Numerical Dispersion in Non-Hydrostatic Modeling of Long-Wave Propagation.

Abstract

Numerical discretization with a finite-difference scheme is known to introduce truncation errors in the form of frequency dispersion in depth-integrated models commonly used in tsunami research and hazard mapping. While prior studies on numerical dispersion have focused on the shallow-water equations, we include the depth-integrated non-hydrostatic pressure and vertical velocity through a Keller box scheme and investigate the properties of the resulting system. Fourier analysis of the discretized governing equations gives rise to a dispersion relation in terms of the time step, grid size, and wave direction. The interworking of the dispersion relation is elucidated by its lead-order approximation, one and two-dimensional numerical experiments, and a case study of the tsunami generated by the 2010 Mentawai Mw 7.8 earthquake. The dispersion relation, aided by its lead-order approximation from the Taylor series expansion, shows that coupling between the spatial discretization and non-hydrostatic terms results in significant reduction of numerical dispersion outside the shallow-water range. The time step, which counteracts numerical dispersion from spatial discretization, only has secondary effects within the applicable range of Courant numbers. Numerical dispersion also decreases for wave propagation oblique to the principal axes of the grid due to effective increase in spatial resolution. A numerical flume experiment of standing waves indicates minor contributions from the implicit solution scheme of the non-hydrostatic terms. A second numerical experiment verifies the properties deduced from the analytical results and demonstrates the effectiveness of discretization in altering progressive waves over a two-dimensional grid. The computational results also demonstrate generation of spurious, short-period trailing waves from hydrostatic model with insufficient numerical dispersion. Since the governing equations for the non-hydrostatic system trend to underestimate dispersion in shoaling water, the numerical effects are complementary in producing a solution closer to Airy wave theory. A case study of the 2010 Mentawai Mw 7.8 earthquake and tsunami event, which has a compact source adjacent to a deep trench, demonstrates the role of dispersion in wave propagation and the implications for the commonly-used source inversion techniques. Non-dispersive models are often used with an initial static sea-surface pulse derived from seafloor deformation in computation of tsunami Green's functions. We compare this conventional approach with more advanced techniques, which use Green's functions computed by a dispersive model with an initial static sea-surface pulse and with the surface waves generated from kinematic seafloor deformation. The fine subfaults needed to resolve the compact rupture results in dispersive waves that require a non-hydrostatic model. The Green's functions from the hydrostatic model are overwhelmed by spurious, grid-dependent short-period oscillations, which are filtered prior to their application. These three sets of tsunami Green's functions are implemented in finite-fault inversions with and without seismic and geodetic data. Seafloor excitation and wave dispersion produce more spread-out waveforms in the Green's functions leading to larger slip with more compact distribution through the inversions. If the hydrostatic Green's functions are not filtered, the resulting slip spreads over a large area to eliminate the numerical artifacts from the lack of dispersion. The fit to the recorded tsunami and the deduced seismic moment, which reflects the displaced water volume, is relatively insensitive to the approach used for computing Green’s functions.