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Entrainment and mixing dynamics of surface-stress-driven stratified flow in a cylinder

Published online by Cambridge University Press:  10 January 2012

A. Shravat ,
C. Cenedese  and
C. P. Caulfield
A. Shravat
Affiliation:
Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK
C. Cenedese
Affiliation:
Woods Hole Oceanographic Institution, MS 21, 360 Woods Hole Road Woods Hole, MA 02543, USA
C. P. Caulfield*
Affiliation:
BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

We extend previous work of Boyer, Davies & Guo (Fluid Dyn. Res., vol. 21, 1997, pp. 381–401) to consider the evolution of an initially two-layer stratified fluid in a cylindrical tank which is driven by a horizontal rotating disk. The turbulent motions induced by the disk drive entrainment at the interface, and similarly to the results of Boyer et al. (1997), the layer nearer to the disk deepens. Through high-frequency conductivity probe measurements, we establish that the deepening layer is very well-mixed, and the thickness of the interface between the two evolving layers appears to be approximately constant. Under certain circumstances, we find that the rate of increase in depth of the deepening layer decreases with time, at variance with the results of Boyer et al. (1997), and implying that the characteristic velocity in the deepening layer decreases as the upper layer deepens. We propose that such time-dependent deepening, and the associated weakening of the upper-layer velocities, occurs naturally because of the combined power requirements of entrainment and layer homogenization which inhibit, when the stratification is very strong, the characteristic velocities of the deepening layer approaching the (constant) velocities of the driving disk, as assumed by Boyer et al. (1997).

JFM classification

Geophysical and Geological Flows: Stratified flows Turbulent Flows: Stratified turbulence Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 691 , 25 January 2012 , pp. 498 - 517
Copyright
Copyright © Cambridge University Press 2012

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References

1. Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 1998 Dynamics of interfaces and layers in a stratified turbulent fluid. J. Fluid Mech. 355, 329–358.CrossRefGoogle Scholar
2. Barenblatt, G. I., Bertsch, M., Dal Passo, R., Prostokishin, V. M. & Ughi, M. 1993 A mathematical model of turbulent heat and mass transfer in stably stratified shear flow. J. Fluid Mech. 253, 341–358.CrossRefGoogle Scholar
3. Boyer, D. L., Davies, P. A. & Guo, Y. 1997 Mixing of a two-layer stratified fluid by a rotating disc. Fluid Dyn. Res. 21, 381–401 (referred to herein as BDG97).CrossRefGoogle Scholar
4. Canuto, V. M., Cheng, Y., Howard, A. M. & Esau, I. N. 2008 Stably stratified flows: A model with no Ri(cr). J. Atmos. Sci. 65, 2437–2447.CrossRefGoogle Scholar
5. Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
6. Crapper, P. F. & Linden, P. F. 1974 The structure of turbulent density interfaces. J. Fluid Mech. 65, 45–63.CrossRefGoogle Scholar
7. Dalziel, S. B., Patterson, M. D., Caulfield, C. P. & Coomaraswamy, I. A. 2008 Mixing efficiency in high-aspect-ratio Rayleigh–Taylor experiments. Phys. Fluids 20, 065106.CrossRefGoogle Scholar
8. Davies, P. A., Guo, Y., Boyer, D. L. & Folkard, A. M. 1995 The flow generated by the rotation of a horizontal disk in a stratified fluid. Fluid Dyn. Res. 17, 27–47.CrossRefGoogle Scholar
9. Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.CrossRefGoogle Scholar
10. Fernando, H. J. S. & Long, R. H. 1988 Experiments on steady buoyancy transfer through turbulent fluid layers separated by density interfaces. Dyn. Atmos. Ocean 12, 233–257.CrossRefGoogle Scholar
11. Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253–282.CrossRefGoogle Scholar
12. Guyez, E., Flor, J.-B. & Hopfinger, E. J. 2007 Turbulent mixing at a stable density interface: the variation of the buoyancy-flux gradient. J. Fluid Mech. 577, 127–136.CrossRefGoogle Scholar
13. Ivey, G. N. & Imberger, J. 1991 On the nature of turbulence in a stratified fluid. Part I: The energetics of mixing. J. Phys. Oceanogr. 21, 650–659.2.0.CO;2>CrossRefGoogle Scholar
14. Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.CrossRefGoogle Scholar
15. Kato, H. & Phillips, O. M. 1969 On the penetration of a turbulent layer into stratified fluid. J. Fluid Mech. 37, 643–655.CrossRefGoogle Scholar
16. Large, W. G., McWilliams, J. C. & Doney, S. C. 1994 Oceanic vertical mixing: a review and a model with a non-local boundary layer parameterization. Rev. Geophys. 32, 363–403.CrossRefGoogle Scholar
17. Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 3–23.CrossRefGoogle Scholar
18. Munro, R. J. & Davies, P. A. 2006 The flow generated in a continuously stratified rotating fluid by the differential rotation of a plane horizontal disc. Fluid Dyn. Res. 38, 522–538.CrossRefGoogle Scholar
19. Munro, R. J., Foster, M. R. & Davies, P. A. 2010 Instabilities in the spin-up of a rotating, stratified fluid. Phys. Fluids 22, 054108.CrossRefGoogle Scholar
20. Niiler, P. P. & Kraus, E. B. 1977 One-dimensional models of the upper ocean. In Modelling and Prediction of the Upper Layers of the Coean (ed. Kraus, E. B. ), pp. 143–172. Pergamon.Google Scholar
21. Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.2.0.CO;2>CrossRefGoogle Scholar
22. Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
23. Phillips, O. M. 1972 Turbulence in a strongly stratified fluid – is it unstable? Deep-Sea Res. 19, 79–81.Google Scholar
24. Pollard, R. T., Rhines, P. B. & Thompson, R. O. R. Y. 1973 The deepening of the wind-mixed layer. Geophys. Fluid Dyn. 3, 381–404.CrossRefGoogle Scholar
25. Posmentier, E. S. 1977 Generation of salinity fine-structure by vertical diffusion. J. Phys. Oceanogr. 7, 298–300.2.0.CO;2>CrossRefGoogle Scholar
26. Sherman, F. S., Imberger, J. & Corcos, G. M. 1978 Turbulence and mixing in stratified waters. Annu. Rev. Fluid Mech. 10, 267–288.CrossRefGoogle Scholar
27. Spigel, R. H., Imberger, J. & Rayner, K. N. 1986 Modelling the diurnal mixed layer. Limnol. Oceanogr. 31, 533–566.CrossRefGoogle Scholar
28. Turner, J. S. 1968 The influence of molecular diffusivity on turbulent entrainment across a density interface. J. Fluid Mech. 33, 639–656.CrossRefGoogle Scholar
29. Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431–471.CrossRefGoogle Scholar
30. Wells, M., Cenedese, C. & Caulfield, C. P. 2010 The relationship between flux coefficient and entrainment ratio in density currents. J. Phys. Oceanogr. 40, 2713–2727.CrossRefGoogle Scholar
31. Woods, A. W., Caulfield, C. P., Landel, J. R. & Kuesters, A. 2010 Non-invasive turbulent mixing across a density interface in a turbulent Taylor–Couette flow. J. Fluid Mech. 663, 347–357.CrossRefGoogle Scholar
32. Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar