3.2.1. Symmetric cases

We first examine the symmetric cases where like-signed vortices lie at equal depth. We vary the ratio $y_{init}/z_{init}$ and we find a systematic trend. If $y_{init}/z_{init}<1$ , the poles of each heton reverse their trajectory by exchanging their horizontal location with the second pole within the same heton, as seen in figure 2(a). For $y_{init}=z_{init}=1$ , the vortices (singularities) reach an (unstable) steady state as they reach the vertices of a cube, see figure 2(b). The situation is unstable in the sense that any perturbation in $y$ imposed on one pole will result in a change of trajectory from reverse (as in $y_{init}/z_{init}<1$ ) to escape at a right angle (as in $y_{init}/z_{init}>1$ ), see figure 2. This statement, based on physical arguments, can be mathematically justified by a linear stability analysis proposed in appendix A. For $y_{init}/z_{init}>1$ , the point vortices exchange partners and escape perpendicularly as new hetons, see figure 2(c). To understand this trend, one can look at the velocity $v_{1}$ of the point vortex 1, lying at the bottom right initially for $x<0$ in the symmetric configuration. In the calculation we rescale ${\rm\Delta}y$ and ${\rm\Delta}z$ by ${\rm\Delta}x$ for convenience (and as the discussion is on ${\rm\Delta}y$ and ${\rm\Delta}z$ ),

Figure 2.

Top view of the trajectories of the ‘symmetric’ vortices for $y_{init}/z_{init}=2/3$ (a), $y_{init}/z_{init}=1$ (b) and $y_{init}/z_{init}=2$ (c). The square symbols label the time by indicating the location of pole 1 every ${\rm\Delta}t=1$ .

We see that the sign of $v_{1}$ depends on the sign of

i.e. only on

Hence, if ${\rm\Delta}y<{\rm\Delta}z$ ( ${\it\alpha}<1$ , which means initially $y_{init}/z_{init}<1$ ), vortex 1 has a velocity $v_{1}>0$ and tends to move in the positive $y$ direction. If ${\rm\Delta}y={\rm\Delta}z$ , $v_{1}=0$ , the vortex does not deviate in the $y$ direction. It is easy to show that when ${\rm\Delta}x={\rm\Delta}y={\rm\Delta}z$ , i.e. when the point vortices are located at the vertices of a cube, $u_{1}$ , the velocity of all vortices in the $x$ direction, becomes zero as well. We recall that the vertical velocity is zero at all times. Consequently, in this case, the singularities are steady, and the four point vortices are in mutual equilibrium. It should be noted, however, that any perturbation in $y$ will make the system bifurcate to one of the two other situations. Finally, if ${\rm\Delta}y>{\rm\Delta}z$ ( ${\it\alpha}>1$ ), $v_{1}<0$ , and vortex 1 goes to further negative $y$ values, escaping the initial hetonic configuration.

This velocity $v_{1}$ comes from the influence of poles 3 and 4 on pole 1. Pole 2 does not induce a velocity in the $y$ direction on pole 1 (as 1 and 2 are aligned along the $x$ direction). The sign of $v_{1}$ depends on which of the two poles (3 or 4) is closest to pole 1.

Figure 3 illustrates the streamlines in three configurations, depending on the respective values of ${\rm\Delta}y$ and ${\rm\Delta}z$ . In the first configuration, we take ${\rm\Delta}y/{\rm\Delta}z=2/3$ , while ${\rm\Delta}x=2{\rm\Delta}z=2\max ({\rm\Delta}y,{\rm\Delta}z)$ . In the second, we have ${\rm\Delta}x={\rm\Delta}y={\rm\Delta}z=1$ . Finally, in the third and last one, we take ${\rm\Delta}y/{\rm\Delta}z=3/2$ , while ${\rm\Delta}x=2{\rm\Delta}y=2\max ({\rm\Delta}y,{\rm\Delta}z)$ . We plot the streamlines in two horizontal planes, $z=-2{\rm\Delta}x$ and $z=0$ . In practice, the streamlines are obtained as curves of isovalue of the streamfunction, rather than (equivalently) integrating $\text{d}x/u=\text{d}y/v$ .

Figure 3.

Streamlines for (a,b) ${\rm\Delta}x=2,{\rm\Delta}y=1,{\rm\Delta}z=1.5$ and $-2\leqslant x,y\leqslant 2$ , $z=-2$ (a) and $z=0$ (b); (c,d) ${\rm\Delta}x=1,{\rm\Delta}y=1,{\rm\Delta}z=1$ and $-1\leqslant x,y\leqslant 1$ , $z=-1$ (c) and $z=0$ (d); (e,f) ${\rm\Delta}x=2,{\rm\Delta}y=1.5,{\rm\Delta}z=1$ and $-2\leqslant x,y\leqslant 2$ , $z=-2$ (e) and $z=0$ (f).

The bottom cross-section, at $z=-2{\rm\Delta}x$ , lies below the hetons. Here, the two bottom singularities have the strongest influence as they are the closest to the cross-section. We recall that the streamfunction is inversely proportional to the distance, ${\it\psi}\propto 1/r$ , where $r$ is the three-dimensional distance between the source and the evaluation points. Hence, we recover closed streamlines surrounding the two bottom negative poles, each pole belonging to a different heton. This renders a classical pattern for two co-rotating vortices. Moving away from the vortices, the influences of the two vortices combine and the streamlines tend to circles around the pair of vortices. It should be noted that this pattern is still influenced by the positive poles located at $z>0$ . The second cross-section at $z=0$ corresponds to a cross-section lying at the plane of symmetry of the structure. Here, the poles are located at equal vertical distance from the plane $z=0$ . Hence, the sign of their contribution to the total streamfunction is only set by their strength (we cannot distinguish from the figure alone a pole above or below $z=0$ ). It should be noted than a third cross-section at $z=+2{\rm\Delta}x$ would give a picture similar to the one for $z=-2{\rm\Delta}x$ , symmetric with respect to the plane $y=0$ , with opposite sign.

Qualitatively similar trajectories were observed in Sokolovskiy & Verron (Reference Sokolovskiy and Verron2000b ), where the authors varied the internal Rossby radius of deformation. However, these similarities remain qualitative, as the problem is fundamentally different. In the two-layer model, the vortices fill the full depth of the layers and the two layers are adjacent. This translates as a different inversion relation between the PV and the streamfunction.

3.2.2. Antisymmetric cases

We now look at the antisymmetric cases where the vortices that are at the same depth have opposite sign. In each case the vortices escape perpendicularly from their original trajectory. It should be noted that they no longer behave as hetons but as barotropic dipoles. By ‘barotropic dipole’ (hereinafter referred to as dipoles for short) we refer to the situation one may think of as a vortex dipole in two dimensions, i.e. here two opposite-signed vortices at the same depth. This is in contrast to a heton which is a baroclinic structure and consists of two opposite-signed vortices at different depths. The trajectory of the vortices is illustrated for three different cases in figure 4. It is shown that they are qualitatively similar.

Figure 4.

Top view of the trajectories of the ‘antisymmetric’ vortices for $y_{init}/z_{init}=2/3$ (a), $y_{init}/z_{init}=1$ (b) and $y_{init}/z_{init}=2$ (c). The square symbols label the time by indicating the location of pole 1 every ${\rm\Delta}t=1$ .

We identify the conditions when the vortices escape by looking at the ‘corner’ in their trajectory. We associate the corner with the condition when $|v_{1}/u_{1}|=1$ , i.e. the threshold between a trajectory mostly in the $x$ direction and a trajectory mostly along  $y$ . The ratio $v_{1}/u_{1}$ is

where here for convenience the horizontal distances ${\rm\Delta}x$ and ${\rm\Delta}y$ are rescaled by ${\rm\Delta}z$ (i.e.  ${\rm\Delta}z=1$ ). To view the limit separating the regions $|v_{1}/u_{1}|>$ or ${<}1$ in the plane ${\rm\Delta}x$ – ${\rm\Delta}y$ , we plot contours of isovalues of $|v_{1}/u_{1}|$ , see figure 5. The limit can only be obtained from the implicit (3.6). It does not correspond to a simpler relation between ${\rm\Delta}x$ and ${\rm\Delta}y$ . In particular, because of the influence of pole 4, the corner does not correspond to the situation when pole 3 becomes as close to pole 1 as pole 2. Indeed, when poles 2 and 3 are equidistant from pole 1, we have $s^{2}=({\rm\Delta}y^{2}+1)/{\rm\Delta}x^{2}=1$ and $|v_{1}/u_{1}|=|{\rm\Delta}x/{\rm\Delta}y|\neq 1$ .

Figure 5.

Isovalues of $|v_{1}/u_{1}|$ for $0.8\leqslant {\rm\Delta}x\leqslant 2.05$ and $0.2\leqslant {\rm\Delta}y\leqslant 1.45$ .

Figure 6 gives an illustration of the topology of the streamlines for antisymmetric pairs of hetons. Here, the pattern is different from the symmetric case. This is due to the fact that the point vortices at the same depth now have opposite sign and behave as vortex dipoles rather than vortex pairs. This topological difference is not apparent in the plane of symmetry $z=0$ . This is due to the fact that the streamfunction depends on the polarity of the singularities and the inverse separation distance between them and the evaluation point only, and not on whether the singularities are above or below the plane $z=0$ . However, the topological difference from the symmetric case becomes clear in a plane closer to one of the dipoles, see figure 6(a). Again, we observed nearly circular streamlines in the vicinity of the vertical axes where the singularities lie. However, between the two centres, the streamlines flatten and are nearly parallel, indicating translation of the structure. This is in contrast to the stagnation point observed for the symmetric case. The pattern shown in the figure 6 is generic for all antisymmetric cases investigated.

Figure 6.

Streamlines for ${\rm\Delta}x=2,{\rm\Delta}y=1,{\rm\Delta}z=1.5$ and $-2\leqslant x,y\leqslant 2$ , $z=-2$ (a) and $z=0$ (b).

3.4.1. Symmetric hetons

We start the investigation with the symmetric hetons. The parameter space is spanned by $d_{y}={\rm\Delta}y/r,\,{\it\varrho}=r/h$ and $\ell _{z}=Dz/h$ . Here, ${\rm\Delta}y$ has the same meaning as in the previous sections; it is the distance in the $y$ direction between the pole centres of a heton. For convenience, its non-dimensional form $d_{y}$ is measured as a fraction of the horizontal radius of the pole, hence $d_{y}\geqslant 2$ would mean that the poles do not overlap. We investigate five values for each of the parameters, $d_{y}=(0.1,\,0.25,\,0.5,\,0.75,\,1)$ , ${\it\varrho}=(1.5,\,2.5,\,3.5,\,4.5,\,5.5)$ and $\ell _{z}=(0.5,\,1,\,1.5,\,2,\,2.5)$ , hence 125 cases in total. For all cases the simulation is started with the poles situated at $x_{0}=\pm 5r$ at least. There are 100 horizontal layers representing the full vertical extent of the flow (i.e. from the bottom of the bottom pole to the top of the top vortex). This includes also the vertical gap between the two poles. The equations are marched in time with the fourth-order Runge–Kutta scheme and the time step is ${\rm\Delta}t=|q|/40$ , where $|q|$ is the PV magnitude. These are the standard set-ups of the method.

Table 1 defines the symbols used to describe the sort of nonlinear interaction that the symmetric pair of hetons undergoes. The range of potential behaviours is as follows. Three main effects are competing. First, the finite core hetons may reproduce the global behaviour predicted by the point vortices. Second, there is the possibility that the hetons break baroclinically. Third, the poles belonging to different hetons, at the same depth, may strongly interact. Different initial conditions can favour one or a combination of such behaviours. The latter two effects are instabilities and are therefore associated with a time scale over which they can develop during the dynamical process.

Table 1.

Meaning of the symbols used to describe the outcome of the interactions between two symmetric hetons.

We know from the point vortex calculation that the global behaviour is influenced by the ratio ${\it\alpha}$ of the horizontal offset of the poles within the heton to their vertical offset. This ratio can be evaluated from the parameters $d_{z},\,{\it\varrho}$ and $\ell _{z}$ as ${\it\alpha}={\it\varrho}(d_{y}/(\ell _{z}+1))$ . Small values of this parameter trigger a reverse motion of the hetons while a large value triggers an escape of the recombined poles perpendicularly to their original trajectory. On the other hand, we know that hetons are more stable when their aspect ratio is small and the poles are separated by either or both large $d_{y}$ and large $\ell _{z}$ . Finally, we know that for two co-rotating vortices, they must be close enough to one another to merge.

Therefore, in our parameter space, we can expect that for small aspect ratios, large vertical offset (for stability) and moderate horizontal offset $d_{y}$ ( ${\it\alpha}<1$ ) the hetons are likely to reverse trajectory. However, in the symmetric case, since the co-rotating vortices are at the same depth, vortex merger is possible. For larger values of the aspect ratio the vortices may merge during the phase when they approach one another. Larger vortices are able to merge from further apart. To merge, the poles must also remain in the vicinity of one another for a sufficiently long period of time. Indeed, the merger process is fundamentally linked to an instability mode of a pair of co-rotating vortices. This means that the instability must have enough time to develop before the dynamics of the hetons (reverse trajectory) takes over and separates the poles. On the other hand, for large aspect ratio and small $d_{y}$ , the translation velocity bringing the hetons together can be too low to prevent the hetons from becoming baroclinically unstable first. In that case each heton may break before reversing and/or being close enough to experience vortex merger with the poles of the other heton.

For larger values of ${\it\alpha}$ , the hetons may recombine and new hetons may escape. However, this also implies that ${\it\varrho}$ must be sufficiently large in our parameter space as $d_{y}$ is taken as a fraction of $r$ . Hence, there is a possibility that in these cases the hetons may undergo a baroclinic breakup (albeit partial) in the process. Finally, the hetons may just be strongly unstable and break up before strongly interacting with one another. All of these scenarios are observed in practice. We summarise the general behaviour in figure 10 using the symbols defined in table 1. This table shows that for five nonlinear regimes out of six, finite core effects are essential, and that these regimes cannot be simply reproduced with point vortices.

Figure 10.

Finite core symmetric hetons: regimes in the plane ${\it\varrho}{-}\ell _{z}$ for (a) $d_{y}={\rm\Delta}y/r=0.1$ , (b) 0.25, (c) 0.5, (d) 0.75 and (e) 1. The solid lines indicate the threshold ${\rm\Delta}y/{\rm\Delta}z=1$ . The axis labels are all identical and are defined in (e). The meaning of all symbols is defined in table 1. An underlined symbol indicates that the vortices produced by the first main interaction experience further strong interaction as they reverse velocity or escape.

We next illustrate examples of the typical evolution of the interacting hetons in each category of behaviour. Figure 11 shows the nonlinear evolution of a symmetric pair of hetons with ${\it\varrho}=1.5$ , $d_{y}=0.5$ , $\ell _{z}=1.5$ , such that ${\it\alpha}=0.2<1$ . Equivalent point vortices would reverse trajectory. Moreover, from the stability results in Reinaud (Reference Reinaud2015), each heton should be baroclinically stable in isolation (and if they were at equilibrium). The result first shows that the hetons indeed do not break baroclinically. Moreover, the minimum separation distance reached by two co-rotating poles is $d/r=4.77$ , which is likely to be larger than the margin of stability for vortex merger, even in a static situation. By static situation we refer to an isolated couple of co-rotating poles. In that case, the separation distance is invariant and the instability has all the time to develop. Here, the poles are usually further apart than this minimum distance. For comparison, two spherical vortices only merge (partially) for $d/r=2.7$ , see Reinaud & Dritschel (Reference Reinaud and Dritschel2002). This case is representative of all cases indicated by the symbol ◃ in figure 10.

Figure 11.

Flow evolution for symmetric finite core hetons with ${\it\varrho}=1.5$ , $d_{y}=0.5$ and $\ell _{z}=1.5$ . The figure is ordered in increasing time (a–d). The times displayed are $t=0$ , $t=50$ , $t=100$ and $t=150$ . The vortices are viewed orthographically at an angle of 60° from the vertical. The front and back panels indicate the side of a virtual viewing square box centred around the vortices. The hetons reverse velocity (label ◃ in table 1).

Figure 12.

Flow evolution for symmetric finite core hetons with ${\it\varrho}=2.5$ , $d_{y}=1$ and $\ell _{z}=1.5$ . The times displayed are $t=0$ , $t=37.5$ , $t=75$ and $t=100$ . The hetons reverse velocity after temporary merger (label ▹ in table 1).

We illustrate in figure 12 the behaviour of the hetons when vorticity poles temporarily merge before the hetons reverse trajectory. Here, the vortices are slightly larger ( ${\it\varrho}=2.5$ ) and we take $d_{y}=1$ while retaining $\ell _{z}=1.5$ , so that ${\it\alpha}=1$ . In this case the hetons are offset enough in the vertical to be baroclinically stable. However, now the vortices can get closer together, and within a limit in $d/r$ where they can merge. The vortex structure thus formed is itself unstable and breaks down back into individual vortices which reverse trajectories. Such behaviour is common for like-signed vortices, see Reinaud & Dritschel (Reference Reinaud and Dritschel2002). These interactions are referred to as partial merger and weak interaction. The breaking up of the vortices formed during the merger process is also forced here by the strain field induced by the opposite-signed vortices at different depths. This is generic to all cases indicated by the symbol ▹ in figure 10.

By decreasing $d_{y}$ from the previous case and increasing ${\it\varrho}$ , we now enter the domain where each heton can destabilise baroclinically. Moreover, small $d_{y}$ means that the translation velocity is small. The baroclinic instability can develop before the two hetons are close to one another. Arguably, one could also start the simulation from hetons at larger separation distances to reach a similar effect. Figure 13 shows the symmetric pair of hetons for ${\it\varrho}=3.5$ , $d_{y}=0.1$ and $\ell _{z}=1.5$ , such that ${\it\alpha}=0.14<1$ . For such a large aspect ratio and small horizontal offset, each heton is strongly unstable. The hetons break baroclinically well before getting close enough to strongly interact with the other heton. During the breaking up of the hetons, the poles reorganise into four hetons. The poles of the hetons are placed in such a way that they reverse (at least temporarily) their trajectories and start to move away from one another. It should be noted that each heton breaks into two main secondary hetons (and some debris). This is consistent with a dominant azimuthal mode of instability $l=2$ . It is also arguable that the shear flow interaction between the two hetons is likely to force a deformation exciting mode 2. The fact that the new hetons move away is probably more related to the nature of their formation rather than the fact that ${\it\alpha}<1$ for the original hetons, as the four hetons produced are fundamentally different from the original ones. This example illustrates all cases marked by the symbol $\times$  in table 1.

Figure 13.

Flow evolution for symmetric finite core hetons with ${\it\varrho}=3.5$ , $d_{y}=0.1$ and $\ell _{z}=1.5$ . The times displayed are $t=0$ , 17.5, 50 and 87.5. The hetons break into four hetons which reverse trajectory (label $\times$ in table 1).

We next increase the parameter ${\it\alpha}$ with ${\it\varrho}=4.5$ , $d_{y}=0.25$ and $\ell _{z}=1$ . This example illustrates all cases indicated by the symbol ▵ in table 1. Results are presented in figure 14. Obviously, ${\it\alpha}$ is not sufficient to qualify the behaviour as ${\it\alpha}$ alone is not sufficient to determine the stability of the hetons. In this case ${\it\alpha}=0.56$ and is still less than 1, but the hetons are strongly unstable. The hetons break asymmetrically. The mode of instability is likely to be a mixture of azimuthal modes $l=2$ and $l=3$ . The nature of the instability modes is important for the evolution of the flow. We saw in the previous example that the azimuthal mode $l=2$ was mainly generating four hetons out of the original two. These hetons were at least initially moving away from one another. Here, more secondary hetons are generated. Six principal hetons can be identified, the larger two near the centre. The new main hetons generated by the destabilisation of the original pair are separated further away in the direction $y$ . Shortly after the breakup the four larger vortices are located at $y\pm 0.8$ and $z\sim \pm 0.3$ , giving ${\it\alpha}\sim 2.6>1$ . It should be pointed out that at this stage these vortices are still surrounded by close secondary vortices having a non-negligible influence on the overall dynamics. Nonetheless, the two main hetons eventually escape perpendicularly to their original trajectory.

Figure 14.

Flow evolution for symmetric finite core hetons with ${\it\varrho}=4.5$ , $d_{y}=0.25$ and $\ell _{z}=1$ at $t=0$ , 15, 50 and 162.5. The hetons break into more than four hetons. The main two hetons escape (label ▵ in table 1).

We now consider a situation where the parameter ${\it\alpha}$ is larger while keeping ${\it\varrho}$ as small as reasonably possible and both $d_{y}$ and $\ell _{z}$ reasonably large. We consider ${\it\varrho}=2.5$ , $d_{y}=1$ and $\ell _{z}=1$ . This example illustrates all cases denoted by the symbol ▫ in table 1. This set-up has ${\it\alpha}=1.25>1$ and keeps the hetons in a region of only weak instability in the parameter space. Results are shown in figure 15. Here, as expected, the hetons only shed a small amount of debris before escaping as hetons in a way rather similar to what equivalent point vortices would do.

Figure 15.

Flow evolution for symmetric finite core hetons with ${\it\varrho}=2.5$ , $d_{y}=1$ and $\ell _{z}=1$ at $t=0$ , 17.5, 40 and 62.5. The hetons shed some debris and escape (label ▫ in table 1).

The four previous examples have illustrated what could be considered as a generic behaviour in the range of parameters investigated. However, due to the richness and complexity of the various effects in competition, the dynamics of a pair of symmetric hetons may prove to be far more complex. This is in particular the case when the original hetons are so strongly unstable that their mutual interaction is no longer one that can be associated with a pair of hetons.

For example, we present the case for ${\it\varrho}=5.5$ , $d_{y}=0.1$ and $d_{z}=0.5$ in figure 16. This corresponds to the largest aspect ratio investigated with smaller vertical and horizontal offset considered, and corresponds to the most baroclinically unstable set-up investigated here. In this case the hetons break violently into many secondary structures. It should be noted that in this context not only are the hetons very strongly unstable but they are only very slowly translating towards each other. By the time some of the PV reaches the centre of the domain nothing really resembles hetonic structures, and the flow is mostly turbulent.

Figure 16.

Flow evolution for symmetric finite core hetons with ${\it\varrho}=5.5$ , $d_{y}=0.1$ and $\ell _{z}=0.5$ at $t=0$ , 50, 125, 200 (label ♢ in table 1).

We next propose a general comment on the flow evolution, considering the parameter ${\it\alpha}={\it\varrho}d_{y}/(1+\ell _{z})$ . Large values of $\ell _{z}$ with small values of ${\it\varrho}d_{y}$ may favour reversal of the heton trajectories. However, small values of the product of ${\it\varrho}$ and $d_{y}$ may be obtained by two limits. Taking $d_{y}$ fixed and making ${\it\varrho}$ smaller increases the stability of the hetons. In this case, the hetons may remain compact and behave in a way that is consistent with the point vortex calculation. On the other hand, setting ${\it\varrho}$ and decreasing $d_{y}$ tends to increase the baroclinic instability (as the growth rate of the instabilities is maximum for $d_{y}=0$ ). The subsequent development of the flow may depend on the dominant mode of instability, in particular the number of secondary hetons generated and their locations. The azimuthal mode $l=2$ may favour the formation of four hetons which initially escape. A combination of azimuthal modes $l=2$ and $l=3$ may favour the formation of six strongly asymmetric hetons. Two of them may remain central in $x$ and escape along $y$ .

On the other hand, large values of ${\it\alpha}$ mean large values for the product ${\it\varrho}d_{y}$ and small values for $\ell _{z}$ . The smaller $\ell _{z}$ is, the more unstable the hetons are. To reach large values of the product of ${\it\varrho}\,d_{y}$ we can increase each of them with different effects as discussed above, large ${\it\varrho}$ favouring instability while large $d_{y}$ favours stability.

We finally compute the volumes ( $V_{1}\geqslant V_{2}\geqslant V_{3}\geqslant V_{4}$ ) of the four largest vortices present in the flow at the end of the simulation ( $t=400$ ), and compare them with the initial volume of the poles (all identical initially). The results are presented for $d_{y}=0.1$ in figure 17.

Figure 17.

Ratio of the volumes of the four largest vortices at $t=400$ to the initial volume for $d_{y}=0.1$ , $1.5\leqslant {\it\varrho}\leqslant 5.5$ and $\ell _{z}=0.5$ ( $+$ ), 1 ( $\star$ ), 1.5 (●), 2 (▪), 2.5 (♦). The figure is organised as follows: (a) largest vortex; (b) second largest vortex; (c) third largest vortex; (d) fourth largest vortex.

Qualitatively, the behaviour is similar for all four vortices, and we can clearly see the influence of the baroclinic instability of the poles. In figure 17, $d_{y}=0.1$ is fixed, and the varying parameters are $r/h$ and $\ell _{z}$ . The regions where the vortices lose the most of their volume correspond to the regions in the parameter space where they are expected to experience strong baroclinic instability: large values of ${\it\varrho}$ and small vertical offset $\ell _{z}$ . On the other hand, for small aspect ratio and large vertical offset, the vortices are expected to be only marginally if even affected by the baroclinic instability and the vortices retain most of their PV.

3.4.2. Antisymmetric hetons

We next turn our attention to antisymmetric pairs of hetons. In this case opposite-signed vortices are at the same depth. The first main difference from the symmetric pair is that we have seen that the point vortices always have qualitatively the same behaviour. The poles of the hetons recombine as dipolar structures. Each dipole consists of the two vortices located at the same depth from the original hetons. Now, the parameter ${\it\alpha}$ no longer segregates two different regimes, one for ${\it\alpha}<1$ and one for ${\it\alpha}>1$ . The second main difference is that vortex merger is no longer possible. Indeed the like-signed vortices are located at different depths and do not share any common horizontal cross-section on which they could touch to merge. However, Reinaud & Dritschel (Reference Reinaud and Dritschel2009) showed that the interaction between counter-rotating vortices could lead to destructive interactions where the vortices break into secondary vortices. Finally, as before, the hetons themselves can be subject to the baroclinic instability provided that the poles are wide enough and their horizontal and/or vertical separation is small enough.

Table 2 summarises the types of interactions observed. As in the previous section, a summary of the evolution is provided in the graphs presented in figure 18.

Table 2.

Meaning of the symbols used to describe the outcome of the interactions between two antisymmetric hetons.

Figure 18.

Finite core antisymmetric hetons: regimes in the plane ${\it\varrho}$ – $\ell _{z}$ for (a) $d_{y}={\rm\Delta}y/r=0.1$ , (b) 0.25, (c) 0.5, (d) 0.75 and (e) 1. The meaning of all symbols is defined in table 2. An underlined symbol indicates that the vortices produced by the first main interaction experience further strong interaction as they escape. A barred symbol indicates that the original baroclinic instability only produces a small amount of debris.

For small aspect ratios ${\it\varrho}$ and large vertical offsets $\ell _{z}$ and/or large horizontal offsets $d_{y}$ the hetons are again expected to be stable to baroclinic modes. They are therefore expected to remain compact and, at least at the beginning, act in a way similar to point vortices. For larger values of the aspect ratio ${\it\varrho}$ and/or smaller values of the separation distances $\ell _{z}$ and $d_{y}$ the hetons may be unstable. If the instability is weak, the hetons are likely to only shed some debris instead of breaking into two main secondary hetons. The poles of the hetons may recombine as dipoles which may break into further secondary structures. In the case of strongly unstable hetons, the evolution of the flow is likely to be linked to the spatial structure of the dominant azimuthal mode (mode $l=2$ or $l=3$ in the present study). The azimuthal mode $l=2$ may break each heton into two main secondary hetons, which may translate away (as seen in the previous case). Mode 3 may break each heton into three structures which would lead to an overall different distribution of PV. Two of the hetons may remain close to the centre of the domain and later escape as dipoles. These dipoles may be unstable and further break as they move away. Finally, in the most unstable configuration, the heton may break baroclinically in such a complex way that the subsequent evolution is turbulent-like, as was observed for the symmetric pair of hetons. We now illustrate all of these cases in our parameter space.

We start with a case where we expect the hetons to be stable. For this we consider a small value for the aspect ratio with ${\it\varrho}=1.5$ and a large value of the vertical separation $\ell _{z}=1$ . Here, we take $d_{y}=0.1$ which is the smallest value considered. The results show that, of course, increasing $d_{y}$ goes in favour of the stability of hetons, hence the same behaviour is observed. The flow evolution is presented in figure 19. The hetons get close enough and the interaction between the vortices at the same depth becomes stronger. The poles of the hetons recombine as same-depth dipoles. The dipoles formed move away perpendicularly from the original trajectory. At $t=50$ , the poles of the dipoles are offset by a distance ${\rm\Delta}x/r\simeq 1.58$ . Similar cases are indicated by the symbol ▫ in figure 18. This behaviour appears to be generic to small values of ${\it\varrho}$ and large values of $\ell _{z}$ , i.e. the left top corners of each graph in figure 18.

Figure 19.

Flow evolution for antisymmetric finite core hetons with ${\it\varrho}=1.5$ , $d_{y}=0.1$ and $\ell _{z}=1$ at $t=0$ , $t=95$ , 110 and 120. The hetons interact and the poles escape as dipoles (label ▫ in table 2).

Increasing the aspect ratio to ${\it\varrho}=5$ makes the heton more unstable. However, compensating this by an increase of the horizontal offset to $d_{y}=1$ we find a case where the hetons are only weakly unstable. The flow evolution is presented in figure 20 (here $\ell _{z}=1$ as previously). The poles of the hetons shed a small amount of debris but the two hetons start to interact and recombine as same-depth dipoles before the hetons have the time to fully destabilise. In other words, the time scale associated with the baroclinic instability is very large compared with the time scale associated with the motion of the poles. At $t=125$ , the poles of the dipoles are at a distance of ${\rm\Delta}x/r\simeq 1.06$ and the dipoles do not strongly destabilise. Such an interaction is referred to by the symbol $\times$  in figure 18 and is found for large values of $d_{y}$ only.

Figure 20.

Flow evolution for antisymmetric finite core hetons with ${\it\varrho}=2.5$ , $d_{y}=1$ and $\ell _{z}=1$ at $t=0$ , 20, 30 and 45. The hetons shed some debris and recombine as dipoles which escape (label $\times$ in table 2).

For smaller values of $d_{y}$ but retaining ${\it\varrho}$ small enough and $\ell _{z}$ large enough, we remain in a region of the parameter space where the hetons are either stable or only weakly unstable. However, the dipolar structures generated by the recombination of the poles of the hetons are now unstable. From the point vortex calculation, and in particular figure 8, we see that a decrease in the original horizontal separation distance between the poles of the hetons denoted there ${\rm\Delta}y$ induces a decrease of the distance ${\rm\Delta}x$ separating the two poles of the dipolar structures in the late evolution of the flow. The smaller this distance is the more likely the dipoles are to be unstable. Indeed, the destructive interaction between two counter-rotating vortices is associated with a critical separation distance between the poles, as discussed in Reinaud & Dritschel (Reference Reinaud and Dritschel2009). Such a type of flow behaviour is indicated by the symbol ◃ in figure 18. This regime is typically found along the diagonal region (or upper band for small $d_{y}$ ) in the graphs of figure 18. A typical example is for ${\it\varrho}=3.5$ , $d_{y}=0.75$ and $\ell _{z}=2$ , and is presented in figure 21. The beginning of the flow evolution is similar to the case ▫; however, the two dipoles consist of two close highly deformed vortices which break asymmetrically, producing further tertiary dipolar structures. Just before breaking, the dipoles are closer than in the previous case shown with ${\rm\Delta}x/r\simeq 0.99$ at $t=60$ .

Figure 21.

Flow evolution for antisymmetric finite core hetons with ${\it\varrho}=3.5$ , $d_{y}=0.75$ and $\ell _{z}=2$ at $t=0$ , 50, 100 and 125. The hetons interact and the poles escape as dipoles which undergo a destructive interaction (label $\bar{{\vartriangleleft}}$ in table 2).

By increasing the value of ${\it\varrho}$ we reach regions of the parameter space where the hetons are more unstable. For intermediate values of ${\it\varrho}$ and $\ell _{z}$ the dominant mode of the baroclinic instability has an azimuthal wavenumber $l=2$ , as expected. In this case the original hetons break into two secondary hetons, in a similar way to how the symmetric pair of hetons does. The newly generated secondary hetons move away (at least initially) from the centre of the domain. An example of such behaviour is presented in figure 22 for ${\it\varrho}=5.5$ , $\ell _{z}=2.5$ and $d_{y}=0.1$ . Such cases are observed in the diagonal region in the graphs of figure 18 for small $d_{y}$ only and are indicated by the symbol *. For larger values of $d_{y}$ the configuration is either not unstable enough or the azimuthal mode $l=3$ becomes dominant (cases discussed below). It should be noted that the subsequent evolution (not shown) can be more complex, with the hetons eventually changing trajectory and strongly interacting with one another. Similar late behaviours are observed in the case of the symmetric pair of hetons.

Figure 22.

Flow evolution for antisymmetric finite core hetons with ${\it\varrho}=5.5$ , $d_{y}=0.1$ and $\ell _{z}=2.5$ at $t=0$ , 50, 100, 150. The hetons break baroclinically into four hetons then move away (label * in table 2).

By taking $\ell _{z}$ smaller than in the last case, the mode with azimuthal wavenumber $l=3$ becomes stronger and each heton initially breaks into three secondary hetons, that is six hetons overall. Two of them (the largest ones) remain at the centre of the domain and recombine as dipoles which move away as point vortices would do. An example is provided in figure 23 with ${\it\varrho}=5.5$ , $\ell _{z}=0.5$ and $d_{y}=0.25$ . The original pair of hetons has broken into six hetons by $t=50$ . The two central ones exhibit poles well separated in the horizontal. They first converge towards one another until the poles at the same depth recombine as dipolar structures which move away mostly in the $y$ direction. The secondary hetons originally move away from the centre but exhibit a more complex later evolution. These interactions are indicated by the symbol ▵ in figure 18 and are generic to small $\ell _{z}$ and large ${\it\varrho}$ , i.e. the bottom right corner of the graphs of figure 18. The loss of symmetry in $x$ and $y$ comes from the low-amplitude numerical noise and the development of turbulence.

Figure 23.

Flow evolution for antisymmetric finite core hetons with ${\it\varrho}=5$ , $d_{y}=0.25$ and $\ell _{z}=0.5$ at $t=0$ , 12.5, 25, 50, 75, 100, 135.5 and 190. The hetons break baroclinically into two dipoles and four hetons. The dipoles eventually escape (label $\bar{\triangle }$ in table 2).

Finally, as was the case for the symmetric pair of hetons, the case with ${\it\varrho}=5.5$ , $d_{y}=0.1$ and $\ell _{z}=0.5$ (not presented), corresponding to the most baroclinic unstable configuration, exhibits a turbulent-like behaviour and is indicated by a ♢ in figure 18.

It should be noted that, in several simulations, the poles of the hetons become tilted vertically, or elongated asymmetrically at the upper and lower surfaces, and that finally, via filamentation, many fragments are shed from these poles. These fragments not only have smaller horizontal size than the original hetons, but also shorter vertical scales.