# N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Problem Formulation

^{th}vortex in layer j (with $j=1$ for the upper layer and $j=2$ for the lower layer). The vortex strengths ${\kappa}_{j}^{\alpha}$ are all equal to ${\kappa}_{2}^{\alpha}=\kappa >0$ in the lower layer and to ${\kappa}_{1}^{\alpha}=-\kappa <0$ in the upper layer, corresponding to ‘warm’ hetons, following [1].

**Remark**

**1.**

## 3. Analysis of Possible Motion Regimes

- (A)
- Unbounded motion within the interval ${\varrho}^{-}\le \varrho \le 1$, when $H<{H}^{+*}$;
- (B)
- Localised (bounded and periodic) motion within the interval ${\varrho}^{-}\le \varrho \le {\varrho}^{+1}$, when $H>{H}^{+*}$;
- (C)
- Unbounded motion within the interval ${\varrho}^{+2}\le \varrho \le 1$, when $H>{H}^{+*}$.

#### 3.1. Examples of Trajectories: Absolute and Relative Choreographies

**Remark**

**2.**

**0**+/+] type, for which the vortex trajectory of the upper layer has a figure-of-eight shape, and panels (a) and (c) show the absolute vortex trajectories of types [

**-**0+/+] and [-0

**+**/+],, respectively, when the upper layer vortex moves along a loop-like trajectory, unwinding in the anticyclonic and cyclonic directions, respectively. This behaviour is found in the black region of Figure 4c.

_{1},c

_{1}). The relative choreographies are the closed trajectories obtained in a reference frame rotating with the average angular velocity $\omega $ of the vortex in the upper layer. One may see that the relative choreographies have loop-like absolute upper-layer trajectories (found inside the black region of Figure 4c) also belong to a type of figure-of-eight trajectory. The figure-of-eight relative choreographies also occur in some parts of the red and yellow areas of the diagram, where the upper-layer vortices move along zigzag trajectories—see Figure 9.

#### 3.2. Direction of Rotation of the External Vortices

**Remark**

**3.**

## 4. Finite-Core Hetons

## 5. Discussion and Concluding Remarks

- ${\varrho}^{+}$ = 0.9655: A regular mixing of warm water in both layers occurs inside an annular region, somewhat wider than the area occupied by the trajectories. The total transfer of warm fluid occurs in the cyclonic direction (type [+/+]).
- ${\varrho}^{+}$ = 0.8954: All the vortices of the lower layer move move along a single trajectory in the cyclonic direction, and all the vortices of the upper layer move in the anticyclonic direction along another peripheral closed curves (type [0/+]). Both sets of vortices carry with them columns of fluid that are warmer than their surroundings.
- ${\varrho}^{+}$ = 0.8960: the twelve vortices form six two-layer pairs with tilted axes, which scatter radially while preserving the N-fold symmetry of the system, and carrying warm water (motion type [∞]).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Some Useful Formulas and Results

## Appendix B. Extreme Properties of the Hamiltonian

## References

- Hogg, N.G.; Stommel, H.M. The heton, an elementary interaction between discrete baroclinic geostrophic vortices, and its implications concerning eddy heat-flow. Proc. R. Soc. Lond. Math. Phys. Sci.
**1985**, 397, 1–20. [Google Scholar] - Hogg, N.G.; Stommel, H.M. Hetonic explosions: The breakup and spread of warm pools as explained by baroclinic point vortices. J. Atmos. Sci.
**1985**, 42, 1465–1476. [Google Scholar] [CrossRef] - Gryanik, V.M. Dynamics of singular geostrophic vortices in a two-level model of atmosphere (ocean). Izvestiya Atmos. Ocean. Phys.
**1983**, 19, 171–179. [Google Scholar] - Chao, S.-Y.; Shaw, P.-T. Eddy maintenance and attrition in a vertically sheared current under Arctic ice. J. Phys. Oceanogr.
**1998**, 28, 2427–2443. [Google Scholar] [CrossRef] - Chao, S.-Y.; Shaw, P.-T. Close interactions between two pairs of heton-like vortices under sea ice. J. Geophys. Res.
**1999**, 104, 23591–23605. [Google Scholar] [CrossRef] - Chao, S.-Y.; Shaw, P.-T. Fission of heton-like vortices under sea ice. J. Oceanogr.
**1999**, 55, 65–78. [Google Scholar] [CrossRef] - Chao, S.-Y.; Shaw, P.-T. Slope-enhanced fission of salty hetons under sea ice. J. Phys. Oceanogr.
**2000**, 30, 2866–2882. [Google Scholar] [CrossRef] - Chao, S.-Y.; Shaw, P.-T. Heton shedding from submarine-canyon plumes in an Arctic boundary current system: Sensitivity to the undercurrent. J. Phys. Oceanogr.
**2003**, 33, 2032–2044. [Google Scholar] [CrossRef] - Fernando, H.J.S.; Chen, R.; Ayotte, B.A. Development of a point plume in the presence of background rotation. Phys. Fluids
**1998**, 10, 2369–2383. [Google Scholar] [CrossRef] - Fernando, H.J.S.; Smith IV, D.C. Vortex structure in geophysical convection. Eur. J. Mech. B/Fluids
**2001**, 20, 437–470. [Google Scholar] [CrossRef] - Gryanik, V.M.; Sokolovskiy, M.A.; Verron, J. Dynamics of heton-like vortices. Regul. Chaotic Dyn.
**2006**, 11, 417–438. [Google Scholar] [CrossRef] - Klinger, B.A.; Marshall, J. Regimes and scaling laws for rotating deep convection in the ocean. Dyn. Atmos. Oceans
**1995**, 21, 221–256. [Google Scholar] [CrossRef] - Legg, S.; Jones, H.; Visbeck, M. A heton perspective of baroclinic eddy transfer in localized open ocean convection. J. Phys. Oceanogr.
**1996**, 26, 2251–2266. [Google Scholar] [CrossRef] - Legg, S.; Marshall, J. A heton model of the spreading phase of open-ocean deep convection. J. Phys. Oceanogr.
**1993**, 23, 1040–1056. [Google Scholar] [CrossRef] - Legg, S.; Marshall, J. The influence of the ambient flow on the spreading of convected water masses. J. Marine Res.
**1998**, 56, 107–139. [Google Scholar] [CrossRef] - Legg, S.; McWilliams, J.; Gao, J. Localization of deep ocean convection by a mesoscale eddy. J. Phys. Oceanogr.
**1998**, 28, 944–970. [Google Scholar] [CrossRef] - Morel, Y.; McWilliams, J. Effect of isopycnal and diapycnal mixing on the stability of oceanic currents. J. Phys. Oceanogr.
**2001**, 31, 2280–2296. [Google Scholar] [CrossRef] - Oliver, K.I.C.; Eldevik, T.; Stevens, D.P.; Watson, A.J. A Greenland Sea perspective on the dynamics of postconvective eddies. J. Phys. Oceanogr.
**2008**, 38, 2755–2771. [Google Scholar] [CrossRef] - Serra, N.; Sadux, S.; Ambar, I. Observations and laboratory modeling of meddy generation of cape St. Vincent. J. Phys. Oceanogr.
**2002**, 32, 3–25. [Google Scholar] [CrossRef] - Shaw, P.-T.; Chao, S.-Y. Effects of a baroclinic current on a sinking dense water plume from a submarine canyon and heton shedding. Deep Sea Res. Part I
**2003**, 50, 357–370. [Google Scholar] [CrossRef] - Messori, G. The Sparadic Nature of Meridional Heat Transport in the Atmosphere. Ph.D. Thesis, Imperial College London, London, UK, 2013; p. 217. [Google Scholar]
- Mokhov, I.I.; Gryanik, V.M.; Doronina, T.N.; Lagun, D.E.; Mokhov, O.I.; Naumov, E.P.; Petukhov, V.K.; Tevs, M.V.; Khairullin, R.R. Vortex Activity in the Atmosphere: Tendencies of Changes; Institute of Atmospheric Physics of RAS: Moscow, Russia, 1993; 97p. [Google Scholar]
- Mokhov, I.I.; Doronina, T.N.; Gryanik, V.M.; Khairullin, R.R.; Korovkina, L.V.; Lagun, V.E.; Mokhov, O.I.; Naumov, E.P.; Petukhov, V.K.; Senatorsky, A.O.; et al. Extratropical cyclones and anticyclones: Tendencies of change. In The Life of Extratropical Cyclones; Gronas, S., Shapiro, M.A., Eds.; Geophysical Institute, University of Bergen: Bergen, Norway, 1994; Volume 2, pp. 56–60. [Google Scholar]
- Pedlosky, J. The instability of continuous heton clouds. J. Atmos. Sci.
**1985**, 42, 1477–1486. [Google Scholar] [CrossRef] - Griffiths, R.W.; Hopfinger, E.J. Experiments with baroclinic vortex pairs in a rotating fluid. J. Fluid Mech.
**1986**, 173, 501–518. [Google Scholar] [CrossRef] - Griffiths, R.W.; Hopfinger, E.J. Coalescing of geostrophic vortices. J. Fluid Mech.
**1987**, 178, 73–97. [Google Scholar] [CrossRef] - Helfrich, K.R.; Battisti, T.M. Experiments on baroclinic vortex shedding from hydrothermal plumes. J. Geophys. Res.
**1991**, 96, 12511–12518. [Google Scholar] [CrossRef] - Thivolle-Cazat, E.; Sommeria, J.; Galmiche, M. Baroclinic instability of two-layer vortices in laboratory experiments. J. Fluid Mech.
**2005**, 544, 69–97. [Google Scholar] [CrossRef] - Danilov, S.; Gryanik, V.; Olbers, D. Equilibration and Lateral Spreading of a Strip-Shaped Convection Region; Report 86; Alfred-Wegener-Institut für Polar- und Meeresforschung: Bremerhaven, Germany, 1998; p. 66. [Google Scholar]
- Danilov, S.; Gryanik, V.; Olbers, D. Equilibration and lateral spreading of a strip-shaped convection region. J. Phys. Oceanogr.
**2001**, 31, 1075–1087. [Google Scholar] [CrossRef] - Doronina, T.; Gryanik, V.; Olbers, D.; Warncke, T. A 3D Heton Mechanism of Lateral Spreading in Lacalized Convection in a Rotating Stratified Fluid; Report 87; Alfred-Wegener-Institut für Polar- und Meeresforschung: Bremerhaven, Germany, 1998; 84p. [Google Scholar]
- Gryanik, V.M.; Doronina, T.N.; Olbers, D.; Warncke, T.H. The theory of three-dimensional hetons and vortex-dominated spreading in localized turbulent convection in a fast rotating stratified fluid. J. Fluid Mech.
**2000**, 423, 71–125. [Google Scholar] [CrossRef] - Gryanik, V.M.; Borth, H.; Olbers, D. The Theory of Quasigeostrophic von Kármán Vortex Streets in Two-Layer Fluids on Beta-Plane and Intermittent Turbulent Jets; Alfred-Wegener-Institut für Polar- und Meeresforschung: Bremerhaven, Germany, 2001; Volume 106, 59p. [Google Scholar]
- Gryanik, V.M.; Borth, H.; Olbers, D. The theory of quasigeostrophic von Kármán vortex streets in two-layer fluids on beta-plane. J. Fluid Mech.
**2004**, 505, 23–57. [Google Scholar] [CrossRef] - Gryanik, V.M.; Doronina, T.N. Advective transport of a conservative solute by baroclinic singular quasigeostrophic vortices in the atmosphere (ocean). Izvestiya Atmos. Ocean. Phys.
**1990**, 26, 1011–1026. [Google Scholar] - Lim, C.C.; Majda, A.J. Point vortex dynamics for coupled surface/interior QG and propagating heton clasters in models for ocean convection. Geophys. Astrophys. Fluid Dyn.
**2001**, 94, 177–220. [Google Scholar] [CrossRef] - Marshall, J.S. Chaotic oscillations and breakup of quasigeostrophic vortices in the N-layer approximation. Phys. Fluids
**1995**, 7, 983–992. [Google Scholar] [CrossRef] - Jamaloodeen, M.I.; Newton, P.K. Two-layer quasigeostrophic potential vorticity model. J. Math. Phys.
**2007**, 48, 48. [Google Scholar] [CrossRef] - Reznik, G.; Kizner, Z. Two-layer quasi-geostrophic singular vortices embedded in a regular flow: 1. Invariants of motion and stability of vortex pairs. J. Fluid Mech.
**2007**, 584, 185–202. [Google Scholar] [CrossRef] - Reznik, G.; Kizner, Z. Two-layer quasi-geostrophic singular vortices embedded in a regular flow: 2. Steady and unsteady drift of individual vortices on a beta plane. J. Fluid Mech.
**2007**, 584, 203–223. [Google Scholar] [CrossRef] - Shteinbuch-Fridman, B.; Makarov, V.; Kizner, Z. Transitions and oscillatory regimes in two-layer geostrophic hetons and tripoles. J. Fluid Mech.
**2017**, 810, 535–553. [Google Scholar] [CrossRef] - Sokolovskiy, M.A.; Verron, J. Finite-core hetons: Stability and interactions. J. Fluid Mech.
**2000**, 423, 127–154. [Google Scholar] [CrossRef] - Young, W.R. Some interactions between small numbers of baroclinic, geostrophic vortices. Geophys. Astrophys. Fluid Dyn.
**1985**, 33, 35–61. [Google Scholar] [CrossRef] - Kozlov, V.F.; Makarov, V.G.; Sokolovskiy, M.A. Numerical model of the baroclinic instability of axially symmetric eddies in two-layer ocean. Izvestiya Atmos. Ocean. Phys.
**1986**, 22, 674–678. [Google Scholar] - Helfrich, K.R.; Send, U. Finite-amplitude evolution of two-layer geostrophic vortices. J. Fluid Mech.
**1988**, 197, 331–348. [Google Scholar] [CrossRef] - Polvani, L.M.; Zabusky, N.J.; Flierl, G.R. Applications of contour dynamics to two-layer quasi-geostrophic flows. Fluid Dyn. Res.
**1988**, 3, 422–424. [Google Scholar] [CrossRef] - Polvani, L.M. Two-layer geostrophic vortex dynamics. 2. Alignment and two-layer V-states. J. Fluid Mech.
**1991**, 225, 241–270. [Google Scholar] [CrossRef] - Reinaud, J.N. On the stability of continuously stratified quasi-geostrophic hetons. Fluid Dyn. Res.
**2015**, 47, 035510. [Google Scholar] [CrossRef] - Kizner, Z. Stability and transitions of hetonic quartets and baroclinic modons. Phys. Fluids
**2006**, 18, 056601. [Google Scholar] [CrossRef] - Makarov, V.G.; Sokolovskiy, M.A.; Kizner, Z. Doubly symmetric finite-core heton equilibria. J. Fluid Mech.
**2012**, 708, 397–417. [Google Scholar] [CrossRef] - Reinaud, J.; Carton, X. The stability and the nonlinear evolution of quasi-geostrophic hetons. J. Fluid Mech.
**2009**, 636, 109–135. [Google Scholar] [CrossRef] - Reinaud, J.N.; Carton, X. Head on collisions between two quasi-geostrophic hetons in a continuously stratified fluid. J. Fluid Mech.
**2015**, 779, 144–180. [Google Scholar] [CrossRef] - Reinaud, J.N.; Carton, X.; Dritschel, D.G. Interaction between a quasi-geostrophic buoyancy filament and a heton. Fluids
**2017**, 2, 37. [Google Scholar] [CrossRef] - Ryzhov, E.A.; Sokolovskiy, M.A. Interaction of two-layer vortex pair with a submerged cylindrical obstacle in a two-layer rotating fluid. Phys. Fluids
**2016**, 28, 056602. [Google Scholar] [CrossRef] - Sokolovskiy, M.A.; Carton, X. Baroclinic multipole formation from heton interaction. Fluid Dyn. Res.
**2010**, 42, 045501. [Google Scholar] [CrossRef] - Sokolovskiy, M.A.; Koshel, K.V.; Verron, J. Three-vortex quasi-geostrophic dynamics in a two-layer fluid. Part I. Analysis of relative and absolute motions. J. Fluid Mech.
**2013**, 717, 232–254. [Google Scholar] [CrossRef] - Sokolovskiy, M.A.; Verron, J. Dynamics of Vortex Structures in a Stratified Rotating Fluid; Atmospheric and Oceanographic Sciences Library; Springer: Cham, Switzerland, 2014; Volume 47, p. 382. [Google Scholar]
- Sokolovskiy, M.; Verron, J.; Carton, X.; Gryanik, V. On instability of elliptical hetons. Theor. Comput. Fluid Dyn.
**2010**, 24, 117–123. [Google Scholar] [CrossRef] - Valcke, S.; Verron, J. On interactions between two finite-core hetons. Phys. Fluids
**1993**, A5, 2058–2060. [Google Scholar] [CrossRef] - Verron, J.; Valcke, S. Scale-dependent merging of baroclinic vortices. J. Fluid Mech.
**1994**, 264, 81–106. [Google Scholar] [CrossRef] - Sokolovskiy, M.A.; Koshel, K.V.; Dritschel, D.G.; Reinaud, J.N. N-symmetric interaction of N hetons. I: Analysis of the case N = 2. Phys. Fluids
**2020**, 32, 096601. [Google Scholar] [CrossRef] - Koshel, K.V.; Reinaud, J.N.; Riccardi, G.; Ryzhov, E.A. Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices. Phys. Fluids
**2018**, 30, 096603. [Google Scholar] [CrossRef] - Reinaud, J.N.; Koshel, K.V.; 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
**2018**, 30, 096604. [Google Scholar] [CrossRef] - Simó, C. New families of solutions to the N-body problems. In European Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I; Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S., Eds.; Progress in Mathematics; Birkhäuser Basel: Basel, Switzerland, 2001; Volume 201, pp. 101–115. [Google Scholar]
- Borisov, A.V.; Mamaev, I.S.; Kilin, A.A. Absolute and relative choreographies in the problem of point vortices moving on a plane. Regul. Chaotic Dyn.
**2004**, 9, 101–111. [Google Scholar] [CrossRef] - Borisov, A.V.; Kilin, A.A.; Mamaev, I.S. The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem. Regul. Chaotic Dyn.
**2013**, 18, 33–62. [Google Scholar] [CrossRef] - Bartsch, T. Periodic solutions of singular first-order Hamiltonian systems of N-vortex type. Arch. Math.
**2016**, 107, 413–422. [Google Scholar] [CrossRef] - Calleja, R.C.; Doedel, E.J.; García-Azpeitia, C. Choreographies in the n-vortex problem. Regul. Chaotic Dyn.
**2018**, 23, 595–612. [Google Scholar] [CrossRef] - Dai, Q.; Gebhard, B.; Bartsch, T. Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary. SIAM J. Appl. Math.
**2018**, 78, 977–995. [Google Scholar] [CrossRef] - Dritschel, D.G. Contour surgery: A topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys.
**1988**, 79, 240–266. [Google Scholar] [CrossRef] - NIST Digital Library of Mathematical Functions. Available online: https://dlmf.nist.gov/.

**Figure 1.**Schematic of the initial vortex system with ${r}_{2}<{r}_{1}$ and $N=6$ (viewed from above): (

**a**) $\Phi =0$; (

**b**) $\Phi =\pi /6$, where $\Phi $ is the angle by which the vortex polygons are offset. The solid lines correspond to polygons with circumscribed radii ${r}_{1}$ and ${r}_{2}$, and the dashed lines correspond to a polygon with $2N=12$ sides joining the vortices in both layers. Red (‘warm’) and blue (‘cold’) colours mark the auxiliary lines and vortices in the upper and lower layers, respectively.

**Figure 2.**Examples of Hamiltonian phase portraits $H(\varrho ,\Phi )$ for $\varrho \in [0;1]$ and $\Phi \in [0;2\pi ]$, at $\tilde{M}=-1.2$: (

**a**) N = 2; (

**b**) N = 3; (

**c**) $N=4$. The thin horizontal lines in each panel indicate $\Phi =n\pi /N$ ($n=0,1,\dots ,2N$). The red dashed lines are the separatrices of field $\Phi \left(\varrho \right){|}_{\tilde{M}=-1.2}$, delimiting regions of localised and unbounded motions.

**Figure 3.**${H}^{+}$ (thick lines) and ${H}^{-}$ (thin lines) as a functions of $\varrho $ for $N=2,3,4,5,6$: (

**a**) $\tilde{M}=-0.001$; (

**a**) the same as (

_{1}**a**), but on the interval $\varrho \in [0.99;1.00]$, i.e., the horizontal axis is scaled by a factor 100; (

**b**) $\tilde{M}=-1$; (

**c**) $\tilde{M}=-100$. In panel (

**d**), $\tilde{M}=-5$ is shown only for $N=2$ and 6. The dashed horizontal line shows the minimum values of ${H}^{+}$, denoted ${H}_{2}^{+*}$ and ${H}_{6}^{+*}$, found at $\varrho ={\varrho}_{2}^{+*}$ and $\varrho ={\varrho}_{6}^{+*}$ respectively. (Here, ${H}_{2}^{+*}\approx 1.8680$ at ${\varrho}_{2}^{+*}=0.5078$ and ${H}_{6}^{+*}\approx 1.0135$ at ${\varrho}_{6}^{+*}=0.9296$.) Two additional lines of constant energy are drawn in solid: $H={H}_{{\mathrm{u}}_{2}}$ lying above ${H}_{2}^{+*}$, and $H={H}_{{\mathrm{d}}_{2}}$ lying below ${H}_{2}^{+*}$ for $N=2$ (blue horizontal lines); also, $H={H}_{{\mathrm{u}}_{6}}$ and $H={H}_{{\mathrm{d}}_{6}}$ for $N=6$ (red horizontal lines). The intersections of either of these lines with ${H}^{-}\left(\varrho \right)$ are denoted by ${\varrho}_{2}^{-}$ and ${\varrho}_{6}^{-}$, respectively. Note, the lines $H={H}_{{\mathrm{u}}_{2}}$ and $H={H}_{{\mathrm{u}}_{6}}$ intersect the curves ${H}^{+}$ at two points, which we denote $\varrho ={\varrho}_{2}^{+1}$ and ${\varrho}_{2}^{+2}$ when $N=2$, and $\varrho ={\varrho}_{6}^{+1}$ and ${\varrho}_{6}^{+2}$ when $N=6$. Note that ${\varrho}_{N}^{+2}>{\varrho}_{N}^{+1}$.

**Figure 4.**(

**a**–

**c**) Regime diagrams showing the possible types of motion in the $({\varrho}^{+},\tilde{M}/N)$-plane, for the initial value ${\Phi}_{0}=\pi /N$ for $N=2,\phantom{\rule{0.166667em}{0ex}}3$ and 6, respectively. The blue [∞] region corresponds to unbounded motion, while the red [$-/+$], black [$-0+/+$] and yellow [$+/+$] regions correspond to localised, bounded motions in which all vortices in both layers move within annular zones bounded by concentric circles. The white lines inside the black regions in these panels and in panel (

**e**) correspond to a families of periodic solutions, the so-called ‘absolute choreographies’. The white markers on these lines correspond to parameters used to generate the blue, green and red lines in Figure 6. The black markers in the upper right corner of the panel (

**c**) correspond to the numerical experiments in Figure 8. The green segment on the white line in panel (

**c**) over the interval ${\varrho}^{+}\in [0.60;0.98]$ corresponds to a family of absolute choreographies shown in Figure 7 below. Horizontal dashed lines are the asymptotes to which the lower boundary of the red region [$-/+$] approaches at $N\to \infty $ (see Equation (26). Vertical lines are asymptotes for the left border of the blue area [∞] as $\tilde{M}\to -\infty $ for each N (see Equation (20)). Panel (

**d**) shows regions for the single regime [$-/+$], for $N=2,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}5$ and 6 (filled with different shades of red). Here, the vertical scale is multiplied by by $\approx 5.4$ for clarity. Panel (

**e**) shows the case $N=3$ (as in panel (

**b**)), but with the vertical scale multiplied by 8. Here, the white markers at $\tilde{M}/N=-0.28$ correspond to parameters for the numerical experiments presented in Figure 5. Panel (

**f**) shows regions for the single regime [∞], for $N=2,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}5$ and 6 (filled with different shades of blue).

**Figure 5.**Trajectories of vortices (here and everywhere below, the absolute and relative trajectories of the vortices are depicted in the Cartesian coordinate system $(x,y)$, the dimensionless horizontal and vertical axis, correspondingly) of the equivalent heton (top view) at early times for $\tilde{M}/N=-0.28$ and $N=3$: (

**a**) ${\varrho}^{+}$ = 0.864: (

**b**) ${\varrho}^{+}$ = 0.878; (

**c**) ${\varrho}^{+}$ = 0.8892; (

**d**) ${\varrho}^{+}$ = 0.900; (

**e**) ${\varrho}^{+}$ = 0.9144; (

**f**) ${\varrho}^{+}$ = 0.9145. The red lines correspond to the upper-layer vortices, and the blue lines correspond to the lower-layer vortices. Legends in brackets indicate the type of motion (see Figure 4); the symbol corresponding to a specific type of trajectory is highlighted in red in panels (

**b**–

**e**). These experiments are for parameters represented by the white markers for $\tilde{M}/N=-0.28$ in Figure 4e.

**Figure 6.**Gallery of the ‘limiting’ periodic trajectories (absolute choreographies) of vortices of an equivalent heton for the following external parameter values (${\varrho}^{+},\tilde{M}/N$): (0.9323; −0.0899) for $N=2$; (0.9543; −0.1223) for $N=3$; (0.9739; −0.1163) for $N=4$; (0.9794; −0.1138) for $N=5$ and (0.9854; −0.1326) for $N=6$.

**Figure 7.**(

**a**,

**b**): Gallery of ‘absolute choreographies’ of the equivalent heton for $N=6$ ($\Phi \left(0\right)={\Phi}_{0}=\pi /6$) along the interval ${\varrho}^{+}\in [0.60;0.98]$ with increment 0.01. Each value of ${\varrho}^{+}$ has its own colour, and is the same for the trajectories of the upper and lower layers. Panel (

**a**) shows the lower-layer trajectories in natural scale. Panel (

**b**) shows the upper-layer trajectories in stretched coordinates: 4 times in x and 16 times in y. Panel (

**c**) shows the limiting simply connected upper-layer trajectories at ${\varrho}^{+}$ = 0.92 (

**left**) and ${\varrho}^{+}$=0.97 (

**right**); here, the x scale is expanded by 8 and the y scale is expanded by 40. Panel (

**d**) shows a family of intermediate doubly connected figure-of-eight trajectories for ${\varrho}^{+}$ = 0.93, 0.94, 0.95 and 0.96. The x scale is the same as in panel (

**c**), but the y scale is doubled.

**Figure 8.**The trajectories of vortices of the equivalent heton (top view) at ${\varrho}^{+}=0.95$ and $N=6$: (

**a**) $\tilde{M}/N=-0.35$; (

**b**) $\tilde{M}/N=-0.35183$ (absolute choreography); (

**c**) $\tilde{M}/N=-0.36$. Relative choreographies in panels (

**a**,

_{1}**c**) into which trajectories from panels (

_{1}**a**,

**c**) are subject to external rotation with positive and negative angular velocity, respectively. The insets to the right of the choreographies show zooms of the upper-layer closed trajectories (here, x is expanded by 5 and y is expanded by 25).

**Figure 9.**(

**a**,

**b**): Absolute vortex trajectories of an equivalent heton for $N=6$, ${\varrho}^{+}=0.95$; (

**a**) $\tilde{M}/N=-0.3$ and (

**b**) $\tilde{M}/N=-0.4$. Panels (

**a**,

_{1}**b**) show their respective figure-of-eight relative choreographies, into which trajectories from panels (

_{1}**a**,

**b**) are subject to external rotation with positive and negative angular velocity, respectively. In the insets, x is expanded by 5 and y is expanded by 25.

**Figure 10.**The same as in Figure 9, but for $\tilde{M}/N=-0.2$ (

**a**) and $\tilde{M}/N=-0.5$ (

**b**). Drop-shaped relative choreographies, into which trajectories from panels (

**a**,

**b**) are subject to external rotation with positive and negative angular velocity and are shown in panels (

**a**,

_{1}**b**), respectively. In inset (

_{1}**a**), x is expanded by 24 and y by 125; in inset (

_{1}**b**), x is expanded by 9 and y by 72.

_{1}**Figure 11.**The trajectories of vortices of the equivalent heton (top view) at ${\varrho}^{+}=0.97$ and $N=6$: (

**a**) $\tilde{M}/N=-0.25$ (type [

**-**0+/+]); (

**b**) $\tilde{M}/N=-0.26767$ (absolute choreography, type [-

**0**+/+]); (

**c**) $\tilde{M}/N=-0.31$ (type [-0

**+**/+]). Relative choreographies in panels (

**a**,

_{1}**c**), into which trajectories from panels (

_{1}**a**,

**c**) are subject to external rotation with positive and negative angular velocity, respectively.

**Figure 12.**The circle choreographies for (

**a**): $N=5$ ($\Phi =\pi /5$), ${\varrho}^{+}=0.78$, $\tilde{M}/N=-3.8923$, ${r}_{1}=4.4598,{r}_{2}=3.4787$ and (

**b**): $N=6$ ($\Phi =\pi /6$), ${\varrho}^{+}=0.82$, $\tilde{M}/N=-3.9796$, ${r}_{1}=4.4930$, ${r}_{2}=4.0427$. Markers indicate the initial position of the equivalent heton vortices.

**Figure 13.**Initial angular velocity profiles of the upper-layer and lower-layer vortices, ${\omega}_{1,2}^{+}={\dot{\phi}}_{1,2}^{+}$, for $N=3$ and $\Phi =\pi /N$: (

**a**) dependence on ${\varrho}^{+}$ for $R=1$ and $R=4$; (

**b**) dependence on R for ${\varrho}^{+}=0.6$ and ${\varrho}^{+}=0.8$. The colours of the curves correspond to the colours of the areas in Figure 4b,e, yellow: regions [$+/+$], and red: region [$-/+$]. The segments of the curves in black correspond to the [$-0+/+$] region. The white markers (on the black segments) correspond to the white curve in Figure 4 with zero values of the angular velocity, $\omega $. The blue marker in panel (

**a**) also corresponds to zero angular velocity—this is the stationary state at the boundary of the region of bounded motions. The monotonically decreasing curves in the upper parts of the panels correspond to ${\omega}_{2}^{+}$; the other curves (in the lower parts of the panels) correspond to ${\omega}_{1}^{+}$. A dashed black line is plotted at zero angular velocity.

**Figure 14.**Upper-layer oscillation to rotation period ratio, ${\delta}_{max}$ (limited to the range $[0,100]$), as a function of ${\varrho}^{+}$ for a hexagonal hetonic array $(N=6)$ for several values of $\tilde{M}/N$ (solid lines). The dashed lines give the asymptotes ${\omega}^{+}=0$. The markers correspond to pairs of values $({\delta}_{max},{\varrho}^{+})$ for the periodic trajectories in Figure 15 (${\delta}_{max}$ = 2, 3, 4 and 5 on the green line) and Figure 16 (${\delta}_{max}$ = 20 on the red and blue lines).

**Figure 15.**Vortex trajectories of an equivalent heton with $N=6$ corresponding to stationary periodic solutions, for integer values ${\delta}_{max}$ = 2, 3, 4 and 5, for $\tilde{M}/N=-1$, and for ${\varrho}^{+}$ = 0.9078 ($R=3.3720,\delta =4/3$), 0.9041 ($R=3.3095,\delta =3/2$), 0.8964 ($R=3.1906,\delta =3$) and 0.8863 ($R=3.0537,\delta =4$), respectively. These trajectories correspond to the black round markers on the green line in Figure 14.

**Figure 16.**The same as in Figure 15, but for the single value ${\delta}_{max}$ = 20. On the left: $\tilde{M}/N=-0.1326$, ${\varrho}^{+}$ = 0.96550 ($R=1.97761,\delta =7/4)$; on the right: $\tilde{M}/N=-0.3$, ${\varrho}^{+}$ = 0.96447 ($R=2.931942$, $\delta =17/4$). These trajectories correspond to the black round markers on the red and blue lines in Figure 14.

**Figure 17.**Periodic trajectories of all six vortices ($N=6$) in each layer (upper/lower row for the upper/lower layer) for ${\delta}_{max}=2,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}4$ and 5 (left to right). Different colours and line widths are used to distinguish the vortices. The initial positions of the vortices are numbered and coloured the same way as the trajectories emanating from them.

**Figure 18.**Superimposed vortex trajectories of different $N=6$ hetons having $\tilde{M}/N=-0.1326$ and (1) ${\varrho}^{+}=0.9655$, $R=1.97761$, $\delta =7/4$, black marker on the red curve in Figure 14 (centre); (2) ${\varrho}^{+}$ = 0.8954, R = 3.024729, $\delta =\infty $, red asymptote in Figure 14 (intermediate structure) and (3) ${\varrho}^{+}=0.8960$, $R=3.088396$ (scattering hetons). Markers indicate the initial positions of the vortices.

**Figure 19.**Trajectories for the first vortices in each layer (black lines) $N=3$, $\tilde{M}/N=-0.28$, (

**a**): $\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.05,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8897$, (

**b**):$\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.1,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8904$, (

**c**):$\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.2,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8935$, (

**d**):$\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.3,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8978$, (

**e**):$\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.4,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.9025$, (

**f**):$\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.5,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.9078$. The blue (respectively, red) contours show the vortex boundaries in the upper (respectively, lower) layer at various times.

**Figure 20.**Evolution of the aspect ratio $a/b$ of vortex 1 in layer 1 for $N=3$, $\tilde{M}/N=-0.28$, and ${R}_{v}=0.05,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8897$ (black), ${R}_{v}=0.1,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8904$ (red) ${R}_{v}=0.2,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8978$ (blue), ${R}_{v}=0.3,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.8978$ (green), ${R}_{v}=0.4,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.9025$ (magenta) ${R}_{v}=0.5,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.9078$ (cyan).

**Figure 21.**Top view on the vortex bounding contours for $N=3$, $\tilde{M}=-0.28$, ${R}_{v}=0.8$, ${\varrho}^{+}=0.9073$ at $t=0$ (

**a**), $t=5$ (

**b**), and $t=12$ (

**c**).

**Figure 22.**Trajectories: $N=3$, $\tilde{M}/N=-0.3$, (

**a**):$\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.2,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.965$, (

**b**):$\phantom{\rule{0.166667em}{0ex}}{R}_{v}=0.5,\phantom{\rule{0.166667em}{0ex}}{\varrho}^{+}=0.9678$ for $0\le t\le 280$. (

**c**): same as (

**b**) but for $280<t<408$.

**Table 1.**Values of ${\varrho}^{+}$ for an absolute choreography for $N=3$ and $\tilde{M}/N=-0.28$ for various initial radii ${R}_{v}$.

${R}_{v}$ | 0.05 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |

${\varrho}^{+}$ | 0.8897 | 0.8904 | 0.8935 | 0.8978 | 0.9025 | 0.9078 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Koshel, K.V.; Sokolovskiy, M.A.; Dritschel, D.G.; Reinaud, J.N.
*N*-Symmetric Interaction of *N* Hetons, II: Analysis of the Case of Arbitrary *N*. *Fluids* **2024**, *9*, 122.
https://doi.org/10.3390/fluids9060122

**AMA Style**

Koshel KV, Sokolovskiy MA, Dritschel DG, Reinaud JN.
*N*-Symmetric Interaction of *N* Hetons, II: Analysis of the Case of Arbitrary *N*. *Fluids*. 2024; 9(6):122.
https://doi.org/10.3390/fluids9060122

**Chicago/Turabian Style**

Koshel, Konstantin V., Mikhail A. Sokolovskiy, David G. Dritschel, and Jean N. Reinaud.
2024. "*N*-Symmetric Interaction of *N* Hetons, II: Analysis of the Case of Arbitrary *N*" *Fluids* 9, no. 6: 122.
https://doi.org/10.3390/fluids9060122