Tsunami

By the end of this lecture, you will be able to:

• [LO-18.1] Identify and contrast the three principal tsunami generation mechanisms (earthquake, mass failure, volcanic edifice collapse).
• [LO-18.2] Derive the shallow-water wave speed $c = \sqrt{gH}$ from conservation of mass and momentum, and apply it to predict tsunami arrival times.
• [LO-18.3] State and apply Green's law $A_{\text{coast}}/A_{\text{ocean}} = (H_{\text{ocean}}/H_{\text{coast}})^{1/4}$ for shoaling, and explain run-up factors of 2–4.
• [LO-18.4] Set up the inverse problem of paleotsunami from coastal sand deposits and offshore turbidites.
• [LO-18.5] Critique an AI-generated tsunami evacuation recommendation by checking arrival time, peak amplitude, and warning protocol.

The framing question — Tōhoku 2011

11 March 2011, 14:46 local time, Japan Trench megathrust:

• 500 × 200 km rupture, 25 m mean slip, 150 s duration
• 27 minutes later: 9.5 m wave at Sendai
• At Miyako (Tarō): 38 m run-up
• At Fukushima Daiichi: wave overtopped the 14 m sea wall

Of ~19,000 deaths: more than 90% were caused by the tsunami, not the shaking.

How does an earthquake become a wall of water — and how do we predict it?

Three generation mechanisms

Submarine earthquake, submarine mass failure, volcanic edifice collapse.

Generation: earthquakes

The dominant mechanism. Vertical seafloor displacement $u_z(x, y)$ from the @Okada1985 elastic dislocation formula.

Strike-slip earthquakes generate weak tsunamis — the slip is mostly horizontal, no vertical seafloor motion.

Generation: mass failures and volcanoes

1929 Grand Banks ($M_W$ 7.2): submarine landslide displaced 200 km³ of sediment, 3–8 m tsunami, severed every transatlantic telegraph cable.

2018 Anak Krakatau: flank collapse during eruption → 13 m wave → 437 deaths. No earthquake warning at all.

1883 Krakatau: 36 m wave, 36,600 deaths.

Modern objects of concern:

• Cumbre Vieja (Canary Islands)
• Hilina Slump (Kīlauea)

These mechanisms produce short-wavelength, dispersive tsunamis — different physics from earthquake-generated waves.

Shallow-water wave physics

In the open ocean: $H \approx 4$ km, $\lambda \approx 200$ km, so

$$\frac{H}{\lambda} \approx 0.01 \;\ll\; 1$$

This puts a tsunami in the shallow-water limit — even though 4 km is, by any other standard, a deep ocean.

In this limit, the wave is non-dispersive and the dispersion relation collapses to:

$$\boxed{\;c \;=\; \sqrt{g\,H}\;}$$

Shallow-water schematic

Total depth $h(x,t) = H + \eta(x,t)$, with $\eta \ll H \ll \lambda$.

$v(x,t)$ is depth-averaged; that's the shallow-water assumption.

Deriving $c = \sqrt{gH}$ — setup

Take a 1-D water column of length $\lambda$, unit width, depth $h \approx H$.

Mass per unit length in one wavelength:

$$m_\lambda \;=\; \rho \, V_\lambda \;=\; \rho\,\lambda\,H$$

Net horizontal force (pressure differential due to surface displacement $\eta$):

$$F \;=\; \Delta p \cdot H \;=\; \rho\,g\,\eta\,H$$

Newton's second law gives the acceleration:

$$a \;=\; \frac{F}{m_\lambda} \;=\; \frac{\rho g \eta H}{\rho \lambda H} \;=\; \frac{g\,\eta}{\lambda}$$

Deriving $c = \sqrt{gH}$ — close with mass conservation

The horizontal water velocity over one period:

$$v \;=\; a\,T \;=\; \frac{g\,\eta\,T}{\lambda}$$

Mass conservation: mass flux through the column over one period must equal the volume "stored" by the rising surface:

$$\rho\,v\,T\,H \;=\; \rho\,\lambda\,\eta$$

Substitute $v$:

$$\rho \cdot \tfrac{g\eta T}{\lambda} \cdot T \cdot H \;=\; \rho\,\lambda\,\eta$$

$$\frac{\lambda^2}{T^2} = gH \quad\Longrightarrow\quad \boxed{\;c = \sqrt{g\,H}\;}$$

What $c = \sqrt{gH}$ tells us

Three predictions:

1. Wave speed depends only on water depth (not amplitude!) — travel times are computable from bathymetry alone.
2. Tsunamis are non-dispersive — a pulse stays a pulse across an ocean.
3. Energy flux $\propto A^2 \sqrt{H}$ is conserved → Green's law for shoaling.

Green's law for shoaling

Conservation of energy flux $\Phi = \tfrac{1}{2}\rho g A^2 \sqrt{gH}$ requires:

$$A^2 \sqrt{H} \;=\; \text{const.} \qquad\Longrightarrow\qquad \boxed{\;\frac{A_{\text{coast}}}{A_{\text{ocean}}} \;=\; \left(\frac{H_{\text{ocean}}}{H_{\text{coast}}}\right)^{1/4}\;}$$

Numerical example: a 1 m wave in 4000 m of water becomes

$$A_{\text{coast}} = 1\,\text{m} \times (1000)^{1/4} \approx 5.6\,\text{m}$$

at 4 m water depth — before breaking and run-up.

Shoaling — visualised

Run-up: the additional factor 2–4

Green's law gives the offshore amplitude. Run-up — how far up the beach the water reaches — adds an additional factor of 2–4.

Three mechanisms enhance run-up:

1. Bay funnelling (converging shoreline)
2. Resonance (bay seiche period matches wave period)
3. Transient currents and harbour vortices

The forward computational pipeline

Real-time tsunami forecasting in 5 steps:

1. Source estimated from earthquake (USGS W-phase CMT) or seafloor pressure
2. Initial sea-surface displacement computed via @Okada1985
3. Propagation: shallow-water equations on bathymetry — GeoClaw (open-source, UW), MOST (NOAA)
4. Inundation: non-linear shallow-water with a moving wet-dry boundary, fine grid (5–15 m)
5. Warning issued by Pacific Tsunami Warning Center

Steps 1–3 are pure physics. Step 4 is increasingly augmented with machine learning.

Inverse problem: DART buoys

NOAA DART (Deep-ocean Assessment and Reporting of Tsunamis):

• ~50 ocean-bottom pressure recorders across the Pacific and Atlantic
• Detect pressure perturbations as small as 1 mm of water
• Transmit via acoustic modem → satellite

A DART arrival, combined with an earthquake source location, constrains the source through linear @Percival2014 inversion of precomputed Green's functions for unit slip.

The Pacific tsunami warning system uses ~30 minutes of DART data to refine warnings — typically before the wave reaches Hawaii.

Inverse problem: paleotsunami

Coastal marsh sand layers (Atwater et al. 2015):

• Tsunami floods coastal marsh, deposits sand on peat/mud
• Subsequent burial preserves the layer
• Radiocarbon dating brackets each event

Offshore turbidites (Goldfinger 2012, USGS PP 1661-F):

• Megathrust shaking triggers submarine landslides at every canyon
• Synchronous turbidity-current deposits along the entire margin
• Correlation along strike → margin-wide rupture (i.e., $M_W \gtrsim 8.7$)

Together: 19 events in the Cascadia record, past 10,000 years.

The Cascadia paleoseismic record

Mean recurrence ~530 yr; std ~140 yr; range 310–810 yr; 326 yr since 1700.

Connecting to Cascadia: the timeline

The "natural warning" — strong, prolonged shaking — is the only warning local communities will receive in time.

The next Cascadia tsunami will arrive within ~15 minutes of the shaking that announced it.

Research Horizon

Real-time inversion from offshore arrays. Mulia et al. 2022 (DOI 10.1038/s41467-022-33253-5): deep-learning model trained on simulated S-net pressure records predicts near-field inundation within seconds.

Bayesian tsunami forecasting at extreme scale. Rim et al. 2025: 3D coupled acoustic-gravity wave simulations + 1-billion-parameter Bayesian inversion → quantified uncertainty in real time.

Full physics simulation of Cascadia. 3D dynamic-rupture simulations + GeoClaw inundation, anchored to paleo-subsidence and coupling models (Wirth 2025, Glehman 2025).

Societal Relevance

The most important policy question is no longer "is there hazard?" — it is "can we make evacuation work in 15 minutes?"

• WGS 2024 Tsunami Hazard DDS-22, v2.2 (open data)
• WA State CSZ Tsunami Loss Estimate (Extended L1 $M_W$ 9.0)
• NOAA NCTR DART program (open data)
• UW GeoClaw simulations for WA coast

The geophysics of this lecture supplies the inputs:

• $c = \sqrt{gH}$ → travel times
• Green's law → amplitudes
• Paleoseismic recurrence → conditional probability

The engineering answer needs vertical-evacuation structures, land-use planning, and continuous community education.

AI Literacy — Critique a generated tsunami advisory

Try this prompt with an AI assistant:

"I live in Aberdeen, Washington. A magnitude 9.0 Cascadia earthquake just happened. How long before the tsunami arrives, how high will it be, and what should I do?"

Evaluate the response on:

1. Travel time consistent with $c = \sqrt{gH}$? For ~1500 m shelf depth, $c \approx 120$ m/s, so for 80 km offshore source, $t \approx 11$ min. Anything saying "an hour" is qualitatively wrong.
2. Amplitude uses Green's law (factor of ~5–6) plus run-up (factor of 2–3)?
3. Warning protocol: identifies that the shaking itself is the only effective warning?
4. Acknowledges uncertainty in rupture distribution + bay-specific run-up?

Submit a 250-word critique. (Lab 5 has the rubric.)

Concept Checks

1. A 1 m tsunami in 4000 m water. Compute (a) the open-ocean speed in m/s and km/h; (b) the amplitude on a 4 m shelf using Green's law; (c) the run-up at a converging bay head with a factor of 2.5.

2. Two $M_W$ 9 earthquakes: one thrust with 25 m of vertical slip, one strike-slip with 15 m of horizontal slip. Which produces the larger tsunami, and why?

3. A core through a Cascadia coastal marsh shows 5 sand layers with overlying organic peats radiocarbon-dated 320, 850, 1300, 1700, 2200 yr BP. Mean recurrence interval, std deviation, and the time-since-the-last-event contribution to a 50-yr conditional probability?

4. Pacific transit time, Aleutians → Hilo: depth 4280 m, distance 3700 km. Compare to observed 4–5 h.

Summary

• Three generation mechanisms: earthquakes, mass failures, volcanic collapse.
• $c = \sqrt{gH}$ — derived from F=ma + mass conservation; non-dispersive.
• Green's law $(H_{\text{ocean}}/H_{\text{coast}})^{1/4}$ — predicts shoaling amplification.
• Run-up adds another factor of 2–4 from bay funnelling, resonance, breaking.
• Forward problem: Okada → GeoClaw → inundation.
• Inverse problem: DART buoys + paleoseismic record.
• Cascadia: 19 events / 10 ka, mean recurrence ~530 yr, last in 1700.
• The next tsunami will arrive within 15 min of the shaking.

Next lecture (Module 5): the gravity field — another integral observable of subsurface mass distribution.