A one parameter family of explicit solutions of the Euler equations is presented comprising a steadily propagating point vortex street situated in a region of uniform vorticity below a periodically deformed vortex jump separating a region of irrotational flow from a uniform shear flow. Various features of the new solutions are described. The limiting solutions are such that the vortex jump develops a periodic sequence of cusps. The stability of the equilibria is investigated numerically using a cylindrical contour dynamics algorithm. The equilibria not too close to the limiting case are found to be structurally robust for a large range of parameter values.

1.
C. H. K.
Williamson
, “
Vortex dynamics in the cylinder wake
,”
Annu. Rev. Fluid Mech.
28
,
477
(
1996
).
2.
P. K.
Newton
, The N-Vortex Problem (
Springer-Verlag
,
New York
,
2001
).
3.
P. G.
Saffman
,
Vortex Dynamics
(
Cambridge University Press
,
Cambridge, England
,
1992
).
4.
H.
Aref
and
M.
Stremler
, “
On the motion of three point vortices in a periodic strip
,”
J. Fluid Mech.
314
,
1
(
1996
).
5.
J.
Montaldi
,
A.
Soulière
, and
T.
Tokieda
, “
Vortex dynamics on a cylinder
,”
SIAM J. Appl. Dyn. Syst.
2
,
417
(
2003
).
6.
H.
Aref
, “
On the equilibrium and stability of a row of point vortices
,”
J. Fluid Mech.
290
,
167
(
1995
).
7.
J. T.
Stuart
, “
On finite amplitude oscillations in laminar mixing layers
,”
J. Fluid Mech.
29
,
417
(
1967
).
8.
G.
Baker
,
P. G.
Saffman
, and
J. S.
Sheffield
, “
Structure of a linear array of hollow vortices of finite cross-section
,”
J. Fluid Mech.
74
,
469
(
1976
).
9.
P. G.
Saffman
and
R.
Szeto
, “
Structure of a linear array of uniform vortices
,”
Stud. Appl. Math.
65
,
223
(
1981
).
10.
R. T.
Pierrehumbert
and
S. E.
Widnall
, “
The structure of organized vortices in a free shear layer
,”
J. Fluid Mech.
102
,
301
(
1981
).
11.
L.
Armi
,
D.
Hebert
,
N.
Oakey
,
J. F.
Price
,
P. L.
Richardson
,
H. T.
Rossby
, and
B.
Ruddick
, “
Two years in the life of a Mediterranean salt lens
,”
J. Phys. Oceanogr.
19
,
354
(
1989
).
12.
J. C.
Gascard
,
A. J.
Watson
,
M.
Messias
,
K. A.
Olsson
,
T.
Johannessen
, and
K.
Simonsen
, “
Long-lived vortices as a mode of deep ventilation in the Greenland Sea
,”
Nature (London)
416
,
525
(
2002
).
13.
M. E.
Stern
and
G. R.
Flierl
, “
On the interaction of a vortex with a shear flow
,”
J. Geophys. Res., [Oceans]
92
,
10733
, doi:10.1029/JC092iC10p10733 (
1987
).
14.
N. R.
McDonald
, “
Steady nonradiating geophysical flow past a cylinder with circulation
,”
Phys. Fluids
14
,
3018
(
2002
).
15.
N. R.
McDonald
, “
A new translating quasigeostrophic V-state
,”
Eur. J. Mech. B/Fluids
23
,
633
(
2004
).
16.
R. B.
Nelson
, “
Modelling vortex-vortex and vortex-boundary interactions
,” Ph.D. thesis,
University College London
,
2009
.
17.
D. I.
Pullin
, “
Contour dynamics methods
,”
Annu. Rev. Fluid Mech.
24
,
89
(
1992
).
18.
D. G.
Dritschel
, “
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
(
1989
).
19.
D. G.
Crowdy
, “
A class of exact multipolar vortices
,”
Phys. Fluids
11
,
2556
(
1999
).
20.
P. J.
Davis
,
The Schwarz Function and Its Applications
,
Carus Mathematical Monographs
(
AMS
,
Rhode Island
,
1978
).
21.
D. G.
Crowdy
, “
Exact solutions for rotating vortex arrays with finite-area cores
,”
J. Fluid Mech.
469
,
209
(
2002
).
22.
D. G.
Crowdy
and
J. S.
Marshall
, “
Analytical solutions for rotating vortex arrays involving multiple vortex patches
,”
J. Fluid Mech.
523
,
307
(
2005
).
23.
E. A.
Overman
, “
Steady-state solutions of the Euler equations in two dimensions II: Local analysis of limiting V-states
,”
SIAM J. Appl. Math.
46
,
765
(
1986
).
24.
D. G.
Crowdy
, “
Explicit solutions for a vortex-wave interaction
,”
J. Fluid Mech.
513
,
161
(
2004
).
25.
M. A.
Stremler
, “
Relative equilibria of singly periodic point vortex arrays
,”
Phys. Fluids
15
,
3767
(
2003
).
26.
H.
Aref
and
D. L.
Vainchtein
, “
Point vortices exhibit asymmetric equilibria
,”
Nature (London)
392
,
769
(
1998
).
27.
D. G.
Crowdy
and
J. S.
Marshall
, “
Growing vortex patches
,”
Phys. Fluids
16
,
3122
(
2004
).
28.
H.
Lamb
,
Hydrodynamics
(
Cambridge University Press
,
Cambridge, England
,
1993
).
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