Non-Hydrostatic Model for Simulating Moving Bottom-Generated Waves: A Shallow Water Extension with Quadratic Vertical Pressure Profile

TL;DR

The paper develops a depth-averaged non-hydrostatic extension of the shallow water equations (SWE) to model waves generated by moving bottoms, addressing dispersion in tsunami-like scenarios. It introduces a pressure split $P = P^{hy} + P^{nh}$ and derives a reformulated bottom pressure relation to avoid time-derivative ambiguities, ensuring equivalence with Boussinesq-type equations while remaining solvable by a projection method. A predictor-corrector discontinuous Galerkin approach is used, with a hydrostatic SWE predictor, an implicit Euler correction, and a coupled elliptic system for the non-hydrostatic pressure and horizontal momentum, solved via LDG fluxes that accommodate non-flat moving bottoms. The method is validated against vertical thrust, sliding landslide, and sloping-beach benchmarks, showing improved dispersion representation and waveform accuracy over hydrostatic SWE and over linear/simplified quadratic pressure models. The work lays a foundation for 2D, bathymetry-rich simulations of landslide tsunamis with potential extensions to friction and viscosity for more realistic applications.

Abstract

We formulate a depth-averaged non-hydrostatic model to solve wave equations with generation by a moving bottom. This model is built upon the shallow water equations, which are widely used in tsunami wave modelling. An extension leads to two additional unknowns to be solved: vertical momentum and non-hydrostatic pressure. We show that a linear vertical velocity assumption turns out to give us a quadratic pressure relation, which is equivalent to Boussinesq-type equations. However, this extension involves a time derivative of an unknown parameter, rendering the solution by a projection method ambiguous. In this study, we derive an alternative form of the elliptic system of equations to avoid such ambiguity. The new set of equations satisfies the desired solubility property, while also consistently representing the non-flat moving topography wave generation. Validations are performed using several test cases based on previous experiments and a high-fidelity simulation. First, we show the efficiency of our model in solving a vertical movement, which represents an undersea earthquake-generated tsunami. Following that, we demonstrate the accuracy of the model for landslide-generated waves. Finally, we compare the performance of our novel set of equations with the linear and simplified quadratic pressure profiles.

Figures (5)

• Figure 1: The illustration of the model.
• Figure 2: Comparison of the measured laboratory data (black) elevation with the simulation using hydrostatic SWE (red) and simulation using non-hydrostatic SWE (blue) for the upthrust case.
• Figure 3: Comparison of the measured laboratory data (black) elevation with the simulation using hydrostatic SWE (red) and simulation using non-hydrostatic SWE (blue) for the down-thrust case.
• Figure 4: Comparison of the measured laboratory data (black) elevation with the simulation using hydrostatic SWE (red) and simulation using non-hydrostatic SWE (blue) for a sliding semi-elliptic plate for $Fr=0.125$ (top), $0.25$ (middle), and $0.375$ (bottom).
• Figure 5: Comparison of the BIEM simulation (black) elevation with our simulation using hydrostatic SWE (red), linear non-hydrostatic SWE (yellow), simplified quadratic non-hydrostatic SWE (purple), and simulation using non-hydrostatic SWE (blue) for a sliding mass over sloping beach.