1. Saleh Abdalla ECMWF, Reading, UK
TSUNAMI
Saleh Abdalla
ECMWF, Reading, UK

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,
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, (freely available at:
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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
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4. INTRODUCTION

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.
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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!)
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7. While Approaching the Coast:
10.6 km
213 km
23 km
metres
10’s cm
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8. Wave Classes and Scales:
1.0
10.0
0.03
3x10-3
2x10-5
1x10-5
Frequency (Hz)
Forcing:
wind
earthquake
moon/sun
Restoring:
gravity
surface tension
Coriolis force 1 2
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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 surface column.
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10. TSUNAMI GENERATION

11. Tsunami Generation
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12. TSUNAMI PROPAGATION

13. 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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14. 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) = 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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15. Tsunami

16. Tsunami Propagation (cont’d):x z zo
-zo
4 ½
2zo D B C
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
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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 + (22/t)]} exp[(83/3t2)-(2 (t-x)/t)]
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18. 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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19. Surface elevation for  = 90.0 (wide generation area)
x = 0
x = 1000 t
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20. Surface elevation for  = 50.0
x = 0
x = 1000 t
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21. Surface elevation for  = 30.0
x = 0
x = 1000 t
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22. Surface elevation for  = 10.0
x = 1000 t
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23. Surface elevation for  = 0.0 (narrow generation area)
x = 1000 t
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24. Surface elevation for  = 10.0 with more realistic ground displacement
F (x,z,t) = Fo(x,z,t) - Fo(x+14,z,t)
higher
second peak 
x = 1000 t
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25. Actual Tsunami Record December 26, 2004 Tsunami
as 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.
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26. Actual
record
trough
crest
Same record
but flipped
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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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28. NUMERICAL MODELLING OF TSUNAMI PROPAGATION

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:
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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  sin
b = bottom friction with components:
and
r = dimensionless bottom friction coefficient (~ 3.3x10-3)
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31. 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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32. 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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33. Boxing-Day Tsunami Simulation: Tsunami Propagation:

34. Boxing-Day Tsunami Simulation: Maximum Wave Height:

35. Boxing-Day Tsunami Simulation: Maximum Wave Height:

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

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

38. Tsunami Detection: Tidal Gauges (Boxing-Day Tsunami)

39. Tsunami Detection: Tidal Gauges (Boxing-Day Tsunami)

40. Tsunami Detection: Tidal Gauges (Boxing-Day Tsunami)

41. Tsunami Detection: Tidal Gauges (Boxing-Day Tsunami)

42. Tsunami Detection and Warning System: Deep-Ocean Assessment and Reporting of Tsunamis (DART)

43. Tsunami

44. seafloor bottom pressure recording (BPR) system
(detects tsunamis of 1 cm)
moored surface buoy for
real-time communications
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45. Tsunami

46. Tsunami Detection System:

47. Use of Satellites: Jason Altimeter (Boxing-Day Tsunami)

48. TSUNAMI DEVASTATION

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)

50. Tsunami

51. Tsunami

52. Tsunami

53. Tsunami

54. Tsunami Generated by Earthquake of December 26, 2004, West of Sumatra (Boxing-Day Tsunami)

55. Incoming Tsunami!
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56. Meluaboh, May 18, and Jan. 7, 2005
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57. Madras Aug. 14, 2002 and Dec. 29, 2004
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58. Banda Aceh Northern Shore June 23, 2004 and Dec. 28, 2004
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59. Banda Aceh Northern Shore June 23, 2004 and Dec. 28, 2004
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60. Srilanka, Kalutara Beach, Dec. 26, 2004 Receding Water from tsunami

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

62. No Tsunami Impact in Deep Open Sea:

63. Tsunami

64. END