SUMMARYThis paper describes the formulation, verification, and validation of a depth-integrated, non-hydrostatic model with a semi-implicit, finite difference scheme. The formulation builds on the nonlinear shallow-water equations and utilizes a non-hydrostatic pressure term to describe weakly dispersive waves. A momentumconserved advection scheme enables modeling of breaking waves without the aid of analytical solutions for bore approximation or empirical equations for energy dissipation. An upwind scheme extrapolates the free-surface elevation instead of the flow depth to provide the flux in the momentum and continuity equations. This greatly improves the model stability, which is essential for computation of energetic breaking waves and run-up. The computed results show very good agreement with laboratory data for wave propagation, transformation, breaking, and run-up. Since the numerical scheme to the momentum and continuity equations remains explicit, the implicit non-hydrostatic solution is directly applicable to existing nonlinear shallow-water models.

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SUMMARYTsunamis generated by earthquakes involve physical processes of different temporal and spatial scales that extend across the ocean to the shore. This paper presents a shock-capturing dispersive wave model in the spherical coordinate system for basin-wide evolution and coastal run-up of tsunamis and discusses the implementation of a two-way grid-nesting scheme to describe the wave dynamics at resolution compatible to the physical processes. The depth-integrated model describes dispersive waves through the non-hydrostatic pressure and vertical velocity, which also account for tsunami generation from dynamic seafloor deformation. The semi-implicit, finite difference model captures flow discontinuities associated with bores or hydraulic jumps through the momentum-conserved advection scheme with an upwind flux approximation. The two-way grid-nesting scheme utilizes the Dirichlet condition of the non-hydrostatic pressure and both the horizontal velocity and surface elevation at the inter-grid boundary to ensure propagation of dispersive waves and discontinuities across computational grids of different resolution. The inter-grid boundary can adapt to bathymetric features to model nearshore wave transformation processes at optimal resolution and computational efficiency. A coordinate transformation enables application of the model to small geographic regions or laboratory experiments with a Cartesian grid. A depth-dependent Gaussian function smoothes localized bottom features in relation to the water depth while retaining the bathymetry important for modeling of tsunami transformation and run-up. Numerical experiments of solitary wave propagation and N -wave run-up verify the implementation of the grid-nesting scheme. The 2009 Samoa Tsunami provides a case study to confirm the validity and effectiveness of the modeling approach for tsunami research and impact assessment.

The 25 October 2010 Mentawai, Indonesia earthquake (Mw 7.8) ruptured the shallow portion of the subduction zone seaward of the Mentawai islands, off‐shore of Sumatra, generating 3 to 9 m tsunami run‐up along southwestern coasts of the Pagai Islands that took at least 431 lives. Analyses of teleseismic P, SH and Rayleigh waves for finite‐fault source rupture characteristics indicate ∼90 s rupture duration with a low rupture velocity of ∼1.5 km/s on the 10° dipping megathrust, with total slip of 2–4 m over an ∼100 km long source region. The seismic moment‐scaled energy release is 1.4 × 10−6, lower than 2.4 × 10−6 found for the 17 July 2006 Java tsunami earthquake (Mw 7.8). The Mentawai event ruptured up‐dip of the slip region of the 12 September 2007 Kepulauan earthquake (Mw 7.9), and together with the 4 January 1907 (M 7.6) tsunami earthquake located seaward of Simeulue Island to the northwest along the arc, demonstrates the significant tsunami generation potential for shallow megathrust ruptures in regions up‐dip of great underthrusting events in Indonesia and elsewhere.

The massive tsunami generated by the 11 March 2011 Tohoku earthquake (Mw 9.0) was widely recorded by GPS buoys, wave gauges, and ocean bottom pressure sensors around the source. Numerous inversions for finite‐fault slip time histories have been performed using seismic and/or geodetic observations, yielding generally consistent patterns of large co‐seismic slip offshore near the hypocenter and/or up‐dip near the trench, where estimated peak slip is ∼60 m. Modeling the tsunami generation and near‐field wave processes using two detailed rupture models obtained from either teleseismic P waves or high‐rate GPS recordings in Japan allows evaluation of how well the finite‐fault models account for the regional tsunami data. By determining sensitivity of the tsunami calculations to rupture model features, we determine model modifications that improve the fit to the diverse tsunami data while retaining the fit to the seismic and geodetic observations.