Hamiltonian dynamics of two same-sign point vortices

The mathematical underpinnings of the dynamics of pointlike vortices in classical fluids were established in the nineteenth century Helmholtz (1867); Kirchhoff (1876); Routh (1880); Lin (1941a, b). In the 1940s, Lars Onsager realized that the equations of motion describing such vortices in two dimensions are mathematically equivalent to Hamilton’s equations of motion for particles moving in one spatial dimension and that a large collection of them could be treated with the machinery of statistical mechanics Onsager (1949); Eyink and Sreenivasan (2006). His motivation was to develop understanding of fluid turbulence by describing the statistical properties of the turbulent fluid in terms of a collection of point-vortex particles. He predicted that in turbulent two-dimensional (2D) fluid flows, the vortices, rather than being randomly distributed throughout the fluid, should arrange into giant clusters of the order of the size of the system Onsager (1949). Although such Onsager vortices appear to be prevalent in many classical fluid flows ranging from large-scale ocean currents and planetary atmospheric flows Miller et al. (1992) to thin liquid films Tabeling et al. (1991); Paret and Tabeling (1998), in the classical context the point-vortex model is often dismissed as an oversimplified description of real fluids.

In the context of quantum mechanics, vortex-filament models in three-dimensional (3D) systems Schwarz (1985) and the corresponding point-vortex models in two dimensions have been applied extensively to superfluid helium, where the quantization of circulation provides justification for treating them within the point-vortex approximation, as already noted by Onsager Onsager (1949). However, quantitative comparisons of vortex dynamics with experiments have been difficult to achieve due to the considerable challenges of imaging individual vortex lines with Ångström-scale core sizes Bewley et al. (2006); Fonda et al. (2014). A renewed interest in the point-vortex models Pointin and Lundgren (1976); Aref (1983); Campbell and O’Neil (1991); Viecelli (1995) has emerged in superfluids of ultracold atomic gases Freilich et al. (2010); Kuopanportti et al. (2011); Middelkamp et al. (2011); Navarro et al. (2013); Moon et al. (2015); Torres et al. (2011a, b) where the vortex cores can be sufficiently large to be resolved optically even in situ Wilson et al. (2015) and their circulation direction might be detected using, e.g., the vortex gyroscope imaging technique Powis et al. (2014).

Recent experiments Freilich et al. (2010); Kuopanportti et al. (2011); Middelkamp et al. (2011); Navarro et al. (2013); Moon et al. (2015) have shown that the vortices in the superfluid gases are amenable to the point-vortex approach, opening up further possibilities for quantitative studies of vortex dynamics such as Kelvin waves Bretin et al. (2003); Fetter (2004); Simula et al. (2008, 2008); Koens et al. (2013), Crow instabilities Crow (1970); Berloff and Roberts (2001); Simula (2011), and Tkachenko vortex waves Tkachenko (1966a, b); Sonin (1987); Coddington et al. (2003); Baym (2003); Simula et al. (2004, 2005); Simula and Machida (2010); Simula (2011, 2013a, 2013b). Understanding such few-vortex phenomena forms the basis for solving more complex problems involving vortices; a topical example is quantum turbulence in 2D systems Kobayashi and Tsubota (2005); Parker and Adams (2005); Bradley and Anderson (2012); Neely et al. (2013); Reeves et al. (2013); Kwon et al. (2014); Reeves et al. (2014); Billam et al. (2014); Simula et al. (2014); Stagg et al. (2015); Billam et al. (2015) and the emergence of Onsager vortices and negative Boltzmann temperatures for vortices in disk-shaped traps Simula et al. (2014); Groszek et al. (2016).

In 2D superfluids, strong turbulence is tantamount to chaotic dynamics of the quantized vortices in the system. For three or more vortices in an effectively 2D Bose–Einstein condensate (BEC) confined by a cylindrically symmetric, highly oblate harmonic trap, the vortex dynamics can become chaotic. However, a two-vortex problem is integrable due to two conservation laws related to the energy and angular momentum of the system. As a precursor to studying the onset of turbulence, we focus here on the problem of two vortices of the same circulation. Our work is motivated by recent BEC experiments that discovered a bifurcation of rigidly rotating stationary states in the two-vortex case Navarro et al. (2013). Here we find that the phase space of the two-vortex system is divided into two topologically distinct regions corresponding to two radically different types of two-vortex motion: In one region the individual orbits of the two vortices overlap, whereas in the other region the orbits never cross each other.

Figure 1 summarizes the main findings. The wall that divides the two-vortex phase space is shown in Fig. 1(a), indicating a sharp transition between the two types of motion as the initial vortex positions are varied. This transition boundary is impenetrable in the sense that any two-vortex state not located on the boundary at any one time will always remain at that side of the boundary as the system evolves in time. Furthermore, we find that the asymmetric rigidly rotating states investigated in Ref. Navarro et al. (2013) and shown here in Fig. 1(b) are not related to this change in the topology of the accessible phase space.

The remainder of this article is organized as follows. In Sec. II, we outline the point-vortex model we use for describing the effectively 2D dynamics of vortices in a highly oblate harmonically trapped BEC. Section III presents our results using two complementary descriptions: the position-space representation and the velocity-space representation. These are used for revealing the phase-space boundary that separates the system dynamics into two topologically distinct classes. The paper is concluded with a discussion in Sec. IV.

We restrict our attention to effectively 2D dynamics where quantized vortices in a harmonically trapped BEC can be modeled as point particles moving in the $xy$ plane. We take the velocity of each vortex to be the sum of two independent contributions: (i) solitary orbital motion due to an axisymmetric harmonic trap, and (ii) motion in the flow field of all other vortices. Thus, we consider the following system of $N$ coupled ordinary differential equations Middelkamp et al. (2011); Navarro et al. (2013); Moon et al. (2015); Torres et al. (2011a):

where $z_{k}=x_{k}+iy_{k}$, $\left(x_{k},y_{k}\right)$ are the position coordinates of the $k$th vortex, $s_{k}\in\mathbb{Z}$ is its circulation number, and $k\in\left\{1,\dots,N\right\}$. The vortices are constrained to move within the Thomas–Fermi radius of the condensate, $R>|z_{k}|$. Furthermore, $\Omega_{0}$ is the orbital angular frequency of a solitary unit-strength vortex infinitesimally close to the trap center Freilich et al. (2010); Fetter and Svidzinsky (2001), and $\Omega_{\mathrm{int}}$ is an angular frequency determining the effective strength of the vortex–vortex interaction Navarro et al. (2013).

Equations (1) are equivalent to Hamilton’s equations of motion,

corresponding to the Hamiltonian

Therefore, the point-vortex model has the peculiar feature that the coordinate space $\{(x_{k},y_{k})\}$ can also be interpreted as the Hamiltonian phase space of the system. In addition to $H$, the model also has another integral of motion, $L=\sum_{k}s_{k}|z_{k}|^{2}$, due to the underlying rotational symmetry. In analogy to point particles, this quantity is referred to as the point-vortex angular momentum; note, however, that it should not be confused with the orbital angular momentum that the vortex induces in the flow of the surrounding superfluid. In fact, whereas the point-vortex angular momentum $s_{k}|z_{k}|^{2}$ of a single vortex increases as the vortex moves away from the symmetry axis, the angular momentum of the superfluid decreases under such circumstances.

By denoting $z_{k}=r_{k}\exp\left(i\theta_{k}\right)$, we obtain the radial and angular vortex velocities $\dot{r}_{k}$ and $\dot{\theta}_{k}$, respectively, from the Cartesian velocities as

The velocity space $\{(\dot{r}_{k},\dot{\theta}_{k})\}$ turns out the be extremely useful for representing the vortex dynamics in subsequent analysis (Sec. III.2).

When is the point-vortex model applicable? In 3D BECs, vortex filaments can be described as point particles as long as they remain straight and parallel to one another, rendering their dynamics effectively 2D. Hence, the validity of Eqs. (1)–(4) extends beyond the regime of quasi-2D BECs confined extremely tightly in the $z$ direction. For instance, although the BECs of Refs. Freilich et al. (2010); Middelkamp et al. (2011); Navarro et al. (2013) were 3D, very good agreement with the point-vortex description was obtained, likely because the trapping along the $z$ axis was strong enough to limit vortex motion to the $xy$ plane and suppress vortex bending and tilting away from the $z$ direction. Aspects of condensate dimensionality in regards to vortices and Kelvin waves were studied theoretically in Ref. Rooney et al. (2011), further indicating that sufficiently oblate yet still 3D BECs may be considered 2D as far as vortex dynamics and quantum turbulence are concerned.

From here on, we focus on a system of two vortices with equal circulations, setting $N=2$ and $s_{1}=s_{2}=1$. We measure lengths in units of $R$ and time in units of $\Omega_{0}^{-1}$. Up to a rotation of the coordinate system, all possible two-vortex configurations are spanned by three variables: the angle $\phi=\textrm{tan}^{-1}\left(r_{2}/r_{1}\right)$, the point-vortex angular momentum $L=r_{1}^{2}+r_{2}^{2}$, and the azimuthal angle $\theta_{21}=\theta_{2}-\theta_{1}$ between the two vortices. Recently, Navarro et al. Navarro et al. (2013) investigated this system both theoretically and experimentally for two-vortex configurations with $\theta_{21}=\pi$. They demonstrated that when $\Omega_{\mathrm{int}}/\Omega_{0}=0.1$ (a value that we adopt throughout this work), the system exhibits a pitchfork bifurcation at $L=L_{\mathrm{cr}}\approx 0.273\,R^{2}$ that induces the emergence and stabilization of asymmetric ($r_{1}\neq r_{2}$) rigidly rotating vortex configurations and the destabilization of symmetric ($r_{1}=r_{2}$, i.e., $\phi=\pi/4$) rigidly rotating states at $L>L_{\mathrm{cr}}$.

We have solved Eqs. (1) numerically using the ode113 function in MATLAB with a relative tolerance of $10^{-13}$, absolute tolerance of $10^{-15}$, and a variable time step. As the initial conditions $\left(\phi,L,\theta_{21}\right)$, we consider 20 equidistant values from $\phi=0.238\,\pi$ to $\phi=0.466\,\pi$ and from $L=0.1\,R^{2}$ to $L=0.955\,R^{2}$, and 10 equidistant values from $\theta_{21}=0.1\,\pi$ to $\theta_{21}=\pi$. For trajectories that have exactly symmetric initial conditions ($\phi=\pi/4$ and $\theta_{21}=\pi$) and exhibit stable rigid-body rotation ($L<L_{\mathrm{cr}}$), our simulations show deviations from the initial radius $r_{k}\left(0\right)$, initial Hamiltonian energy $H$, and initial $L$ of at most $10^{-6}$ in the respective units of each over time intervals under consideration.

In this section, we present our numerical results on the dynamics of two same-sign point vortices and, in particular, describe the emergence of the two distinct classes of motion in the system. To relate our results to the findings of Ref. Navarro et al. (2013), we limit the specific examples to the case $\theta_{21}=\pi$ corresponding to vortices that are initially located on opposite sides of the center of the harmonic trap. However, we emphasize that the two distinct phase-space regions persist for all values of $\theta_{21}$ [Fig. 1(a)]. We first consider the position-coordinate representation (Sec. III.1) in order to provide a physically intuitive picture but subsequently switch to using the radial and angular velocities as our coordinates (Sec. III.2) because the emergence of the two types of motion is most apparent in this representation.

In conclusion, we have numerically studied the dynamics of two same-sign point vortices in a harmonically trapped superfluid. We discovered an impenetrable wall in the 3D phase space of possible two-vortex configurations that divides the ensuing vortex dynamics into two distinct types. In the first type, the two vortices move inside the same annular regions in the trap, whereas in the second type their orbits never intersect. The two types are particularly easy to distinguish in the 2D parameter space spanned by the angular and radial velocities of the vortices, where the first type results in one closed curve along which both vortices travel and the second type yields separate loops for each vortex. This phase-space wall is distinct from the bifurcation of rigidly rotating two-vortex configurations found by Navarro et al. Navarro et al. (2013). Importantly, the phase-space wall also persists for configurations where the two vortices are not initially at opposite sides of the trap center, unlike the rigidly rotating states.

Introducing the velocity-space representation opens a number of ways to extend the investigations of point-vortex dynamics in future studies. One obvious question is how the introduction of asymmetry between the vortices, i.e., $s_{1}\neq s_{2}$, would affect the transition phenomena in the phase space; the archetypal example of such a configuration is the vortex–antivortex pair ($s_{2}=-s_{1}$), which is known to exhibit stationary solutions in the harmonically trapped system Freilich et al. (2010); Kuopanportti et al. (2011); Middelkamp et al. (2011); Torres et al. (2011a). On the other hand, increasing the number of vortices to three in Eqs. (1) results in the emergence of chaotic vortex dynamics in a particularly simple yet experimentally relevant setup; in the absence of the trap [i.e., for $\Omega_{0}=0$ in Eqs. (1)], chaos can reign only if $N\geq 4$. In fact, already the two-vortex case is likely to exhibit chaotic advection Aref (2002): If one formally introduces a third vortex with $s_{3}=0$, its motion in the flow field of the two genuine vortices may be chaotic. Chaotic advection is known to exist in the presence of three genuine vortices for $\Omega_{0}=0$ Aref (1983).

Ultimately, the point-vortex model will serve as an efficient model for 2D quantum turbulence that corresponds to highly chaotic motion of a large number of point vortices. As such, it shows promise in further elucidating such phenomena as the inverse energy cascade, the emergence of Onsager vortices, and negative absolute Boltzmann temperatures associated with 2D turbulence in superfluids.