OPTIMAL INITIAL CONDITIONS
FOR SIMULATION OF
SEISMOTECTONIC TSUNAMIS
M.A. Nosov, S.V. Kolesov
Faculty of Physics
M.V.Lomonosov Moscow State University, Russia
OPTIMAL INITIAL CONDITIONS
FOR SIMULATION OF
SEISMOTECTONIC TSUNAMIS
OPTIMAL INITIAL CONDITIONS
FOR SIMULATION OF
SEISMOTECTONIC TSUNAMIS
[WinITDB, 2007]:
1547
1940
 80%
OPTIMAL INITIAL CONDITIONS
FOR SIMULATION OF
SEISMOTECTONIC TSUNAMIS
INITIAL CONDITIONS or “roundabout manoeuvre”
1. Earthquake focal mechanism:
 Fault plane orientation and depth
 Burgers vector
2. Slip distribution
Central Kuril Islands, 15.11.2006
[http://earthquake.usgs.gov/]
INITIAL CONDITIONS or “roundabout manoeuvre”
3. Permanent vertical bottom deformations:
 the Yoshimitsu Okada analytical formulae
 numerical models
Central Kuril Islands, 15.11.2006
4. Long wave theory
OPTIMAL INITIAL CONDITIONS
FOR SIMULATION OF
SEISMOTECTONIC TSUNAMIS
OPTIMAL INITIAL CONDITIONS
FOR SIMULATION OF
SEISMOTECTONIC TSUNAMIS
The “roundabout manoeuvre” means
Initial Elevation = Vertical Bottom Deformation
???
There are a few reasons why…
Dynamic bottom deformation (Mw=8)
[Andrey Babeyko, PhD, GeoForschungsZentrum, Potsdam]
Dynamic bottom deformation (Mw=8)
permanent
bottom
deformation
duration ~10-100 s
[Andrey Babeyko, PhD, GeoForschungsZentrum, Potsdam]
Period of bottom oscillations
Time-scales for
tsunami generation
H / g  L / gH
Tsunami generation is
an instant process if
T  H/g
However, if
T  4H / c
ocean behaves as
a compressible
medium
L is the horizontal size of tsunami source;
H is the ocean depth
g is the acceleration due to gravity
c is the sound velocity in water
finite duration
H/g
instant
Time-scales for
tsunami generation
H / g  L / gH
Tsunami generation is
an instant process if
T  H/g
L is the horizontal size of tsunami source;
Htraditional
is the ocean depth
assumptions
the acceleration due to gravity
(i.e.gc isisinstant
& incompressible)
the sound velocity in water
are valid
finite duration
H/g
However, if
T  4H / c
ocean behaves as
a compressible
medium
4H / c
1.Elastic oscillations do not propagate upslope
2.Elastic oscillations and gravitational waves are
not coupled (in linear case)
Linear = Incompressible!
Initial Elevation = Vertical Bottom Deformation
???
“Smoothing”: min~H
Tmin ~ H / g
1
cosh kH 
( x , y, t ) 
s  i


8  3 i s  i


1
 dp  dm  dn
exponentially
decreasing function
p 2 exp( pt  imx  iny ) (p, m, n )

cosh( kH) gk tanh( kH)  p 2
where



0


 (p, m, n )   dt  dx  dy exp( pt  imx  iny ) ( x, y, t )
k 2  m2  n 2

Initial Elevation = Vertical Bottom Deformation
Due to “smoothing”
Permanent bottom deformations
vertical
horizontal
Central Kuril Islands, 15.11.2006
Sloping bottom and 3-component bottom deformation:
contribution to tsunami

n(x, y, t )  n x , n y , n z 

(x, y, t )  x , y , z 
Bottom
deformation
vector
Normal to bottom
Sloping bottom and 3-component bottom deformation:
contribution to tsunami
nx  0
ny  0
nz  1
 
n  (n, )  n x x  n yy  n z z
traditionally
neglected
n
traditionally
under
consideration
Tsunami generation problem: Incompressible = Linear
Linear potential theory (3D model)

1) Dynamicbottom deformation (DBD)

Not instant!
 F  0

(
x
,
y
,
t
)
  2 F
F
2) Phase dispersion is
 2  g , z  0
z
taken into account
 t
 F   
Disadvantages:
   (, n ), z   H( x , y).
 n t
1) Inapplicable under near-shore
 x , y , t   
1 F
g t


v ( x , y, z , t )   F
z 0
conditions due to nonlinearity,
bottom friction etc.;
2) Numerical solution requires huge
computational capability;
3) Problem with reliable DBD data.
Simple way out for practice Instant
generation

If
you
can’t
have
the
dt
,

is
bottom
displaceme
nt duration

0
best make the best of
what you have
  H / g
ˆ
F  0   F  0,

ˆ
where F   Fdt


at bottom z  H( x, y) : 0  ( )
0
F   
  , n  
 n t
ˆ
F  
  0 , n 
n
at water surface z  0 :
 F
2
 g
F
 Fˆ
2

 g
t
z
t
nondimensi onal variables :
2
2
Fˆ
z
t*  t / , z*  z / H
  H / g
2ˆ
2
ˆ
 F
 F

 Fˆ  0
2
t *
H / g z *
initial
elevation :


 0   w z 0 dt  
0
0
F
z
dt 
z 0
Fˆ
z
z 0
Simple way out for practice Instant
generation
ˆ
F  0
Not only vertical but also
bottom deformations
ˆF  0, z  0 Permanent
horizontal bottom deformation
(all 3 components!)
is taken into account
ˆ
F  




,
n
,
z


H
(
x
,
y
)
0

“Smoothing”, i.e. removing of
n
shortwave components which are
ˆ
not peculiar to real tsunamis
F
0 
initial elevation
z
Linear shallow water theory
 2
1
 
 
1
 
 
 gH
  2
 gH cos  
 2
2
2
t
R cos   
  R cos   
 
Initial conditions:
(, ,0)  0 (, )
Boundary conditions:

  0 at shoreline
n

0
t

t
Initial
elevation

t


gH 
R

R

at external
gH  boundary
15.11.2006
Initial Elevation=Vertical Bottom Deformation
15.11.2006
Smoothing: Initial Elevation from Laplace Problem
13.01.2007
Initial Elevation=Vertical Bottom Deformation
13.01.2007
Smoothing: Initial Elevation from Laplace Problem
Comparison of runup heights calculated using
traditional (pure Z) and optimal (Laplace XYZ)
approach
10
Runup heights, m (pure Z)
Runup heights, m (pure Z)
10
15.11.2006
1
0.1
13.01.2007
1
0.1
0.1
1
10
Runup heights, m (Laplace XYZ)
0.1
1
10
Runup heights, m (Laplace XYZ)
Conclusions:
1. Optimal method for the specification of initial
conditions in the tsunami problem is suggested and
proved;
2. The initial elevation is determined from 3D problem
in the framework of linear potential theory;
3. Both horizontal and vertical components of the
bottom deformation and bathymetry in the vicinity of
the source is taken into account;
4. Short wave components which are not peculiar to
gravitational waves generated by bottom motions are
removed from tsunami spectrum.
15 Nov 2006
Volume, km
3
13 Jan 2007
10
8
6
4
9.0
6.1
6.1
2
0
-2
-6.4
-4
-5.2
-5.2
-6
-8
Laplace, Laplace, Z
XYZ
Pure, Z
Laplace, Laplace, Z
XYZ
Pure, Z
15 Nov 2006
14
Energy, 10 J
13 Jan 2007
3
2
2.36
1
1.00
0.84
0.97
1.23
1.36
0
Laplace, Laplace, Z
XYZ
Pure, Z
Laplace, Laplace, Z
XYZ
Pure, Z