Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
CHAPTER 4: TSUNAMI MODELLING
4.1 GOVERNING EQUATIONS AND ASSUMPTIONS BEHIND
Nonlinear Form of Shallow Water Equations in Cartesian Coordinates for Near Field
Tsunami Modeling
To describe the tsunami wave propagation, the nonlinear form of long wave equations are
used. Due to relative small size of the Black Sea, the Coriolis effect is neglected and
Cartesian coordinates are applied. These equations are,
  u h +    vh +  


0
t
x
 y
(1)
u
u
u
  x
u
v
g

0
t
x
 y
x 
(2)
v
v
v
  y
u
v
g

0
t
x
 y
 y 
(3)
where  is the water elevation, u and v are components of the horizontal velocities averaged
on depth in x and y directions, x and y are the bottom shear stress components, x and у
are horizontal coordinates in zonal and meridional directions, t is time, h(x,y) is unperturbed
basin depth, g is the gravitational acceleration. By using D, total depth D=h+, the discharge
fluxes M and N in x, y directions respectively can be described in relation to u and v by


M  u (h   )  uD , N  v(h   )  vD
(4)
If the shear stress is defined in terms of Manning’s coefficient n, then the equations (1-3)
can be written in the following form for the discharge fluxes M and N.
  M  N


0
t x  y
(5)
 M   M 2    MN 
 gn 2

 

 7/3 M M 2  N 2  0

  gD
 t  x D   y D 
 x D
(6)
 N   MN    N 2 
 gn 2

  gD

 7/3 N M 2  N 2  0


 t  x D   y D 
 y D
(7)
We use the numerical model TUNAMI-N2 for the simulation of the propagation and coastal
amplification of long waves. The model was originally authored by Professor Fumihiko
Imamura in Disaster Control Research Center in Tohoku University (Japan) through the
Tsunami Inundation Modeling Exchange (TIME) program [Goto et. al., 1997]. TUNAMI-N2 is
one of the key tools for developing studies for propagation and coastal amplification of
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
tsunamis in relation to different initial conditions. It solves Eqs. 5-7 using a leap-frog scheme
in finite difference technique for the basins of irregular shape and topography [Yalçıner et.
al., 2001b 2001c; 2002a, 2002b, 2003; Goto et. al., 1997; Nagano et. al., 1991; Shuto et. al.,
1990]. The program can compute the water surface fluctuations and velocities at all
locations, even at shallow and land regions (Imamura, 1996) within the limitations of grid
size.
As an example, the model is applied to the Black sea in the boundaries of the longitudes
27.33ºE and 42.00ºE and the latitudes 40.50ºN and 47.33ºN. The bathymetry of the Black
Sea was digitized and used with a mesh resolution of 1.0 km, (i.e. 1171 nodes in E-W
direction and 757 nodes in S-N direction). A typical value of 0.025 is used as Manning’s
coefficient [Munson et. al., 1998]. The time step was chosen 4 sec. to satisfy the CFD
condition. The evolution of the wave with respect to the assumed initial wave form is
computed by the model. The water surface fluctuations in 8 hours real time duration at all
grid points are computed and stored for comparison with the observations and instrumental
data.
In order to analyze the tsunamis in the Black sea, the model is applied to the 1939 and 1966
tsunamis.
4.1.1 Modeling of the 1966 Anapa Tsunami in Black Sea and Comparison with the
Measurements at some locations on the Russian Coast
The Anapa earthquake occurred on July 12, 1966 at 18:53 of GMT with magnitude 5.8
and the intensity 6 [Nikonov, 1997; KOERI, 2004]. the epicenter was located at the
coordinates 44.70N and 37.30E, approximately 10 km away from the shore. Travel time
chart of this tsunami was calculated by [Dotsenko, 1995, 1996] using the ray tracing
method. Since the sea level fluctuations recorded by tide-gauges are exist for this
event, they are used for validating the numerical simulation in this study.
By using the available information about the epicenter coordinates, the fault line (axis
of the initial form of the tsunami wave) is assumed in between the coordinates 37.50oE,
44.53oN and 37.90oE, 44.38oN with a length of 36km and angle of 105o from North.
Since there is no sufficient information available about the source parameters of this
earthquake, the width of the source area (minor axis of the ellipse) is assumed as
17km. The dip and slip angles of the fault are selected as 60o and -90o, respectively,
the displacement and focal depth have been selected as 1.5m and 1 km. The crest and
trough amplitudes are 0.3 m and -0.9 m. respectively [Okada, 1985] (see Fig. 4.1).
For modelling this event, we have used the propagation model and computed the sea
state at different times, and time histories of water surface oscillations and the
maximum positive amplitudes at every grid point. Snapshots (at t=0, 30, 60, 90 min) of
the tsunami wave propagation are shown in Figure 4.2. According to simulation results,
the tsunami arrives 45 min. later at the south coast and disturbance remains
observable for 120 min.
Figure 4.3 shows the distributions of the computed maximum elevations of the sea
level (tsunami directivity) in the sea and along the north and south coasts. It is clearly
seen from this figure that the wave energy propagates at right angles to the major axis
of the tsunami source towards south as it is predicted in the theoretical models of wave
propagation in the basin of constant depth [Pelinovsky, 1996].
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
The distribution of the maximum positive amplitudes along the north and south coasts
of the Black sea are also shown in Fig. 4.2. The distributions are non-uniform because
of the coastal topography. The maximum positive amplitudes were not too high
comparing to the amplitudes of the common storm waves. Possibly this is why this
tsunami was observed only near the source area.
The numerical and instrumental time histories of water surface fluctuations at the tidegauge stations; (Gelendzhik, Tuapse, Kerch Strait from the Azov Sea, and Feodossiya
for 1966 Anapa tsunami are shown in Figure 4.3 for comparison. The recorded surface
elevations also contain the weak tidal wave. As seen that the computed arrival time of
the wave at Gelendzhik is in well agreement with the measurements. The shape and
first polarity of the instrumental and numerical records in Gelendzhik are also in
sufficient agreement. The first recorded oscillation in Tuapse shows a small uplift in
contrary to the numerical results. If the direction of the major axis of initial wave is
slightly changed, the better agreement may be obtained. Since the weak tidal wave is
also superposed on the recorded small amplitude of tsunami waves, the comparison of
these records (especially in Feodosia and Kerch) with simulation becomes difficult. The
period of the wave is about 15 minutes in measurements and but it is in average 11
minutes in the computed record for Gelendzhik. The difference in the wave period
between measurements and computations provides that the width of the source area
assumed in the computation might be somewhat wider. The higher amplitudes
observed in the computed wave record suggest that the amplitude of the initial wave
assumed in the model is somewhat higher than the actual amplitude of the initial wave
occurred in this event.
Figure 4.1: Initial Wave and Sea State at Different Time Steps (t=30, 60, 90 min) of the
1966 Anapa Tsunami
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
52
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
Figure 4.2. Directivity of the wave energy during the 1966 Anapa event and the distribution
of maximum positive tsunami amplitudes along north and south coasts of the Black sea.
Figure 4.3. The comparison of the instrumental and numerical records of water surface
fluctuations at the tide-gauge stations; a) Gelendzhik, b) Tuapse, c) Kerch Strait from the
Azov Sea, d) Feodossiya for 1966 Anapa tsunami
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
4.1.2. Modeling of 1939 Tsunami in the Black Sea
Strong earthquake occurred on December 26, 1939 and aftershocks continued up to
January 2, 1940 in Turkey near Erzincan, and 39000 people died. This earthquake is
remarkable not only because of its devastating casualties, but also because of the
tsunami generation in the Black sea [Richter, 1958]; the epicenter was on land and far
from the shore. The earthquake occurred along the North Anatolian Fault Zone (NAF)
with strike slip fault motion [Kuran and Yalçıner, 1993]. There is no sufficient data
describing the source for this tsunami. From the available information about the wave
motion near the south coast, the source of tsunami (axis of the initial form of the wave)
is assumed to be situated near Fatsa town between the coordinates 38.40oE, 40.90oN
and 37.20oE, 40.70oN with a length of 120km. The major axis lies parallel to shore
oriented as 105o clockwise from North. The width of the source area (minor axis of the
ellipse) is assumed to be 25km. The uplift of the water surface of the initial wave is
assumed to be at the north of major axis, at the deeper part of the source region. This
assumption is valid if i) there is a submarine landslide at offshore of Fatsa town, ii)
there is a secondary fault at offshore of Fatsa town, iii) the north part of NAF (near
Fatsa town) uplifted partly and/or moved west by the strike slip motion of NAF. Since
there is not any data about the submarine fault mechanism and/or submarine landslide,
an initial wave is assumed at offshore of the coastline from Fatsa to Giresun towns with
the crest and trough amplitudes as 0.30m and -0.90 m respectively (Figure 4.4). This
shape of the wave is selected in order to satisfy the withdrawal of shoreline
(subsidence of water surface) at nearest shoreline (south coast). This type of the
shoreline motion by tsunamis can be observed either by the submarine landslide
generated tsunamis or by the fault break which uplifts the land near the coast [Liu,
Synolakis, Yeh, 1991 and Yeh, Liu, Synolakis, 1996, Yalciner et. al. 2003].
Figure 4.4 also shows the snapshots of tsunami propagation at 0, 30, 60 and 90
minutes after its generation. Tsunami approaches North coast in between 25 min and 2
hours later, sea level oscillations are triggered in entire basin of the Black Sea. The
tsunami propagates mainly in the direction at right angles to the major axis of the
source northward. Fig. 4.5 shows the distribution of the wave heights along the north
and south coasts. The wave disturbance is stronger in the eastern Black Sea. The
mean amplitude of tsunami on the northern coast is about 10 cm. On the south coast
maximum wave amplitudes are extreme in Sinop-Fatsa (reading up to 2.5m). The wave
heights at east and west of the extreme are significantly less along south coast.
Tsunami waves related the 1939 Erzincan event were recorded by several Soviet tidegauges (Sevastopol, Yalta, Feodossiya, Mariupol (Azov Sea), Kerch, Novorossiysk,
Tuapse, Poti, Batumi) [Grigorash and Korneva, 1972]. Some of the records are helpful
for the comparison. Maximal tsunami waves are evident on records at Feodossiya,
Kerch and Novorossiysk; these cities are in the “projector” zone of the directivity
diagram obtained with the simulation in this study. Figure 4.6 shows records for several
coastal locations on the north coast for comparison with numerical results. As seen
from Figure 10, the arrival time of the wave at these locations are in qualitative
agreement with the measurements. The shape of the measured wave is a leading
elevation wave at the locations along North coast. Unfortunately Tuapse record has an
interruption just after the tsunami arrival. Nonetheless it shows an about 20 cm halfoscillation that is not too far from the computed half-wave. The boundary in the
passage between the Azov Sea and the Black Sea is narrow in the depth data of the
simulation. This causes less passage of the wave in simulation than the actual
situation. The reason why the smaller amplitudes of waves at Mariupol near Azov sea
is computed in simulation. The instrumental data at Novorossiysk and also at
Sebastopol are higher than the computed data.The period of the waves can not be
determined accurately from the measurements, but it is seen from Figure 10 that the
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
computed wave period is smaller than the observations. This comparison brings that
the assumed amplitude of the initial wave and width of the source may be larger in the
actual event.
Figure 4.4. Initial Wave and Sea State at Different Time Steps of the 1939 Erzincan Tsunami
Figure 4.5. Directivity of the wave energy during the 1939 tsunami and the distribution of
maximum positive tsunami amplitudes along north and south coasts of the Black sea.
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
Figure 4.6. The comparison of the instrumental and numerical records of water surface
fluctuations at the tide-gauge stations at Sevastopol, Yalta, Feodossiya, Mariupol (Azov
Sea), Kerch, Novorossiysk, Tuapse, Poti, Batumi for 1939 Erzincan tsunami.
4.2 LINEAR FORM OF SHALLOW WATER EQUATIONS IN SPHERICAL
COORDINATES FOR FAR FIELD TSUNAMI MODELING
In computation of a far-field tsunami, the dispersion term becomes important because the
long travel distance acts to disperse wave components. The linear Boussinesq equation
including the physical dispersion term is considered appropriate to add this effect. In the
following, another simple method is used to replace the physical dispersion term by the
numerical dispersion term that is inevitably resulted as the truncation error of a numerical
scheme. This replacement becomes possible, if the appropriate grid size is selected. Then,
the linear long wave theory of lower order of approximation becomes almost equivalent to the
linear Boussinesq equation of higher order of approximation. In addition, this replacement
saves CPU time and the computer resources.
Long waves have small relative depth (the ratio of water depth to wave length). The vertical
acceleration of water particles can be omitted when compared to the gravitational
acceleration. It can be assumed that velocity of water particles is vertically uniform. Based on
these approximations, the linear long wave theory is given by the following expressions with
the spherical coordinates (Shuto and Imamura, 1992).

1

t R cos
 M 

    ( N cos )  0


M
gh 

 fN
t R cos 
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
(3.1)
(3.2)
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
N gh 

  fM
t
R 
Chapter 4
(3.3)
1
1
n
n
M n 1  M n 1
N n 1 cos  1  N n 1 cos  1 
j

,
m
j

,
m
j ,m 
m
j ,m 
m 
 j ,m2   j ,m2
1

2
2
2
2
2
2


  0 (3.4)
t
R cos  m 






Figure 4.7. Spherical Co-ordinates
Where; R is the radius of the earth,  is the water level, M and N are discharge fluxes in the 
(along a parallel of latitude) and  (along a circle of longitude) directions, g is the gravitational
acceleration, and f=(2sin) is the Coriolis coefficient.
When equations (3.1-3.3) are applied to the Leap-Frog Scheme (3.5), (3.6), and (3.9) are
obtained.
M n 11  M n
1
j  ,m
2
j  ,m
2
t
N n 11  N n
Where ;
t

1
j  ,m
2
1
2
j 1, m
n

1
2
j ,m
n
 fN 
(3.5)
 j , m21   j , m2

  fM 
R sin  m

(3.6)

R cos  m
gh
1
j  ,m
2
j  ,m
2
gh
1
j  ,m
2

n
1
n
1

1 n
n
n
n
 N j 1, m  1  N j 1, m  1  N j , m  1  N j , m  1 
4
2
2
2
2
(3.7)

1
M   M n 1 1  M n 1  M n 1  M n 1 
j  ,m
j  , m 1
j  ,m
4  j  2 ,m 2
2
2
2 
(3.8)
N 
Here unknowns, , M and N are given by the following explicit expressions.

1
2
j ,m
n

1
2
j ,m
n


 R1  M n 1  M n 1  N n 1 cos  1  N n 1 cos  1 
j  ,m
j ,m
m
j ,m
m
2
2
2
2
2
 j  2 ,m
M n11  M n 1  R2  h
j  ,m
2
N n1
j ,m 
1
2
j  ,m
2
 Nn
j ,m 
1
2
 R4  h
1
n 
 n 12
2




1  j 1,m
j ,m   R3 N
j  ,m
2 

1
n 
 n 12
2




1  j ,m1
j ,m   R5 M
j ,m 
2

© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
(3.9)
(3.10)
(3.11)
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
Computation points are given in Figure 4.8 as follows;
Figure 4.8. Computation Points for Water Level and Discharge
Where;
R1 = t/(Rcosm)
R2 = g.t/(Rcosm)
R3 = 2tsinm
R4 = gt/(R)
R5  2t sin 
m
1
2
A point of calculation is numbered as (j,m,n) in the (, ,t) directions and the grid lengths
are(,,t). These grid lengths are taken equal ((=) and,  is the angular velocity of
earth.
In the solution of the problem, the Leap-Frog Scheme is used (Goto and Ogawa, 1982). It is
a central difference scheme with the truncation error of second order and it assumes that the
calculation point for  does not coincide the calculation point for discharge fluxes M and N.
4.3 MODELING OF TWO LAYER FLOW AS FOR SUBMARINE LANDSLIDE
Two-layer long waves or flows (like the surface wave and mudslide) can also be applied in
the case of underwater landslides accompanying underwater earthquakes generating
tsunamis (Imamura and Imteaz, (1995)). Therefore, the interaction between each layer
should also be considered as different from the one-layer models.
Two-layer flow may result from density differences within the fluid. This density difference
may be due to temperature differences or dissolved substances as well as suspended
material within the liquid. In two-layer flow both layers interact and play a significant role in
the establishment of control of the flow. The effect of the mixing or entrainment process at a
front or an interface becomes important (Imamura and Imteaz, (1995)).
In the past years a number of analytical and experimental studies have been carried out on
two-layer flow. The reliability and accuracy of the two-layer model is verified through the
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
comparison with the analytical solution under simplified conditions by Imamura and Imteaz
(1995).
4.3.1. Theoretical Approach
Two-layer flows that occur due to an underwater landslide can be modeled using a
non-horizontal bottom with a hydrostatic pressure distribution, uniform density
distribution, uniform velocity distribution and negligible interfacial mixing in each layer.
The schematic view of two-layer flow is shown in Fig. 4.9. Conservation of mass and
momentum can be integrated in each layer, with the kinetic and dynamic boundary
conditions at the free surface and interface surface (Imamura and Imteaz (1995)).
Figure 4.9. Definition sketch for two-layer profile (Imamura and Imteaz (1995)).
Governing equations for the upper layer are given in the following and the derivation
procedure is summarized in Appendix A.
 (1   2) M 1 N 1


0
t
x
y
(3.1)
 M 12   M 1N 1 
 


M 1  D1   D1 
1
 2 g 2


 gD1
 gD1

M 1 M 12  N 12  0
t
x
y
x
x D17 / 3
(3.2)
 N 12   M 1 N 1 
 


N 1  D1   D1 
1
 2 g 2


 gD1
 gD1
 7 / 3 N 1 M 1 2  N 12  0
t
y
x
y
y D1
(3.3)
and those for the lower layer are:
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
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Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
 2 M 2 N 2


0
t
x
y
Chapter 4
(3.4)
 M 22   M 2 N 2 
 


  1 h1  2   2 h1  g 2
M 2  D 2   D 2 


 gD 2 


   7 / 3 M 2 M 22  N 22  0

t
x
y
  x x x  x x  D 2
 N 22   M 2 N 2 
 


  1 h1  2   2 h1  g 2
N 2  D 2   D 2 
 


 gD 2 


   7 / 3 N 2 M 22  N 22  0
t
y
y
  y y y  y y  D 2
(3.5)
(3.6)
Where η is the surface elevation, D=h+ η the total depth, h is the still water depth, M
the discharge, ρ the density of the fluid, α=ρ1/ρ2, and subscripts 1 and 2 indicate the
upper and lower layer respectively (Imamura and Imteaz, (1995)). M1, N1, η1, η2, M2, N2
are solved from the above 6 equations numerically. At the open boundary the total
derivative of surface elevation is set to be equal to zero as:
D
0
Dt
4.3.2 Numerical Approach
The staggered leap-frog scheme (Shuto, Goto, Imamura, (1990)) has been used to
solve the governing equations for long waves numerically. This scheme is one of
explicit central difference schemes. The staggered scheme considers that the
computation point for one variable, η, does not coincide with the computation point for
other variable, M. There are half step differences, 1/2∆t and 1/2∆x between
computation points of two variables as can be seen in Fig. 4.10. Thus one variable, η,
is placed at the middle of ∆t∆x rectangle, placing other variables at the four corner of
rectangle and vice versa (see Fig. 4.10). In the finite difference formulation ‘n’ denotes
the temporal grid points and ‘i’ and ‘j’ denote the spatial grid points along x and y
directions. ∆x, ∆y, and ∆t are spatial grid spacings and time step respectively. Using
this scheme, the finite difference equations for the governing equations are obtained
(Imamura and Imteaz, (1995)).
.
Figure 4.10. Points schematics of the staggered leap-frog scheme (Imamura, Imteaz (1995))
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In spatial direction, all of η1, η2 at step ‘n+ ½’ and all of M1, M2 at step ‘n’ are given as
initial conditions. For all later time steps at left and right boundaries, all values of either
discharge or water elevation would be calculated by using the values of previous time
step or estimated wave celerity. This solution contains two progressive waves with
different celerities and one reflective wave. By using the mass continuity equation for a
lower layer, all η2 at step ‘n+3/2’ are calculated and then all η1 at step ‘n+3/2’ for an
upper layer are calculated using latest values of η2. Then, using the momentum
equation for an upper and a lower layer, all values of M1, N1, M2, N2 at step ‘n+1’ are
simultaneously calculated. Similarly, using new values of η1, η2, M1, N1, M2, N2 as initial
values for the next time step, the calculations proceeds in time up to desired step.
While doing these calculations, it is very difficult for the model to derive a stability
condition analytically due to the interactions between two layers. For this purpose
Courant-Friedrichs-Lewys (CFL) condition is applied where two celerities exist as one
for a progressive wave and one for reflected wave. Stability is initially investigated for
some arbitrary Δx and Δt. This result suggests that the model is stable up to a certain
limit of Δx/Δt and this limit varies with the variation of α and β (Imamura and Imteaz
(1995)) as verified with different test results obtained during the preparation of this
thesis for different values of α (ratio of density of fluid in an upper layer to a lower one)
and β (ratio of water depth in a lower layer to an upper one). It is suggested by
Imamura and Imteaz (1995) that as for lower ‘α’ and for higher ‘β’, an amplification of a
top surface increases and vice versa. According to Imamura and Imteaz (1995), for
α=0.5 and β=4.0, celerity of top surface calculated through analytical expression,
c2=
 gh (1   ) /(1  )  controls the stability criteria, while for α=0.4 and β=1.0, celerity
2

gh1(1   )

of interface c1 =
corresponds to the stability criteria. It is suggested by
Imamura and Imteaz (1995) to consider the maximum of c1 and c2, to satisfy the
stability condition Δt ≤ Δx/max(c1, c2).
The numerical model TWO-LAYER is developed in Tohoku University, Disaster Control
Research Center by Prof. Imamura. The model is more realistic than the previous
models in the sense that it combines the effects of an underwater earthquake and an
underwater landslide on the generation and propagation of tsunami waves, which is the
case in reality as observed and studied for years for different cases of tsunami events.
4.3.3. Test of the Model in a Regular Shaped Basin
In the present chapter, the model TWO-LAYER is tested by using a regular shaped
basin for modeling of generation and propagation of water waves due to underwater
mass failure mechanisms. According to the results obtained and observations made
throughout the test applications, in order to obtain accurate results the duration and
domain of simulation as well as the characteristics of the mass failure mechanism must
be chosen accurately and described very precisely. For stability the time step and grid
size should also be selected properly.
4.3.3.1. Test Basin and Dimensional Parameters
The test basin selected here is a rectangular basin with 150 km. width along East West
direction and 125 km. length along South North direction. Three boundaries of this
basin (at East, North and West) are set as open boundaries (outgoing boundaries) to
avoid wave reflection and unexpected amplification inside the basin The land is
located at the South of this domain. The sloping bottom is chosen to start with -100m.
elevation as land at South boundary and uniformly deepen up to 2000 m. water depth
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at North boundary, satisfying 1:60 uniform bottom slope. The grid spacings are 400 m.
with 375 nodes along East-West direction and 313 nodes along South-North direction.
The computational domain is shown in Figure 4.11.
Although 22 stations were selected in the domain to observe the water surface
fluctuations for checking the performance of the program and observing the wave
motion during the test processes, the results at 6 stations are presented in the following
sections. The locations of these 6 stations are also shown in Fig. 4.11 with their
numbers. The first three stations are located between the mass failure area and the
coastline, while the other three (4,5 and 6) are located inside the failure area.
120.00
South-North Direction (km.)
100.00
80.00
60.00
40.00
Deposited
region
6
5
20.00
0.00
0.00
Slided
region
4
3
2
1
20.00
40.00
60.00
80.00
100.00 120.00 140.00
East-West Direction (km.)
Figure 4.11. Computational Domain for the Test of the Model and the Failure Area
Optimum number of time steps and thus an optimum time of simulation is determined
by observing the duration in which the tsunami waves reach the shore line. If the time
of simulation is chosen more than the optimum time than the reflected waves dissipate
from the boundaries of the simulation domain and the values of maximum water
surface elevations obtained near the shoreline are very small. According to this
information the simulation time for the test in the domain shown in Fig. 4.1 is selected
as 30 minutes. For comparison a flat bottom with depth of 500 m. domain with the
same size as in Fig. 4. 1. is also used in some of the test runs. In this flat bottom
domain the simulation duration was selected as 15 minutes.
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4.3.3.2. The Mass Failure Parameters
The program solves the generation of the tsunami wave due to the mass failure
mechanism at the source area and calculates the water surface elevations at each grid
point while propagating the wave in the basin. During the process, the time histories of
the water surface elevation at all grid points are obtained and they are stored at 22
selected stations. The data of maximum water surface elevation during the simulation
at grid points along the coastline is also stored.
Inside the basin, a smaller region is chosen as the region of mass failure (Fig. 4.11).
The mass failure mechanism is generated at this region which is a rectangular area
with a width of 20 km. along E-W direction (in between 65. and 85. km) and a length of
40 km along S-N direction (in between 15. and 55. km).
The initial and final profile of the sea bottom in the mass failure area is also shown in
Figure 4.12. with corresponding parameters. The physical meaning of these
parameters are; (h+) is the increase of water depth in the eroded area due to the mass
failure; (h-) is the decrease of water depth in the accreted area due to the mass failure;
(L+) is the length of the eroded area; (L-) is the length of the accreted area respectively.
The conservation of the moved volume of sediment before and after the failure is
satisfied by the following relation for constant width of the failure:
h+ L+ = h- LThis conservation of volume can be observed in Figure 4.12. where a sectional view of
the test basin and the failure area is shown.
Sea bottom before mass failure
h+
Eroded
volume
LL+
Accreted
Sea bottom
after mass failure
volume
h-
Figure 4.12. Sectional View of the Mass Failure Area
The parameters that are changed during the tests are h+, L+, h-, α (which is equal to
the ratio of the densities of the two layers) and β (which is equal to the ratio of the
water depths of the two layers) and slope of the bottom of the test basin. Plenty of
these mass failure parameters are used during different applications in order to find out
relative effects on wave generation and maximum water surface elevations near the
coastline for each parameter. The different values of these parameters in some of the
chosen test applications in order to investigate their effects on the wave form and
maximum water surface elevations near the coastline values along the coast are listed.
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The value of α has been used as 0.68 for the application to 1998 Papua New Guinea
tsunami (Smith, 2000).
4.3.4. Application of the Model to the Sea of Marmara
Until recently, it was common to choose a tsunami-wave-like water surface as the initial
condition for mathematical solution of the long wave equations (Yalç ner et. al., 1995;
Titov and Synolakis, 1998, Yalç ner et. al., 2001c). However, the kinematic
characteristics of mass flows at the sea bottom due to landslides or slumps necessitate
different type of initial wave profiles. Therefore, it is important to incorporate the
movement of the sea bottom. The model TWO LAYER (Imamura and Imteaz, 1995)
which was created in Tohoku University Disaster Control Research Centre in Japan is
used in this study. TWO LAYER solves the non-linear long wave equations
simultaneously by using the finite difference technique and following the Leap-Frog
solution procedure (Shuto et al., 1990) within two interfacing layers with appropriate
kinematic and dynamic boundary conditions at the sea bed, interface and water
surface. The two interfacing layers are the water body in the sea and the moving mass
at the bottom. After the sensitivity analysis for different input parameters such as the
geometry of the basin, the volume and size of the slided mass, and the bottom slope
tested in a regular shaped basin (Özbay, 2000, Yaciner et. al. 2001c), TWO LAYER
has been applied to the sea of Marmara for three different tsunami scenarios.
The computation domain of the sea of Marmara for these applications is chosen as
bounded by the longitudes 26.55ºE and 29.96ºE and the latitudes 40.30ºN and
41.11ºN. The grid size is taken as 300 m. The computational domain and the bottom
topography used for the model applications are shown in Figure 4.13. A composition of
multibeam bathymetry data with the main active structures and subbasins are also
shown in Figure: 4.14.
Figure 4.13. The computational domain and the bottom topography used for the mode
and the sub-basins in the Sea of Marmara. The full line follows 500 m contour.
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Figure 4.14. A composition of multibeam bathymetry data with the main active structures
and subbasins (modified from Alpar, 2000b; EcevitoElu, 2000; Demirbag E, 2000; Akkargan
and Alpar, 2000; ÜGümezsoy, 2001; Le Pichon et al., 2001). WMSB : West Marmara subbasin, MMSB : Middle Marmara sub-basin, EMSB : East Marmara sub-basin.
Three different scenarios have been applied for hypothetical case studies. They are;
i) The occurrence of a landslide at offshore of Yenikap coasts of 'stanbul (CASE 1),
ii) The occurrence of a landslide at offshore of Tuzla (CASE 2),
iii) The occurrence of an earthquake at Armutlu Fault and two accompanying landslides
along this fault (CASE 3).
4.3.4.1.Tsunami due to underwater landslide occurrence at offshore Yenikapi (CASE 1)
In the first case (CASE 1), an underwater landslide scenario is assumed to occur at the
North West part of East Marmara sub-basin at offshore Yenikap around the 28.72ºE
and 40.90ºN co-ordinates. The water depth at the assumed failure area is 300 m. The
failure area is assumed as 10 km. wide along East-West direction and 1.8 km long
along North South direction. The failure is assumed to move as an underwater
landslide of a layer of 6 m height towards South. It is also assumed that the slided
material is deposited as a layer of 3 m height to a 10 km wide and 3.6 km long area to
an approximate depth of 900 m at South (Fig. 4.15).
The model is applied to the underwater failure scenario of CASE 1 choosing a
simulation duration of 90 minutes in real time. The sea state at different time steps, the
water surface fluctuations at different selected station, the maximum positive tsunami
amplitudes are computed along the coastline. The initial waveform due to this scenario
and the distribution of maximum positive tsunami amplitudes in the sea as a layout,
near the North and South shores are shown together in Figure 4.15.
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Figure 4.15. The initial tsunami waveform and the distributions of maximum positive tsunami
amplitudes along north and south coasts due to the underwater landslide scenario at
offshore Yenikapı (CASE 1).
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The simulated arrival time of tsunami waves are approximately 25 min to Tekirda0
coasts, 16 minutes to Marmara Ere0lisi, less than 5 minutes to Yenikap , 10 minutes to
Armutlu Peninsula, and 30 minutes to Kocadere. The water surface can exceed 3 m at
North and 2.5 m at South along approximately 8 km of coastline (Fig. 4.15).
4.3.4.2. Tsunami due to underwater landslide occurrence at offshore Tuzla (CASE 2)
Although the stability of the slopes offshore Tuzla Cape is uncertain (Fig. 4.16), there
are suspicious parts on this slope, which might fail together or in parts with a seismic
trigger. As the case (CASE 2), a part of this slope is assumed to slide. According to this
analogy, an underwater landslide scenario is assumed to occur at the west part of
Tuzla Cape at around the 29.20ºE and 40.79ºN co-ordinates. The water depth at the
assumed failure area is 250 m. The failure area is assumed as 4.5 km wide along EastWest direction and 2 km long along North-South direction. The failure is assumed to
move as an underwater landslide of a layer of 5 m height towards South. It is also
assumed that the slide material is deposited as a layer of 2.8 m height to a 4.5 km wide
and 3.6 km long area to an approximate depth of 600 m at South (Fig. 4.16). The
model is applied to the underwater failure scenario of CASE 2 as in CASE 1.
The simulated arrival time of tsunami waves due to the failure scenario of CASE 2 are
about 35 min to Tekirdağ coasts, 20 minutes to Marmara Ere0lisi, less than 8 minutes
to Yenikapı , 3 minutes to Çınarcık and Yalova, and 25 minutes to Kocadere. It is also
seen that the water surface will exceed 3 m at North and South along approximately 5
km of coastline (Fig. 4.16).
4.3.4.3. Tsunami due to the scenario of fault break and two accompanying underwater
landslides at offshore Armutlu Peninsula (CASE 3)
The model is applied to a scenario by assuming a fault break at offshore Armutlu
Peninsula with 5 m vertical displacement and two accompanying underwater landslides
along it. In this case, it is assumed that Armutlu fault will break from 29.27°E, 40.66°N
to 28.60°E, 40.74°N with uplift at South of 2.87 m and subsidence at North of 1.97 m.
The initial waveform related to this fault break scenario is shown in Figure 4.17 . It is
also assumed that the Armutlu Fault triggered two different landslides simultaneously.
One of these landslides is selected to occur in between the co-ordinates 28.82°E,
40.71°N and 29.14°E, 40.68°N (25 km wide) with a slide of 1.3 km long and 6 m thick
material. Its deposition at the deeper location is assumed as 2 m thickness and 4.9 km
width. The other landslide is selected to occur in between the co-ordinates 28.59°E,
40.69°N and 28.74°E, 40.68°N (12 km wide) with a slide of 2.6 km long and 12 m thick
material. Its deposition at the deeper location is assumed as 4 m thickness and 7.8 km
long. The change of sea bottom topography related to these landslides is shown in
Figure 4.17.
The model is applied to the scenario of CASE 3. In application, the initial waveform is
inputted and the landslides are let to go to generate wave while the initial wave is
propagating. The distribution of maximum positive tsunami amplitudes along the
shorelines throughout the simulation duration and snapshots at different time steps (1,
5, 10 and 15th minutes) of the sea state are also shown in Figures 4.17 and 4.18 for
this scenario.
The arrival time of tsunami according to the scenario of CASE 3 is about 25 minutes to
Tekirdag, 0, 18 minutes to Marmara Ereglisi, 15 minutes to Yenikapi , a few minutes to
Çİnarc k, 6 minutes to Armutlu Peninsula and 25 minutes to Kocadere. It is also seen
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that the water surface will exceed 3 m at North and South along approximately 25 km
of coastline and exceeds 5 m at some locations (Fig. 4.18).
Figure 4.16. The initial tsunami waveform and the distributions of maximum positive tsunami
amplitudes along north and south coasts due to the underwater landslide scenario at
offshore Tuzla Cape (CASE 2). The vertical scale is exaggerated 7 times of horizontal scale
in 3D view of sea bottom at offshore Tuzla. The view is from the point at 1600 m above sea
level and form SW to NE direction.
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Figure 4.17. The initial tsunami waveforms due to the fault break and underwater landslide
scenario at offshore Armutlu Peninsula and the distributions of maximum positive tsunami
amplitudes along north and south coasts (CASE 3).
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Figure 4.18. Sea state at 1, 5, 10, 15th minutes of tsunami propagation according to fault
break and underwater landslide scenario at offshore Armutlu Peninsula (CASE 3).
4.4.
A COMPARATIVE STUDY ON AMPLITUDE EVOLUTION AND RUNUP OF
SOLITARY WAVES ON A SLOPING PLANE
The motions of long waves at shallower depths near the shoreline, run-up and the following
inundation have been studied using theoretical, experimental and numerical approaches.
Various analytical solutions for runup of nonlinear waves on plane slopes have been given by
Shuto (1967); Gjevik & Pedersen (1981); Pedersen and Gjevik, (1983); Kim et. al., (1983);
Synolakis (1987); Pelinovsky and Mazova (1992); Synolakis and Skjelbreia, (1993);
Pelinovsky et al (1996); Kanoglu, (1996); Kanoglu and Synolakis, (1997); Lin et. al., (1999);
Carrier and Yeh, (2002). In analytical approaches the runup problem is studied either by
using empirical formulae or by solving the governing equations for specific initial and
boundary conditions. Experimental data on runup of solitary waves are given among others
by Hall and Watts, (1953), Pedersen and Gjevik, (1983) and Synolakis (1987), Shankar and
Jayaratne, (2002), Lee and Raichlen, (2002).
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A detailed analytical and experimental study on the runup and amplitude evolution of solitary
waves on plane beaches is given in Synolakis, (1987). An exact solution to an approximate
theory for non-breaking solitary waves was introduced to derive the maximum runup
asymptotically. Laboratory experiments had been performed to support the theory and the
satisfactory prediction of the climb of the wave on the slope and maximum runup by linear
theory has been determined. Pelinovsky and Mazova (1992) investigated tsunami runup on a
beach with two different parameters; the angle of bottom slope and the breaking parameter.
Titov, Synolakis, (1995a, 1995b, 1998), Imamura (1995), Yalciner et, al., (2001), Hubbard,
Dodd, (2002), Lynett et. al., (2002), Lee and Raichlen, (2002), Maiti and Sen, (1999) are
some references of numerical studies on long wave runup. The computer program, TUNAMIN2, used for the simulation of the propagation of long waves is developed by Prof. Imamura
in Disaster Control Research Center in Tohoku University, Japan. TUNAMI-N2 is one of the
key tools for developing studies for propagation and coastal amplification of tsunamis in
relation to different initial conditions. It solves the nonlinear form of long-wave equations and
depth averaged velocities with bottom friction by finite difference technique for the basins of
irregular shape and bathymetry and provides us a very convenient tool to simulate tsunamis.
Shuto, Goto and Imamura (1990), Goto and Ogawa, (1992), Imamura, (1995), Goto et. al.
(1997), Yalciner et. al., (2001), Yalciner et. al., (2002) are some of the studies used TUNAMIN2.
In this study particularly, the behavior of solitary wave on a sloping beach and the runup
phenomenon by the non-linear numerical modeling (TUNAMI-N2) is studied. The shape of
the solitary wave on the plane slope, the maximum positive amplitudes near the coastline are
computed and compared with the analytical and experimental results (Demirbas, 2002).
Numerical Application with Solitary Wave
A solitary wave centered at a location x = X1 when t = 0 has the following surface profile:
H
 ( x,0)  sec h 2 γ( x  X 1 )
d
(1)
(2)
3H 1 / 2
γ(
)
4d
Where H is the amplitude of solitary wave, d is the water depth at the toe of the sloping
plane, X1 is the distance from the specified location.
The linearized long wave equations for the canonical problem are solved by Synolakis,
(1987) and the runup law is derived for the nonbreaking solitary waves.
1
5
(3)
R
H
 2.831(cot β) 2 ( ) 4
d
d
where R is the runup of solitary waves, β is the angle of sloping plane with horizontal.
The breaking condition of solitary waves on a sloping plane is presented by Gjevik &
Pedersen (1981):
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H
 0.479(cot β)
d
Chapter 4
10
9
(4)
This criterion has been reported to be in excellent agreement with laboratory data for solitary
waves by Synolakis, (1987).
The canonical problem named by Tadepalli and Synolakis (1994), in wave runup is the
determination of the runup of a long wave propagating over a constant depth region and then
climbing up a sloping beach of constant slope. There are a few numbers of studies for
different wave profiles on the canonical problem. We selected the canonical problem with a
regular shaped basin of 10 km length and width. The water depth of the horizontal bottom is
chosen as 30 m. On one side of the basin the plane beach is located with bottom slope of
1/20. The other boundaries are selected as open boundaries. The cross section of the basin
along x direction is shown in Fig. 4.19. The initial wave is inputted near the center of the
basin where the wave crest is parallel to the shoreline (along z axis) and thus the wave
propagation is forced along x direction towards shore without dispersion.
y
z
x
H initial
Gauges
d
Toe
β


Figure 4.19: Cross Section of the Basin, Location of the Initial Solitary Wave and the Gauge
Locations where the Water Surface Elevations are Computed
The location of the calculated maximum water elevation near the shoreline obtained by the
numerical model with finite difference method does not coincide with the location of the
actual runup. In the numerical model, the elevation of the water is computed at the fixed
locations of grid points. Obviously the smaller grid sizes result nearer maximum elevations to
actual runup. In this application the smallest possible grid size is selected to obtain optimum
run time and to obtain best possible comparison between experimental/analytical and
numerical results. The grid size and time step are selected as 20 m and 0.25 seconds
respectively in order to satisfy stability in computation. The time histories of water surface
elevations at different locations, the sea state at different time steps, the snapshots of the
surface profile along the axis of wave propagation at specified time step, the maximum water
elevation reached at every grid point throughout the domain during the simulation are
computed and stored. By using the stored data the shape of the wave at different locations
on the slope and the water surface along the axis of wave direction at specified time steps
are presented in the following. The results are compared with the analytical and experimental
data of Synolakis (1987).
There are two cases presented in Synolakis (1987). They are also selected in this
application. In these cases, the normalized height of incoming wave, (H/d) is 0.019 (nonbreaking) and 0.040 (breaking).
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The normalized water surface elevation (η) representing the climb of solitary wave at the toe
of the slope, and at the shoreline for the nonbreaking case on the 1:19.85 slope are shown in
Fig. 4.20 as function of the dimensionless time. As seen from this Figure that the numerical
model computes fairly consistent water surface fluctuation with experimental and analytical
data.
The comparison is extended to check the water surface profile along the axis of propagation
at different dimensionless time steps. The water surface profiles at different dimensionless
time steps are given in Fig. 4.21 and 4.22. As seen from these figures that the numerical
model provides fully consistent shape of the wave and amplitude evolution on the plane
slope with the analytical and experimental results, especially on the wet part of the slope for
the both breaking and non-breaking conditions of incoming solitary wave.
The numerical experiments are repeated by using different incoming solitary waves. The
normalized maximum positive wave amplitudes near the shoreline are computed for each
experiment. The comparison of numerical data with the runup law and experimental data of
Synolakis (1987) is given in Fig 4.23. This figure shows that the distribution of the data points
of numerical results show similar trend with the analytical and experimental data, but the
numerical results stay below the others. The underestimation of maximum amplitude in
numerical results comes from fixed grid size of the numerical solution.
Numerical results
Numerical results
Figure 4.20: The normalized water surface elevation representing the climb of solitary wave
at x=19.85 (at the toe of the slope), and at x=0.25 (at the shoreline) with H/d = 0.019 up a
1:19.85 slope as function of the dimensionless time. [----------, experimental (Synolakis,
1987); ------, analytical (Synolakis, 1987 ; ----------- , numerical (this study)]
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0,04
0,02
0
Numerical results
-0,02
-2
0
2
4
6
8
10
12
14
16
18
20
6
8
10
12
14
16
18
20
8
10
12
14
16
18
20
8
10
12
14
16
18
20
0,04
0,02
0
Numerical results
-0,02
-2
0
2
4
0,1
0,08
0,06
0,04
Numerical results
0,02
0
-0,02
-2
0
2
4
6
0,06
0,04
0,02
0
-0,02
Numerical results
-0,04
-2
0
2
4
6
Figure 4.21. The normalized water surface profile representing the climb of solitary wave
along the wave direction with H/d = 0.019 up a 1:19.85 slope as function of the normalized
distance at different dimensionless time steps, (a) t=25, (d) t=40, (g) t=55, (i) t=65. [....,
experimental (Synolakis, 1987); _____-, analytical (Synolakis, 1987 ; ----------- , numerical
(this study)]
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
74
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
0,15
Numerical results
0,1
0,05
0
-0,05
-5
0
5
10
15
20
10
15
20
10
15
20
10
15
20
0,15
Numerical results
0,1
0,05
0
-0,05
-5
0
5
0,25
0,2
0,15
Numerical results
0,1
0,05
0
-0,05
-5
0
5
0,25
0,2
0,15
Numerical results
0,1
0,05
0
-0,05
-5
0
5
Figure 4..22: The normalized water surface profile representing the climb of solitary wave
along the wave direction with H/d = 0.040 up a 1:19.85 slope as function of the normalized
distance at different dimensionless time steps, (a) t=20, (c) t=32, (e) t=44, (g) t=56. [....,
experimental (Synolakis, 1987); _____-, analytical (Synolakis, 1987 ; ----------- , numerical
(this study)]
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
75
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
1,0E+00
H/d
Run-up Law (1/20)
(numerical) (1/20) Non-Breaking
Lab Data Non-Breaking (Synolakis, (1987)
(numerical) (1/20) Breaking
Lab Data Breaking (Synolakis, (1987)
Figure 4.23: The Comparison of Numerically Computed Maximum Positive Wave
Amplitudes near the Shoreline with the Runup law and Experimental Data Given in
Synolakis, (1987)
The analytical results do not cover the non-linear terms of the long wave equations. Lin et al
(1999) states that the numerical results of the depth averaged equations models predict
smaller value of runup tongue. However analytical approach is consistent with the
experimental results. Therefore it is shown that the numerical approach computes
satisfactory agreement of water motion when the wave climbs on the slope. But the
computation gives smaller runup on the slope at land.
© A.C Yalciner, H. Karakus, C. Ozer, G. Ozyurt
76
Understanding the Generation, Propagation, Near- and Far-Field Impacts of TSUNAMIS
and Planning Strategies to Prepare for Future Events
Chapter 4
Acknowledgements
This research is supported by INTAS (01-2156) and RFBR (02-05-64426 and 03-0564975) for Russian co-authors. METU Basic Research Fund is acknowledged by its support
to Turkish author for developing and application of the model with the research number AFP2001-03-03-06. The depth data is developed by TUBITAK-YDABCAG-60 Basic Research
Grant in Turkey. The authors thank Prof. Costas Synolakis for his valuable discussions,
cooperation and helps during the revising process of this manuscript. The sincere gratitudes
are also extended to both Prof. Fumihiko Imamura and Prof. Costas E. Synolakis for their
long term endless collaboration with the principal authors Yalciner and Pelinovsky. The
tsunami propagation model TUNAMI-N2 is a registered copyright of Professors Imamura,
Yalciner and Synolakis.
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