propagation of finite strip sources over a flat bottom utku kânoğlu vasily v. titov baran aydın...
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PROPAGATION OF FINITE STRIP SOURCES OVER A FLAT BOTTOM
Utku KânoğluVasily V. TitovBaran AydınCostas E. Synolakis
ITS-2009Novosibirsk, Russia14-17 July 2009
Focusing phenomenon
During the 1989 ITS meeting in Novosibirsk, Drs. Marchuk and Titov numerically verified that a plus-minus source focuses at a point where abnormal wave height is observed.
Dr. Titov suggested analytical investigation of the focusing phenomenon.
Analytical Model
Physical description of the problem
Analytical ModelCarrier & Yeh (CMES, 2005) approach
(Axisymmetric Problem)
Initial conditions
(FINITE-CRESTED initial waveform)
(Zero initial velocity)
0( , 0) ( )r t r
( , 0) 0t r t
2 2 1( ) 0 ( ) 0r x y
tt xx yy tt r rgd rr
0 0
0 0
( , ) ( ) ( , ) ( , ) ( ) ( , )k t r J kr r t dr r t k J kr k t dk
(Fourier-Bessel transform)
Solution for Gaussian hump
(Self-similar solution)
2 /40
0
( , ) ( ) cos( ) kr t k J kr kt e dk
Analytical ModelCarrier & Yeh (CMES, 2005) approach
Extension to strip sources
2
0 0 0( , ) [erf ( ) erf ( )] yx y x L x x x e
Analytical ModelCarrier & Yeh (CMES, 2005) approach
Analytical ModelCarrier & Yeh (CMES, 2005) approach
Drawbacks of Carrier & Yeh approach
Elliptic integral in the solution results in
SINGULARITY Trial-error approximation for the integrand
Limited application
New Analytical Model
Governing partial differential equation
(Linear Shallow-water Wave Equation)
Initial conditions
(FINITE-CRESTED initial waveform)
(Zero initial velocity)
( ) 0tt xx yygd
0( , , 0) ( , )x y t x y
( , , 0) 0t x y t
η: water elevation above still water level
g: gravitational acceleration
d: basin depth (constant)
New Analytical Model
Solution technique
(Fourier integral transform over space variables)
(Inverse Fourier transform)
( )( , , ) ( , , ) i kx lyk l t x y t e dxdy
( )2
1( , , ) ( , , )
(2 )i kx lyx y t k l t e dkdl
New Analytical Model
Solution in Fourier space
Solution in physical space
2 20( , , ) ( , ) cos( )k l t k l t k l
( ) 2 202
1( , , ) ( , ) cos( )
(2 )i kx lyx y t k l e t k l dkdl
New Analytical Model
Features of the new approach
No approximations involved
Direct integration can be performed
Different initial waveforms can be imposed
Results of Carrier&Yeh (CMES, 2005) reproduced with direct integration
New Analytical Model
New Analytical Model
Solitary initial condition
0 ( , ) ( ) ( )x y f x g y
0 0
1( ) [tanh( ( )) tanh( ( ( )))]
2f x x x x x L
21( ) sech ( ( ))g y H y y
03
4
Hp Steepness parameter
Scaling parameter
New Analytical Model
N-wave initial condition
0 ( , ) ( ) ( )x y f x g y
0 0
1( ) [tanh( ( )) tanh( ( ( )))]
2f x x x x x L
22 1( ) ( )sech ( ( ))g y H y y y y
0( ) ( 1) cosech( ); / 2ikxikLf k i e e k
1998 PNG Event
(Figure taken from Synolakis et al. 2002)
• Earthquake: M = 7
• Casualties: 2100+
• Maximum tsunami waveheight: ~ 30 m
Analytical Model Results
Snapshots
Analytical Model Results
Snapshots
Analytical Model Results
Snapshots
Analytical Model Results
Maximum wave height envelope
Analytical Model Results
Maximum wave height envelope
Analytical Model Results
Maximum wave height envelope
Analytical Model Results
Maximum wave height envelope
Analytical Model Results
Maximum wave height envelope
(Figure taken from Synolakis et al. 2002)
Conclusions
We presented a new analytical solution for wave propagation over a constant depth basin.
Our solution Does not involve approximations Versatile in different initial waveforms
New solution Can be used to explain some extreme runup observations on
the field Can be used as a benchmark analytical solution for numerical
models
PROPAGATION OF FINITE STRIP SOURCES OVER A FLAT BOTTOM
Utku KânoğluVasily V. TitovBaran Aydın [email protected] E. Synolakis
ITS-2009Novosibirsk, Russia14-17 July 2009