A Kirchhoff elliptic vortex is a two-dimensional elliptical region of uniform vorticity embedded in an inviscid, incompressible, and irrotational fluid. By using analytic theory and contour dynamics simulations, we describe the evolution of perturbed Kirchhoff vortices by decomposing solutions into constituent linear eigenmodes. With small amplitude perturbations, we find excellent agreement between the short time dynamics and the predictions of linear analytic theory. Elliptical vortices must have aspect ratios less than to be completely stable. At late times, unstable perturbations evolve to states consisting of filaments surrounding and connecting one or more separate vortex core regions. Even modes have two different evolution paths accessible to them, dependent on the initial phase. Ellipses can first fission into more than one separate region when , from the negative branch mode. Increasing the perturbation amplitude can result in nonlinear instability, while the perturbation is still small relative to any vortex dimension. For the lowest modes, we quantify the transition from linear to nonlinear behavior.
REFERENCES
1.
2.
S.
Taneda
, “Downstream development of wakes behind cylinders
,” J. Phys. Soc. Jpn.
14
, 843
(1959
).3.
D. S.
Dosanjh
and T. M.
Weeks
, “Interaction of a starting vortex as well as a vortex street with a traveling shock wave
,” AIAA J.
3
, 216
(1965
).4.
P.
Freymuth
, “On transition in a separated laminar boundary layer
,” J. Fluid Mech.
25
, 683
(1966
).5.
J. P.
Christiansen
and N. J.
Zabusky
, “Instability, coalescense and fission of finite-area vortex structures
,” J. Fluid Mech.
61
, 219
(1973
).6.
J. C.
McWilliams
, “The emergence of isolated coherent vortices in turbulent flow
,” J. Fluid Mech.
146
, 21
(1984
).7.
8.
D. G.
Dritschel
, “On the stabilization of a two-dimensional vortex strip by adverse shear
,” J. Fluid Mech.
206
, 193
(1989
).9.
C.
Cerretelli
and C. H. K.
Williamson
, “The physical mechanism for vortex merging
,” J. Fluid Mech.
475
, 41
(2003
).10.
T. B.
Mitchell
and C. F.
Driscoll
, “Electron vortex orbits and merger
,” Phys. Fluids
8
, 1828
(1996
).11.
S.
Kida
, “Motion of an elliptical vortex in a uniform shear flow
,” J. Phys. Soc. Jpn.
50
, 3517
(1981
).12.
V. V.
Meleshko
and G. J. F.
van Heijst
, “On Chaplygin’s investigations of two-dimensional vortex structures in an inviscid fluid
,” J. Fluid Mech.
272
, 157
(1994
).13.
B.
Legras
, D. G.
Dritschel
, and P.
Caillol
, “The erosion of a distributed two-dimentional vortex in a background straining flow
,” J. Fluid Mech.
441
, 369
(2001
).14.
A. E. H.
Love
, “On the stability of certain vortex motions
,” Proc. London Math. Soc.
25
, 18
(1893
).15.
G.
Kirchhoff
, Vorlesungen über matematische Physic: Mechanik
(Teubner
, Leipzig
, 1876
).16.
W.
Thomson
, “On the vibrations of a columnar vortex
,” Philos. Mag.
10
, 155
(1880
).17.
J.
Burbea
and M.
Landau
, “The Kelvin waves in vortex dynamics and their stability
,” J. Comput. Phys.
45
, 127
(1982
).18.
Y.
Guo
, C.
Hallstrom
, and D.
Spirn
, “Dynamics near an unstable Kirchhoff ellipse
,” Commun. Math. Phys.
245
, 297
(2004
).19.
Y. H.
Wan
, “The stability of rotating vortex patches
,” Commun. Math. Phys.
107
, 1
(1986
).20.
Y.
Tang
, “Nonlinear stability of vortex patches
,” Trans. Am. Math. Soc.
304
, 617
(1987
).21.
D. G.
Dritschel
, “The stability and energetics of corotating uniform vortices
,” J. Fluid Mech.
157
, 95
(1985
).22.
J. R.
Kamm
, “Shape and stability of two-dimensional vortex regions
,” Ph.D. thesis, California Institute of Technology
, 1987
(http://resolver.caltech.edu/CaltechETD:etd-06302004-093810).23.
J.-Y.
Chemin
, “Existence globale pour le problém des poches de tourbillon
,” C. R. Acad. Sci., Ser. I: Math.
312
, 803
(1991
).24.
A. L.
Bertozzi
and P.
Constantin
, “Global regularity for vortex patches
,” Commun. Math. Phys.
152
, 19
(1993
).25.
N. J.
Zabusky
, M. H.
Hughes
, and K. V.
Roberts
, “Contour dynamics for the Euler equations in two dimensions
,” J. Comput. Phys.
30
, 96
(1979
).26.
G. S.
Deem
and N. J.
Zabusky
, “Vortex waves: Stationary ‘V states,’ interactions, recurrance, and breaking
,” Phys. Rev. Lett.
40
, 859
(1978
).27.
G. S.
Deem
and N. J.
Zabusky
, “Erratum: Vortex waves: Stationary ‘V states,’ interactions, recurrence, and breaking
,” Phys. Rev. Lett.
41
, 518
(1978
).28.
L. M.
Polvani
, G. R.
Flierl
, and N. J.
Zabusky
, “Filamentation of unstable vortex structures via separatrix crossing: A quantitative estimate of onset time
,” Phys. Fluids A
1
, 181
(1989
).29.
D. G.
Dritschel
, “The nonlinear evolution of rotating configurations of uniform vorticity
,” J. Fluid Mech.
172
, 157
(1986
).30.
P. W. C.
Vosbeek
, G. J. F.
van Heijst
, and V. P.
Mogendorff
, “The strain rate in evolutions of (elliptical) vortices in inviscid two-dimensional flows
,” Phys. Fluids
13
, 3699
(2001
).31.
G. K.
Batchelor
, An Introduction to Fluid Dynamics
(Cambridge University Press
, New York
, 1967
).32.
P. G.
Saffman
and R.
Szeto
, “Equilibrium shapes of a pair of equal uniform vortices
,” Phys. Fluids
23
, 2339
(1980
).33.
D. I.
Pullin
, “The nonlinear behaviour of a constant vorticity layer at a wall
,” J. Fluid Mech.
108
, 401
(1981
).34.
L. F.
Shampine
and M. W.
Reichelt
, “The MATLAB ODE suite
,” SIAM J. Sci. Comput. (USA)
18
, 1
(1997
).35.
E. A.
Overman
and N. J.
Zabusky
, “Evolution and merger of isolated vortex structures
,” Phys. Fluids
25
, 1297
(1982
).36.
M. V.
Melander
, J. C.
McWilliams
, and N. J.
Zabusky
, “Axisymmetrization and vorticity-gradient intensification of an isolated two-dimentional vortex through filamentation
,” J. Fluid Mech.
178
, 137
(1987
).37.
N.
Mattor
, B. T.
Chang
, and T. B.
Mitchell
, “Beat-wave resonant down scattering of diocotron and Kelvin modes
,” Phys. Rev. Lett.
96
, 045003
(2006
).38.
W. M.
Chan
, K.
Shariff
, and T. H.
Pulliam
, “Instabilities of two-dimensional inviscid compressible vortices
,” J. Fluid Mech.
253
, 173
(1993
).39.
E. G.
Broadbent
and D. W.
Moore
, “The two-dimensional instability of an incompressible vortex in a tube with energy-absorbent walls
,” Proc. R. Soc. London, Ser. A
446
, 39
(1994
).40.
D.
Crowdy
and A.
Surana
, “Contour dynamics in complex domains
,” J. Fluid Mech.
593
, 235
(2007
).41.
See EPAPS Document No. E-PHFLE6-20-063804 for MATLAB routines for CD simulations and analysis. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.
© 2008 American Institute of Physics.
2008
American Institute of Physics
You do not currently have access to this content.