Conference on Tsunami and Nonlinear Waves How well can we predict tsunamis?

Harvey Segur, University of Colorado, USA

Overall objective

Identify and specify design parameters for an early warning system for tsunamis.

a) System must be reliable, fast enough to provide time to respond, and must minimize both unidentified tsunamis and false alarms.

b) Compare with Pacific Tsunami Warning System.

General picture of tsunami dynamics, near India

• Initiated by an underwater seismic event

– earthquake or landslide

– not by tropical cyclones (which can

create a storm surge)

General picture of tsunami dynamics, near India

• Initiated by an underwater seismic event – earthquake or landslide – not by tropical cyclones

• For short times, water wave has: – small amplitude (compared to fluid depth) – long wavelength (compared to fluid depth) – surface shape might be 1-D or 2-D

Linear wave equation, with variable depth

Linear wave equation,variable depth

In 1-D,

Note: mass is conserved.

€

z = ζ (x, y, t)

€

(u,v) →

€

z = −h(x,y)

€

∂tζ + ∂x (u ⋅h) = 0,

∂t (u ⋅h) + c 2∂xζ = 0,

c 2 = g ⋅h(x)

€

g↓

Linear wave equation,variable depth

In 1-D,

Exact massconservation

€

z = ζ (x, y, t)

€

(u,v) →

€

z = −h(x,y)

€

∂tζ + ∂x (u ⋅h) = 0,

∂t (u ⋅h) + c 2∂xζ = 0,

c 2 = g ⋅h(x)

€

g↓

€

∂tζ + ∂x u ⋅dz = 0−h

ζ

∫

Implications for early warning system

• Design measuring system to provide initial data for linear wave equation.

• Need quick information (10-20 minutes)

• Need accurate information only for crucial quantities.

• Which quantities are crucial?

Crucial measured quantities(make a list)

• Time and location of epicenter

• Spatial extent of rupture (?)

• Volume of displaced water

• Other?

Models for tsunami propagation and evolution

• Linear wave equation for short times• And then what?• 2 cases

– KdV-type evolution for long times

– Wave equation fails in shallow

coastal waters

Case 1: KdV model (or KP, or Boussinesq, or …)

• Includes nonlinearity, frequency dispersion and (perhaps) 2-D surface patterns

• Requires (nearly) uniform depth• Requires long distances

with , need propagation distance

€

ε =amplitude

depth

€

D ~1

ε⋅depth

Case 2: Failure of linear wave equation

• As a long wave with small amplitude enters shallow coastal waters, the solution contradicts the assumptions of model:– wavelengths shorten – wave amplitude grows, while fluid

depth shrinks

What is new governing equation?

Case 1: KdV wave evolution

Experimental equipment (Hammack)

References: Hammack & Segur, 1974, 1978a,b

KdV - Negative initial

data

(no solitons)

KdV -Positive

initial wave

(solitons!)

KdV - mixed initial datawith wave volume = 0

Case 2: wave evolution near shore

• Leading-order eq’ns

€

ζ (x, t)

€

h(x)

€

u(x, t) →

€

∂tζ + ∂x (u ⋅h) = 0,

€

∂t (u ⋅h) + c 2∂xζ = 0,

€

c 2(x) = g ⋅h(x)

€

g↓

Case 2: wave evolution near shore

• Leading-order eq’ns

• 2 regimes

€

ζ (x, t)

€

h(x)

€

u(x, t) →

€

∂tζ + ∂x (u ⋅h) = 0,

€

∂t (u ⋅h) + c 2∂xζ = 0,

€

c 2(x) = g ⋅h(x)

€

→ x

€

|

|

x = 0

€

x = L

€

z = −H

€

g↓

€

h(x) = H,

€

x > L

€

h(x) = s ⋅ x,

€

0 < x < L

Case 2: wave evolution near shore

Equations:

For x > L, ,

incoming outgoing (known) (unknown)

€

∂tζ + ∂x (u ⋅h) = 0,

€

∂t (u ⋅h) + c 2∂xζ = 0,

€

c 2(x) = g ⋅h(x)

€

ζ (x, t) = G(t +x

c∞

) + F(t −x

c∞

)€

c = c∞ = gH

Case 2: wave evolution near shore

Equations:

For 0 < x < L, .

€

∂tζ + ∂x (u ⋅h) = 0,

€

∂t (u ⋅h) + c 2∂xζ = 0,

€

c 2(x) = g ⋅h(x)

€

c 2 = gs ⋅ x,

€

∂t2ζ = ∂x (gsx ⋅∂xζ )

Case 2: jump to the (partial) answer

Equations:

For 0 < x < L, .

As x –> 0, ζ(x,t) approaches a self-similar form:

For p > 0, ζ(x,t) blows up even if

Z() is bounded!

€

∂tζ + ∂x (u ⋅h) = 0,

€

∂t (u ⋅h) + c 2∂xζ = 0,

€

c 2(x) = g ⋅h(x)

€

c 2 = gs ⋅ x,

€

∂t2ζ = ∂x (gsx ⋅∂xζ )

€

ζ (x, t) = (gs

x)p ⋅Z(η ),

€

=tgs

x

€

d2Z

dη 2= p2Z + pη

dZ

dη+

η

4

d

dη(η

dZ

dη).

Case 2: detailed analysis

Equations:

For 0 < x < L, .

Set

€

∂tζ + ∂x (u ⋅h) = 0,

€

∂t (u ⋅h) + c 2∂xζ = 0,

€

c 2(x) = g ⋅h(x)

€

c 2 = gs ⋅ x,

€

∂t2ζ = ∂x (gsx ⋅∂xζ )

€

⇒

€

∂x (x∂xˆ ζ ) +

ω2

gsˆ ζ = 0.

€

ζ (x, t) = ˆ ζ (x,ω) ⋅e iωt

−∞

∞

∫ dω.

Case 2: analysis

Equations:

For 0 < x < L, .

Set

Set ,

€

∂tζ + ∂x (u ⋅h) = 0,

€

∂t (u ⋅h) + c 2∂xζ = 0,

€

c 2(x) = g ⋅h(x)

€

c 2 = gs ⋅ x,

€

⇒

€

∂x (x∂xˆ ζ ) +

ω2

gsˆ ζ = 0.

€

y = 2 |ω |x

gs

€

ˆ ζ (x,ω) = ˜ ζ (y)

€

⇒

€

d2 ˜ ζ

dy 2+

1

y

d ˜ ζ

dy+ ˜ ζ = 0

€

ζ (x, t) = ˆ ζ (x,ω) ⋅e iωt

−∞

∞

∫ dω.

€

∂t2ζ = ∂x (gsx ⋅∂xζ )

Aha!Bessel’s eq’n - order 0

€

d2 ˜ ζ

dy 2+

1

y

d ˜ ζ

dy+ ˜ ζ = 0

Aha!Bessel’s eq’n - order 0

Wavelengths shorten as x –> 0

€

d2 ˜ ζ

dy 2+

1

y

d ˜ ζ

dy+ ˜ ζ = 0

Linear, long-wave modelin variable depth - Conclusions

For all real , with ,

Features:

(i) (0) encodes the wave volume; (0) = 0;

(ii) Y0(y) is singular at y = 0 ’(0) = 0;

iii) Find self-similar solution, with p = 3/2 blow up!

€

y = 2 |ω |x

gs

€

ˆ ζ (x,ω) = α (ω)J0(y) + β (ω)Y0(y),€

ζ (x, t) = ˆ ζ −∞

∞

∫ (x,ω) ⋅e iωtdω

Proposal #1

Q9: As a tsunami begins to evolve into a large-amplitude wave near shore, what controls the wave evolution?

A: (i) for a long nonlinear wave

(ii) conservation of wave volume

(iii) wave reflection by the changing bathymetry.

€

c = g(h(x) + ζ (x, t))

Proposal/Question #2

For an early warning system that acts quickly enough to be effective, details of the nonlinear, complicated evolution near shore might be less valuable than the time saved by not computing this evolution. (Parameterize it instead.)

Questions & Answers(tsunami generation)

Q1: What causes tsunamis?A: Underwater seismic events, with significant

movement of the sea bed- earthquakes- underwater landslides- NOT wind, storms or tropical cyclones

(But a tropical cyclone can create a storm surge.)

Questions & Answers(tsunami generation)

Q2: Not all underwater earthquakes create tsunamis. What information about an earthquake determines whether it generates a tsunami?

A: • The time and place of the earthquake.• Claim: The volume of water displaced by

the earthquake is the next most important piece of information about the quake.

(To be demonstrated.)

Questions and Answers(tsunami generation)

Q3: Are there “immediate” seismic measurements of the earthquake that determine the volume of water displaced?

A: First answer (i) A “strike-slip fault” displaces very little water. (ii) A “thrust fault” or a “normal fault” can

displace much more water. (iii) From historical records, geologists can

classify known fault regions into one of these types. (Is this reliable?)

Questions and Answers(tsunami generation)

Q3: Are there “immediate” seismic measurements of the earthquake that determine the volume of water displaced?

A: Roger.Bilham@colorado.edu answers:• The displacement of the sea floor can be determined

quite accurately from the magnitude, focal mechanism and depth of the earthquake.

• The determination takes about 25 minutes after the mainshock - the time taken for seismic waves to travel to the world's seismic array, plus about 10 seconds of computer time.

• Start with the seismic determination and then confirm the amplitude of the tsunami with a tide gage measurement.

Questions and Answers(tsunami propagation)

Q4: If a tsunami occurs, where and when will it reach shore?

A: Simplest approximate answer

Where? If there is a straight line from the epicenter of the quake to your beach, then you will experience some part of the tsunami. (Sufficient, not necessary)

When? Until the tsunami reaches shallow coastal waters, locally.

Total time: along each path.€

c = gh(x)

€

T(x) =ds

gh(s)0

x

∫

Questions and Answers(tsunami propagation - away from shore)

Q4: If a tsunami occurs, where and when will it reach shore?

A: More accurate answer: Either solve the wave eq’n in 2-D, or use geometric

optics with a spatially varying “index of refraction”,

. Along each curve from the epicenter to your beach,

the total propagation time along that path is:

Minimize this for the warning system. Note that the tsunami can diffract around objects.€

T(x) =ds

gh(s)0

x

∫€

gh(x)

An example of tsunami diffraction

The tsunami, on the western side of Sri Lanka

Questions and Answers(tsunami propagation - away from shore)

Q5: The earthquake fault of Dec. 2004 occurred over a 900 km-long curve. Where and when does the tsunami reach shore?

A: Draw curves from each point along the fault to your beach. Along each curve,

. Repeat the previous calculation, possible with different starting times.

(This is Huygens’ Principle from optics!)€

c = gh(x)

Questions and Answers(tsunami propagation)

Q6: The formula is wrong in shallow coastal waters, where the wave changes its shape and its amplitude grows large. How to compute an arrival time that builds in this effect?

A: The objection is valid. Correcting for the evolution in shallow coastal waters is important to predict the size and shape of the wave that arrives, but it might not matter in estimating the arrival time (only).

€

c = gh(x)

Questions and answers(tsunami propagation)

Q7: In 2004, the westward propagating tsunami reached India in just under 2 hours. The eastward propagating tsunami reached Bandeh Aceh in a few minutes. Would an early warning system have helped in Bandeh Aceh?

A. I don’t see how. 15-25 minutes is required to receive the seismic information on the world’s seismic array.

Questions and Answers(What happens near shore?)

Q8: The linear,

long-wave model

assumes both: , .

As the wave approaches shore, , and a(x,t), (x,t) both change. What happens to the assumptions underlying the linear, long-wave model?

Claim : Typically, both assumptions fail. Even though , grows.

(To be demonstrated.)

€

a

€

h€

€

a << h

€

h2 << λ2

€

h(x) → 0

€

h(x) → 0

€

h2

λ2

Issues to be resolved

• How important is the wave volume?

• In shallow coastal waters, how does the linear, long-wave model break down?

• How does tsunami evolve in shallow coastal waters?

Significance of wave volume

• Recall– KdV: constant fluid depth

wave eq’n for short times KdV for longer times

– tsunami: variable depth wave eq’n for short times, away from shore

evolution (somehow?) in shallow coastal waters

• Wave evolution in KdV regime(Hammack & Segur, 1974, 1978a,b)