Tsunami Shallow-Water Wave Model (2D)

Wave & Shelf Parameters

Offshore wave height H₀ 2.0 m

Wave period T 600 s

Shelf depth h₀ 60 m

Beach slope β 3.0°

Live Readouts (measured from the solver)

Calm — building up

Wave height at monitor point

– m

Depth at monitor point

– m

Local shallow-water speed c=√(gH)

– m/s

Max vertical runup so far

– m

Shoreline vs still-water line

– m

Simulated time

0 s

How this 2D model differs from the 3D version

The 3D companion simulator applies closed-form formulas (Green's Law, a fixed breaker index, the Synolakis runup law) directly to a kinematic wave shape. This 2D version instead numerically solves the actual governing equations — the nonlinear shallow-water (Saint-Venant) equations, the same PDE system used by real tsunami models such as GeoClaw, COMCOT and MOST:

∂H/∂t + ∂q/∂x = 0                         (mass)
∂q/∂t + ∂/∂x[q²/H + ½gH²] = −gH·∂z_b/∂x   (momentum)
H = total water depth, q = H·u = discharge, z_b(x) = bed elevation

It is integrated on a finite-volume grid with a Rusanov (local Lax–Friedrichs) flux and an Audusse hydrostatic reconstruction for the bed-slope source term, which keeps a flat "lake at rest" exactly at rest despite the shelf/beach bathymetry step — a well-known failure mode of naive shallow-water solvers. A simple wetting/drying threshold lets the shoreline itself move as a genuine emergent output of the simulation.

Because the scheme is a shock-capturing finite-volume method, shoaling amplification, wave breaking (a steep bore-like front forming in H) and runup all emerge from the numerics instead of being imposed — this is a different computational mechanism from the 3D version's closed-form kinematics, though both target the same physical phenomenon and were checked to agree with Green's Law and the shallow-water wave speed c=√(gh) in the pre-breaking regime.

H₀, T — offshore wave height and period, applied as a characteristic (non-reflecting) boundary forcing at the left edge of the grid.
Shelf depth h₀ — still-water depth of the flat shelf before the slope begins.
Beach slope β — the single planar slope carrying the profile from the shelf edge, through sea level, up onto dry land.

Real-world relevance: this is the same class of numerical method — a well-balanced shallow-water finite-volume scheme — that coastal engineers run at production scale for tsunami inundation and hazard mapping.

Sea surface η(x,t) Steep front (breaking) Seafloor & beach z_b(x) Monitor point