Numerical modelling of tsunami waves: Application to the simulation of an earthquake generated tsunami

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1 Numerical modelling of tsunami waves: Application to the simulation of an earthquake generated tsunami Evangelia T. Flouri 1,2, Nikos Kaligeris 1,2, George Alexandrakis 1,3, Nikolaos A. Kampanis 1, and Costas E. Synolakis 1,2 1 Institute of Applied and Computational Mathematics, FORTH, Heraklion, Greece 2 Department of Environmental Engineering, Technical University of Crete, Chania, Greece 3 Department of Geology and Geoenvironment, University of Athens, Zographou, Greece flouri@iacm.forth.gr, n-kalligeris@hotmail.gr, g alex@geol.uoa.gr, kampanis@iacm.forth.gr, Costas.synolakis@enveng.tuc.gr Abstract. Tsunami waves (long waves) are effectively modelled by the nonlinear shallow water equations. These are often solved numerically by appropriate finite difference or finite volume techniques. For the applications presented herein, a dispersive, splitting direction based, finite difference method is applied for the construction of an approximate solution. All three phases of the evolution of the tsunami wave, the generation, propagation and runup are reproduced, providing a complete simulation capability of the tsunami wave. A case study considering an earthquake generated tsunami that strikes Crete island is considered. Seismic fault parameters are reconstructed from historical description of the specific earthquake and related geological and seismological references. Thus realistic initial and boundary conditions were generated, for the numerical simulation of the wave field of the induced tsunami. Actual bathymetric and topographic data of the coastal area considered are used for the description of the complex computational domain and grid generation. Inundation maps for the wave field and plots of damage metrics of the tsunami, like the flow depth and current speed, are also shown. Key words: earthquake generated tsunamis, propagation and runup, method of splitting tsunami, non-linear shallow-water equations, dispersive discretization scheme, tsunami damage metrics 1 Introduction Tsunami waves are long waves propagating in the sea. The actual free surface flow is turbulent and of geographical size, and its full description is given by the Navier-Stokes equations, in complex 3D domains. Appropriate turbulence closure models may be incorporated for more effective modeling of the cascade of length and time scales present. This makes the problem, expressed usually in the Reynolds-Averaged (RANS) form, computationally very demanding. Of particular difficulty is the discretization of the equations, the handling of the air-water interface movement and the poorly described turbulence characteristics.

2 A simpler, although reliable model, is obtained by depth averaging the NS equations and using the shallow water approximation. The water is considered as an incompressible fluid with depth-averaged horizontal particle velocities, with some variants as regards to friction, and Coriolis force (usually neglected). Pressure is assumed hydrostatic, vertical accelerations negligible and depth averaged viscous and turbulent stress are applied. As an alternative, the Bussinesq approximation is often used when dispersion is or is assumed to be relevant in modifying the waveforms. Such models are usually referred to as depth averaged models. They provide qualitative approximate solutions efficiently and are therefore widely used in engineering applications. In the present work the MOST model of NOAA has been used for the numerical simulation of the tsunami wave generated by an earthquake of approximate magnitude 8.5 placed in the maritime area SW of Crete. Details on the method of computing inundation are found in [15]; cf. also [16], [17], [18]. In the first section the numerical method used for the tsunami simulation is briefly described. In the section that follows, numerical results from the simulations performed are shown. In the closing section conclusions and a short discussion are presented. 2 Method of Splitting Tsunami (MOST) The evolution of earthquake-generated tsunami waves has three distinctive stages: generation, propagation, and run-up. The MOST numerical model computes all three stages, providing a complete tsunami simulation capability. The generation stage of tsunami evolution includes the formation of the initial disturbance of the ocean surface due to the earthquake-triggered deformation of the seafloor. This initial water surface disturbance evolves into a long gravity wave radiating from the earthquake source. Modelling of the initial stage of tsunami generation is therefore closely linked to studies of earthquake source mechanisms. The tsunami generation process, in the numerical model used, is based on a fault plane model of the earthquake source [6], [9], which assumes an incompressible liquid layer on an underlying elastic half space to characterize the ocean and Earth s crust. The implementation of this elastic fault plane model, [15], utilizes a formula for static sea-floor deformation to calculate the initial conditions required for subsequent computations of tsunami propagation and inundation. A tsunami can propagate long distances before it strikes a shoreline hundreds or thousands of kilometers from the earthquake source. To accurately model tsunami propagation over such large distances, Earth s curvature should be taken into account. Other factors, such as Coriolis forces and dispersion, may also be important. Dispersion changes the wave shape due to slightly different propagation speeds of waves with different frequencies. This effect can be taken into account even without the explicit use of dispersive terms in the governing equations; it was suggested in [11] that this process could be simulated by exploiting the numerical dispersion inherent in finite-difference algorithms. This method accounts for dispersive effects, but allows the use of non-dispersive linear or non-linear equations for wave propagation modelling. 2

3 The MOST propagation model uses a numerical dispersion scheme and the nonlinear shallow-water wave equations in spherical coordinates, with Coriolis terms, h t + (uh) λ + (νh cos Φ) Φ = 0, R cos Φ u t + uu λ R cos Φ + νu Φ R + gh λ R cos Φ = gd λ R cos Φ + fν, ν t + uν λ R cos Φ + νν Φ R + gh Φ R = gd Φ R fν, where λ is longitude, Φ is latitude, h = η(λ, Φ, t) + d(λ, Φ, t), η(λ, Φ, t) is the amplitude, d(λ, Φ, t) is the undisturbed water depth, u(λ, Φ, t) and v(λ, Φ, t) are the depthaveraged velocities in the directions of longitude and latitude, respectively, g is the gravity acceleration, f is the Coriolis parameter (f = 2ω sin Φ), and R is Earth s radius. In the MOST model, these equations are solved numerically using the splitting method described in [15]. Run-up of a tsunami onto dry land is probably the most underdeveloped part of any tsunami simulation model, primarily because of a serious lack of two major types of data, high quality field measurements for testing of the models, and fine resolution bathymetric and topographic data. It should be noted that improvements of the numerical simulations of the inundation process, has been supported by a series of large scale run up experiments conducted at the Coastal Engineering Research Center (CERC) of the U.S. Corps of Engineers, [3], and by several post tsunami surveys which provided high quality field data, cf. e.g. [13], [7], [2]. In [5] a finite volume based numerical model for the solution of the shallow water equations has been developed and has been successfully tested against experimental results. 3 A case study In the present study, the ground motion produced by an earthquake of magnitude 8.5 that had taken place in 365A.D in SW Crete. This event the largest know earthquake in the Mediterranean for the past 2000 years. The earthquake location and fault mechanism parameters used in the present report to derive ground motion prediction maps are taken from Shaw et al. (2008). Table 1: Discretization parameters used for the numerical model (MOST). Spatial resolution of bathymetry-topography grids Grid A coarse λ = Φ = 0.01 = m Grid B intermediate λ = Φ = = m Grid C fine λ = Φ = = m The impact of the selected earthquake induced tsunamis is evaluated for the city of Heraklion that is located in the central part of the Island of Crete. Figure 1 Simulation of the generation process in the MOST model is based on elastic deformation theory ([6], [9]) as described in [18]. 3

4 Fig. 1: The grids A, B and C Grid used for the tsunami simulations in the present study for the study area. For the numerical experiments a set of three nested grids (A, B and C) was used for each test site, with resolution as presented in Table 1. The nested grids used for the major sites of Heraklion are shown in Figure 1. 4 Numerical Results At first, numerical simulations were performed to model propagation (only) of the tsunami waves in the Eastern Mediterranean (an area much wider than the study area), using coarse bathymetry/topography data obtained by GEBCO. The aim was to test the initial source parameters and check the geographic extend of the propagation field. In Figure 10, we present the calculated maximum tsunami wave height in the wider area of SE Mediterranean. The results show strong impact in areas like North Africa and Peloponnesus, consistent with historical documents. In order to show the evolution of tsunami propagation with time, a video was produced. In Figure 2 we present some snapshots showing the wave amplitude in time +10sec, +12min and +50min, respectively, where T denotes the time of the seismic rupture (initial conditions, and assuming that the sea-floor deformation due to the earthquake is transferred to the free surface instantaneously). We have to note that a tsunami is not a single wave but rather a series of waves. The wave heights versus time for the city the of Heraklion are shown in Figure 3. A leading depression-n wave that causes the shoreline to withdraw appears in Heraklion, the first tsunami wave reaches the coast first after 23 min (0.38 h). Here the run-up in certain areas reaches 6m. The production of inundation maps plays an important role to the mitigation of tsunami hazards. Such maps are certainly useful in assessing the population, structures and facilities exposed at risk and may be proved useful in plans for emergency response. In Figure 4 is presented the outcome of the simulations in the form of inundation line superimposed on a satellite image for the area of Heraklion. Based on the simulation, 4

5 Fig. 2: Tsunami wave amplitude in +10sec, +12min and +50min. Fig. 3: Tsunami wave amplitude versus time (in hours) for Heraklion in a depth of 23 m. most of the coastal areas are flooded. The commercial harbor and the beaches located at the western side are the mostly affected in this region. Tsunamis can generate large onshore currents and cause dramatic damage to structures and move large objects far inland. In order to identify zones of potentially high tsunami hazard risk, it is important to introduce metrics that characterize the tsunami impact and reflect the distribution of forces over the entire impacted area. It is worth emphasizing that planning based solely on inundation maps of maximum wave height could be dangerously misleading since regions of high currents do not always correspond to regions of high wave height. For example, as the tsunami evolves over dry land, the flow depth decreases up to the point of maximum run-up, and the velocity of the shoreline tip becomes zero. During rundown the flow depth remains small, but the velocity can be substantial. We refer to [4] for a relevant example in a simple one dimensional setting. Also, it was confirmed in [8] that even for propagation over a simple geometry of a sloping beach, the highest velocity does not always occur close to the highest inundation depth location. For a discussion about metrics of interest that address the estimation of impact forces and currents we refer to [14] and the references therein. Next, we calculate some useful parameters that are of interest in assessing tsunami impact. 5

6 Fig. 4: The tsunami inundation line for the main city of Heraklion and the eastern part, superimposed on GoogleEarth satellite image. Fig. 5: Maximum wave height and run-up distribution for Heraklion. 6

7 Fig. 6: Maximum flow depth for Heraklion. Let d(x, y), in cartesian coordinates, denote the function determining the bathymetry and topography information with respect to an undisturbed mean sea level. We follow the convention that d(x, y) > 0 in the sea, while d(x, y) < 0 on land. The elevation of the water above the undisturbed mean sea level at each point (x, y) and at time t is denoted by η(x, y, t), and flow depth is defined as h(x, y, t) := η(x, y, t) + d(x, y). Let, also, V(x, y, t) denote the velocity vector, and v(x, y, t), u(x, y, t) denote its x and y components, respectively. Then by V (x, y, t) we denote the magnitude of V, i.e. V (x, y, t) := u 2 (x, y, t) + v 2 (x, y, t). Water elevation η(x, y, t), and flow depth h(x, y, t) := η(x, y, t) + d(x, y), and velocities v(x, y, t) and u(x, y, t) are computed by the numerical model; in the sequel their maxima in time are plotted. In Figure 5 we present the maximum wave height and the runup distribution for Heraklion, while in Figure 6 the maximum flow depth is shown for the same area. The areas that seem to be most vulnerable, in the sense that the maximum flow depth exceeds 2m, are the western coastal area and the area of Karteros bay in the east. The maximum calculated flow depth value is 4.63m and occurs along the eastern coast. In Figure 7 we plot the velocity vectors for time T = 0h, T = 0.5h, T = 0.6h T = 0.8h, T = 0.9h, T = 1.0h in Giofyros, a coastal area west of the city of Heraklion. 5 Conclusions In the present technical report, we presented in detail all scientific and technical work done under the task of numerical modeling of a tsunami generated by a seismic event with the characteristics of the selected seismic scenario (of 365 A.D.). Several numerical simulations were performed using the MOST model, which included full tsunami simulations (involving all three phases: generation, propagation and inundation) and highly accurate bathymetry and topography data were used for the correct representation of the tsunami inland penetration in the selected sites. 7

8 8 Fig. 7: Velocity vectors for time T = 0h, T = 0.5h, T = 0.6h, T = 0.8h, T = 0.9h, T = 1.0h in the area of Giofyros in Heraklion.

9 The results of the numerical simulations were post-processed and critical damage metrics were calculated and presented. Tsunami run-up distribution was calculated and presented for the major sites. Further, detailed tsunami inundation maps were produced (inundation lines superimposed on satellite images) for all sites. These maps play a significant role for tsunami hazard mitigation and form the necessary tool for the Civil Protection authorities in assessing the elements at risk (population, infrastructures, etc.) and in emergency response planning. Also, more tsunami damage metrics were defined and calculated for the areas of the major sites of Heraklion. Thematic maps for useful parameters such as tsunami flow depth, current speed and velocities, the momentum flux and the Froude number were produced and presented. All the above results and maps aim in providing up-to-date scientific results on the potential impact of a major tsunami event as the one in the selected scenario, and in providing all the necessary scientific information needed for the design of the exercise. References 1. Bernard, E.N, F.I. Gonzalez, and V.V. Titov (1997): Pacific Disaster Center Tsunami Forecasting Capabilities via the Tsunami Community Modeling Facility. A Proposal to the Office of the Assistant Deputy Under-Secretary of Defense for Space Integration, August, Borrero, J., M. Ortiz, V. Titov, and C.E. Synolakis (1997): Field survey of Mexican tsunami produces new data, unusual photos. Eos Trans. AGU, 78, 85, Briggs, M.J., C.E. Synolakis, G.S. Harkins, and D.R. Green (1995): Laboratory experiments of tsunami runup on circular island. Pure Appl. Geophys., 144(3/4), Carrier, G. F., Wu, T. T., Yeh, H., (2003), Tsunami runup and drawdown on a plane beach, J. Fluid Mech., 475, A. I. Delis, M. Kazolea, N. A. Kampanis, A robust high resolution finite volume scheme for the simulation of long waves over complex domains, Int. J. Num. Meth. In Fluids 56 (2008), Gusiakov, V.K. (1978): Static displacement on the surface of an elastic space. Ill-posed problems of mathematical physics and interpretation of geophysical data, Novosibirsk, VC SOAN SSSR, (in Russian). 7. Imamura, F., C.E. Synolakis, E. Gica, V. Titov, E. Listanco, and H.G. Lee (1995): Field survey of the 1994 Mindoro Island, Philippines tsunami. Pure Appl. Geophys., 144, Kanoglu, U., and C. E. Synolakis, (2006), Initial value problem solution of nonlinear a shallow water-wave equations, Phys. Rev. Lett., 97, Okada, Y. (1985): Surface deformation due to shear and tensile faults in a half space. Bull. Seism. Soc. Am., 75, Shaw, B., Ambraseys, N. N., England, P. C., Floyd, M. A., Gorman, G.J., Higham, T.F.G., Jackson, J.A., Nocquet, J.-M., Pain, C.C. & Piggott, M.D. (2008), Eastern Mediterranean tectonics and tsunami hazard inferred from the AD 365 earthquake. Nature. 11. Shuto, N. (1991): Numerical simulation of tsunamis. In Tsunami Hazard, Bernard, E. (ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, Smith, W.H.F., and D.T. Sandwell (1994): Bathymetric prediction from dense satellite altimetry and sparse shipboard bathymetry. J. Geophys. Res., 99, 21,803 21,824. 9

10 13. Synolakis, C.E., F. Imamura, S. Tinti, Y. Tsuji, H. Matsutomi, B. Cooke, and M. Usman (1995): The East Java tsunami of July 4, Eos Trans. AGU, 76, 257, C. E. Synolakis, Tsunami and Seiche, in Earthquake Engineering Handbook, W.-F. Chen, and C. Scawthorn (Eds.), CRC Press, Titov, V.V. (1997): Numerical modeling of long wave runup. Ph.D. thesis, University of Southern California, Los Angeles, StateCalifornia, 141 pp. 16. Titov, V.V., and C.E. Synolakis (1995): Modeling of breaking and nonbreaking long wave evolution and runup using VTCS-2. J. Waterways, Ports, Coastal and Ocean Engineering, 121, Titov, V.V., and C.E. Synolakis (1996): Numerical modeling of 3-D long wave runup using VTCS-3. In Long Wave Runup Models, P. Liu, H. Yeh, and C. Synolakis (eds.), World Scientific Publishing Co. Pte. Ltd., Singapore, Titov, V.V., and C. E. Synolakis (1997): Extreme inundation flows during the Hokkaido- Nansei-Oki tsunami. Geophys. Res. Lett, 24,