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  80 % 1940

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 4. Long wave theory

Central Kuril Islands, 15.11.2006

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 (M w =8)

[Andrey Babeyko, PhD, GeoForschungsZentrum, Potsdam]

Dynamic bottom deformation (M w =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  4 H / 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

However, if

T  4 H / c

ocean behaves as a compressible medium L

is the horizontal size of tsunami source;

H traditional assumptions

is the acceleration due to gravity

c

is the sound velocity in water

are valid finite duration

H / g 4 H / 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

T min ~ H / g 1  ( x , y , t ) 

exponentially decreasing function

cosh 1 8  3 i s s    i  i  dp     dm     dn p 2 exp( cosh( pt kH  ) imx  gk  iny tanh( )  ( p , kH )  p m 2 ,  n ) where  ( p , m , n )   0  dt      dx      dy exp(  pt  imx k 2  m 2  n 2  iny )  ( x , y , t )

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

 ( x , y , t )   n ( x ,   x ,  y ,  z  y , t )   n x , n y , n z 

Normal to bottom Bottom deformation vector

Sloping bottom and 3-component bottom deformation:

 n  (  n ,

contribution to tsunami

n x  0 n y  0  )  n x  x n y  y  n n z z  1  z

traditionally neglected

 n

traditionally under consideration

Tsunami generation problem: Incompressible = Linear

             t 2 F  n F 2 F      x  v ( x , , y , y , t  0

Linear potential theory (3D model)

  t z ,  g t  ( )    , F z  n , ),  1   g    F t F z z z   0  0  ( x , y , t )

Not instant!

2) Phase dispersion is taken into account

 H ( x , y ).

Disadvantages:

1) Inapplicable under near-shore 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

 0  dt ,  is bottom

best make the best of

nt

Instant generation

duration  

what you have

H / g  F  0 at bottom    0 , z   H ( x , y ) : where  0   0    (  ) Fdt   F  n    t  ,     n    0 ,  n 

at water surface z  0 :  2 F  t 2   g  F  z   2 Fˆ  t 2   g  Fˆ  z nondimensi t *  t /  , onal variables z *  z / H :    2 Fˆ  t * 2    2  Fˆ H / g  z *  Fˆ  0 initial elevation : H / g  0   0  w z  0 dt   0   F  z z  0 dt   Fˆ  z z  0

Simple way out for practice Instant generation

 

0

  

n

 0 

0 ,

    

z z



0

Not only vertical but also Permanent bottom deformations horizontal bottom deformation (all 3 components!) is taken into account

0

,



n



, z

 

H ( x , y )

“Smoothing”, i.e. removing of shortwave components which are not peculiar to real tsunamis

initial elevation

Linear shallow water theory

 2   t 2  R 2 1 cos 2       gH        R 2 1 cos       gH cos       

Initial conditions:

 (  ,  , 0 )   0 (  ,  )    t  0

Initial elevation Boundary conditions:

    n  0

at shoreline

     t      t gH R     gH R    

at external 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

10

Comparison of runup heights calculated using traditional (pure Z) and optimal (Laplace XYZ) approach 15.11.2006

13.01.2007

10 1 1 0.1

0.1

1 Runup heights, m (Laplace XYZ) 10 0.1

0.1

1 Runup heights, m (Laplace XYZ) 10

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 13 Jan 2007 Volume, km 3 10 8 6 -2 -4 -6 -8 4 2 0 9.0

6.1

6.1

-6.4

-5.2

-5.2

Laplace, XYZ Laplace, Z Pure, Z Laplace, XYZ Laplace, Z Pure, Z

3 15 Nov 2006 Energy, 10 14 J 13 Jan 2007 2 2.36

1 1.23

1.36

1.00

0.84

0.97

0 Laplace, XYZ Laplace, Z Pure, Z Laplace, XYZ Laplace, Z Pure, Z