Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T12:16:17.163Z Has data issue: false hasContentIssue false
  • English
  • Français

Bottom-trapped currents as statistical equilibrium states above topographic anomalies

Published online by Cambridge University Press:  17 April 2012

A. Venaille
A. Venaille*
Affiliation:
Laboratoire de Physique, ENS-Lyon, 46 Allée d’Italie, 69007 Lyon, France
*
†Email address for correspondence: antoine.venaille@ens-lyon.org
Get access Rights & Permissions [Opens in a new window]

Abstract

Oceanic geostrophic turbulence is mostly forced at the surface, yet strong bottom-trapped flows are commonly observed along topographic anomalies. Here we consider the case of a freely evolving, initially surface-intensified velocity field above a topographic bump, and show that the self-organization into a bottom-trapped current can result from its turbulent dynamics. Using equilibrium statistical mechanics, we explain this phenomenon as the most probable outcome of turbulent stirring. We compute explicitly a class of solutions characterized by a linear relation between potential vorticity and streamfunction, and predict when the bottom intensification is expected. Using direct numerical simulations, we provide an illustration of this phenomenon that agrees qualitatively with theory, although the ergodicity hypothesis is not strictly fulfilled.

JFM classification

Geophysical and Geological Flows: Geostrophic turbulence Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 699 , 25 May 2012 , pp. 500 - 510
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bouchet, F. 2008 Simpler variational problems for statistical equilibria of the 2D Euler equation and other systems with long range interactions. Physica D: Nonlinear Phenom. 237, 1976–1981.CrossRefGoogle Scholar
2. Bouchet, F. & Simonnet, E. 2009 Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102 (9), 094504.CrossRefGoogle ScholarPubMed
3. Bouchet, F. & Venaille, A. 2012 Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep., doi:10.1016/j.physrep.2012.02.001.Google Scholar
4. Bretherton, F. P. & Haidvogel, D. B. 1976 Two-dimensional turbulence above topography. J. Fluid Mech. 78, 129–154.CrossRefGoogle Scholar
5. Carnevale, G. F. & Frederiksen, J. S. 1987 Nonlinear stability and statistical mechanics of flow over topography. J. Fluid Mech. 175, 157–181.CrossRefGoogle Scholar
6. Dewar, W. K. 1998 Topography and barotropic transport control by bottom friction. J. Mar. Res. 56, 295–328.Google Scholar
7. Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253–282.CrossRefGoogle Scholar
8. Gill, A. E., Green, J. S. A. & Simmons, A. J. 1974 Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep-Sea Res. 21, 499–528.Google Scholar
9. Held, I. M., Pierrehumbert, R. T., Garner, S. T. & Swanson, K. L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.CrossRefGoogle Scholar
10. Kraichnan, R. H. & Montgomery, D. 1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619.Google Scholar
11. Majda, A. & Wang, X. 2006 Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press.CrossRefGoogle Scholar
12. Maltrud, M. & Vallis, G. K. 1991 Energy spectra and coherent structures in forced two-dimensional and geostrophic turbulence. J. Fluid Mech. 228, 321–342.Google Scholar
13. Merryfield, W. J. 1998 Effects of stratification on quasi-geostrophic inviscid equilibria. J. Fluid Mech. 354, 345–356.CrossRefGoogle Scholar
14. Miller, J., Weichman, P. B. & Cross, M. C. 1992 Statistical mechanics, Euler’s equation, and Jupiter’s red spot. Phys. Rev. A 45, 2328–2359.Google Scholar
15. de Miranda, A. P., Barnier, B. & Dewar, W. K. 1999 On the dynamics of the Zapiola anticyclone. J. Geophys. Res. 104, 21137–21150.CrossRefGoogle Scholar
16. Robert, R. & Sommeria, J. 1992 Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics. Phys. Rev. Lett. 69, 2776–2779.CrossRefGoogle Scholar
17. Salmon, R., Holloway, G. & Hendershott, M. C. 1976 The equilibrium statistical mechanics of simple quasi-geostrophic models. J. Fluid Mech. 75, 691–703.CrossRefGoogle Scholar
18. Saunders, P. M. & King, B. A. 1995 Bottom currents derived from a shipborne ADCP on WOCE cruise A11 in the South Atlantic. J. Phys. Oceanogr. 25, 329–347.2.0.CO;2>CrossRefGoogle Scholar
19. Smith, K. S. 2007 The geography of linear baroclinic instability in Earth’s oceans. J. Mar. Res. 65, 655–683.CrossRefGoogle Scholar
20. Smith, K. S. & Vallis, G. K. 2001 The scales and equilibration of midocean eddies: freely evolving flow. J. Phys. Oceanogr. 31, 554–571.2.0.CO;2>CrossRefGoogle Scholar
21. Tulloch, R., Marshall, J., Hill, C. & Smith, K. S. 2011 Scales, growth rates and spectral fluxes of baroclinic instability in the ocean. J. Phys. Oceanogr. 41, 1057–1076.Google Scholar
22. Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
23. Vallis, G. K. & Maltrud, M. E. 1993 Generation of mean flows and jets on a beta plane and over topography. J. Phys. Oceanogr. 23, 1346–1362.Google Scholar
24. Venaille, A. & Bouchet, F. 2011 Solvable phase diagrams and ensemble inequivalence for two-dimensional and geophysical flows. J. Stat. Phys. 143 (2), 346–380.Google Scholar
25. Venaille, A., Le Sommer, J., Molines, J. M. & Barnier, B. 2011a Stochastic variability of oceanic flows above topography anomalies. Geophys. Res. Lett. 38 (16611).Google Scholar
26. Venaille, A., Vallis, G. K. & Griffies, S. 2012 The catalytic role of beta effect in barotropization process. J. Fluid Mech. (submitted) arXiv:1201.0657.Google Scholar
27. Venaille, A., Vallis, G. K. & Smith, K. S. 2011b Baroclinic turbulence in the ocean: analysis with primitive equation and quasi-geostrophic simulations. J. Phys. Oceanogr. 41 (9), 1605–1623.CrossRefGoogle Scholar