Multi-scale Methods for Geophysical Flows

  • Christian L. E. Franzke
  • Marcel OliverEmail author
  • Jens D. M. Rademacher
  • Gualtiero Badin
Part of the Mathematics of Planet Earth book series (MPE, volume 1)


Geophysical flows comprise a broad range of spatial and temporal scales, from planetary- to meso-scale and microscopic turbulence regimes. The relation of scales and flow phenomena is essential in order to validate and improve current numerical weather and climate prediction models. While regime separation is often possible on a formal level via multi-scale analysis, the systematic exploration, structure preservation, and mathematical details remain challenging. This chapter provides an entry to the literature and reviews fundamental notions as background for the later chapters in this collection and as a departure point for original research in the field.



We thank Sergey Danilov for helpful advice on several aspects of this paper, and Federica Gugole and Gözde Özden for careful proof reading of parts of the manuscript. CF was supported by the German Research Foundation (DFG) through the Cluster of Excellence CliSAP (EXC177) at the University of Hamburg and DFG grant FR3515/3-1. GB was partially supported by DFG grants FOR1740 and BA5068/8-1.


  1. Allen, J.S., Holm, D.D., Newberger, P.A.: Toward an extended-geostrophic Euler–Poincaré model for mesoscale oceanographic flow. In: Norbury, J., Roulstone, I. (eds.) Large-Scale Atmosphere–Ocean Dynamics, vol. 1, pp. 101–125. Cambridge University Press (2002)Google Scholar
  2. Ansorge, C., Mellado, J.P.: Analyses of external and global intermittency in the logarithmic layer of Ekman flow. J. Fluid Mech. 805, 611–635 (2016)MathSciNetCrossRefGoogle Scholar
  3. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Springer (1999)Google Scholar
  4. Babin, A., Mahalov, A., Nicolaenko, B.: Global splitting, integrability and regularity of \(3\)D Euler and Navier-Stokes equations for uniformly rotating fluids. Eur. J. Mech. B Fluids 15(3), 291–300 (1996)MathSciNetzbMATHGoogle Scholar
  5. Babin, A., Mahalov, A., Nicolaenko, B.: Global splitting and regularity of rotating shallow-water equations. Eur. J. Mech. B Fluids 16(5), 725–754 (1997)MathSciNetzbMATHGoogle Scholar
  6. Babin, A., Mahalov, A., Nicolaenko, B.: Fast singular oscillating limits of stably-stratified 3D Euler and Navier–Stokes equations and ageostrophic wave fronts. In: Norbury, J., Roulstone, I. (eds.) Large-Scale Atmosphere–Ocean Dynamics, vol. 1, pp. 126–201. Cambridge University Press (2002)Google Scholar
  7. Badin, G.: On the role of non-uniform stratification and short-wave instabilities in three-layer quasi-geostrophic turbulence. Phys. Fluids 26(9), 096603 (2014)CrossRefGoogle Scholar
  8. Badin, G., Crisciani, F.: Variational Formulation of Fluid and Geophysical Fluid Dynamics—Mechanics, Symmetries and Conservation Laws. Springer (2018)Google Scholar
  9. Balmforth, N.J., Morrison, P.J., Thiffeault, J.-L.: Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model (2013). arXiv:1303.0065
  10. Bates, M.L., Grimshaw, R.H.J.: An extended equatorial plane: linear spectrum and resonant triads. Geophys. Astrophys. Fluid Dyn. 108(1), 1–19 (2014)MathSciNetCrossRefGoogle Scholar
  11. Beck, M., Sandstede, B., Zumbrun, K.: Nonlinear stability of time-periodic viscous shocks. Arch. Ration. Mech. Anal. 196(3), 1011–1076 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Becker, E.: Frictional heating in global climate models. Mon. Weather Rev. 131, 508–520 (2003)CrossRefGoogle Scholar
  13. Benamou, J.D., Brenier, Y.: Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem. SIAM J. Appl. Math. 58(5), 1450–1461 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Blender, R., Badin, G.: Hydrodynamic Nambu brackets derived by geometric constraints. J. Phys. A: Math. Theor. 48(10), 105501 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Blender, R., Badin, G.: Construction of Hamiltonian and Nambu forms for the shallow water equations. Fluids 2(2), 24 (2017)CrossRefGoogle Scholar
  16. Blumen, W., Wu, R.: Geostrophic adjustment: Frontogenesis and energy conversion. J. Phys. Oceanogr. 25(3), 428–438 (1995)CrossRefGoogle Scholar
  17. Bokhove, O., Oliver, M.: Parcel Eulerian-Lagrangian fluid dynamics of rotating geophysical flows. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462, 2563–2573 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. Bokhove, O., Vanneste, J., Warn, J.: A variational formulation for barotropic quasi-geostrophic flows. Geophys. Astrophys. Fluid Dyn. 88(1–4), 67–79 (1998)MathSciNetCrossRefGoogle Scholar
  19. Bouchut, F., Le Sommer, J., Zeitlin, V.: Breaking of balanced and unbalanced equatorial waves. Chaos 15(1), 013503 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. Boyd, J.P.: Equatorial solitary waves. Part I: Rossby solitons. J. Phys. Oceanogr. 10(11), 1699–1717 (1980)CrossRefGoogle Scholar
  21. Boyd, J.P.: Equatorial solitary waves. Part V: initial value experiments, coexisting branches, and tilted-pair instability. J. Phys. Oceanogr. 32(9), 2589–2602 (2002)CrossRefGoogle Scholar
  22. Boyd, J.P., Zhou, C.: Kelvin waves in the nonlinear shallow water equations on the sphere: nonlinear travelling waves and the corner wave bifurcation. J. Fluid Mech. 617, 187–205 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  23. Brethouwer, G., Duguet, Y., Schlatter, P.: Turbulent-laminar coexistence in wall flows with Coriolis, buoyancy or Lorentz forces. J. Fluid Mech. 704, 137–172 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  24. Burkhardt, U., Becker, E.: A consistent diffusion-dissipation parameterization in the ECHAM climate model. Mon. Weather Rev. 134, 1194–1204 (2006)CrossRefGoogle Scholar
  25. Çalık, M., Oliver, M., Vasylkevych, S.: Global well-posedness for the generalized large-scale semigeostrophic equations. Arch. Ration. Mech. Anal. 207(3), 969–990 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  26. Callaghan, T.G., Forbes, L.K.: Computing large-amplitude progressive Rossby waves on a sphere. J. Comput. Phys. 217(2), 845–865 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  27. Cao, C., Li, J., Titi, E.S.: Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion. Commun. Pure Appl. Math. 69(8), 1492–1531 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  28. Chan, I.H., Shepherd, T.G.: Balance model for equatorial long waves. J. Fluid Mech. 725, 55–90 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  29. Charve, F.: Asymptotics and vortex patches for the quasigeostrophic approximation. J. Math. Pures Appl. 85(4), 493–539 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  30. Charve, F.: Asymptotics and lower bound for the lifespan of solutions to the primitive equations. Acta Appl. Math. 1–37 (2018)Google Scholar
  31. Charve, F., Ngo, V.-S.: Global existence for the primitive equations with small anisotropic viscosity. Rev. Mat. Iberoam. 27(1), 1–38 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. Chatterjee, R.: Dynamical symmetries and Nambu mechanics. Lett. Math. Phys. 36, 117–126 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  33. Chemin, J., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier–Stokes Equations. Clarendon Press (2006)Google Scholar
  34. Cheng, B., Mahalov, A.: Time-averages of fast oscillatory systems. Discrete Contin. Dyn. Syst. Ser. S 6(5), 1151–1162 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  35. Chorin, A.J., Hald, O.H., Kupferman, R.: Optimal prediction and the Mori-Zwanzig representation of irreversible processes. Proc. Natl. Acad. Sci. 97(7), 2968–2973 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  36. Constantin, A.: Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr. 43(1), 165–175 (2013)CrossRefGoogle Scholar
  37. Crommelin, D., Majda, A.: Strategies for model reduction: comparing different optimal bases. J. Atmos. Sci. 61(17), 2206–2217 (2004)MathSciNetCrossRefGoogle Scholar
  38. Cullen, M.J.P.: A comparison of numerical solutions to the Eady frontogenesis problem. Q. J. R. Meteorol. Soc. 134(637), 2143–2155 (2008)CrossRefGoogle Scholar
  39. Cullen, M.J.P., Purser, R.J.: An extended Lagrangian theory of semi-geostrophic frontogenesis. J. Atmos. Sci. 41(9), 1477–1497 (1984)CrossRefGoogle Scholar
  40. Dalibard, A.-L., Saint-Raymond, L.: Mathematical study of the \(\beta \)-plane model for rotating fluids in a thin layer. J. Math. Pures Appl. (9), 94(2), 131–169 (2010)Google Scholar
  41. Danilov, S., Juricke, S., Kutsenko, A., Oliver, M.: Toward consistent subgrid momentum closures in ocean models. This volume, Chapter 5 (2019)Google Scholar
  42. Dijkstra, H.A., Wubs, F.W., Cliffe, A.K., Doedel, E., Dragomirescu, I.F., Eckhardt, B., Gelfgat, A.Y., Hazel, A.L., Lucarini, V., Salinger, A.G., et al.: Numerical bifurcation methods and their application to fluid dynamics: analysis beyond simulation. Commun. Comput. Phys. 15(1), 1–45 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  43. Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: The dynamics of modulated wave trains. Mem. Am. Math. Soc. 199(934), viii+105 (2009)Google Scholar
  44. Dolaptchiev, S.I., Klein, R.: A multiscale model for the planetary and synoptic motions in the atmosphere. J. Atmos. Sci. 70, 2963–2981 (2013)CrossRefGoogle Scholar
  45. Dolaptchiev, S.I., Timofeyev, I., Achatz, U.: Subgrid-scale closure for the inviscid Burgers-Hopf equation. Commun. Math. Sci. 11, 757–777 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  46. Dritschel, D.G., Gottwald, G.A., Oliver, M.: Comparison of variational balance models for the rotating shallow water equations. J. Fluid Mech. 822, 689–716 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  47. Dritschel, D.G., Viúdez, A.: A balanced approach to modelling rotating stably stratified geophysical flows. J. Fluid Mech. 488, 123–150 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  48. Dutrifoy, A., Majda, A.J., Schochet, S.: A simple justification of the singular limit for equatorial shallow-water dynamics. Commun. Pure Appl. Math. 62(3), 322–333 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  49. Eliassen, A.: The quasi-static equations of motion with pressure as independent variable. Geofys. Publ. 17, 1–44 (1948)MathSciNetGoogle Scholar
  50. Emanuel, K.A.: Inertial instability and mesoscale convective systems. Part I: linear theory of inertial instability in rotating viscous fluids. J. Atmos. Sci. 36(12), 2425–2449 (1979)CrossRefGoogle Scholar
  51. Emanuel, K.A.: Comments on “inertial instability and mesoscale convective systems. Part I: linear theory of inertial instability in rotating viscous fluids”. J. Atmos. Sci. 42(7), 747–752 (1984)CrossRefGoogle Scholar
  52. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  53. Ford, R., McIntyre, M.E., Norton, W.A.: Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57(9), 1236–1254 (2000)MathSciNetCrossRefGoogle Scholar
  54. Franzke, C., Majda, A.J.: Low-order stochastic mode reduction for a prototype atmospheric GCM. J. Atmos. Sci. 63, 457–479 (2006)MathSciNetCrossRefGoogle Scholar
  55. Franzke, C., Majda, A.J., Vanden-Eijnden, E.: Low-order stochastic mode reduction for a realistic barotropic model climate. J. Atmos. Sci. 62, 1722–1745 (2005)MathSciNetCrossRefGoogle Scholar
  56. Franzke, C., O’Kane, T., Berner, J., Williams, P., Lucarini, V.: Stochastic climate theory and modelling. WIREs Clim. Change 6, 63–78 (2015)CrossRefGoogle Scholar
  57. Franzke, C.L.E.: Predictions of critical transitions with non-stationary reduced order models. Phys. D 262, 35–47 (2013)MathSciNetCrossRefGoogle Scholar
  58. Fringer, O.B.:. Towards nonhydrostatic ocean modeling with large-eddy simulation. In: Glickson, D. (ed.) Oceanography in 2025, pp. 81–83. National Academies Press (2009)Google Scholar
  59. Gardiner, C.W.: Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer (2009)Google Scholar
  60. Gerkema, T., Shrira, V.I.: Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195–219 (2005a)MathSciNetzbMATHCrossRefGoogle Scholar
  61. Gerkema, T., Shrira, V.I.: Near-inertial waves on the “nontraditional” \(\beta \)-plane. J. Geophys. Res. Oceans 110(C1) (2005b)Google Scholar
  62. Gill, A.E.: Atmosphere-Ocean Dynamics. Academic Press (1982)Google Scholar
  63. Givon, D., Kupferman, R., Stuart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17(6), R55 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  64. Goh, R., Wayne, C.E.: Vortices in stably-stratified rapidly rotating Boussinesq convection (2018). arXiv:1802.05369
  65. Gottwald, G., Crommelin, D., Franzke, C.: Stochastic climate theory. In: Franzke, C., O’Kane, T. (eds.) Nonlinear and Stochastic Climate Dynamics, pp. 209–240. Cambridge University Press (2017)Google Scholar
  66. Gottwald, G.A., Oliver, M.: Slow dynamics via degenerate variational asymptotics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470(2170), 20140460 (2014)MathSciNetCrossRefGoogle Scholar
  67. Grenier, E., Masmoudi, N.: Ekman layers of rotating fluids, the case of well prepared initial data. Commun. Partial Differ. Equ. 22(5–6), 953–975 (1997)MathSciNetzbMATHGoogle Scholar
  68. Griffies, S.M., Pacanowski, R.C., Schmidt, M., Balaji, V.: Tracer conservation with an explicit free surface method for \(z\)-coordinate ocean models. Mon. Weather Rev. 129(5), 1081–1098 (2001)CrossRefGoogle Scholar
  69. Grooms, I., Julien, K., Fox-Kemper, B.: On the interactions between planetary geostrophy and mesoscale eddies. Dyn. Atmos. Oceans 51(3), 109–136 (2011)CrossRefGoogle Scholar
  70. Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-dimensional Dynamical Systems. Springer (2011)Google Scholar
  71. Hasselmann, K.: Stochastic climate models. Part I. Theory. Tellus 28(6), 473–485 (1976)CrossRefGoogle Scholar
  72. Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 471(2176), 20140963, 19 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  73. Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137(1), 1–81 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  74. Holm, D.D., Schmah, T., Stoica, C.: Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press (2009)Google Scholar
  75. Holm, D.D., Zeitlin, V.: Hamilton’s principle for quasigeostrophic motion. Phys. Fluids 10(4), 800–806 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  76. Holton, J.R., Hakim, G.J.: An Introduction To Dynamic Meteorology. Academic Press (2012)Google Scholar
  77. Hoskins, B.J.: The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32(2), 233–242 (1975)CrossRefGoogle Scholar
  78. Hsia, C.-H., Ma, T., Wang, S.: Stratified rotating Boussinesq equations in geophysical fluid dynamics: dynamic bifurcation and periodic solutions. J. Math. Phys. 48(6), 065602, 20 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  79. Hsu, H.-C.: An exact solution for nonlinear internal equatorial waves in the \(f\)-plane approximation. J. Math. Fluid Mech. 16(3), 463–471 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  80. Ibragimov, R.N.: Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications. Phys. Fluids 23(12), 123102 (2011)zbMATHCrossRefGoogle Scholar
  81. Julien, K., Knobloch, E.: Reduced models for fluid flows with strong constraints. J. Math. Phys. 48(6) (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  82. Julien, K., Knobloch, E., Milliff, R., Werne, J.: Generalized quasi-geostrophy for spatially anisotropic rotationally constrained flows. J. Fluid Mech. 555, 233–274 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  83. Kamenkovich, V.M., Koshlyakov, M.N., Monin, A.S.: Synoptic Eddies in the Ocean. D. Reidel Publishing Company (1986)Google Scholar
  84. Kapitula, T., Promislow, K.: Spectral and Dynamical Stability of Nonlinear Waves. Springer (2013)Google Scholar
  85. Kasahara, A., Gary, J.M.: Studies of inertio-gravity waves on midlatitude beta-plane without the traditional approximation. Q. J. R. Meteorol. Soc. 136(647), 517–536 (2010)Google Scholar
  86. Kawahara, G., Uhlmann, M., van Veen, L.: The significance of simple invariant solutions in turbulent flows. Ann. Rev. Fluid Mech. 44(1), 203–225 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  87. Khouider, B., Majda, A.J., Stechmann, S.N.: Climate science in the tropics: waves, vortices and PDEs. Nonlinearity 26(1), R1 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  88. Klein, R.: Scale-dependent models for atmospheric flows. Ann. Rev. Fluid Mech. 42, 249–274 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  89. Klingbeil, K., Burchard, H.: Implementation of a direct nonhydrostatic pressure gradient discretisation into a layered ocean model. Ocean Model. 65, 64–77 (2013)CrossRefGoogle Scholar
  90. Koide, T., Kodama, T.: Navier–Stokes, Gross–Pitaevskii and generalized diffusion equations using the stochastic variational method. J. Phys. A 45(25), 255204, 18 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  91. Kurtz, T.G.: A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12(1), 55–67 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  92. Kwasniok, F.: Empirical low-order models of barotropic flow. J. Atmos. Sci. 61(2), 235–245 (2004)CrossRefGoogle Scholar
  93. Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. American Mathematical Society (2003)Google Scholar
  94. Majda, A., Franzke, C., Crommelin, D.: Normal forms for reduced stochastic climate models. Proc. Natl. Acad. Sci. USA 106, 3649–3653 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  95. Majda, A., Timofeyev, I., Vanden-Eijnden, E.: A priori tests of a stochastic mode reduction strategy. Phys. D 170, 206–252 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  96. Majda, A.J., Franzke, C., Khouider, B.: An applied mathematics perspective on stochastic modelling for climate. Philos. Trans. R. Soc. A 366, 2429–2455 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  97. Majda, A.J., Klein, R.: Systematic multiscale models for the tropics. J. Atmos. Sci. 60(2), 393–408 (2003)CrossRefGoogle Scholar
  98. Majda, A.J., Timofeyev, I., Vanden-Eijnden, E.: Models for stochastic climate prediction. Proc. Natl. Acad. Sci. USA 96(26), 14687–14691 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  99. Majda, A.J., Timofeyev, I., Vanden-Eijnden, E.: A mathematical framework for stochastic climate models. Commun. Pure Appl. Math. 54(8), 891–974 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  100. Majda, A.J., Timofeyev, I., Vanden-Eijnden, E.: Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci. 60(14), 1705–1722 (2003)MathSciNetCrossRefGoogle Scholar
  101. Marsden, J.E., Ratiu, T.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer (2013)Google Scholar
  102. McIntyre, M.: Dynamical meteorology—balanced flow. In: Pyle, J., Zhang, F. (eds.) Encyclopedia of Atmospheric Sciences, pp. 298–303, 2nd edn. Academic Press, Oxford (2015)Google Scholar
  103. McIntyre, M.E., Norton, W.A.: Potential vorticity inversion on a hemisphere. J. Atmos. Sci. 57(9), 1214–1235 (2000)MathSciNetCrossRefGoogle Scholar
  104. McWilliams, J.C.: A note on a consistent quasigeostrophic model in a multiply connected domain. Dyn. Atmos. Oceans 1(5), 427–441 (1977)CrossRefGoogle Scholar
  105. Meyries, M., Rademacher, J., Siero, E.: Quasilinear parabolic reaction-diffusion systems: User’s guide to well-posedness, spectra and stability of travelling waves. SIAM J. Appl. Dyn. Sys. 13, 249–275 (2014)zbMATHCrossRefGoogle Scholar
  106. Mohebalhojeh, A.R., Dritschel, D.G.: Hierarchies of balance conditions for the \(f\)-plane shallow-water equations. J. Atmos. Sci. 58(16), 2411–2426 (2001)MathSciNetCrossRefGoogle Scholar
  107. Monahan, A.H., Culina, J.: Stochastic averaging of idealized climate models. J. Clim. 24(12), 3068–3088 (2011)CrossRefGoogle Scholar
  108. Moon, W., Wettlaufer, J.S.: On the interpretation of Stratonovich calculus. New J. Phys. 16(5), 055017 (2014)CrossRefGoogle Scholar
  109. Mori, H.: Transport, collective motion, and Brownian motion. Prog. Theor. Phys. 33(3), 423–455 (1965)zbMATHCrossRefGoogle Scholar
  110. Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7(8), 2405 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  111. Olbers, D., Eden, C.: A global model for the diapycnal diffusivity induced by internal gravity waves. J. Phys. Oceanogr. 43(8), 1759–1779 (2013)CrossRefGoogle Scholar
  112. Olbers, D., Willebrand, J., Eden, C.: Ocean Dynamics. Springer (2012)Google Scholar
  113. Oliver, M.: Variational asymptotics for rotating shallow water near geostrophy: a transformational approach. J. Fluid Mech. 551, 197–234 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  114. Oliver, M., Vasylkevych, S.: Generalized LSG models with spatially varying Coriolis parameter. Geophys. Astrophys. Fluid Dyn. 107, 259–276 (2013)MathSciNetCrossRefGoogle Scholar
  115. Oliver, M., Vasylkevych, S.: Generalized large-scale semigeostrophic approximations for the \(f\)-plane primitive equations. J. Phys. A: Math. Theor. 49, 184001 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  116. Palmer, T., Buizza, R., Doblas-Reyes, F., Jung, T., Leutbecher, M., Shutts, G., Steinheimer, M., Weisheimer, A.: Stochastic parametrization and model uncertainty. Technical report, ECMWF (2009)Google Scholar
  117. Papanicolaou, G.C.: Some probabilistic problems and methods in singular perturbations. Rocky Mt. J. Math. 6(4), 653–674 (1976). Summer Research Conference on Singular Perturbations: Theory and Applications. Northern Arizona University, Flagstaff, Arizona (1975)Google Scholar
  118. Pavliotis, G.A., Stuart, A.: Multiscale Methods: Averaging and Homogenization. Springer (2008)Google Scholar
  119. Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn. Springer (1987)Google Scholar
  120. Plougonven, R., Zeitlin, V.: Lagrangian approach to geostrophic adjustment of frontal anomalies in a stratified fluid. Geophys. Astrophys. Fluid Dyn. 99(2), 101–135 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  121. Ragone, F., Badin, G.: A study of surface semi-geostrophic turbulence: freely decaying dynamics. J. Fluid Mech. 792, 740–774 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  122. Reznik, G.M.: Wave adjustment: general concept and examples. J. Fluid Mech. 779, 514–543 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  123. Risken, H.: The Fokker–Planck Equation. Springer (1996)Google Scholar
  124. Roberts, A.J.: Normal form transforms separate slow and fast modes in stochastic dynamical systems. Phys. D 387(1), 12–38 (2008)MathSciNetGoogle Scholar
  125. Sagaut, P.: Large Eddy Simulation for Incompressible Flows: An Introduction. Springer (2006)Google Scholar
  126. Saint-Raymond, L.: Lecture notes: Mathematical study of singular perturbation problems. Applications to large-scale oceanography. Journées Eq. Deriv. Part. 1–49 (2010)Google Scholar
  127. Saito, K., Ishida, J., Aranami, K., Hara, T., Segawa, T., Narita, M., Honda, Y.: Nonhydrostatic atmospheric models and operational development at JMA. J. Meteorol. Soc. Jpn. 85B, 271–304 (2007)CrossRefGoogle Scholar
  128. Salmon, R.: The shape of the main thermocline. J. Phys. Oceanogr. 12, 1458–1479 (1982)CrossRefGoogle Scholar
  129. Salmon, R.: Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431–444 (1983)zbMATHCrossRefGoogle Scholar
  130. Salmon, R.: New equations for nearly geostrophic flow. J. Fluid Mech. 153, 461–477 (1985)zbMATHCrossRefGoogle Scholar
  131. Salmon, R.: Large-scale semigeostrophic equations for use in ocean circulation models. J. Fluid Mech. 318, 85–105 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  132. Salmon, R.: Lectures on Geophysical Fluid Dynamics. Oxford University Press (1998)Google Scholar
  133. Salmon, R.: A general method for conserving quantities related to potential vorticity in numerical models. Nonlinearity 18(5), R1 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  134. Salmon, R.: A general method for conserving energy and potential enstrophy in shallow water models. J. Atmos. Sci. 64, 515–531 (2007)CrossRefGoogle Scholar
  135. Sandstede, B.: Stability of travelling waves. In: Handbook of Dynamical Systems, vol. 2, pp. 983–1055. North-Holland, Amsterdam (2002)zbMATHGoogle Scholar
  136. Schaefer-Rolffs, U., Becker, E.: Horizontal momentum diffusion in GCMs using the dynamic Smagorinsky model. Mon. Weather Rev. 141(3), 887–899 (2013)CrossRefGoogle Scholar
  137. Schaefer-Rolffs, U., Knöpfel, R., Becker, E.: A scale invariance criterion for LES parametrizations. Meteorol. Z. 24(1), 3–13 (2015)CrossRefGoogle Scholar
  138. Schneider, G.: Error estimates for the Ginzburg-Landau approximation. Z. Angew. Math. Phys. 45(3), 433–457 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  139. Schubert, W.H., Taft, R.K., Silvers, L.G.: Shallow water quasi-geostrophic theory on the sphere. J. Adv. Model. Earth Syst. 1(2), 2 (2009)Google Scholar
  140. Shaw, T.A., Shepherd, T.G.: A theoretical framework for energy and momentum consistency in subgrid-scale parameterization for climate models. J. Atmos. Sci. 66, 3095–3114 (2009)CrossRefGoogle Scholar
  141. Shepherd, T.G.: Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287–338 (1990)CrossRefGoogle Scholar
  142. Simonnet, E., Dijkstra, H.A., Ghil, M.: Bifurcation analysis of ocean, atmosphere, and climate models. In: Ciarlet, P. (ed.) Handbook of Numerical Analysis, vol. 14, pp. 187–229. Elsevier (2009)Google Scholar
  143. Smagorinsky, J.: General circulation experiments with the primitive equations. Mon. Weather Rev. 91(3), 99–164 (1963)CrossRefGoogle Scholar
  144. Smith, R.K., Dritschel, D.G.: Revisiting the Rossby-Haurwitz wave test case with contour advection. J. Comput. Phys. 217(2), 473–484 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  145. Stern, A., Tong, Y., Desbrun, M., Marsden, J.E.:. Geometric computational electrodynamics with variational integrators and discrete differential forms. In: Chang, D.E., Holm, D.D., Patrick, G., Ratiu, T. (eds.) Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, pp. 437–475. Springer (2015)Google Scholar
  146. Stewart, A.L., Dellar, P.J.: Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech. 651, 387 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  147. Stewart, A.L., Dellar, P.J.: Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. J. Fluid Mech. 690, 16–50 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  148. Stinis, P.: A comparative study of two stochastic mode reduction methods. Phys. D 213(2), 197–213 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  149. Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160(2), 295–315 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  150. Temam, R., Wang, S.H.: Inertial forms of Navier-Stokes equations on the sphere. J. Funct. Anal. 117(1), 215–242 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  151. Temam, R., Wirosoetisno, D.: Slow manifolds and invariant sets of the primitive equations. J. Atmos. Sci. 68(3), 675–682 (2010)zbMATHCrossRefGoogle Scholar
  152. Theiss, J., Mohebalhojeh, A.R.: The equatorial counterpart of the quasi-geostrophic model. J. Fluid Mech. 637, 327–356 (2009)zbMATHCrossRefGoogle Scholar
  153. Thuburn, J., Li, Y.: Numerical simulations of Rossby-Haurwitz waves. Tellus A 52(2), 181–189 (2000)CrossRefGoogle Scholar
  154. Tort, M., Dubos, T.: Usual approximations to the equations of atmospheric motion: a variational perspective. J. Atmos. Sci. 71(7), 2452–2466 (2014)CrossRefGoogle Scholar
  155. Tort, M., Dubos, T., Bouchut, F., Zeitlin, V.: Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography. J. Fluid Mech. 748, 789–821 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  156. Tort, M., Ribstein, B., Zeitlin, V.: Symmetric and asymmetric inertial instability of zonal jets on the \(f\)-plane with complete Coriolis force. J. Fluid Mech. 788, 274–302 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  157. Trias, F.X., Folch, D., Gorobets, A., Oliva, A.: Building proper invariants for eddy-viscosity subgrid-scale models. Phys. Fluids 27(6) (2015)CrossRefGoogle Scholar
  158. Vallis, G.K.: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press (2006)Google Scholar
  159. Vanderbauwhede, A.: Lyapunov–Schmidt method for dynamical systems. In: Mathematics of Complexity and Dynamical Systems, vol. 1–3, pp. 937–952. Springer, New York (2012)CrossRefGoogle Scholar
  160. Vanneste, J.: Balance and spontaneous wave generation in geophysical flows. Ann. Rev. Fluid Mech. 45(1), 147–172 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  161. Verkley, W., van der Velde, I.: Balanced dynamics in the tropics. Q. J. R. Meteorol. Soc. 136(646), 41–49 (2010)CrossRefGoogle Scholar
  162. von Storch, J.-S., Badin, G., Oliver, M.: The interior energy pathway: inertial gravity wave emission by oceanic flows. This volume, Chapter 2 (2019)Google Scholar
  163. Wang, W., Roberts, A.J.: Slow manifold and averaging for slow-fast stochastic differential system. J. Math. Anal. Appl. 398(2), 822–839 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  164. Warn, T., Bokhove, O., Shepherd, T., Vallis, G.: Rossby number expansions, slaving principles, and balance dynamics. Q. J. R. Meteorol. Soc. 121(523), 723–739 (1995)CrossRefGoogle Scholar
  165. White, A.A.: A view of the equations of meteorological dynamics and various approximations. In: Norbury, J., Roulstone, I. (eds.) Large-Scale Atmosphere–ocean Dynamics, vol. 1, pp. 1–100. Cambridge University Press (2002)Google Scholar
  166. Whitehead, J.P., Wingate, B.A.: The influence of fast waves and fluctuations on the evolution of the dynamics on the slow manifold. J. Fluid Mech. 757, 155–178 (2014)MathSciNetCrossRefGoogle Scholar
  167. Wouters, J., Lucarini, V.: Multi-level dynamical systems: connecting the Ruelle response theory and the Mori-Zwanzig approach. J. Stat. Phys. 151(5), 850–860 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  168. Zeitlin, V., Medvedev, S.B., Plougonven, R.: Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory. J. Fluid Mech. 481, 269–290 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  169. Zwanzig, R.: Nonlinear generalized Langevin equations. J. Stat. Phys. 9(3), 215–220 (1973)CrossRefGoogle Scholar
  170. Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press (2001)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Christian L. E. Franzke
    • 1
  • Marcel Oliver
    • 2
    Email author
  • Jens D. M. Rademacher
    • 3
  • Gualtiero Badin
    • 1
  1. 1.Center for Earth System Research and Sustainability (CEN), Universität HamburgHamburgGermany
  2. 2.Jacobs UniversityBremenGermany
  3. 3.Fachbereich MathematikUniversität BremenBremenGermany

Personalised recommendations