G. MAZE, X. CARTON, G.

LAPEYRE
Laboratoire de Physique des Océans
IFREMER, BP70 - 29280 Plouzané
UBO 6, avenue Le Gorgeu
BP809 - 29285 Brest Cedex, France
E-mail: gmaze@univ-brest.fr
E-mail: Xavier.Carton@ifremer.fr
E-mail: glapeyre@ifremer.fr

DYNAMICS OF A 2D VORTEX DOUBLET

UNDER EXTERNAL DEFORMATION
Received October 4, 2004 DOI: 10.1070/RD2004v009n04ABEH000291

The influence of an external strain (or shear) field on the evolution of two identical vortices is investigated in a two-
dimensional incompressible fluid. Using point vortex modeling, two regimes of the vortex doublet (co-rotation and
irreversible separation) are determined; the critical intensity of the large scale flow separating these two regimes for a
given initial separation of vortices, is calculated. Finite-area effects are then considered for the vortices. The steady
states of piecewise constant vortices are computed algebraically and numerically; positive strain (or shear) favors vortex
deformation. This deformation has a dominant elliptical component. An elliptical model of two vortices confirms the
point vortex model results for centroid trajectories, and the steady state model results concerning the influence of positive
strain on vortex deformation. It also provides an estimate of critical merger distance in the presence of large scale flow.
Finally, the finite-time, nonlinear evolution of the vortex doublet is simulated with a numerical code of the 2D vorticity
equation. The various regimes (stationarity, merger, co-rotation, ejection) are classified in the plane of initial vortex
separation and of external deformation. These regimes are analyzed, and the critical merger distance is evaluated for
negative and positive external strain; the results are in agreement with the elliptical model prediction. Merger efficiency,
defined as the ratio of final to initial vortex circulation, is computed; for the same initial distance, it is smaller for
negative strain. It also depends in a more complex way of the initial vortex distance.

1. Introduction
Vortices are robust features of geophysical and experimental flows. The long life of geophysical vortices
is now widely documented (see [5] for a review of observations and of mathematical theories of oceanic
vortices). This robustness is due to strong internal dynamics, prevailing on large scale advection and
deformation effects. But in-situ observations have also shown that oceanic vortices can exchange fluid
when they are in close vicinity (see [19]). In the ocean, this interaction is rapid, and most often partial,
insofar as only a small part of the vortex fluid is exchanged and as the two vortices still exist after the
interaction, though with different sizes than initially.
In a different context (two-dimensional incompressible turbulence), vortex merger has been recog-
nized as a key process for the growth of coherent vortices [16], [17]. Vortex merger is a fast, nonlinear
process by which two vortices join and form a larger one which is initially elliptical [22]. Merger is
defined as “complete” if its end product is a single vortex surrounded by filaments; it is partial if two
distinct vortices eventually survive. In two-dimensional, free-decay turbulence, the inverse energy cas-
cade is materialized by vortex merger, while the direct enstrophy cascade corresponds in physical space
to the ejection of vorticity filaments during the merging process. Depending on viscosity and on the
initial vortex structure, these filaments can be dissipated, allowing the final vortex to axisymmetrize,
or the filaments can be wrapped around the vortex boundary and prevent its final axisymmetrization.
In turbulent flows, merger catalysis by a third vortex has also been shown numerically.
Mathematics Subject Classification 76U05

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 477

 G. MAZE, X. CARTON, G. LAPEYRE

These observations have prompted specific studies of vortex merger. For two identical Rankine 1
vortices, a critical distance has been found, beyond which vortices do not merge. This distance
was assessed by several studies to lie in the range [3.16, 3.35] radii for two equal vortices [22], [8],
[13], [20], [15]. But, both in turbulence and in nature, such symmetric initial conditions are rarely
obtained and the two vortices differ in size or in intensity. The merger of two unequal vortices can be
incomplete and lead to the formation of several end-products, among which vortices smaller than any
of the former two [14]. A critical merger distance was calculated for two such vortices also [10], [23].
A weak efficiency of the merging process can also be due to other environmental factors, such as
spatially-variable fluid thickness [2], anisotropic rate of global rotation (the so-called beta-effect of
geophysical fluid dynamics) [1], barotropic instability of the vortices [4] or divergent flows which lead
to cyclone-anticyclone asymmetry [7].
The merger of two vortices in a fluid free of large-scale currents is also an idealized situation:
both in turbulence and in nature, vortices evolve in a large-scale deformation field (a shear or strain
field due to non uniform currents or to neighboring vortices). The evolution of a single vortex in
a large-scale deformation field is now well known: it becomes increasingly elliptical with imposed
deformation, and limits to its robustness have been set, in terms of ratio of external deformation over
internal vorticity [9].
The present paper is therefore concerned with the influence of such external deformation field on
the evolution of two identical Rankine vortices (a preliminary investigation can be fond in [6]). This
problem is addressed here both with mathematical and numerical methods, and after recalling the
framework of the study (section 2), the motion of the vortex centers is quantified via point vortex
dynamics (section 3). The influence of finite vortex area on their merger are then investigated; steady
states of the vortex doublet are computed and the idealization of the vortex as an ellipse provides
a quasi-analytical description of its evolution (section 4). A numerical code of the nonlinear two-
dimensional vorticity equation is then used to classify the various evolutions of finite-area vortex
doublet and to calculate the physical characteristics of merger (critical merger distance, efficiency. . . ;
section 5). The influence of external deformation on merger efficiency and the localization of most
deformed vortex regions are computed with an analytical criterion. Conclusions are finally drawn.

2. Mathematical and physical setting

Two dimensional incompressible, inviscid flows are governed by the vorticity equation:

∂t ζ + J(ψ, ζ) = 0 (2.1)

where ζ = ∇2 ψ and ψ are the relative vorticity and streamfunction, and J(a, b) = ∂ x a∂y b − ∂x b∂y a is
the Jacobian operator (x, y are Cartesian coordinates referenced to the center of the plane).
The initial conditions are two identical vortices (called “vortex doublet”), with unit vorticity and
radius, located initially at x = ±d/2, y = 0. The streamfunction is the sum of their contributions, of
external strain and of solid-body rotation:
Γ 2 Ω
ψ = ψv − (x − y 2 ) + (x2 + y 2 )
2 2
The external flow field is a pure (linear) shear when Γ = ±Ω.
In section 5, Eq. 2.1 is implemented numerically to study the nonlinear evolution of the vortex
doublet; the numerical scheme for this implementation is a pseudo-spectral decomposition in space
(using fast Fourier transforms); horizontal resolution in physical space is 256 2 , with a square domain of
length 2π. This numerical model performs time advection with a mixed Euler-leapfrog scheme. In this
code, the null right-hand side of Eq. 2.1 is replaced by a hyper-viscosity operator −(−1) n/2 νn ∇n ζ ; this
1
A Rankine vortex is a circular patch of constant relative vorticity in a 2D incompressible fluid

478 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

operator is a numerical procedure to eliminate the enstrophy accumulation at small scales, resulting
from the turbulent cascade. This hyper viscosity does not alter the physical outcome provided n is
large enough (here n = 4) and νn small enough (here ν4 = 10−8 ). To avoid Gibbs instability with this
numerical code, its initial conditions are slightly smoothed in vorticity (over two meshes) and the large
scale deformation field is introduced only where vorticity exceeds a small threshold (5% of maximum
vorticity).

3. Point vortex modeling

In the case of inviscid dynamics, we first assess the trajectory of the vortex centers by point vortex
modeling.

3.1. Hamiltonian
Let us consider the case of two vortices with identical circulation κ. Their instantaneous positions
are (ρ, θ) and (ρ, θ + π) in polar coordinates. Since the vorticity equation (2.1) conserves total energy
because dissipation is neglected, vortex motion is determined by the Hamiltonian

κ2
H = − [ln(2ρ) − 2γρ2 cos(2θ) + 2ωρ2 ] (3.1)
4π
where we have set γ = πΓ/κ and ω = πΩ/κ. We assume γ > 0 since a change of sign of γ is equivalent
to a rotation of the axes. The radial and azimuthal velocities of each point vortex relatively to the
center of the plane are

1 κ
vr = ρ̇ = ∂θ H = − γρ sin(2θ)
κρ π
(3.2)
1 κ 1
vθ = ρθ̇ = − ∂ρ H = [ − 2γρ cos(2θ) + 2ωρ]
κ 2π 2ρ

Figure 1 shows three plots of the Hamiltonian isolines in polar coordinate plane for three values
of γ and ω. These three diagrams are themselves presented in the (ρ, θ) plane. For ω > γ, the planar
representation of the Hamiltonian shows that all trajectories are closed. Otherwise, there exists open
or closed trajectories, separated by a fixed point in the polar coordinate plane.
Fixed points are given by ρ̇ = θ̇ = 0, leading to

1
θs = 0, ρs = √
2 γ−ω

for γ − ω > 0. There also exists the solution

1
θc = π/2, ρc = √
2 −γ − ω

when γ + ω < 0. Recall that, owing to the symmetry, θ is defined in ] − π/2, π/2].
Stability of these fixed points is easily obtained from Eq.3.2. The eigenvalues of the Jacobian
matrix
∂ ρ̇ 1 ∂ ρ̇
!
∂ρ ρ ∂θ
J= ∂ρθ̇ 1 ∂ρθ̇
∂ρ ρ ∂θ

are, for θ = 0,
2κ p
λs,± = ± γ(γ − ω).
π

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 479

480
REGULAR AND CHAOTIC DYNAMICS, V. 9,

G. MAZE, X. CARTON, G. LAPEYRE

4, 2004

Fig. 1. Three plots of Hamiltonian isolines (each of them in the polar coordinate plane) for a vortex doublet (with κ = π), in the (ω, γ) parameter plane.
The three plots of H(ρ, θ) correspond respectively to ω = 0.07, γ = 0.03, ω = 0, γ = 0.1, ω = −0.07, γ = 0.03
 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

They are of opposite sign so that this fixed point is a saddle point. It is easily shown (either by
computing the associated eigenvectors or by calculating the second order derivatives of H in ρ and θ)
that this fixed point is stable/unstable with respect to radial/angular disturbances.
For θ = π/2, the eigenvalues are
2κ p
λc,± = ±i γ(−γ − ω)
π
so that this fixed point is a center.
This analysis is illustrated by figure 1: for 0 < γ < ω, no fixed point is observed. For γ > |ω|,
only the saddle point is present; it separates open and closed trajectories. Conversely, for a given
initial separation 2ρf between the vortices, γ − ω will be the critical value of external deformation
separating open and closed trajectories. For 0 < γ < −ω, both the saddle point and the centers exist
in the (ρ, θ) plane.

3.2. Trajectories
For zero deformation field, the vortex trajectories are circles:
κt
ρ = ρ0 , θ = θ0 (t) = .
4πρ2
Assuming a weak deformation flow and setting 0 < γ  1, 0 < ω  1, the vortex trajectories can be
obtained in perturbation expansion in γ and ω:

ρ = ρ0 + γf (t) = ρ0 + 2γρ30 cos(2θ0 (t)),

κρ0 t
θ = θ0 (t) + γg(t) = θ0 (t) − 3γρ30 sin(2θ0 (t)) + ω .
π
The assumption of a weak deformation imposes small distances from the center of the plane. Therefore,
internal trajectories (i.e. trajectories bounded by the fixed point) are ellipses at first order in γ (see
again figure 1). Their aspect ratio and ellipticity are

1 − 2γρ20
λ = , e = 4γρ20 .
1 + 2γρ20

This result can also be obtained directly from the Hamiltonian by setting ρ = ρ 0 /(1 + e cos(2θ)) and
by cancelling the 2θ part, leading to e = 4γρ 20 .

4. Steady states of piecewise-constant vortices

After assessing the vortex trajectories, we determine their deformation under stationary conditions,
both analytically and numerically.

4.1. Approximate analytical solution

Under the hypothesis of stationarity, each vortex boundary is computed by perturbation expansion,
assuming that the vortex deformation is weak. In steady configuration, the vortex boundary is a
streamline of the total flow (induced by the companion vortex and the large-scale deformation field).
The companion vortex is idealized here as a point vortex located at its center and having its total
circulation. The mathematical technique used to compute the stationary contour deformation is
extensively explained in [8] and is not recalled in detail here.
The vortex radius is r = 1 + η(α) where η is the contour deformation from a circle, and α is the
local angle (relative to the vortex center; see figure 2). The contour deformation η is expanded in

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 481

 G. MAZE, X. CARTON, G. LAPEYRE

powers of  = 1/d and the order n term is assumed of the form cos(nα) − 1. Let γ and ω be at most
of order O(2 ). Stationarity is given by
∂ψ dη ∂ψ
=− . (4.1)
∂r dα ∂α

Ω
y

γ >0

ρ α
r
x

d/2 d/2

Fig. 2. Idealized flow configuration for the calculation of steady states: a piecewise-constant vortex is influenced
by the large-scale field and by a distant point vortex

At order , the condition for stationarity of the vortex center (already obtained with point vortices)
is recovered
1 − (γ − ω) d2 = 0 (4.2)
assuming γ > 0, γ > ω.
At order 2 , the elliptical deformation of the vortex contour is
1 + 2γd2
η2 = η0 [cos(2α) − 1], η0 = . (4.3)
1 + 2ω
We note that under zero external deformation the classical result η 0 = 1 is recovered [8].

4.2. Numerical solution

The steady states of the vortex doublet are now computed numerically; the algorithm is based on
the same principle as above: the steady vortex contour is a streamline of a total field and it has zero
total rotation rate. This steady vortex contour is computed iteratively with two Newton-Raphson
procedures.
Starting from a first guess of circular vortices with rotation rate given by point vortex theory,
√
the vortex boundaries on the x-axis are fixed as abscissae ρ f − 1 and ρf + 1, with ρf = 1/(2 γ − ω).
The N − 2 other ordinates of the vortex contour, the rotation rate and the (constant) value of the
streamfunction on the contour represent the other unknowns. The derivative of the total streamfunc-
tion with respect to these variables is used in a Newton-Raphson procedure to obtain a first steady
state (with non zero rotation rate). A second steady state is obtained with center ρ f − 0.05 with

482 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

a different rotation rate. With these two seeds, a second Newton-Raphson procedure, based on the
displacement of the vortex center, converges towards zero rotation rate. The final position of the
vortex center ρnum is then obtained. The area of the steady vortex is A num (remember that, assuming
ellipticity the theoretical value is A th = π(1 − 22 η0 ) using the formula hereabove).

4.3. Analysis of numerical solutions and comparison with analytical solutions

First, the theoretical and the numerical positions of steady vortex centers are compared for various
values of γ and ω (see figure 3). In the absence of external deformation, the relative difference between
theory and numerics is at most 1% (the numerical value being slightly larger than the theoretical one).
When external deformation is applied, discrepancies can rise to 23% for maximum values of γ and ω
allowing the existence of steady states.

comparison of steady vortex center between theory and numerics

7

5
/2
num
d

γ=ω=0
γ=0.01,0.09, ω=0
γ=0, ω=−0.01, −0.09
γ=0.05, ω=−0.01,−0.05
1
1 3 5 7
dth / 2

Fig. 3. Comparison of stationary vortex center position between theory and numerical experiments (d th /2 = ρf ),
for various values of γ and ω

Secondly, the theoretical and the numerical areas of steady vortices are compared for the same
values of γ and ω (see figure 4). In the absence of external deformation, the relative difference between
theory and numerics is at most 2% (the numerical value being slightly smaller than the theoretical one).
When external deformation is applied, differences can also rise to 23% for limiting states (discrepancies
being larger for pure strain than in presence of external rotation).
Finally, the steady vortex contours are Fourier analyzed in α and the amplitude of each mode
(m = 1 − 4) is plotted versus ρf = dth /2 (see figure 5a), in the case of pure strain (other cases are
similar). This analysis shows that the elliptical mode (m = 2) dominates the other angular modes by
far. In this case, the square mode (m = 4) is five times less intense, and is followed in amplitude by
the asymmetric mode (m = 1) and by the triangular one (m = 3). This can be related to the two-fold
symmetry of the large-scale strain field. Indeed, in the absence of deformation, modes m = 1, 3, 4 have
similar amplitudes. In the case of pure rotation, mode m = 1 is naturally favored.

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 483

 G. MAZE, X. CARTON, G. LAPEYRE

comparison of steady vortex area between theory and numerics

2.5
Anum

γ=ω=0
1.5 γ=0.01,0.09, ω=0
γ=0, ω=−0.01, −0.09
γ=0.05, ω=−0.01,−0.05
1.5 2 2.5 3
A
th

Fig. 4. Comparison of stationary vortex area between theory and numerical experiments (see text for definition
of Ath ), for various values of γ and ω

The amplitude of the elliptical mode is now compared between the various cases (no deformation,
pure strain, pure rotation, mixed deformation). Figure 5b shows a 40% increase in elliptical mode
amplitude when rotation is added. This increase is even more drastic (110%) in the presence of strain.
Mixed deformation (strain+rotation) does not substantially change this increase from the pure strain
case, but favors the growth of the asymmetric mode.

4.4. The elliptical model

The previous section has shown that the elliptical mode prevails in the deformation of the vortex
contours at equilibrium. An elliptical vortex model, which retains only four degrees of freedom for
each vortex (centroid position, aspect ratio, angle), should therefore be appropriate to describe the
dynamics of the system as long as merger has not occurred. We apply this model to our problem
following the algebra given by [13].
Calling Z = xc + iyc the complex coordinate of the vortex centroid, λ = b/a the aspect ratio of
each vortex, and e = [a2 − b2 ]/2 exp(2iφ) its ellipticity, with φ the angle between the major axis and
the x-axis, the dynamical equations of the elliptical vortex are

dZ
= −i(γZ ∗ − ωZ) − [ZBC]∗ (4.4)
dτ

and
de 1
= i[e( + 2ω) − F (2γ + i[C(B + e)]∗ )] (4.5)
dτ ab + F

484 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION
REGULAR AND CHAOTIC DYNAMICS, V. 9,

Amplitudes of contour modes 1-4 for gamma=0.01,0.09, omega=0 Amplitude of contour mode 2 for various cases
0.25 0.22
mode m=1 gamma=omega=0
mode m=2 0.2 gamma=0.01,0.09, omega=0
mode m=3 gamma=0, omega=-0.01,-0.09
mode m=4 gamma=0.05, omega=-0.01,-0.05
0.18
0.2

0.16

0.14
0.15
Amp m=1-4

Amp m=2
0.12

0.1
0.1
0.08

0.06

0.05
0.04

0.02

0 0
1.5 2 2.5 3 3.5 4 4.5 5 1.5 2 2.5 3 3.5 4 4.5 5
1/2 dth 1/2 dth
4, 2004

Fig. 5. (a) Amplitude of contour modes m = 1 − 4 versus theoretical distance of vortex center for pure strain case; (b) amplitude of elliptical mode
m = 2 versus dth /2 for various values of γ and ω
485
 G. MAZE, X. CARTON, G. LAPEYRE

where the time scale has been normalized as τ = 4πκt, with the following parameters
i
B = 4Z 2 − e; C =
B 2 − e2
p a2 + b 2
F = a2 b2 + |e|2 =
2
Firstly, we use the elliptical model to confirm our analytical calculation of steady states; assuming
that the vortices are weakly elliptical (with average radius unity and centroids located at ±d/2, 0),
the equations become: i
B = d2 , C = 2 , F = 1
d
The equation for the motion of centroids is in stationary form:

(γ − ω) d2 − 1 = 0

which is the condition of no motion for point vortices, and the ellipticity equation is, again in stationary
form: 1   1
e +ω − γ+ 2 = 0
2 d
2
and using e = 2η0 /d , we obtain
1 + 2γd2
η0 =
1 + 2ω
the value of contour deformation calculated in section 4.1.
Secondly, the elliptical model allows the computation of a critical distance between vortex centers
under which merger should occur. At this distance, elliptical vortices exhibit a change in direction
of rotation of their main axis [13]. The results are shown in table 1. Adding positive external strain
reduces the domain of existence of steady states (thus favoring merger), but only up to a given strain
amplitude. Beyond this value, merger is not favored any more; we will observe this behavior again in
the nonlinear results of the spectral code (see section 5).

5. Nonlinear evolutions of the vortices

5.1. Nonlinear regimes: classification and analysis
The nonlinear pseudospectral code is now run with the initial conditions of two piecewise constant
vortices in the large-scale deformation field. Two parameters are varied: the intensity of the strain
(scaled by the unit vorticity of each vortex) and the initial intercentroid distance (scaled by the radius
of each vortex here chosen as r0 = 0.3 to avoid spurious periodicity effects).
Previous studies of vortex merger are used to determine the appropriate values of these param-
eters. In the absence of strain, two piecewise constant vortices merge if their intercentroid distance
is smaller than a critical value; several estimates of this critical value, ranging between 3.16 and 3.35
radii, can be found in the literature [22], [8], [15], [13], [20]. Point vortex theory shows that vortices
at this distance are stationary if γ ∼ 0.1. Therefore, our choice for ranges of initial intercentroid
distances will span [2, 5], and γ will most often vary between −0.2 and 0.2 (and exceptionally rise up
to 0.8).
The time-series of vorticity plots obtained after each simulation allow the identification of four
different regimes in the [γ, d/r0 ] plane:
— stationarity: if the initial distance between the vortex centers d is equal to 2ρ f , the two vortices
become slightly deformed but do not move away from their initial position (see figure 6). In practice,
this regime is difficult to obtain numerically since the two vortices have finite extent here, and since
the fixed point position is unstable to azimuthal disturbances. High-resolution simulations (512x512)
were necessary to locate this regime precisely in the parameter plane.

486 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

2π/3

π/3

−π/3

−2π/3

−π
−π −2π/3 −π/3 0 π/3 2π/3 π

Fig. 6. Vortex stationarity. The Rankine vortices are bounded by the thick solid lines. Their initial positions
are given by the small squares. Dashed lines denote the instantaneous streamlines. The two strain axes are
indicated (circles on axis of compression and stars on axis of expansion)

— strain-induced expulsion: if the initial distance d between the vortex centers is larger than 2ρ f ,
the strain field is dominant in the vortex motion. The two vortices then drift apart, following open
trajectories slightly different from the initial Hamiltonian isolines due to finite-size effects and to
motion of the vortices in the fixed external flow (see figure 7).
— co-rotation: if the initial distance d between the vortex centers is smaller than 2ρ f but larger
than the critical distance for merger, the two vortices follow close, elliptical trajectories (see figure 8)
as predicted by point vortex theory. When varying the intensity of the strain, the aspect ratio of
these trajectories is nearly that given by the point vortex and elliptical models (see table 2). As
expected, the stronger the strain, the more elongated the trajectory is. Each vortex shape is also close
to elliptical with an aspect ratio predicted by the analytical solution (see table 3). Indeed, the relative
difference in vortex ellipticity between the elliptical and spectral models is at most 9%.
— merger: since vortex merger is a fast nonlinear process, the long term effect of viscous dissipa-
tion can be neglected and the efficiency of vortex merger can be defined as the ratio of the circulation
of the final vortex to the total circulation of the two initial vortices. To neglect viscous effects, we
impose that merging process takes place within a rotation period of the vortex doublet.
Figure 9 shows the time series of vorticity plots for merger in a weak strain. The strain elongates
the two vortices meridionally and slows down global rotation, at least initially; this latter point was
already predicted by point vortex theory. Nevertheless, the global process of vortex merger with weak

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 487

 G. MAZE, X. CARTON, G. LAPEYRE

2π/3

π/3

−π/3

−2π/3

−π
−π −2π/3 −π/3 0 π/3 2π/3 π

Fig. 7. Strain-induced ejection of vortices. The Rankine vortices are bounded by the thick solid lines. Their
initial positions are given by the small squares and their trajectory by the thin solid line. Dashed lines denote
the instantaneous streamlines. The two strain axes are indicated (circles on axis of compression and stars on
axis of expansion)

large-scale strain is quite comparable to that without strain. Fluid is exchanged between the two
structures near the center of the plane. Their collapse increases and the two vortex cores spiral down
towards the center: their thinning cross-section allows fluid mixing and the formation of a single vortex
at the center. Here, the strain rate is not sufficient to break this structure apart. Indeed, we recall
that [9] has shown that a strain rate on order of 15% of the vorticity is required for an elliptical vortex
to split apart.
Figure 10 presents the distribution of regimes in the (γ, d/r 0 ) plane. Only two regimes (merger
and co-rotation) exist in the absence of strain. Two other regimes are specific of strain effects: strain-
induced expulsion and equilibria. In the presence of weak strain, vortices need to be initially very
distant to be advected outwards. The curve separating the regimes of co-rotation and of strain-induced
√
expulsion has d/r0 ∼ 1/ γ. This relation was already obtained by point vortex theory as a separation
between open and close trajectories.
For weak positive strain, merger is favored. Still, this favorable influence of strain on critical
merger distance reaches a maximum for γ ∼ 0.4. Then, strain-induced vortex expulsion becomes the
dominant process. This optimum value of strain can be considered as the limit of a background effect.

488 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

2π/3

π/3

−π/3

−2π/3

−π
−π −2π/3 −π/3 0 π/3 2π/3 π

Fig. 8. Vortex co-rotation. The Rankine vortices are bounded by the thick solid lines. Their initial positions
are given by the small squares and their trajectory by the thin solid line. The two strain axes are indicated
(circles on axis of compression and stars on axis of expansion)

Indeed, strong strains have disruptive effects on coherent structures, among which that of breaking
vortices.

5.2. Critical merger distance

Merger occurs only when vortices are initially close enough. This critical distance is influenced by strain
as previous sections have indicated. The variation of this critical distance with strain is summarized
in tables 4 and 5. With the specific orientation of the strain chosen here, a positive strain favors
merger while a negative strain acts against it. Again, this is in agreement with the deformation of the
vortices induced by their companion and by the large scale strain, computed with the steady state
and elliptical models. Values of critical merger distance obtained with the spectral code show good
correspondence with minimum distances for the existence of elliptical vortex doublets (3 % relative
error).
The weaker influence of a negative strain than that of a positive strain on vortex merger can
be explained by the variation of angular velocity with γ (see section 3.2). As recalled above, the
doublet rotates more slowly in the first quadrant under positive strain and thus it can fully undergo
the favorable influence of this strain on merger (in particular, via an increased deformation). On

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 489

 G. MAZE, X. CARTON, G. LAPEYRE

0τ 0.16τ 0.32τ 0.48τ

0.64τ 0.8τ 0.95τ 1.11τ

1.27τ 1.43τ 1.59τ 1.75τ

1.91τ 2.07τ 2.23τ 2.39τ

2.55τ 2.71τ 2.86τ 3.02τ

3.18τ 3.34τ 3.5τ 3.66τ

Fig. 9. Vortex merger. Time series of vorticity plots for merger of two vortices with uniform vorticity ζ = 1,
initially separated by a distance d = 3.4 r0 in a weak strain (γ = 0.18). Evolution is from left to right and from
top to bottom

490 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

0.8

0.7

0.6

0.5

0.4

0.3
γ

0.2

0.1

−0.1

−0.2
0 2 4 6 8 10 12 14 16 18 20
d/r0

Fig. 10. Distribution of regimes in the (γ, d/r0 ) plane. Square: merger; cross: co-rotation; circle: stationarity;
star: strain-induced expulsion. Dashed lines separate different regimes and solid line is point vortex stationarity
(remember that r0 = 0.3 in the spectral code)

the contrary, the vortex doublet rotates more rapidly in the first quadrant with negative strain, and
therefore the negative effects of this strain on merger are felt for a shorter duration.
Finally, the influence of a non-zero rotation rate (ω) is assessed. Point vortex and steady state
calculations have indicated that positive rotation has an effect opposite to that of positive strain on
vortex doublet evolution: it favors elliptical trajectories and decreases each vortex ellipticity. Table 6
shows the variation of the critical merger distance for ω = ±γ, obtained with the spectral code and the
elliptical model. The two models indicate the same tendency, namely the increase of critical merger
distance with increasing external rotation. This positive influence of ω on co-rotation against merger
is in agreement with the point-vortex and steady state results.

5.3. Filamentation analysis

The role of vortex deformation during merger and the resulting formation of filaments are evaluated
here with the Lagrangian criterion on vortex filamentation, derived by [11]. This deformation is due to
the combined action of the external strain γ and of the strain induced by the companion vortex. This
criterion is based on a Lagrangian approach of tracer gradients; it predicts the exponential growth
rate of tracer gradients and therefore the stretching and straining undergone by vortices. We recall
that this rate is q
ef f
σ = σ 2 − (ζ + 2φ̇)2

(or zero if imaginary) where σ is the rate of deformation, ζ is vorticity and φ̇ is the rotation rate of
the principal axes of deformation.

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 491

 G. MAZE, X. CARTON, G. LAPEYRE

Firstly, we investigate the case with weak external deformation, using the criterion derived by [11]
(see figure 11). Initially, the deformation is oriented along the doublet axis. It is more intense between
the vortices and less intense on the outer side 2 . This induces an asymmetry in vortex deformation
by stretching the inner side more than the outer side of the vortices (as seen at t = 0.64τ ). The
vortex centroids will then collapse towards the center of the plane with maximum deformation still
aligned along the doublet axis, and concentrated near the cusps of their contours. This concentration
will induce vortex splitting, accompanied by the ejection of peripheral filaments (cf. t = 1.11τ and
t = 1.43τ ). During merger, the low-vorticity region between vortex cores is strongly deformed and
elongated; it will rapidly disappear (at t = 2.55τ ). Merger thus seems favored by this stronger
deformation on the inner side of the vortices: more fluid will be advected inwards than outwards
leading to the collapse of centroids.

Fig. 11. Effective rate of deformation by Lapeyre et al (1999) criterion at time t = 0, 0.64τ, 1.11τ, 1.43τ for
the simulation of figure 9. In black, values larger than 0.9, in dark gray, 0.5, in light gray 0.3, and 0.25

This case is also analyzed with the Okubo-Weiss [18], [21] criterion which does not take the
rotation of deformation axes (i.e. φ̇) into account. Clearly, this criterion cannot locate the regions
of strong deformation as accurately as the Lagrangian criterion [11]. In particular, according to
the Okubo-Weiss criterion, deformation would be relatively isotropic around the vortices at t = 0
(figure 12). As seen on figure 9, this is not the case. Therefore, taking into account the rotation of
deformation axes is essential: it strongly diminishes the effective deformation which is then localized
in specific regions (compare figures 11 and 12). Furthermore, at t = 1.11τ , the Okubo-Weiss criterion
does not predict the strong deformation at the cusp of the separating filament.

2
inner and outer must be understood here with reference to the center of the plane

492 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

Fig. 12. Effective rate of deformation by Okubo-Weiss criterion at time t = 0, 0.64τ, 1.11τ, 1.43τ for the
simulation of figure 9. In black, values larger than 0.9, in dark gray, 0.5, in light gray 0.3, and 0.25

Secondly, we examine the influence of various external strains on mergerusing the Lagrangian
criterion derived by [11]. Table 7 shows, for several values of γ, the critical merger distance, the merger
efficiency for the same initial distance (d/r 0 = 2.83) at time t = 2.5τ , and the average deformation σ ef f
in the vortex core and outside the vortices (average is taken in space and time before filamentation,
i.e. for t < 1.75τ ).
As explained previously, the critical merger distance increases from negative to positive external
strains (γ). But a larger merger efficiency for positive strain may seem paradoxical: indeed positive γ
corresponds to larger total deformation and thus stronger filamentation could be expected. In fact,
the paradox can be resolved by separating the role of deformation inside and outside the vortices.
The deformation exerted around the vortices tends to deform the vortex boundary which will elongate
and promote merger while the deformation exerted inside the vortices tends to split the vortices and
favor filamentation. Table 7 shows that the deformation exerted on the vortex boundaries is larger
for positive γ than for negative γ; vortex elongation and collapse vary accordingly. This variation is
consistent with the observed dependence of the critical merger distance on γ. On the other hand, the
deformation exerted on the vortex cores is larger for negative γ leading to more filamentation; this is
also consistent with the observed efficiency.

5.4. Merger efficiency

We now determine merger efficiency when the initial distance between vortices is not held constant.
Different processes can affect efficiency, such as external strain, initial distance between the vortices and
small scale dissipation. For equal initial distance, a positive external strain increases the efficiency while

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 493

 G. MAZE, X. CARTON, G. LAPEYRE

a negative one decreases it. Increasing the distance between vortices will decrease the deformation
exerted on them but, at the same time, will delay merger. The vortices will experience a smaller
deformation but on a longer period, and the result on the filamentation is not straightforward to
predict [12]. Also the small scale dissipation will constantly erode the vortices and decrease the
efficiency of vortex merger.
Table 8 shows the merger efficiency taking into account all these effects. It is computed for
an initial vortex distance equal to 90% of the critical merger distance and after a very long time
compared to merger. Since filaments are created during merger to accommodate angular momentum
conservation and are finally wiped out by viscosity, merger will never be a 100% efficient process.
Merger efficiency is clearly maximum for zero external strain. We see that when varying accordingly
the initial distance, the efficiency is almost independent of the sign of the external strain. Thus
increasing the initial distance between vortices has an effect opposite to that of the sole external
strain. Our interpretation is that a deformation applied longer elongates the vortices more efficiently.

6. Conclusions and openings

The influence of a large scale deformation field on the evolution of a vortex doublet in a two-dimensional
incompressible fluid has been studied with analytical and numerical models. Equal and positive
vorticity was chosen for the vortices (negative vorticity leads to symmetrical results with a change in
sign of strain and global rotation).
Point vortex theory has evidenced that two identical vortices possess an equilibrium position in
an external strain (if this strain amplitude is larger than external rotation). The nature of vortex
trajectories depend on the initial location of the vortices (inside or outside of the fixed point). Point
vortices initially close to each other rotate along an elliptical trajectory which was also observed for
finite-area vortices, while initially distant vortices undergo strain-induced expulsion along open stream-
lines. This equilibrium position, unstable to orthoradial perturbation, was also observed numerically
in the interaction of two finite-area vortices; its analytical characteristics were retrieved and confirmed
by numerical simulations with a spectral model.
In an external strain, interacting vortices exhibit a dominant elliptical deformation of their con-
tours, both in steady state calculations and in the spectral model. The elliptical model was therefore
used to compute the doublet dynamics. The steady state, elliptical and spectral models confirmed
that each vortex ellipticity is stronger with positive strain than in its absence. Moreover, the point
vortex, elliptical and spectral models gave very consistent results concerning the ellipticity of the
doublet trajectory for initially close vortices.
The spectral code evidenced four different nonlinear evolutions for the vortex doublet in the
presence of strain: co-rotation, merger, stationarity and strain induced expulsion. Only the former
two are found in the absence of external strain. Co-rotation of vortices along elliptical trajectories
was already described hereabove. Stationary vortices are mostly deformed elliptically. Strain-induced
expulsion of finite-area vortices obeys the rule obtained by point vortex theory (variation of critical
distance with external strain and rotation). For merger, the critical distance increases with positive
strain (and decreases with negative strain). As expected, external rotation favors co-rotation of the
doublet against its merger.
These variations of the critical merger distance and of merger efficiency with external parameters,
have lead to a closer inspection of vortex deformation and of the filamentation process using a La-
grangian criterion in the spectral code. Indeed, the elliptical deformation is not the only one induced
on each vortex by strain and by its companion. The Lagrangian analysis indicates that each vortex
is submitted to an asymmetric deformation, stronger on the side facing the other vortex than on the
opposite side. This asymmetry elongates the inner side of each vortex more than their outer side and
promotes merger. Moreover, the influence of the external flow on the vortex doublet depends on the

494 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

sign of γ: for a given initial distance, positive γ promotes merger with a larger efficiency than for
negative γ, by increasing the deformation of the outer vortex boundary. On the contrary, when the
initial distance between vortices is not held constant, the influence of the mutual strain also intervenes
and merger efficiency decreases for both signs of external strain. Finally, it is observed that merger
efficiency also depends on the duration of the merging process.
Merger is therefore not an all powerful mechanism for vortex growth as soon as external factors are
involved. Large scale strain or shear has an essential influence on merger and filamentation efficiency.
For an application to two-dimensional turbulence, the present study provides useful information on
the ability of small vortices (formed by the wrap-up of filaments) to merge, in the strain or shear
flow imposed by the large coherent vortices. This shear or strain rate can most often be computed
numerically. Such calculations should be performed with numerical simulations of turbulence in the
future, to validate our simple approach. For an application to oceanic vortex dynamics, the present
study should be extended to stratified flows by using a finite deformation radius. Recently, two deep
vortices were observed to interact (but not merge) southwest of Portugal. Application of our criteria to
the data will help determine the relative influence of large scale strain (due to surrounding currents),
of beta effect (variable Coriolis parameter) and of stratification on this interaction.

Acknowledgements
This work is a contribution to the SEMANE research program supported by IFREMER and by
SHOM/CMO (French Navy Oceanographic Center). Thanks are due to Dr Bernard Legras (LMD
Paris) and to Pr David Dritschel (Maths Dpt, Univ. St Andrews, Scotland) for fine suggestions and
comments.

Tables
Table 1. Critical distance between vortex centers (below which merger should occur) versus γ in elliptical model
γ 0 0.044 0.089 0.133 0.178 0.356 0.534 0.712
d/r 3.25 3.30 3.43 3.53 3.63 4.05 3.85 3.45

Table 2. Aspect ratio of co-rotating vortex trajectories versus γ. sp stands for spectral code, em for elliptical
model and pv for point vortex theory
γ 0 0.044 0.089 0.133 0.178
λsp 1 0.91 0.84 0.76 0.70
λem 1 0.88 0.79 0.70 0.63
λpv 1 0.90 0.80 0.70 0.60

Table 3. Aspect ratio of vortex versus γ. sp stands for spectral code and em for elliptical model

γ 0 0.044 0.089 0.133 0.178

λsp 0.82 0.81 0.80 0.80 0.79
λem 0.89 0.87 0.84 0.83 0.81

Table 4. Critical merger distance in the spectral code for negative strain

γ −0.178 −0.133 −0.089 −0.044 0

d/rsp 3.15 3.15 3.20 3.30 3.35

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 495

 G. MAZE, X. CARTON, G. LAPEYRE

Table 5. Comparison of critical merger distance between spectral code sp and the elliptical model em

γ 0 0.044 0.089 0.133 0.178

d/rsp 3.35 3.4 3.5 3.6 3.7
d/rem 3.25 3.3 3.43 3.53 3.63

Table 6. Critical merger distance for ω = ±γ. sp stands for spectral code and em for elliptical model

γ=ω 0.044 0.089 0.133 0.178

d/rsp 3.45 3.6 3,7 3.9
d/rem 3.32 3.48 3.58 3.68
γ = −ω 0.044 0.089 0.133 0.178
d/rsp 3.35 3.4 3.45 3.5
d/rem 3.25 3.35 3.45 3.55

Table 7. Filamentation analysis for d/r0 = 2.83

γ −0.178 −0.133 0.133 0.178

d/r 3.15 3.15 3.65 3.70
κf inal /κinitial 0.798 0.800 0.847 0.855
ef f
σoutside 0.0182 0.0179 0.0193 0.0200
ef f
σinside 0.0763 0.0763 0.0746 0.0743

Table 8. Merger efficiency for d/r0 = 0.9 dcritical /r0

γ −0.178 −0.133 −0.089 −0.044 0

κf inal /κinitial 0.797 0.820 0.835 0.846 0.855
γ 0 0.044 0.089 0.133 0.178
κf inal /κinitial 0.855 0.851 0.846 0.820 0.798

References

[1] C. Bertrand, X. Carton. Vortex merger on the β-plane. rotating-tank experiments. Phys. Fluids. 1991.
C. R. Acad. Sci. Paris. 1993. T. 316, II. P. 1201–1206. V. A3. P. 1411–1415.

[2] P. Capéran, J. Verron. Numerical simulation of a [4] X. Carton. On the merger of shielded vortices. Euro-
physical experiment on two-dimensional vortex merg- phys. Lett. 1992. V. 18. P. 697–703.
ing. Fluid Dyn. Res. 1988. V. 3. P. 87–92. [5] X. Carton. Hydrodynamical modeling of oceanic vor-
tices. Surveys in Geophys. 2001. V. 22. P. 179–263.
[3] G. F. Carnevale, P. Cavazza, P. Orlandi, R. Purini.
An explanation for anomalous vortex merger in [6] X. Carton, B. Legras, G. Maze. Two-dimensional vor-

496 REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004

 DYNAMICS OF A 2D VORTEX DOUBLET UNDER EXTERNAL DEFORMATION

tex merger in an external strain field. Journal of Tur- [15] M. V. Melander, N. J. Zabusky, J. C. McWilliams.
bulence. 2002. V. 3. Article 045 (IoP electronic pub- Symmetric vortex merger in two dimensions. J. Fluid
lication). Mech. 1988. V. 195. P. 303–340.
[7] B. Cushman-Roisin, B. Tang. Geostrophic turbulence [16] J. C. McWilliams. The emergence of isolated coher-
and the emergence of eddies beyond the radius of ent vortices in turbulent flow. J. Fluid Mech. 1984.
deformation. J. Phys. Oceanogr. 1990. V. 20. V. 146. P. 21–43.
P. 97–113.
[17] J. C. McWilliams. The vortices of two-dimensional
[8] D. G. Dritschel. The stability and energetics of coro-
turbulence. J. Fluid Mech. 1990. V. 219. P. 361–385.
tating uniform vortices. J. Fluid Mech. 1985. V. 157.
P. 95–134. [18] A. Okubo. Horizontal dispersion of floatable particles
[9] D. G. Dritschel. The stability of elliptical vortices in in the vicinity of velocity singularities such as conver-
an external straining flow. J. Fluid Mech. 1990. gences. Deep-Sea Res. 1970. V. 17. P. 445–454.
V. 210. P. 223–261. [19] K. Schultz-Tokos, H. H. Hinrichsen, W. Zenk. Merg-
[10] D. G. Dritschel, D. Waugh. Quantification of the ing and migration of two meddies. J. Phys. Oceanogr.
inelastic interaction of unequal vortices in two- 1994. V. 24. P. 2129–2141.
dimensional vortex dynamics. Phys. Fluids. 1992.
[20] D. Waugh. The efficiency of symmetric vortex merger.
V. A4. P. 1737–1744.
Phys. Fluids. 1992. V. A4. P. 1745–1758.
[11] G. Lapeyre, P. Klein, B. L. Hua. Does the tracer gradi-
ent vector align with the strain eigenvectors in 2D tur- [21] J. Weiss. The dynamics of enstrophy transfer in two-
bulence? Phys. Fluids. 1999. V. 11. P. 3729–3737. dimensional hydrodynamics. LJI-Tn-81-121 report.
La Jolla Institute, La Jolla, CA. 1981.
[12] G. Lapeyre, B. L. Hua, B. Legras. Comment on ”Find-
ing finite-time invariant manifold in two-dimensional [22] H. M. Wu, E. A. Overman II, N. J. Zabusky. Steady
velocity fields”. Chaos. 2001. V. 11. P. 427–430. states of the Euler equations in two dimensions. Rotat-
[13] B. Legras, D. G. Dritschel. The elliptical model of two- ing and translating V-States with limiting cases. I. Nu-
dimensional vortex dynamics. Phys. Fluids. 1991. merical algorithms and results. J. Comp. Phys. 1984.
V. A3. P. 845–869. V. 53. P. 42–71.
[14] M. V. Melander, N. J. Zabusky, J. C. McWilliams. [23] I. Yasuda, G. R. Flierl. Two-dimensional asymmet-
Asymmetric vortex merger in two dimensions: which ric vortex merger: merger dynamics and critical
vortex is ”victorious”? Phys. Fluids. 1987. V. 30. merger distance. Dyn. Atmos. Oceans. 1997. V. 26.
P. 2604–2610. P. 159–181.

REGULAR AND CHAOTIC DYNAMICS, V. 9, 4, 2004 497