- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Simulations of eddy kinetic energy transport in barotropic turbulence

Ian Grooms*

  • Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA

PDF

Phys. Rev. Fluids 2, 113801 – Published 27 November, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.113801

Abstract

Eddy energy transport in rotating two-dimensional turbulence is investigated using numerical simulation. Stochastic forcing is used to generate an inhomogeneous field of turbulence and the time-mean energy profile is diagnosed. An advective-diffusive model for the transport is fit to the simulation data by requiring the model to accurately predict the observed time-mean energy distribution. Isotropic harmonic diffusion of energy is found to be an accurate model in the case of uniform, solid-body background rotation (the f plane), with a diffusivity that scales reasonably well with a mixing-length law κ∝Vℓ, where V and ℓ are characteristic eddy velocity and length scales. Passive tracer dynamics are added and it is found that the energy diffusivity is 75% of the tracer diffusivity. The addition of a differential background rotation with constant vorticity gradient β leads to significant changes to the energy transport. The eddies generate and interact with a mean flow that advects the eddy energy. Mean advection plus anisotropic diffusion (with reduced diffusivity in the direction of the background vorticity gradient) is moderately accurate for flows with scale separation between the eddies and mean flow, but anisotropic diffusion becomes a much less accurate model of the transport when scale separation breaks down. Finally, it is observed that the time-mean eddy energy does not look like the actual eddy energy distribution at any instant of time. In the future, stochastic models of the eddy energy transport may prove more useful than models of the mean transport for predicting realistic eddy energy distributions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (49)

  1. D. B. Chelton, M. G. Schlax, and R. M. Samelson, Global observations of nonlinear mesoscale eddies, Prog. Oceanogr. 91, 167 (2011).
  2. S. M. Griffies, M. Winton, W. G. Anderson, R. Benson, T. L. Delworth, C. O. Dufour, J. P. Dunne, P. Goddard, A. K. Morrison, A. Rosati, A. Wittenberg, J. Yin, and R. Zhang, Impacts on ocean heat from transient mesoscale eddies in a hierarchy of climate models, J. Climate 28, 952 (2015).
  3. J. Kay et al., The Community Earth System Model (CESM) large ensemble project: A community resource for studying climate change in the presence of internal climate variability, Bull. Am. Meteorol. Soc. 96, 1333 (2015).
  4. S. Griffies et al., Coordinated ocean-ice reference experiments (COREs), Ocean Model. 26, 1 (2009).
  5. A. R. Karspeck, S. Yeager, G. Danabasoglu, T. Hoar, N. Collins, K. Raeder, J. Anderson, and J. Tribbia, An ensemble adjustment Kalman filter for the CCSM4 ocean component, J. Climate 26, 7392 (2013).
  6. P. R. Gent and J. C. McWilliams, Isopycnal mixing in ocean circulation models, J. Phys. Ocean. 20, 150 (1990).
  7. P. R. Gent, J. Willebrand, T. J. McDougall, and J. C. McWilliams, Parameterizing eddy-induced tracer transports in ocean circulation models, J. Phys. Ocean. 25, 463 (1995).
  8. M. H. Redi, Oceanic isopycnal mixing by coordinate rotation, J. Phys. Ocean. 12, 1154 (1982).
  9. I. Held and V. Larichev, A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane, J. Atmos. Sci. 53, 946 (1996).
  10. M. Visbeck, J. Marshall, T. Haine, and M. Spall, Specification of eddy transfer coefficients in coarse-resolution ocean circulation models, J. Phys. Ocean. 27, 381 (1997).
  11. P. H. Stone, A simplified radiative-dynamical model for the static stability of rotating atmospheres, J. Atmos. Sci. 29, 405 (1972).
  12. K. S. Smith, The geography of linear baroclinic instability in earth's oceans, J. Marine Res. 65, 655 (2007).
  13. R. Tulloch, J. Marshall, C. Hill, and K. S. Smith, Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean, J. Phys. Ocean. 41, 1057 (2011).
  14. A. Klocker and R. Abernathey, Global patterns of mesoscale eddy properties and diffusivities, J. Phys. Ocean. 44, 1030 (2014).
  15. P. Cessi, An energy-constrained parameterization of eddy buoyancy flux, J. Phys. Ocean. 38, 1807 (2008).
  16. C. Eden and R. J. Greatbatch, Towards a mesoscale eddy closure, Ocean Model. 20, 223 (2008).
  17. D. P. Marshall and A. J. Adcroft, Parameterization of ocean eddies: Potential vorticity mixing, energetics and Arnold's first stability theorem, Ocean Model. 32, 188 (2010).
  18. M. F. Jansen, A. J. Adcroft, R. Hallberg, and I. M. Held, Parameterization of eddy fluxes based on a mesoscale energy budget, Ocean Model. 92, 28 (2015).
  19. D. P. Marshall, J. R. Maddison, and P. S. Berloff, A framework for parameterizing eddy potential vorticity fluxes, J. Phys. Ocean. 42, 539 (2012).
  20. S. Bachman and B. Fox-Kemper, Eddy parametrization challenge suite I: Eady spindown, Ocean Model. 64, 12 (2013).
  21. S. Bachman, D. Marshall, J. Maddison, and J. Mak, Evaluation of a scalar eddy transport coefficient based on geometric constraints, Ocean Model. 109, 44 (2017).
  22. D. P. Marshall, M. H. Ambaum, J. R. Maddison, D. R. Munday, and L. Novak, Eddy saturation and frictional control of the antarctic circumpolar current, Geophys. Res. Lett. 44, 286 (2017).
  23. J. Mak, D. Marshall, J. Maddison, and S. D. Bachman, Emergent eddy saturation from an energy constrained eddy parameterisation, Ocean Model. 112, 125 (2017).
  24. S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000).
  25. G. L. Mellor and T. Yamada, Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys. 20, 851 (1982).
  26. I. Grooms, L.-P. Nadeau, and K. S. Smith, Mesoscale eddy energy locality in an idealized ocean model, J. Phys. Ocean. 43, 1911 (2013).
  27. R. Chen, G. R. Flierl, and C. Wunsch, A description of local and nonlocal eddy-mean flow interaction in a global eddy-permitting state estimate, J. Phys. Ocean. 44, 2336 (2014).
  28. I. Grooms, K. S. Smith, and A. J. Majda, Multiscale models for synoptic-mesoscale interactions in the oceans, Dyn. Atmos. Oceans 58, 95 (2012).
  29. J. C. McWilliams and G. R. Flierl, On the evolution of isolated, nonlinear vortices, J. Phys. Ocean. 9, 1155 (1979).
  30. G. Reznik, Dynamics of localized vortices on the beta plane, Izv. Atmos. Ocean. Phys. 46, 784 (2010).
  31. X. Zhai, H. Johnson, and D. Marshall, Significant sink of ocean-eddy energy near western boundaries, Nat. Geosci. 3, 608 (2010).
  32. J. J. Early, R. Samelson, and D. B. Chelton, The evolution and propagation of quasigeostrophic ocean eddies, J. Phys. Ocean. 41, 1535 (2011).
  33. A. Klocker and D. P. Marshall, Advection of baroclinic eddies by depth mean flow, Geophys. Res. Lett. 41, 3517 (2014).
  34. R. Abernathey and G. Haller, Transport by Lagrangian vortices in the Eastern Pacific, arXiv:1705.08487.
  35. D. K. Lilly, in Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences, edited by H. H. Goldstine (IBM, Yorktown Heights, 1967), pp. 195–210.
  36. A. Yoshizawa, A statistically-derived subgrid model for the large-eddy simulation of turbulence, Phys. Fluids 25, 1532 (1982).
  37. A. Yoshizawa and K. Horiuti, A statistically-derived subgrid-scale kinetic energy model for the large-eddy simulation of turbulent flows, J. Phys. Soc. Jpn. 54, 2834 (1985).
  38. V. Yakhot and S. A. Orszag, Renormalization group analysis of turbulence. I. Basic theory, J. Sci. Comput. 1, 3 (1986).
  39. I. Grooms, A computational study of turbulent kinetic energy transport in barotropic turbulence on the f-plane, Phys. Fluids 27, 101701 (2015).
  40. G. Denk and S. Schäffler, Adams methods for the efficient solution of stochastic differential equations with additive noise, Computing 59, 153 (1997).
  41. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, Philadelphia, 2003).
  42. G. I. Taylor, Diffusion by continuous movements, Proc. London Math. Soc. 20, 196 (1921).
  43. A. Monin and A. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1 (MIT Press, Cambridge, 1971).
  44. A. Provenzale, Transport by coherent barotropic vortices, Annu. Rev. Fluid Mech. 31, 55 (1999).
  45. O. Bühler, Waves and Mean Flows (Cambridge University Press, Cambridge, 2014).
  46. N. Hogg and H. Stommel, The heton: An elementary interaction between discrete baroclinic geostrophic vortices, and its implications concerning eddy heat-flow, Proc. R. Soc. London Ser. A 397, 1 (1985).
  47. D. Olbers and C. Eden, A global model for the diapycnal diffusivity induced by internal gravity waves, J. Phys. Ocean. 43, 1759 (2013).
  48. J. Weiss, A. Provenzale, and J. McWilliams, Lagrangian dynamics in high-dimensional point-vortex systems, Phys. Fluids 10, 1929 (1998).
  49. I. Grooms, A Gaussian-product stochastic Gent-McWilliams parameterization, Ocean Model. 106, 27 (2016).

Outline

Information

Phys. Rev. Fluids 2, 113801– Published 27 November, 2017
  • Received 23 June 2017

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation