# Coupling of a nonlinear dispersive water wave model with sediment transport and seabed morphodynamic models for application in near-shore areas

2021-04-30

Kazhyken, Kazbek

## Abstract

Sediment transport and bed morphodynamic processes in near-shore regions pose significant risks to coastal infrastructure and environment, such as compromised structural integrity of coastal structures due to excessive scouring, roles of sediment deposits as sinks, vessels, and sources of dangerous pollutants, natural habitat degradation due to beach and shoreline erosion. Therefore, a mathematical modeling of these processes carries a clear engineering and scientific relevance. The sediment transport and bed morphodynamic processes are primarily forced by water waves and motion, which, in their turn, are affected by changes in the sediment bed surface driven by the sediment transport processes. Therefore, any mathematical modeling of these hydro-sediment-morphodynamic processes involves a coupling between a hydrodynamic model, which describes water waves and motion, and a sediment transport and bed evolution model, which resolves changes in the sediment bed surface driven by sediment erosion, transport, and deposition rates. One of the most widespread mathematical models used to resolve these processes is formed by the shallow water hydro-sediment-morphodynamic equations (SHSM), where the nonlinear shallow water equations (NSWE) are coupled with a sediment advection model and the sediment continuity Exner equation. If the shallowness parameter is defined as μ=H₀²/L₀², where H₀ and L₀ are the depth and length scales of the water flow, respectively, then NSWE provide a depth-averaged hydrodynamic model that is Ο(μ) consistent with the incompressible Euler equations. Therefore, SHSM provide a suitable computationally efficient model for the shallow water flow regimes where μ<<1. One of the drawbacks of using NSWE is the model's lack of capacity to capture wave dispersion effects that can be remedied by replacing NSWE with a nonlinear dispersive wave hydrodynamic model. One such model is formed by the Green-Naghdi equations (GN), a depth-averaged hydrodynamic model that is Ο(μ²) consistent with the incompressible Euler equations. In the presented work a new dispersive wave hydro-sediment-morpho-dynamic model is presented, where NSWE in the hydrodynamic part of SHSM is replaced with a single parameter variation of GN introduced by Bonneton et al. The new model is Ο(μ²) consistent with the incompressible Euler equations in its hydrodynamic part, and can be applied in coastal areas where wave dispersion effects are prevalent. The model is further subdivided into two classes: (1) a model that does not take into account the suspended load transport and considers the effects of the bed load transport only, (2) a model that resolves the effects of both the suspended and bed load transport. The first model is referred to as a dispersive wave hydro-morphodynamic model, and the second one as a dispersive wave hydro-sediment-morphodynamic model. The models are treated numerically with Strang operator splitting, and discontinuous Galerkin finite element methods. For the first model two modes of discretization are proposed: (1) the decoupled mode where GN and the Exner equations are solved separately, (2) the coupled mode where both of the equations are solve simultaneously at each time step. For the second model a fully coupled mode of discretization is employed. The developed numerical solution algorithms are validated with benchmark numerical examples. The results of the validation simulations indicate that the developed model has the ability to accurately resolve hydrodynamics of regular and solitary waves along with their dispersive properties, and sediment transport and bed morphodynamic processes provided that the empirical models for the suspended and bed load transport are properly calibrated. The results indicate that the model has the potential to be used in studies of coastal morphodynamics. As an example application of the model, the Ria Formosa lagoon is selected for a simulation of hydro-sediment-morphodynamic processes. An unstructured finite element mesh representation of the western circulation cell of the lagoon is generated, and data to parametrize sediment transport, bottom friction, and tidal wave models is gathered. The example application indicates that the developed dispersive wave model can be used to simulate the processes in coastal areas with complex irregular geometries. Moreover, the terms of the model that are Ο(μ²) consistent with the incompressible Euler equations introduce new features into the flow parameters that have the potential to significantly affect the resulting sediment transport and bed morphodynamic process simulations