Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T14:51:07.365Z Has data issue: false hasContentIssue false

Stability of a vortex in radial density stratification: role of wave interactions

Published online by Cambridge University Press:  25 May 2011

HARISH N. DIXIT
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
RAMA GOVINDARAJAN*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
*
Email address for correspondence: rama@jncasr.ac.in

Abstract

We study the stability of a vortex in an axisymmetric density distribution. It is shown that a light-cored vortex can be unstable in spite of the ‘stable stratification’ of density. Using a model flow consisting of step jumps in vorticity and density, we show that a wave interaction mediated by shear is the mechanism for the instability. The requirement is for the density gradient to be placed outside the vortex core but within the critical radius of the Kelvin mode. Conversely, a heavy-cored vortex, found in other studies to be unstable in the centrifugal Rayleigh–Taylor sense, is stabilized when the density jump is placed in this region. Asymptotic solutions at small Atwood number At show growth rates scaling as At1/3 close to the critical radius, and At1/2 further away. By considering a family of vorticity and density profiles of progressively increasing smoothness, going from a step to a Gaussian, it is shown that sharp gradients are necessary for the instability of the light-cored vortex, consistent with recent work which found Gaussian profiles to be stable. For sharp gradients, it is argued that wave interaction can be supported due to the presence of quasi-modes. Probably for the first time, a quasi-mode which decays exponentially is shown to interact with a neutral wave to give exponential growth in the combined case. We finally study the nonlinear stages using viscous direct numerical simulations. The initial exponential instability of light-cored vortices is arrested due to a restoring centrifugal buoyancy force, leading to stable non-axisymmetric structures, such as a tripolar state for an azimuthal wavenumber of 2. The study is restricted to two dimensions, and neglects gravity.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Present address: Department of Mathematics, University of British Columbia, Vancouver, Canada.

References

REFERENCES

Abramowitz, M. & Stegun, I. A. (Ed.) 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327342.CrossRefGoogle Scholar
Bassom, A. P. & Gilbert, A. D. 1998 The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109140.CrossRefGoogle Scholar
Birman, V. K., Martin, J. E. & Meiburg, E. 2005 The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations. J. Fluid Mech. 537, 125144.CrossRefGoogle Scholar
Briggs, R. J., Daugherty, J. D. & Levy, R. H. 1970 Role of Landau damping in crossed-field electron beams and inviscid shear flow. Phys. Fluids 13 (2), 421432.CrossRefGoogle Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 114.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Carpenter, J. R., Tedford, E. W., Rahmani, M. & Lawrence, G. A. 2010 Holmboe wave fields in simulation and experiment. J. Fluid Mech. 648, 205223.CrossRefGoogle Scholar
Carton, X. J., Flierl, G. R. & Polvani, L. M. 1989 The generation of tripoles from unstable axisymmetric isolated vortex structures. Europhys. Lett. 9, 339344.CrossRefGoogle Scholar
Case, K. M. 1960 Stability of an inviscid plane Couette flow. Phys. Fluids 3, 143148.CrossRefGoogle Scholar
Coquart, L., Sipp, D. & Jacquin, L. 2005 Mixing induced by Rayleigh–Taylor instability in a vortex. Phys. Fluids 17, 021703.CrossRefGoogle Scholar
Craik, A. D. D. 1985 Wave Interactions and Fluid Flow. Cambridge University Press.Google Scholar
Dixit, H. N. & Govindarajan, R. 2010 Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification. J. Fluid Mech. 646, 415439.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Eckhoff, K. S. & Storesletten, L. 1978 A note on the stability of steady inviscid helical gas flows. J. Fluid Mech. 89, 401411.CrossRefGoogle Scholar
Eckhoff, K. S. & Storesletten, L. 1980 On the stability of rotating compressible and inviscid fluids. J. Fluid Mech. 99, 433448.CrossRefGoogle Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235274.CrossRefGoogle Scholar
Fabrikant, A. L. & Stepanyants, Yu. A. 1998 Propagation of Waves in Shear Flows. World Scientific Series on Nonlinear Science.CrossRefGoogle Scholar
Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287318.CrossRefGoogle Scholar
Fung, Y. T. 1983 Non-axisymmetric instability of a rotating layer of fluid. J. Fluid Mech. 127, 8390.CrossRefGoogle Scholar
Fung, Y. T. & Kurzweg, U. H. 1975 Stability of swirling flows with radius-dependent density. J. Fluid Mech. 72, 243255.CrossRefGoogle Scholar
Gans, R. F. 1975 On the stability of a shear flow in a rotating gas. J. Fluid Mech. 68, 403412.CrossRefGoogle Scholar
Govindarajan, R. 2004 Effect of miscibility on the linear stability of a two-fluid channel. Intl. J. Multiphase Flow 30, 11771192.CrossRefGoogle Scholar
Hall, I. M., Bassom, A. P. & Gilbert, A. D. 2003 The effect of fine structure on the stability of planar vortices. Eur. J. Mech. (B/Fluids) 22, 179198.CrossRefGoogle Scholar
Harnik, N., Heifetz, E., Umurhan, O. M. & Lott, F. 2008 A buoyancy–vorticity wave interaction approach to stratified shear flow. J. Atmos. Sci. 65, 26152630.CrossRefGoogle Scholar
van Heijst, G. J. F. & Kloosterziel, R. C. 1989 Tripolar vortices in a rotating fluid. Nature 338, 569571.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, 301331.CrossRefGoogle Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111, 877946.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509512.CrossRefGoogle Scholar
Howard, L. N. & Maslowe, S. A. 1973 Stability of stratified shear flows. Boundary-Layer Meteorol. 4, 511523.CrossRefGoogle Scholar
Joly, L., Fontane, J. & Chassaing, P. 2005 The Rayleigh–Taylor instability of two-dimensional high-density vortices. J. Fluid Mech. 537, 415431.CrossRefGoogle Scholar
Kochar, G. T. & Jain, R. K. 1979 Note on Howard's semicircle theorem. J. Fluid Mech. 91, 489491.CrossRefGoogle Scholar
Kurzweg, U. H. 1969 A criteria for the stability of heterogenous swirling flows. Z. Angew. Math. Phys. 20, 141143.CrossRefGoogle Scholar
Le Dizès, S. 2008 Inviscid waves on a Lamb–Oseen vortex in a rotating stratified fluid: consequences for the elliptic instability. J. Fluid Mech. 597, 283303.CrossRefGoogle Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602 (1–8).CrossRefGoogle Scholar
Lees, L. 1958 Note on the stabilizing effect of centrifugal forces on the laminar boundary layer over convex surfaces. J. Aero. Sci. 25, 407408.Google Scholar
McWilliams, J. C., Graves, L. P. & Montgomery, M. T. 2003 A formal theory for vortex Rossby waves and vortex evolution. Geophys. Astrophys. Fluid Dyn. 97, 275309.CrossRefGoogle Scholar
Nadiga, B. T. & Aurnou, J. M. 2008 A tabletop demonstration of atmospheric dynamics: Baroclinic instability. Oceanography 21, 196201.CrossRefGoogle Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press.Google Scholar
Romanova, N. N. 2008 Resonant interaction of waves of continuous and discrete spectra in the simplest model of a stratified shear flow. Izvestiya, Atmospheric and Oceanic Physics 44, 5363.CrossRefGoogle Scholar
Romanova, N. N. & Shrira, V. I. 1988 Explosive generation of surface waves by wind. Izv. Atmos. Ocean. Phys. 24, 528535.Google Scholar
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, 149176.CrossRefGoogle Scholar
Saunders, P. M. 1973 The instability of a baroclinic vortex. J. Phys. Oceanogr. 3, 6165.2.0.CO;2>CrossRefGoogle Scholar
Sazonov, I. A. 1989 Interaction of continuous-spectrum waves with each other and with discrete-spectrum waves. Fluid Dyn. (translated from Izv. Akad. Nauk SSSR) 24, 586592.Google Scholar
Schecter, D. A., Dubin, D. H. E., Cass, A. C., Driscoll, C. F., Lansky, I. M. & O'Neil, T. M. 2000 Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 12, 23972412.CrossRefGoogle Scholar
Schecter, D. A. & Montgomery, M. T. 2003 On the symmetrization rate of an intense geophysical vortex. Dyn. Atmos. Oceans 37, 5588.CrossRefGoogle Scholar
Shrira, V. I. & Sazonov, I. A. 2001 Quasi-modes in boundary-layer-type-flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations. J. Fluid Mech. 446, 133171.CrossRefGoogle Scholar
Sipp, D., Fabre, D., Michelin, S. & Jacquin, L. 2005 Stability of a vortex with a heavy core. J. Fluid Mech. 526, 6776.CrossRefGoogle Scholar
Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15, 18611874.CrossRefGoogle Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A. 132, 499523.Google Scholar
Turner, M. R. & Gilbert, A. D. 2007 Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles. J. Fluid Mech. 593, 255279.CrossRefGoogle Scholar
Uberoi, M. S., Chow, C. Y. & Narain, J. P. 1972 Stability of coaxial rotating jet and vortex of different densities. Phys. Fluids 15, 17181727.CrossRefGoogle Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Vanneste, J. 1996 Rossby wave interaction in a shear flow with critical levels. J. Fluid Mech. 323, 317338.CrossRefGoogle Scholar
Voronovich, A. G. & Rybak, S. A. 1978 Explosive instability of stratified currents. Dokl. Acad. Sci. USSR 239, 14571460.Google Scholar
Voronovich, V. V., Pelinovsky, D. E. & Shrira, V. I. 1998 a On the internal waves shear flow resonance in shallow water. J. Fluid Mech. 354, 209237.CrossRefGoogle Scholar
Voronovich, V. V., Shrira, V. I. & Stepanyants, Yu. A. 1998 b Two-dimensional modes for nonlinear vorticity waves in shear flows. Stud. Appl. Maths 100, 132.CrossRefGoogle Scholar