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Numerical Study of Tsunami Propagation in Mentawai Islands West Sumatra Universal Journal of Geoscience 5(4): 112-116, 2017 http://www.hrpub.org DOI: 10.13189/ujg.2017.050404 Numerical Study of Tsunami Propagation in Mentawai Islands West Sumatra Noverina Alfiany*, Kazuhiro Yamamoto, Masaji Watanabe Graduate School of Environmental and Life Science, Okayama University, Japan Copyright©2017 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Shallow water equations were analyzed available from U.S Geological website. Satake, et. al [2] numerically for simulation of tsunami propagation from the computed 25th, October 2010, Mentawai Islands coastal source area to coastal zone of Mentawai Islands along West tsunami heights from the tsunami source model which was Sumatra Island. A triangular mesh was set for spatial proposed from a field survey and waveform recording with discretization and a system of partial differential equation is linear equations. The slip distributions on the fault planes reduced to a system of ordinary differential equations. were estimated through the inversion of tsunami waveform, Initial displacement based on Okada model was applied. whereas the strikes and rakes were estimated from USGS W Our numerical techniques were demonstrated with a phase solution. simulation of tsunami generated from an earthquake with In this study the propagation of tsunami waves generated epicenter near South Pagai Island, Mentawai Islands, in Mentawai Islands segment were studied numerically. A Indonesia. triangular mesh was employed for discretization of the governing equations, a system of partial differential Keywords Tsunami Simulation, Okada Model, Finite equations, to a system of ordinary differential equations. Element Method, ODE Solver, Mentawai Islands The earthquake fault plane parameters based on Satake, et. al [2] were used for computation with Okada theory [4,5]. Numerical results illustrate the tsunami waves propagation generated by the 2010 Mentawai Islands earthquake. 1. Introduction On the West Sumatra off coast, there are Mentawai 2. Materials and Methods Islands segment, which is a part of the Sumatra subduction zone, which results from the under-thrusting of the A system of partial differential equations consisting of Indo-Australian plate beneath the overriding Eurasian plate, momentum equations and a continuity equation were used, stretches over 5600 km [1]. In the last decade, these and reduced by using discretization based on a triangular Mentawai Islands segment produced many great earthquake mesh to a system of ordinary differential equations. with potential tsunami events, including the October 25th, 2010 Mentawai Islands tsunami earthquake. The earthquake 2.1. Governing Equations occurred at 21:42:22.0 with the epicenter 3.487ºS, 100.082ºE and 20 km depth (NOAA Significant Earthquake In the theory of long waves, the vertical acceleration of Data), and generated a tsunami within estimated 8 minutes water particles are negligible compared to the gravitational off the earthquake. Seismological analyses indicate that the acceleration except for an oceanic propagation of tsunami. Mentawai earthquake was a “tsunami earthquake”, which Consequently, the vertical motion of water particles has no produced a much larger tsunami than expected from the effect on the pressure distribution [6] which leads to two seismic magnitude [2]. dimensional equations with dynamic and kinetic conditions, Ulutaş [3] applied two kinds of tsunami numerical this is called the shallow water theory, can be obtained. models based on the shallow water equations to simulate the The numerical model is based on the nonlinear shallow South Pagai Island, Sumatra earthquake tsunami which water equations [7]. The following system of partial occurred on 25th, October 2010. The numerical models were differential equations (1)-(3) were solve numerically for analyzed by using nonlinear and dispersive long wave simulation of the tsunami propagation tsunami models, and earthquake parameters which are Universal Journal of Geoscience 5(4): 112-116, 2017 113 + + = 0, = + , (15) (1) 푑휂푖 휕푀 휕푁 휕휂 휕푀 휕푁 푑푡 휕푥 휕푦 휕푡 휕푥 휕푦 = − � �( + ) , + + +� +� + = 0, ( 2 ) ( ) 2 2 푑푀푖 휕 푀 휕 푀푁 휕휂 휕푀 휕 푀 휕 푀푁 휕휂 푔푛 (16) 7 2 2 푑푡 휕푥 휂+ℎ 휕푦 휂+ℎ 휕푥 푖 푖 푥 휕푡 휕푥 퐻 휕푦 퐻 휕푥 (2) − � � − � � − 푔 휂 ℎ − 퐹 � � � � 푔퐻 퐻3 √푀 푁 = ( + ) , ( ) ( 2 ) + + + + + = 0. 푖 2 2 푑푁 휕 푀푁 휕 푁 휕휂 휕푁 휕 푀푁 휕 푁 휕휂 푔푛 푖 푖 (17)푦 7 2 2 푑푡 − 휕푥 � 휂+ℎ � − 휕푦 � 휂+ℎ � − 푔 휕푦 휂 ℎ − 퐹 휕푡 휕푥 퐻 휕푦 퐻 휕푦 (3) � � � � 푔퐻 퐻3 √푀 푁 where Variables and are the flux of - and = + , (18) -components, respectively which are defined by 2 푀 푁 푥 ( 푔푛 ) 2 2 푥 7 푖 푖 푖 = , = (4) 푦 퐹 휂푖+ℎ푖 3 푀 �푀 푁 휂 휂 = + . (19) 2 where and푀 ∫−ℎ represent푢 푑푧 푁 the∫ −ℎ 푣 푑푧-component and ( 푔푛 ) 2 2 푦 7 푖 푖 푖 -component of velocity respectively, and = + is 3 � The partial derivatives퐹 휂푖+ℎ 푖 on 푁the 푀right-hand푁 sides of the the total푢 depth with푣 sea depth and푥 the water surface system equation (16)-(17) become elevation푦 from the mean sea level . The constant퐻 휂 isℎ the ℎ ( ) gravitational acceleration and is the Manning roughness = 2 , (20) ( 2 ) ( ) ( 2 ) coefficient. 휂 푔 휕 푀 푀푖 휕푀 푀푖 휕 휂+ℎ 2 푛 휕푥 � 휂+ℎ � 휂푖+ℎ푖 휕푥 − 휂푖+ℎ푖 (휕푥 ) = 2 , (21) 2.1.1. Finite Element Analysis ( 2 ) ( ) ( 2 ) 휕 푁 푁푖 휕푁 푁푖 휕 휂+ℎ 2 Let , ( = 1, 2, … , ) be nodal points of a 휕푦 휂+ℎ 휂푖+ℎ푖 휕푦 휂푖+ℎ푖 (휕푦 ) =� � + − , (22) ( ) ( ) ( ) ( ) ( = triangular 푗 mesh푗 in -plane. A basis function 휕 푀푁 푁푖 휕푀 푀푖 휕푁 푀푖푁푖 휕 휂+ℎ 2 1, 2, … , �푥) is푦 a� piecewise푗 linear푚 function that satisfies 휕푥 휂+ℎ 휂푖+ℎ푖 휕푥 휂푖+ℎ푖 휕푥 휂푖+ℎ푖 (휕푥 ) 푖 � � = + − . (23) 푥푦 ϕ 푖 ( ) ( ) ( ) ( ) 1 if = , 휕 푀푁 푁푖 휕푀 푀푖 휕푁 푀푖푁푖 휕 휂+ℎ 푚 , = (5) 2 0 otherwise, 휕푦 � 휂+ℎ � 휂푖+ℎ푖 휕푦 휂푖+ℎ푖 휕푦 − 휂푖+ℎ푖 휕푦 푖 푗 푗 푖 푗 ( ) ( ) for , = 1, 2,ϕ …�푥 , 푦 .� For� the analysis, functions are The partial derivatives and ear approximated by linear combinations of the basis functions 휕 휂+ℎ 휕 휂+ℎ 푖 푗 푚 approximated by and respectively.휕푥 휕푦 ( , , ) ( ) ( , ), (6) 휕휂 휕휂 푚 Values of partial휕푥 derivatives휕푦 at a nodal point are 푀( 푥, 푦, 푡) ≈ ∑푗=1 푀(푗 )푡 ϕ(푗 푥, 푦), (7) approximated by weighted averages of partial derivatives 푚 over elements that have the nodal point as vertices. For 푁( 푥, 푦, )푡 ≈ ∑푗=1 푁(푗 )푡 ϕ(푗 푥, 푦), (8) 푚 example, ( ) 푗=1 푗 (푗 ) 휂 푥 푦 푡, ≈ ∑ 휂 푡 ϕ 푥, 푦. (9) ( ) 푚 = , (24) Then the governing equations푗=1 (1)-푗(3)푗 become system of 푘 ℎ 푥 푦 ≈ ∑ ℎ ϕ 푥 푦 휕휂 1 푙 휕휂 푙 푘=1 푘 ( ) equations (10)-(12). �휕푥�푖 ∑푘=1 퐴푘 ∑ 퐴 �휕푥� = , (25) 푘 + + = 0, (10) 휕휂 1 푙 휕휂 푙 휕휂푗 푘=1 푘 푚 휕푀 휕푁 where , ,…,�휕푦�푖 are∑ 푘=1the 퐴areas푘 ∑ of퐴 the�휕푦 elements� which 푗=1 푗 +∑ 휕푡 ϕ + � 휕푥 휕푦�+ + ( ) ( 2 ) ( ) have the 1 -th2 node 푙as a common vertex, and is an 휕푀푗 휕 푀 휕 푀푁 휕휂 퐴 퐴 퐴 푚 푚 휕휂 푘 푗=1 휕푡+ 푗 =휕푥 0, 휂+ℎ 휕푦 휂+ℎ 휕푥 푗=1 (푗11) ∑ ϕ � � � � 푔 ∑ �휂 approximate푖 value of the partial derivative � 휕푥in� the -th 휕휂 ℎ푗 � ϕ푗 퐹푥+ + + + elements [7]. The collocation method was implemented in ( ) ( 2 ) �휕푥� 푘 푚 휕푁푗 휕 푀푁 휕 푁 휕휂 푚 analysis of the system of equations above for discretization 푗=1 + 푗 = 0, 푗=1 (12)푗 ∑ 휕푡 ϕ 휕푥 � 휂+ℎ � 휕푦 � 휂+ℎ � 푔 휕푦 ∑ �휂 on a triangular mesh, and the system of partial differential equations was transformed to a system of ordinary whereℎ푗 � ϕ푗 퐹푦 differential equations. The system of those ordinary 2 2 2 differential equations were solved numerically using a = 7 =1 + , (13) 푔푛 3 standard ODE solver i.e the fourth order =1 + 푚 푥 푗 푗 푗 푗 Adam-Bashforth-Moulton predictor-corrector in conjunction 퐹 푛 �∑ �푀 푁 �Φ ∑푗 �휂푗 2ℎ푗� Φ푗 2 2 with the fourth order Runge-Kutta method to generate values = 7 =1 + . (14) of approximate solutions at the first three steps. 푔푛 3 =1 + 푚 푦 푗 푗 푗 푗 퐹 푛 �∑ �푀 푁 �Φ System of ∑equations푗 �휂푗 ℎ푗� Φ(10)푗 -(12) become the system 2.2. Okada Model [4, 5] equation (15)-(17) Numerous theoretical formulations describing the 114 Numerical Study of Tsunami Propagation in Mentawai Islands West Sumatra deformation of an isotropic homogeneous semi-infinite 101º respectively, were estimated from USGS W phase medium have been developed with increasing completeness solution. The dip angles were assumed to be 7.5º for 2-5.92 and generality of source type and geometry, and range from km depths subfaults and 12º for 9.83-16.07 km depths the derivation of the surface displacement due to a point subfaults [2]. All those parameter values were employed in source to the strain fields at depth [4]. The closed analytical the Okada formula. expressions was checked and reviewed by Okada [4], which The vertical seafloor deformation for each subfaults are already published to describe the surface deformation which were computed using Okada model, was set as an due to shear fault in a half-space. Furthermore, an unknown initial condition to generate the initial tsunami wave. Figure solution for the displacements strains and tilts arising from 1 shows the initial surface deformation for the 28 subfaults opening-mode dislocations. in the fault plane source area.
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