Formation of columnar baroclinic vortices in thermally stratified nonlinear spin-up

J. R. Pacheco; R. Verzicco

doi:10.1017/jfm.2012.175

Formation of columnar baroclinic vortices in thermally stratified nonlinear spin-up

Published online by Cambridge University Press: 29 May 2012

J. R. Pacheco and

R. Verzicco

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J. R. Pacheco*: Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA Environmental Fluid Dynamics Laboratories, Department of Civil Engineering and Geological Sciences, The University of Notre Dame, South Bend, IN 46556, USA PANT&R Engineering, Gilbert, AZ 85296, USA
R. Verzicco: Affiliation:
Dipartimento di Ingegneria Meccanica, Universita’ di Roma ‘Tor Vergata’, via del Politecnico 1, 00133, Roma, Italy PoF, University of Twente, 7500 AE Enschede, The Netherlands
*: †Email address for correspondence: rpacheco@asu.edu

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Abstract

We investigate the mechanisms that affect the formation of columnar vortices for spin-up in a cylinder where the temperatures at the horizontal walls are prescribed. Numerical results from the three-dimensional Navier–Stokes equations show that a short-lived instability, suppressed by the combined effect of rotation and stratification, generates small temperature variations in the azimuthal direction. Temperature-gradient anomalies produce vorticity, and these vortices stir the fluid at the interface of the central vortex core thus reinforcing the temperature gradients. For sufficiently strong temperature gradients, the central vortex core breaks up into several columnar vortices. It is found, in particular, that small aspect ratios (height over radius of the cylindrical fluid layer) tend to inhibit the instability, while larger ones, , have the opposite effect. The main source of instability is the baroclinic vorticity production and not the presence of a solid sidewall since, counter-intuitively, the flow is more unstable for a free-slip boundary than for a no-slip one. Finally the effect of the temperature boundary conditions (isothermal versus adiabatic) on the horizontal boundaries has been investigated. The adiabatic boundaries help to preserve for longer times the sloping density interfaces that feed, with their potential energy, the baroclinic vorticity production; this results in more unstable flows.

JFM classification

Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex dynamics

Type: Papers
Information: Journal of Fluid Mechanics , Volume 702 , 10 July 2012 , pp. 265 - 285

DOI: https://doi.org/10.1017/jfm.2012.175 [Opens in a new window]
Copyright: Copyright © Cambridge University Press 2012

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References

1. Allen, J. S. 1973 Upwelling and coastal jets in a continuously stratified ocean. J. Phys. Oceanogr. 3, 245–257.2.0.CO;2>CrossRef Google Scholar

2. Benton, E. R. & Clark, A. 1974 Spin-up. Annu. Rev. Fluid Mech. 6, 257–280.CrossRef Google Scholar

3. Boyer, D. L., Sous, D. & Sommeria, J. 2009 Laboratory experiments on along-slope flows in homogeneous and stratified rotating fluids. Dyn. Atmos. Oceans 46, 19–35.CrossRef Google Scholar

4. Caldwell, D. R. & Van Atta, C. W. 1970 Characteristics of Ekman boundary layer instabilities. J. Fluid Mech. 44, 79–95.CrossRef Google Scholar

5. Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar

6. Duck, P. W. & Foster, M. R. 2001 Spin-up of homogeneous and stratified fluids. Annu. Rev. Fluid Mech. 33, 231–263.Google Scholar

7. Flór, J.-B., Bush, J. W. M. & Ungarish, M. 2004 An experimental investigation of spin-up from rest of a stratified fluid. Geophys. Fluid Dyn. 98, 277–296.CrossRef Google Scholar

8. Flór, J.-B., Scolan, H. & Gula, J. 2011 Frontal instabilities and waves in a differentially rotating fluid. J. Fluid Mech. 685, 240–257.CrossRef Google Scholar

9. Flór, J.-B., Ungarish, M. & Bush, J. W. M. 2002 Spin-up from rest in a stratified fluid: boundary flows. J. Fluid Mech. 472, 51–82.Google Scholar

10. Garrett, C., MacCready, P. & Rhines, P. 1993 Boundary mixing and arrested Ekman layers: rotating stratified flow near a sloping boundary. Annu. Rev. Fluid Mech. 25, 291–323.CrossRef Google Scholar

11. Greenspan, H. P. 1980 A note on the spin-up from rest of a stratified fluid. Geophys. Fluid Dyn. 15, 1–5.CrossRef Google Scholar

12. Guimbard, D., S Le Dizès, M., Le Bars, P. L. G. & Leblanc, S. 2010 Elliptic instability of a stratified fluid in a rotating cylinder. J. Fluid Mech. 660, 240–257.Google Scholar

13. Gula, J., Plougonven, R. & Zeitlin, V. 2009a Ageostrophic instabilities of fronts in a channel in a stratified rotating fluid. J. Fluid Mech. 627, 485–507.Google Scholar

14. Gula, J., Zeitlin, V. & Plougonven, R. 2009b Instabilities of two-layer shallow-water flows with vertical shear in the rotating annulus. J. Fluid Mech. 638, 27–47.CrossRef Google Scholar

15. Hallworth, M. A., Huppert, H. E. & Ungarish, M. 2001 Axisymmetric gravity currents in a rotating system: experimental and numerical investigations. J. Fluid Mech. 447, 1–29.Google Scholar

16. Hunt, J. C. R., Pacheco, J. R., Mahalov, A. & Fernando, H. J. S. 2005 Effects of rotation and sloping terrain on the fronts of density currents. J. Fluid Mech. 537, 285–315.Google Scholar

17. Hunt, J. C. R., Wray, A. A. & Moin, P. 1988 Eddies, stream, and convergence zones in turbulent flows. In Proceedings of the 1988 CTR Summer Program, pp. 193–208. Centre for Turbulence Research, Stanford, CA.Google Scholar

18. Hyun, J. M., Fowlis, W. W. & Warn-Varnas, A. 1982 Numerical solutions for the spin-up of a stratified fluid. J. Fluid Mech. 117, 71–90.CrossRef Google Scholar

19. Kanda, I. 2004 A laboratory study of columnar baroclinic vortices in a continuously stratified fluid. Dyn. Atmos. Oceans 38, 69–92.Google Scholar

20. Lilly, D. K. 1966 On the instability of Ekman boundary flow. J. Atmos. Sci. 23, 481–494.2.0.CO;2>CrossRef Google Scholar

21. Linden, P. F. & van Heijst, G. J. F. 1984 Two-layer spin-up and frontogenesis. J. Fluid Mech. 143, 69–94.CrossRef Google Scholar

22. Lynch, R. E., Rice, J. R. & Thomas, D. H. 1964 Tensor product analysis of partial difference equations. Bull. Am. Math. Soc. 70, 378–384.Google Scholar

23. Mahalov, A., Pacheco, J. R., Voropayev, S. I., Fernando, H. J. S. & Hunt, J. C. R. 2000 Effects of rotation on fronts of density currents. Phys. Lett. A 270, 149–156.Google Scholar

24. Manley, T. O. & Hunkins, H. 1985 Mesoscale eddies of the Arctic Ocean. J. Geophys. Res. 90, 4911–4930.CrossRef Google Scholar

25. McWilliams, J. C. 1985 Submesoscale, coherent vortices in the ocean. Geophys. Rev. 23, 165–182.CrossRef Google Scholar

26. Moulin, F. Y. & Flór, J.-B. 2004 On the spin-up by a rotating disk in a rotating stratified fluid. J. Fluid Mech. 516, 155–180.Google Scholar

27. Munro, R. J., Foster, M. R. & Davies, P. A. 2010 Instabilities in the spin-up of a rotating, stratified fluid. Phys. Fluids 22 (5), 054108.Google Scholar

28. Neitzel, G. P. & Davis, S. H. 1981 Centrifugal instabilities during spin-down to rest in finite cylinders. Numerical experiments. J. Fluid Mech. 102, 329–352.Google Scholar

29. Nicolle, A. & Eames, I. 2011 Numerical study of flow through and around a circular array of cylinders. J. Fluid Mech. 679, 1–31.Google Scholar

30. Olson, D. B. 1991 Rings in the ocean. Annu. Rev. Earth Planet. Sci. 19, 283–311.Google Scholar

31. Pacheco, J. R. 2008 Mixing enhancement in electro-osmotic flows via modulation of electric fields. Phys. Fluids 20, 093603.Google Scholar

32. Pacheco, J. R., Lopez, J. M. & Marques, F. 2011a Pinning of rotating waves to defects in finite Taylor–Couette flow. J. Fluid Mech. 666, 254–272.Google Scholar

33. Pacheco, J. R., Pacheco-Vega, A. & Chen, K. P. 2011b Mixing-dynamics of a passive scalar in a three-dimensional microchannel. Intl J. Heat Mass Transfer 54 (4), 959–966.Google Scholar

34. Pacheco, J. R., Ruiz-Angulo, A., Zenit, R. & Verzicco, R. 2011c Fluid velocity fluctuations in a collision of a sphere with a wall. Phys. Fluids 23 (6), 063301.Google Scholar

35. Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.Google Scholar

36. Sakai, S. 1989 Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176.Google Scholar

37. Smirnov, S. A., Baines, P. G., Boyer, D. L., Voropayev, S. I. & Srdić-Mitrović, A. N. 2005a Long-time evolution of linearly stratified spin-up flows in axisymmetric geometries. Phys. Fluids 17, 016601.CrossRef Google Scholar

38. Smirnov, S. A., Boyer, D. L. & Baines, P. G. 2005b Non-axisymmetric effects of stratified spin-up in an axisymmetric annular channel. Phys. Fluids 17, 086601.Google Scholar

39. Smirnov, S. A., Pacheco, J. R. & Verzicco, R. 2010a Numerical simulations of nonlinear thermally stratified spin-up in a circular cylinder. Phys. Fluids 22 (11), 116602.Google Scholar

40. Smirnov, S. A., Pacheco, J. R. & Verzicco, R. 2010b Three-dimensional vortex visualization in stratified spin-up. J. Vis. 13 (2), 81–84.Google Scholar

41. Ungarish, M. & Mang, J. 2003 The flow field and bare-spot formation in spin-up from rest of a two-layer fluid about a vertical axis. J. Fluid Mech. 474, 117–145.Google Scholar

42. Verzicco, R. & Camussi, R. 1997 Transitional regimes of low-Prandtl thermal convection in a cylindrical cell. Phys. Fluids 9, 1287–1295.Google Scholar

43. Verzicco, R., Lalli, F. & Campana, E. 1997 Dynamics of baroclinic vortices in a rotating stratified fluid: a numerical study. Phys. Fluids 9, 419–432.CrossRef Google Scholar

44. Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402–414.CrossRef Google Scholar

45. Warn-Varnas, A., Fowlis, W. W., Piacsek, S. & Lee, S. M. 1978 Numerical solutions and laser-Dopler measurements of spin-up. J. Fluid Mech. 85, 609–639.Google Scholar