Modeling of Tsunami Generation and Propagation by a Spreading Curvilinear Seismic Faulting in Linearized Shallow-Water Wave Theory

Affiliation(s)

Department of Basic and Applied Science, College of Engineering & Technology, Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt.

Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt.

Department of Basic and Applied Science, College of Engineering & Technology, Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt.

Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt.

ABSTRACT

The processes of tsunami evolution during its generation in search for possible amplification mechanisms resulting from unilateral spreading of the sea floor uplift is investigated. We study the nature of the tsunami build up and propagation during and after realistic curvilinear source models represented by a slowly uplift faulting and a spreading slip-fault model. The models are used to study the tsunami amplitude amplification as a function of the spreading velocity and rise time. Tsunami waveforms within the frame of the linearized shallow water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space) for the movable source models. We analyzed the normalized peak amplitude as a function of the propagated uplift length, width and the average depth of the ocean along the propagation path.

The processes of tsunami evolution during its generation in search for possible amplification mechanisms resulting from unilateral spreading of the sea floor uplift is investigated. We study the nature of the tsunami build up and propagation during and after realistic curvilinear source models represented by a slowly uplift faulting and a spreading slip-fault model. The models are used to study the tsunami amplitude amplification as a function of the spreading velocity and rise time. Tsunami waveforms within the frame of the linearized shallow water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space) for the movable source models. We analyzed the normalized peak amplitude as a function of the propagated uplift length, width and the average depth of the ocean along the propagation path.

KEYWORDS

Tsunami Modeling, Shallow Water Theory, Water Wave, Bottom Topography, Laplace and Fourier Transforms

Tsunami Modeling, Shallow Water Theory, Water Wave, Bottom Topography, Laplace and Fourier Transforms

Cite this paper

nullH. Hassan, K. Ramadan and S. Hanna, "Modeling of Tsunami Generation and Propagation by a Spreading Curvilinear Seismic Faulting in Linearized Shallow-Water Wave Theory,"*Applied Mathematics*, Vol. 1 No. 1, 2010, pp. 44-64. doi: 10.4236/am.2010.11007.

nullH. Hassan, K. Ramadan and S. Hanna, "Modeling of Tsunami Generation and Propagation by a Spreading Curvilinear Seismic Faulting in Linearized Shallow-Water Wave Theory,"

References

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[3] C. E. Synolakis and E. N. Bernard, “Tsunami Science Be-fore and Beyond Boxing Day 2004 Phil.,” Philosophi-cal Transactions of the Royal Society of London A, Vol. 364, No. 1845, 2006, pp. 2231-2265.

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[6] D. Dutykh, F. Dias and Y. Kervella, “Linear Theory of Wave Generation by a Moving Bottom,” Comptes Rendus Mathematique, Vol. 343, No. 7, 2006, pp. 499-504.

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[9] Y. Okada, “Surface Deformation Due to Shear and Tensile Faults in a Half Space,” Bulletin of the Seismological So- ciety of America, Vol. 75, No. 4, 1985, pp. 1135-1154.

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[11] T. Momoi, “Tsunami in the Vicinity of a Wave Origin,” Bulletin Earthquake Research Institute, Tokyo University, Vol. 42, 1964, pp. 133-146.

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[14] Y. Kervella, D. Dutykh and F. Dias, “Comparison between Three-Dimensional Linear and Nonlinear Tsunami Gener- ation Models,” Theoretical and Computational Fluid Dy- namics, Vol. 21, No. 4, 2007, pp. 245-269.

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[17] J. L. Bona, W. G. Pritchard and L. R. Scott, “An Evalua- tion of a Model Equation for Water Waves,” Philosophical Transactions of the Royal Society of London A, Vol. 302, No. 1471, 1981, pp. 457-510.

[18] M. S. Abou-Dina and F. M. Hassan, “Generation and Propagation of Nonlinear Tsunamis in Shallow Water by a Moving Topography,” Applied Mathematics and Compu- tation, Vol. 177, No. 2, 2006, pp. 785-806.

[19] F. M. Hassan, “Boundary Integral Method Applied to the Propagation of Non-Linear Gravity Waves Generated by a Moving Bottom,” Applied Mathematical Modeling, Vol. 33, No. 1, 2009, pp. 451-466.

[20] N. Zahibo, E. Pelinovsky, T. Talipova, A. Kozelkov and A. Kurkin, “Analytical and Numerical Study of Nonlinear Ef- fects at Tsunami Modeling,” Applied Mathematics and Computation, Vol. 174, No. 2, 2006, pp. 795-809.

[21] M. I. Todorovsk and M. D. Trifunac, “Generation of Tsu- namis by a Slowly Spreading Uplift of the Sea Floor,” Soil Dynamics and Earthquake Engineering, Vol. 21, No. 2, 2001, pp. 151-167.

[22] M. D. Trifunac and M. I. Todorovska, “A Note on Differ-ences in Tsunami Source Parameters for Submarine Slides and Earthquakes,” Soil Dynamics and Earthquake Engi- neering, Vol. 22, No. 2, 2002, pp. 143-155.

[23] M. I. Todorovsk, M. D. Trifunac and A. Hayir, “A Note on Tsunami Amplitudes above Submarine Slides and Slumps,” Soil Dynamics and Earthquake Engineering, Vol. 22, No. 2, 2002, pp. 129-141.

[24] M. D. Trifunac, A. Hayir and M. I. Todorovska, “A Note on the Effects of Nonuniform Spreading Velocity of Sub-ma- rine Slumps and Slides on the Near-Field Tsunami Ampli- tudes,” Soil Dynamics and Earthquake Engineering, Vol. 22, No. 3, 2002, pp. 167-180..

[25] M. D. Trifunaca, A. Hayira and M. I. Todorovska, “Was Grand Banks Event of 1929 a Slump Spreading in Two Directions,” Soil Dynamics and Earthquake Engineering, Vol. 22, No. 5, 2002, pp. 349-360..

[26] J. L. Hammack, “A Note on Tsunamis: Their Generation and Propagation in an Ocean of Uniform Depth,” Journal of Fluid Mechanics, Vol. 60, No. 4, 1973, pp. 769-799.

[27] D. Dutykh and F. Dias, “Water Waves Generated by a Moving Bottom,” In K. Anjan, Ed. Tsunami and Nonlinear waves, Springer-Verlag, Berlin, 2007, pp. 63-94.

[28] N. A. Haskell, “Elastic Displacements in the Near-Field of a Propagating Fault,” Bulletin of the Seismological Society of America, Vol. 59, No. 2, 1969, pp. 865-908.

[29] V. V. Titov and F. I. Gonzalez, “Implementation and Test- ing of the Method of Splitting Tsunami (MOST) Model,” NOAA/Pacific Marine Environmental Laboratory, No. 1927, 1997.

[30] A. Y. Bezhaev, M. M. Lavrentiev, A. G. Marchuk and V. V. Titov, “Determination of Tsunami Sources Using Deep Ocean Wave Records,” Center Mathematical Models in Geophysics, Bull. Nov. Comp., Vol. 11, 2006, pp. 53-63.

[31] D. R. Fuhrman and P. A. Madsen, “Tsunami Generation, Propagation, and Run-up with a High-Order Boussinesq Model,” Coastal Engineering, Vol. 56, No. 7, 2009, pp. 747-758.

[32] X. Zhao, B. Wang and H. Liu, “Modeling the Submarine Mass Failure Induced Tsunamis by Boussinesq Equations,” Journal of Asian Earth Sciences, Vol. 36, No. 4, 2009, pp. 47-55.

[33] H. Benioff and F. Pess, “Progress Report on Long Period Seismographs,” Geophysical Journal International, Vol. 1, No. 3, 1958, pp. 208-215.

[34] H. Kanamori and G. S. Stewart, “A Slowly Earthquake,” Physics of the Earth and Planetary Interiors, Vol. 18, No. 3, 1972, pp. 167-175.

[35] P. G. Silver and T. H. Jordan, “Total-Moment Spectra of Fourteen Large Earthquakes,” Journal of Geophysical Re- search, Vol. 88, No. B4, 1983, pp. 3273-3293.

[36] A. Hayir, “Ocean Depth Effects on Tsunami Amplitudes Used in Source Models in Linearized Shallow-Water Wave Theory,” Ocean Engineering, Vol. 31, No. 3-4, 2004, pp. 353-361.

[1] A. Ben-Menahem and M. Rosenman, “Amplitude Pat- terns of Tsunami Waves from Submarine Earthquakes,” Journal of Geophysical Research, Vol. 77, No. 17, 1972, pp. 3097-3128.

[2] E. O. Tuck and L. S. Hwang, “Long Wave Generation on a Sloping Beach,” Journal of Fluid Mechanics, Vol. 51, No. 3, pp. 449-461, 1972.

[3] C. E. Synolakis and E. N. Bernard, “Tsunami Science Be-fore and Beyond Boxing Day 2004 Phil.,” Philosophi-cal Transactions of the Royal Society of London A, Vol. 364, No. 1845, 2006, pp. 2231-2265.

[4] J. R. Houston and A. W. Garcia, “Type 16 Flood Insurance Study,” Usace Waterways Experiment Station Report, No. H-74-3, Vicksburg, 1974.

[5] S. Tinti and E. Bortolucci, “Analytical Investigation on Tsunamis Generated by Submarine Slides,” Annali Di Geofisica, Vol. 43, No. 3, 2000, pp. 519-536.

[6] D. Dutykh, F. Dias and Y. Kervella, “Linear Theory of Wave Generation by a Moving Bottom,” Comptes Rendus Mathematique, Vol. 343, No. 7, 2006, pp. 499-504.

[7] R. Takahasi and T. Hatori, “A Model Experiment on the Tsunami Generation from a Bottom Deformation Area of Elliptic Shape,” Bulletin Earthquake Research Institute, Tokyo University, Vol. 40, 1962, pp. 873-883.

[8] R. Takahasi, “On Some Model Experiments on Tsunami Generation,” International Union of Geodesy and Geo- physics, Vol. 24, 1963, pp. 235-248.

[9] Y. Okada, “Surface Deformation Due to Shear and Tensile Faults in a Half Space,” Bulletin of the Seismological So- ciety of America, Vol. 75, No. 4, 1985, pp. 1135-1154.

[10] K. Nakamura, “On the Waves Caused by the Deformation of the Bottom of the Sea I.,” Science Reports of the Tohoku University, Vol. 5, 1953, pp. 167-176.

[11] T. Momoi, “Tsunami in the Vicinity of a Wave Origin,” Bulletin Earthquake Research Institute, Tokyo University, Vol. 42, 1964, pp. 133-146.

[12] K. Kajiura, “Leading Wave of a Tsunami,” Bulletin Earth- quake Research Institute, Tokyo University, Vol.41, 1963, pp. 535-571.

[13] J. B. Keller, “Tsunamis: Water Waves Produced by Earth- quakes,” International Union of Geodesy and Geophysics, Vol. 24, 1963, pp. 150-166.

[14] Y. Kervella, D. Dutykh and F. Dias, “Comparison between Three-Dimensional Linear and Nonlinear Tsunami Gener- ation Models,” Theoretical and Computational Fluid Dy- namics, Vol. 21, No. 4, 2007, pp. 245-269.

[15] M. Villeneuve, “Nonlinear, Dispersive, Shallow-Water Waves Developed by a Moving Bed,” Journal of Hydrau- lic Research, Vol. 31, No. 2, 1993, pp. 249-266.

[16] P. L. Liu and J. A. Liggett, “Applications of Boundary Element Methods to Problems of Water Waves,” In P. K. Banerjee and R. P. Shaw Eds., Developments in Boundary Element Methods, 2nd Edition, Applied Science Publishers, England, Chapter 3, 1983, pp. 37-67.

[17] J. L. Bona, W. G. Pritchard and L. R. Scott, “An Evalua- tion of a Model Equation for Water Waves,” Philosophical Transactions of the Royal Society of London A, Vol. 302, No. 1471, 1981, pp. 457-510.

[18] M. S. Abou-Dina and F. M. Hassan, “Generation and Propagation of Nonlinear Tsunamis in Shallow Water by a Moving Topography,” Applied Mathematics and Compu- tation, Vol. 177, No. 2, 2006, pp. 785-806.

[19] F. M. Hassan, “Boundary Integral Method Applied to the Propagation of Non-Linear Gravity Waves Generated by a Moving Bottom,” Applied Mathematical Modeling, Vol. 33, No. 1, 2009, pp. 451-466.

[20] N. Zahibo, E. Pelinovsky, T. Talipova, A. Kozelkov and A. Kurkin, “Analytical and Numerical Study of Nonlinear Ef- fects at Tsunami Modeling,” Applied Mathematics and Computation, Vol. 174, No. 2, 2006, pp. 795-809.

[21] M. I. Todorovsk and M. D. Trifunac, “Generation of Tsu- namis by a Slowly Spreading Uplift of the Sea Floor,” Soil Dynamics and Earthquake Engineering, Vol. 21, No. 2, 2001, pp. 151-167.

[22] M. D. Trifunac and M. I. Todorovska, “A Note on Differ-ences in Tsunami Source Parameters for Submarine Slides and Earthquakes,” Soil Dynamics and Earthquake Engi- neering, Vol. 22, No. 2, 2002, pp. 143-155.

[23] M. I. Todorovsk, M. D. Trifunac and A. Hayir, “A Note on Tsunami Amplitudes above Submarine Slides and Slumps,” Soil Dynamics and Earthquake Engineering, Vol. 22, No. 2, 2002, pp. 129-141.

[24] M. D. Trifunac, A. Hayir and M. I. Todorovska, “A Note on the Effects of Nonuniform Spreading Velocity of Sub-ma- rine Slumps and Slides on the Near-Field Tsunami Ampli- tudes,” Soil Dynamics and Earthquake Engineering, Vol. 22, No. 3, 2002, pp. 167-180..

[25] M. D. Trifunaca, A. Hayira and M. I. Todorovska, “Was Grand Banks Event of 1929 a Slump Spreading in Two Directions,” Soil Dynamics and Earthquake Engineering, Vol. 22, No. 5, 2002, pp. 349-360..

[26] J. L. Hammack, “A Note on Tsunamis: Their Generation and Propagation in an Ocean of Uniform Depth,” Journal of Fluid Mechanics, Vol. 60, No. 4, 1973, pp. 769-799.

[27] D. Dutykh and F. Dias, “Water Waves Generated by a Moving Bottom,” In K. Anjan, Ed. Tsunami and Nonlinear waves, Springer-Verlag, Berlin, 2007, pp. 63-94.

[28] N. A. Haskell, “Elastic Displacements in the Near-Field of a Propagating Fault,” Bulletin of the Seismological Society of America, Vol. 59, No. 2, 1969, pp. 865-908.

[29] V. V. Titov and F. I. Gonzalez, “Implementation and Test- ing of the Method of Splitting Tsunami (MOST) Model,” NOAA/Pacific Marine Environmental Laboratory, No. 1927, 1997.

[30] A. Y. Bezhaev, M. M. Lavrentiev, A. G. Marchuk and V. V. Titov, “Determination of Tsunami Sources Using Deep Ocean Wave Records,” Center Mathematical Models in Geophysics, Bull. Nov. Comp., Vol. 11, 2006, pp. 53-63.

[31] D. R. Fuhrman and P. A. Madsen, “Tsunami Generation, Propagation, and Run-up with a High-Order Boussinesq Model,” Coastal Engineering, Vol. 56, No. 7, 2009, pp. 747-758.

[32] X. Zhao, B. Wang and H. Liu, “Modeling the Submarine Mass Failure Induced Tsunamis by Boussinesq Equations,” Journal of Asian Earth Sciences, Vol. 36, No. 4, 2009, pp. 47-55.

[33] H. Benioff and F. Pess, “Progress Report on Long Period Seismographs,” Geophysical Journal International, Vol. 1, No. 3, 1958, pp. 208-215.

[34] H. Kanamori and G. S. Stewart, “A Slowly Earthquake,” Physics of the Earth and Planetary Interiors, Vol. 18, No. 3, 1972, pp. 167-175.

[35] P. G. Silver and T. H. Jordan, “Total-Moment Spectra of Fourteen Large Earthquakes,” Journal of Geophysical Re- search, Vol. 88, No. B4, 1983, pp. 3273-3293.

[36] A. Hayir, “Ocean Depth Effects on Tsunami Amplitudes Used in Source Models in Linearized Shallow-Water Wave Theory,” Ocean Engineering, Vol. 31, No. 3-4, 2004, pp. 353-361.