Governing equations
Tsunamis are long waves (wavelength 10–300 km, ocean depth 4 km), so the water column moves as one slab. The nonlinear shallow-water equations used here are
h_t + (D u)_x + (D v)_y = 0
u_t + u u_x + v u_y + g ∂(η)/∂x + g n² |U| u / D^(4/3) = 0 (v likewise)
with free surface η above mean sea level, bed elevation z, total depth D = η − z, gravity g = 9.81 m/s² and Manning roughness n. The linear limit gives the wave speed c = √(g h): about 200 m/s (720 km/h) at 4 km depth, 31 m/s at 100 m, 10 m/s at 10 m.
Numerics
- Staggered (Arakawa C) grid, forward–backward time stepping (stable for cΔt/Δx < 1/√2; the step is chosen from the live fastest signal speed, CFL 0.45), first-order upwind advection, semi-implicit Manning friction.
- Continuity is in flux form with a donor-cell limiter, so water volume is conserved to round-off and depth never goes negative; a face carries flow only if D > 1 cm (wetting and drying: the shoreline moves across land).
- Nested one-way coupling: the 2 km ocean grid (256×256, 512 km) drives the 25 m coastal grid (240×240, 6 km) through Flather radiation boundaries, un = uref + √(g/D)(η − ηref), interpolated in space and time. Reflections leave the coastal grid; they are not fed back to the ocean grid.
- What the checks show: in Experiment A the solver's pulse travels about 0.8 % slower than √(g h) (numerical dispersion of the scheme, the same for every depth); in B the amplitude ratio is about 4 % above Green's law because the ramp has a finite slope, the wave is only weakly nonlinear and a little energy reflects; in D the simulated 5 cm front trails the ideal ray by several per cent on the 2 km grid (threshold detection plus numerical dispersion), and arrives slightly early at the 25 m grid.
- Travel-time map: fast-marching solution of the eikonal equation |∇T| = 1/√(g h), the same ray idea used by warning centres for an instant ETA.
Source
The seafloor displacement is a smooth, compact dipole (a cos² bump): uplift over the shallow part of the megathrust and subsidence landward of it, copied instantly to the sea surface (the rupture takes minutes, the wave crosses it in the same time, so this is a first approximation). Rupture length and width follow the subduction-interface scaling of Strasser, Arango and Bommer (2010); seismic moment M0 = 10^(1.5 Mw + 9.1) N m (Hanks and Kanamori, 1979); average slip D = M0/(μ L W) with μ = 30 GPa. Peak vertical seafloor motion is taken as 0.8 · 1.5 · D sin(15°) where the factor 1.5 mimics the large near-trench slip seen in the 2011 Tohoku event. This is a parameterisation, not an Okada (1985) elastic dislocation, and no Kajiura low-pass filter for the water depth is applied. Length is capped at 400 km to fit the domain (slip is increased to keep M0).
Shoaling and run-up
As depth decreases the wave slows and, to conserve energy flux, grows: Green's law H ∝ h^(−1/4). A 1 m open-ocean wave in 4 km of water becomes about 2.8 m at 55 m depth before it reaches the beach, and run-up can exceed that again (Synolakis 1987). Experiment B measures the amplitude ratio in the solver against this law.
What is simplified (honest list)
- No frequency dispersion: real ocean tsunamis disperse slightly over thousands of km; here the domain is only 500 km. No tides, no earth elasticity or gravity loading.
- 25 m cells cannot resolve single buildings, so buildings enter as increased Manning n and as coloured damage markers by flood depth. The town is generic and invented; no real place is modelled.
- Breaking waves, sediment, debris and structural failure are not simulated. Foam in the picture is a visual hint from the Froude number |u|/√(g D) and from the shallow swash edge.
- One-way nesting, first-order advection and the 2 km ocean grid damp very short waves; the numbers are for learning, not for evacuation planning.
- Wave heights for a given Mw depend strongly on slip distribution; the presets are illustrative scenarios, not forecasts. Warning-time numbers use a simple walking model.
References
Green (1838); Synolakis (1987) run-up of solitary waves; Titov and Gonzalez (1997), the MOST model behind NOAA forecasts; Imamura, the TUNAMI nested staggered-grid models; Strasser, Arango and Bommer (2010); Hanks and Kanamori (1979); Okada (1985); Kajiura (1963); DART deep-ocean buoys; reports of the 2010 Chile and 2011 Tohoku tsunamis for scale.