Elastic Interaction between a Vortex Dipole and an Axisymmetrical Vortex in Quasi-Geostrophic Ocean Dynamics

We investigate numerically the elastic interaction between a dipole and an axisymmetrical vortex in inviscid isochoric twodimensional (2D), as well as in three-dimensional (3D) flows under the quasi-geostrophic (QG) approximation. The dipole is a straight moving Lamb-Chaplygin (L-C) vortex such that the absolute value of either its positive or negative amount of vorticity equals the vorticity of the axisymmetrical vortex. The results for the 2D and 3D cases show that, when the L-C dipole approaches the vortex, their respective potential flows interact, the dipole’s trajectory acquires curvature and the dipole’s vorticity poles separate. In the QG dynamics, the vortices suffer little vertical deformation, being the barotropic effects dominant. At the moment of highest interaction, the negative vorticity pole elongates, simultaneously, the positive vorticity pole evolves towards spherical geometry and the axisymmetrical vortex acquires prolate ellipsoidal geometry in the vertically stretched QG space. Once the L-C dipole moves away from the vortex, its poles close, returning the vortices to their original geometry, and the dipole continues with a straight trajectory but along a direction different from the initial one. The vortices preserve, to a large extent, their amount of vorticity and the resulting interaction may be practically qualified as an elastic interaction. The interaction 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.

Ocean swirls, also known as eddies or vortices are ubiquitous in all oceans. Often 35 they drift as two vortices together, rotating in opposite directions, known as eddy-pairs. 36 The eddy-pair can encounter different structures as well as with other ocean vortices. 37 Here we prove that elastic interactions between two vortices are possible, meaning 38 that the interaction does not change the vorticity properties of the vortices. We use 39 the quasi-geostrophic three-dimensional approximation as well as a two-dimensional 40 model. We also describe numerically inelastic interactions, where the dipole (vortex-41 pair) separates or loses part of its vorticity in two-dimensions.  The dipole structure and its stability has been subject of many experimental, 57 laboratory, and numerical studies (Couder & Basdevant, 1986   We use the Lamb-Chaplygin dipole model whose vorticity distribution ζ d (r, θ) in 141 polar coordinates (r, θ) is a piecewise function given by k1r cos θ e r − 1 2 (J 0 (k 1 r) − J 2 (k 1 r)) sin θ e θ 0 ≤ k 1 r ≤ j 1,1 J0(j1,1) 2k 2 1 r 2 (k 2 1 r 2 − j 2 1,1 ) cos θ e r − (k 2 1 r 2 + j 2 1,1 ) sin θ e θ j 1,1 < k 1 r , 147 where e r and e θ are the radial and azimuthal unit basis vectors, respectively.

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In order to provide flow solutions with vanishing velocity at infinity we must add where u d (r, θ) is the velocity field (7) in the steady state, r(x, y) = x 2 + y 2 and 155 θ(x, y) = arctan(y/x). Thus, the dipole moves, in absence of background velocity, 156 straight along the x-axis with a constant speed equal to u 0 = −C d J 0 (j 1,1 )/(2k 1 ).

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The vorticity distribution ζ v (r, θ) of the axisymmetrical vortex is given by the 158 Bessel function of order 0 (Figure 1), truncated at a radius r = j 0,1 /k 2 , that is

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where C v is a constant vorticity amplitude and k 2 is the vortex's wavenumber. The 161 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 172 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 (11), we obtain the wavenumber ratio 178 where H 1 (x) is the Struve function of order 1. This is consistent with the circulation 179 of one-half of the Lamb dipole obtained by (Kloosterziel et al., 1993). The circulation 180 of the vortex is 182 and therefore applying (11) we obtain the vorticity amplitudes ratio The initial vorticity distribution is represented in figure 2. The dipole's poles 185 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 The volume of the positive part of the dipole is V +

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-6-manuscript submitted to JGR: Oceans 229 where A ± d are the time dependent regions of points (x, y, t) where ±ζ(x, y, t) > 0. The 230 time dependent center of the whole dipole r d is given by

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In this case, due to the large north-south initial distance between the dipole 247 and the axisymmetrical vortex, there is no vorticity exchange between the vortices.

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After the time of largest interaction (t 123, Figure 5), the dipole's poles close and 249 the dipole acquires a rigid vorticity distribution which is similar to its initial one but 250 rotated positively (Figure 3).

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The mechanism of the dipole's trajectory change, due to the interaction between  the approaching dipole even before any vorticity interaction can take place. 293 We have also analyzed 2D interactions similar to the one described before but 294 changing the initial positions of the dipole along the y-axis (video referenced in Figure   295 9). In this video, the dipole with green vorticity contours at the top simulates the  The next dipole, with yellow vorticity contours, is located at the same y-coordinate 305 as the axisymmetrical vortex (y = 0). In this case the vortices collide and merging 306 occurs (video in Figure 9). The next two dipoles, with white and red vorticity con-307 tours, are situated at the same distance as the vortices black and green, respectively,

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-11-manuscript submitted to JGR: Oceans   trajectory with a small radius δr 0.6 (too small to be appreciated in Figure 10) and 319 returns, after the interaction time, to a new location very close to the initial one. Given an initial vorticity field ζ(x, y, t 0 ) the vorticity time integration is done in 360 four steps.