The structure of a two-dimensional viscous dipole is accurately analyzed using both numerical simulations and theoretical analyses. First, a model is proposed, which computes the dipole velocity and the vortex ellipticity based on a heuristic relation between a vortex patch and a vortex with distributed vorticity profile. Second, during the stage where vortices are close to each other, a generalized self-similar solution is postulated to describe the vorticity profiles observed during the viscous spreading of the dipole. Numerical as well as theoretical considerations are given, which demonstrate the adequacy of such a hypothesis. Finally the structure of the tail that is generated behind the dipole is given in an analytical form, which favorably compares to numerical results.

1.
K.
Ahlnas
,
T. C.
Royer
, and
T. H.
George
, “
Multiple dipole eddies in the Alaska coastal current detected with Landsat thematic mapper data
,”
J. Geophys. Res.
92
,
13041
, DOI: 10.1029/JC092iC12p13041 (
1987
).
2.
Y.
Couder
,
J. -M.
Chomaz
, and
M.
Rabaud
, “
On the hydrodynamics of soap films
,”
Physica D
37
,
384
(
1989
).
3.
J. B.
Flor
and
G. J. F.
van Heijst
, “
An experimental study of a dipolar vortex structure in a stratified fluid
,”
J. Fluid Mech.
279
,
101
(
1994
).
4.
R. C.
Kloosterziel
, “
On the evolution and saturation of instabilities of two-dimensional isolated circular vortices
,”
J. Fluid Mech.
388
,
217
(
1999
).
5.
A.
Provenzale
, “
Transport by coherent barotropic vortices
,”
Annu. Rev. Fluid Mech.
31
,
55
(
1999
).
6.
S.
Le Dizès
, “
Non-axisymmetric vortices in two-dimensional flows
,”
J. Fluid Mech.
406
,
175
(
2000
).
7.
J.
Jiménez
,
H. K.
Moffatt
, and
C.
Vasco
, “
The structure of the vortices in freely decaying two-dimensional turbulence
,”
J. Fluid Mech.
313
,
209
(
1996
).
8.
L.
Ting
and
C.
Tung
, “
Motion and decay of a vortex in a nonuniform stream
,”
Phys. Fluids
8
,
1039
(
1965
).
9.
P.
Meunier
,
U.
Ehrenstein
,
Th.
Leweke
, and
M.
Rossi
, “
A merging criterion for two-dimensional co-rotating vortices
,”
Phys. Fluids
14
,
2757
(
2002
).
10.
Ch.
Josserand
and
M.
Rossi
, “
The merging of two co-rotating vortices: A numerical study
,”
Eur. J. Mech. B/Fluids
26
,
779
(
2007
).
11.
J. H. G. M.
van Geffen
and
G. J. F.
van Heijst
, “
Viscous evolution of 2D dipolar vortices
,”
Fluid Dyn. Res.
22
,
191
(
1998
).
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.
J. Juul
Rasmussen
,
J. S.
Hesthaven
,
J. P.
Lynov
,
A. H.
Nielsen
, and
M. R.
Schmidt
, “
Dipolar vortices in two-dimensional flows
,”
Math. Comput. Simul.
40
,
207
(
1996
).
14.
R. T.
Pierrehumbert
, “
A family of steady, translating vortex pairs with distributed vorticity
,”
J. Fluid Mech.
99
,
129
(
1980
).
15.
D.
Montgomery
and
G.
Joyce
, “
Statistical mechanics of negative temperature states
,”
Phys. Fluids
17
,
1139
(
1974
).
16.
D.
Sipp
,
L.
Jacquin
, and
C.
Cossu
, “
Self-adaptation and viscous selection in concentrated two-dimensional vortex dipoles
,”
Phys. Fluids
12
,
245
(
2000
).
17.
A. H.
Nielsen
and
J. J.
Rasmussen
, “
Formation and temporal evolution of the Lamb dipole
,”
Phys. Fluids
9
,
982
(
1997
).
18.
L.
Ting
and
F.
Bauer
, “
Viscous vortices in two-and three-dimensional space
,”
Comput. Fluids
22
,
565
(
1993
).
19.
B.
Cantwell
and
N.
Rott
, “
The decay of a viscous vortex pair
,”
Phys. Fluids
31
,
3213
(
1988
).
20.
S.
Kida
,
M.
Takaoka
, and
F.
Hussain
, “
Formation of head-tail structure in a two-dimensional uniform straining flow
,”
Phys. Fluids A
3
,
2688
(
1991
).
21.
R. R.
Trieling
,
J. M. A.
van Wesenbeeck
, and
G. J. F.
van Heijst
, “
Dipolar vortices in an strain flow
,”
Phys. Fluids
10
,
144
(
1998
).
22.
S.
Le Dizès
and
A.
Verga
, “
Viscous interactions of two co-rotating vortices before merging
,”
J. Fluid Mech.
467
,
389
(
2002
).
23.
A.
Mariotti
,
B.
Legras
, and
D. G.
Dritschel
, “
Vortex stripping and the erosion of coherent structures in two-dimensional flows
,”
Phys. Fluids
6
,
3954
(
1994
).
24.
I.
Delbende
and
M.
Rossi
, “
Nonlinear evolution of a swirling jet instability
,”
Phys. Fluids
17
,
044103
(
2005
).
25.
H. K.
Moffatt
,
S.
Kida
, and
K.
Okhitani
, “
Stretched vortices—the sinews of turbulence; large-Reynolds-number asymptotics
,”
J. Fluid Mech.
259
,
241
(
1994
).
26.
H.
Lamb
,
Hydrodynamics
(
Cambridge University Press
,
Cambridge
,
1932
).
27.
P. G.
Saffman
,
Vortex Dynamics
(
Cambridge University Press
,
Cambridge
,
1992
).
28.
S.
Kida
, “
Motion of an elliptic vortex in a uniform shear flow
,”
J. Phys. Soc. Jpn.
50
,
3517
(
1981
).
You do not currently have access to this content.