Orbits of a system of three point vortices and the associated chaotic mixing

Vortex motion is a fundamental aspect of high Reynolds number fluid flows. In two spatial dimensions, the situation is considerably simpler due to material conservation of vorticity in (ideal) inviscid, incompressible flows. Yet, such flows can still exhibit highly complex nonlinear behavior, whether chaotic or fully turbulent (e.g., Dritschel , 2008). Even in relatively simple flows, many bulk properties, such as (enstrophy) dissipation rates, tracer transport, and mixing efficiency, remain hard to quantify.

One of the simplest and well-studied problems going back to Kirchhoff (1876) is the dynamics of a system of “point vortices,” singular distributions of vorticity having vanishing area but finite circulation. This ansatz reduces the dynamical system for the flow from an infinite-dimensional to a finite-dimensional Hamiltonian system. Excellent reviews may be found in Newton (2001), Aref (2009), and Newton (2014). The representation allows analytic progress in many cases and has also inspired a whole class of numerical point vortex methods (e.g., Harlow, 1964; Christiansen and Zabusky, 1973; Cottet and Koumoutsakos, 2000; and Aref, 2010).

Here, we study the dynamics of three point vortices together with passive particles to study the associated mixing. The dynamics of three point vortices is relatively simple (Aref, 1979), since almost all orbits (for vortices having arbitrary circulations) are periodic in an appropriate rotating frame of reference (Kuznetsov and Zaslavsky, 1998). The exception is “vortex collapse” (Gröbli, 1877 and Aref, 1992) in which the three vortices approach a common position in finite time. This only occurs for certain special conditions on the vortex circulations and initial positions. The three-vortex system is special, lying as it does between the essentially trivial dynamics of the two-vortex system and the generally chaotic dynamics of the $N$-vortex system with $N≥4$ (e.g., Neufeld and Tél, 1998).

While the three-vortex problem is relatively simple, to the authors’ knowledge, no complete description of the motion for arbitrary vortex circulations exists in the literature. Here, we show that the motion can be surprisingly complex and varied, despite being “simply” periodic. Vortices may approach either co-linear or equilateral relative equilibria, thereby greatly increasing the orbital period of the vortices. The stability properties of such equilibria are key to understanding the observed complex behavior across parameter space.

The fluid mixing induced by a general time-dependent periodic flow is often chaotic in nature (Ottino, 1989), with the separation of passive particles diverging exponentially and exhibiting a sensitive dependence on initial conditions. This too is a well-studied problem, but to the authors’ knowledge has not been examined in the context of the motion of three point vortices, except for one special case (Kuznetsov and Zaslavsky, 1998). Here, we show that, away from the immediate vicinity of each vortex, three vortices in general provide an efficient mixing mechanism for the fluid.

The structure of the paper is as follows. Section II briefly reviews the dynamical system and the specific setup of the problem studied, namely, an initially co-linear array of vortices with arbitrary circulations. Section III studies relative equilibria and their stability properties, including possible transitions between different equilibria (when slightly perturbed). Section IV illustrates a sample of the varied periodic orbits exhibited by the vortices. Section V examines the mixing of one or more passive particles, utilizing Poincaré sections and finite-time Lyapunov exponents to quantify the mixing. Finally, Sec. VI offers a few conclusions and suggestions for future research.

Steady states of system (1) exist for a suitable choice of reference frame $Ω$ and have been documented in Aref (2009). They fall into two classes: co-linear configurations in which the vortices lie on a straight line and configurations in which the vortices lie on the vertices of an equilateral triangle. Equilateral triangular equilibria occur for any combination of the vortex strengths.

Figure 1 shows how the vortex strengths required for an equilibrium state vary with the parameters $Ω$ and $a$⁠. Over the range plotted (and generally for $Ω<4$⁠), $κ 2$ is everywhere positive. When $Ω=4$ and $a=1$⁠, $κ 2$ vanishes. In general, $κ 2<1$ and limits to $1$ as $a→∞$⁠. The other two vortices can have both negative and positive strengths. On the line $Ω=1$⁠, we have $κ 0=0$⁠, and the minimum value of $κ 0=− 1 3$ occurs when $Ω=0$ and $a=1$⁠, while the maximum, $κ 0=1$⁠, occurs when $Ω=4$ and $a=1$ (larger values occur for $Ω>4$⁠). The situation for $κ 1$ is more complicated. It reaches a maximum of $2 3$ when $Ω=0$ and $a=1$⁠, and a minimum of approximately $−0.3848$ when $Ω=4$ and $a≈3$⁠. For large $a$⁠, both $κ 1$ and $κ 0$ tend to zero.

Linear stability of the above vortex equilibrium is determined by linearizing (1) about the steady co-linear array above. Again, since $∑ j κ j x j=0$⁠, we need only the two pairs of equations for $j=0$ and $1$⁠. Linearization results in a $4×4$ matrix eigenvalue problem (details omitted) for the eigenvalue $σ$⁠. If the real part, $σ r$⁠, is positive, the array is linearly unstable.

Note that $Ω c(a)$ decreases monotonically from $4$ when $a=1$ to $3$ as $a→∞$⁠. Then, the growth rate is given by $σ r=σ= c 2$⁠. This is plotted in Fig. 2 over a moderate range of $Ω$ and $a$⁠. The maximum growth rate occurs when $Ω=0$ and $a=1$⁠, and there $σ r=4$⁠. For these parameter values, both $κ 1$ and $κ 2$ vanish (i.e., vortices 1 and 2 become passive particles). In general, none of the vortex strengths vanish along the margin of stability, $Ω= Ω c(a)$⁠, for $a>1$⁠.

All co-linear states are unstable when $Q>0$⁠. This is consistent with Aref (2009) (see Sec. VII B and Table 4 therein).

We next consider the conditions for which a general (unsteady) co-linear array may evolve asymptotically toward a triangular equilibrium. Such a transition will be possible only if both the angular impulse and the energy of the two states are the same.

One can show that $F=0$ if $κ 0=0$⁠, $κ 1=0$⁠, or $κ 2=0$⁠. Then, one vortex (at least) is passive, a simple case that we do not consider further. The general case for non-vanishing circulations cannot be solved analytically, so a numerical method was developed to determine all non-trivial solutions of $F( κ 1; κ 2,a)=0$ for $κ 1$⁠, given $κ 2$ and $a$⁠. The results are shown in Fig. 3; each black curve corresponds to a single value of $a$ and gives a relation between $κ 1$ and $κ 2$ for which $F=0$⁠.

Also shown in the figure is an ellipse (red) denoting values of $κ i$ for which $Q=0$⁠, with $Q>0$ in the interior, $Q<0$ in the exterior. Both $J=0$ and $E=0$ when $Q=0$⁠.

The curves denoting $F=0$ are not straight lines, despite appearances, except when $a=1$⁠. In this special case, the transition curve is $κ 2=− κ 1$⁠, and moreover, $J=Q=− κ 1 2$ and $E=0$ (for both the co-linear and triangular states). Otherwise, for $a>1$⁠, the curves bend, especially near the ellipse $Q=0$⁠.

None of the transition curves cross inside the ellipse, where $Q>0$⁠. Only the curve for $a=1$ is tangent to the ellipse at the exceptional point $κ 1= κ 2=0$ where the problem reduces to a single vortex and two passive particles. Otherwise, for each $a>1$⁠, there are two disjoint transition curves that appear to originate at infinity and end on the ellipse $Q=0$⁠. An analysis for $| κ 2 |≫1$⁠, taking $κ 1∼c κ 2$ for some constant $c$⁠, shows that $J$⁠, $Q$⁠, and $E$ are all proportional to $κ 2 2$ (neglecting lower order terms). Then, requiring $J/Q=exp⁡(−E/Q)$ yields an implicit equation for $c$ in terms of $a$ alone. This is expected to be the slope seen in Fig. 3 for each value of $a$⁠.

The conclusion is that transitions between steady co-linear states and triangular states can only occur if $Q=0$⁠, i.e., if $J=E=0$⁠. Of course, if both states are steady, it is impossible to evolve from one to the other, but if one allows small perturbations, a near transition may occur.

The dependence of $Ω$ and the orbital frequency $Ω p=2π/T$ on $κ 1$ and $κ 2$ are shown in Fig. 4 for a selection of values of parameter $a$⁠. While both $Ω$ and $Ω p$ vary smoothly with $κ i$ over much of the parameter space, there are many lines across which one or both exhibit a sudden change in gradient. Some of these can be identified with the lines of equilibria already discussed, such as the unstable branch of co-linear equilibria (red line), but not the stable branch (green line). Note also the sharp variation across the yellow line where transitions are possible between co-linear and triangular states (see Sec. IV for details). On the whole, the dependence on the vortex strengths is remarkably complicated. As elaborated below, this complexity is partly due to orbits passing close to other co-linear equilibria, different in form from the generally unsteady co-linear initial conditions.

When either $κ 1=0$ or $κ 2=0$⁠, one may notice that $Ω$ is not constant [i.e., either $( 1 + a ) 2/ a 2$ or $( 1 + a ) 2$] for all values of the other vortex strength, yet $Ω=1$ when $κ 0=0$ (corresponding to $κ 1+ κ 2=1$⁠). In these cases, the analytical solution is known, suggesting there is an error in the numerical solution. However, it turns out that there is an infinite multiplicity of periodic orbits for every set of parameters $(a, κ 1, κ 2)$⁠: for every solution with rotation rate $Ω$⁠, there are countably infinite others with rotation rates $Ω+n Ω p$ for any integer $n$ (all have the same $Ω p$⁠). Recall that $Ω=θ/T$⁠, where $θ$ is the angle through which the line connecting vortex 1 and vortex 2 rotates by time $t=T$⁠. We can also rotate by $θ+2πn$ for any integer $n$ to obtain a different closed orbit. If we choose $n=1$ in a certain region of parameter space [e.g., outside the “petal” region lying mainly in the first quadrant for $a=1$ in Fig. 4(a)], we can ensure $Ω= ( 1 + a ) 2/ a 2$ along $κ 1=0$ and $Ω= ( 1 + a ) 2$ along $κ 2=0$⁠. However, we then lose the property $Ω=1$ along $κ 0=0$⁠. Since the lines $κ 1=0$ or $κ 2=0$ intersect $κ 0=0$ (i.e., $κ 1+ κ 2=1$⁠), the analytical solutions are themselves inconsistent at this point. (In the analytical solutions, only the passive vortex moves in the rotating frame of reference, while the other two are stationary.) The conclusion is that orbits are not unique.

This is illustrated in Fig. 5 for the special case $a=1$⁠, $κ 1=0$⁠, and values of $κ 2$ straddling the edge of the “petal” region. Between $κ 2=0.306155$ and $0.306156$⁠, the orbit changes discontinuously (due to the existence of an unstable co-linear equilibrium in this gap, see below). The middle panel adds $Ω p$ to $Ω$ at $κ 2=0.306156$ to maintain $Ω=4$ across the discontinuity. The right panel, having a much smaller $Ω$⁠, is the result of using the rotation of the line connecting vortex 1 and vortex 2 to determine $Ω$ (the default approach here). While the middle panel might be preferred over the right one, adding $Ω p$ to $Ω$ in this region is inconsistent with the analytical solution for $κ 0=0$⁠. Moreover, it is difficult to do numerically, especially for $a>1$⁠, since the lines of discontinuity are not always straightforward to identify.

The vicinity of the margin of stability of co-linear equilibria is shown in Fig. 6. The period of the orbits and background rotation depends sensitively on the direction of approach in the $κ 1$– $κ 2$ plane. Along the (red/green) line of equilibria, $Ω p$ in fact drops to zero, the steady state corresponding to an orbit of infinite period. It does so smoothly, but this only becomes apparent at a still higher magnification.

To illustrate the complexity of the orbital frequencies, we show in Fig. 7 a cross section along the line $κ 1=−0.1$ for the case $a=1.5$⁠. The unstable equilibrium at $κ 2=0.37192982$ is indicated by the dashed red line, while the point where a transition may occur to an equilateral triangular equilibrium at $κ 2=−0.16504433$ is indicated by the dashed magenta line. At both of these points, $Ω p=0$⁠. However, numerically, it is impossible to reach this limit (this would require an infinite integration time). There are three other points where $Ω p$ appears to dip toward zero. These occur at $κ 2=0.10748024$⁠, $0.12356649$⁠, and $0.49467082$⁠, referred subsequently as points “A,” “B,” and “C.” There is also a local maximum in $Ω p$ at $κ 2=0.11187662$⁠, but despite appearances, this is a smooth maximum with continuous $∂ Ω p/∂ κ 2$⁠. Moreover, the maximum does not coincide with a nearby maximum in rotation frequency $Ω$⁠.

In summary, while all initially unsteady co-linear states exhibit periodic orbits, their properties can be surprisingly rich. Depending on parameters, these orbits may pass close to triangular equilibria or other co-linear equilibria, leading to sharp dips in the particle rotation frequency $Ω p$ and related sharp maxima or minima in $Ω$⁠. This explains the non-smooth variation of $Ω$ and $Ω p$ seen in Figs. 4 and 6. In particular, the edge of the small lobe seen for small positive $κ 1$ and $κ 2$ in the plot for $a=1$ coincides with co-linear equilibria. The lobe is distorted for $a>1$ but continues to be associated with co-linear equilibria.

Figure 9 shows the vortex trajectories for a selection of orbits for the case $a=1.5$ and $κ 1=−0.1$⁠, illustrating, in particular, how the behavior changes across the values of $κ 2$ discussed. A video starting from $κ 2=−0.9995$ shown in Fig. 10 (Multimedia available online) scans through $κ 2$ at variable speed to focus on the sharp transitions in behavior seen in Fig. 7.

The first two panels (reading from the upper left, across, then downward) correspond to cases on either side of the point where a transition to an equilateral equilibrium is possible. The orbit changes discontinuously across this point and has a completely different topology. For $κ 2=0$ (upper right panel), the topology remains the same but now the two active vortices move on circular trajectories, while the passive vortex moves on a complicated trajectory. Increasing $κ 2$ further (left panel of second row), the trajectory of vortex 2 (red) simplifies to a banana-shaped loop, while that of vortex 1 (blue) exhibits a fold. A slight increase in $κ 2$ brings us across point A where a transition to a distinct co-linear equilibrium is possible, and the trajectories change discontinuously. The inner portions of the trajectories are closely similar but at the higher value of $κ 2$ there is a wide excursion of vortices 1 and 2 to the left. Moving across to $κ 2=0.11187662$ (a smooth local maximum in $Ω$ not associated with any transition), the trajectories generally simplify, with vortex 1 (blue) moving on a simple loop. In the next row, first two panels, we show trajectories either side of point B, where another transition to a distinct co-linear equilibrium is possible. The topology changes dramatically, with the vortices exhibiting many loops before forming a closed orbit. Note also the change in the axis scale: the vortex trajectories here become much more extensive in space at this point. A further increase to $κ 2=0.1387$⁠, where $Ω$ and $Ω p$ both exhibit a strong minimum, results in simpler trajectories though they are still remarkably complicated. Further increasing $κ 2$ brings us across point C where another transition to a distinct co-linear equilibrium is possible (bottom left two panels). Again, there is an abrupt change in topology, simplifying with increasing $κ 2$⁠. This continues to the final panel for $κ 2=0.5$⁠, which is not a special point. Now vortex 1 (blue) exhibits a single fold in its trajectory, while the other vortices exhibit simple closed orbits.

We next turn to the motion of one or more passive particles moving in the flow field of the three vortices. It is well known that particle motion can be chaotic when subject to periodic forcing (see, e.g., Ottino, 1989). In the present context, the periodic motion of the three vortices serves as the “forcing.”

We next use this to study the dynamics of one or a collection of passive particles and to understand how efficiently the periodic motion of three point vortices can mix a tracer. We employ two common methods of analysis: Poincaré sections and Finite-Time Lyapunov Exponents (FTLEs) (see, e.g., Ottino, 1989, and references therein).

The Finite-Time Lyapunov Exponent (FTLE) provides a complementary, quantitative measure of mixing. This is here obtained by evolving initially closely separated particle pairs, located initially on a regular grid, for a moderate number of periods. Here, we take a grid of size $500×500$⁠, giving $500000$ passive particles overall. The extent of the grid is chosen to correspond to that shown in the Poincaré sections to facilitate comparison. The particle pairs are initially separated by a distance $d$ of $d(0)= 10 − 7$ and centered on grid points. Relative to each grid point, we place the two particles at positions $±d(0)(cos⁡ϕ,sin⁡ϕ)/2$ and consider 18 values of $ϕ$ separated by $10 °$⁠. For each value of $ϕ$⁠, and for each particle pair, we next integrate one period $T$ and determine $S=log⁡(d(T)/d(0))$⁠. The separation distance of the pair is then reduced back to $d(0)$ while keeping the same orientation and center of the pair. We then integrate a further period and accumulate the value of the “stretch” $S$⁠. We repeat this until we reach $t= t max=10 T$ (a limit imposed by the high cost of evolving so many particles). Finally, we define the FTLE as the maximum over $ϕ$ of $S/ t max$⁠, denoted $λ$⁠. This measures the maximum stretching rate of a particle pair initiated from some points in the space. Note that the regions within a distance $d i=0.04 | κ i | / Ω max$ to each point vortex are masked out of the images shown below, to enable a reasonable time step for the remaining particles.

We begin by considering two cases near a stable co-linear equilibrium. Then, the underlying periodic motion is weak in the sense that the vortices remain close to their original positions (in the rotating frame of reference). We may regard this as a weakly forced “oscillator,” and we can anticipate that any chaotic motion will be limited to regions around the separatrices connecting stagnation points (see Polvani and Wisdom, 1990 for a related fluid dynamics example). The two cases chosen lie either side of a stable equilibrium point at $a=1.5$⁠, $κ 1=−0.24$⁠, and $κ 2=0.2785965$⁠, approximately. The Poincaré sections and maps of the corresponding FTLEs are shown in Fig. 12. Unmixed (regular) regions in the Poincaré sections exhibit distinct curves along which the motion is quasi-periodic. The mixing regions on the other hand exhibit a blurry appearance, indicative of chaotic motion. The FTLEs, in particular, indicate that the regions around the separatrices are associated with the most intense stretching. Note, the regions outside the separatrices appear to exhibit some mixing as $λ>0$ there, but this is spurious, as the integration time $t max$ is finite. Only in the limit $t max→∞$ would one expect $λ→0$ in these regions.

We next study mixing when the vortices are far from equilibrium. One may anticipate that mixing will be more vigorous in such cases due to the complex time dependence of the flow induced by the vortex trajectories. We focus on three pairs of cases, each involving point vortex trajectories that pass close to an equilibrium configuration.

The first pair of cases straddle the transition point to an equilateral equilibrium at $κ 2=−0.16504433$⁠, approximately. The Poincaré sections and FTLE maps for these cases are shown in Fig. 13. As seen previously in Fig. 9, the vortex trajectories in these two cases are dramatically different. Chaotic mixing is widespread in both cases, in stark contrast to the previous cases involving vortex motion close to a stable equilibrium. Mixing is more vigorous at the slightly larger value of $κ 2$⁠, evidently related to the more complex vortex trajectories in this case. Though more prominent at the smaller value of $κ 2$⁠, in both cases crescent-shaped islands of regularity appear in the chaotic sea, as well as more circular regions surrounding the initial positions of each vortex. However, overall, the wide excursions of the vortex trajectories in these cases enable much more efficient mixing than seen previously for vortices near a stable equilibrium configuration.

The second pair of cases straddle the transition point “A” to a distinct co-linear equilibrium at $κ 2=0.107480245$⁠, approximately. The Poincaré sections and FTLE maps for these cases are shown in Fig. 14. Referring back to Fig. 9, the trajectories differ mainly by a large excursion to the left of vortices 1 and 2 (a discontinuous change in topology crossing the transition point). Mixing is again widespread, and even more vigorous than in the previous pair of cases, but the details differ. The islands of regularity appear in different places. For example, the large island to the right of the initial position of vortex 2 evident for the smaller value of $κ 2$ splits into two smaller islands for the larger value of $κ 2$⁠. Mixing appears more intense for the smaller value of $κ 2$⁠, but there could be more mixing beyond the domain of view for the larger value of $κ 2$ due to the wide excursions of the vortex trajectories in this case.

The third and final pair of cases straddle the transition point “C” to a distinct co-linear equilibrium at $κ 2=0.494670825$⁠, approximately. The Poincaré sections and FTLE maps for these cases are shown in Fig. 15. In these cases, the Poincaré sections show virtually no crescent-shaped islands of regularity, except at the very edge of the section for the larger value of $κ 2$ (and these are small). The vortex trajectories differ substantially in these two cases (see Fig. 9), yet the mixing pattern is largely similar. The main difference occurs near the vortices: no regular zones are evident near the vortices at the smaller value of $κ 2$ (recall the near-vortex zones are masked out), but substantial regular zones surround the vortices at the larger value of $κ 2$⁠. This is likely due to the simpler form of the trajectories in this case, and the fact that the trajectories remain closer to the original vortex positions over a greater portion of the orbit.

One of the simplest problems in vortex dynamics concerns the motion of point vortices in planar, two-dimensional flows. This is a well-studied problem going back to Kirchhoff (1876) and is especially attractive as the equations of motion constitute a finite Hamiltonian system. A single vortex does not move, two vortices co-rotate and exceptionally translate, three vortices move periodically on closed orbits (and may exceptionally collapse to a single point in finite time; Gröbli, 1877 and Aref, 1979, 1992), and more than three vortices can exhibit chaotic motion (Newton, 2001, 2014). The problem clearly gets more challenging with increasing numbers of vortices, yet to some extent the three-vortex problem has been overlooked, perhaps due to the lack of chaotic motion, or due to the fact that vortex collapse may be analyzed readily. Some partial results are available (Aref, 1979; Aref, 1992; Kuznetsov and Zaslavsky, 1998; Newton, 2001; and Newton, 2014), but a complete exploration of the rich parameter space appears not to have been attempted before.

The present work has demonstrated that just three vortices can exhibit surprisingly complex behavior, even if it is well known that the motion must be periodic (barring vortex collapse). The complexity arises due to the existence of surfaces of equilibria slicing through the three-dimensional parameter space. These equilibria take the form of either a co-linear array or an equilateral triangle, irrespective of the vortex circulations. Passing through parameter space, their existence results in bifurcations in the orbit patterns of the vortices, when starting from a general, unsteady co-linear configuration.

We have further explored the mixing of passive particles surrounding the vortices, extending the work of Kuznetsov and Zaslavsky (1998) who studied mixing around vortices having identical circulations. We used both Poincaré sections and finite-time Lyapunov exponents to explore the mixing both qualitatively and quantitatively. In general, we find that mixing is widespread and efficient over a region comparable in size to the maximum excursion of the vortices. Only when vortices are near a stable equilibrium is mixing tightly confined to the separatrices connecting the stagnation points of the flow, as has been found previously (Polvani and Wisdom, 1990). In conclusion, the periodic motion of three vortices often provides an efficient way to mix tracers in their local environment.

There are several avenues to explore in future work. It would be straightforward to consider related dynamical systems more relevant to vortex motion in geophysical fluid dynamics. One such model is the quasi-geostrophic shallow-water model, applicable to a thin layer of fluid with a free surface subject to rotation and gravity (see Vallis, 2017, and references therein). This model involves a single additional parameter $γ=f/c$⁠, where $f$ is the Coriolis frequency and $c= g H$⁠, with $g$ being the gravity and $H$ the mean fluid depth. Another model one could study is the surface quasi-geostrophic model, describing the surface buoyancy field (in the oceans) or the surface (potential) temperature field (in the atmosphere), in certain asymptotic limits (see Reinaud , 2022, and references therein). This model has no additional parameter but involves a shorter-range Green function similar to that found in three-dimensional flows. Both models admit point vortex formulations. Moreover, there are extensions of point vortex dynamics to the three-dimensional quasi-geostrophic equations (see, e.g., Reinaud (2022)). This involves a much richer parameter space since the vortices can be offset vertically (yet remain forever in their original horizontal plane). All the models described above have the property that three vortices move on periodic orbits. To date, we know very little about these orbits, and even less about the mixing of passive tracers in such flows.

The study of mixing induced by such vortex systems is more than of purely academic interest. In many geophysical situations, mixing and transport is dominated by the motions of a small number of coherent, quasi-two-dimensional vortices, possibly interacting with a larger-scale, externally forced background flow. Examples range from sub-mesoscale oceanic vortices to the coherent vortices on Jupiter, co-existing with a stable background jet flow in midlatitudes, or in relative isolation in the polar cap regions (Adriani , 2018). Understanding the behavior of mixing in the simpler point vortex system, particularly near stable equilibrium configurations, could shed light on the underlying mixing properties of such more complex systems, without the need for a complete representation of the flow field.

The authors have no conflicts to disclose.

All authors played an equal part in the development of the research, its analysis and interpretation, and in preparing the manuscript.

David G. Dritschel: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Gregory N. Dritschel: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Richard K. Scott: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.