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Three-dimensionalization of barotropic vortices on the f-plane

Published online by Cambridge University Press:  26 April 2006

W. D. Smyth  and
W. R. Peltier
W. D. Smyth
Affiliation:
College of Oceanography and Atmospheric Sciences, Oregon State University, Corvallis OR 97331, USA
W. R. Peltier
Affiliation:
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
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Abstract

We examine the stability characteristics of a two-dimensional flow which consists initially of an inflexionally unstable shear layer on an f-plane. Under the action of the primary instability, the vorticity in the shear-layer initially coalesces into two Kelvin–Helmholtz vortices which subsequently merge to form a single coherent vortex. At a sequence of times during this process, we test the stability of the two-dimensional flow to fully three-dimensional perturbations. A somewhat novel approach is developed which removes inconsistencies in the secondary stability analyses which might otherwise arise owing to the time-dependence of the two-dimensional flow.

In the non-rotating case, and before the onset of pairing, we obtain a spectrum of unstable longitudinal modes which is similar to that obtained previously by Pierrehumbert & Widnall (1982) for the Stuart vortex, and by Klaassen & Peltier (1985, 1989, 1991) for more realistic flows. In addition, we demonstrate the existence of a new sequence of three-dimensional subharmonic (and therefore ‘helical’) instabilities. After pairing is complete, the secondary instability spectrum is essentially unaltered except for a doubling of length- and timescales that is consistent with the notion of spatial and temporal self-similarity. Once pairing begins, the spectrum quickly becomes dominated by the unstable modes of the emerging subharmonic Kelvin–Helmholtz vortex, and is therefore similar to that which is characteristic of the post-pairing regime. Also in the context of non-rotating flow, we demonstrate that the direct transfer of energy into the dissipative subrange via secondary instability is possible only if the background flow is stationary, since even slow time-dependence acts to decorrelate small-scale modes and thereby to impose a short-wave cutoff on the spectrum.

The stability of the merged vortex state is assessed for various values of the planetary vorticity f. Slow rotation may either stabilize or destabilize the columnar vortices, depending upon the sign of f, while fast rotation of either sign tends to be stabilizing. When f has opposite sign to the relative vorticity of the two-dimensional basic state, the flow becomes unstable to new mode of instability that has not been previously identified. Modes whose energy is concentrated in the vortex cores are shown to be associated, even at non-zero f, with Pierrehumbert's (1986) elliptical instability. Through detailed consideration of the vortex interaction mechanisms which drive instability, we are able to provide physical explanations for many aspects of the three-dimensionalization process.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 265 , 25 April 1994 , pp. 25 - 64
Copyright
© 1994 Cambridge University Press

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References

Bartello, P., Métais, M. & Lesieur, M. 1994 Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech. (submitted).Google Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.Google Scholar
Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359–391.Google Scholar
Benzi, R., Patarnello, S. & Santangelo, P. 1988 Self-similar coherent structures in two-dimensional decaying turbulence. J. Phys. A: Math. Gen. 21, 1221–1237.Google Scholar
Bernal, L. P. & Roshko, A. 1986 Streamwise vortex structure in plane mixing layers. J. Fluid Mech. 170, 499–522.Google Scholar
Breidenthal, R. E. 1981 Structure in turbulent mixing layers and wakes using a chemical reaction. J. Fluid Mech. 109, 1–24.Google Scholar
Browand, F. K. & Ho, C. M. 1983 The mixing layer: an example of quasi two-dimensional turbulence. J. Mec. Num. Spec. 99–120.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large-scale structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Comte, P., Lesieur, M. & Lamballais, E. 1992 Large and small-scale stirring of vorticity and a passive scalar in a 3D temporal mixing layer. Phys. Fluids A 4, 2761–2778.Google Scholar
Corcos, G. M. & Lin, S. J. 1984 The mixing layer: deterministic models of a turbulent flow. Part 2. The origin of the three-dimensional motion. J. Fluid Mech. 139, 67–95.Google Scholar
Farrell, B. J. 1989 Optimal excitation of baroclinic waves. J. Atmos. Sci. 46, 1193–1206.Google Scholar
Gent, P. R. & McWilliams, J. C. 1986 The instability of barotropic circular vortices. Geophys. Astrophys. Fluid Dyn. 35, 209–233.Google Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic. 662 pp.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press. 327 pp.
Johnson, J. A. 1963 The stability of shearing motion in a rotating fluid. J. Fluid Mech. 17, 337–352.Google Scholar
Klaassen, G. P. & Peltier, W. R. 1985 The onset of turbulence in finite-amplitude Kelvin–Helmholtz billows. J. Fluid Mech. 155, 1–35.Google Scholar
Klaassen, G. P. & Peltier, W. R. 1989 The role of trasverse secondary instabilities in the evolution of free shear layers. J. Fluid Mech. 202, 367–402.Google Scholar
Klaassen, G. P. & Peltier, W. R. 1991 The influence of stratification on secondary instability in free shear layers. J. Fluid Mech. 227, 71–106.Google Scholar
Kloosterzeil, R. C. & van Heijst, G. J. E. 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.Google Scholar
Lasheras, J. C., Cho, J. S. & Maxworthy, T. 1986 On the origin and evolution of streamwise vortical structures in a plane, free shear layer. J. Fluid Mech. 172, 231–258.Google Scholar
Lasheras, J. C. & Choi, H. 1988 Three-dimensional instability of a plane, free shear layer: an experimental study of the formation and evolution of streamwise vortices. J. Fluid Mech. 189, 53–86.CrossRefGoogle Scholar
Lesieur, M. 1991 Turbulence in Fluids. 2nd rev. edn. Dordrecht: Kluwer.
Lesieur, M., Staquet, C., Le Roy, P. & Comte, P. 1988 The mixing layer and its coherence examined from the point of view of two-dimensional turbulence. J. Fluid Mech. 192, 511–534.Google Scholar
Lesieur, M., Yanase, S. & Métais, O. 1991 Stabilizing and destabilizing effects of a solid-body rotation on quasi-two-dimensional shear layers. Phys. Fluids A 3, 403–407.Google Scholar
McWilliams, J. C. 1983 On the relevance of two-dimensional turbulence to geophysical fluid motions. J. Mec. Num. Spec. 83–98.Google Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Marcus, P. 1990 Vortex dynamics in a shearing zonal flow. J. Fluid Mech. 215, 393–430.Google Scholar
Melander, M. V., Zabusky, N. J. & McWilliams, J. C. 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340.Google Scholar
Métais, O., Yanase, S., Flores, C., Bartello, P. & Lesieur, M. 1991 Reorganization of coherent vortices in shear layers under the action of solid-body rotation. In Turbulent Shear Flows, Munich.
Metcalfe, R. W., Orszag, S. A., Brachet, M. E., Menon, S. & Riley, J. J. 1987 Secondary instability of a temporally-growing mixing layer. J. Fluid Mech. 184, 207–243.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Shepherd, T. G. 1987 Rossby waves and two-dimensional turbulence in a large-scale zonal jet. J. Fluid Mech. 183, 467–509.Google Scholar
Smyth, W. D. 1992 Spectral transfers in two-dimensional anisotropic flow. Phys. Fluids A 4, 340–349.Google Scholar
Smyth, W. D. & Peltier, W. R. 1989 The transition between Kelvin–Helmholtz and Holmboe instability: an investigation of the overreflection hypothesis. J. Atmos. Sci. 46, 3698–3720.Google Scholar
Smyth, W. D. & Peltier, W. R. 1990 Three-dimensional primary instabilities of a stratified, dissipative, parallel flow. Geophys. Astrophys. Fluid Dyn. 52, 249–261.Google Scholar
Smyth, W. D. & Peltier, W. R. 1991 Instability and transition in finite-amplitude Kelvin–Helmholtz and Holmboe waves. J. Fluid Mech. 228, 387–415.Google Scholar
Smyth, W. D. & Peltier, W. R. 1993 Two-dimensional turbulence in homogeneous and stratified shear layers. Geophys. Astrophys. Fluid Dyn. 69, 1–32.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.Google Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294.Google Scholar
Wygnanski, I., Oster, D., Fiedler, H. & Dziomba, B. 1979 On the perserverance of quasi-two-dimensional eddy structures in a turbulent mixing layer. J. Fluid Mech. 93, 325–335.Google Scholar
Yanase, S., Flores, C., Métais, O. & Riley, J. 1993 Rotating free shear flows: Part 1. Linear stability analyses. Phys. Fluids A 5, 2725–2737.Google Scholar