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Balanced and unbalanced routes to dissipation in an equilibrated Eady flow

Published online by Cambridge University Press:  17 June 2010

M. JEROEN MOLEMAKER ,
JAMES C. MCWILLIAMS  and
XAVIER CAPET
M. JEROEN MOLEMAKER*
Affiliation:
IGPP, UCLA, 405 Hilgard Avenue, Los Angeles, CA 90066, USA
JAMES C. MCWILLIAMS
Affiliation:
IGPP, UCLA, 405 Hilgard Avenue, Los Angeles, CA 90066, USA
XAVIER CAPET
Affiliation:
Laboratoire de Physique des Océans, IFREMER/CNRS, Brest, France
*
†Email address for correspondence: [email protected]
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Abstract

The oceanic general circulation is forced at large scales and is unstable to mesoscale eddies. Large-scale currents and eddy flows are approximately in geostrophic balance. Geostrophic dynamics is characterized by an inverse energy cascade except for dissipation near the boundaries. In this paper, we confront the dilemma of how the general circulation may achieve dynamical equilibrium in the presence of continuous large-scale forcing and the absence of boundary dissipation. We do this with a forced horizontal flow with spatially uniform rotation, vertical stratification and vertical shear in a horizontally periodic domain, i.e. a version of Eady's flow carried to turbulent equilibrium. A direct route to interior dissipation is presented that is essentially non-geostrophic in its dynamics, with significant submesoscale frontogenesis, frontal instability and breakdown, and forward kinetic energy cascade to dissipation. To support this conclusion, a series of simulations is made with both quasigeostrophic and Boussinesq models. The quasigeostrophic model is shown as increasingly inefficient in achieving equilibration through viscous dissipation at increasingly higher numerical resolution (hence Reynolds number), whereas the non-geostrophic Boussinesq model equilibrates with only weak dependence on resolution and Rossby number.

JFM classification

Geophysical and Geological Flows: Geostrophic turbulence Geophysical and Geological Flows: Ocean processes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 35 - 63
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Blumen, B. 1978 Uniform potential vorticity flow. Part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35, 774–783.2.0.CO;2>CrossRefGoogle Scholar
Boccaletti, G., Ferrari, R. & Fox-Kemper, B. 2007 Mixed layer instabilities and restratification. J. Phys. Ocean. 37, 2228–2250.CrossRefGoogle Scholar
Capet, X., Klein, P., Bach, L.-H., Lapeyre, G. & McWilliams, J. 2008 a Surface kinetic energy transfer in surface quasi-geostrophic flows. J. Fluid Mech. 604, 165–174.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, M. & Shchepetkin, A. F. 2008 b Mesoscale to submesoscale transition in the California current system: frontal processes. J. Phys. Ocean. 38, 44–69.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, M. M. J. & Shchepetkin, A. F. 2008 c Mesoscale to submesoscale transition in the California current system. Part III: energy balance and flux. J. Phys. Ocean. 38, 2256–2269.CrossRefGoogle Scholar
Charney, J. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
Eady, E. 1949 Long waves and cyclone waves. Tellus 1, 33–52.CrossRefGoogle Scholar
Held, I., Pierrehumbert, R., Garner, S. & Swanson, K. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.CrossRefGoogle Scholar
Hoskins, B. & Bretherton, F. 1972 Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29, 11–37.2.0.CO;2>CrossRefGoogle Scholar
Hoyer, J. & Sadourny, R. 1982 Closure modelling of fully developed baroclinic instability. J. Atmos. Sci. 39, 707–721.2.0.CO;2>CrossRefGoogle Scholar
Klein, P., Bach, L., Lapeyre, G., Capet, X., LE Gentil, S. & Sasaki, H. 2008 Upper ocean turbulence from high resolution 3-D simulation. J. Phys. Ocean. (in press).CrossRefGoogle Scholar
Leonard, B. P. 1979 A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Meth. Appl. Mech. Engng 19, 59–98.CrossRefGoogle Scholar
Lindborg, E. 2005 The effect of rotation on the mesoscale energy cascade in the free atmosphere. Geophys. Res. Lett. 19, 59–98.Google Scholar
Lorenz, E. 1955 Available energy and the maintenance of the general circulation. Tellus 7, 157–167.CrossRefGoogle Scholar
McWilliams, J. 1985 A note on a uniformly valid model spanning the regimes of geostrophic and isotropic, stratified turbulence: balanced turbulence. J. Atmos. Sci. 42, 1773–1774.2.0.CO;2>CrossRefGoogle Scholar
McWilliams, J. 2003 Diagnostic force balance and its limits. In Nonlinear Processes in Geophysical Fluid Dynamics (ed. Fuentes, O. Velasco, Sheinbaum, J. & Ochoa, J.), pp. 287–304. Kluwer.CrossRefGoogle Scholar
Molemaker, M. & Dijkstra, H. 2000 Stability of a cold-core eddy in the presence of convection: hydrostatic versus non-hydrostatic modelling. J. Phys. Ocean. 30, 475–494.2.0.CO;2>CrossRefGoogle Scholar
Molemaker, M. J. & McWilliams, J. C. 2010 Local balance and cross-scale flux of available potential energy. J. Fluid Mech. 645, 295–314.CrossRefGoogle Scholar
Molemaker, M. & Vilá-Guerau de Arellano, J. 1998 Control of chemical reactions by convective turbulence in the boundary layer. J. Atmos. Sci. 55, 568–579.2.0.CO;2>CrossRefGoogle Scholar
Molemaker, M., McWilliams, J. & Yavneh, I. 2000 Instability and equilibration of centrifugally stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 5270–5273.CrossRefGoogle Scholar
Molemaker, M., McWilliams, J. & Yavneh, I. 2005 Baroclinic instability and loss of balance. J. Phys. Ocean. 35, 1505–1517.CrossRefGoogle Scholar
Muller, P., McWilliams, J. & Molemaker, M. 2005 Routes to dissipation in the ocean: the 2d/3d turbulence conundrum. In Marine Turbulence: Theories, Observations and Models (ed. Baumert, H., Simpson, J. & Sundermann, J.), pp. 397–405. Cambridge University Press.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Press, W., Flannery, B., Teukolsky, S. & Vetterling, W. 1986 Numerical Recipes. Cambridge University Press.Google Scholar
Rhines, P. & Young, W. 1982 Homogenization of potential vorticity in planetary gyres. J. Fluid Mech. 122, 347–367.CrossRefGoogle Scholar
Shchepetkin, A. & McWilliams, J. 1998 Quasi-monotone advection schemes based on explicit locally adaptive dissipation. Monthly Weather Rev. 126, 1541–1580.2.0.CO;2>CrossRefGoogle Scholar
Stone, P. 1966 On non-geostrophic baroclinic instability. J. Atmos. Sci. 23, 390–400.2.0.CO;2>CrossRefGoogle Scholar
Stone, P. 1970 On non-geostrophic baroclinic instability. Part II. J. Atmos. Sci. 27, 721–726.2.0.CO;2>CrossRefGoogle Scholar
Tulloch, R. & Smith, K. 2009 Quasigeostrophic turbulence with explicit surface dynamics: Application to the atmospheric energy spectrum. J. Atmos. Sci. 66, 450–467.CrossRefGoogle Scholar
Waite, M. & Bartello, P. 2006 The transition from geostrophic to stratified turbulence. J. Fluid Mech. 568, 89–108.CrossRefGoogle Scholar
Winters, K., Lombard, P., Riley, J. & D'Asaro, E. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–128.CrossRefGoogle Scholar