Numerical Modelling of Dispersive Waves Generated by Landslide Motion

The authors study the surface waves which are generated by the submarine landsliding on a curvilinear bottom slope of a deep reservoir. The shallow water models of the first and second approximations are used to describe the surface waves. An underwater landslide is described by the model of motion of a 'quasi-deformed' body on curvilinear surface under the effect of mass and external forces. The numerical algorithm for solving the nonlinear dispersive equations is based on the finite differential approximation of the hyperbolic system, which is similar to the shallow water equations of the first hydrodynamic approximation and the elliptic equation for the depth-average dispersive pressure component. The comparison of the numerical results obtained in the framework of the dispersion-free shallow water model and the nonlinear dispersive model is done.

Keywords

underwater landslide; irregular bottom; surface waves; shallow water equations; nonlinear dispersive equations; landslide motion law; numerical algorithm.

References

1. Harbitz C.B., Lovholt F., Pedersen G., Glimsdal S., Masson D.G. Mechanisms of Tsunami Generation by Submarine Landslides - a Short Review. Norwegian Journal of Geology, 2006, vol. 86, no. 3, pp. 255-264.
2. Shokin Yu.I., Fedotova Z.I., Khakimzyanov G.S., Chubarov L.B., Beisel S.A. Modelling Surfaces Waves of Generated by a Moving Landslide with Allowance for Vertical Flow Structure. Russ. J. Numer. Anal. Math. Modelling, 2007, vol. 22, no. 1, pp. 63-85. DOI: 10.1515/RNAM.2007.22.1.63
3. Beisel S.A., Chubarov L.B., Khakimzyanov G.S. Simulation of Surface Waves Generated by an Underwater Landslide Moving over an Uneven Slope. Russ. J. Numer. Anal. Math. Modelling, 2011, vol. 26, no. 1, pp. 17-38. DOI: 10.1515/RJNAMM.2011.002
4. Tappin D.R., Watts P., Grilli S.T. The Papua New Guinea Tsunami of 17 July 1998: Anatomy of a Catastrophic Event. Nat. Hazards Earth Syst. Sci., 2008, vol. 8, pp. 243-266. DOI: 10.5194/nhess-8-243-2008
5. Eletskij S.V., Maiorov Yu.B., Maksimov V.V., Nudner I.S., Fedotova Z.I., Khazhoyan M.G., Khakimzyanov G.S., Chubarov L.B. Simulation of Surface Waves Generation by a Moving Part of the Bottom Down the Coastal Slope [Modelirovanie generatsii poverkhnostnykh voln peremeshcheniem fragmenta dna po beregovomu sklonu]. Vychislitel'nye tekhnologii [Computational Technologies], 2004, vol. 9, pp. 194-206.
6. Grilli S.T., Watts P. Tsunami Generation by Submarine Mass Failure. I: Modeling, Experimental Validation, and Sensitivity Analyses. J. Waterway, Port, Coastal, Ocean Eng., 2005, vol. 131, no. 6, pp. 283-297. DOI: 10.1061/(ASCE)0733-950X(2005)131:6(283)
7. Enet F., Grilli S.T. Experimental Study of Tsunami Generation by Three-Dimensional Rigid Underwater Landslides. J. Waterway, Port, Coastal, Ocean Eng., 2007, vol. 133, no. 6, pp. 442-454. DOI: 10.1061/(ASCE)0733-950X(2007)133:6(442)
8. Pelinovsky E., Poplavsky A. Simplified Model of Tsunami Generation by Submarine Landslides. J. Phys. Chem. Earth, 1996, vol. 21, no. 12, pp. 13-17.
9. Grilli S.T., Watts P. Modeling of Waves Generated by a Moving Submerged Body: Applications to Underwater Landslides. Eng. Anal. Boundary Elem., 1999, vol. 23, no. 8, pp. 645-656. DOI: 10.1016/S0955-7997(99)00021-1
10. Beisel S.A., Chubarov L.B., Dutykh D., Khakimzyanov G.S., Shokina N.Yu. Simulation of Surface Waves Generated by an Underwater Landslide in a Bounded Reservoir. Russ. J. Numer. Anal. Math. Modelling, 2012, vol. 27, no. 6, pp. 539-558. DOI: 10.1515/rnam-2012-0031
11. Chubarov L.B., Eletskij S.V., Fedotova Z.I., Khakimzyanov G.S. Simulation of Surface Waves Generation by an Underwater Landslide. Russ. J. Numer. Anal. Math. Modelling, 2005, vol. 20, no. 5, pp. 425-437. DOI: 10.1515/156939805775186668
12. Fedotova Z.I., Khakimzyanov G.S. On Analysis of Conditions for Derivation of Nonlinear-Dispersive Equations [Analiz usloviy vyvoda NLD-uravneniy]. Vychislitel'nye tekhnologii [Computational Technologies], 2012, vol. 17, no. 5, pp. 94-108.
13. Yanenko N.N. Izbrannye trudy [Selected Works]. Moscow, Nauka, 1991. 416 p.
14. Roache P.J. Computational Fluid Dynamics. Albuquerque, Hermosa Publishers, 1976. 446 p.
15. Khakimzyanov G.S., Yaushev I.K. Numerical Calculation of Steady Subsonic Axisymmetric Flows of Ideal Compressible Fluid in the Channels of Complex Shape [O chislennom raschete dozvukovykh ustanovivshikhsya osesimmetrichnykh teceniy ideal'noy szhimaemoy zhidkosti v kanalakh slozhnoy formy]. Izvestiya Sibirskogo otdeleniya Akademii nauk SSSR. Seriya: Tekhnicheskie nauki [Proceedings of the Siberian Branch of the USSR Academy of Sciences. Ser.: Technical Sciences], 1981, no. 13, issue 3, pp. 50-57.
16. Khakimzyanov G.S., Yaushev I.K. Calculation of the Pressure in Stationary Problems of Ideal Fluid Dynamics. USSR Computational Mathematics and Mathematical Physics, 1984, vol. 24, issue 5, pp. 170-175. DOI: 10.1016/0041-5553(84)90175-7
17. Gusev O.I. On an Algorithm for Surface Waves Calculation within the Framework of Nonlinear Dispersive Model with a Movable Bottom [Ob algoritme rascheta poverkhnostnykh voln v ramkakh nelineyno-dispersionnoy modeli na podvizhnom dne]. Vychislitel'nye tekhnologii [Computational Technologies], 2012, vol. 17, no. 5, pp. 46-64.
18. Kuropatenko V.F. On Finite-Difference Methods for Fluid Dynamic Equations [O raznostnykh metodakh dlya uravneniy gidrodinamiki]. Tr. Mat. Inst. im. V.A. Steklova Akad. Nauk SSSR [Proceedings of the Steklov Mathematical Institute USSR AS], 1966, vol. 74, no. 1, pp. 107-137.
19. Rozhdestvenskiy B.L., Yanenko N.N. Sistemy kvazilineynykh uravneniy i ikh prilozheniya k gazovoy dinamike [Systems of Quasilinear Equations and Their Application to Gas Dynamics]. Moscow, Nauka, 1978. 687 p.
20. Kuropatenko V.F. Local Conservativeness of Difference Schemes for the Equations of Gas Dynamics. USSR Computational Mathematics and Mathematical Physics, 1985, vol. 25, issue 4, pp. 134-142. DOI: 10.1016/0041-5553(85)90157-0
21. Shokin Yu.I., Yanenko N.N. Metod differentsial'nogo priblizheniya. Primenenie k gazovoy dinamike [Method of Differential Approximation. Application to Gas Dynamics]. Novosibirsk, Nauka, 1985. 364 p.
22. Shokina N.Yu. To the Problem of Construction of Difference Schemes on Movable Grids. Russ. J. Numer. Anal. Math. Modelling, 2012, vol. 27, no. 6, pp. 603-626. DOI: 10.1515/rnam-2012-0035