Tsunami shoaling

Why does a wave that a ship at sea never notices rise into a wall of water at the shore?

A crew in deep water, directly over a passing tsunami, records nothing. Hours later the same wave destroys a coastline. The instinctive reading is that the wave grew, as if something fed it — but nothing offshore adds energy to it, so whatever changes is a redistribution of what it already had.

The question sharpens: what is the wave holding constant, and what is it allowed to trade?

Start with why the ship feels nothing, because that is measurable rather than mysterious. A tsunami's period is long — minutes to about an hour, set by the seafloor displacement that made it. Take twenty minutes. In water four kilometres deep its speed is the square root of gravity times depth: the square root of 9.8 times 4000, about 198 metres per second or 700 kilometres per hour. Twenty minutes at that speed is a wavelength of 198 times 1200 seconds, near 240 kilometres. Half a metre of height spread over 240 kilometres is a surface slope of one in half a million; the ship rises half a metre over ten minutes and calls it swell.

Now pause on what that calculation smuggled in. Why the shallow-water speed in four kilometres of ocean? Because "shallow" means shallow compared with the wavelength, and 4 km against 240 km is shallow by any measure. That is what makes a tsunami a different animal from a wind wave: a ten-second swell has a wavelength of a couple of hundred metres and its motion dies out within about half a wavelength of the surface, while a tsunami moves the entire column, seabed to surface, across an ocean.

Two things are then nearly fixed. The period is fixed — the source set the clock, and no stretch of coast can change how often crests arrive. And along a ray, the energy flux is approximately conserved, since dissipation over a smooth deep basin is small over a few hours.

Follow those two through. Energy per unit area of surface goes as the square of the amplitude. Flux is that energy times the speed at which it travels, which for a long wave is the wave speed itself, the square root of gravity times depth. So amplitude squared times the square root of depth stays constant, meaning amplitude scales as depth to the minus one quarter. Put in the numbers: going from 4000 metres to 10 metres is a depth ratio of 400, and 400 to the one-quarter power is 4.47. Half a metre becomes about 2.2 metres.

The wavelength does something more dramatic. Speed falls from 198 metres per second to the square root of 98, near 9.9 — a factor of twenty. With the period unchanged, the wavelength must fall by the same factor, from 240 kilometres to about 12. The front of the wave is in slower water than the back, so the train compresses. That, physically, is what shoaling is: the same energy packed into a fifth of its former length, forced to show itself as height.

And here honesty is required, because 2.2 metres is not the wall of water in the photographs. Green's law above is a linear, non-breaking, straight-in estimate, and its shortfall against observed run-up is real. Four things fill the gap: refraction, which bends rays toward banks and headlands and concentrates the flux into a narrower front; resonance in a bay whose natural period is near the wave's; nonlinear steepening in the last shallow kilometres, which can turn the front into a bore; and run-up itself, the conversion of forward momentum into height as the water climbs the land. All four depend on local bathymetry and coastline shape, and I will not quote a general multiplier, because published run-up ratios differ enormously between sites within a single event.

What convinces me the core of this is right is a prediction that could fail daily and does not. Arrival-time forecasts across an entire ocean are computed from bathymetry alone using the square root of gravity times depth, and they land within minutes on coasts thousands of kilometres away. If the speed law were wrong, those forecasts would be hours out. The complementary falsifier is the period: compare a deep-ocean bottom-pressure record with a coastal gauge for the same event, and the period must match while the amplitude grows. Growth with an altered period would mean this is not one wave train being transformed but something else entirely.

Two neighbours are worth setting straight against this one. A rogue wave and a tsunami are opposite kinds of event: a rogue wave is a rare draw from the ordinary statistics of a wind sea, unremarkable physics producing an extreme value, essentially unforecastable. A tsunami is a single deterministic wave whose whole transformation — speed, arrival time, wavelength — is computable from bathymetry before it arrives. The old name "tidal wave" is wrong about the cause but revealing about the physics, since tides are also long waves that feel the whole depth.

A signal-and-noise problem also hides in the offshore measurement. Half a metre is far smaller than the swell and the tide, so how is it detected at all? By seabed pressure sensors, which sit below the depth wind-wave motion reaches, and by subtracting a tide whose periods are known in advance — exploiting exactly the property identified above, that the tsunami is the only signal moving the whole column at a period of minutes.

The genuine uncertainty is recurrence, not mechanism. Instrumental records cover roughly a century while the interval between great tsunamis on a coast is often several centuries, so recurrence estimates rest on buried sand sheets in coastal marshes and historical accounts — and on telling those deposits from storm deposits, which is not always clean.

• Why the leading edge sometimes withdraws the sea before the first crest arrives, and what that says about the motion at the source.

Every term the collection defines is gathered in the glossary.