Skip to main content
SpringerLink
Log in

Solvability of Vorticity Equation on a Sphere

  • Chapter
  • First Online:
Mathematical Problems of the Dynamics of Incompressible Fluid on a Rotating Sphere

  • 617 Accesses

Abstract

The chapter is devoted to the questions of existence, uniqueness, and asymptotic stability of solutions to the barotropic vorticity equation (BVE) for a viscous and forced fluid on a rotating sphere. Unlike the other works devoted to the vorticity equation on a sphere, we generalize the theorems on the unique solvability and global asymptotic stability of the BVE solution to the case of a more general form of the turbulent viscosity term; to wit, instead of the classical Navier-Stokes form \(\nu \Delta ^{2}\psi\) (Dymnikov and Filatov, Mathematics of Climate Modeling, Birkhäuser, Boston (1997); Ilyin and Filatov, Mathematical Physics, pp. 128–145, Leningrad Pedagogical Institute, Leningrad (1987); Ilyin and Filatov, Dokl. Akad. Nauk SSSR 301(1), 18–22 (1988)), we consider the term \(\nu (-\Delta )^{s}\psi\) where s ≥ 2 is a real number. Hyperdiffusion of this sort is used because ordinary linear diffusion is often too dissipative for many applications. The forcing is also considered as a function of fractional degree of smoothness on the sphere.

In Sect. 3.1, a non-stationary problem for the nonlinear barotropic vorticity equation on the sphere is formulated. Some important properties of the nonlinear term (Jacobian) are established in Sect. 3.2. The existence and uniqueness of a weak solution of non-stationary problem is proved in Sect. 3.3. In the particular case when a weak solution is sufficiently smooth, it is a classical solution. In Sect. 3.4, the existence of weak stationary solutions for the barotropic vorticity equation on the sphere is proved, and a condition that guarantees the uniqueness of a solution of this problem is obtained. It is shown that the degree of smoothness of stationary solution depends on real degree s of the Laplace operator in the term of turbulent viscosity \(\nu (-\Delta )^{s+1}\psi\). The boundedness of attractive sets and the asymptotic behavior of solutions (as time tends to infinity) as well as sufficient conditions for the global asymptotic stability of classical and weak solutions are examined in Sect. 3.5. The role of the structure and smoothness of forcing is also analyzed. Sufficient conditions for the global asymptotic stability of smooth and weak solutions are obtained in Sect. 3.6.

The dimension of the global vorticity equation attractor is evaluated in Sect. 3.7 provided that the external forcing is a quasi-periodic (in time) homogeneous spherical polynomial of degree n. It is known that in the case of a stationary forcing, the Hausdorff dimension of the global BVE attractor is limited by the Grashof number. We show that in the case of a non-stationary quasi-periodic forcing, the dimension of global BVE attractor depends not only on the generalized Grashof number, but also on the spatial and temporal structure of BVE forcing. Moreover, we show that the Hausdorff dimension of a globally attractive spiral solution may become arbitrarily large as the degree n of the quasi-periodic forcing grows, even if the generalized Grashof number is bounded. It should be noted that a quasi-periodic forcing, as compared with a time-invariant forcing, more adequately describes the effects of small-scale baroclinic processes which are not explicitly described by the barotropic vorticity equation. This result has a meteorological interest, since it shows that the search of a finite-dimensional global attractor in the barotropic atmosphere is not well justified.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Andrews, D.G.: An Introduction to Atmospheric Physics, 2nd edn. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  2. Arakawa, A., Lamb, V.R.: A potential enstrophy and energy conserving scheme for the shallow-water equation. Mon. Weather Rev. 109, 18–36 (1981)

    Article  Google Scholar 

  3. Arnold, V.I.: Ordinary Differential Equations. Springer, Berlin (2006)

    Google Scholar 

  4. Arrowsmith, D.K., Place, C.M.: Ordinary Differential Equations. A Qualitative Approach with Applications. Chapman and Hall, London (1990)

    MATH  Google Scholar 

  5. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North Holland, Amsterdam (1992)

    MATH  Google Scholar 

  6. Baklanovskaya, V.F.: Investigation of the method of nets for two-dimensional equations of the Navier-Stokes type with nonnegative viscosity. USSR Comput. Math. Math. Phys. 24, 145–156 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben-Artzi, M.: Global solutions of two-dimensional Navier-Stokes and Euler equations. Arch. Ration. Mech. Anal. 128(4), 329–358 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezis, H.: Remarks on the preceding paper by M. Ben-Artzi: “Global solutions of two-dimensional Navier-Stokes and Euler equations”. Arch. Ration. Mech. Anal. 128(4), 359–360 (1994)

    Google Scholar 

  9. Cao, C., Rammaha, M.A., Titi, E.S.: The Navier–Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom. Z. Angew. Math. Phys. 50, 341–360 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carpio, A.: Asymptotic behavior for the vorticity equations in dimensions two and three. Commun. Part. Diff. Equ. 19, 827–872 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chern, I-L., Colin, T., Kaper, H.G.: Classical solutions of the nondivergent barotropic equations on the sphere. Commun. Part. Diff. Equ. 17(5–6), 1001–1019 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Constantin, P., Foias, C., Temam, R.: On the dimension of the attractors in two-dimensional turbulence. Physica D 30, 284–296 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Daletskii, Yu.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs, vol. 43. American Mathematical Society, Providence, RI (1974)

    Google Scholar 

  14. Doering, C.R., Gibbon, J.D.: Note on the Constantin-Foias-Temam attractor dimension estimate for two dimensional turbulence. Physica D 48, 471–480 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dunford, N., Schwartz, J.T.: Linear Operators. Part II. Wiley, New York (1963)

    MATH  Google Scholar 

  16. Dymnikov, V.P., Filatov, A.N.: Mathematics of Climate Modeling. Birkhäuser, Boston (1997)

    MATH  Google Scholar 

  17. Dymnikov, V.P., Skiba, Yu.N.: Barotropic instability of zonally asymmetric atmospheric flows. In: Computational Processes and Systems, Issue 4, pp. 63–104. Nauka, Moscow (1986) (in Russian)

    Google Scholar 

  18. Dymnikov, V.P., Skiba Yu.N.: Spectral criteria for stability of barotropic atmospheric flows. Izv. Atmos. Oceanic Phys. 23(12), 943–951 (1987)

    MATH  Google Scholar 

  19. Gallagher, I., Gallay, T.: Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity. Math. Ann. 332(2), 287–327 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gallay, Th., Wayne, C.E.: Global Stability of vortex solutions of the two dimensional Navier-Stokes equation. Commun. Math. Phys. 255, 97–129 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gallay, Th., Wayne, C.E.: Existence and stability of asymmetric Burgers vortices. J. Math. Fluid. Mech. 9, 243–261 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gibbon, J.D.: A voyage around the Navier-Stokes equations. Physica D 92, 133–139 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giga, Y., Miyakawa, T., Osada, H.: Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Ration. Mech. Anal. 104(3), 223–250 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gorelov, A.S., Filatov, A.N.: Inertial manifolds for equations of barotropic atmosphere. Russ. J. Numer. Anal. Math. Model. 7(1), 25–43 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Grassberger, P.: Do climatic attractors exist? Nature 323, 609–612 (1986)

    Article  Google Scholar 

  26. Haurwitz, B.: The motion of atmospheric disturbances on the spherical earth. J. Mar. Res. 3, 254–267 (1940)

    Google Scholar 

  27. Helgason, S.: Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions. Academic Press, Orlando (1984)

    MATH  Google Scholar 

  28. Ilyin, A.A.: Euler equations with dissipation. Math. USSR-Sbornik 74(2), 475–485 (1993)

    Article  MathSciNet  Google Scholar 

  29. Ilyin, A.A.: On the dimension of attractors for Navier-Stokes equations on two-dimensional compact manifolds. Diff. Integr. Equ. 6(1), 183–214 (1993)

    MathSciNet  MATH  Google Scholar 

  30. Ilyin, A.A.: Navier-Stokes equations on the rotating sphere. A simple proof of the attractor dimension estimate. Nonlinearity 7, 31–39 (1994)

    MATH  Google Scholar 

  31. Ilyin, A.A., Filatov, A.N.: Navier-Stokes equations on a sphere and their unique solvability. The stability of steady solutions. In: Mathematical Physics, pp. 128–145. Leningrad Pedagogical Institute, Leningrad (1987) (in Russian)

    Google Scholar 

  32. Ilyin, A.A., Filatov, A.N.: On unique solvability of Navier-Stokes equations on two-dimensional sphere. Dokl. Akad. Nauk SSSR 301(1), 18–22 (1988)

    MathSciNet  Google Scholar 

  33. Ivanov, V.A., Skiba Yu.N.: Some Embedding Theorems on Compact Homogeneous Spaces and Their Application to the Study of the Stability of Barotropic Fluid on a Sphere, vol. 242. Department of Numerical Mathematics, Akad. Nauk SSSR. VINITI, Moscow, 38 pp (1990)

    Google Scholar 

  34. Kato, T.: The Navier-Stokes equation for an incompressible fluid in \(\mathbb{R}^{2}\) with a measure as the initial vorticity. Diff. Integr. Equ. 7, 949–966 (1994)

    MathSciNet  MATH  Google Scholar 

  35. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  36. Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Courier Dover Publications, New York (1999)

    MATH  Google Scholar 

  37. Korneichuk, N.P.: Exact Constants in Approximation Theory. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  38. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)

    MATH  Google Scholar 

  39. Ladyzhenskaya, O.A.: Minimal global B-attractors of semigroups and of initial-boundary value problems for nonlinear partial differential equations. Dokl. Akad. Nauk SSSR 294(1), 33–37 (1987) (in Russian)

    MathSciNet  Google Scholar 

  40. Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)

    MATH  Google Scholar 

  41. Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  42. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, New York (1972)

    Book  MATH  Google Scholar 

  43. Marchioro, C.: An example of absence of turbulence for the Reynolds number. Commun. Math. Phys. 105, 99–106 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  44. Neven, E.C.: Quadrupole modons on a sphere. Geophys. Astrophys. Fluid Dyn. 65, 105–126 (1992)

    Article  Google Scholar 

  45. Neven, E.C.: Modons in shear flow on a sphere. Geophys. Astrophys. Fluid Dyn. 74, 51–71 (1994)

    Article  Google Scholar 

  46. Nezlin, M.V., Snezhkin, E.N.: Nonlinear vortex structures and selforganization in unstable flows in rotating shallow water. In: Nonlinear Waves. Structures and Bifurcations, pp. 67–85. Moscow, Nauka (1987) (in Russian)

    Google Scholar 

  47. Nicolis, C., Nicolis, G.: Evidence for climatic attractors. Nature 326, 64–66 (1987)

    Article  Google Scholar 

  48. Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn. Springer, New York (2008)

    MATH  Google Scholar 

  49. Rossby, C.-G.: Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. Mar. Res. 2, 38–55 (1939)

    Article  Google Scholar 

  50. Rossby, C.-G.: Planetary flow patterns in the atmosphere. Q. J. R. Meteorol. Soc. 66, 68–87 (1940)

    Google Scholar 

  51. Sakuma, H., Ghil, M.: Stability of stationary barotropic modons by Lyapunov’s direct method. J. Fluid Mech. 211, 393–416 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  52. Salmon, R.: Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations. J. Atmos. Sci. 61, 2016–2036 (2004)

    Article  MathSciNet  Google Scholar 

  53. Samoilenko, A.M.: Elements of the Mathematical Theory of Multi-Frequency Oscillations. Kluwer, Dordrecht (1991)

    Book  Google Scholar 

  54. Simmons, A.J., Wallace, J.M., Branstator, G.W.: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci. 40(6), 1363–1392 (1983)

    Article  Google Scholar 

  55. Skiba, Yu.N.: Unique Solvability of the Barotropic Vorticity Equation for a Viscous Fluid in Classes of Generalized Functions on a Sphere, vol. 194, 56 pp., Department of Numerical Mathematics, Akad. Nauk SSSR. VINITI, Moscow (1988) (in Russian)

    Google Scholar 

  56. Skiba, Yu.N.: Mathematical Questions of the Dynamics of Viscous Barotropic Fluid on a Rotating Sphere. USSR Academy of Sciences, Moscow (in Russian) (1989)

    Google Scholar 

  57. Skiba, Yu.N.: On sufficient conditions for global asymptotic stability of the barotropic fluid on a sphere. Il Nuovo Cimento C 17, 359–368 (1994)

    Article  Google Scholar 

  58. Skiba, Yu.N.: On the long-time behavior of solutions to the barotropic atmosphere model. Geophys. Astrophys. Fluid Dyn. 78 (1–4), 143–167 (1994)

    Article  MathSciNet  Google Scholar 

  59. Skiba, Yu.N.: Total energy and mass conserving finite-difference schemes for the shallow water equations. Russ. Meteorol. Hydrol. 2, 35–43 (1995)

    Google Scholar 

  60. Skiba, Yu.N.: Asymptotic regimes of the barotropic vorticity equation on a sphere. Russ. Meteorol. Hydrol. 3, 37–45 (1996)

    Google Scholar 

  61. Skiba, Yu.N.: On dimension of attractive sets of viscous fluids on a sphere under quasi-periodic forcing. Geophys. Astrophys. Fluid Dyn. 85(3–4), 233–242 (1997)

    Article  MathSciNet  Google Scholar 

  62. Skiba, Yu.N.: On the existence and uniqueness of solution to problems of fluid dynamics on a sphere. J. Math. Analys. Appl. 388, 627–644 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  63. Skiba, Yu.N.: Asymptotic behavior and stability of solutions to barotropic vorticity equation on a sphere. Commun. Math. Anal. 14(2), 143–162 (2013)

    MathSciNet  MATH  Google Scholar 

  64. Skiba, Yu.N.: Role of forcing in large-time behavior of vorticity equation solutions on a sphere. Atmósfera 28(4), 283–296 (2015)

    Article  Google Scholar 

  65. Skiba, Yu.N., Filatov, D.: On splitting-based mass and total energy conserving arbitrary order shallow-water schemes. Numer. Methods Partial Differ. Equ. 23(3), 534–552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  66. Skiba, Yu.N., Filatov, D.: Conservative arbitrary order finite difference schemes for shallow-water flows. J. Comput. Appl. Math. 218(2), 579–591 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  67. Sobolevskiy, P.E.: The existence of solutions for the mathematical model of a nonlinear viscous fluid. Dokl. Akad. Nauk SSSR 285(1), 44–48 (1985)

    MathSciNet  Google Scholar 

  68. Sundstrëm, A.: Stability theorems for the barotropic vorticity equations. Mon. Weather Rev. 97 (4), 340–345 (1969)

    Article  Google Scholar 

  69. Szeptycki, P.: Equations of hydrodynamics on compact Riemannian manifolds. Bull. L’acad. Pol. Sci. Ser. Sci. Math. Astr. Phys. 21(4), 335–339 (1973)

    MathSciNet  MATH  Google Scholar 

  70. Szeptycki, P.: Equations of hydrodynamics on manifold diffeomorfic to the sphere. Bull. L’acad. Pol. Sci. Ser. Sci. Math. Astr. Phys. 21(4), 341–344 (1973)

    MathSciNet  MATH  Google Scholar 

  71. Temam, R.: Attractors for Navier-Stokes equations. Res. Not. Mat. 122, 272–292 (1985)

    MathSciNet  MATH  Google Scholar 

  72. Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, RI (2000)

    MATH  Google Scholar 

  73. Thompson, Ph.D.: A generalized class of exact time-dependent solutions of the vorticity equation for nondivergent barotropic flow. Mon. Weather Rev. 110, 1321–1324 (1982)

    Article  Google Scholar 

  74. Tribbia, J.J.: Modons in spherical geometry. Geophys. Astrophys. Fluid Dyn. 30, 131–168 (1984)

    Article  Google Scholar 

  75. Verkley, W.T.M.: The construction of barotropic modons on a sphere. J. Atmos. Sci. 41(16), 2492–2504 (1984)

    Article  MathSciNet  Google Scholar 

  76. Verkley, W.T.M.: Stationary barotropic modons in westerly background flows. J. Atmos. Sci. 44, 2383–2398 (1987)

    Article  Google Scholar 

  77. Verkley, W.T.M.: Modons with uniform absolute vorticity. J. Atmos. Sci. 47, 727–745 (1990)

    Article  MathSciNet  Google Scholar 

  78. Vreugdenhil, C.B.: Numerical Methods for Shallow-Water Flow. Springer, Berlin (2016)

    Google Scholar 

  79. Wang, Y.-P.: Analytical solution of nonlinear barotropic vorticity equation. J. Hydrodyn. 20(4), 530–535 (2008)

    Article  Google Scholar 

  80. Wei, M., Frederiksen, J.S.: Finite-time normal mode disturbances and error growth during Southern Hemisphere blocking. Adv. Atmos. Sci. 22(1), 69–89 (2005)

    Article  Google Scholar 

  81. Yu, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Commun. Math. Phys. 255, 161–181 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Atmospheric Sciences, National Autonomous University of Mexico, Mexico, Distrito Federal, Mexico

    Yuri N. Skiba

Authors
  1. Yuri N. Skiba
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Skiba, Y.N. (2017). Solvability of Vorticity Equation on a Sphere. In: Mathematical Problems of the Dynamics of Incompressible Fluid on a Rotating Sphere. Springer, Cham. https://doi.org/10.1007/978-3-319-65412-6_3

Download citation

Publish with us

Policies and ethics

Search

Navigation