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Bauhaus Summer School in Forecast Engineering: Global Climate change and the challenge for built environment
17-29 August 2014, Weimar, Germany
Evaluating The Performance of Tsunami Propagation Models
KIAN, Rozita
METU Department of Civil Engineering, Ocean Engineering Research, Ankara Turkey,
kian.rozita@metu.edu.tr
YALCINER, Ahmet Cevdet
METU Department of Civil Engineering, Ocean Engineering Research, Ankara Turkey,
yalciner@metu.edu.tr
ZAYTSEV, Andrey
Special Research Bureau for Automation of Marine Researches, Far Eastern Branch of Russian
Academy of Sciences, 693013 Russia Uzhno-Sakhalinsk, Russia, aizaytsev@mail.ru
Abstract
There are several numerical models computing the behavior of long waves and tsunamis under
different input wave and bathymetric conditions. Two of the applied models in this study are NAMI
DANCE (developed in collaboration with METU, Turkey and Special Bureau of Automation of
Research Russian Academy of Sciences, Russia) and FUNWAVE (developed by James T. Kirby et al.
(1998), University of Delaware) with the capability of modeling the waves considering the
hydrodynamic characteristic such as velocity and direction of the waves. The models consider the
dispersion effect in the simulating the tsunami propagation. The numerical simulations are performed
for various cases of uniform water depths, wave amplitudes and grid sizes and time steps using
momentum equations with and without dispersion. Comparisons show that in the both cases of using
nonlinear shallow water equations (without dispersion) and Boussinesq equations, the results are in
agreement for both models. The suggestions for the effect of grid size and time step selection to have
optimal simulation results in the long waves are presented and discussed.
1. Introduction
There are many numerical models in order to simulate tsunami propagation. Most of tsunamis are
studied as linear long wave theory in deep water where they are generated. However, in the shallow
water near the shorelines they are studied as nonlinear long wave theory since their height increase
while at the same time their wave length decrease. The long wave theory of Shuto (1991) is used as
base in the tsunami computations. Effects of frequency dispersion mostly in shallow water are
described by Boussinesq-tpe equations (Nwogo, 1993; Wei and Kirby, 1995; Kirby et al., 1998; Lynett
et al., 2002; Lynett and Liu, 2004; Lynett, 2007). Since the frequency dispersion effect is accumulative
it plays an important role in transoceanic tsunamis (Mei, 1989).
There are three most commonly used numerical models; COMCOT (Liu et al, 1994; 1998), TUNAMI-
N2 (Imamura, 1996) and MOST (Titov and Synolakis, 1998) which Non-Linear Shallow Water
Equations (NSWE) are applied with finite difference method. Then the TUNAMI-N2 code was
modified, improved and registered in USA granting copyright to Professors Imamura, Yalciner and
Synolakis in 2000 (Yalciner et al, 2001, 2002, 2003 and 2004; Kurkin et al, 2003; Zaitsev et al, 2002;
Yalciner and Pelinovsky, 2007). Then NAMI DANCE code is developed in order to do the
computational procedures of TUNAMI N2 in C++ language and is applied for tsunami simulations
and visualizations. The program has been applied to several tsunami events (Zaitsev et al, 2008; Ozer
et al, 2008, 2011; Yalciner et al, 2010).
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 2
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NAMI DANCE is developed by C++ programming language by following the staggered leap frog
scheme numerical solution procedures based on the calculation principals of TUNAMI-N2
(TUNAMI-N2, 2001). The added modules of NAMI DANCE made it an improved form of TUNAMI
N2 while providing direct simulations in nested domains with selective coordinate system (Cartesian
and spherical) and with selective equation type (as linear or non-linear), and efficient visualization in
multiprocessor environment. NAMI DANCE can perform the calculations by using all the processors
of the executed computer and increase the simulation speed by means of inputting discharge fluxes in
s and y direction through the domain (Ozer, 2012).
The Accuracy of numerical model used in this study, NAMI DANCE, is tested by performing a
simulation for the tsunami propagation in several uniform bathymetries similar to Yoon (2002). The
displacement results are then compared with linearized Nwogu’s Boussinesq equations (1993)
implemented in the FUNWAVE code (Wei and Kirby 1995; Kirby et al. 1998). FUNWAVE is a fully
nonlinear Boussinesq wave model with improved dispersion relationships for short waves. The
accuracy of FUNWAVE has been verified for various coastal problems such as shoaling, refraction,
diffraction and breaking of waves in Yoon et al.(2007). In this study the numerical simulations are
performed for various cases of uniform water depths, wave amplitudes and grid sizes and time steps
using momentum equations with and without dispersion in both models of NAMI DANCE and
FUNWAVE with comparisons. The suggestions for the effect of grid size and time step selection to
have optimal simulation results in the long waves are presented and discussed.
2. Numerical Models
In this section tsunami propagation in flat bathymetries are simulated and then the grid size and time
step effects are investigated via comparing the two models of NAMI DANCE and FUNWAVE.
2.1. NAMI DANCE
Tsunami numerical modeling by NAMI DANCE is based on the solution of nonlinear form of the long
wave equations with respect to related initial and boundary conditions. There are several numerical
solutions of long wave equations for tsunamis. In general the explicit numerical solution of Nonlinear
Shallow Water (NSW) equations is preferable for the use since it uses reasonable computer time and
memory, and also provides the results in acceptable error limit. The NAMI DANCE program for
simulating tsunami propagation has been applied to several tsunami events and used in many institutes
(Zaitsev et al, 2008; Ozer et al, 2008, 2011; Yalciner et al, 2010, 2012).
2.2. FUNWAVE
FUNWAVE is a fully nonlinear Boussinesq wave model with improved dispersion relationships for
short waves. Peregrine (1967) derives the standard Boussinesq equations for variable water depths.
Numerical models which use Peregrine’s equation as their base are comparable well with field data
(Elgar and Guza 1985). Assuming the effects of weak frequency dispersion in deep or intermediate
water leads to invalidity in Boussinesq equations application. The dispersion effect is approximated
polynomially and is only proper to be considered in shallow water regions. It is possible to select the
simulation type for different Boussinesq equations to consider dispersion effect or not by choosing ibe
values (control parameter for different types of Boussinesq equations). ibe=0 is for linearized Nwogu’s
equation, ibe=1 is for Nwogu’s (1993) extended Boussinesq equations, ibe=2 for fully nonlinear
Boussinesq equations of Wei et al. (1995); ibe=3 for Peregrine’s (1967) Boussinesq equations and
ibe=4 for nonlinear shallow water equations (Kirby 1998). In this paper the bathymetries are selected
flat and deep so ibe=4 is satisfactory.
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 3
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2.3. SIMULATIONS
Tsunami propagation in a uniform flat bathymetry with a Gaussian hump as an initial free surface
profile similar to Yoon (2002) and Yoon et al. (2007) is used to be simulated in NAMI DANCE and
FUNWAVE. The Gaussian function in Figure1 represents the displacement of the initial free surface.
Figure 1. Profile of initial free surface and the coordinate system [Yoon, 2002]
( ) ( ) (1)
( )
(2)
Where α denotes the characteristic radius of Gaussian function, and ( ) is the distance of
the Gaussian hump from the center and represents the angle from the x axis.
( ) ∫ ( ) [
√
√ ( )
] ( )
(3)
Where is the zero order Bessel function. By applying the parameters as ,
and water depth, , for 200, 300, 400, 600, 800, 1000, 1200 and , grid size , 400,
800, 1200, 1600m, and finally the time step is selected as 1, 3, 6 and 9sec for the tsunami
simulations.
Figure 2. Schematic picture of wave propagation containing initial wave as a source and gauge points
in horizontal direction in a domain. (not scaled)
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 4
4
The schematic top view of the bathymetry, initial source and the gauges are shown in the Figure 2.
This initial wave propagates radially in all directions, but in this paper only the x direction is studied,
however the waves propagate simetrically. Figures 3 shows the initial wave propagation in the domain
in several times. The grid size is 800m water depth is 400m and time step is 3sec. Figure 3(a) shows
the initial wave; Figure 3(b-d) represent the the tsunami propagation after 300, 900, 1800, 3000 and
4800sec in 12000m away from the source. The domain is . Figure 3 indicates
that tsunami in simulations propagate radially symetrical.
a) b)
c) d)
e) f)
Figure 3. Wave propagation snapshots for , and (a) initial wave,
after (b) 300sec, (c) 900sec, (d) 1800sec, (e) 3000sec, (f) 4800sec
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 5
5
In order to show that in deep water dispersion effect is neglegible, three simulations are depicted in
Figure 4. The parameters are as grid size equal to 800m, water depth 200m and time step 3sec. The
models are simulated once by considering only NLSWE in NAMI DANCE program and FUNWAVE
(ibe=4), then including dispersion effect with applying Boussinesq Equations (ibe=1). In the shallow
into intermediate water depths dispersion effect is important to be considered because without
considering the wave height is taller and shifted forward Kian et al. (2013). But as we see in Figure 4
the time history results for both methods either considering the dispersion effects or not they are very
close to each other. Hence, NLSWE is used in rest of simulations in this paper.
Figure 4. Time history comparison for FUNWAVE and NAMI DANCE models at gauge in 12000m
away from the initial wave center. ( , )
Also four models with different time steps ( ) are simulated in NAMI DANCE in
order to investigate the time step effect. Figure 5 represents that the results for several time steps are
very close to each other and NAMI DANCE program is not sensitive to the time step; therefore, for
the rest of the modelings is chosen.
Figure 5. Time history comparison in the gauge 12000m away from the initial wave center. (
, ) in NAMI DANCE
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
NLSWE-dx-800m-h200m NAMI DANCE-dt-3sec
FUNWAVE-dt-3sec-ibe4
FUNWAVE-dt-3sec-ibe1
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
NLSWE-dx-800m-h200m dt-1secdt-3secdt-6secdt-9sec
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 6
6
The grid size effect in NAMI DANCE model is studied in two different ways here. In the first one by
fixing the water depth 400m and changing the grid size (400m, 800m, 1600m), then the resluts are
compared with the FUNWAVE results (see Figure 6). In the second method the grid size is fixed as
800m in Figure 7 and 1200m in Figure 8 then the water depth is changed.
Figure 6. Time history comparison in the gauge 12000m away from the initial wave center.
, , (a) , (b) , (c)
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
a) NLSWE-dx-400m-h400m NAMI DANCE-dt-3sec
FUNWAVE-dt-3sec
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
b) NLSWE-dx-800m-h400m NAMI DANCE-dt-3sec
FUNWAVE-dt-3sec
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
c) NLSWE-dx-1600m-h400m NAMI DANCE-dt-3sec
FUNWAVE-dt-3sec
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 7
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Figure 7. Time history comparison in the gauge 12000m away from the initial wave center.
, , (a) h , (b) h , (c)
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
a) NLSWE-dx-800m-h600m NAMI DANCE-dt-3sec
FUNWAVE-dt-3sec
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
b) NLSWE-dx-800m-h800m NAMI DANCE-dt-3sec
FUNWAVE-dt-3sec
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel (
m)
Time (min)
c )NLSWE-dx-800m-h1000m NAMI DANCE-dt-3sec
FUNWAVE-dt-3sec
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 8
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Figure 8. Time history comparison in the gauge 12000m away from the initial wave center.
, , (a) h , (b) , (c) , (d) , (e)
-0.1
-0.05
0
0.05
0.1
0.15
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel
(m)
Time (min)
a) NLSWE-dx-1200m-h300m NAMI DANCE-dt-3secFUNWAVE-dt-3sec
-0.1
-0.05
0
0.05
0.1
0.15
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel
(m)
Time (min)
b) NLSWE-dx-1200m-h600m NAMI DANCE-dt-3sec
-0.1
-0.05
0
0.05
0.1
0.15
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel
(m)
Time (min)
c) NLSWE-dx-1200m-h800m NAMI DANCE-dt-3secFUNWAVE-dt-3sec
-0.1
-0.05
0
0.05
0.1
0.15
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel
(m)
Time (min)
d) NLSWE-dx-1200m-h1200m NAMI DANCE-dt-…FUNWAVE-dt-3sec
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100
Wat
er
Surf
ace
Le
vel
(m)
Time (min)
e) NLSWE-dx-1200m-h2400m NAMI DANCE-dt-3secFUNWAVE-dt-3sec
KIAN, Rozita, YALCINER, Ahmet Cevdet, ZAYTSEV, Andrey / FE 2014 9
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Comparison of simulation results of FUNWAVE and NAMI DANCE in Figure 6 indicates that when
grid size is chosen equal to the water depth they fit fairy well. Thus this situation is the best way for
NAMI DANCE to achive the results close to the FUNWAVE results. Figure 7 and 8 show that when
grid size is chosen greater than water depth value then NAMI DANCE obtains smaller wave height in
comparison to FUNWAVE’s results. In contrast, for grid size smaller than water depth, the NAMI
DANCE program attains greater values for water wave heights. The best fit is obtained when the
values are selected equally.
3. Results and Conclusion
As mentioned before there are several numerical models to simulate tsunami propagation in tsunami
hazard research area. In this paper we have employed two modeling programs including NAMI
DANCE and FUNWAVE. The effect of two parameters including the grid size and time step have
been investigated in these two programs by using NLSWE. Several grid sizes and time steps were
used in the preliminary simulations and it is concluded that NAMI DANCE is not sensitive to time
step selection; then was used in the rest of the models. The dispersion effect is negligible in
the deep water (see Figure 4), hence we used NLSWE in our simulations. Comparisons show that if
grid size is selected greater than water depth then NAMI DANCE results in smaller water wave height
than FUNWAVE. In the same fashion, when grid size is selected smaller than the water wave height,
NAMI DANCE leads to greater wave height. Furthermore, when they are set in equal values then the
results of NAMI DANCE fit very close to the FUNWAVE’s. Thus NAMI DANCE could be used as a
proper alternative to FUNWAVE in tsunami simulations provided that the grid size is set equally to
the water depth.
Acknowledgments
The authors thank Prof. James T. Kirby, Ge Wei, Qin Chen, Andrew B. Kennedy and Robert A.
Dalrymple for providing FUNWAVE for this study. This study is supported by TUBITAK 2215 Grant
for PhD Fellowship for Foreign Citizens, B.02.1.TBT.0.06.01.00-215.01-82/16157
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