Skip to main content
SpringerLink
Log in

Numerical simulation of wave motions on a rotating attracting spherical zone

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The Riemann problem for the shallow water model on a rotating attracting spherical zone numerically is solved. A shock-capturing difference scheme is constructed that approximates the system of conservation laws describing discontinuous solutions of the given model. The Riemann problem is formulated as one of developing a wave process from initial data representing a spherical zone covered by various equilibria and zonal flows. Two Riemann problems are numerically simulated: the breakdown of water “ridges” of various shapes at equilibrium and propagation of contact discontinuity perturbations between an equilibrium and a zonal flow. The general properties of such solutions independent of the geometric configuration of the domains occupied by elementary solutions in the initial data are demonstrated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. E. Kochin, Collected Works (Akad. Nauk SSSR, Moscow, 1949), Vol. 1 [in Russian].

    Google Scholar 

  2. I. A. Kibel’, Introduction to Hydrodynamic Methods of Short-Range Weather Forecasts (Gostekhizdat, Moscow, 1957) [in Russian].

    Google Scholar 

  3. H. P. Greenspan, The Theory of Rotating Fluids (Cambridge Univ. Press, Cambridge, 1968).

    MATH  Google Scholar 

  4. L. N. Sretenskii, Theory of Wave Motions in Fluid (ONTI, Moscow, 1936) [in Russian].

    Google Scholar 

  5. G. I. Marchuk, Numerical Solution of Problems in Atmosphere and Ocean Dynamics (Gidrometeoizdat, Leningrad, 1974), [in Russian].

    Google Scholar 

  6. G. I. Marchuk, V. P. Dymnikov, V. B. Zalesnyi, V. N. Lykosov, and V. Ya. Galin, Mathematical Modeling of Atmosphere and Ocean General Circulation (Gidrometeoizdat, Leningrad, 1984) [in Russian].

    Google Scholar 

  7. A. Gill, Atmosphere-Ocean Dynamics (Academic, New York, 1982; Mir, Moscow, 1986).

    Google Scholar 

  8. J. Pedlosky, Geophysical Fluid Dynamics (Springer-Verlag, Heidelberg, 1981; Mir, Moscow, 1984).

    Google Scholar 

  9. J. R. Holton, An Introduction to Dynamic Meteorology (Academic, San Diego, 1992).

    Google Scholar 

  10. A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean (Courant Inst. Math. Sci., New York, 2003).

    MATH  Google Scholar 

  11. H. A. Dijkstra, Nonlinear Physical Oceanography (Springer, New York, 2001; Inst. Komp’yuternykh Issledovanii, Moscow, 2007).

    Google Scholar 

  12. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics (Springer, New York, 1999; MTsNMO, Moscow, 2007).

    Google Scholar 

  13. L. V. Ovsyannikov, “The Cauchy-Poisson problem on a sphere,” Vestn. Leningr. Univ. 3(13), 146–153 (1976).

    Google Scholar 

  14. A. A. Cherevko and A. P. Chupakhin, “Shallow water equations on a rotating attracting sphere: I. Derivation and basic properties,” Prikl. Mekh. Tekh. Fiz. 50(2), 24–36 (2009).

    MathSciNet  Google Scholar 

  15. J. J. Stoker, Water Waves (Wiley, New York, 1957; Inostrannaya Literatura, Moscow, 1959).

    MATH  Google Scholar 

  16. L. V. Ovsyannikov, N. I. Makarenko, V. I. Nalimov, et al., Nonlinear Problems in the Theory of Surface and Internal Waves (Nauka, Novosibirsk, 1985) [in Russian].

    Google Scholar 

  17. A. A. Cherevko and A. P. Chupakhin, “Shallow water equations on a rotating attracting sphere: II. Simple steady waves and sound characteristics,” Prikl. Mekh. Tekh. Fiz. 50(3), 82–96 (2009).

    MathSciNet  Google Scholar 

  18. N. Flyer, E. Lehto, S. Blaise, G. B. Wright, and A. St-Cyr, “A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere,” J. Comput. Phys. 231, 4078–4095 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  19. L. J. G. Blom and J. G. Verwer, “Spatial discretization of the shallow water equations in spherical geometry using Osher’s scheme,” J. Comp. Phys. 165(2), 542–565 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  20. M. I. Ivanov, “One-dimensional steady-state flows of a rotating gas and zonal wind,” Fluid Dyn. 47(1), 114–119 (2012).

    Article  MATH  Google Scholar 

  21. M. I. Ivanov, “Natural oscillations of a rotating spherical fluid layer of variable depth,” Fluid Dyn. 45(1), 121–125 (2010).

    Article  MATH  Google Scholar 

  22. N. H. Ibragimov and R. N. Ibragimov, “Integration by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell,” Phys. Lett. A 375, 3858–3865 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  23. A. V. Ivanova, V. V. Ostapenko, and A. P. Chupakhin, “Numerical simulation of shallow water flows on a rotating attracting sphere,” Vestn. Novosib. Gos. Univ. Ser. Mat. Mekh. Inf. 10(3), 30–45 (2010).

    MATH  Google Scholar 

  24. Z. I. Fedotova and G. S. Khakimzyanov, “Nonlinear dispersive shallow water equations on a rotating sphere,” Vychisl. Tekhnol. 15(3), 135–145 (2010).

    MATH  Google Scholar 

  25. V. V. Ostapenko, A. A. Cherevko, and A. P. Chupakhin, “Discontinuous solutions of the shallow water equations on a rotatable attracting sphere,” Fluid Dyn. 46(2), 196–213 (2011).

    Article  MATH  Google Scholar 

  26. B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Am. Math. Soc., Providence, 1983).

    Google Scholar 

  27. A. A. Simon-Miller, J. H. Rogers, P. J. Gierasch, D. Choi, M. D. Allison, A. Adamoli, and H. J. Mettig, “Longitudinal variation and waves in Jupiter’s south equatorial wind jet,” Icarus 218, 817–830 (2012).

    Article  Google Scholar 

  28. L. Galen, R. Gisler, R. P. Weaver, Ch. L. Mader, and M. Gittings, “Two- and three-dimensional asteroid impact simulations,” J. Comput. Sci. Eng. 6(3), 46–55 (2004).

    Article  Google Scholar 

  29. S. A. Stewart and P. J. Allen, “A 20-km-diameter multi-ringed impact structure in the North Sea,” Nature 418(6897), 520–523 (2002).

    Article  Google Scholar 

  30. S. N. Ward and E. Asphaug, “Impact tsunami-Eltanin,” Deep-Sea Res. Part II Topical Stud. Oceanogr. 49(6), 1073–1079 (2002).

    Article  Google Scholar 

  31. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).

    MATH  Google Scholar 

  32. V. V. Ostapenko, “Convergence of finite-difference schemes behind a shock front,” Comput. Math. Math. Phys. 37(10), 1161–1172 (1997).

    MathSciNet  Google Scholar 

  33. J. Casper and M. N. Carpenter, “Computational consideration for the simulation of shock-induced sound,” SIAM J. Sci. Comput. 19(3), 813–828 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  34. V. V. Ostapenko, “Construction of high-order accurate shock-capturing finite-difference schemes for unsteady shock waves,” Comput. Math. Math. Phys. 40(12), 1784–1800 (2000).

    MATH  MathSciNet  Google Scholar 

  35. W. Schacht, E. V. Vorozhtsov, A. F. Voevodin, and V. V. Ostapenko, “Numerical modeling of hydraulic jumps in a spiral channel with rectangular cross section,” Fluid Dyn. Res. 31, 185–213 (2002).

    Article  Google Scholar 

  36. V. V. Ostapenko, Hyperbolic Systems of Conservation Laws and Their Application to Shallow Water Theory (Novosib. Gos. Univ., Novosibirsk, 2004) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lavrent’ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent’eva 15, Novosibirsk, 630090, Russia

    V. V. Ostapenko, A. V. Speshilova, A. A. Cherevko & A. P. Chupakhin

  2. Novosibirsk State University, Universitetskii pr. 2, Novosibirsk, 630090, Russia

    V. V. Ostapenko, A. A. Cherevko & A. P. Chupakhin

Authors
  1. V. V. Ostapenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Speshilova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Cherevko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. P. Chupakhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Ostapenko.

Additional information

Original Russian Text © V.V. Ostapenko, A.V. Speshilova, A.A. Cherevko, A.P. Chupakhin, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 3, pp. 469–487.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ostapenko, V.V., Speshilova, A.V., Cherevko, A.A. et al. Numerical simulation of wave motions on a rotating attracting spherical zone. Comput. Math. and Math. Phys. 55, 470–486 (2015). https://doi.org/10.1134/S0965542515030124

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542515030124

Keywords

Search

Navigation