Инд. авторы:	Гусев О.И., Хакимзянов Г.С.
Заглавие:	Численное моделирование распространения длинных поверхностных волн по вращающейся сфере в рамках полной нелинейно-дисперсионной модели
Библ. ссылка:	Гусев О.И., Хакимзянов Г.С. Численное моделирование распространения длинных поверхностных волн по вращающейся сфере в рамках полной нелинейно-дисперсионной модели // Вычислительные технологии. - 2015. - Т.20. - № 3. - С.3-32. - ISSN 1560-7534. - EISSN 2313-691X.
Внешние системы:	РИНЦ: 23757837;
Реферат:	rus: Для численного моделирования процесса распространения длинных поверхностных волн предложен алгоритм, основанный на расщеплении системы нелинейно-дисперсионных уравнений на вращающейся сфере на равномерно эллиптическое уравнение для дисперсионной составляющей давления и гиперболическую систему уравнений мелкой воды первого приближения с модифицированным источниковым членом в правой части уравнения импульса. Алгоритм реализован в виде явной двухшаговой схемы предиктор-корректор, на каждом шаге которой поочередно решаются задачи, полученные в результате расщепления. На модельных задачах о распространении волн над ровным дном дана оценка важности учета эффектов, связанных со сферичностью Земли и ее вращением, зависимости дисперсионных эффектов от дальности распространения волн и размеров области начального возмущения свободной границы. eng: The paper examines the sensitivity of long surface wave propagation to Earth sphericity, Coriolis force, centrifugal force and frequency dispersion. For a numerical simulation, we propose the algorithm based on the partitioning of fully nonlinear dispersive equations on a rotating sphere based on a uniformly elliptic equation for the dispersion component of pressure and the hyperbolic system of shallow water equations for the momentum. The momentum equations yield the modified source term in the first approximation in the right hand side. These subproblems resulting from the partitioning are solved on each step of the explicit implemented two-step predictor-corrector scheme. The system of difference equations approximating the elliptic subproblem with the second order is constructed with use of integro-interpolation method and is solved by the SOR method. A model domain includes the most part of the Pacific Ocean. The function, which defines distribution of depth from the still water surface was set to be constant. Gaussian perturbations of free surface with different effective width served as idealized sources of waves. Numerical results show that in terrestrial conditions centrifugal force can be neglected for all the considered sources, but other effects can change the wave pattern considerably. Thus, sphericity increases the maximum wave amplitude, but Coriolis force and dispersion decrease. Besides that, the influence of sphericity and Coriolis force increases with an effective source width, while the influence of dispersion decreases. Wave propagation distance enhances the influence of all these effects. The approximate formula for the quick assessment of the propagation distance sufficient for demonstration of the dispersion effects is derived and its good agreement with calculations is shown.
Ключевые слова:	numerical simulation; Nonlinear dispersive equations; long surface waves; shallow water; rotating sphere; сила Кориолиса; дисперсия; численное моделирование; нелинейно-дисперсионные уравнения; длинные поверхностные волны; мелкая вода; вращающаяся сфера; Coriolis force; dispersion;
Издано:	2015
Физ. характеристика:	с.3-32
Цитирование:	1. Imamura, F. Simulation of wave-packet propagation along sloping beach by TUNAMI-code. Long-wave Runup Models. Eds. H. Yeh, P. Liu and C. Synolakis. Singapore: World Scientific; 1996: 231-241. 2. Zaitsev, A.I., Kurkin, A.A., Levin, B.V., Pelinovsky, E.N., Yalciner, A., Troitskaya, Yu.I., Ermakov, S.A. Numerical simulation of catastrophic tsunami propagation in the Indian Ocean. Doklady Earth Sciences. 2005; 402(4):614-618. 3. Titov, V.V., Synolakis, C.E. Numerical modeling of long wave runup using VTCS-3. Long-wave Runup Models. Eds. H. Yeh, P. Liu and C. Synolakis. Singapore: World Scientific; 1996: 242-248. 4. Wei, Y., Bernard, E., Tang, L., Weiss, R., Titov, V., Moore, C., Spillane, M., Hopkins, M., Kanoglu, U. Real-time experimental forecast of the Peruvian tsunami of August 2007 for U.S. coastlines. Geophysical Research Letters. 2008; (35):L04609. 5. Tang, L., Titov, V.V., Bernard, E., Wei, Y., Chamberlin, C., Newman, J.C., Mofjeld, H., Arcas, D., Eble, M., Moore, C., Uslu, B., Pells, C., Spillane, M.C., Wright, L.M., Gica, E. Direct energy estimation of the 2011 Japan tsunami using deep-ocean pressure measurements. Journal of Geophysical Research. 2012; (117):C08008. 6. Shokin, Yu.I., Babailov, V.V., Beisel, S.A., Chubarov, L.B., Eletsky, S.V., Fedotova, Z.I., Gusyakov, V.K. Mathematical modeling in application to regional tsunami warning systems operations. Notes on Numerical Fluid Mechanics and Multidisciplinary Design. Computational Science and High Performance Computing III. Berlin: Springer-Verlag; 2007: (101):52-69. 7. Fedotova, Z.I. On the application of the MacCormack scheme for problems of long wave hydrodynamics. Computational Technologies. 2006; 11(Special issue, devoted to N.N. Yanenko’s 85-th anniversary, part 2):53-63. (In Russ.) 8. Gusyakov, V.K., Fedotova, Z.I., Khakimzyanov, G.S., Chubarov, L.B., Shokin, Yu.I. Some approaches to local modelling of tsunami wave runup on a coast. Russian Journal of Numerical Analysis and Mathematical Modelling. 2008; 23(6):551-565. 9. 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. Russian Journal of Numerical Analysis and Mathematical Modelling. 2012; 27(6):539-558. 10. Pelinovsky, E.N. Tsunami Wave Hydrodynamics. Nizhny Novgorod: Institute Applied Physics Press; 1996: 276. (In Russ.) 11. Dalrymple, R.A., Grilli, S.T., Kirby, J.T., Watts, P. Tsunamis and challenges for accurate modeling. Oceanography. 2006; 19(1): 142-151. 12. Murty, T.S., Rao, A.D., Nirupama, N., Nistor, I. Numerical modelling concepts for tsunami warning systems. Current Science. 2006; 90(8):1073-1081. 13. Glimsdal, S., Pedersen, G.K., Atakan, K., Harbitz, C.B., Langtangen, H.P., Lovholt, F. Propagation of the Dec. 26, 2004, Indian Ocean Tsunami: Effects of dispersion and source characteristics. International Journal of Fluid Mechanics Research. 2006; 33(1):15-43. 14. Horrillo, J., Kowalik, Z., Shigihara, Y. Wave dispersion study in the Indian Ocean-tsunami of December 26, 2004. Marine Geodesy. 2006; (29):149-166. 15. Lovholt, F., Pedersen, G., Gisler, G. Oceanic propagation of a potential tsunami from the La Palma Island. Journal of Geophysical Research. 2008; (113):C09026. 16. Lovholt, F., Pedersen, G., Glimsdal, S. Coupling of dispersive tsunami propagation and shallow water coastal response. Open Oceanography Journal. 2010; (4):71-82. 17. Grue, J., Pelinovsky, E.N., Fructus, D., Talipova, T., Kharif, C. Formation of undular bores and solitary waves in the Strait of Malacca caused by the 26 December 2004 Indian Ocean tsunami. Journal of Geophysical Research. 2008; (113):C05008. 18. Peregrine, D.H. Calculations of the development of an undular bore. Journal of Fluid Mechanics. 1966; 25(part 2):321-331. 19. Glimsdal, S., Pedersen, G.K., Harbitz, C.B., Lovholt, F. Dispersion of tsunamis: does it really matter? Natural Hazards and Earth System Science. 2013; (13):1507-1526. 20. Kirby, J.T., Shi, F., Tehranirad, B., Harris, J.C., Grilli, S.T. Dispersive tsunami waves in the ocean: Model equations and sensitivity to dispersion and Coriolis effects. Ocean Modelling. 2013; (62):39-55. 21. Grilli, S.T., Harris, J.C., Tajalli Bakhsh, T., Masterlark, T.L., Kyriakopoulos, C., Kirby, J.T., Shi, F. Numerical simulation of the 2011 Tohoku tsunami based on a new transient FEM co-seismic source: comparison to far- and near-field observations. Pure and Applied Geophysics. 2013; 170(6-8):1333-1359. 22. Nwogu, O. Alternative form of Boussinesq equations for nearshore wave propagation. Journal of Waterway, Port, Coastal, and Ocean Engineering. 1993; 119(6):618-638. 23. Lovholt, F., Pedersen, G. Instabilities of Boussinesq models in non-uniform depth. International Journal for Numerical Methods in Fluids. 2009; (61):606-637. 24. Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., Grilli, S.T. A high-order adaptive time-stepping TVD solver for Boussinesq modelling of breaking waves and coastal inundation. Ocean Modelling. 2012; (43-44):36-51. 25. Shi, F., Kirby, J.T., Tehranirad, B. Tsunami benchmark results for spherical coordinate version of FUNWAVE-TVD (Version 2.0). Research Report No. CACR-12-02. Center for Applied Coastal Research: University of Delaware; 2012: 39. 26. Fedotova, Z.I., Khakimzyanov, G.S. Nonlinear-dispersive shallow water equations on a rotating sphere. Russian Journal of Numerical Analysis and Mathematical Modelling. 2010; 25(1):15-26. 27. Fedotova, Z.I., Khakimzyanov, G.S. Nonlinear-dispersive shallow water equations on a rotating sphere and conservation laws // Journal of Applied Mechanics and Technical Physics. 2014; 55(3):404-416. 28. Shokin, Yu.I., Fedotova, Z.I., Khakimzyanov, G.S. Hierarchy of nonlinear models of the hydrodynamics of long surface waves. Doklady Physics. 2015; 60(5):224-228. 29. Gusev, O.I. Modul' rascheta poverkhnostnykh voln NLDSW_sphere.Svidetel'stvo o gosudarstvennoy registratsii programmy dlya EVM № 2015616421 ot 09.06.2015. (In Russ.) 30. Fedotova, Z., Khakimzyanov, G. Full nonlinear dispersion model of shallow water equations on a rotating sphere. Journal of Applied Mechanics and Technical Physics. 2011; 52(6):865-876. 31. Lynett, P.J., Liu, P.L.-F. A numerical study of submarine-landslide-generated waves and run-up. Proceedings of the Royal Society of London. Series A. 2002; (458):2885-2910. 32. Chubarov, L.B., Eletskij, S.V., Fedotova, Z.I., Khakimzyanov, G.S. Simulation of surface waves generation by an underwater landslide. Russian Journal of Numerical Analysis and Mathematical Modelling. 2005; 20(5):425-437. 33. 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. Russian Journal of Numerical Analysis and Mathematical Modelling. 2007; 22(1):63-85. 34. Cherevko, A.A., Chupakhin, A.P. Equations of the shallow water model on a rotating attracting sphere. 1. Derivation and General Properties. Journal of Applied Mechanics and Technical Physics. 2009; 50(2):188-198. 35. Fedotova, Z.I., Khakimzyanov, G.S. Shallow water equations on a movable bottom. Russian Journal of Numerical Analysis and Mathematical Modelling. 2009; 24(1):31-41. 36. Gusev, O.I. Algorithm for surface waves calculation above a movable bottom within the frame of plane nonlinear dispersive model. Computational Technologies. 2014; 19(6):19-40. (In Russ.) 37. Gusev, O.I., Shokina, N.Yu., Kutergin, V.A., Khakimzyanov, G. S. Numerical modelling of surface waves generated by underwater landslide in a reservoir. Computational Technologies. 2013; 18(5):74-90. (In Russ.) 38. Shokin, Yu.I., Beisel, S.A., Gusev, O.I., Khakimzyanov, G.S., Chubarov, L.B., Shokina, N.Yu. Numerical modelling of dispersive waves generated by landslide motion. Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming, Computer Software”. 2014; 7(1):121-133. (In Russ.) 39. Khakimzyanov, G.S., Gusev, O.I., Beisel, S.A., Chubarov, L.B., Shokina, N.Yu. Modelling of tsunami generated by submarine landslides in the Black Sea. Russian Journal of Numerical Analysis and Mathematical Modelling. 2015; 30(4). (In print) 40. Ladyzhenskaya, O.A., Ural’tseva, N.N. Linear and Quasilinear Elliptic Equations. Moscow: Nauka; 1973: 576. (In Russ.) 41. Gusev, O.I. On an algorithm for surface waves calculation within the framework of nonlinear dispersive model with a movable bottom. Computational Technologies. 2012; 17(5):46-64. (In Russ.) 42. Khakimzyanov, G.S., Shokin, Yu.I., Barakhnin, V.B., Shokina, N.Yu. Numerical Simulation of Fluid Flows with Surface Waves. Novosibirsk: FUE «Publishing House SB RAS»; 2001: 394. (In Russ.) 43. Khakimzyanov, G.S., Shokina, N.Yu. Adaptive grid method for one-dimensional shallow water equations. Computational Technologies. 2013; 18(3):54-79. (In Russ.) 44. Shokina, N.Yu. To the problem of construction of difference schemes on movable grids. Russian Journal of Numerical Analysis and Mathematical Modelling. 2012; 27(6):603-626. 45. Synolakis, C.E., Bernard, E.N., Titov, V.V., Kanoglu, U., Gonzalez, F.I. Validation and verification of tsunami numerical models. Pure and Applied Geophysics. 2008; (165):2197-2228. 46. Horrillo, J., Grilli, S.T., Nicolsky, D., Roeber, V., Zhang, J. Performance benchmarking tsunami models for NTHMP’s inundation mapping activities. Pure and Applied Geophysics. 2015; 170(3-4):1333-1359. 47. Dao, M.H., Tkalich, P. Tsunami propagation modelling - a sensitivity study. Natural Hazards and Earth System Science. 2007; (7):741-754. 48. Nosov, M.A., Nurislamova, G.N., Moshenceva, A.V., Kolesov, S.V. Residual hydrodynamic fields after tsunami generation by an earthquake. Izvestiya RAN. Fizika atmosfery i okeana. 2014; 50(5):591-603. (In Russ.) 49. Shokin, Yu.I., Chubarov, L.B. The numerical modelling of long wave propagation in the framework of non-linear dispersion models. Computers and Fluids. 1987; 15(3):229-249. 50. Grilli, S.T., Ioualalen, M., Asavanant, J., Shi, F., Kirby, J.T., Watts, P. Source constraints and model simulation of the December 26, 2004, Indian Ocean tsunami. Journal of Waterway, Port, Coastal, and Ocean Engineering. 2007; (133):414-428. 51. Fedotova, Z.I., Khakimzyanov, G.S. Nonlinear dispersive equations of the shallow water on the rotating sphere. Computational Technologies. 2010; 15(3):135-145. (In Russ.)