave solutions are generated, which include periodic wave, solitary wave, periodic-like wave, soliton-like wave, Jacobi elliptic function periodic wave, combined non-degenerative Jacobi elliptic function-like wave, Weierstrass elliptic function periodic and rational solutions. The presented approach can be also applied to nonlinear wave equations with variable coefficients in mathematical physics and mechanics.

Nodal DG-FEM solution of high-order Boussinesq-type equations

A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.

Higher order Boussinesq-type equations for water waves on uneven bottom

Higher order Boussinesq-type equations for wave propagation over variable bathymetry were derived. The time dependent free surface boundary conditions were used to compute the change of the free surface in time domain. The free surface velocities and the bottom velocities were connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth were retained in the inverse series expansion of the exact solution, with which the problem was closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model was extended to the slope bottom which is not so mild. For linear properties, some validation computations of linear shoaling and Booij's tests were carried out. The problems of wave-current interactions were also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results confirm perfectly to the theoretical solution as well as other numerical solutions of the full potential problem available.

Two sets of higher-order Boussinesq-type equations for water waves

Two sets of higher-order Boussinesq-type equations for water waves

Two sets of higher-order Boussinesq-type equations for water waves

Based on the classical Boussinesq model by Peregrine [Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech. 27 (4), 815–827], two parameters are introduced to improve dispersion and linear shoaling characteristics. The higher order non-linear terms are added to the modified Boussinesq equations. The non-linearity of the Boussinesq model is analyzed. A parameter related to h/ L 0 is used to improve the quadratic transfer function in relatively deep water. Since the dispersion characteristic of the modified Boussinesq equations with two parameters is only equal to the second-order Padé expansion of the linear dispersion relation, further improvement is done by introducing a new velocity vector to replace the depth-averaged one in the modified Boussinesq equations. The dispersion characteristic of the further modified Boussinesq equations is accurate to the fourth-order Padé approximation of the linear dispersion relation. Compared to the modified Boussinesq equations, the accuracy of quadratic transfer functions is improved and the shoaling characteristic of the equations has higher accuracy from shallow water to deep water.

Resonant response of harbours to short-wave groups

A non-linear short-wave propagation model is used for the simulation of low-frequency waves induced by short-wave groups. The model is based on the higher order Boussinesq-type equations for the propagation of breaking and non-breaking waves in nearshore areas. The 2DH (two horizontal dimensions) Boussinesq model is used to calculate the resonant response of harbours due to bound waves generated by short-wave groups. The results are compared with experimental data available in the literature. In these experiments, bound long waves are generated by carrier bichromatic short waves, which form regular wave groups. The results of the model were found to be in good agreement with the experimental data. The 1DH Boussinesq model is applied in the surf zone to simulate surf beat generation by a time-varying breakpoint and the reflected long wave induced by bichromatic wave groups. A new breaking criterion is introduced, based on the relative trough Froude number. Model generation of the surf beat is also shown to be in good agreement with experimental data.

Nonlinear periodic waves on a beach

High-order Boussinesq-type equations for long periodic nonlinear waves climbing a sloping beach are derived and investigated in shallow water approximation. Potential and surface elevation for wave motion are expanded in Fourier series up to the third harmonic inclusively. Coefficients of these series are explicitly presented as polynomials of Bessel functions. One may speculate that the obtained expression are the first terms of the expanded exact solution to Euler equation for surface waves.

Long periodic waves on an even beach

High-order Boussinesq-type equations for long waves over an even slope are derived and investigated. Potential and surface elevation for periodic wave motion are expanded in Fourier series up to the third harmonic inclusively. Coefficients of this series are expressed as polynomials of Bessel functions.