An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves

In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to kH ≈ 10, that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method

A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies

A two-layer shallow water model for bedload sediment transport: convergence to Saint-Venant-Exner model

A two-layer shallow water type model is proposed to describe bedload sediment transport. The upper layer is filled by water and the lower one by sediment. The key point falls on the definition of the friction laws between the two layers, which are a generalization of those introduced in Fern\'andez-Nieto et al. (ESAIM: M2AN, 51:115-145, 2017). This definition allows to apply properly the two-layer shallow water model for the case of intense and slow bedload sediment transport. Moreover, we prove that the two-layer model converges to a Saint-Venant-Exner system (SVE) including gravitational effects when the ratio between the hydrodynamic and morphodynamic time scales is small. The SVE with gravitational effects is a degenerated nonlinear parabolic system. This means that its numerical approximation is very expensive from a computational point of view, see for example T. Morales de Luna et al. (J. Sci. Comp., 48(1): 258-273, 2011). In this work, gravitational effects are introduced into the two-layer system without such extra computational cost. Finally, we also consider a generalization of the model that includes a non-hydrostatic pressure correction for the fluid layer and the boundary condition at the sediment surface. Numerical tests show that the model provides promising results and behave well in low transport rate regimes as well as in many other situations

Internal wave generation in a non-hydrostatic wave model

In this work, internal wave generation techniques are developed in an open source non-hydrostatic wave model (Simulating WAves till SHore, SWASH) for accurate generation of regular and irregular long-crested waves. Two different internal wave generation techniques are examined: a source term addition method where additional surface elevation is added to the calculated surface elevation in a specific location in the domain and a spatially distributed source function where a spatially distributed mass is added in the continuity equation. These internal wave generation techniques in combination with numerical wave absorbing sponge layers are proposed as an alternative to the weakly reflective wave generation boundary to avoid re-reflections in case of dispersive and directional waves. The implemented techniques are validated against analytical solutions and experimental data including water surface elevations, orbital velocities, frequency spectra and wave heights. The numerical results show a very good agreement with the analytical solution and the experimental data indicating that SWASH with the addition of the proposed internal wave generation technique can be used to study coastal areas and wave energy converter (WEC) farms even under highly dispersive and directional waves without any spurious reflection from the wave generator

3D free surface flow simulations based on the integral form of the equations of motion

This work deals with a novel three-dimensional finite-volume non-hydrostatic shock-capturing model for the simulation of wave transformation processes and wave-structure interaction. The model is based on an integral formulation of the Navier-Stokes equations solved on a coordinate system in which the vertical coordinate is varying in time. A finite-volume shock-capturing numerical technique based on high order WENO reconstructions is adopted in order to discretize the fluid motion equations

Ocean Rossby waves as a triggering mechanism for primary Madden-Julian events

The Madden–Julian Oscillation (MJO) is sporadic, with episodes of cyclical activity interspersed with inactive periods. However, it remains unclear what may trigger a Madden–Julian (MJ) event which is not immediately preceded by any MJO activity: a ‘primary’ MJ event. A combination of case-studies and composite analysis is used to examine the extent to which the triggering of primary MJ events might occur in response to ocean dynamics. The case-studies show that such events can be triggered by the arrival of a downwelling oceanic equatorial Rossby wave, which is shown to be associated with a deepening of the mixed layer and positive sea-surface temperature (SST) anomalies of the order of 0.5–1 °C. These SST anomalies are not attributable to forcing by surface fluxes which are weak for the case-studies analysed. Furthermore, composite analysis suggests that such forcing is consistently important for triggering primary events. The relationship is much weaker for successive events, due to the many other triggering mechanisms which operate during periods of cyclical MJO activity. This oceanic feedback mechanism is a viable explanation for the sporadic and broadband nature of the MJO. Additionally, it provides hope for forecasting MJ events during periods of inactivity, when MJO forecasts generally exhibit low skill

Development and Optimization of Non-Hydrostatic Models for Water Waves and Fluid-Vegetation Interaction

The primary objective of this study is twofold: 1) to develop an efficient and accurate non-hydrostatic wave model for fully dispersive highly nonlinear waves, and 2) to investigate the interaction between waves and submerged flexible vegetation using a fully coupled wave-vegetation model. This research consists of three parts. Firstly, an analytical dispersion relationship is derived for waves simulated by models utilizing Keller-box scheme and central differencing for vertical discretization. The phase speed can be expressed as a rational polynomial function of the dimensionless water depth, $kh$, and the layer distribution in water column becomes an optimizable parameter in this function. For a given tolerance dispersion error, the range of $kh$ is extended and the layer thicknesses are optimally selected. The derived theoretical dispersion relationship is tested with linear and nonlinear standing waves generated by an Euler model. The optimization method is applicable to other non-hydrostatic models for water waves. Secondly, an efficient and accurate approach is developed to solve Euler equations for fully dispersive and highly nonlinear water waves. Discontinuous Galerkin, finite difference, and spectral element formulations are used for horizontal discretization, vertical discretization, and the Poisson equation, respectively. The Keller-box scheme is adopted for its capability of resolving frequency dispersion accurately with low vertical resolution (two or three layers). A three-stage optimal Strong Stability-Preserving Runge-Kutta (SSP-RK) scheme is employed for time integration. Thirdly, a fully coupled wave-vegetation model for simulating the interaction between water waves and submerged flexible plants is presented. The complete governing equation for vegetation motion is solved with a high-order finite element method and an implicit time differencing scheme. The vegetation model is fully coupled with a wave model to explore the relationship between displacement of water particle and plant stem, as well as the effect of vegetation flexibility on wave attenuation. This vegetation deformation model can be coupled with other wave models to simulate wave-vegetation interactions

Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model

We investigate here the ability of a Green-Naghdi model to reproduce strongly nonlinear and dispersive wave propagation. We test in particular the behavior of the new hybrid finite-volume and finite-difference splitting approach recently developed by the authors and collaborators on the challenging benchmark of waves propagating over a submerged bar. Such a configuration requires a model with very good dispersive properties, because of the high-order harmonics generated by topography-induced nonlinear interactions. We thus depart from the aforementioned work and choose to use a new Green-Naghdi system with improved frequency dispersion characteristics. The absence of dry areas also allows us to improve the treatment of the hyperbolic part of the equations. This leads to very satisfying results for the demanding benchmarks under consideration