Slide 1

Saleh Abdalla

ECMWF, Reading, UK

TSUNAMI

Tsunami Slide 2

Main References:

G.F. Carrier (1971): “The Dynamics of Tsunamis”, in

“Mathematical Problems in the Geophysical Sciences”

by W.H. Reid (Ed.), American Mathematical Society,

157-187.

Z. Kowalik, W. Knight, T. Logan, and P. Whitmore

(2005): “Numerical Modeling of the Global Tsunami:

Indonesian Tsunami of 26 December 2004”, Science of

Tsunami Hazards, Vol. 23, No. 1, 40-56.

(freely available at: http://www.sthjournal.org/)

Tsunami Slide 3

Program of the lecture:

4. Tsunamis

4.1. Introduction- Tsunami main characteristics.- Differences with respect to wind waves.

4.2. Generation and Propagation- Basic principles.- Propagation characteristics.- Numerical simulation.

4.3. Examples- Boxing-Day (26 Dec. 2004) Tsunami.- 1 April 1946 Tsunami that hit Hawaii

Slide 4

INTRODUCTION

Tsunami Slide 5

Introduction:

Tsunami: A natural phenomenon of a series of waves generated when water in a lake or the sea is rapidly displaced on a massive scale.

The term “tsunami” comes from the Japanese language meaning “harbour (tsu) wave (nami)”.

Causes of tsunami:- Bottom movement (earthquake).- Submarine landslide.- Marine volcanic eruption.- Coastal landslide.- Meteorite.

Tsunami Slide 6

Introduction (cont’d):

Tsunami typical dimensions:- wave length of several hundreds of kilometres,- wave period of several minutes (up to an hour),- wave height of several tens of centimetres.

Tsunami is a shallow water with wave speed of(g is acceleration due to gravity and D is the water depth) typical tsunami speed in an ocean with 4000 m depth

is 720 km/hr (200 m/s)

Near the coast, the tsunami speed is much lower (e.g. for 40 m depth, the speed is 20 m/s) shoaling (piling-up) while approaching the coast wave height increases (can reach as high as 30 m!)

Dgc

Tsunami Slide 7

While Approaching the Coast:

213 km 23 km

10.6 km

10’s cm

metres

Tsunami Slide 8

Wave Classes and Scales:

1.010.0 0.03 3x10-3 2x10-5 1x10-5

Frequency (Hz)

Forcing:wind

earthquakemoon/sun

Restoring:gravity

surface tension Coriolis force

12

Tsunami Slide 9

Differences between Tsunami and Wind Waves (in deep ocean):

Tsunami Wind-Wave

Generation: earthquake wind

Restoration: gravity + Coriolis gravity

Typical length: several 100’s km few cm’s-few 100 m’s

Typical period: several minutes several seconds

Typical speed: few 100 m/s few 10 m/s

Water motion: almost linear motion orbital motion over a rather

over the whole water “thin” layer at surfacecolumn.

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TSUNAMI GENERATION

Tsunami Slide 11

Tsunami Generation

Slide 12

TSUNAMI PROPAGATION

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Tsunami Propagation:

Linear, inviscid theory governing the propagation of gravity waves: v =

Continuity (conservation of mass): xx + yy + zz = 0

Linearised boundary condition at the free surface:at y = 0 y(x,0,z,t) + tt(x,0,z,t) = 0

(gravity is in negative y).

Unit time satisfies: g/L = 1, L is the unit of length.Take: L as the water depth the bottom is at: y = -1.

Free surface displacement, , (x,z,t) = y(x,0,z,t)

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Tsunami Propagation (cont’d):

At the bottom: y(x,-1,z,t) = F(x,z,t)

F(x,z,t) is the prescribed vertical component of velocity of ground motion.

Adopt a simple “fundamental source”, F, as:Fo(x,z,t) = ½ ()-½ exp(-x2/4) g(t)

g(t) = is any function such that:

-0 dt g(t) = 1

g(t 0) = 0 g(t < -To) = 0 ,

To < time for wave to cross generation area.

Following discussion is a simulation for tsunami generated in the Gulf of Alaska during the earthquake of 28 March 1964.

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Tsunami Slide 16

Tsunami Propagation (cont’d):Area over which early wave motion takes place:

width = 4 ½ (in x-direction) length = 2 zo = 800 miles

Motion starts at -zo and

takes 5 min. to reach zo

Crest-line aligns with CD and proceeds as if the ground motion was simultaneous at all z in (-zo, zo)

but along CD line.

Ground motion did not occur for - < zo < equivalent to ground motion for the

whole range but with lower amplitude proportional to(A½/x)½ , A=generation area

x

zzo

-zo

4 ½

2zo

D

BC

Tsunami Slide 17

Tsunami Propagation (cont’d):Solution of the 3-D problem can be approximated as:

3D

(x,y,z,t) = (A½/x)½ 2D

(x,y,t)

for z in the wedge-shaped region z < x/3 , x > 2A½

This solution is adequate for tsunami motion.(not so for the whole radiation pattern).

Solution for: F (x,z,t) = Fo(x,z,t) - Fo(x+b,z,t)

with suitable and b values.

Using Fourier transform, by choosing g(t) = (t), by adopting the Boussinesque theory and by sacrificing some of the accuracy, it is possible to find:

where Ai denotes the Airy function.

= ½ (2/t)1/3 Ai {(2/t)1/3[x - t + (22/t)]} exp[(83/3t2)-(2 (t-x)/t)]

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Tsunami Propagation (cont’d):Assessment of the effect of the lateral scale of the generating

ground motion: For a Gaussian(-like) initial ground displacement,

(0, t),

(1000, t) for = 90, 50, 30, 10, 0.

(for 3000 m depth, they correspond to widths of generating area of 114, 85, 65, 38 and 0 km, respectively).

For wide generating areas, the initial disturbance propagates without much of change.

For narrow generating areas, (directional) dispersion plays an important role in producing extra crests.

A more realistic forcing with: F (x,z,t) = Fo(x,z,t) - Fo(x+14,z,t)

produces a second crest which is higher than the first (consistent with observations at Hawaii).

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Surface elevation for = 90.0(wide generation area)

x = 0 x = 1000

t

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Surface elevation for = 50.0

x = 0 x = 1000

t

Tsunami Slide 21

Surface elevation for = 30.0

x = 0 x = 1000

t

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Surface elevation for = 10.0

x = 1000

t

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Surface elevation for = 0.0 (narrow generation area)

x = 1000

t

Tsunami Slide 24

Surface elevation for = 10.0 with more realistic ground displacement

x = 1000

highersecond peak

t

F (x,z,t) = Fo(x,z,t) - Fo(x+14,z,t)

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Actual Tsunami Record

December 26, 2004 Tsunamias Recorded on the Depth Gauge of the Ship Mercator

anchored 1 mile from coast of Phuket, Thailand.The time is in Hours and the depth in Meters.

Tsunami Slide 26

Actualrecord

Same recordbut flipped

trou

gh

crest

Tsunami Slide 27

Tsunami Propagation (cont’d):

Variable depth effects:

scattering by bottom irregularities may not be important for tsunami propagation.

submerged ridges may act as wave guides.

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NUMERICAL MODELLING OF TSUNAMI PROPAGATION

Tsunami Slide 29

Numerical Modelling of Tsunami Propagation:

Navier-Stokes.

Spherical coordinates: = latitude, = longitude and R = distance from the Earth’s centre.

Define: z = R - Ro (Ro is the Earth radius = 6370 km)

Vertically averaged continuity and equations of motion:

0)cos(cos

1)(

cos

1

vDR

uDRtt oo

DR

gv

R

uu

R

vu

R

u

t

u

o

b

oooo

cossin

cos2

cos

DR

gu

R

uv

R

vv

R

u

t

v

o

b

oooo

sincos

2cos

Tsunami Slide 30

Numerical Modelling of Tsunami Propagation (cont’d):

u = velocity in the (E-W) direction.

v = velocity in the (N-S) direction.

= sea level

= bottom displacement

g = acceleration due to gravity

= water density

D = total water depth …. D = H + -

= Earth’s angular velocity

Coriolis parameter f = 2 sinb = bottom friction with components:

and

r = dimensionless bottom friction coefficient (~ 3.3x10-3)

22 vuurb 22 vuvrb

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Solution:

Numerical solution (e.g. finite difference).

Boundary conditions:- Dry points (land boundary):

normal velocity is zero.- Wet points (edge of computational domain):

radiation condition.

Treatment of dynamic wetting and drying.

For numerical considerations, time step is few seconds! excessive computational CPU time.

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Source Function:

Dislocation formulae require:- fault plane location, depth, strike, dip, slip, length & width- seismic moment and rigidity

Several approaches to estimate the total rupture extent.(early estimates which are usually refined later).

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Boxing-Day Tsunami Simulation:Tsunami Propagation:

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Boxing-Day Tsunami Simulation:Maximum Wave Height:

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Boxing-Day Tsunami Simulation:Maximum Wave Height:

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Boxing-Day Tsunami Simulation: Arrival Time(in hours)

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Boxing-Day Tsunami: Observed Arrival Time(and Locations of Deep Ocean Buoys)

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Tsunami Detection: Tidal Gauges(Boxing-Day Tsunami)

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Tsunami Detection: Tidal Gauges(Boxing-Day Tsunami)

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Tsunami Detection: Tidal Gauges(Boxing-Day Tsunami)

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Tsunami Detection: Tidal Gauges(Boxing-Day Tsunami)

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Tsunami Detectionand Warning System:

Deep-Ocean Assessment and Reporting of Tsunamis (DART)

Tsunami Slide 43

Tsunami Slide 44

seafloor bottom pressure recording (BPR) system

(detects tsunamis of 1 cm) moored surface buoy forreal-time communications

Tsunami Slide 45

Tsunami Slide 46

Tsunami Detection System:

Tsunami Slide 47

Use of Satellites:

Jason Altimeter

(Boxing-Day Tsunami)

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TSUNAMI DEVASTATION

Slide 49

Tsunami Generated by Earthquake of April

1, 1946, Aleutian Islands, Alaska

and hit Hawaii

(maximum rise of water was ~8 m in Hilo

and as much as 12 m in other areas on the

island of Hawaii)

Tsunami Slide 50

Tsunami Slide 51

Tsunami Slide 52

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Tsunami Generated by Earthquake of

December 26, 2004, West of Sumatra

(Boxing-Day Tsunami)

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Incoming Tsunami!

Tsunami Slide 56

Meluaboh, May 18, 2004 and Jan. 7, 2005

Tsunami Slide 57

Madras

Aug. 14, 2002andDec. 29, 2004

Tsunami Slide 58

Banda AcehNorthern Shore

June 23, 2004

and

Dec. 28, 2004

Tsunami Slide 59

Banda AcehNorthern Shore

June 23, 2004

and

Dec. 28, 2004

Tsunami Slide 60

Srilanka, Kalutara Beach, Dec. 26, 2004Receding Water from tsunami

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Srilanka, Kalutara Beach, Dec. 26, 2004Receding Water from tsunami

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No Tsunami Impact in Deep Open Sea:

Tsunami Slide 63

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END