Evaluating The Performance of Tsunami Propagation Models

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), TUNAMIN2 (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

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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. 2


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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 ( the Gaussian hump from the center and represents the angle from the x axis. (

)

∫

(

)

[

√ √

(

)

] (

)

)

is the distance of

(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.

Water Surface Level (m)

NAMI DANCE-dt-3sec FUNWAVE-dt-3sec-ibe4 FUNWAVE-dt-3sec-ibe1

NLSWE-dx-800m-h200m

0.2 0.15 0.1 0.05 0 -0.05 -0.1 0

20

40

60

80

100

Time (min)

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.


0.2

NLSWE-dx-800m-h200m

dt-1sec dt-3sec dt-6sec dt-9sec

0.15 0.1 0.05 0 -0.05 -0.1 0

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Time (min)

60

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Figure 5. Time history comparison in the gauge 12000m away from the initial wave center. ( , ) in NAMI DANCE

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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.


a) NLSWE-dx-400m-h400m

NAMI DANCE-dt-3sec

0.2

FUNWAVE-dt-3sec

0.15 0.1 0.05 0 -0.05 -0.1 0

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Time (min) b) NLSWE-dx-800m-h400m

0.2

FUNWAVE-dt-3sec

0.15 0.1 0.05 0 -0.05 -0.1 0 0.2


NAMI DANCE-dt-3sec

20

40

Time (min)

60

c) NLSWE-dx-1600m-h400m

80

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NAMI DANCE-dt-3sec

0.15

FUNWAVE-dt-3sec

0.1 0.05 0 -0.05 -0.1 -0.15 0

20

40

60

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100

Time (min)

Figure 6. Time history comparison in the gauge 12000m away from the initial wave center. , , (a) , (b) , (c)

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0.2

7

NAMI DANCE-dt-3sec


FUNWAVE-dt-3sec

0.15 0.1 0.05 0 -0.05 -0.1 0

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Time (min) b) NLSWE-dx-800m-h800m

0.2

NAMI DANCE-dt-3sec FUNWAVE-dt-3sec

0.15 0.1 0.05 0 -0.05 -0.1 0

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40

60

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Time (min) c )NLSWE-dx-800m-h1000m


0.2


0.15 0.1 0.05 0 -0.05 -0.1 0

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Time (min)

60

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Figure 7. Time history comparison in the gauge 12000m away from the initial wave center. , , (a) h , (b) h , (c)

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0.15

8



0.1 0.05 0

-0.05 -0.1 0

40 Time (min) 60

80

b) NLSWE-dx-1200m-h600m

0.15


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NAMI DANCE-dt-3sec

0.1

0.05 0

-0.05 -0.1 0

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c) NLSWE-dx-1200m-h800m

0.15 Water Surface Level (m)

20

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0.05 0

-0.05 -0.1 0


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40 Time (min) 60

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NAMI DANCE-dt-… FUNWAVE-dt-3sec

d) NLSWE-dx-1200m-h1200m

0.1

0.05 0

-0.05 -0.1 0


0.2

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e) NLSWE-dx-1200m-h2400m

0.15 0.1

0.05 0

-0.05 -0.1 0

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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) 8


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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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