On the open sea propagation of 2004 global tsunami generated by the sea bed deformation

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1 On the open sea propagation of 00 global tsunami generated by the sea bed deformation Md. Fazlul Karim Principal Lecturer, Engineering Mathematics Unit, Faculty of Engineering, Universiti Teknologi Brunei, Jalan Tungku Link, Gadong, Brunei. D. S. Sankar Professor, Engineering Mathematics Unit, Faculty of Engineering, Universiti Teknologi Brunei, Jalan Tungku Link, Gadong, Brunei. Esa Yunus Department of Mechanical Engineering, Faculty of Engineering, Universiti Teknologi Brunei, Jalan Tungku Link, Gadong, Brunei. Abstract Over the past years, large effort has been done to model the 00 global tsunami along the coastal belts of North Sumatra in Indonesia and Penang Island in Peninsular Malaysia (see, Roy et al. 007, Thanh et al. 008). All these studies were based on the assumption that the initial displacement of the water surface at the source zone in the form of rise and fall is equal to the static shift of the sea floor deformation in the rupture zone, which is not entirely accurate. These studies have neglected the dynamics of seafloor displacement over a short period of time. The main factor, which determines the initial size of a tsunami, is the amount of vertical sea floor deformation (Iguchi, 0). The properties of Indonesian tsunami 00 are related to the magnitude of the bottom displacement and directional properties of the source (Kowalik et al. 00). In this paper, we present a reappraisal of the initial tsunami source of 00 Indonesian tsunami taking the amount of vertical sea floor deformation and the dynamics of seafloor displacement over a short period as the initial condition of tsunami generation. A non-linear shallow water model in the finite difference framework is used to simulate the 00 global tsunami propagation into the Indian Ocean and its effect on the coastal belts of North Sumatra and Peninsular Malaysia. The model results presented in this paper compared with the historical observation along the coastal belts of North Sumatra and Peninsular Malaysia. Keywords: Polar Coordinates; Shallow water model; Sumatra Island, Penang Island; Indonesian tsunami of 00; Tsunami propagation and surge. Introduction The earthquake generated tsunami of 00 originated at Sumatra is a global event as it propagated into three of the four global oceans. It propagated into the Pacific and Atlantic Oceans, in addition to covering most of the Indian Ocean (Murty et al. 00). The accuracy of the tsunami computation from generation to inundation obviously depends on the initial condition imposed in the model. In numerical simulations of tsunamis due to earthquake, tsunami modelers frequently assumed that the initial static sea surface deformation at initiation is the same as the seabed deformation after the occurrence of the earthquake (see, Karim et al. 008). The amount of vertical sea floor deformation determines the initial size of a tsunami (Kowalik et al. 00, Iguchi, 0). The initial tsunami waveform from the seafloor depends on a number of factors with the duration of rupture being an important factor. The assumption of initial tsunami generation that the initial sea surface deformation is equal to the vertical displacement of the sea floor is justified if the earthquake rupture occurs very rapidly (Denys, 008). However, in spite of extensive research work on the 00 global tsunami by a considerable number of researchers, the amount of free-water surface elevation is still not well understood, implying a great uncertainty in the tsunami propagation modelling. Even though there are constraints to the accurate prediction of tsunami height and arrival time at a particular location, we believe that sufficient attention has not been paied by previous researchers to the crucial subject of initial condition of tsunami generation. In the previous studies mentioned above, modelers take instant sea surface enhancement due to seabed deformations to obtain the initial source generation. Computations of Kowalik et al. (00) show that to consider the source generation as the time dependent sea bed rupture is indeed a very good approximation. While several modellers mentioned above have used the initial displacement of water surface is equal to the amount of the shift of seabed, this paper uses the amount of vertical seafloor displacement as an initial condition of tsunami propagation, which is the main factor of generation mechanism. The main feature of this paper is the use of more realistic initial condition of tsunami source generation. The vertically averaged shallow water equations are used to simulate the propagation of tsunami. The motivation of this study pertains to the attempt of incorporating a realistic source of tsunami to simulate its propagation and its impact along the coastal belts of North 686

2 Sumatra and Peninsular Malaysia. This paper is organized in 6 sections. In section, we present the model formulation of shallow water equations in polar coordinates. Section contains the numerical technique to solve shallow water equations and grid generation. The initial condition of tsunami source is validated in section. Section provides the results of this study and comparison between the propagation of tsunami in which the static sea surface elevation and the more realistic approach of dynamic seafloor deformation in which the wave s generated by the motion of the seabed. In section 6, results are summarized by offering a conclusion where we revel that the tsunami generated by the dynamic deformation of the seabed gives the accuracy of the numerical computation. Similar to Roy et al. (007), the depth averaged equations of continuity and momentums are r( h ) v r t t r r ( h) v 0 r vr vr v v v r r fv t r r F g r r ( h ) v v v v v r fvr t r r g F r ( h) where, vr is the component of radial velocity and v is the component of tangential velocity, is the sea level, is the sea bed displacement (Figure ), g is the earth gravity, and D h is the total depth of the ocean. The Coriolis parameter is taken as f sin, is the angular speed of the earth and is the latitude of a location in the analysis area. To simplify the sea bed friction terms in eqs. () and () the following notations is introduced: Fr C f vr vr v, F C fv vr v where, C f is the friction coefficient and is the water density, kg m -. () Figure : Map of the model domain with pole of the coordinate system Model Formulation Vertically integrated shallow water equations Shallow water equations are extensively used in tsunami modeling (Cho Y-S, Sohna 007). As the horizontal scale of the flow is much bigger than the depth of the fluid, tsunami waves can be characterized by shallow water equations. To study tsunami events due to sea floor deformation (uplift and subsidence), the shallow water equations are derived from equations of conservation of mass and conservation of momentum of fluid dynamics. These equations are commonly used in modeling of long wave, turbulent flow and other physical and biological flow problems (Gaur et al. 0, Roy et al. 007, Rahimi and Karim 0, Sankar and Karim 0, Tanahashi et al. 999 (a, b)). Figure : Schematic diagram which shows the sea bed deformation and sea surface elevation from the mean sea level (MSL) Boundary conditions and transformation to stretch along radial direction In this study, a model domain similar to Roy et al. (007) is considered. The pole is set at O (. N, 00 0 E). At the boundary of the coast, the normal velocity is set to zero. For open sea boundaries, similar to Johns et al. (98), the radiation types of boundary conditions are used to allow the disturbance to go out of the model area. The analysis area is bounded by the radial lines = 0, = = 9 through O and the circular arc r = R (Figure ). Following Roy et al. (007), to incorporate the bending of the coastline and the offshore 687

3 islands, the mesh size should be small near the coastal belt. The polar coordinate system ensures the finer mesh along tangential direction. If one takes the angle between two successive grid lines constant to keep the mesh size uniform along tangential direction that gives us physically an uneven grid system in the sense that the arc distance between two successive grid lines is finer near the coast than far away from the coast. Similar transformation of Roy et al. (007) is taken to perform the stretching along radial direction so that the mesh becomes finer near the coast. Grid Generation and Numerical Scheme In the transformed domain, similar to Roy et al. (007), the discrete grid points ( i, θ j ) are defined by i i ; i,,,......,m j j ; j,,,......, n where, m = 778 and n = 77; also = / and = /077 so that in the computational domain ranges from 0 to = 9 and ranges from 0 to 777/077. Although is a constant, r varies from 0.8 km to.9 km and the total radial distance is approximately 60 km. A sequence of time is given by t tk kt ; k,,,... Since the pole is considered at the land, where no computation is done, there will be no problem of instability during simulation. In the computational domain a staggered grid (Figure ) system is utilized where there are three distinct types u,v, of computational points. In this simulation, the velocity is discretized in a semi-implicit finite difference manner similar to Roy et al (007). At the coast of the boundary, the normal velocity is considered as zero, and this is achieved through proper stair step model. The time step is chosen secs to satisfy stability of the numerical simulation. In the solution process, following Kowalik et al (00), the value of the friction coefficient is chosen to be uniform (C f = 0.00) throughout the physical domain. The ocean depth data are taken from the Admiralty bathymetric charts Figure : Staggered grid system (roughly) Initial Condition Waves of the event Indonesian tsunami 00 at the sea surface are generated by the moving disturbances of sea bed. Most studies in -D of tsunami computation assume that the initial sea surface enhancement is equal to the vertical displacement of the sea bottom (see. Piatanesi and Lorito 007). The details of wave motions are neglected during the time that the rupture operates. While this is often justified because the earthquake rupture occurs rapidly, there are some specific causes where the duration of rupture plays an important role. During the event of December 00, there was in the northern extent of rupture relatively slow faulting that lead to significant vertical bottom motion (Denys 008). Several investigations are devoted for determining rupture zone associated with the earthquake event of 00 (see, Ni et al. 00; Ammon et al. 00; Ishii et al., 006; Lay et al.00). This study takes into account the dynamics of seabed displacement over a short period and do not consider the waveform as suddenly generated. All these studies suggest that the total rupture length is approximately 00 to 00 km, aftershock zone by propagating northward at ~.8 km s - for ~ 8 min. Thus, the initial disturbance of the sea surface along the source zone is time dependent. Moreover, the distribution of aftershocks and seafloor deformations indicate a time progression of the rupture with - separate subzones (Ammon et al. 00; Lay et al.00; Tanioka et al. 006). Based on elastic deformation theory of Okada (98), Kowalik (00) estimated the extent of rupture from southeast to north-west which is between 9-97E and -N of the seabed. Based on rupture parameters estimated by several authors, a reasonable tsunami source (length ~ 00 km and width ~ 00 km) has been constructed for the global tsunami of 00 in terms of magnitude and timing of seabed displacement. This rectangular zone has been divided into segments. Thus the source has been activated gradually from south to north and from west to east. Based on the information s outlined in section and, a reasonable dynamic tsunami source with a maximum seabed rise of m and subsidence of.7m is considered as the initial condition of tsunami. The initial sea floor displacement and the velocity components are taken as zero everywhere. Results and Discussions Tsunami arrival time and maximum amplitude of surge towards North Sumatra We consider the 0. m sea level rise as the arrival of tsunami. Figure depicts the arrival time contours, along the coast from North Sumatra. It is seen that after generation at source the surge propagates towards the coastal belts of North Sumatra and reaches at Banda Ache (north-west coast) in 0 min. Tsunami propagated around the North Sumatra and arrived along north-west coast within 0 to 80 min and the same along the north-east coast within 0 to 60 min. The local bathymetric effects near the shore of east coast could have played a role. Estimated tsunami arrival times reported in Alexander and Richard (007) are in good agreement with the computed arrival time obtained in this study. 688

4 Computed results of this model are close to the field measurements by a post tsunami Turkish-Indonesian-USA (Yalciner et al. 00) survey team. Figure 6: Elevation of Tsunami propagation towards Penang Island at 0 minutes Figure : Tsunami arrival time towards North Sumatra Figure 7: Elevation of Tsunami propagation towards Penang at 68 minutes Figure : Maximum water level computed around North Sumatra Tsunami propagation along Penang Island The tsunami propagation towards the Penang Island in Peninsular Malaysia can also be seen in Figs. 6-8, where the sea surface disturbance pattern is shown at three different times. Tsunamis from the source zone propagated around the Indian Ocean and struck the west coast of Penang Island 0 min after the earthquake and surged for inland (Figure 6). At 68 min the tsunami has proceeded considerably towards Penang Island after flooding the west of Penang Island (Figure 7). In 7 min the disturbance propagates further towards island and flooded the whole coastal belts of Penang (Figure 8). Figure 8: Elevation of Tsunami propagation towards Sumatra at 7 minutes Maximum surge towards Penang Island and arrival time Figure 9 depicts the maximum water elevation along Penang coast. The maximum elevation at Penang Island ranges from m (south-east) to m (maximum at north-west) along the coastal belts of Penang. The west coast shows an increasing trend towards the north. On the east coast, the calculated tsunami heights show a decreasing trend towards the north. 689

5 The coastal surge estimated by this model for Penang region is close to the field survey reported in Roy et al. (007). We computed the times of attaining maximum water elevations along these regions. Figure 0 shows that the time of attaining maximum surges at the north-west coast of Penang is 0 min and within 80 min surge reached in east coast. Satake et al. (00) reported that tsunami waves reached at Penang Island in 0 minute after the earthquake. Thus, the model shows slightly different results than observation. The reason for this inconsistency needs to be investigated. Since the wave celerity is very sensitive to the ocean topography, any error in the local topographic data could have influenced the computation. Figure 9: Maximum water level computed around Penang Island one (maximum). Similar pattern is observed in some other places. In general the surge amplitude has gradually decreased as one goes to east of north Sumatra. There emerges to be no regular deviation, and local topographic effects could have played a role. This model results is very close to the field measurement on Sumatra made by the Turkish-Indonesian-USA team (Yalciner et al. 00) during January -9, 00. Similar patterns are observed for the two coastal locations Penang Island in west Malaysia (Figure ). At south-east coast of Penang, the maximum water level is. m and the water level continues to oscillate for long time (Figure a) and the same is. at north-west (Figure b). In north-west, the second wave was the height wave. All the coastal locations in both the islands the variation of tsunami height is found to be irregular. This happens due to local topographic and bathymetric effects. Simulated results of Murty et al. (00) show the similar patterns in cases of near field in the Indian Ocean and far field in the Pacific Ocean. Thus, our model results consistent with the model of Murty et al. (00). The model results with particular reference to the tsunami arrival time, maximum surge and time histories of sea surface elevation show qualitatively similar with quantitatively differences to the previous studies of Roy et al. (006). Although the computed results we have obtained using the new reappraisal of the initial tsunami source with reference to arrival time, maximum surge and the time series of sea surface elevation are qualitatively similar to the computed results of previous studies of Roy et al. (006), results of this study show quantitative differences and better agreement with the field measurements. Figure (a). Figure 0: Tsunami arrival time towards Phuket Island Computed water levels along the Coastal belts of Sumatra and Penang Island We used the 0 secs sea surface enhancement record in this simulation. The time series of sea surface fluctuations at two offshore coastal locations of Sumatra Island are shown in figure. The maximum amplitude near the coastline of Banda Aceh (north-west Sumatra) is approximately 9.8 m (Figure a) and the oscillation continues for several hours with gradually decreased amplitude. At Medan (east coast of Sumatra) time series of sea level shows similar pattern as that of Banda Aceh where the maximum elevation is found to be.8 (Figure b). It is important to note that tsunami height of the first wave is found to be lower than the second and fourth Figure (b). Figure : Time histories of computed elevation at two coastal locations of North Sumatra: (a) North-west (b) East coast 690

6 elevation in m elevation in m time in hr Figure (a) time in hr Figure (b). Figure : Time histories of computed elevation at two coastal locations of Penang Island: (a) South-east coast (b) North-west Conclusion We studied the propagation of the global tsunami of 00 in Indian Ocean in Polar coordinates. In this study, a more realistic time dependent source is incorporated to generate initial tsunami. We studied the characteristics of this tsunami, with particular reference to the arrival time, maximum surge and time histories of sea surface elevation. It is interesting to note that first wave was having smaller amplitude then the second or third waves in some places. References [] Alexander, B., and Richard, E., The 6 December 00 Sumatra Tsunami: Analysis of tide gauge data from the world ocean part. Indian Ocean and South Africa, Pure and Applied Geophysics, 6, 6-08, 007. [] Ammon, J.C., Ji, C., Thio, H., Robinson, D., Ni, S., Hjorleifsdottir, V., Kanamori, H., Lay, T., Das, S., Helmberger, D., Ichionose, G., Polet, J., and Wald, D., Rupture Process of the 00 Sumatra-Andaman Earthquake, Science 08, -9, 00. [] Gaur, P. C., Ismail, A. I. M., Karim, M. F., Implementation of method of lines to predict water levels due to a storm along the coastal region of Bangladesh, Journal of Oceanography, The Oceanographic Society of Japan and Springer Japan, DOI 0.007/s x, 99-0, 0. [] Cho, Y. S., Sohna, D. H., and Lee, S., Practical modified scheme of linear shallow-water equations for distant propagation of tsunamis, Ocean Eng V: , 007. [] Denys, D., Modelisation mathematique des tsunamis, Ph.D thesis, Centre de mathematiques et de leurs applications, ENS CACHAN/CNRS/UMR 86, France, version -, 008. [6] Iguchi, T., A mathematical analysis of tsunami generation in shallow water due to sea bed deformation, Proceedings of the Royal Society of Edinburgh, A, -608, 0. [7] Johns, B., Rao, A. D., Dube, S. K., Sinha P. C., Numerical modeling of tide surges interaction in the Bay of Bengal, Phil. Trans. R. Sco. of London A, 07-, 98. [8] Karim, M. F., Roy, G.D., Ismail, A. I. M., A Study of Open Boundary Conditions for Far Field Tsunami Computation, WSEAS Transactions on Environment and Development, (), -9, 008. [9] Kowalik, Z., Knight, W., Whitmore, P. M., Numerical Modeling of the Tsunami: Indonesian Tsunami of 6 December 00, Journal of Science of Tsunami Hazards, (), 0-6, 00. [0] Lay, T., Kanamori, H., Ammon, J. C., Nettles, M., Steven., N. W., Aster, R., Susan, L. B., Michael, R., B., and Butler, B., The great Sumatra-Andaman Earthquake of 6 December 00, Science 08, 7-, 00. [] Ni, S., Kanamori, H., and Helmberger, D., Energy radiation from the Sumatra Earthquake, Nature (70), 8-8, 00. [] Murty, T. S., Nirupama, N., Nistor, I., and Hamdi, S., Far-field characteristics of the tsunami of 6 December 00, ISET Journal of Earthquake Technology, Technical Note, vol., No., - 7, 00. [] Okeda, Y., Surface Deformation due to Shear and Tensile Faults in a Half Space, Bull. Seism. Soc. Am. 7, -, 98. [] Piatanesi, A., and Lorito, S., Rupture process of the 00 Sumatra-Andaman Earthquake from Tsunami waveform inversion, Bull. Seismol. Soc. Am, 97, -, doi 0. 78/000067, 007. [] Roy, G. D., Karim, M. F., and Ismail, A. M., A Non-Linear Polar Coordinate Shallow Water Model for Tsunami Computation along North Sumatra and 69

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