The effect of fine structure on the stability of planar vortices. (English) Zbl 1119.76313
Summary: This study considers the linear, inviscid response to an external strain field of classes of planar vortices. The case of a Gaussian vortex has been considered elsewhere, and an enstrophy rebound phenomenon was noted: after the vortex is disturbed enstrophy feeds from the non-axisymmetric to mean flow. At the same time an irreversible spiral wind-up of vorticity fluctuations takes place. A top-hat or Rankine vortex, on the other hand, can support a non-decaying normal mode.
In vortex dynamics processes such as stripping and collisions generate vortices with sharp edges and often with bands or rings of fine scale vorticity at their periphery, rather than smooth profiles. This paper considers the stability and response of a family of vortices that vary from a broad profile to a top-hat vortex. As the edge of the vortex becomes sharper, a quasi-mode emerges and vorticity winds up in a critical layer, at the radius where the angular velocity of the fluid matches that of a normal mode on a top-hat vortex. The decay rate of these quasi-modes is proportional to the vorticity gradient at the critical layer, in agreement with theory. As the vortex edge becomes sharper it is found that the rebound of enstrophy becomes stronger but slower.
The stability and linear behaviour of coherent vortices is then studied for distributions which exhibit additional fine structure within the critical layer. In particular we consider vorticity profiles with ‘bumps’, ‘troughs’ or ‘steps’ as this fine structure. The modified evolution equation that governs the critical layer is studied using numerical simulations and asymptotic analysis. It is shown that depending on the form of the short-scale vorticity distribution, this can stabilise or destabilise quasi-modes, and it may also lead to oscillatory behaviour.
In vortex dynamics processes such as stripping and collisions generate vortices with sharp edges and often with bands or rings of fine scale vorticity at their periphery, rather than smooth profiles. This paper considers the stability and response of a family of vortices that vary from a broad profile to a top-hat vortex. As the edge of the vortex becomes sharper, a quasi-mode emerges and vorticity winds up in a critical layer, at the radius where the angular velocity of the fluid matches that of a normal mode on a top-hat vortex. The decay rate of these quasi-modes is proportional to the vorticity gradient at the critical layer, in agreement with theory. As the vortex edge becomes sharper it is found that the rebound of enstrophy becomes stronger but slower.
The stability and linear behaviour of coherent vortices is then studied for distributions which exhibit additional fine structure within the critical layer. In particular we consider vorticity profiles with ‘bumps’, ‘troughs’ or ‘steps’ as this fine structure. The modified evolution equation that governs the critical layer is studied using numerical simulations and asymptotic analysis. It is shown that depending on the form of the short-scale vorticity distribution, this can stabilise or destabilise quasi-modes, and it may also lead to oscillatory behaviour.
MSC:
| 76B47 | Vortex flows for incompressible inviscid fluids |
References:
| [1] | Brachet, M. E.; Meneguzzi, M.; Politano, H.; Sulem, P.-L., The dynamics of freely decaying two-dimensional turbulence, J. Fluid Mech., 194, 333-349 (1988) |
| [2] | Fornberg, B., A numerical study of 2-d turbulence, J. Comput. Phys., 25, 1-31 (1977) · Zbl 0461.76040 |
| [3] | McWilliams, J. C., The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech., 146, 21-43 (1984) · Zbl 0561.76059 |
| [4] | Ting, L.; Klein, R., Viscous Vortical Flows, Lecture Notes in Phys., 374 (1991), Springer: Springer New York · Zbl 0748.76007 |
| [5] | Legras, B.; Dritschel, D., Vortex stripping and the generation of high vorticity gradients in two-dimensional flows, Appl. Sci. Res., 51, 445-455 (1993) · Zbl 0778.76041 |
| [6] | Legras, B.; Dritschel, D. G.; Caillol, P., The erosion of a distributed two-dimensional vortex in a background straining flow, J. Fluid Mech., 441, 369-398 (2001) · Zbl 1097.76548 |
| [7] | Jiménez, J.; Wray, A. A.; Saffman, P. G.; Rogallo, R. S., The structure of intense vorticity in isotropic turbulence, J. Fluid Mech., 255, 65-90 (1993) · Zbl 0800.76156 |
| [8] | Montgomery, M. T.; Kallenbach, R. J., A theory for vortex Rossby waves and its application to spiral bands and intensity changes in hurricanes, Q. J. R. Meteorol. Soc., 123, 435-465 (1997) |
| [9] | Guinn, T. A.; Schubert, W. H., Hurricane spiral bands, J. Atmos. Sci., 50, 3380-3403 (1993) |
| [10] | Smith, G. B.; Montgomery, M. T., Vortex axisymmetrization: dependence on azimuthal wave-number or asymmetric radial structure changes, Q. J. R. Meteorol. Soc., 121, 1615-1650 (1995) |
| [11] | Haynes, P., Transport, stirring and mixing in the atmosphere, (Chaté, H.; Villermaux, E.; Chomaz, J.-M., Mixing, Chaos and Turbulence (1999), Kluwer Academic) |
| [12] | Bassom, A. P.; Gilbert, A. D., The spiral wind-up of vorticity in an inviscid planar vortex, J. Fluid Mech., 371, 109-140 (1998) · Zbl 0918.76009 |
| [13] | Lundgren, T. S., Strained spiral vortex model for turbulent fine structure, Phys. Fluids, 25, 2193-2203 (1982) · Zbl 0536.76034 |
| [14] | Bassom, A. P.; Gilbert, A. D., The spiral wind-up and dissipation of vorticity and of a passive scalar in a strained planar vortex, J. Fluid Mech., 398, 245-270 (1999) · Zbl 0965.76018 |
| [15] | Macaskill, C.; Bassom, A. P.; Gilbert, A. D., Nonlinear wind-up in a strained planar vortex, Eur. J. Mech. B Fluids, 21, 293-306 (2002) · Zbl 1006.76023 |
| [16] | Balmforth, N. J.; Llewellyn Smith, S. G.; Young, W. R., Disturbing vortices, J. Fluid Mech., 426, 95-133 (2001) · Zbl 0988.76016 |
| [17] | Le Dizès, S., Non-axisymmetric vortices in two-dimensional flows, J. Fluid Mech., 406, 175-198 (1999) · Zbl 0990.76016 |
| [18] | Schecter, D. A.; Dubin, D. H.E.; Cass, A. C.; Driscoll, C. F.; Lansky, I. M.; O’Neil, T. M., The damping of asymmetries on a two-dimensional vortex, Phys. Fluids, 12, 2397-2412 (2000) · Zbl 1184.76483 |
| [19] | Briggs, R. J.; Daugherty, J. D.; Levy, R. H., Role of Landau damping in cross-field electron beams and inviscid shear flow, Phys. Fluids, 13, 421-432 (1970) |
| [20] | Kida, S., Motion of an elliptic vortex in a uniform shear flow, J. Phys. Soc. Japan, 50, 3512-3520 (1981) |
| [21] | Mariotti, A.; Legras, B.; Dritschel, D. G., Vortex stripping and the erosion of coherent structures in two-dimensional flows, Phys. Fluids, 6, 3954-3962 (1994) |
| [22] | Driscoll, C. F.; Fine, K. S., Experiments on vortex dynamics in pure electron plasmas, Phys. Fluids B, 2, 1359-1366 (1990) |
| [23] | Koumoutsakos, P., Inviscid axisymmetrization of an elliptical vortex, J. Comput. Phys., 138, 821-857 (1997) · Zbl 0902.76080 |
| [24] | Melander, M. V.; McWilliams, J. C.; Zabusky, N. J., Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation, J. Fluid Mech., 178, 137-159 (1987) · Zbl 0633.76023 |
| [25] | Dritschel, D. G.; Waugh, D. W., Quantification of the inelastic interaction of unequal vortices in 2-dimensional vortex dynamics, Phys. Fluids, 4, 1737-1744 (1992) |
| [26] | Kevlahan, N. K.-R.; Farge, M., Vorticity filaments in two-dimensional turbulence: creation, stability and effect, J. Fluid Mech., 346, 49-76 (1997) · Zbl 0910.76026 |
| [27] | Nielsen, A. H.; He, X.; Rasmussen, J. J.; Bohr, T., Vortex merging and spectral cascade in two-dimensional flows, Phys. Fluids, 3, 2263-2265 (1996) · Zbl 1027.76516 |
| [28] | Yao, H. B.; Zabusky, N. J.; Dritschel, D. G., High-gradient phenomena in 2-dimensional vortex interactions, Phys. Fluids, 7, 539-548 (1995) · Zbl 1039.76504 |
| [29] | Balmforth, N. J.; Del Castillo Negrete, D.; Young, W. R., Dynamics of vorticity defects in shear, J. Fluid Mech., 333, 197-230 (1996) · Zbl 0896.76020 |
| [30] | Balmforth, N. J., Stability of vorticity defects in viscous shear, J. Fluid Mech., 357, 199-224 (1998) · Zbl 0906.76016 |
| [31] | Sipp, D.; Jacquin, L.; Cossu, C., Self-adaptation and viscous selection in concentrated two-dimensional vortex dipoles, Phys. Fluids, 12, 245-248 (2000) · Zbl 1149.76539 |
| [32] | Bajer, K.; Bassom, A. P.; Gilbert, A. D., Accelerated diffusion in the centre of a vortex, J. Fluid Mech., 437, 395-411 (2001) · Zbl 0981.76023 |
| [33] | Bernoff, A. J.; Lingevitch, J. F., Rapid relaxation of an axisymmetric vortex, Phys. Fluids, 6, 3717-3723 (1994) · Zbl 0838.76024 |
| [34] | Moffatt, H. K.; Kamkar, H., The time-scale associated with flux explusion, (Soward, A. M., Stellar and Planetary Magnetism (1981), Gordon & Breach), 91-97 |
| [35] | Rhines, P. B.; Young, W. R., How rapidly is a passive scalar mixed within closed streamlines?, J. Fluid Mech., 133, 133-145 (1983) · Zbl 0576.76088 |
| [36] | Drazin, P. G.; Reid, W. H., Hydrodynamic Stability (1981), Cambridge University Press · Zbl 0449.76027 |
| [37] | Maslowe, S. A., Critical layers in shear flows, Ann. Rev. Fluid Mech., 18, 405-432 (1986) · Zbl 0634.76046 |
| [38] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1965), Dover |
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