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Dynamics of passive scalars and tracers advected by a two-dimensional tripolar vortex

Published online by Cambridge University Press:  26 August 2009

PAULO J. S. A. FERREIRA de SOUSA  and
JOSÉ C. F. PEREIRA
PAULO J. S. A. FERREIRA de SOUSA*
Affiliation:
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA Department of Mechanical Engineering/LASEF, Instituto Superior Técnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
JOSÉ C. F. PEREIRA
Affiliation:
Department of Mechanical Engineering/LASEF, Instituto Superior Técnico (IST), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
*
†Email address for correspondence: [email protected]
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Abstract

The dynamics of passive scalars and tracers during the formation and subsequent persistence of a laminar tripolar vortex, obtained through an unstable monopolar vortex seeded with a k = 2 azimuthal perturbation, is investigated. Two-dimensional direct numerical simulations of passive scalars with Schmidt numbers Sc = 0.1, 1, 10 and 100 are performed. The scalar variance for the four cases is analysed, as well as the different dispersion patterns up to 10 times greater than the time for formation of the tripolar vortex. During the formation of the tripole, an accelerated scalar dissipation is observed. That dissipation is connected to the advection-dominated processes associated with the growth of the perturbation mode. During that process, the patterns of mixing of the different passive scalars are very much the same as for vorticity. This stage of accelerated dissipation is preceded and followed by stages of diffusion-dominated scalar dissipation. Passive Lagrangian tracers are used to explore the transport of fluid elements during the evolution, and to provide a detailed view of the tripolar vortex formation and behaviour for longer times. Chaotic mixing was studied by examining patterns of spatial variation of finite-time Lyapunov exponents. As the perturbation grows and the tripolar vortex is formed, two large regions of regular flow, divided by a region of chaotic flow, form for each satellite. When the tripole is fully formed, it is composed of three distinct regular regions, corresponding to the core of negative vorticity and the two satellites of positive vorticity. The comparison between the evolution for vorticity, concentration, randomly distributed particles and Lyapunov exponents shows that transport occurs mainly in the regions of chaotic flow that surround the tripolar vortex after its formation. For longer times, both the chaotic/regular flow interfaces and the vorticity gradients are responsible for the integrity of the tripolar system.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 634 , 10 September 2009 , pp. 41 - 60
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Abraham, E. R. & Bowen, M. M. 2002 Chaotic stirring by a mesoscale surface-ocean flow. Chaos 12, 373–381.CrossRefGoogle ScholarPubMed
Babiano, A., Basdevant, C., Legras, B. & Sadourny, R. 1987 Vorticity and passive scalar dynamics in two-dimensional turbulence. J. Fluid Mech. 183, 379–397.CrossRefGoogle Scholar
Bajer, K., Bassom, A. P. & Gilbert, A. D. 2001 Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411.CrossRefGoogle Scholar
Bassom, A. P. & Gilbert, A. D. 1998 The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109–140.CrossRefGoogle Scholar
Bassom, A. P. & Gilbert, A. D. 1999 The spiral wind-up and dissipation of vorticity and a passive scalar in a strained planar vortex. J. Fluid Mech. 398, 245–270.CrossRefGoogle Scholar
Boatto, S. & Pierrehumbert, R. T. 1999 Dynamics of a passive tracer in a velocity field of identical point vortices. J. Fluid Mech. 394, 137–174.CrossRefGoogle Scholar
Bogucki, D., Domaradzki, J. A. & Yeung, P. K. 1997 Direct numerical simulations of passive scalars with Pr > 1 advected by turbulent flow. J. Fluid Mech. 343, 111–130.CrossRefGoogle Scholar
Carton, X., Flierl, G. R. & Polvani, L. 1989 The generation of tripoles from unstable axisymmetric isolated vortex structures. Europhys. Lett. 9, 339–344.CrossRefGoogle Scholar
Elhmaidi, D., Provenzale, A., Lili, T. & Babiano, A. 2004 Stability of two-dimensional vorticity filaments. Phys. Fluids A 333, 85–90.Google Scholar
Ferreira de Sousa, P. J. S. A. & Pereira, J. C. F. 2005 Fourth and tenth order compact finite difference solutions of perturbed circular vortex flows. Intl J. Numer. Fluids 49, 603–618.CrossRefGoogle Scholar
Flohr, P. & Vassilicos, J. C. 1997 Accelerated scalar dissipation in a vortex. J. Fluid Mech. 348, 295–317.CrossRefGoogle Scholar
Fuentes, O. U. V., van Heijst, G. J. F. & van Lipzig, N. P. M. 1996 Unsteady behaviour of a topography-modulated tripole. J. Fluid Mech. 307, 11–41.CrossRefGoogle Scholar
von Hardenberg, J., McWilliams, J. C., Provenzale, A., Shchepetkin, A. & Weiss, J. B. 2000 Vortex merging in quasi-geostrophic flows. J. Fluid Mech. 412, 331–353.CrossRefGoogle Scholar
van Heijst, G. J. F. & Kloosterziel, R. C. 1989 Tripolar vortices in a rotating fluid. Nature 338, 569–571.CrossRefGoogle Scholar
van Heijst, G. J. F., Kloosterziel, R. C. & Williams, C. W. M. 1991 Laboratory experiments on the tripolar vortex in a rotating fluid. J. Fluid Mech. 225, 301–331.CrossRefGoogle Scholar
Kevlahan, N. K.-R. & Farge, M. 1997 Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346, 49–76.CrossRefGoogle Scholar
Kloosterziel, R. C. & Carnevale, G. F. 1999 On the evolution and saturation of instabilities of two-dimensional isolated circular vortices. J. Fluid Mech. 388, 217–257.CrossRefGoogle Scholar
Kuznetsov, L. & Zaslavsky, G. M. 1998 Regular and chaotic advection in the flow field of a three-vortex system. Phys. Rev. E 58, 7330–7349.CrossRefGoogle Scholar
Kuznetsov, L. & Zaslavsky, G. M. 2000 Passive particle transport in three-vortex flow. Phys. Rev. E 61, 3777–3792.CrossRefGoogle ScholarPubMed
Lapeyre, G. 2002 Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence. Chaos 5, 688–698.CrossRefGoogle Scholar
Legras, B., Santangelo, P. & Benzi, R. 1988 High-resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett. 5, 37–42.Google Scholar
Leith, C. E. 1984 Minimum enstrophy vortices. Phys. Fluids 27, 1388–1395.CrossRefGoogle Scholar
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42. J. Fluid Mech. 346, 49–76.CrossRefGoogle Scholar
Meunier, P. & Villermaux, E. 2003 How vortices mix. J. Fluid Mech. 476, 213–222.CrossRefGoogle Scholar
Ottino, J. M. 1990 Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech. 22, 207–254.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1991 Large-scale horizontal mixing in planetary atmospheres. Phys. Fluids A 3, 1250–1260.CrossRefGoogle Scholar
Pierrehumbert, R. T. & Yang, H. 1993 Global chaotic mixing on isentropic surfaces. J. Atmos. Sci. 50, 2462–2480.2.0.CO;2>CrossRefGoogle Scholar
Pingree, R. D. & LeCann, B. 1992 Anticyclonic eddy x91 in the southern bay of Biscay, May 1991 to February 1992. J. Geophys. Res. 97, 14353–14367.CrossRefGoogle Scholar
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31, 55–93.CrossRefGoogle Scholar
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines. J. Fluid Mech. 133, 133–145.CrossRefGoogle Scholar
Rossi, L. F., Lingevitch, L. F. & Bernoff, A. J. 1997 Quasi-steady monopole and tripole attractors in relaxing vortices. Phys. Fluids 9, 2329–2338.CrossRefGoogle Scholar
Vranjes, J. 1999 Tripolar vortex in plasma flow. Planet. Space Sci. 47, 1531–1535.CrossRefGoogle Scholar
Vranjes, J., Maric, G. & Shukla, P. K. 1999 a Tripolar vortices and vortex chains in dusty plasma Phys. Lett. A 258, 317–322.CrossRefGoogle Scholar
Vranjes, J., Stenflo, L. & Shukla, P. K. 1999 b Tripolar vortices and vortex chains in a shallow atmosphere. Phys. Lett. A 267, 184–187.CrossRefGoogle Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar
Waugh, D., Abraham, E. R. & Bowen, M. M. 2006 Spatial variations of stirring in the surface ocean: a case study of the Tasman sea. J. Phys. Oceanogr. 36, 526–542.CrossRefGoogle Scholar
Wonhas, A. & Vassilicos, J. C. 2001 Mixing in frozen and time-periodic two-dimensional vortical flows. J. Fluid Mech. 442, 359–385.CrossRefGoogle Scholar
Yeung, P. K. 2002 Lagrangian investigations of turbulence. Annu. Rev. Fluid Mech. 34, 115–142.CrossRefGoogle Scholar