Elastic Interaction between a Mesoscale Eddy-Pair and a Cyclonic Eddy

We investigate numerically the elastic interaction between an eddy-pair and an axisymmetrical cyclonic eddy in inviscid isochoric two-dimensional (2D), as well as in three-dimensional (3D) flows under the quasi-geostrophic (QG) approximation. The eddypair is a straight moving Lamb-Chaplygin dipole where the absolute value of either its positive or negative amount of vorticity equals the vorticity of the axisymmetrical eddy. The results for the 2D and 3D cases show that interactions with almost no vorticity exchange or vorticity loss to the background field between ocean eddies, but changing their displacement velocity, are possible. When the eddy-pair approaches the axisymmetrical eddy, their respective potential flows interact, the eddypair’s trajectory acquires curvature and their vorticity poles separate. In the QG dynamics, the eddies suffer little vertical deformation, being the barotropic effects dominant. At the moment of highest interaction, the anticyclonic eddy of the pair elongates, simultaneously, the cyclonic eddy of the pair evolves towards spherical geometry, and the axisymmetrical eddy acquires prolate ellipsoidal geometry in the vertically stretched QG space. Once the eddy-pair moves away from the axisymmetrical eddy, its poles close, returning to their original geometry, and the anticyclonic and cyclonic eddy continue as an eddy-pair with a straight trajectory but along a new direction. The interaction is sensitive to the initial conditions and, depending on the initial position of the eddy-pair, as well as on small changes in the vorticity distribution of the axisymmetrical eddy, 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 interact with different vortical structures. Here we prove that elastic 37 interactions between two vortices are possible, meaning that the interaction does not 38 change the vorticity properties of the vortices. In particular, the interaction is between 39 an eddy-pair and an axisymmetrical cyclonic eddy. We use the quasi-geostrophic (1) 120 and vertical vorticity ζ(x, t)

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where k is the vertical unit vector and ∇ is the 2D gradient operator. The basic 123 dynamical equation is the material conservation of vorticity Equation (3) is numerically integrated (a brief description is given in Appendix A) to 126 evolve in time the vorticity field from prescribed initial vorticity conditions ζ(x, t 0 ).

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The results of these numerical simulations are described in section 6.  To describe the eddy-pair, we use the Lamb-Chaplygin dipole model whose vor-146 ticity distribution ζ d (r, θ) in polar coordinates (r, θ) is a piecewise function given by

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The vorticity distribution ζ v (r, θ) of the axisymmetrical cyclonic eddy is given 163 by the Bessel function of order 0 (Figure 1), truncated at a radius r = j 0,1 /k 2 , that is 165 where C v is a constant vorticity amplitude and k 2 is the cyclone's wavenumber. The 166 velocity of the cyclonic eddy u v (r) = v(r)e θ is azimuthal and is given by 168 When both, the eddy-pair and the axisymmetrical cyclonic eddy, are present they 169 interact due to their exterior potential flows. This interaction depends on the vortices 170 amplitudes (C d , C v ) and vortices extension given by the wavenumbers (k 1 , k 2 ). Since Since the radius of the eddy-pair is R d = j 1,1 /k 1 , the area is A + d = π(j 1,1 /k 1 ) 2 /2 and 178 applying (11), we obtain the wavenumber ratio The amplitudes ratio C v /C d is obtained equating the circulation of the target 181 eddy to the positive circulation of the eddy-pair. The positive circulation of the eddy-182 pair is 188 and therefore applying (11) we obtain the vorticity amplitudes ratio The initial vorticity distribution is represented in figure 2. The eddy-pair's poles 191 are close together and have the same vorticity contours as the axisymmetrical eddy.

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The initial interaction between both vortices, as inferred from the stream function is  cyclonic eddy in the QG space is given by The volume of the cyclonic eddy of the eddy-pair is boundary radius of the eddy-pair. While the positive circulation of the eddy-pair in The volume of the target axisymmetrical eddy is boundary of the cyclone and the circulation of the cyclone is 236 where A ± d are the time dependent regions of points (x, y, t) where ±ζ(x, y, t) > 0. The 237 time dependent center of the eddy-pair r d is given by  In this case, due to the large North-South initial distance between the eddy-254 pair and the axisymmetrical cyclonic eddy, there is no vorticity exchange between 255 the vortices. After the time of largest interaction (t 123, Figure 5), the poles of 256 the eddy-pair close and the eddy-pair acquires a rigid vorticity distribution which is 257 similar to its initial one but rotated positively (Figure 3).

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The mechanism of the eddy-pair's trajectory change, due to the interaction be-  We have also analyzed 2D interactions similar to the one described before but 303 changing the initial positions of the eddy-pair along the y-axis (video referenced in Fig-304 ure 9). In this video, the eddy-pair with green vorticity contours at the top simulates 305 the elastic interaction described above. The eddy-pair with black vorticity contours 306 is located half the way of the green eddy-pair. This interaction is really similar to 307 the interaction described in Appendix B where the eddy-pair is scattered by the ax-308 isymmetrical eddy and the eddy-pair's poles separate. When the negative pole is close 309 to the positive axisymmetrical eddy, these two vortices join together, giving rise to 310 partner exchange and formation of a new eddy-pair (video in Figure 9). The positive 311 vorticity pole of the eddy-pair is left behind and evolves towards an axisymmetrical 312 cyclonic eddy close to the initial position of the initial axisymmetrical cyclonic eddy.

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The next eddy-pair, with yellow vorticity contours, is located at the same y-314 coordinate as the axisymmetrical cyclonic eddy (y = 0). In this case, the vortices 315 collide and merging occurs (video in Figure 9). The next two eddy-pairs, with white 316 and red vorticity contours, are situated at the same distance as the vortices black and 317 green, respectively, but with reversed sign, so that the positive pole of the eddy-pair is 318 the closest pole to the axisymmetrical cyclonic eddy. In these cases, a positive-positive 319 pole interaction occurs. The white eddy-pair, closer to the axisymmetrical cyclonic 320 eddy, suffers straining out vorticity processes, while the red eddy-pair, far from the 321 axisymmetrical eddy, experiences also an elastic interaction but weaker than the one 322 experienced by the green eddy-pair (video in Figure 9). In this case, the red eddy-pair flow of the eddy-pair and describes an almost semi-circular trajectory with a small 328 radius δr 0.6 (too small to be appreciated in Figure 10) and returns, after the 329 interaction time, to a new location very close to the initial one.  ζ(x, y, t 0 + δt) = −δt u · ∇ζ + ζ(x, y, t 0 ) .

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After the step 4 the loop returns to step 1 for the next time integration (t 0 + δt).

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The numerical simulations were carried out using a 2D pseudospectral code where the 386 vorticity field ζ(x, y, t) is numerically integrated, following the steps described above, In this case the axisymmetrical eddy boundary R v is extended to the first zero 393 R v = j 1,1 /k 2 , instead of j 0,1 /k 2 (Figure 1), and its vorticity distribution is given by 394 ζ(r)/C v = J 0 (k 2 r) − J 0 (j 1,1 ) so that ζ(j 1,1 /k 2 ) = 0 with no vorticity jump at the eddy 395 boundary. The other initial conditions described in section 4 remain unchanged. In 396 the general case where the eddy boundary is taken at k 2 r = j m,n the external flow 397 u 0 (r), is given by 398 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 eddy-pair moves initially with a straight trajectory approaching