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

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

• Harvey Segur, University of Colorado, USA

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Overall objective

• Identify and specify design parameters for an
  early warning system for tsunamis.
• System must be reliable, fast enough to provide
  time to respond, and must minimize both
  unidentified tsunamis and false alarms.
• Compare with Pacific Tsunami Warning System.

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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)

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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

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Linear wave equation,variable depth

• In 1-D,
• Note mass is conserved.

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Linear wave equation,variable depth

• In 1-D,
• Exact mass
• conservation

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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?

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Crucial measured quantities(make a list)

• Time and location of epicenter
• Spatial extent of rupture (?)
• Volume of displaced water
• Other?

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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

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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

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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?

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Case 1 KdV wave evolution

• Experimental equipment (Hammack)
• References Hammack Segur, 1974, 1978a,b

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• KdV -
• Negative initial
• data
• (no solitons)

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• KdV -
• Positive
• initial wave
• (solitons!)

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KdV - mixed initial datawith wave volume 0

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Case 2 wave evolution near shore

• Leading-order eqns

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Case 2 wave evolution near shore

• Leading-order eqns
• 2 regimes

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Case 2 wave evolution near shore

• Equations
• For x gt L, ,
• incoming outgoing (known) (unknown)

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Case 2 wave evolution near shore

• Equations
• For 0 lt x lt L, .

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Case 2 jump to the (partial) answer

• Equations
• For 0 lt x lt L,
• .
• As x gt 0, ?(x,t) approaches a self-similar form
• For p gt 0, ?(x,t)
• blows up even if
• ? Z(?) is bounded!

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Case 2 detailed analysis

• Equations
• For 0 lt x lt L, .
• Set

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Case 2 analysis

• Equations
• For 0 lt x lt L, .
• Set
• Set ,

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Aha!

• Bessels eqn - order 0

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Aha!

• Bessels eqn - order 0
• Wavelengths shorten as x gt 0

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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!

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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.

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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.)

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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.)

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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.)

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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?)

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Questions and Answers(tsunami generation)

• Q3 Are there immediate seismic measurements
  of the earthquake that determine the volume of
  water displaced?
• A Roger.Bilham_at_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.

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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.

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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 eqn 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.

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An example of tsunami diffraction

• The tsunami, on the western side of Sri Lanka

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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!)

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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).

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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 dont see how. 15-25 minutes is required
  to receive the seismic information on the worlds
  seismic array.

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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.)

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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?

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Significance of wave volume

• Recall
• KdV constant fluid depth
• wave eqn for short times
• KdV for longer times
• tsunami variable depth
• wave eqn for short times, away from shore
  evolution (somehow?) in shallow coastal waters
• Wave evolution in KdV regime
• (Hammack Segur, 1974, 1978a,b)