slide 1 saleh abdalla ecmwf, reading, uk tsunami
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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
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 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.
Tsunami Slide 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)
Tsunami Slide 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) = 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.
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)]
Tsunami Slide 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).
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)
Tsunami Slide 25
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 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.
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
Tsunami Slide 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.
Tsunami Slide 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).
Tsunami Slide 42
Tsunami Detectionand Warning System:
Deep-Ocean Assessment and Reporting of Tsunamis (DART)
Tsunami Slide 44
seafloor bottom pressure recording (BPR) system
(detects tsunamis of 1 cm) moored surface buoy forreal-time communications
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)
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