This paper examines coherent vortices and spatial distributions of energy density in asymptotic states of numerically simulated, horizontally homogeneous, doubly periodic, quasi two-dimensional f-plane turbulence. With geophysical applications in mind, the paper progresses from freely decaying two-dimensional flow to freely decaying equivalent barotropic flow, freely decaying two-layer quasi-geostrophic (QG) flow, and, finally, statistically steady two-layer QG turbulence forced by a baroclinically unstable mean flow and damped by bottom Ekman friction. It is demonstrated here that, with suitable elaborations, a two-vortex state having a sinh-like potential vorticity/streamfunction (q/ψ) scatter plot arises in all of these systems. This extends, at least qualitatively, previous work in inviscid and freely decaying two-dimensional flows to flows having stratification, forcing, and dissipation present simultaneously. Potential vorticity steps and ribbons of kinetic energy are shown to form in freely decaying equivalent barotropic flow and in the equivalent barotropic limit of baroclinically unstable flow, which occurs when Ekman damping is strong. Thus, contrary to expectations, strong friction can under some circumstances create rather than hinder the formation of sharp features. The ribbons are present, albeit less dramatically, in moderately damped baroclinically unstable turbulence, which is arguably a reasonable model for mid-ocean mesoscale eddies.

1.
P. B.
Rhines
, “
Waves and turbulence on a beta plane
,”
J. Fluid Mech.
69
,
417
(
1975
).
2.
P. B. Rhines, “The dynamics of unsteady currents,” in The Sea, edited by E. D. Goldberg, I. N. McCave, J. J. O’Brien, and J. H. Steele (Wiley, New York, 1977), Vol. 6, pp. 189–318.
3.
R.
Salmon
, “
Two-layer quasi-geostrophic turbulence in a simple special case
,”
Geophys. Astrophys. Fluid Dyn.
10
,
25
(
1978
).
4.
G.
Joyce
and
D.
Montgomery
, “
Negative temperature states for a two-dimensional guiding center plasma
,”
J. Plasma Phys.
10
,
107
(
1973
).
5.
D.
Montgomery
and
G.
Joyce
, “
Statistical mechanics of negative temperature states
,”
Phys. Fluids
17
,
1139
(
1974
).
6.
D.
Montgomery
,
W.
Matthaeus
,
W.
Stribling
,
D.
Martinez
, and
S.
Oughton
, “
Relaxation in two dimensions and the sinh-Poisson equation
,”
Phys. Fluids A
4
,
3
(
1992
).
7.
J. G.
Charney
, “
The dynamics of long waves in a baroclinic westerly current
,”
J. Meteorol.
4
,
135
(
1947
).
8.
E. T.
Eady
, “
Long waves and cyclone waves
,”
Tellus
1
,
33
(
1949
).
9.
R.
Salmon
, “
Baroclinic instability and geostrophic turbulence
,”
Geophys. Astrophys. Fluid Dyn.
15
,
167
(
1980
).
10.
D. B.
Haidvogel
and
I. M.
Held
, “
Homogeneous quasigeostrophic turbulence driven by a uniform temperature gradient
,”
J. Atmos. Sci.
37
,
2644
(
1980
).
11.
G. K.
Vallis
, “
On the predictability of quasi-geostrophic flow: The effects of beta and baroclinicity
,”
J. Atmos. Sci.
40
,
10
(
1983
).
12.
B. L.
Hua
and
D. B.
Haidvogel
, “
Numerical simulations of the vertical structure of quasi-geostrophic turbulence
,”
J. Atmos. Sci.
43
,
2923
(
1986
).
13.
R. L.
Panetta
, “
Zonal jets in wide baroclinically unstable regions: Persistence and scale selection
,”
J. Atmos. Sci.
50
,
2073
(
1993
).
14.
V. D.
Larichev
and
I. M.
Held
, “
Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability
,”
J. Phys. Oceanogr.
25
,
2285
(
1995
).
15.
I. M.
Held
and
V. D.
Larichev
, “
A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane
,”
J. Atmos. Sci.
53
,
946
(
1996
).
16.
K. S.
Smith
and
G. K.
Vallis
, “
The scales and equilibration of midocean eddies: Forced-dissipated flow
,”
J. Phys. Oceanogr.
32
,
1699
(
2002
).
17.
B. K. Arbic, “Generation of mid-ocean eddies: The local baroclinic instability hypothesis,” Ph.D. thesis, Massachusetts Institute of Technology–Woods Hole Oceanographic Institution Joint Program in Oceanography, 2000.
18.
B. K. Arbic and G. R. Flierl, “Effects of mean flow direction on energy, isotropy, and coherence of baroclinically unstable beta-plane geostrophic turbulence,” J. Phys. Oceanogr. (to be published).
19.
G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge University Press, Cambridge, 1953).
20.
R.
Fjortoft
, “
On the changes in the spectral distributions of kinetic energy for two-dimensional nondivergent flow
,”
Tellus
5
,
225
(
1953
).
21.
J. C.
McWilliams
, “
The emergence of isolated coherent vortices in turbulent flow
,”
J. Fluid Mech.
146
,
21
(
1984
).
22.
C. C. Lin, On the Motion of Vortices in Two Dimensions (University of Toronto Press, Toronto, 1943).
23.
L.
Onsager
, “
Statistical hydrodynamics
,”
Nuovo Cimento, Suppl.
6
,
279
(
1949
).
24.
J. B.
Flor
and
G. J. F.
van Heijst
, “
An experimental study of dipolar vortex structures in a stratified fluid
,”
J. Fluid Mech.
279
,
101
(
1994
).
25.
R. A.
Pasmanter
, “
On long-lived vortices in 2D viscous flows, most probable states of inviscid 2D flows and a soliton equation
,”
Phys. Fluids
6
,
1236
(
1994
).
26.
R.
Robert
and
J.
Sommeria
, “
Statistical equilibrium states for two-dimensional flows
,”
J. Fluid Mech.
229
,
291
(
1991
).
27.
J.
Sommeria
,
C.
Staquet
, and
R.
Robert
, “
Final equilibrium state of a two-dimensional shear layer
,”
J. Fluid Mech.
233
,
661
(
1991
).
28.
B.
Turkington
, “
Statistical equilibrium measures and coherent states in two-dimensional turbulence
,”
Commun. Pure Appl. Math.
52
,
781
(
1999
).
29.
H.
Brands
,
J.
Stulemeyer
,
R. A.
Pasmanter
, and
T. J.
Schep
, “
A mean field prediction of the asymptotic state of decaying 2D turbulence
,”
Phys. Fluids
9
,
2815
(
1997
).
30.
P.
Tabeling
, “
Two-dimensional turbulence: A physicist approach
,”
Phys. Rep.
362
,
1
(
2002
).
31.
V. D.
Larichev
and
J. C.
McWilliams
, “
Weakly decaying turbulence in an equivalent-barotropic fluid
,”
Phys. Fluids A
3
,
938
(
1991
).
32.
A.
Hasegawa
and
K.
Mima
, “
Pseudo-three-dimensional turbulence in magnetized nonuniform plasma
,”
Phys. Fluids
21
,
87
(
1978
).
33.
D.
Fyfe
and
D.
Montgomery
, “
Possible inverse cascade behavior for drift-wave turbulence
,”
Phys. Fluids
22
,
246
(
1979
).
34.
S.
Yanase
and
M.
Yamada
, “
The effect of the finite Rossby radius on two-dimensional isotropic turbulence
,”
J. Phys. Soc. Jpn.
53
,
2513
(
1984
).
35.
M.
Ottaviani
and
J. A.
Krommes
, “
Weak- and strong-turbulence regimes of the forced Hasegawa–Mima equation
,”
Phys. Rev. Lett.
69
,
2923
(
1992
).
36.
N.
Kukharkin
,
S. A.
Orszag
, and
V.
Yakhot
, “
Quasicrystallization of vortices in drift-wave turbulence
,”
Phys. Rev. Lett.
75
,
2486
(
1995
).
37.
T.
Watanabe
,
H.
Fujisaka
, and
T.
Iwayama
, “
Dynamical scaling law in the development of drift wave turbulence
,”
Phys. Rev. E
55
,
5575
(
1997
).
38.
T.
Watanabe
,
T.
Iwayama
, and
H.
Fujisaka
, “
Scaling law for coherent vortices in decaying Rossby wave turbulence
,”
Phys. Rev. E
57
,
1636
(
1998
).
39.
T.
Iwayama
,
T.
Watanabe
, and
T. G.
Shepherd
, “
Infrared dynamics of decaying two-dimensional turbulence governed by the Charney–Hasegawa–Mima equation
,”
J. Phys. Soc. Jpn.
70
,
376
(
2001
).
40.
T.
Iwayama
,
T. G.
Shepherd
, and
T.
Watanabe
, “
An ideal form of decaying two-dimensional turbulence
,”
J. Fluid Mech.
456
,
183
(
2002
).
41.
K. S.
Smith
,
G.
Boccaletti
,
C. C.
Henning
,
I.
Marinov
,
C. Y.
Tam
,
I. M.
Held
, and
G. K.
Vallis
, “
Turbulent diffusion in the geostrophic inverse cascade
,”
J. Fluid Mech.
469
,
13
(
2002
).
42.
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Mechanics (Springer-Verlag, New York, 1988).
43.
J. H. LaCasce, “Baroclinic vortices over a sloping bottom,” Ph.D. thesis, Massachusetts Institute of Technology–Woods Hole Oceanographic Institution Joint Program in Oceanography, 1996.
44.
M. E.
Stern
, “
Minimal properties of planetary eddies
,”
J. Mar. Res.
33
,
1
(
1975
).
45.
V.
Larichev
and
G.
Reznik
, “
Two-dimensional Rossby soliton: An exact solution
,”
Rep. U.S.S.R. Acad. Sci.
,
231
(
5
),
Also
POLYMODE
News
,
19
(
1976
).
46.
R. Salmon, Lectures on Geophysical Fluid Dynamics (Oxford University Press, Oxford, 1998).
47.
P. B.
Rhines
and
W. R.
Young
, “
Homogenization of potential vorticity in planetary gyres
,”
J. Fluid Mech.
122
,
347
(
1982
).
48.
X. J.
Carton
,
G. R.
Flierl
, and
L. M.
Polvani
, “
The emergence of tripoles from unstable axisymmetric vortex structures
,”
Europhys. Lett.
9
,
339
(
1989
).
49.
L. M.
Polvani
and
X. J.
Carton
, “
The tripole: A new coherent vortex structure of inviscid two-dimensional flows
,”
Geophys. Astrophys. Fluid Dyn.
51
,
87
(
1990
).
50.
G. R.
Flierl
, “
On the instability of geostrophic vortices
,”
J. Fluid Mech.
197
,
349
(
1988
).
51.
G. R.
Flierl
, “
Models of vertical structure and the calibration of two-layer models
,”
Dyn. Atmos. Oceans
2
,
341
(
1978
).
52.
J. G.
Charney
, “
Geostrophic turbulence
,”
J. Atmos. Sci.
28
,
1087
(
1971
).
53.
We thank Jim McWilliams for suggesting this name.
54.
L. M.
Polvani
,
J. C.
McWilliams
,
M. A.
Spall
, and
R.
Ford
, “
The coherent structures of shallow-water turbulence: Deformation radius effects, cyclone/anticyclone asymmetry and gravity-wave generation
,”
Chaos
4
,
177
(
1994
).
55.
Y. G.
Morel
and
X. J.
Carton
, “
Multipolar vortices in two-dimensional incompressible flows
,”
J. Fluid Mech.
267
,
23
(
1994
).
56.
K. S.
Fine
,
A. C.
Cass
, and
W. G.
Flynn
, “
Relaxation of 2D turbulence to vortex crystals
,”
Phys. Rev. Lett.
75
,
3277
(
1995
).
57.
S.
Li
and
D.
Montgomery
, “
Decaying two-dimensional turbulence with rigid walls
,”
Phys. Lett. A
218
,
281
(
1996
).
58.
H. J. H.
Clercx
,
S. R.
Maassen
, and
G. J. F.
van Heijst
, “
Spontaneous spin-up during the decay of 2D turbulence in a square container with rigid boundaries
,”
Phys. Rev. Lett.
80
,
5129
(
1998
).
59.
S. R.
Maassen
,
H. J. H.
Clercx
, and
G. J. F.
van Heijst
, “
Self-organization of quasi-two-dimensional turbulence in stratified fluids in square and circular containers
,”
Phys. Fluids
14
,
2150
(
2002
).
60.
F.
Paparella
,
A.
Babiano
,
C.
Basdevant
,
A.
Provenzale
, and
P.
Tanga
, “
A Lagrangian study of the Antarctic polar vortex
,”
J. Geophys. Res., [Atmos.]
102
,
6765
(
1997
).
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