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A generalized Osborn–Cox relation

Published online by Cambridge University Press:  27 July 2009

CARSTEN EDEN,
DIRK OLBERS and
RICHARD J. GREATBATCH
CARSTEN EDEN*
Affiliation:
Leibniz Institute of Marine Sciences at Kiel University, 24105 Kiel, Germany
DIRK OLBERS
Affiliation:
Alfred-Wegener-Institute for Polar and Marine Research, 27515 Bremerhaven, Germany
RICHARD J. GREATBATCH
Affiliation:
Leibniz Institute of Marine Sciences at Kiel University, 24105 Kiel, Germany
*
†Email address for correspondence: ceden@ifm-geomar.de
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Abstract

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The generalized temporal residual mean (TRM-G) framework is reviewed and illustrated using a numerical simulation of vertical shear instability. It is shown how TRM-G reveals the physically relevant amount of diapycnal eddy fluxes and implied diapycnal mixing, and how TRM-G relates to the Osborn–Cox relation, which is often used to obtain observational estimates of the diapycnal diffusivity. An exact expression for the diapycnal diffusivity in the TRM-G is given in the presence of molecular diffusion, based on acknowledging and summing up an entire hierarchy of eddy buoyancy moments. In this revised form of the Osborn–Cox relation, diapycnal diffusivity is related only to irreversible mixing of buoyancy, since all advective and molecular flux terms are converted to dissipation of variance and higher order moments. An approximate but closed analytical expression can be given for the revised Osborn–Cox relation with the caveat that this closed expression implies unphysical cross-boundary rotational fluxes.

It is demonstrated that the original Osborn–Cox relation, in which advective and molecular flux terms are simply neglected, is an approximation to the full form valid to first order. In the numerical simulation the original Osborn–Cox relation holds to a surprisingly good approximation despite large advective fluxes of variance and large lateral inhomogeneity in the turbulent mixing.

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009, pp. 457 - 474
Copyright
Copyright © Cambridge University Press 2009

References

REFERENCES

Andrews, D. G. & McIntyre, M. E. 1976 Planetary waves in horizontal and vertical shear: the generalized Eliassen–Palm relation and the zonal mean accelaration. J. Atmos. Sci. 33, 2031–2048.2.0.CO;2>CrossRefGoogle Scholar
Andrews, D. G. & McIntyre, M. E. 1978 Generalized Eliassen–Palm and Charney–Drazin theorems for waves on axisymmetric mean flows in compressible atmosphere. J. Atmos. Sci. 35, 175–185.Google Scholar
De Szoeke, R. A. & Bennett, A. F. 1993 Microstructure fluxes across density surfaces. J. Phys. Oceanogr. 23, 2254–2264.Google Scholar
Dillon, T. M. & Park, M. M. 1987 The available potential energy of overturns as an indicator of mixing in the seasonal thermocline. J. Geophys. Res. 92 (C5) 5345–5353.Google Scholar
Eden, C., Greatbatch, R. J. & Olbers, D. 2007 Interpreting eddy fluxes. J. Phys. Oceanogr. 37, 1282–1296.Google Scholar
Gregg, M. C. 1998 Estimation and geography of diapycnal mixing in the stratified ocean. In Physical Processes in Lakes and Oceans (ed. J. Imberger), Coastal and Estuarine Studies, vol. 54, pp. 305–338. Am. Geophys. Union.CrossRefGoogle Scholar
McDougall, T. J. & McIntosh, P. C. 1996 The temporal-residual-mean velocity. Part I. Derivation and the scalar conservation equation. J. Phys. Oceanogr. 26, 2653–2665.Google Scholar
McDougall, T. J. & McIntosh, P. C. 2001 The temporal-residual-mean velocity. Part II. Isopycnal interpretation and the tracer and momentum equations. J. Phys. Oceanogr. 31 (5), 1222–1246.2.0.CO;2>CrossRefGoogle Scholar
Medvedev, A. S. & Greatbatch, R. J. 2004 On advection and diffusion in the mesosphere and lower thermosphere: the role of rotational fluxes. J. Geophys. Res. 109 (D07104, 10.1029/2003JD003931).Google Scholar
Nakamura, N. 1996 Two-dimensional mixing, edge formation, and permeability diagnosed in an area coordinate. J. Atmos. Sci. 53, 1524–1537.Google Scholar
Osborn, T. R. & Cox, C. S. 1972 Oceanic fine structure. Geophys. Astrophys. Fluid Dyn. 3 (1), 321–345.Google Scholar
Plumb, R. A. 1990 A nonacceleration theorem for transient quasi-geostrophic eddies on a three-dimensional time-mean flow. J. Atmos. Sci. 47 (15), 1825–1836.2.0.CO;2>CrossRefGoogle Scholar
Plumb, R. A & Ferrari, R. 2005 Transformed Eulerian-mean theory. Part I. Nonquasigeostrophic theory for eddies on a zonal-mean flow. J. Phys. Oceanogr. 35 (2), 165–174.CrossRefGoogle Scholar
Schmitt, R. W., Ledwell, J. R., Montgomery, E. T., Polzin, K. L. & Toole, J. M. 2005 Enhanced diapycnal mixing by salt fingers in the thermocline of the tropical Atlantic. Science 308 (5722), 685–688.CrossRefGoogle ScholarPubMed
Thorpe, S. A. 1977 Turbulence and mixing in a Scottish loch. Phil. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 286 (1334), 125–181.Google Scholar
Winters, K. B. & D'Asaro, E. A. 1995 Diascalar flux and the rate of fluid mixing. J. Fluid. Mech. 317, 179–193.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar