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Theoretical and Computational Fluid Dynamics

, Volume 32, Issue 6, pp 713–732 | Cite as

The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: variations from quasi-geostrophy

  • Jared P. WhiteheadEmail author
  • Terry Haut
  • Beth A. Wingate
Original Article

Abstract

Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system.

Keywords

Wave-mean flow interaction Multi-scale asymptotics Slow manifold Reduced models 

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References

  1. 1.
    Ariel, G., Engquist, B., Kim, S.J., Tsai, R.: Iterated averaging of three-scale oscillatory systems. Technical Report ICES REPORT 13-01, The Institute for Computational Engineering and Sciences, The University of Texas (2013)Google Scholar
  2. 2.
    Ariel, G., Engquist, B., Tsai, R.: A multi scale method for highly oscillatory ordinary differential equations with resonance. Math. Comput. 78(266), 929–956 (2009)CrossRefGoogle Scholar
  3. 3.
    Bogoliubov, N.N., Mitropolsky, Y.A.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach, New York (1961)Google Scholar
  4. 4.
    Bourgeois, A., Beale, J.T.: Validity of the quasigeostrophic model for large scale flow in the atmosphere and ocean. SIAM J. Math. Anal. 25, 1023–1068 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bresch, D., Gérard-Varet, D., Grenier, E.: Derivation of the planetary geostrophic equations. Arch. Ration. Mech. Anal. 182, 387–413 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Charney, J.G.: On the scale of atmospheric motions. Geophys. Publ. 17(2), 17 (1948)MathSciNetGoogle Scholar
  7. 7.
    Charney, J.G.: On a physical basis for numerical prediction of large-scale motions in the atmosphere. J. Meteorol. 6(6), 371–385 (1949)CrossRefGoogle Scholar
  8. 8.
    Charney, J.G., Fjortoft, R., von Neumann, J.: Numerical integration of the barotropic vorticity equation. Tellus 2, 237–254 (1950)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chemin, J.Y.: A propos d’un probléme de pénalisation de type anti symétrique. Comptes Rendus de l’Académie des Sceinces - Series I 321(7), 861–864 (1995)zbMATHGoogle Scholar
  10. 10.
    Chen, L.-Y., Goldenfeld, N., Oono, Y.: Erratum: “Renormalization group theory for global asymptotic analysis” [Phys. Rev. Lett. 73(10), 1311–1315 (1994); MR1289625 (95d:81089)]. Phys. Rev. Lett. 74(10), 1889 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, Lin-Yuan, Goldenfeld, Nigel, Oono, Y.: The Renormalization group and singular perturbations: multiple scales, boundary layers and reductive perturbation theory. Phys. Rev. E54, 376–394 (1996)Google Scholar
  12. 12.
    Cheverry, C.: Propagation of oscillations in real vanishing viscosity limit. Commun. Math. Phys. 247, 655–695 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Embid, P.F., Majda, A.J.: Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Commun. Partial Differ. Equ. 21(3–4), 619–658 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Embid, P.F., Majda, A.J.: Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers. Geophys. Astrophys. Fluid Dyn. 87(1–2), 1–50 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Engquist, B., Tsai, R.: Heterogeneous multi scale methods for stiff ordinary differential equations. Math. Comput. 74(252), 1707–1742 (2005)CrossRefGoogle Scholar
  16. 16.
    Feireisl, E., Novotny, A.: Multiple scales and singular limits for compressible rotating fluids with general initial data. Commun. Partial Differ. Equ. 39, 1104–1127 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gander, M.J.: 50 years of time parallel time integration. In: Carraro, T., Geiger, M., Korkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition. Springer, Berlin (2015)Google Scholar
  18. 18.
    Gander, M.J., Hairer, E.: Analysis for parareal algorithms applied to Hamiltonian differential equations. J. Comput. Appl. Math. 259, 2–13 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Grenier, E.: Oscillatory perturbations of the Navier Stokes equations. Journal de Matheématiques Pures et Appliquées 76(6), 477–498 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Haut, T., Wingate, B.A.: An asymptotic parallel-in-time method for highly oscillatory PDES. SIAM J. Numer. Anal. 36(2), A643–A713 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Holm, D.D., Zeitlin, Vladimir: Hamilton’s princple for quasigeostrophic motion. Phys. Fluids 10, 800–806 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Julien, K., Knobloch, E., Milliff, R., Werne, J.: Generalized quasi-geostrophy for spatially anisotropic rotationally constrained flows. J. Fluid Mech. 555, 233–274 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Klainerman, S., Majda, A.J.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34(4), 481–524 (1981)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lelong, P., Riley, J.J.: Internal wave-vortical mode interactions in strongly stratified flows. J. Fluid Mech. 232, 1–19 (1991)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lions, J., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. Comptes Rendus de l’Academie des Sciences Series I Mathematics 332(7), 661–668 (2001)zbMATHGoogle Scholar
  26. 26.
    Lorenz, E.N.: Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci. 37, 1685–1699 (1980)CrossRefGoogle Scholar
  27. 27.
    Majda, A.J., Grote, M.J.: Model dynamics and vertical collapse in decaying strongly stratified flows. Phys. Fluids 9(10), 2932–2940 (1997)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Moise, I., Ziane, M.: Renormalization group method. Applications to partial differential equations. J. Dyn. Differ. Equ. 13(2), 275–321 (2001)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Muraki, D.J., Snyder, C., Rotunno, R.: The next-order corrections to quasigeostrophic theory. J. Atmos. Sci. 56, 1547–1560 (1999)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Pedlosky, J.: Geophysical Fluid Dynamics. Springer, Berlin (1987)CrossRefGoogle Scholar
  31. 31.
    Petcu, M., Temam, R.M., Wirosoetisno, D.: Renormalization group method applied to the primitive equations. J. Differ. Equ. 208, 215–257 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Remmel, M., Smith, L.: New intermediate models for rotating shallow water and an investigation of the preference for anticyclones. J. Fluid Mech. 635, 321–359 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Riley, J.J., DeBruynKops, S.M.: Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15(7), 2047–2059 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Schochet, S.: Fast singular limits of hyperbolic PDE’s. J. Differ. Equ. 114, 476–512 (1994)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Schochet, S.: The mathematical theory of low Mach number flows. ESAIM Math. Model. Numer. Anal. 39(3), 441–458 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Smith, L.M., Waleffe, F.: Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11(6), 1608–1622 (1999)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Smith, L.M., Waleffe, F.: Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168 (2002)CrossRefGoogle Scholar
  38. 38.
    Sukhatme, J., Smith, Leslie M.: Vortical and wave modes in 3D rotating stratified flows: random large-scale forcing. Geophys. Astrophys. Fluid Dyn. 102(5), 437–455 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Temam, R.M., Wirosoetisno, D.: On the solutions of the renormalized equations at all orders. Adv. Differ. Equ. 8(8), 1005–1024 (2003)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Temam, R.M., Wirosoetisno, D.: Exponential approximations for the primitive equations of the ocean. Discrete Contin. Dyn. Syst. Ser. B 7(2), 425–440 (2007)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Temam, R.M., Wirosoetisno, D.: Stability of the slow manifold in the primitive equations. SIAM J. Math. Anal. 42(1), 427–458 (2010)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Warn, T., Bokhove, O., Shepherd, T.G., Vallis, G.K.: Rossby number expansions, slaving principles, and balance dynamics. Q. J. R. Meteorol. Soc. 121, 723–739 (1995)CrossRefGoogle Scholar
  43. 43.
    Wingate, B.A., Embid, P., Holmes-Cerfon, M., Taylor, M.A.: Low Rossby limiting dynamics for stably stratified flow with finite Froude number. J. Fluid Mech. 676, 546–571 (2011)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wirosoetisno, D., Shepherd, T.G., Temam, R.M.: Free gravity waves and balanced dynamics. J. Atmos. Sci. 59(23), 3382–3398 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Jared P. Whitehead
    • 1
    Email author
  • Terry Haut
    • 2
  • Beth A. Wingate
    • 3
  1. 1.Department of MathematicsBrigham Young UniversityProvoUSA
  2. 2.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA
  3. 3.Department of MathematicsUniversity of ExeterExeterUK

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