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How rapidly is a passive scalar mixed within closed streamlines?

Published online by Cambridge University Press:  20 April 2006

P. B. Rhines  and
W. R. Young
P. B. Rhines
Affiliation:
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543
W. R. Young
Affiliation:
University of California, San Diego, Marine Physical Laboratory of the Scripps Institution of Oceanography, La Jolla, California 92093
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Abstract

The homogenization of a passive ‘tracer’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈P1/3(L/U), where the Péclet number P=LU/κ (L,U and κ are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L2/κ. The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 133 , August 1983 , pp. 133 - 145
Copyright
© 1983 Cambridge University Press

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References

Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube Proc. R. Soc. Lond. A235, 67–77.Google Scholar
Batchelor, G. K. 1956 Steady laminar flow with closed streamlines at large Reynolds number J. Fluid Mech. 1, 177–190.Google Scholar
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in a turbulent field J. Fluid Mech. 5, 113–133.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benney, D. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows Stud. Appl. Maths 48, 181–204.Google Scholar
Carter, H. & Okubo, A. 1965 A study of the physical processes and movement and dispersion in the Cape Kennedy area. Chesapeake Bay Inst., Ref. 65–2, Johns Hopkins University.Google Scholar
Kraichnan, R. H. 1974 Convection of a passive scalar by a quasi-uniform random straining field J. Fluid Mech. 64, 737–762.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Moffatt, H. K. & Kamkar, H. 1982 The time-scale associated with flux expulsion. To appear in Proc. Workshop on The Theory of Stellar and Planetary Magnetism. Budapest.
Okubo, A. 1967 The effect of shear in an oscillatory current in horizontal diffusion from an instantaneous source Int. J. Oceanogr. Limnol. 1, 194–204.Google Scholar
Parker, R. L. 1966 Reconnexion of lines of force in rotating spheres and cylinders Proc. R. Soc. Lond. A291, 60–72.Google Scholar
Redekopp, L. 1980 Solitary waves with critical layers. G.F.D. Lectures, Woods Hole Oceanographic Institution, pp. 55–72.
Rhines, P. B. 1983 Lectures in geophysical fluid dynamics. In Mathematical Problems in the Geosciences. American Math. Soc., Prov. R.I., U.S.A.
Rhines, P. B. & Young, W. R. 1982a Homogenization of potential vorticity in planetary gyres J. Fluid Mech. 122, 347–367.Google Scholar
Rhines, P. B. & Young, W. R. 1982b A theory of wind-driven ocean circulation. I. Mid-ocean gyres. J. Mar. Res., 40 (suppl.), 559–596.Google Scholar
Salmon, R. 1980 Baroclinic instability and geostrophic turbulence Geophys. Astrophys. Fluid Dyn. 15, 167–211.Google Scholar
Saffman, P. 1962 The effect of wind shear on horizontal spreas from an instantaneous ground source Q. J. R. Met. Soc. 88, 382–393.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube Proc. R. Soc. Lond. A219, 186–203.Google Scholar
Weiss, J. 1981 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. La Jolla Institute preprint.
Weiss, N. O. 1966 The expulsion of magnetic flux by eddies Proc. R. Soc. Lond. A293, 310–328.Google Scholar
Young, W. R. 1981 On the vertical structure of the wind-driven circulation. Ph.D. dissertation, WHOI/MIT graduate program in oceanography.
Young, W. R. & Rhines, P. B. 1982 A theory of wind-driven ocean circulation. II. Western Boundary layer J. Mar. Res. 40, 849–872.Google Scholar
Young, W. R., Rhines, P. B. & Garrett, C. J. R. 1982 Shear-flow dispersion, internal waves and horizontal mixing in the ocean J. Phys. Oceanogr. 12, 515–527.Google Scholar