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Calculation of the Initial Elevation of the Water Surface at the Source of a Tsunami in a Basin with Arbitrary Bottom Topography

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Mathematical Models and Computer Simulations Aims and scope

Abstract

A two-dimensional (0xz) numerical model that makes it possible to calculate the initial elevation of the water surface in the source of a tsunami in a basin of variable depth is developed in the potential theory of an incompressible fluid under the approximation of an instantaneous deformation of the sea bottom. The model makes it possible to take into account the contribution of the horizontal component of the bottom deformation and the smoothing effect of the water layer through the use of the σ coordinate. To test the numerical model, we obtained an analytical solution to the problem of the initial elevation in a basin with a flat sloping bottom with a bottom deformation of a triangular shape. The results of the test show that for a spatial step typical for numerical tsunami models, there is close agreement between the numerical and analytical solutions. Using the developed σ model, we calculate the initial elevations of the water surface during the Kuril earthquake on January 13, 2007 and the Great East Japan Tohoku earthquake on March 11, 2011 (along the selected 2D sections). The results obtained are used to test an approximate method for calculating the initial elevation, known as the Kajiura filter, in which the ocean depth is assumed to be constant throughout the area of the source of the tsunami.

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REFERENCES

  1. M. A. Nosov, “Tsunami waves of seismic origin: The modern state of knowledge,” Izv. Atmos. Ocean. Phys. 50 (5), 474–484 (2014). https://doi.org/10.1134/S0001433814030098

    Article  Google Scholar 

  2. A. R. Gusman, I. E. Mulia, K. Satake, S. Watada, M. Heidarzadeh, and A. F. Sheehan, “Estimate of tsunami source using optimized unit sources and including dispersion effects during tsunami propagation: The 2012 Haida Gwaii earthquake,” Geophys. Res. Lett. 43 (18), 9819–9828 (2016). https://doi.org/10.1002/2016GL070140

    Article  Google Scholar 

  3. G. C. Lotto, G. Nava, and E. M. Dunham, “Should tsunami simulations include a nonzero initial horizontal velocity?,” Earth, Planets Space 69 (1), 117, 1–14 (2017). https://doi.org/10.1186/s40623-017-0701-8

  4. T. Maeda and T. Furumura, “FDM simulation of seismic waves, ocean acoustic waves, and tsunamis based on tsunami-coupled equations of motion,” Pure Appl. Geophys. 170 (1), 109–127 (2013). https://doi.org/10.1007/s00024-011-0430-z

    Article  Google Scholar 

  5. J. E. Kozdon and E. M. Dunham, “Constraining shallow slip and tsunami excitation in megathrust ruptures using seismic and ocean acoustic waves recorded on ocean-bottom sensor networks,” Earth Planet. Sci. Lett. 396, 56–65 (2014). https://doi.org/10.1016/j.epsl.2014.04.001

    Article  Google Scholar 

  6. G. C. Lotto, T. N. Jeppson, and E. M. Dunham, “Fully coupled simulations of megathrust earthquakes and tsunamis in the Japan Trench, Nankai Trough, and Cascadia Subduction Zone,” Pure Appl. Geophys. 176, 4009–4041 (2019). https://doi.org/10.1007/s00024-018-1990-y

    Article  Google Scholar 

  7. S. Popinet, “Adaptive modelling of long-distance wave propagation and fine-scale flooding during the Tohoku tsunami,” Nat. Hazards Earth Syst. Sci. 12 (4), 1213–1227 (2012). https://doi.org/10.5194/nhess-12-1213-2012

    Article  Google Scholar 

  8. S. Iwasaki, “Experimental study of a tsunami generated by a horizontal motion of a sloping bottom,” Bull. Earthquake Res. Inst. 57, 239–262 (1982).

    Google Scholar 

  9. Y. Tanioka and K. Satake, “Tsunami generation by horizontal displacement of ocean bottom,” Geophys. Res. Lett. 23 (8), 861–864 (1996). https://doi.org/10.1029/96GL00736

    Article  Google Scholar 

  10. M. A. Nosov, A. V. Bolshakova, and S. V. Kolesov, “Displaced water volume, potential energy of initial elevation, and tsunami intensity: Analysis of recent tsunami events,” Pure Appl. Geophys. 171 (12). 3515–3525 (2014). https://doi.org/10.1007/s00024-013-0730-6

    Article  Google Scholar 

  11. A. V. Bolshakova, M. A. Nosov, and S. V. Kolesov, “The properties of co-seismic deformations of the ocean bottom as indicated by the slip-distribution data in tsunamigenic earthquake sources,” Moscow Univ. Phys. Bull. 70 (1), 62–67 (2015). https://doi.org/10.3103/S0027134915010038

    Article  Google Scholar 

  12. M. A. Nosov, A. V. Bolshakova, and K. A. Sementsov, “Energy characteristics of tsunami sources and the mechanism of wave generation by seismic movements of the ocean floor,” Moscow Univ. Phys. Bull. 76, S136–S142 (2021). https://doi.org/10.3103/S0027134922010076

    Article  Google Scholar 

  13. R. Takahashi, “On seismic sea waves caused by deformations of the sea bottom,” Bull. Earthquake Res. Inst. 20, 357–400 (1942).

    MathSciNet  MATH  Google Scholar 

  14. K. Kajiura, “The leading wave of a tsunami,” Bull. Earthquake Res. Inst. 41, 535–571 (1963).

    Google Scholar 

  15. M. A. Nosov and S. V. Kolesov, “Optimal initial conditions for simulation of seismotectonic tsunamis,” Pure Appl. Geophys. 168 (6–7), 1223–1237 (2011). https://doi.org/10.1007/s00024-010-0226-6

    Article  Google Scholar 

  16. M. A. Nosov and K. A. Sementsov, “Calculation of the initial elevation at the tsunami source using analytical solutions,” Izv. Atmos. Ocean. Phys. 50 (5), 539–546 (2014). https://doi.org/10.1134/S0001433814050089

    Article  Google Scholar 

  17. M. A. Nosov, Introduction to Tsunami Wave Theory (Yanus-K, Moscow, 2019) [in Russian].

    Google Scholar 

  18. K. Aki and P. G. Richards, Quantitative Seismology, 2nd ed. (Univ. Sci. Books, Sausalito, CA, 2002).

  19. C. Ji, D. J. Wald, and D. V. Helmberger, “Source description of the 1999 Hector Mine, California, earthquake, Part I: Wavelet domain inversion theory and resolution analysis,” Bull. Seismol. Soc. Am. 92 (4), 1192–1207 (2002). https://doi.org/10.1785/0120000916

    Article  Google Scholar 

  20. S. E. Minson, J. R. Murray, J. O. Langbein, and J. S. Gomberg, “Real-time inversions for finite fault slip models and rupture geometry based on high-rate GPS data,” J. Geophys. Res.: Solid Earth 119, 3201–3231 (2014). https://doi.org/10.1002/2013JB010622

    Article  Google Scholar 

  21. S. Kawamoto, Y. Ohta, Y. Hiyama, M. Todoriki, T. Nishimura, T. Furuya, Y. Sato, T. Yahagi, and K. Miyagawa, “REGARD: A new GNSS-based real-time finite fault modeling system for GEONET,” J. Geophys. Res.: Solid Earth 122, 1324–1349 (2017).

    Article  Google Scholar 

  22. S. Watada, S. Kusumoto, and K. Satake, “Traveltime delay and initial phase reversal of distant tsunamis coupled with the self-gravitating elastic Earth,” J. Geophys. Res.: Solid Earth 119, 4287–4310 (2014).

    Article  Google Scholar 

  23. T.-C. Ho, K. Satake, and S. Watada, “Improved phase corrections for transoceanic tsunami data in spatial and temporal source estimation: Application to the 2011 Tohoku earthquake,” J. Geophys. Res.: Solid Earth 122 (12), 10155–10175 (2017). https://doi.org/10.1002/2017JB015070

    Article  Google Scholar 

  24. J. L. Hammack, “A note on tsunamis: their generation and propagation in an ocean of uniform depth,” J. Fluid Mech. 60 (4), 769–799 (1973). https://doi.org/10.1017/S0022112073000479

    Article  MATH  Google Scholar 

  25. S. N. Ward, “Tsunamis,” in The Encyclopedia of Physical Science and Technology, 3rd ed., Ed. by R. A. Meyers (Academic Press, San Diego, 2001), Vol. 17, pp. 175–191.

    Google Scholar 

  26. Y. Kervella, D. Dutykh, and F. Dias, “Comparison between three-dimensional linear and nonlinear tsunami generation models,” Theor. Comput. Fluid Dyn. 21 (4), 245–269 (2007). https://doi.org/10.1007/s00162-007-0047-0

    Article  MATH  Google Scholar 

  27. T. Saito and T. Furumura, “Three-dimensional tsunami generation simulation due to sea-bottom deformation and its interpretation based on the linear theory,” Geophys. J. Int. 178 (2), 877–888 (2009). https://doi.org/10.1111/j.1365-246X.2009.04206.x

    Article  Google Scholar 

  28. Y. Tanioka and T. Seno, “Sediment effect on tsunami generation of the 1896 Sanriku tsunami earthquake,” Geophys. Res. Lett. 28 (17), 3389–3392 (2001). https://doi.org/10.1029/2001GL013149

    Article  Google Scholar 

  29. H. Tsushima, R. Hino, Y. Tanioka, F. Imamura, and H. Fujimoto, “Tsunami waveform inversion incorporating permanent seafloor deformation and its application to tsunami forecasting,” J. Geophys. Res. 117, B03311 (2012). https://doi.org/10.1029/2011JB008877

    Article  Google Scholar 

  30. T. Baba, Y. Gon, K. Imai, K. Yamashita, T. Matsuno, M. Hayashi, and H. Ichihara, “Modeling of a dispersive tsunami caused by a submarine landslide based on detailed bathymetry of the continental slope in the Nankai trough, southwest Japan,” Tectonophys. 768, 228182 (2019). https://doi.org/10.1016/j.tecto.2019.228182

    Article  Google Scholar 

  31. T. Saito, “Tsunami generation: validity and limitations of conventional theories,” Geophys. J. Int. 210 (3), 1888–1900 (2017). https://doi.org/10.1093/gji/ggx275

    Article  Google Scholar 

  32. T. Baba, S. Allgeyer, J. Hossen, P. R. Cummins, H. Tsushima, K. Imai, K. Yamashita, and T. Kato, “Accurate numerical simulation of the far-field tsunami caused by the 2011 Tohoku earthquake, including the effects of Boussinesq dispersion, seawater density stratification, elastic loading, and gravitational potential change,” Ocean Modell. 111, 46–54 (2017). https://doi.org/10.1016/j.ocemod.2017.01.002

    Article  Google Scholar 

  33. I. V. Fine and E. A. Kulikov, “Calculation of sea surface displacements in a tsunami source area caused by instantaneous vertical deformation of the seabed due to an underwater earthquake,” Vychisl. Tekhnol. 16 (2), 111–118 (2011).

    Google Scholar 

  34. T. Saito, “Dynamic tsunami generation due to sea-bottom deformation: Analytical representation based on linear potential theory,” Earth, Planets Space 65 (12), 1411–1423 (2013). https://doi.org/10.5047/eps.2013.07.004

    Article  Google Scholar 

  35. E. A. Kulikov, V. K. Gusiakov, A. A. Ivanova, and B. V. Baranov, “Numerical tsunami modelling and the bottom relief,” Moscow Univ. Phys. Bull. 71 (6), 527–536 (2016). https://doi.org/10.3103/S002713491605012X

    Article  Google Scholar 

  36. E. A. Kulikov, A. Yu. Medvedeva, and I. V. Fain, “Tsunami hazard assessment in the Caspian Sea,” Okeanol. Issled. (Oceanol. Res.) 47 (5), 74–88 (2019). https://doi.org/10.29006/1564-2291.JOR-2019.47(5).6

  37. T. Saito, Tsunami Generation and Propagation (Springer, Tokyo, 2019). https://doi.org/10.1007/978-4-431-56850-6

  38. M. A. Nosov and S. V. Kolesov, “Method of specification of the initial conditions for numerical tsunami modeling,” Moscow Univ. Phys. Bull. 64 (2), 208–213 (2009). https://doi.org/10.3103/S0027134909020222

    Article  MATH  Google Scholar 

  39. O. M. Phillips, “On the generation of waves by turbulent wind,” J. Fluid Mech. 2 (5), 417–445 (1957). https://doi.org/10.1017/S0022112057000233

    Article  MathSciNet  MATH  Google Scholar 

  40. Y.-Y. Li, B. Wang, and D.-H. Wang, “Characteristics of a terrain-following sigma coordinate,” Atmos. Ocean. Sci. Lett. 4 (3), 157–161 (2011). https://doi.org/10.1080/16742834.2011.11446922

    Article  Google Scholar 

  41. S. M. Griffies, C. Böning et al., “Developments in ocean climate modelling,” Ocean Modell. 2 (3-4), 123–192 (2000). https://doi.org/10.1016/S1463-5003(00)00014-7

    Article  Google Scholar 

  42. A. V. Gusev, “Numerical model of ocean hydrodynamics in curvilinear coordinates for reproducing the circulation of the world ocean and its separate water areas,” Candidate’s Dissertation in Physics and Mathematics (Inst. Vychisl. Mat., Moscow, 2009).

  43. N. A. Dianskii, Modeling Ocean Circulation and Studying Its Response to Short-Term and Long-Term Atmospheric Impacts (Fizmatlit, Moscow, 2013) [in Russian].

    Google Scholar 

  44. M. A. Nosov and S. V. Kolesov, “Combined numerical model of a tsunami,” Math. Models Comput. Simul. 11 (5), 679–689 (2019). https://doi.org/10.1134/S2070048219050156

    Article  MathSciNet  MATH  Google Scholar 

  45. K. A. Sementsov, M. A. Nosov, S. V. Kolesov, V. A. Karpov, H. Matsumoto, and Y. Kaneda, “Free gravity waves in the ocean excited by seismic surface waves: Observations and numerical simulations,” J. Geophys. Res. Oceans 124 (11), 8468–8484 (2019). https://doi.org/10.1029/2019JC015115

    Article  Google Scholar 

  46. R. L. Haney, “On the pressure gradient force over steep topography in sigma coordinate ocean models,” J. Phys. Oceanogr. 21 (4), 610–619 (1991). https://doi.org/10.1175/1520-0485(1991)021<0610:OTPGFO>2.0.CO;2

    Article  Google Scholar 

  47. A. Beckmann and D. B. Haidvogel, “Numerical simulation of flow around a tall isolated seamount. Part I: Problem formulation and model accuracy,” J. Phys. Oceanogr. 23 (8), 1736–1753 (1993). https://doi.org/10.1175/1520-0485(1993)023<1736:NSOFAA>2.0.CO;2

    Article  Google Scholar 

  48. K. A. Sementsov and A. V. Bolshakova, “A model for the generation of waves in the ocean by seismic bottom movements in sigma-coordinates,” Moscow Univ. Phys. Bull. 75 (1), 87–94 (2020). https://doi.org/10.3103/S0027134920010129

    Article  Google Scholar 

  49. A. A. Samarskii, Introduction to the Theory of Difference Schemes (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  50. K. A. Sementsov, “Dynamic and static models of generation of surface gravity waves in the ocean by earthquakes,” Candidate’s Dissertation in Physics and Mathematics (Mosk. Gos. Univ. im. M. V. Lomonosova, Moscow, 2017).

  51. Y. Okada, “Surface deformation due to shear and tensile faults in a half-space,” Bull. Seismol. Soc. Am. 75 (4), 1135–1154 (1985). https://doi.org/10.1785/BSSA0750041135

    Article  Google Scholar 

  52. B. W. Levin and M. A. Nosov, Physics of Tsunamis, 2nd ed. (Springer, Cham, 2016). https://doi.org/10.1007/978-3-319-24037-4

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ACKNOWLEDGMENTS

The authors thank the reviewer for his numerous valuable comments that contributed to a significant improvement of the manuscript.

Funding

This study was supported by the Russian Science Foundation, grant no. 22-27-00415; https://rscf.ru/project/22-27-00415/.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    K. A. Semenstov & M. A. Nosov

  2. Institute of Marine Geology and Geophysics, Far East Branch, Russian Academy of Sciences, Yuzhno-Sakhalinsk, Russia

    M. A. Nosov

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  1. K. A. Semenstov
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Correspondence to K. A. Semenstov.

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Semenstov, K.A., Nosov, M.A. Calculation of the Initial Elevation of the Water Surface at the Source of a Tsunami in a Basin with Arbitrary Bottom Topography. Math Models Comput Simul 15, 746–758 (2023). https://doi.org/10.1134/S2070048223040166

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