Elastic Interaction between a Vortex Dipole and an Axisymmetrical Vortex in Two-Dimensional Flows

We investigate numerically the elastic interaction between a dipole and an axisymmetrical vortex in inviscid isochoric twodimensional flows satisfying Euler’s vorticity conservation equation. This work contributes to previous studies addressing inelastic vortex interactions. The dipole is a straight moving Lamb-Chaplygin (L-C) vortex, where the absolute value of either the positive or the negative amount of vorticity equals the amount of vorticity of the target vortex. The results show that, when the straight moving L-C dipole approaches the axisymmetrical vortex, the potential flows of both vortices interact, the dipole’s trajectory acquires curvature and the dipole’s vorticity poles separate. Once the L-C dipole moves away from the target vortex, the poles close and the dipole continues with a straight trajectory but along a direction different from the initial one. Though there is very small vorticity exchange between the dipole’s poles and a small vorticity leakage to the background field, the vortices preserve, to a large extent, their amount of vorticity and the resulting interaction may be practically qualified as an elastic interaction. This process is sensitive to the initial conditions and, depending on the initial position of the dipole as well as on small changes in the vorticity distribution of the axisymmetrical vortex, inelastic interactions may instead occur. Since the initial vorticity distributions are based on the eigenfunctions of the two-dimensional Laplacian operator in circular geometry these results are directly applicable to three-dimensional baroclinic geophysical flows under the quasi-geostrophic approximation.

to be inelastic, in the sense that the vorticity dipole suffers irreversible changes during 69 the interaction, for example during vortex merging or partial or complete straining 70 out processes (Dritschel, 1995;Dritschel & Waugh, 1992;Dubosq & Viúdez, 2007;71 McWilliams & Zabusky, 1982;Voropayev & Afanasyev, 1992). However, in many in-72 stances ocean vortices do not interact strongly with one another for long time periods 73 (Carton, 2001). Consequently, elastic interactions, where vorticity exchange does not 74 occur, are also possible between ocean vortices. In this study we investigate numeri-75 cally, as a particular kind of elastic dipole-vortex interaction, the interaction between 76 a translating dipole and an axisymmetrical vortex.

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In view of the complexity of baroclinic three-dimensional (3D) vortices, it is 78 more practical to investigate first the barotropic two-dimensional (2D) case, assum- 108 and vertical vorticity ζ(x, t) We use the Lamb-Chaplygin dipole model whose vorticity distribution ζ d (r, θ) in 119 polar coordinates (r, θ) is a piecewise function given by where e r and e θ are the radial and azimuthal unit basis vectors.

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In order to provide flow solutions with vanishing velocity at infinity we must  134 where u d (r, θ) is the velocity field (5) in the steady state, r(x, y) = x 2 + y 2 and 135 θ(x, y) = arctan(y/x).

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The vorticity distribution ζ v (r, θ) of the axisymmetrical vortex is given by the 137 Bessel function of order 0 (Figure 1), truncated at a radius r = j 0,1 /k 2 , that is 139 where C v is a constant vorticity amplitude and k 2 is the vortex's wavenumber. The 140 vortex velocity u v (r) = v(r)e θ is azimuthal and is given by The radius of the vortex is R v = j 0,1 /k 2 , which implies A v = π(j 0,1 /k 2 ) 2 . Since the 150 radius of the dipole is R d = j 1,1 /k 1 , the area is A + d = π(j 1,1 /k 1 ) 2 /2 and applying (9), 151 we obtain the wavenumber ratio 152 k 1 k 2 = 1 √ 2 j 1,1 j 0,1 1.127 . 153 The amplitudes ratio C v /C d is obtained from the circulation of the vortex and 154 the positive part of the dipole. The positive circulation of the dipole is

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where H 1 (x) is the Struve function of order 1. The circulation of the vortex is 158 and therefore applying (9) we obtain the vorticity amplitudes ratio 160 The initial vorticity distribution is represented in figure 2. The dipole's poles 161 are close together and have the same vorticity contours as the axisymmetrical vortex.

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The initial interaction between both vortices, as inferred from the stream function is parts of the L-C dipole (r + d (t) and r − d (t), respectively) are defined as where A ± d are the time-dependent regions of points (x, y, t) where ±ζ(x, y, t) > 0. The 173 time-dependent center of the whole dipole r d is given by whole interaction process is shown in the video referenced in Figure 6.

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The dipole's speed of displacement after the interaction is very close to its orig-

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We have also analyzed interactions similar to the one described before but chang-220 ing the initial positions of the dipole along the y-axis (video referenced in Figure 7).

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In this video, the dipole with green vorticity contours at the top simulates the elas-222 tic interaction described above. The dipole with black vorticity contours is located  trajectory with a small radius δr 0.6 (too small to be appreciated in Figure 8) and 245 returns, after the interaction time, to a new location very close to the initial one.

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Though this is a simple two-dimensional flow study, the application to three- u 0 (r), is given by 318 u 0 (r) C v /k 2 = J 0 (j m,n ) 2 j 2 m,n k 2 r − J 1 (j m,n ) j m,n k 2 r e θ .

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In this case, the dipole moves initially with a straight trajectory approaching the