Evaluating The Performance of Tsunami Propagation Models · KIAN, Rozita, YALCINER, Ahmet Cevdet,...

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,

[email protected]

YALCINER, Ahmet Cevdet

METU Department of Civil Engineering, Ocean Engineering Research, Ankara Turkey,

[email protected]

ZAYTSEV, Andrey

Special Research Bureau for Automation of Marine Researches, Far Eastern Branch of Russian

Academy of Sciences, 693013 Russia Uzhno-Sakhalinsk, Russia, [email protected]

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).

mailto:[email protected]



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.


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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)


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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


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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


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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


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, , (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)


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


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, , (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)


-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


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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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Evaluating The Performance of Tsunami Propagation Models · KIAN, Rozita, YALCINER, Ahmet Cevdet,...

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