Simulation of landslide tsunami in the Russian Far East based on 3D Navier–Stokes equations

   The paper presents the results of modeling landslide tsunamis near the Kamchatka Peninsula in part of the Pacific Ocean. The paper describes the model based on the three-dimensional (3D) Navier–Stokes equations. The model is supplemented with the rheological relation based on the Bingham model to account for the rheology of landslide masses. The paper proposes a modification of the classical Bingham model with a non-zero yield strength, which implies that the medium is at rest or it moves as a solid (body) in the absence of a tension in the medium exceeding this limit. The application of the classical model is impossible within the framework of the used equation system. The paper proposes its modification, which consists in the possibility of changing the yield strength to zero value by adding a linear function to a given shear rate, until which the fluid flows as Newtonian and after reaching it, the flow conditions of the substance obeys Bingham’s law. An original algorithm is used to simulate waves in real water areas, it implements open boundary conditions. The algorithm is based on the use of a damping boundary layer that absorbs the kinetic energy of the incoming wave, which is taken into account using an additional source in the angular momentum equation. A method is proposed for determining the resistance coefficient, the value of which determines the intensity of absorption of the kinetic energy of the wave. The used mathematical model makes it possible to model in a single way the occurrence, propagation and rolling on shore of tsunami waves of landslide origin. The results of modeling of an underwater landslide in the waters of the Kamchatka Bay near the city of Ust-Kamchatsk are presented, taking into account bathymetric date. The paper includes the analysis of dependence of wave heights on the volume of landslide in the zone of its initial position and at several points of the coast, as well as sections of the coast (in particular on Bering Island), which may be most severely affected by the occurrence of landslide tsunamis in this water area.

1. Pelinovsky E.N. Hydrodynamics of Tsunami Waves. N. Novgorod, Institute of Applied Physics of the Russian Academy of Sciences,1996. 276 p. (in Russian).

2. Levin B.V., Nosov М.А. Physics of Tsunami and Similar Phenomena in the Ocean. Мoscow, Yanus-K, 2005. 360 p. (in Russian).

3. Goto C., Ogawa Y., Shuto N., Imamura N. Numerical method of tsunami simulation with the leap-frog scheme (IUGG/IOC Time Project). IOC Manuals and Guides. 1997, 35, 130.

4. Wei G., Kirby J., Grilli S., Subramanya R. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. Journal of Fluid Mechanics. 1995, 294, 71–92. doi: 10.1017/S0022112095002813

5. Pelinovsky E., Zahibo N., Dunkly P., Edmonds M., Herd R., Talipova T., Kozelkov A.S., Nikolkina I. Tsunami generated by the volcano eruption on July 12–13, 2003 at Montserrat, Lesser Antilles. Science of Tsunami Hazards. 2004, 22, 1, 44–57.

6. Zahibo N., Pelinovsky E., Yalciner A., Kurkin A., Kozelkov A.S., Zaitsev A. Modelling the 1867 Virgin Island Tsunami. Natural Hazards and Earth System Sciences. 2003, 3, 5, 367–376. doi: 10.5194/nhess-3-367-2003

7. Mader C.H., Gittings M.L. Modeling the 1958 Lituya Bay mega tsunami, II. Science of Tsunami Hazards. 2002, 20, 5, 241–250.

8. Kozelkov A.S. The numerical technique for the landslide tsunami simulations based on Navier-Stokes equations. Journal of Applied Mechanics and Technical Physics. 2017, 58, 7, 1192–1210. doi: 10.1134/S0021894417070057

9. Kozelkov A.S., Kurkin A.A., Pelinovsky E.N., Tyatyushkina E.S., Kurulin V.V., Tarasova N.V. Landslide-type tsunami modelling based on the Navier-Stokes Equations. Science of tsunami Hazards, Journal of Tsunami Society International. 2016, 35, 3, 106–144.

10. Landau L.D., Lifshits E.M. Theoretical Physics: Textbook. In 10 vol. Vol. 6. Hydrodynamics. 3^rd ed. Moscow, Nauka, 1986. 736 p.

11. Horrillo J., Wood A., Kim G.B., Parambath A. A simplified 3-D Navier-Stokes numerical model for landslide-tsunami: Application to the Gulf of Mexico. Journal of Geophysical Research: Oceans. 2013, 118, 6934–6950. doi: 10.1002/2012JC008689

12. Kozelkov A.S., Kurkin A.A., Pelinovsky E.N., Kurulin V.V. Modeling the Cosmogenic Tsunami within the Framework of the Navier-Stokes Equations with Sources of Different Types. Fluid Dynamics. 2015, 50(2), 306–313. doi: 10.1134/S0015462815020143

13. Qin X., Motley M., LeVeque R., Gonzalez F., Mueller K. A comparison of a two-dimensional depth-averaged flow model and a three-dimensional RANS model for predicting tsunami inundation and fluid forces. Natural Hazards and Earth System Sciences. 2018, 18(9). doi: 10.5194/nhess-18–2489–2018

14. Delemen I.F., Konstantinova T.G. Assessment of the landslide risk on the Petropavlovsk-Kamchatsky territory during an expected large earthquake. Proceedings of the Second Regional Scientific and Technical Conference “Problems of complex geophysical monitoring of the Russian Far East”. Petropavlovsk-Kamchatsky, 11–17 October 2009; Obninsk, FRC “Unified Geophysical Service of RAS”. 2010, 116–120 (in Russian).

15. Lomtev V.L. Features of the structure and formation of the Kamchatka Canyon (The Pacific margin of Kamchatka). Geodynamics & Tectonophysics. 2018, 9(1), 177–197. doi: 10.5800/GT-2018-9-1-0344 (in Russian).

16. Franco A., Moernaut J., Schneider-Muntau B., Aufleger M., Strasser M., Gems B. Lituya Bay 1958 Tsunami — detailed pre-event bathymetry reconstruction and 3D-numerical modelling utilizing the CFD software Flow-3D. Natural Hazards and Earth System Science. 2020, 20, 2255–2279. doi: 10.5194/nhess-2019-285

17. Hirt C.W., Nichols B.D. Volume of Fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics. 1981, 39, 201–225. doi: 10.1016/0021-9991(81)90145-5

18. Gerbeau J.F., Vidrascu M. A quasi-newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. ESAIM Mathematical Modelling and Numerical Analysis. 2003, 37, 4, 631–647. doi: 10.1051/m2an:2003049

19. Ogibalov P.М., Mirzadzhanzade А.H. Unsteady flows of viscoplastic media. Moscow, MSU, 1977, 372 p. (in Russian).

20. Hrabry А.I., Zaytsev D.К., Smirnov Е.М. Numerical simulation of flows with a free surface based on the VOF method. Scientific Works of the Krylov State Scientific Center. SPb., Krylovsky State Scientific Center. 2003, 78 (362), 53–64 (in Russian).

21. Kozelkov A.S., Lashkin S.V., Efremov V.R., Volkov K.N., Tsibereva Yu.A., Tarasova N.V. An implicit algorithm of solving Navier–Stokes equations to simulate flows in anisotropic porous media. Computers and Fluids. 2018, 160, 164–174. doi: 10.1016/j.compfluid.2017.10.029

22. Chen Z.J., Przekwas A.J. A coupled pressure-based computational method for incompressible/compressible flows. Journal of Computational Physics. 2010, 229, 9150–9165. doi: 10.1016/j.jcp.2010.08.029

23. Hrabry А.I. Numerical simulation of unsteady turbulent flows of a liquid with a free surface. Thesis for the degree of Ph.D. in Physics and Mathematics, St. Petersburg, 2014, 154 p. (in Russian).

24. Ilgamov М.А., Gilmanov А.N. Non-reflective conditions at the boundaries of the computational domain. Moscow, Fizmatlit, 2003, 242 p. (in Russian).

25. Kar S.K., Turco R.P. Formulation of a lateral sponge layer for limited-area shallow-water models and an extension for the vertically stratified case. Monthly Weather Review. 1994, 123, 1542–1559. doi: 10.1175/1520-0493(1995)123<1542:FOALSL>2.0.CO;2

26. Podgornova О.V. Construction of discrete transparent boundary conditions for anisotropic and inhomogeneous media. Thesis for the degree of Ph.D. in Physics and Mathematics. Moscow, 2008, 109 p. (in Russian).

27. Givoli D., Neta B. High-order nonreflecting boundary conditions for the dispersive shallow water equations. Journal of Computational and Applied Mathematics. 2003, 158, 49–60. doi: 10.1080/1061856031000113608

28. Оvcharova А.S. A computational method for steady flows of a viscous fluid with a free boundary in stream function-vorticity variables. Journal of Applied Mechanics and Technical Physics. 1998, 39(2), 59–68 (in Russian).

29. Fürst J., Musil J. Development of non-reflective boundary conditions for free-surface flows. Proc. Topical problems of fluid mechanics, Prague, 21–23 February, 2018, 97–104. doi: 10.14311/TPFM.2018.013

30. Tyatyushkina E.S. The use of three-dimensional Navier-Stokes Equations, offended by Reynolds, for modeling the waves of the tsunami. Thesis for the degree of Ph.D. in Physics and Mathematics, N. Novgorod, 2021, 139 p. (in Russian).

31. Kozelkov A.S., Kurkin A.A., Sharipova I.L., Kurulin V.V., Pelinovsky E.N., Tyatyushkina E.S., Meleshkina D.P., Lashkin S.V., Tarasova N.V. Minimal Basis of Problems for Validating Methods of Numerical Simulation of Turbulent Viscous Incompressible Flows. Proceedings of the Nizhny Novgorod State Technical University n. a. R.E. Alekseev. 2015, 2(109), 49–69 (in Russian).

32. Tyatyushkina E.S., Kozelkov A.S., Kurkin A.A., Pelinovsky E.N., Kurulin V.V., Plygunova К.S., Utkin D.A. Verification of the LOGOS Software Package for Tsunami Simulations. Geosciences. 2020, 10, 385. doi: 10.3390/geosciences10100385

33. Kozelkov A.S., Kurulin V.V. Eddy-resolving numerical scheme for simulation of turbulent incompressible flows. Computational Mathematics and Mathematical Physics. 2015, 55, 7, 1232–1241. doi: 10.1134/S096554251507009X

34. Kozelkov A.S., Kurulin V.V., Puchkova О.L., Tyatyushkina E.S. Detached Eddy Simulation of Turbulent Viscous Incompressible Flows on Unstructured Grids. Mathematical Modeling. 2014, 26(8), 81–96 (in Russian).

35. Kozelkov A.S., Kurkin A.A., Krutyakova О.L., Kurulin V.V., Tyatyushkina E.S. Zonal RANS–LES approach based on an algebraic Reynolds stress model. RAS Proceedings. Mechanics of Liquid and Gas, 2015, 5, 24–33 (in Russian).

36. Kozelkov A.S., Kurkin A.A., Pelinovsky E.N. Effect of the angle of water entry of a body on the generated wave heights. Fluid Dynamics. 2016, 51, 2, 288–298. doi: 10.1134/S0015462816020162

37. Kozelkov A.S., Kurkin A.A., Pelinovsky E.N., Kurulin V.V., Tyatyushkina E.S. Modeling the disturbances in the Lake Chebarkul caused by the fall of the meteorite in 2013. Fluid Dynamics. 2015, 50(6), 828–840. doi:10.1134/S0015462815060137

38. Kozelkov A.S. The numerical technology for the landslide tsunami simulations based on Navier-Stokes equations. Computational Continuum Mechanics. 2016, 9(2), 218–236. doi: 10.7242/1999–6691/2016.9.2.19 (in Russian).

39. General Bathymetric Chart of the Oceans. URL: https://www.gebco.net/ (date of access: 15. 05. 2021).

40. Satellite map of the Kamchatka Krai — Yandex map. URL: https://yandex.ru/maps/11398/kamchatka-krai/sputnik/?ll=169.782981%2C61.350179&z=8 (date of access: 15. 05. 2021).

41. Watts P. Wavemaker curves for tsunamis generated by underwater landslides. Journal of Waterway, Port, Coastal, and Ocean Engineering. 1998, 124, 3, 127–137.

42. Watts P. Tsunami features of solid block underwater landslides. Journal of Waterway, Port, Coastal, and Ocean Engineering. 2000, 126, 3, 144–152. doi: 10.1061/(ASCE)0733–950X(2000)126:3(144)

43. Watts P., Grilli St.T., Tappin D.R., Fryer G.J. Tsunami generation by submarine mass failure. II: Predictive equations and case studies. Journal of Waterway, Port, Coastal, and Ocean Engineering. 2005, 131, 6, 298–310. doi: 10.1061/(ASCE)0733–950X(2005)131:6(298)