Accuracy of depth-integrated nonhydrostatic wave models

Depth-integrated nonhydrostatic models have been wildly used to simulate propagation of waves. Yet, there lacks a well-documented theoretical framework that can be used to assess the accuracy and scope of applications of these models and the related numerical approaches. In this work, we carry out Stokes-type Fourier and shoaling analyses to examine the linear and nonlinear properties of a popular one-layer depth-integrated nonhydrostatic model derived by Stelling and Zijlema (2003). The theoretical analysis shows that the model can satisfactorily interpret the dispersity for linear waves but presents evident divergence for nonlinear solutions even when kd → 0. A generalized depth-integrated nonhydrostatic formulation using arbitrary elevation as a variable is then derived and analyzed to examine the effects of neglecting advective and diffusive nonlinear terms in the previous studies and explore possible improvements in numerical solutions for wave propagation. Compared with the previous studies, the new generalized formulation exhibits similar dispersion relationship and improved shoaling effect. However, no significant improvement is presented for the nonlinear properties, indicating that retaining neglected nonlinear terms may not significantly improve the nonlinear performance of the nonhydrostatic model. Further analysis shows that the nonlinear properties of the depth-integrated nonhydrostatic formulation may be improved by defining variables at one-third of the still water level. However, such an improvement comes at the price of decreasing accuracy in describing dispersion and shoaling properties

A new multilayer nonhydrostatic formulation for surface water waves

This work presents a new multilayer nonhydrostatic formulation for surface water waves. The new governing equations define velocities and pressure at an arbitrary location of a vertical layer and only contain spatial derivatives of maximum second order. Stoke-type Fourier and shoaling analyses are carried out to scrutinize the mathematical properties of the new formulation, subsequently optimizing the representative interface and the location to define variables in each layer to improve model accuracy. Following the analysis, the one-layer model exhibits accurate linear and nonlinear characteristics up to kd = I, demonstrating similar solution accuracy to the existing second-order Boussinesq-type models. The two-layer model with optimized coefficients can maintain its linear and nonlinear accuracy up to kd = 4I, which boasts of better solution accuracy a larger application range than most existing fourth-order Boussinesq model and two-layer Boussinesq models. The three-layer model presents accurate linear and nonlinear characteristics up to kd = 10Ï, effectively removing any shallow water limitation. The current multilayer nonhydrostatic water wave model does not predefine the vertical flow structures, and more accurate vertical velocity distributions can be obtained by considering the velocity profiles in coefficient optimization

Analytic study on long wave transformation over a seamount with a pit

In this paper, an analytic solution is derived for linear long waves scattering over a submarine seamount landform with a pit. The seamount is axisymmetric with a pit on the top. The water depth is defined by a trinomial function in the radial direction. The governing linear shallow water equation for long waves is expressed in the polar coordination, which is solved through separation of variables. As the topography is axisymmetric, solutions can be written as Fourier-cosine series. Waves over the seamount are expressed using Frobenius series expansion, while the water surface elevation in the outer region is expressed as Fourier-Bessel series, and the final solution is obtained by matching them at the conjunction. The solution can be degenerated into the previous analytic solutions for waves propagation over an axisymmetric pit or a submerged hump by adjusting the topography parameters

Analytical and numerical investigation of trapped ocean waves along a submerged ridge

Based on the linear shallow-water equations, new analytical solutions are derived for trapped waves over a ridge with a hyperbolic-cosine squared cross-sectional profile which may be used to idealize many real-world ocean ridges. In the new analytical formulation, the free surface of the trapped waves is described using the combination of the first and second kinds of the associated Legendre functions, which is further analysed to reveal the existence of both symmetrical and anti-symmetrical trapped waves on the ridge under consideration. New algebraic equations are also derived to depict the wave dispersion relationships, allowing explicit quantification of their sensitivity to the topographic profile. Furthermore, a ray-tracing method is applied to interpret the propagation paths of trapped waves over the ridge and better understand the excitation mechanisms. Finally, an extensively validated Boussinesq wave model is used to conduct numerical experiments for trapped waves induced by tsunamis. The numerical predictions are consistent with the new analytical solutions, which effectively confirms the validity of the new analytical framework for trapped waves over a more general type of oceanic ridges

A fully coupled hydrodynamic-DEM model for simulating debris dynamics and impact forces

Many post-event field investigations suggest that dramatic flood hydrodynamics and impact of large floating objects contribute significantly to building damage during a tsunami event. However, the interactions between the transient flood waves, floating debris, and structures have neither been well explored nor understood, and few modelling tools have been developed to simulate these complex interactive processes, especially when multiple debris are involved. This paper introduces a novel fully coupled modelling system, based on a high-performance 2D hydrodynamic model and a 3D discrete element method (DEM) model, for simulating the movement of multiple debris carried along by highly convective flows and directly quantifying the induced impact forces on structures. The proposed model is applied to reproduce a series of physical experiments, and the results agree well with the experimental measurements. It is demonstrated that the newly coupled modelling system can capture the interaction between the fluid, debris of different shapes and sizes and structures, making it suitable for real-world applications. The model provides a new robust tool for simulating the extreme hazards caused by tsunamis or flash flooding and assessing their risk, and therefore has the potential to be useful for planning and designing disaster risk reduction schemes in those at-risk areas.</p

Graphics processing unit (GPU)-enhanced nonhydrostatic model with grid nesting for global tsunami propagation and coastal inundation

Nonhydrostatic models have proven their superiority in describing tsunami propagation over trans-oceanic distances and nearshore transformation because of their good dispersion and nonlinearity properties. The novel one-layer nonhydrostatic formulations proposed by Wang et al. [Phys. Fluids 35, 076610 (2023)] have been rederived in the spherical coordinate system incorporating Coriolis effects to enable the application of basin-wide tsunami propagation. The model was implemented using the fractional step method, where the hydrostatic step was solved by a Godunov-type finite-volume scheme, and the nonhydrostatic step was obtained with the finite-difference method. Additionally, a two-way grid-nesting scheme was employed to adapt the topographic features for efficient computation of tsunami propagation in deep ocean and coastal inundation. Furthermore, graphics processing unit (GPU)-parallelism technique was incorporated to further optimize the model performance. An idealized benchmark test as well as three experiments of regular and irregular waves, solitary, and N-waves transformations have been simulated to demonstrate the superior performance of the current GPU-accelerated grid-nesting nonhydrostatic model. Finally, the model has been applied to reproduce the 1964 Prince William Sound Tsunami, its propagation across the North Pacific and induced inundation in the Seaside.</p