This study develops a finite-difference numerical formulation to describe the motion of a singular monopole in a quasigeostrophic β-channel model with scale-selective frictional damping, using parameter values typical for the middle-latitude atmosphere and a wide range of viscosities. In this model, the “theoretical” singular vortex is replaced by the equivalent nonsingular vortex of a finite amplitude, consistent with the finite spatial resolution of the model. Numerical experiments demonstrate that at initial stages of the singular-vortex (SV) evolution, this model accurately reproduces the behavior expected from the theoretical considerations of the inviscid case. The numerical model also approximately conserves several invariants of motion derived from the continuous equations and accurately represents their modifications in the presence of friction. The evolution of a singular cyclone in the Northern Hemisphere starts with the development of the dipolar β gyres in the regular component of the flow; these gyres induce initial northward displacement and subsequent westward bending of the cyclone trajectory. At larger times, the β gyres gradually disintegrate, and the singular cyclone in the Northern Hemisphere continues to move northwestward by forming a dipolelike system with the localized secondary regular-field anticyclone northeast of the singular-cyclone center resulting eventually in a friction-assisted steady-state regime. The SV trajectories tend to become more zonally elongated for large vortices and small values of viscosity. These results lay a foundation for numerical consideration of systems of multiple singular vortices, which could provide further insights into our fundamental understanding of the processes underlying the multiscale atmospheric and oceanic variability.

1.
Arakawa
,
A.
, “
Computational design for long-term numerical integrations of the equations of atmospheric motion
,”
J. Comput. Phys.
1
,
119
143
(
1966
).
2.
Carnevale
,
G. F.
,
Kloosterziel
,
R. C.
, and
van Heijst
,
G. J. F.
, “
Propagation of barotropic vortices over topography in a rotating tank
,”
J. Fluid Mech.
233
,
119
139
(
1991
).
3.
Driscoll
,
C. F.
,
Jin
,
D. Z.
,
Schecter
,
D. A.
, and
Dubin
,
D. H. E.
, “
Vortex dynamics of 2D electron plasmas
,”
Physica C
369
,
21
27
(
2002
).
4.
Dubin
,
D. H. E.
, “
Collisional diffusion in a two-dimensional point vortex gas or a two-dimensional plasma
,”
Phys. Plasmas
10
(
5
),
1338
1350
(
2003
).
5.
Dunn
,
D. C.
,
McDonald
,
N. R.
, and
Johnson
,
E. R.
, “
The motion of a singular vortex near an escarpment
,”
J. Fluid Mech.
448
,
335
365
(
2001
).
6.
Early
,
J. J.
,
Samelson
,
R. M.
, and
Chelton
,
D. B.
, “
The evolution and propagation of quasigeostrophic ocean eddies
,”
J. Phys. Oceanogr.
41
,
1535
1554
(
2011
).
7.
Evans
,
C.
,
Wood
,
K. M.
,
Aberson
,
S. D.
 et al., “
The extratropical transition of tropical cyclones. Part I: Cyclone evolution and direct impacts
,”
Mon. Weather Rev.
145
,
4317
4344
(
2017
).
8.
Gryanik
,
V. M.
, “
The dynamics of localized vortices: ‘Vortex charges’ in baroclinic fluid
,”
Izv., Acad. Sci., USSR, Atmos. Oceanic
19
,
467
(
1983a
).
9.
Gryanik
,
V. M.
, “
The dynamics of singular geostrophic vortices in two-level model of the atmosphere (ocean)
,”
Izv., Acad. Sci., USSR, Atmos. Oceanic
19
,
227
(
1983b
).
10.
Gryanik
,
V. M.
, “
Singular geostrophic vortices on a β-plane as a model of synoptic eddies
,”
Okeanologiya
26
,
174
179
(
1986
) [
Oceanology
26,
126
(
1986
)].
11.
Gryanik
,
V. M.
,
Borth
,
H.
, and
Olbers
,
D.
, “
The theory of quasi-geostrophic von Karman vortex streets in two-layer fluids on a beta-plane
,”
J. Fluid Mech.
505
,
23
57
(
2004
).
12.
Gryanik
,
V. M.
and
Tevs
,
M. V.
, “
Dynamics of singular geostrophical vortices in a N-layer model of the atmosphere (ocean)
,”
Izv., Atmos. Oceanic Phys.
25
,
179
188
(
1989
).
13.
Kiwamoto
,
Y.
,
Ito
,
K.
,
Sanpei
,
A.
, and
Mohri
,
A.
, “
Dynamics of electron-plasma vortex in background vorticity distribution
,”
Phys. Rev. Lett.
85
(
15
),
3173
3176
(
2000
).
14.
Kono
,
M.
,
Shibahara
,
H.
, and
Yabuki
,
K.
, “
Point vortex dynamics in a magnetized plasma
,”
AIP Conf. Proc.
284
,
559
(
1994
).
15.
Koshel
,
K. V.
,
Reinaud
,
J. N.
,
Riccardi
,
G.
, and
Ryzhov
,
E. A.
, “
Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices
,”
Phys. Fluids
30
,
096603
(
2018
).
16.
Koshel
,
K. V.
,
Ryzhov
,
E. A.
, and
Carton
,
X.
, “
Vortex interactions subjected to deformation flows: A review
,”
Fluids
4
,
14
(
2019
).
17.
Kravtsov
,
S.
and
Gulev
,
S.
, “
Kinematics of eddy–mean-flow interaction in an idealized atmospheric model
,”
J. Atmos. Sci.
70
,
2574
2595
(
2013
).
18.
Kravtsov
,
S.
,
Rudeva
,
I.
, and
Gulev
,
S.
, “
Reconstructing sea-level pressure variability via a feature tracking approach
,”
J. Atmos. Sci.
72
,
487
506
(
2015
).
19.
Kuhlbrodt
,
T.
and
Nevir
,
P.
, “
Low-order point vortex models of atmospheric blocking
,”
Meteorol. Atmos. Phys.
73
,
127
138
(
2000
).
20.
Lam
,
J. S.
and
Dritshel
,
D. G.
, “
On the beta-drift of an initially circular vortex patch
,”
J. Fluid Mech.
436
,
107
129
(
2001
).
21.
Leoncini
,
X.
and
Verga
,
A.
, “
Dynamics of vortices and drift waves: A point vortex model
,”
Eur. Phys. J. B
86
,
95
(
2013
).
22.
Löptien
,
U.
and
Ruprecht
,
E.
, “
Effect of synoptic systems on the variability of the North Atlantic oscillation
,”
Mon. Weather Rev.
133
,
2894
2904
(
2005
).
23.
Mailier
,
P. J.
,
Stephenson
,
D. B.
,
Ferro
,
C. A. T.
, and
Hodges
,
K. I.
, “
Serial clustering of extratropical cyclones
,”
Mon. Weather Rev.
134
,
2224
2240
(
2006
).
24.
McDonald
,
N. R.
, “
The motion of geophysical vortices
,”
Philos. Trans. R. Soc., A
357
,
3427
3444
(
1999
).
25.
McWilliams
,
J. C.
, “
A note on a consistent quasi-geostrophic model in a multiply connected domain
,”
Dyn. Atmos. Oceans
1
,
427
441
(
1977
).
26.
Morikawa
,
G. K.
, “
Geostrophic vortex motion
,”
J. Meteorol.
17
,
148
(
1960
).
27.
Obukhov
,
A. M.
, “
On the geostrophic wind
,”
Izv. Acad. Nauk SSSR, Geogr.
8
,
281
(
1949
).
28.
Reinaud
,
J. N.
,
Koshel
,
K. V.
, and
Ryzhov
,
E. A.
, “
Entrapping of a vortex pair interacting with a fixed point vortex revisited. II. Finite size vortices and the effect of deformation
,”
Phys. Fluids
30
,
096604
(
2018
).
29.
Reznik
,
G. M.
, “
Motion of a point vortex on the β-plane
,”
Oceanology
30
,
523
528
(
1990
).
30.
Reznik
,
G. M.
, “
Dynamics of singular vortices on a β-plane
,”
J. Fluid Mech.
240
,
405
432
(
1992
).
31.
Reznik
,
G. M.
, “
Dynamics of localized vortices on a β-plane
,”
Izv., Atmos. Oceanic Phys.
46
,
784
797
(
2010
).
32.
Reznik
,
G. M.
and
Dewar
,
W.
, “
An analytical theory of distributed axisymmetric barotropic vortices on the β-plane
,”
J. Fluid Mech.
269
,
301
321
(
1994
).
33.
Reznik
,
G. M.
,
Grimshaw
,
R.
, and
Benilov
,
E. S.
, “
On the long-term evolution of an intense localized divergent vortex on the beta-plane
,”
J. Fluid Mech.
422
,
249
280
(
2000
).
34.
Reznik
,
G. M.
and
Kizner
,
Z.
, “
Two-layer quasigeostrophic singular vortices embedded in a regular flow. Part I: Invariants of motion and stability of vortex pairs
,”
J. Fluid Mech.
584
,
185
202
(
2007a
).
35.
Reznik
,
G. M.
and
Kizner
,
Z.
, “
Two-layer quasigeostrophic singular vortices embedded in a regular flow. Part 2. Steady and unsteady drift of individual vortices on a beta-plane
,”
J. Fluid Mech.
584
,
203
223
(
2007b
).
36.
Reznik
,
G. M.
and
Kizner
,
Z.
, “
Singular vortices in regular flows
,”
Theor. Comput. Fluid Dyn.
24
,
65
75
(
2010
).
37.
Ryzhov
,
E. A.
and
Koshel
,
K. V.
, “
Resonance phenomena in a two-layer two-vortex shear flow
,”
Chaos
26
,
113116
(
2016
).
38.
Ryzhov
,
E. A.
,
Koshel
,
K. V.
,
Sokolovskiy
M. A.
, and
Carton
,
X.
, “
Interaction of an along-shore propagating vortex with a vortex enclosed in a circular bay
,”
Phys. Fluids
30
,
016602
(
2018
).
39.
Ryzhov
,
E. A.
and
Sokolovskiy
,
M. A.
, “
Interaction of a two-layer vortex pair with a submerged cylindrical obstacle in a two layer rotating fluid
,”
Phys. Fluids
28
,
056602
(
2016
).
40.
Saffman
,
P. G.
,
Vortex Dynamics
(
Cambridge University Press
,
1992
).
41.
Schecter
,
D. A.
and
Dubin
,
D. H. E.
, “
Theory and simulations of two-dimensional vortex motion driven by a background vorticity gradient
,”
Phys. Fluids
13
(
6
),
1704
1723
(
2001
).
42.
Southwick
,
O. R.
,
Johnson
,
E. R.
, and
McDonald
,
N. R.
, “
A point vortex model for the formation of ocean eddies by flow separation
,”
Phys. Fluids
27
,
016604
(
2015
).
43.
Sutyrin
,
G. G.
and
Flierl
,
G. R.
, “
Intense vortex motion on the beta plane: Development of the beta gyres
,”
J. Atmos. Sci.
51
(
5
),
773
790
(
1994
).
44.
Sutyrin
,
G. G.
,
Hesthaven
,
J. S.
,
Lynov
,
J. P.
, and
Rasmussen
,
J. J.
, “
Dynamical properties of vortical structures on the beta_plane
,”
J. Fluid Mech.
268
,
103
131
(
1994
).
45.
Sutyrin
,
G. G.
and
Morel
,
Y. G.
, “
Intense vortex motion in a stratified fluid on the beta-plane: An analytical theory and its validation
,”
J. Fluid Mech.
366
,
203
220
(
1997
).
46.
Tamarin
,
T.
and
Kaspi
,
Y.
, “
Mechanisms controlling the downstream poleward deflection of midlatitude storm tracks
,”
J. Atmos. Sci.
74
,
553
572
(
2016
).
47.
Tur
,
A.
and
Yanovsky
,
V.
,
Coherent Vortex Structures in Fluids and Plasmas
(
Springer International Publishing AG
,
2017
), p.
306
.
48.
Wu
,
C.-C.
and
Emanuel
,
K. A.
, “
Interaction of a baroclinic vortex with background shear: Application to hurricane movement
,”
J. Atmos. Sci.
50
,
62
76
(
1993
).
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