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The IDEMIX Model: Parameterization of Internal Gravity Waves for Circulation Models of Ocean and Atmosphere

  • Dirk OlbersEmail author
  • Carsten Eden
  • Erich Becker
  • Friederike Pollmann
  • Johann Jungclaus
Chapter
Part of the Mathematics of Planet Earth book series (MPE, volume 1)

Abstract

The IDEMIX concept is an energetically consistent framework to describe wave effects in circulation models of ocean and atmosphere. It is based on the radiative transfer equation for an internal gravity wave field in physical and wavenumber space and was shown to be successful for ocean applications. An improved IDEMIX model for the ocean will be constructed and extended by a new high-frequency, high vertical wavenumber compartment, forcing by mesoscale eddy dissipation, anisotropic tidal forcing, and wave–mean flow interaction. It will be validated using observational and model estimates. A novel concept for gravity wave parameterization in atmospheric circulation models is developed. As for the ocean, the wave field is represented by the wave energy density in physical and wavenumber space, and its prognostic computation is performed by the radiative transfer equation. This new concept goes far beyond conventional gravity wave schemes which are based on the single-column approximation and, in particular, on the strong assumptions of a stationary mean flow and a stationary wave energy equation. The radiative transfer equation has—to our knowledge—never been considered in the atmospheric community as a framework for subgrid-scale parameterization. The proposed parameterization will, for the first time, (1) include all relevant sources continuously in space and time and (2) accommodate all gravity wave sources (orography, fronts, and convection) in a single parameterization framework. Moreover, the new scheme is formulated in a precisely energy-preserving fashion. The project will contribute to a transfer of knowledge from the oceanic community to the atmospheric community and vice versa. We give a brief description of the oceanic and atmospheric internal wave fields, the most important processes of generation and interactions, and the ingredients of the model IDEMIX.

References

  1. Alexander, M., Dunkerton, T.: A spectral parameterization of mean-flow forcing due to breaking gravity waves. J. Atmos. Sci. 56(24), 4167–4182 (1999)MathSciNetCrossRefGoogle Scholar
  2. Alexander, M., Geller, M., McLandress, C., Polavarapu, S., Preusse, P., Sassi, F., Sato, K., Eckermann, S., Ern, M., Hertzog, A., et al.: Recent developments in gravity-wave effects in climate models and the global distribution of gravity-wave momentum flux from observations and models. Q. J. R. Meteorol. Soc. 136(650), 1103–1124 (2010)Google Scholar
  3. Alford, M.: Internal swell generation: The spatial distribution of energy flux from the wind to mixed layer near-inertial motions. J. Phys. Oceanogr. 31(8), 2359–2368 (2001)CrossRefGoogle Scholar
  4. Andrews, D.G., McIntyre, M.E.: Planetary waves in horizontal and vertical shear: The generalized Eliassen-Palm relation and the zonal mean accelaration. J. Atmos. Sci. 33, 2031–2048 (1976)CrossRefGoogle Scholar
  5. Arbic, B.K., Shriver, J.F., Hogan, P.J., Hurlburt, H.E., McClean, J.L., Metzger, E.J., Scott, R.B., Sen, A., Smedstad, O.M., Wallcraft, A.J.: Estimates of bottom flows and bottom boundary layer dissipation of the oceanic general circulation from global high-resolution models. J. Geophys. Res, 114(C2) (2009)Google Scholar
  6. Becker, E.: Direct heating rates associated with gravity wave saturation. J. Atmos. Sol. Terr. Phys. 66(6), 683–696 (2004)CrossRefGoogle Scholar
  7. Becker, E.: Dynamical control of the middle atmosphere. Space Sci. Rev. 168(1–4), 283–314 (2012)CrossRefGoogle Scholar
  8. Becker, E., McLandress, C.: Consistent scale interaction of gravity waves in the Doppler spread parameterization. J. Atmos. Sci. 66(5), 1434–1449 (2009)CrossRefGoogle Scholar
  9. Bell, T.H.: Topographically generated internal waves in the open ocean. J. Geophys. Res. 80(3), 320–327 (1975)CrossRefGoogle Scholar
  10. Bell, T.H.: Radiation damping of inertial oscillations in the upper ocean. J. Fluid Mech. 88(02), 289–308 (1978)zbMATHCrossRefGoogle Scholar
  11. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. McGraw-Hill Book Company, New York (1978)Google Scholar
  12. Berry, M.V., Mount, K.: Semiclassical approximations in wave mechanics. Rep. Prog. Phys. 35(1), 315–397 (1972)CrossRefGoogle Scholar
  13. Bretherton, F.P., Garrett, C.J.: Wavetrains in inhomogeneous moving media. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 302, pp. 529–554. The Royal Society (1968)Google Scholar
  14. Brüggemann, N., Eden, C.: Routes to dissipation under different dynamical conditions. J. Phys. Oceanogr. 45(8), 2149–2168 (2015)CrossRefGoogle Scholar
  15. Bryan, K., Lewis, L.: A water mass model of circulation cells and the source waters of the Pacific equatorial undercurrent. J. Phys. Oceanogr. 28, 62–84 (1979)Google Scholar
  16. Bühler, O., Mcintyre, M.E.: Wave capture and wave-vortex duality. J. Fluid Mech. 534, 67–95 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. Cairns, J.L., Williams, G.O.: Internal wave observations from a midwater float, 2. J. Geophys. Res. 81(12), 1943–1950 (1976)CrossRefGoogle Scholar
  18. Charron, M., Manzini, E.: Gravity waves from fronts: Parameterization and middle atmosphere response in a general circulation model. J. Atmos. Sci. 59(5), 923–941 (2002)MathSciNetCrossRefGoogle Scholar
  19. Cummins, P.F., Holloway, G., Gargett, A.E.: Sensitivity of the GFDL ocean general circulation model to a parameterization of vertical diffusion. J. Phys. Oceanogr. 20, 817–830 (1990)CrossRefGoogle Scholar
  20. Desaubies, Y.J.: Analytical representation of internal wave spectra. J. Geophys. Res. 6(6), 976–981 (1976)Google Scholar
  21. Dewar, W.K., Hogg, A.M.: Topographic inviscid dissipation of balanced flow. Ocean Model. 32(1), 1–13 (2010)CrossRefGoogle Scholar
  22. Dunkerton, T.J., Fritts, D.C.: Transient gravity wave-critical layer interaction. Part I: convective adjustment and the mean zonal acceleration. J. Atmos. Sci. 41(6), 992–1007 (1984)CrossRefGoogle Scholar
  23. Dushaw, B.D., Worcester, P.F., Dzieciuch, M.A.: On the predictability of mode-1 internal tides. Deep-Sea Res. 58(6), 677–698 (2011)CrossRefGoogle Scholar
  24. Eden, C.: Revisiting the energetics of the ocean in Boussinesq approximation. J. Phys. Oceanogr. 45(3), 630–637 (2015)CrossRefGoogle Scholar
  25. Eden, C., Czeschel, L., Olbers, D.: Towards energetically consistent ocean models. J. Phys. Oceanogr. 44(8), 2093–2106 (2014)CrossRefGoogle Scholar
  26. Eden, C., Greatbatch, R.J.: Diapycnal mixing by mesoscale eddies. Ocean Model. 23, 113–120 (2008a)CrossRefGoogle Scholar
  27. Eden, C., Greatbatch, R.J.: Towards a mesoscale eddy closure. Ocean Model. 20, 223–239 (2008b)CrossRefGoogle Scholar
  28. Eden, C., Olbers, D.: An energy compartment model for propagation, non-linear interaction and dissipation of internal gravity waves. J. Phys. Oceanogr. 44, 2093–2106 (2014)CrossRefGoogle Scholar
  29. Eden, C., Olbers, D.: Interaction of internal waves with a mean flow. Part II: wave drag. J. Phys. Oceanogr. 47, 1403–1412 (2017)CrossRefGoogle Scholar
  30. Egbert, G.D., Erofeeva, S.Y., Ray, R.D.: Assimilation of altimetry data for nonlinear shallow-water tides: quarter-diurnal tides of the Northwest European Shelf. Cont. Shelf Res. 30(6), 668–679 (2010)CrossRefGoogle Scholar
  31. Ford, R., McIntyre, M., Norton, W.: Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57(9), 1236–1254 (2000)MathSciNetCrossRefGoogle Scholar
  32. Fritts, D.C., Alexander, M.J.: Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41(1) (2003)Google Scholar
  33. Fritts, D.C., Dunkerton, T.J.: A quasi-linear study of gravity-wave saturation and self-acceleration. J. Atmos. Sci. 41(22), 3272–3289 (1984)CrossRefGoogle Scholar
  34. Fritts, D.C., Vadas, S.L., Wan, K., Werne, J.A.: Mean and variable forcing of the middle atmosphere by gravity waves. J. Atmos. Sol. Terr. Phys. 68(3), 247–265 (2006)CrossRefGoogle Scholar
  35. Fritts, D.C., VanZandt, T.E.: Spectral estimates of gravity wave energy and momentum fluxes. Part I: energy dissipation, acceleration, and constraints. J. Atmos. Sci. 50(22), 3685–3694 (1993)CrossRefGoogle Scholar
  36. Garcia, R., Marsh, D., Kinnison, D., Boville, B., Sassi, F.: Simulation of secular trends in the middle atmosphere, 1950–2003. J. Geophys. Res. 112(D9) (2007)Google Scholar
  37. Garrett, C.: What is the near-inertial wave band and why is it different from the rest of the internal wave spectrum? J. Phys. Oceanogr. 31, 962–71 (2001)MathSciNetCrossRefGoogle Scholar
  38. Garrett, C., Munk, W.: Space-time scales of internal waves. Geophys. Astrophys. Fluid Dyn. 3(1), 225–264 (1972)CrossRefGoogle Scholar
  39. Garrett, C., Munk, W.: Space-time scales of internal waves: a progress report. J. Geophys. Res. 80, 291–297 (1975)CrossRefGoogle Scholar
  40. Gaspar, P., Gregoris, Y., Lefevre, J.-M.: A simple eddy kinetic energy model for simulations of the oceanic vertical mixing: tests at station PAPA and Long-Term Upper Ocean Study site. J. Geophys. Res. 95, 16179–16193 (1990)CrossRefGoogle Scholar
  41. Goff, J.A., Arbic, B.K.: Global prediction of abyssal hill roughness statistics for use in ocean models from digital maps of paleo-spreading rate, paleo-ridge orientation, and sediment thickness. Ocean Model. 32(1), 36–43 (2010)CrossRefGoogle Scholar
  42. Gregg, M.C.: Scaling turbulent dissipation in the thermocline. J. Geophys. Res. 94(C7), 9686–9698 (1989)CrossRefGoogle Scholar
  43. Hasselmann, K.: Weak-interaction theory of ocean waves. Basic Dev. Fluid Dyn. 2, 117–182 (1968)CrossRefGoogle Scholar
  44. Hasselmann, K., Barnett, T., Bouws, E., Carlson, H., Cartwright, D., Enke, K., Ewing, J., Gienapp, H., Hasselmann, D., Krusemann, P., Meerburg, A., Müller, P., Olbers, D., Richter, K., Sell, W., Walden, H.: Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z. 12, 1–95 (1973)Google Scholar
  45. Hines, C.O.: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere. Part 1: basic formulation. J. Atmos. Sol. Terr. Phys. 59(4), 371–386 (1997)CrossRefGoogle Scholar
  46. Jayne, S., St. Laurent, L.: Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett. 28(5), 811–814 (2001)CrossRefGoogle Scholar
  47. Jones, W.L.: Ray tracing for internal gravity waves. J. Geophys. Res. 74(8), 2028–2033 (1969)CrossRefGoogle Scholar
  48. Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G., Woollen, J., Zhu, Y., Chelliah, M., Ebiszusaki, W., Higgins, W., Janowiak, J., Mo, K., Ropelewski, C., Wang, J., Leetmaa, A., Reynolds, R., Jenne, R., Joseph, D.: The NCEP/NCAR 40-years reanalysis project. Bull. Am. Meteorol. Soc. 77, 437–471 (1996)CrossRefGoogle Scholar
  49. Kunze, E., Firing, E., Hummon, J.M., Chereskin, T.K., Thurnherr, A.M.: Global abyssal mixing inferred from lowered ADCP shear and CTD strain profiles. J. Phys. Oceanogr. 36(8), 1553–1576 (2006)CrossRefGoogle Scholar
  50. Kunze, E., Smith, S.L.: The role of small-scale topography in turbulent mixing of the global ocean. Oceanography 17(1), 55–64 (2004)CrossRefGoogle Scholar
  51. Landau, L.D., Lifshitz, E.M.: Mechanics, Course of Theoretical Physics, vol. 1. Pergamon Press Ltd. (1982)Google Scholar
  52. Lighthill, M.J.: On sound generated aerodynamically. I. General theory. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 211, pp. 564–587. The Royal Society (1952)Google Scholar
  53. Lindzen, R.S.: Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res. 86(C10), 9707–9714 (1981)CrossRefGoogle Scholar
  54. McComas, C., Müller, P.: Time scales of resonant interactions among oceanic internal waves. J. Phys. Oceanogr. 11, 139–147 (1981)CrossRefGoogle Scholar
  55. McComas, C.H.: Equilibrium mechanisms within the oceanic internal wave field. J. Phys. Oceanogr. 7, 836–845 (1977)CrossRefGoogle Scholar
  56. McDougall, T.J.: Potential enthalpy: a conservative oceanic variable for evaluating heat content and heat fluxes. J. Phys. Oceanogr. 33(5), 945–963 (2003)MathSciNetCrossRefGoogle Scholar
  57. McFarlane, N.: The effect of orographically excited gravity wave drag on the general circulation of the lower stratosphere and troposphere. J. Atmos. Sci. 44(14), 1775–1800 (1987)CrossRefGoogle Scholar
  58. McLandress, C., Shepherd, T.G.: Simulated anthropogenic changes in the brewer-dobson circulation, including its extension to high latitudes. J. Clim. 22(6), 1516–1540 (2009)CrossRefGoogle Scholar
  59. Miller, M., Palmer, T., Swinbank, R.: Parametrization and influence of subgridscale orography in general circulation and numerical weather prediction models. Meteorol. Atmos. Phys. 40(1–3), 84–109 (1989)CrossRefGoogle Scholar
  60. Molemaker, M.J., McWilliams, J.C., Yavneh, I.: Baroclinic instability and loss of balance. J. Phys. Oceanogr. 35, 1505 (2005)MathSciNetCrossRefGoogle Scholar
  61. Molemaker, M.J., McWilliams, J. C., Capet, X.: Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech. 654, 35–63 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  62. Müller, M., Cherniawsky, J., Foreman, M., Storch, J.-S.: Global M2 internal tide and its seasonal variability from high resolution ocean circulation and tide modeling. Geophys. Res. Lett. 39(19) (2012)CrossRefGoogle Scholar
  63. Müller, P.: On the diffusion of momentum and mass by internal gravity waves. J. Fluid Mech. 77(04), 789–823 (1976)zbMATHCrossRefGoogle Scholar
  64. Müller, P., Holloway, G., Henyey, F., Pomphrey, N.: Nonlinear interactions among internal gravity waves. Rev. Geophys. 24(3), 493–536 (1986)CrossRefGoogle Scholar
  65. Müller, P., Natarov, A.: The internal wave action model (IWAM). In: Proceedings, 13th Aha Huliko’a Hawaiian Winter Workshop, School of Ocean and Earth Science and Technology, Special Publication (2003)Google Scholar
  66. Müller, P., Olbers, D.J.: On the dynamics of internal waves in the deep ocean. J. Geophys. Res. 80, 3848–3860 (1975)CrossRefGoogle Scholar
  67. Müller, P., Olbers, D.J., Willebrand, J.: The IWEX spectrum. J. Geophys. Res. 83, 479–500 (1978)CrossRefGoogle Scholar
  68. Munk, W.: Internal waves and small-scale processes. In: Warren, B.A., Wunsch, C. (eds.) Evolution of Physical Oceanography, pp. 264–291. MIT Press, Cambridge, MA (1981)Google Scholar
  69. Nastrom, G., Gage, K.S.: A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42(9), 950–960 (1985)CrossRefGoogle Scholar
  70. Nikurashin, M., Ferrari, R.: Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean. Geophys. Res. Lett. 38, L08610 (2011)CrossRefGoogle Scholar
  71. Nycander, J.: Generation of internal waves in the deep ocean by tides. J. Geophys. Res. 110(C9), C10028 (2005)CrossRefGoogle Scholar
  72. Olbers, D., Eden, C.: A global model for the diapycnal diffusivity induced by internal gravity waves. J. Phys. Oceanogr. 43, 1759–1779 (2013)CrossRefGoogle Scholar
  73. Olbers, D., Eden, C.: Revisiting the generation of internal waves by resonant interaction with surface waves. J. Phys. Oceanogr. 46, 2335–2350 (2016)CrossRefGoogle Scholar
  74. Olbers, D., Eden, C.: Interaction of internal waves with a mean flow. Part I: energy conversion. J. Phys. Oceanogr. 47, 1389–1401 (2017)CrossRefGoogle Scholar
  75. Olbers, D., Willebrand, J., Eden, C.: Ocean Dynamics. Springer, Heidelberg (2012)zbMATHCrossRefGoogle Scholar
  76. Olbers, D.J.: Nonlinear energy transfer and the energy balance of the internal wave field in the deep ocean. J. Fluid Mech. 74, 375–399 (1976)zbMATHCrossRefGoogle Scholar
  77. Olbers, D.J.: Models of the oceanic internal wave field. Rev. Geophys. Space Phys. 21, 1567–1606 (1983)CrossRefGoogle Scholar
  78. Olbers, D.J.: Internal gravity waves. In: Sündermann, J. (ed.) Landolt-Börnstein—Numerical Data and Functional Relationships in Science and Technology—New Series, Group V, vol. 3a, pp. 37–82. Springer, Berlin (1986)Google Scholar
  79. Olbers, D.J., Herterich, K.: The spectral energy transfer from surface waves to internal waves. J. Fluid Mech. 92, 349–379 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  80. Osborn, T.R., Cox, C.S.: Oceanic fine structure. Geophys. Astrophys. Fluid Dyn. 3, 321–345 (1972)CrossRefGoogle Scholar
  81. Pollmann, F., Eden, C., Olbers, D.: Evaluating the global internal wave model IDEMIX using a finescale strain approximation. J. Phys. Oceanogr. 47, 2267–2289 (2017)CrossRefGoogle Scholar
  82. Polton, J.A., Smith, J.A., MacKinnon, J.A., Tejada-Martínez, A.E.: Rapid generation of high-frequency internal waves beneath a wind and wave forced oceanic surface mixed layer. Geophys. Res. Lett. 35(13) (2008)Google Scholar
  83. Polzin, K.L.: Mesoscale eddy-internal wave coupling. Part I: symmetry, wave capture, and results from the mid-ocean dynamics experiment. J. Phys. Oceanogr. 38(11), 2556–2574 (2008)CrossRefGoogle Scholar
  84. Polzin, K.L., Lvov, Y.V.: Toward regional characterizations of the oceanic internal wavefield. Rev. Geophys. 49(4), RG4003 (2011)CrossRefGoogle Scholar
  85. Polzin, K.L., Toole, J.M., Schmitt, R.W.: Finescale parameterizations of turbulent dissipation. J. Phys. Oceanogr. 25(3), 306–328 (1995)CrossRefGoogle Scholar
  86. Preusse, P., Eckermann, S.D., Ern, M., Oberheide, J., Picard, R.H., Roble, R.G., Riese, M., Russell, J.M., Mlynczak, M.G.: Global ray tracing simulations of the saber gravity wave climatology. J. Geophys. Res. 114(D8) (2009)Google Scholar
  87. Richter, J.H., Sassi, F., Garcia, R.R.: Toward a physically based gravity wave source parameterization in a general circulation model. J. Atmos. Sci. 67(1), 136–156 (2010)CrossRefGoogle Scholar
  88. Rimac, A., von Storch, J.-S., Eden, C., Haak, H.: The influence of high-resolution wind stress field on the power input to near-inertial motions in the ocean. Geophys. Res. Lett. 40(18), 4882–4886 (2013)CrossRefGoogle Scholar
  89. Scinocca, J.F.: An accurate spectral nonorographic gravity wave drag parameterization for general circulation models. J. Atmos. Sci. 60(4), 667–682 (2003)MathSciNetCrossRefGoogle Scholar
  90. Scott, R., Goff, J., Naveira Garabato, A., Nurser, A.: Global rate and spectral characteristics of internal gravity wave generation by geostrophic flow over topography. J. Geophys. Res. 116(C9) (2011)Google Scholar
  91. Senf, F., Achatz, U.: On the impact of middle-atmosphere thermal tides on the propagation and dissipation of gravity waves. J. Geophys. Res. 116(D24) (2011)CrossRefGoogle Scholar
  92. 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(10), 3095 (2009)CrossRefGoogle Scholar
  93. Smith, W.H.F., Sandwell, D.T.: Global seafloor topography from satellite altimetry and ship depth soundings. Science 277, 1957–1962 (1997)Google Scholar
  94. Tandon, A., Garrett, C.: On a recent parameterization of mesoscale eddies. J. Phys. Oceanogr. 26, 406–416 (1996)CrossRefGoogle Scholar
  95. Thorpe, S.: The excitation, dissipation, and interaction of internal waves in the deep ocean. J. Geophys. Res. 80(3), 328–338 (1975)CrossRefGoogle Scholar
  96. von Storch, J.-S., Eden, C., Fast, I., Haak, H., Hernández-Deckers, D., Maier-Reimer, E., Marotzke, J., Stammer, D.: An estimate of the Lorenz energy cycle for the world ocean based on the STORM/NCEP simulation. J. Phys. Oceanogr. 42(12), 2185–2205 (2012)CrossRefGoogle Scholar
  97. Wang, S., Zhang, F.: Source of gravity waves within a vortex-dipole jet revealed by a linear model. J. Atmos. Sci. 67(5), 1438–1455 (2010)CrossRefGoogle Scholar
  98. Warner, C., McIntyre, M.: On the propagation and dissipation of gravity wave spectra through a realistic middle atmosphere. J. Atmos. Sci. 53(22), 3213–3235 (1996)CrossRefGoogle Scholar
  99. Warner, C., McIntyre, M.: An ultrasimple spectral parameterization for nonorographic gravity waves. J. Atmos. Sci. 58(14), 1837–1857 (2001)CrossRefGoogle Scholar
  100. Watson, K.M.: The coupling of surface and internal gravity waves: revisited. J. Phys. Oceanogr. 20, 1233–1248 (1990)CrossRefGoogle Scholar
  101. Whalen, C., Talley, L., MacKinnon, J.: Spatial and temporal variability of global ocean mixing inferred from argo profiles. Geophys. Res. Lett. 39, L18612 (2012)CrossRefGoogle Scholar
  102. Whitham, G.: Two-timing, variational principles and waves. J. Fluid Mech. 44(02), 373–395 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  103. Williams, P.D., Haine, T.W., Read, P.L.: Inertia–gravity waves emitted from balanced flow: observations, properties, and consequences. J. Atmos. Sci. 65(11) (2008)CrossRefGoogle Scholar
  104. Wunsch, C., Ferrari, R.: Vertical mixing, energy and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  105. Yiğit, E., Aylward, A.D., Medvedev, A.S.: Parameterization of the effects of vertically propagating gravity waves for thermosphere general circulation models: Sensitivity study. J. Geophys. Res. 113(D19) (2008)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dirk Olbers
    • 1
    • 2
    Email author
  • Carsten Eden
    • 3
  • Erich Becker
    • 4
  • Friederike Pollmann
    • 3
  • Johann Jungclaus
    • 5
  1. 1.Alfred Wegener Institute for Polar and Marine Research (AWI)BremerhavenGermany
  2. 2.MARUM/Institut für UmweltphysikUniversität BremenBremenGermany
  3. 3.Center for Earth System Research and Sustainability (CEN), Universität HamburgHamburgGermany
  4. 4.Leibniz-Institut für Atmosphärenphysik e.V., Universität RostockRostockGermany
  5. 5.Max-Planck-Institut für Meteorologie (MPI-M)HamburgGermany

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