Tsunami Early Warning Lab

Why a tsunami is a shallow-water wave

The wavelength of a tsunami (tens to hundreds of kilometres) is much larger than the ocean depth, so the water moves as a whole column and the wave speed is c = √(g h): about 200 m/s (720 km/h) over a 4 km deep ocean and only 16 m/s over a 25 m shelf. The linear shallow-water equations are ∂η/∂t = -∂(h u)/∂x and ∂u/∂t = -g ∂η/∂x, where η is the surface height and u the depth-averaged velocity.

Slowing and growing

As the water becomes shallower the front slows and the energy flux ½ ρ g η2 c is squeezed, so the height grows by Green's law η ∝ h-1/4. In open water the wave may be only 0.5 m tall, which a ship does not notice; on a shelf it grows several times, and breaking and run-up add more.

Warning time

A bottom-pressure buoy detects the wave at tb = ∫ dx/c. After an analysis and alert delay the warning goes out, and the lead time is tcoast - tb - delay. A buoy placed farther out gives more time, but must be close enough to the source to see it quickly.

Honest simplifications

• One-dimensional transect, linear equations, no spreading loss, so amplitudes do not decay with distance as a real circular wave does.
• The source height and width are a schematic function of magnitude, not a fault model; the coast is a reflecting wall without run-up or inundation.
• No dispersion and no bottom friction; real tsunamis show weak dispersion over long distances.

This is an educational model, not a forecasting tool.

• Load Pacific megathrust and compare the buoy trace with the coast trace; the later pulse is the reflection from the wall.
• Move the buoy farther from the source and watch the lead-time curve in the last graph.
• Widen the shelf: the wave slows for longer so the arrival is later, while Green's law lifts the height.
• Load Slow alert, late buoy to see the lead time fall below zero.