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The stability of a two-dimensional vorticity filament under uniform strain

Published online by Cambridge University Press:  26 April 2006

D. G. Dritschel ,
P. H. Haynes ,
M. N. Juckes  and
T. G. Shepherd
D. G. Dritschel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
P. H. Haynes
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
M. N. Juckes
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present affiliation: Department of Meteorology, University of Reading, RG6 2AU, UK.
T. G. Shepherd
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present affiliation: Department of Physics, University of Toronto, Toronto M5S 1A7, Canada.
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Abstract

The quantitative effects of uniform strain and background rotation on the stability of a strip of constant vorticity (a simple shear layer) are examined. The thickness of the strip decreases in time under the strain, so it is necessary to formulate the linear stability analysis for a time-dependent basic flow. The results show that even a strain rate γ (scaled with the vorticity of the strip) as small as 0.25 suppresses the conventional Rayleigh shear instability mechanism, in the sense that the r.m.s. wave steepness cannot amplify by more than a certain factor, and must eventually decay. For γ < 0.25 the amplification factor increases as γ decreases; however, it is only 3 when γ ė 0.065. Numerical simulations confirm the predictions of linear theory at small steepness and predict a threshold value necessary for the formation of coherent vortices. The results help to explain the impression from numerous simulations of two-dimensional turbulence reported in the literature that filaments of vorticity infrequently roll up into vortices. The stabilization effect may be expected to extend to two- and three-dimensional quasi-geostrophic flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 230 , September 1991 , pp. 647 - 665
Copyright
© 1991 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. 1960 Handbook of Mathematical Functions. US National Bureau of Standards.
Babiano, A., Basdevant, C., Legras, B. & Sadourny, R. 1987 Vorticity and passive-tracer dynamics in two-dimensional turbulence. J. Fluid Mech. 183, 379–397.Google Scholar
Batchelor, G. K. 1987 The stability of a large gas bubble rising through liquid. J. Fluid Mech.184, 399–422.Google Scholar
Brachet, M. E., Meneguzzi, M., Politano, H. & Sulem, P. L. 1988 The dynamics of freely decaying two-dimensional turbulence. J. Fluid Mech.194, 333–349.Google Scholar
Dhanakv, M. R. 1981 The stability of an expanding circular vortex layer. Proc. R. Soc. Lond. A 375, 443–451.Google Scholar
Dritschel, D. G. 1988 The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid Mech. 194, 511–547.Google Scholar
Dritschel, D. G. 1989a On the stabilization of a two-dimensional vortex strip by adverse shear. J. Fluid Mech. 206, 193–221.Google Scholar
Dritschel, D. G. 1989b Contour dynamics and contour surgery: numerical algorithms for extended high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comput. Phys. Rep. 10, 77–146.Google Scholar
Dritschel, D. G. 1990 The stability of elliptical vortices in an external straining flow. J. Fluid Mech. 210, 223–261.Google Scholar
Dritschel, D. G. & Polvani, L. M. 1991 The roll-up of vortex strips on a sphere. J. Fluid Mech. (in press).Google Scholar
Juckes, M. N. & Mclntyre, M. E. 1987 A high-resolution, one-layer model of breaking planetary waves in the stratosphere. Nature 328, 590–596.Google Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.Google Scholar
Legras, B., Santangelo, P. & Benzi, R. 1988 High-resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett. 5, 37–42.Google Scholar
Mcwilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Moore, D. W. 1976 The stability of an evolving two-dimensional vortex sheet. Mathematika 23, 128–133.Google Scholar
Moore, D. W. & Griffith-Jones, R. 1974 The stability of an expanding circular vortex sheet. Mathematika 21, 128–133.Google Scholar
Moore, D. W. & Saffman, P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence and its Detection (ed. J. H. Olsen, A. Goldburg & M. Rogers), pp. 339–354. New York: Plenum.
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math.Soc. 11, 57–70. (Also in Rayleigh's Theory of Sound, Vol. 2, 368, Dover, 1945.)Google Scholar
Rayleigh, Lord 1887 On the stability, or instability, of certain fluid motions, II. Proc. Lond. Math. Soc. 19, 67–74. (Also in Rayleigh's Theory of Sound, Vol. 2, 367, Dover, 1945.)Google Scholar
Santangelo, R., Benzi, R. & Legras, B. 1989 The generation of vortices in high-resolution, two-dimensional decaying turbulence and the influence of initial conditions on the breaking of self-similarity. Phys. FluidsA 1, 1027–1034.Google Scholar
Shelley, M. J. & Baker, G. R. 1990 On the connection between thin vortex layers and vortex sheets. J. Fluid Mech. 215, 161–194.Google Scholar
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733.Google Scholar
Waugh, D. W. & Dritschel, D. G. 1991 The stability of filamentary vorticity in two-dimensional geophysical vortex dynamics models J. Fluid Mech. 231, 575–598.Google Scholar