2.1 Quasi-classical approximations

But how can the continuum Gross–Pitaevskii equations provide us with details about the Lagrangian dynamics and reconnections that underlie the vortex tangle of quantum fluids?

This can be done by writing the wave function as $\unicode[STIX]{x1D713}=\sqrt{\unicode[STIX]{x1D70C}}\text{e}^{\text{i}\unicode[STIX]{x1D719}}$ , where $\unicode[STIX]{x1D70C}$ is the density and $\unicode[STIX]{x1D719}$ is the complex phase. The phase velocity $\boldsymbol{v}_{\unicode[STIX]{x1D719}}$ and quantised circulation $\unicode[STIX]{x1D6E4}$ around the zero-density line defects are then defined as:

(2.5a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{v}_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}=\text{Im}(\unicode[STIX]{x1D713}^{\dagger }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713})/\unicode[STIX]{x1D70C},\quad \unicode[STIX]{x1D6E4}=\int \boldsymbol{v}_{\unicode[STIX]{x1D719}}\boldsymbol{\cdot }\boldsymbol{s}=2\unicode[STIX]{x03C0}. & & \displaystyle\end{eqnarray}$$

Even though these lines cannot represent a true vorticity field because the vorticity $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{v}_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D719}\equiv 0$ . In this picture, vortex reconnection appears naturally as the instantaneous re-alignment of these lines and exchange of circulation when the line defects meet. If dimensions were added, the quantised circulation has the classical units of circulation: $\unicode[STIX]{x1D6E4}\sim \unicode[STIX]{x1D70C}L^{2}/T$ . Note these two differences with classical vortices governed by the Navier–Stokes equation: classical circulation is not quantised and viscous reconnection is never 100 %.

To extract the Lagrangian motion of quantised vortices from fields defined on three-dimensional meshes these issues must be addressed:

1. (i) As the state relaxes from its initial form, it should not be dominated by either internal waves (phonons) or strong fluctuations along the vortex trajectories (Kelvin waves).

2. (ii) Second, a method is needed for identifying the direction and positions that the vortices follow as they pass through the three-dimensional mesh.

The two innovations introduced in this paper, (2.7) for the initial density and (3.1) for tracking vortices, resolve both problems and allow us to extract smooth motion for the vortices from the calculated solutions of the Gross–Pitaevskii equations on Eulerian meshes.

To complete the discussion of the Gross–Pitaevskii equations (2.1), the full analogy to classical hydrodynamic equations comes from inserting $\unicode[STIX]{x1D713}=\sqrt{\unicode[STIX]{x1D70C}}\exp (\text{i}\unicode[STIX]{x1D719})$ into (2.1) to get the standard equation for $\unicode[STIX]{x1D70C}$ and a Bernoulli equation for $\unicode[STIX]{x1D719}$ :

(2.6a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{v}_{\unicode[STIX]{x1D719}})=0,\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D719}+(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})^{2}=0.5(1-\unicode[STIX]{x1D70C})+\unicode[STIX]{x1D6FB}^{2}\sqrt{\unicode[STIX]{x1D70C}}/\sqrt{\unicode[STIX]{x1D70C}}. & & \displaystyle\end{eqnarray}$$

The $\boldsymbol{v}_{\unicode[STIX]{x1D719}}$ velocity equation can then be formed by taking the gradient of the $\unicode[STIX]{x1D719}$ equation.

As in Kerr (Reference Kerr2011), the numerics are a standard semi-implicit spectral algorithm where the nonlinear terms are calculated in physical space, then transformed to Fourier space to calculate the linear terms. In Fourier space, the linear part of the complex equation is solved through integrating factors with the Fourier transformed nonlinearity added as a third-order Runge–Kutta explicit forcing. The domain is imposed by using no-stress cosine transforms in all three directions. For all of the calculations the domain size is $L_{x}\times L_{y}\times L_{z}=(16\unicode[STIX]{x03C0})^{3}\approx 50^{3}$ or $(32\unicode[STIX]{x03C0})^{3}$ . Both $128^{3}$ and $256^{3}$ grids were used, with the $256^{3}$ grid giving smoother temporal evolution. Most of the analysis and graphics will use the $\unicode[STIX]{x1D6FF}_{0}=3$ , $256^{3}$ calculation.

Figure 1.

Anti-parallel case: density, kinetic energy $K_{\unicode[STIX]{x1D713}}=|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}$ and pseudo-vorticity $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}_{r}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}_{i}|$ isosurfaces plus vortex lines at these times: (a) $t=0.125$ , with $\max (K_{\unicode[STIX]{x1D713}})=1.3$ and (b) $t=4$ with $\max (K_{\unicode[STIX]{x1D713}})=0.83$ . $\max |\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}|=0.4$ for all the times. The vortex lines show that the pseudo-vorticity method not only follows the lines $\unicode[STIX]{x1D70C}=0$ well, but the $t=4$ frame also shows that it can be used to follow isosurfaces of fixed $\unicode[STIX]{x1D70C}$ . The $t=0.125$ frame shows that initially the kinetic energy is largest between the elbows of the two vortices, with large $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$ at the elbows. The $t=4$ frame shows the newly reconnected vortices as they are separating with undulations on the vortex lines that will form additional reconnections and vortex rings at later times.

Figure 2.

Overview of the evolution of the $\unicode[STIX]{x1D6FF}_{0}=\text{3, 256}^{3}$ orthogonal calculation from a three-dimensional perspective for two times: $t=6$ as reconnection is beginning and $t=14$ after it has ended, where $t_{r}\approx 8.9$ . Green $\unicode[STIX]{x1D70C}=0.05$ isosurfaces encase the vortex cores, blue and red lines. The Frenet–Serret frames for $t=8.75$ are given in figure 6. A twist in the post-reconnection $t=14$ right ( $x>0$ ) vortex is visible.

Figure 3.

The same $\unicode[STIX]{x1D6FF}_{0}=3$ fields and times as in figure 2 from the Nazarenko perspective, which looks down the $45^{\circ }$ propagation plane in the $y{-}z$ -plane in figure 8(a). This perspective is useful because the pre- and post-reconnection symmetries can be seen most clearly, which is why it is the basis for comparison of reconnection angles in figure 8(b) and § 4.6. The arms are extending away from the reconnection plane, either towards or away from the viewer. Also note that from this perspective, loops are clearly not forming.

2.2 Choice of initial configurations and profiles

Configurations. Two initial vortex configurations are used in this paper, anti-parallel vortices with a perturbation and orthogonal vortices. Both configurations have been used many times for both classical (Navier–Stokes) and quantum fluids, including the first calculations using the three-dimensional Gross–Pitaevskii equations (Koplik & Levine Reference Koplik and Levine1993).

The advantage of focusing on these configurations is that the interactions leading to reconnection for most other configurations, for example colliding or initially linked vortex rings, can be reduced to either anti-parallel or orthogonal dynamics, both of which can resolve the reconnection events in smaller global domains. This is because the initial reconnection events require, effectively, only half of each ring.

These two configurations also represent the two extremes for the initial chirality or linking number of the vortex lines in a quantum fluid. In classical fluids, the corresponding global property is the helicity, $h=\int (\unicode[STIX]{x1D735}\times \boldsymbol{u})\boldsymbol{\cdot }\boldsymbol{u}\,\text{d}V$ . When $h$ is large, it tends to suppress nonlinear interactions. Anti-parallel initial conditions have zero net helicity and are sensitive to initial instabilities while orthogonal initial conditions have a large helicity, so reconnection can be delayed (Boratav et al. Reference Boratav, Pelz and Zabusky1992).

Density profile. The density profiles for all the vortex cores in this paper are determined by the following Padé approximate:

(2.7) $$\begin{eqnarray}\displaystyle & & \displaystyle |\unicode[STIX]{x1D713}_{sb}|=\sqrt{\unicode[STIX]{x1D70C}_{sb}}={\displaystyle \frac{c_{1}r^{2}+c_{2}r^{4}}{1+d_{1}r^{2}+d_{2}r^{4}}}\nonumber\\ \displaystyle & & \displaystyle \quad \text{with }c_{1}=0.3437,c_{2}=0.0286,d_{1}=0.3333,\text{ and }d_{2}=0.02864.\end{eqnarray}$$

Note that $c_{2}\lesssim d_{2}$ , which implies that as $r\rightarrow \infty$ , the density approaches the usual background of $\unicode[STIX]{x1D70C}=1$ from below more slowly than the true Padé of this order does. The true Padé, derived by Berloff (Reference Berloff2004), has $c_{2}=d_{2}=11/384\approx 0.02864$ . Therefore, the profile with $c_{2}\lesssim d_{2}$ is designated as $\unicode[STIX]{x1D713}_{sb}$ because it is a sub-Berloff (Reference Berloff2004) profile. Furthermore, because the calculations are in finite domains, to ensure that the Neumann boundary conditions are met, a set of up to 24 mirror images of the vortices are multiplied together. This multiplication process takes the slight, original $\unicode[STIX]{x1D70C}_{sb}<1$ values as $r\rightarrow \infty$ and generates stronger differences. At the boundaries, this gives $(1-\unicode[STIX]{x1D70C})\approx 0.02-0.03$ .

For all of the configurations discussed here, using this $|\unicode[STIX]{x1D713}|\lesssim 1$ initial profile appears to be crucial in allowing us to obtain clear scaling laws for the pre- and post-reconnection separation of the vortices. As discussed in § 4.2, further tests have confirmed that the temporal separations of all of the true Padé approximates had significant fluctuations. This includes the new Padé’s from Rorai et al. (Reference Rorai, Sreenivasan and Fisher2013), which are close to what one gets using a diffusion steady-state equation as in Meichle et al. (Reference Meichle, Rorai, Fisher and Lathrop2012).

Once the best profile has been chosen, then one must choose the trajectories of the interacting vortices. The anti-parallel global states are shown first in figure 1 as they illustrate the use of all of the three-dimensional diagnostics.

The width of the vortex cores, the healing length $a$ , is determined by the distance over which the vortex-core profile $\sqrt{\unicode[STIX]{x1D70C}_{sb}(r)}$ rises from zero to $\sqrt{\unicode[STIX]{x1D70C}_{sb}(1)}\approx 0.52$ . For our profiles, $a\approx 1$ in simulation units.

2.3 Anti-parallel: initial trajectory and global development

Based upon past experience with classical vortices and Kerr (Reference Kerr2011), the positions $\boldsymbol{s}_{\pm }(y)$ of the two nearly anti-parallel $y$ -vortices, with a perturbation in the direction of propagation $x$ , were

(2.8) $$\begin{eqnarray}\displaystyle \boldsymbol{s}_{\pm }(y)=\left(\unicode[STIX]{x1D6FF}_{x}\frac{2}{\cosh ((y/\unicode[STIX]{x1D6FF}_{y})^{1.8})}-x_{c},y,\pm z_{c}\right). & & \displaystyle\end{eqnarray}$$

The parameters used were $\unicode[STIX]{x1D6FF}_{x}=-1.6$ , $\unicode[STIX]{x1D6FF}_{y}=1.25$ , $x_{c}=21.04$ and $z_{c}=2.35$ . The power of 1.8 on the normalized position was chosen to help localise the perturbation near the $y=0$ symmetry plane. The density profiles were applied perpendicular to this trajectory, and not perpendicular to the $y$ -axis.

As in Kerr (Reference Kerr2011), two of the Neumann boundaries act as symmetry planes to increase the effective domain size. These planes are the $y=0$ , $x{-}z$ perturbation plane and $z=0$ , $x{-}y$ dividing plane. Because the goal of this calculation was to focus upon the scaling around the first reconnection, the long domain used in Kerr (Reference Kerr2011) to generate a chain of vortices is unnecessary and $L_{y}$ is less. In addition, based upon recent experience with Navier–Stokes reconnection (Kerr Reference Kerr2013), $L_{z}$ was increased to ensure that the evolving vortices do not see their mirror images across the upper $z$ Neumann boundary condition.

Figure 1 shows the state at $t=0.125$ , essentially the initial condition, and the state at $t=4$ , after the first reconnection event at $t_{r}\approx 2.4$ . Three isosurfaces are given. Low-density isosurfaces ( $\unicode[STIX]{x1D70C}=0.05$ ), isosurfaces of the kinetic energy $K_{\unicode[STIX]{x1D713}}$ (2.4) and isosurfaces of $|\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D713}}|$ (3.1), a pseudo-vorticity that is introduced in the next section. The vortex lines defined by $\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D713}}$ and Proposition 1 in § 3 are shown using thickened curves. The structure at the time of reconnection is discussed in § 5 using figure 12 and how the flow would develop later has already been documented by Kerr (Reference Kerr2011), which shows several reconnections forming a stack of vortex rings.

Both $K_{\unicode[STIX]{x1D713}}$ and $\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D713}}$ are functions of the first derivatives of the wave function, but show different aspects of the flow. The $K_{\unicode[STIX]{x1D713}}$ isosurfaces show where the momentum is large. Initially, the momentum is dominated by forward motion between the perturbations, as shown for $t=0.125$ . Post-reconnection, at $t=4$ , the $K_{\unicode[STIX]{x1D713}}$ surfaces show that the primary motion is around the vortices. $|\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D713}}|$ is large where the vortex cores bulge and have the greatest curvature.

2.4 Orthogonal: initial separations and global development

The orthogonal vortices are respectively parallel to the $y$ axis and parallel to the $z$ axis through two points on either side of $x=y=z=0$ on the line through $y=z=0$ . The five separations and other details of the simulations are given in table 1. Because all of the orthogonal cases with $\unicode[STIX]{x1D6FF}_{0}\geqslant 2$ behave qualitatively in the same manner, all of the orthogonal three-dimensional images will be taken from the $256^{3}$ , $\unicode[STIX]{x1D6FF}_{0}=3$ calculation, whose estimated reconnection occurs at time $t_{r}=8.9$ .

Table 1.

Cases: type is orthogonal $\bot$ or anti-parallel $\Vert$ , initial separations, approximate reconnection times, initial kinetic and interaction energies.

The first two sets of three-dimensional isosurfaces and vortex lines in figures 2 and 3 are used to show the global evolution of the vortices through reconnection, with figure 2 providing a true three-dimensional perspective of the orthogonality and figure 3, defined by figure 8(b), providing a perspective down the $y=z$ axis, which is also used for the determination of three-dimensional angles in § 4.4. The three times chosen for each are $t=4$ , pre-reconnection, $t=8.5$ as reconnection begins and $t=19$ , post-reconnection.

The two isosurfaces are for a low density of $\unicode[STIX]{x1D70C}=0.05$ and kinetic energy of $K_{\unicode[STIX]{x1D713}}=0.5$ , where $\max (K_{\unicode[STIX]{x1D713}})=0.58$ . There are two pseudo-vortex lines (3.1) in each frame, one that originates on the $y=0$ plane and the other on the $z=0$ plane. The $t=8.5$ frame also shows some additional orientation vectors that will be discussed in § 4.

Some qualitative features are:

1. (i) The initially orthogonal vortices are attracted towards each other at their points of closest approach, asymmetrically bending out towards each other.

2. (ii) During this stage there is a some loss of the kinetic energy between $t=8$ and $t=11$ , $\unicode[STIX]{x1D6E5}_{t}K_{\unicode[STIX]{x1D713}}<10\,\%K_{\unicode[STIX]{x1D713}}(t=0)$ . This is converted into interaction energy $I$ (2.4). There are no further noticeable changes in $K_{\unicode[STIX]{x1D713}}$ for $t>12$ .

3. (iii) After reconnection, from one perspective there is a slight twist on one vortex, but it is not twisted enough for the vortices to loop back upon themselves and reconnect again. Instead, the two new vortices pull back from one another, as shown by the $t=19$ frame in figure 2. This is consistent with experimental observations of vortex interactions using solid hydrogen particles (Bewley et al. Reference Bewley, Paoletti, Sreenivasan and Lathrop2008; Schwarzschild Reference Schwarzschild2010) in the sense that post-reconnection filaments simply pull back from one another and do not loop. Section 4.3 provides further structural information using the axes of the respective Frenet–Serret frames.

3.1 Curvature obtained from the $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$ lines

The curvature of the lines identified by the pseudo-vorticity algorithm will be found by applying the Frenet–Serret relations to the derivatives of the trajectories $\boldsymbol{r}(s)$ of the vortex lines.

Definition 3.1. The Frenet–Serret frame for any smooth curve $\boldsymbol{r}(s):[0,1]\rightarrow \mathbb{R}^{3}$ has an orthonormal triple of unit vectors $(\boldsymbol{T},\boldsymbol{N},\boldsymbol{B})$ at each point $\boldsymbol{r}(s)$ where $\boldsymbol{T}(s)$ is the tangent, $\boldsymbol{N}(s)$ is the normal and $\boldsymbol{B}(s)$ is the bi-normal. The following relations between $(\boldsymbol{T},\boldsymbol{N},\boldsymbol{B})$ define the curvature $\unicode[STIX]{x1D705}$ and torsion $\unicode[STIX]{x1D70F}$ :

(3.9a ) $$\begin{eqnarray}\displaystyle & \boldsymbol{T}(s)=\unicode[STIX]{x2202}_{s}\boldsymbol{r}(s), & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x2202}_{s}\boldsymbol{T}=\unicode[STIX]{x1D705}\boldsymbol{N}, & \displaystyle\end{eqnarray}$$

(3.9c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x2202}_{s}\boldsymbol{N}=\unicode[STIX]{x1D70F}\boldsymbol{B}-\unicode[STIX]{x1D705}\boldsymbol{T}, & \displaystyle\end{eqnarray}$$

(3.9d ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x2202}_{s}\boldsymbol{B}=-\unicode[STIX]{x1D70F}\boldsymbol{N}. & \displaystyle\end{eqnarray}$$

The numerical algorithm for calculating the curvature $\unicode[STIX]{x1D705}$ and its normal eigenvector $\boldsymbol{N}$ uses the function gradient in Matlab twice. That is, first $\boldsymbol{r}_{,s}$ and then $\boldsymbol{r}_{,ss}$ are generated. Next, normalising $\boldsymbol{r}_{,s}$ gives the tangent vector $\boldsymbol{T}$ , the direction vector between points on the vortex lines. Finally, the derivative of $\boldsymbol{T}$ gives us both the curvature, $\unicode[STIX]{x1D705}=|\unicode[STIX]{x2202}_{s}\boldsymbol{T}|$ and the normal $\boldsymbol{N}=\unicode[STIX]{x2202}_{s}\boldsymbol{T}/\unicode[STIX]{x1D705}$ . In practice it is better to calculate the curvature using:

(3.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}=|\boldsymbol{r}_{,s}\times \boldsymbol{r}_{,ss}|/|\boldsymbol{r}_{,s}|^{3}. & & \displaystyle\end{eqnarray}$$

4.1 Approach and separation

To determine the separation scaling laws before and after reconnection, these steps are followed.

1. (i) First, the trajectories of the $y$ and $z$ vortex lines, originating respectively on the $y=0$ and $z=0$ planes, are identified by applying the Matlab streamline algorithm to the pseudo-vorticity field $\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D713}}$ .

2. (ii) Next, points $\boldsymbol{x}_{y}$ and $\boldsymbol{x}_{z}$ are the points of closest approach and the time dependence of the minimum separation distance $\unicode[STIX]{x1D6FF}_{yz}(t)=|\boldsymbol{x}_{y}-\boldsymbol{x}_{z}|$ were identified and plotted, as in the inset of figure 5.

3. (iii) Once it was clear that neither the incoming nor outgoing separations obeyed the dimensional expectation (1.1), several alternative scaling laws were applied to the separations. Only the $1/3$ incoming power law and outgoing $2/3$ law working well for every case. Then, by using these power laws, refined estimates of $\tilde{t}_{r}(\unicode[STIX]{x1D6FF}_{0})$ were made by cubing the $\unicode[STIX]{x1D6FF}$ separations for $t<\tilde{t}_{r}(\unicode[STIX]{x1D6FF}_{0})$ ) and taking the $3/2$ power of $\unicode[STIX]{x1D6FF}$ for $t>\tilde{t}_{r}(\unicode[STIX]{x1D6FF}_{0})$ .

4. (iv) For all of the initial $\unicode[STIX]{x1D6FF}_{0}$ , the estimates of $t_{r}$ using the $t<t_{r}$ (cubic) and $t>t_{r}$ ( $3/2$ ) fits gave nearly identical $t_{r}$ , which were then combined and used to give the fit $t_{r}(\unicode[STIX]{x1D6FF}_{0})=(0.67\unicode[STIX]{x1D6FF}_{0}+0.064)^{3}$ shown in figure 4.

5. (v) Using these $t_{r}(\unicode[STIX]{x1D6FF}_{0})$ , figure 5 compares the scaled pre- and post-reconnection separations $\unicode[STIX]{x1D6FF}(t)$ by overlaying all five cases from Rorai (Reference Rorai2012) using

   (4.1a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{in}\sim |t_{r}-t|^{1/3}\quad \text{for }t<t_{r}(\unicode[STIX]{x1D6FF}_{0})\quad \text{and}\quad \unicode[STIX]{x1D6FF}_{out}\sim |t-t_{r}|^{2/3}\quad \text{for }t>t_{r}(\unicode[STIX]{x1D6FF}_{0}).\quad & & \displaystyle\end{eqnarray}$$
   The inset illustrates the strength of the new scaling laws compared to the dimensional prediction (1.1) by scaling the separations $\unicode[STIX]{x1D6FF}_{yz}(t)$ using (1.1).

Figure 4.

Reconnection times as a function of the initial separation. Reconnection times are estimated by using times immediately before and after reconnection plus the empirical $1/3$ and $2/3$ scaling laws.

Figure 5.

Pre- and post-reconnection separations for five cases: $\unicode[STIX]{x1D6FF}_{0}=2,3,4,5,6$ for calculations with $256^{3}$ points. ●: $\unicode[STIX]{x1D6FF}_{0}=2$ , $+$ : $\unicode[STIX]{x1D6FF}_{0}=3$ , ○: $\unicode[STIX]{x1D6FF}_{0}=4$ , $\divideontimes$ : $\unicode[STIX]{x1D6FF}_{0}=5$ , ▵: $\unicode[STIX]{x1D6FF}_{0}=6$ . Pre-reconnection distances are raised to the power 3, while post-reconnection distances are raised to the power $3/2$ . This scaling is visibly better than the dimensional prediction ( $\unicode[STIX]{x1D6FF}^{2}$ versus time to reconnection) shown in the inset. The scaled separations for $t<t_{r}$ fluctuate more strongly than the $t>t_{r}$ scaled separations.

4.2 Sub-Berloff profile

To obtain the clear scaling laws of figure 5, it was necessary to apply both the sub-Berloff profile (2.7) and the mirror images, as discussed above. To demonstrate the value of the sub-Berloff profile, additional calculations were done using the best full Padé approximate profiles in the same geometries and either used mirror images or did not. This included, the true Berloff profile, that is (2.7) with $c_{2}=d_{2}=0.02864$ , the low-order $1\times 1$ Padé approximate of the Fetter (Reference Fetter, Mahanthappa and Brittin1969), (2.7) with $d_{1}=1$ and $c_{2}=d_{2}=0$ , and the Padé approximates from Rorai et al. (Reference Rorai, Sreenivasan and Fisher2013), solutions that are very close to the ideal diffusive solution. All gave approach and separation curves similar to those for the orthogonal case in Zuccher et al. (Reference Zuccher, Caliari, Baggaley and Barenghi2012), including temporal oscillations in the separations $\unicode[STIX]{x1D6FF}_{yz}$ as well as weak signatures of the clear and distinctly different approach and separation scaling laws that we are reporting. Further work will be needed to identify why instabilities generated on the vortex lines are either suppressed by the sub-Berloff profile or absorbed by it.

4.3 Evolution of the orthogonal geometry during reconnection

Figure 6.

Isosurfaces, quantum vortex lines and orientation vectors in three dimensions for the $\unicode[STIX]{x1D6FF}_{0}=3$ , $256^{3}$ calculation at $t=6$ (a), $t=8.75$ (b) and $t=12$ (c) where $t_{r}\approx 8.9$ . Lines and vectors at the closest points on each vortex are the separation $D$ and Frenet–Serret vectors (3.9): $\boldsymbol{T}$ , $\boldsymbol{N}$ and $\boldsymbol{B}$ .

The three-dimensional evolution of the vortices is illustrated in figures 2, 3 and 6 and the alignments at or near reconnection are illustrated with two sketches taken from different perspectives in figure 8(a,b). Figures 2 and 3 illustrate the global changes in structure from two perspectives. One a general perspective and another that shows the symmetries by looking down the $y=z$ $45^{\circ }$ direction. This view was inspired by the linear reconnection model Nazarenko & West (Reference Nazarenko and West2003) and is designated the NP perspective. Each figure has a $t=6$ frame, long before the reconnection at $t_{r}\approx 8.9$ , and a frame at $t=14$ , long after reconnection.

The local in time and space evolution around the reconnection is illustrated using the sketches and these three times in figure 6: $t=6$ is at the beginning of reconnection, $t=8.75$ is just before the reconnection time of $t_{r}\approx 8.9$ and $t=12$ shows the end of reconnection. Over this period the $\unicode[STIX]{x1D70C}=0.05$ isosurfaces change slowly while the $\unicode[STIX]{x1D70C}\equiv 0$ pseudo-vorticity lines within them move rapidly towards one another. The Frenet–Serret frames around the points of closest approach are discussed in §§ 4.4 and 4.5.

Figure 8(a,b) provides two planar sketches at or near the reconnection time, with the best reference point for each being the mid-point between the closest points on the two vortices:

(4.2) $$\begin{eqnarray}\displaystyle \boldsymbol{x}_{r}(\unicode[STIX]{x1D6FF}_{0},t)=0.5(\boldsymbol{x}_{y}(t)+\boldsymbol{x}_{z}(t)). & & \displaystyle\end{eqnarray}$$

The sketch in figure 8(a) projects the vortices at the reconnection time $t_{r}$ onto the $y{-}z$ plane around the point of reconnection: $\boldsymbol{x}_{r}(\unicode[STIX]{x1D6FF}_{0},t_{r})$ , along with the two planes of interest, the reconnection or osculating plane and the propagation/symmetry plane. These are defined in terms of the average Frenet–Serret basis vectors (4.3) in § 4.5. The second sketch in figure 8(b) looks down the $x=0$ , $y=z$   $45^{\circ }$ direction of propagation of $\boldsymbol{x}_{r}(\unicode[STIX]{x1D6FF}_{0},t)$ onto the $\boldsymbol{T}_{av}\times \boldsymbol{N}_{av}$ (4.3) reconnection plane. Important features include:

1. (i) Because figure 8(a) is at $t=t_{r}$ , the projections of the planes and vortices all cross at $\boldsymbol{x}=\boldsymbol{x}_{r}(\unicode[STIX]{x1D6FF}_{0},t_{r})$ , which means that the red and blue curves trace both the pre- and post-reconnection trajectories of the vortices, as follows:

   1. (1) The trajectories before reconnection follow the curves parallel to the $y=0$ and $z=0$ axes that are half blue and half red. These are projections of the red and blue lines in figure 8(b).

   2. (2) The trajectories immediately after reconnection are indicated by the red and blue curves coming out of the reconnection plane.

2. (ii) The two orthogonal lines through $\boldsymbol{x}_{r}$ represent the reconnection and propagation planes:

   1. (1) The reconnection plane, defined by $\boldsymbol{T}_{av}=0.5(\boldsymbol{T}_{y}-\boldsymbol{T}_{z})$ and $\boldsymbol{N}_{av}=0.5(\boldsymbol{N}_{y}-\boldsymbol{N}_{z})$ (4.3). Before or after reconnection, the separation vector $\boldsymbol{D}=(\boldsymbol{x}_{z}-\boldsymbol{x}_{y})/|\boldsymbol{x}_{z}-\boldsymbol{x}_{y}|\neq 0$ is also in this plane.

      1. (I) It is shown below that $\boldsymbol{T}_{y,z}$ and $\boldsymbol{N}_{y,z}$ swap at reconnection, so all of these basis vectors stay in this plane after reconnection.

   2. (2) And the propagation plane, which contains the velocity of $\boldsymbol{x}_{r}(\unicode[STIX]{x1D6FF}_{0},t)$ and the average bi-normal $\boldsymbol{B}_{av}=0.5(\boldsymbol{B}_{y}+\boldsymbol{B}_{z})$ (4.3).

3. (iii) The $x=0$ , $y=z$ projection in figure 8(b) is the NP perspective inspired by the linear model of Nazarenko & West (Reference Nazarenko and West2003), which tells us that $\boldsymbol{x}_{r}=0.5(\boldsymbol{x}_{y}+\boldsymbol{x}_{z})$ translates in the $(-y,-z)$ direction. This motion tells us that the reconnection does not occur at the centre of the computational box.

   1. (1) Note that for the points on either side of $\boldsymbol{x}_{r}$ , the tangents $\boldsymbol{T}_{y,z}$ and curvature vectors $\boldsymbol{N}_{y,z}$ are anti-parallel. The components that are not anti-parallel are directed out of the Nazarenko perspective. This also holds for the lines across the central, green $\unicode[STIX]{x1D70C}=0.05$ isosurface in figure 6(b). Figure 9(a) shows how $\boldsymbol{T}_{y}\boldsymbol{\cdot }\boldsymbol{T}_{z}$ , $\boldsymbol{N}_{y}\boldsymbol{\cdot }\boldsymbol{N}_{z}$ and $\boldsymbol{B}_{y}\boldsymbol{\cdot }\boldsymbol{B}_{z}$ converge to this state as $t\rightarrow t_{r}$ .

   2. (2) The lines drawn across the centre of figure 8(b) are used to determine the long-range angles discussed in § 4.6.

Figure 7.

Curvature of the vortex lines is given only for the $\unicode[STIX]{x1D6FF}_{0}=\text{3, 256}^{3}$ calculation with $t_{r}\approx 8.9$ as the curves about $t_{r}(\unicode[STIX]{x1D6FF}_{0})$ are similar for all $\unicode[STIX]{x1D6FF}_{0}$ . (a) Curvatures against arclength $s$ at four times: $t=6$ , 8.75, 9.5 and 12. The profiles for the other line are similar, with their maximum peaks at $t=9.5$ , both off the plotted scale. (b) Time evolution of the maximum curvatures of the two lines. For each line there is a sudden jump to the large post-reconnection maximum at $t=t_{r}$ .

Figure 8.

(a) Sketch of the orthogonal reconnection around the reconnection point $\boldsymbol{x}_{r}$ at the reconnection time $t_{r}$ of the vortices projected onto the $y{-}z$ plane. The pre-reconnection trajectories approximately follow the $y{-}y_{mid}=0$ and $z{-}y_{mid}=0$ axes and the post-reconnection trajectories are the red and blue lines. Two projected planes are indicated in black. The reconnection plane $(z{-}z_{r})=-(y{-}y_{r})$ contains the tangents to the vortex lines $\boldsymbol{T}_{y,z}$ and their curvature vectors $\boldsymbol{N}_{y,z}$ both before and after reconnection, as well at the separation vector between the closest points $\boldsymbol{x}_{y,z}$ . The propagation/symmetry plane $(z{-}z_{r})=(y{-}y_{r})$ which contains the bi-normals $\boldsymbol{B}_{y,z}$ and the direction of propagation of the mid-point $\boldsymbol{x}_{r}(t)$ between the closest points of the vortices: $\boldsymbol{x}_{y}$ and $\boldsymbol{x}_{z}$ . From this perspective the angles along the reconnection plane before reconnection are shallow, and those after reconnection are sharp. (b) Sketch just before reconnection using the Nazarenko perspective, that is looking down on the reconnection plane along the propagation/symmetry plane. Four line segments (in three-dimensions) have been added that connect $\boldsymbol{x}_{r}$ , the mid-point between the vortices at closest approach, to four points, each located $\unicode[STIX]{x0394}s$ units along the vortices from their respective closest points. Angles generated between adjacent segments as a function of $\unicode[STIX]{x0394}s$ are plotted in figure 11.

4.4 Curvature

Curvature has played a central role in our understanding of quantum turbulence due to its use in predicting velocities in the law of Biot–Savart and the local induction approximation. The connection between these approximations for the velocities and the true dynamics of quantum fluids, as modelled by the Gross–Pitaevskii equations, would be in how the gradient of the phase $\unicode[STIX]{x1D719}$ of the wave function $\unicode[STIX]{x1D713}$ is modified by the curvature of the vortex lines.

In that context, could curvature profiles provide clues for the origins of the anomalous scaling exponents of the orthogonal separations? For example, if the local induction approximation is relevant, then a sudden increase in the maximum curvature of the lines, such as in figure 7, might explain the change in scaling.

Is that increase large enough, or is more needed? Two primary features should be noted. First, the profiles of the curvature along the vortex lines in figure 7(a) are very symmetric in $s$ about the points of closest approach, both before and after reconnection. Second, while there by a sharp jump in $\unicode[STIX]{x1D705}_{max}$ at $t=t_{r}$ in the time-evolving curvature maxima $\unicode[STIX]{x1D705}_{max}$ in figure 7(b) ( $\unicode[STIX]{x1D6FF}_{0}=3$ case), at all other times $\unicode[STIX]{x1D705}_{max}$ is modest. That is, for $t<t_{r}$ , there is only modest growth, which for $t>t_{r}$ relaxes rapidly to the pre-reconnection values of $\unicode[STIX]{x1D705}_{max}$ . All other cases have similar behaviour, including one slightly under-resolved $\unicode[STIX]{x1D6FF}_{0}=4$ case in a large $(32\unicode[STIX]{x03C0})^{3}$ domain.

Therefore, one must conclude that a bigger picture is needed. Our proposal is to look at the alignments of their respective Frenet–Serret frames as another reason for the changes in scaling. Using analysis from Rorai (Reference Rorai2012), the next section will look first at the alignments of the points of closest approach, then at alignments between points further along the vortices in § 4.6.

4.5 Orthogonal: Frenet–Serret orientation

Besides allowing us to calculate the curvature of the vortex lines, knowing their trajectories allows us to calculate the Frenet–Serret frame (3.9). For the $256^{3}$ , $\unicode[STIX]{x1D6FF}_{0}=3$ calculation, figures 9 and 10 show the evolution of the Frenet–Serret frames at the closest points $\boldsymbol{x}_{y,z}$ . These alignments provide quantitative support for describing the local frame at the reconnection point $\boldsymbol{x}_{r}$ in terms of the reconnection plane and the propagation plane sketched in figure 8(a). First $t<t_{r}$ evolution is given, then the post-reconnection $t>t_{r}$ flip.

1. (i) For $t<t_{r}$ , pre-reconnection: figure 9(a) shows that the vorticity direction or tangent vectors $\boldsymbol{T}_{y,z}$ , the curvature vectors $\boldsymbol{N}_{y,z}$ and the bi-normals $\boldsymbol{B}_{y,z}$ converge to their opposites as $t\rightarrow t_{r}$ , both before and after reconnection.

2. (ii) Figure 9(b) then uses following averages of the Frenet–Serret frames between $\boldsymbol{x}_{y}$ and $\boldsymbol{x}_{z}$

   (4.3) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}lc@{}}\displaystyle \circ & \boldsymbol{N}_{av}=0.5(\boldsymbol{N}_{y}-\boldsymbol{N}_{z})\quad \text{and is parallel to }\boldsymbol{D}(t),\\ \displaystyle \circ & \boldsymbol{T}_{av}=0.5(\boldsymbol{T}_{y}-\boldsymbol{T}_{z})\quad \text{and is perpendicular to both }\boldsymbol{N}_{av}\text{ and }\boldsymbol{D},\\ \displaystyle \circ & \boldsymbol{B}_{av}=0.5(\boldsymbol{B}_{y}+\boldsymbol{B}_{z})\quad \text{and is perpendicular to }\boldsymbol{T}_{av},\boldsymbol{N}_{av}\text{ and }\boldsymbol{D}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

   1. (1) That is: subtract the tangent and curvature vectors because they become anti-parallel as $t{\nearrow}t_{r}$ and add the bi-normals because they are parallel as $t{\nearrow}t_{r}$ .

   2. (2) Comparing these averages shows that $\boldsymbol{T}_{y,z}$ , $\boldsymbol{N}_{y,z}$ and their unit separation vector $\boldsymbol{D}=(\boldsymbol{x}_{z}-\boldsymbol{x}_{y})/|\boldsymbol{x}_{z}-\boldsymbol{x}_{y}|$ all lie in the same plane, the reconnection plane in figure 8(a).

   3. (3) These alignments between $\boldsymbol{T}_{av}$ , $\boldsymbol{N}_{av}$ and $\boldsymbol{B}_{av}$ with $\boldsymbol{D}$ form in the early stages, long before the reconnection at $t_{r}=8.9$ .

3. (iii) Post-reconnection Frenet–Serret frames flip:

   1. (1) Figure 10 shows that the directions of $\boldsymbol{T}_{av}$ and $\boldsymbol{N}_{av}$ swap and $\boldsymbol{D}$ rotates by $90^{\circ }$ so that all three are still in the reconnection plane with the same relations to one another.

   2. (2) Note that $\boldsymbol{B}_{av}$ remains orthogonal to the reconnection plane.

Figure 9.

These two frames show how the basis vectors of the Frenet–Serret frames (3.9) at the closest points $\boldsymbol{x}_{y,z}$ of the $\unicode[STIX]{x1D6FF}_{0}=3$ , $256^{3}$ calculation are aligned and help define the reconnection plane and the propagation plane used in figure 8(a,b). (a) The alignments between the Frenet–Serret components at $\boldsymbol{x}_{y,z}$ . For $t<t_{r}$ the tangent vectors $\boldsymbol{T}_{y,z}$ and curvature vectors $\boldsymbol{N}_{y,z}$ become increasingly anti-parallel as the reconnection time is approached while the bi-normal vectors $\boldsymbol{B}_{y,z}$ become increasingly aligned. These trends are reversed for $t>t_{r}$ . (b) The alignments between the averages over $\boldsymbol{x}_{y,z}$ of the Frenet–Serret frames defined by (4.3) with the separation vector $\boldsymbol{D}$ between $\boldsymbol{x}_{y}$ and $\boldsymbol{x}_{z}$ .

Figure 10.

For the $\unicode[STIX]{x1D6FF}_{0}=3$ , $256^{3}$ calculation, the alignments over time of $\boldsymbol{D}(t_{r}^{-})=\boldsymbol{D}(t_{r}-\text{d}t)$ with $\boldsymbol{T}_{av}$ and $\boldsymbol{N}_{av}$ , the averages (4.3) over the tangent and curvature components of the Frenet–Serret frames at the closest points $\boldsymbol{x}_{y,z}$ . $\boldsymbol{B}_{av}\boldsymbol{\cdot }\boldsymbol{D}(t_{r}^{-})\approx 0$ for all times. $\boldsymbol{D}(t_{r}^{-})$ is the separation direction at $t=8.9$ , just before reconnection.

The alignments of the components of the Frenet–Serret frames at $\boldsymbol{x}_{y,z}$ and the separation of these points $\boldsymbol{D}$ are significantly different than their alignments for the anti-parallel case in § 5. Comparisons are discussed in § 6.

4.6 Angles at reconnection

Figure 11.

Angles between the segments in three-dimensional space displayed in figure 8(b) as a function of $\unicode[STIX]{x0394}s$ for $\unicode[STIX]{x1D6FF}_{0}=5$ and $t=40$ (pre-reconnection) and $t=41$ (post-reconnection). The angles between the arms of the same vortex, triplets [ $\boldsymbol{x}_{1,v1},\boldsymbol{x}_{r},\boldsymbol{x}_{2,v1}$ ] and [ $\boldsymbol{x}_{1v2},\boldsymbol{x}_{r},\boldsymbol{x}_{2v2}$ ], are indicated by circles (blue and red) before reconnection, and stars (blue and red) after reconnection. The angles sum to $360^{\circ }$ as $\unicode[STIX]{x0394}s\rightarrow 0$ , implying that the segments near the reconnection point $\boldsymbol{x}_{r}$ lie on a plane, specifically, the reconnection plane in figure 8(b) The sum of the angles grows with $\unicode[STIX]{x0394}s$ and is about ${\sim}362^{\circ }$ for $\unicode[STIX]{x0394}s\sim 2.7$ (vertical black line), which is consistent with the arms of reconnecting vortices becoming progressively more convex and hyperbolic as they move away from $\boldsymbol{x}_{r}$ , both before and after reconnection. The rapidly changing behaviour for small $\unicode[STIX]{x0394}s$ is a kinematic result of how the lines and angles indicated in figure 8(b) were chosen and is not significant. What could be more significant are the differences in the angles for intermediate $\unicode[STIX]{x0394}s$ and their influence upon any Biot–Savart contributions to the velocities.

What additional dynamics might help us identify the differences between the orthogonal and anti-parallel separation scaling laws?

One place to look is larger-scale alignments and long-range interactions. While the underlying Gross–Pitaevskii equations are local, the existence of the vortex structures means that such alignments should exist and should influence the local motion of the $\unicode[STIX]{x1D70C}=0$ lines to the degree that the law of Biot–Savart can be applied. With the goal of identifying any such long-range interactions, this section determines the evolution of pre- and post-reconnection angles between points on the extended structures.

Relevant angles can be defined between the arms of the reconnecting vortices around the reconnection point as follows.

1. (i) From the points of closest approach $\boldsymbol{x}_{y,z}(t)$ , define $\boldsymbol{x}_{r}(t)=0.5(\boldsymbol{x}_{y}(t)+\boldsymbol{x}_{z}(t))$ .

2. (ii) Move $\pm \unicode[STIX]{x0394}s$ along the arms of the vortices from $\boldsymbol{x}_{y,z}(t)$ and identify four new points: $\boldsymbol{x}_{1,y}$ , $\boldsymbol{x}_{2,y}$ , $\boldsymbol{x}_{1,z}$ and $\boldsymbol{x}_{2,z}$ , illustrated in the Nazarenko perspective sketch in figure 2.

3. (iii) To get a fully three-dimensional perspective, note that these points lie on outstretched arms such as in figure 2(b).

4. (iv) Connect the four points with $\boldsymbol{x}_{r}$ to form an extended three-dimensional frame then calculate the angles $\unicode[STIX]{x1D703}_{i}$ between these four vectors.

5. (v) The plots of $\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x0394}s)$ all show these qualitatively similar features:

   1. (1) The sum of the $\unicode[STIX]{x1D703}_{i}$ grows as $\unicode[STIX]{x0394}s$ increases, starting from $\sum _{i}\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x0394}s=0)=360^{\circ }$ , showing that the inner ( $\unicode[STIX]{x0394}s\approx 0$ ) structure is a plane.

   2. (2) The vertical line in figure 11 represents the $\unicode[STIX]{x0394}s_{p}$ for which $\sum _{i}\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x0394}s)=362^{\circ }>360^{\circ }$ , indicating that the structure is mildly hyperbolic, the opposite of the anti-parallel pyramid structure with acute angles.

   3. (3) The inner angles, that is the angles between the triplets [ $\boldsymbol{x}_{1,v1},\boldsymbol{x}_{r},\boldsymbol{x}_{2,v1}$ ] and [ $\boldsymbol{x}_{1v2},\boldsymbol{x}_{r},\boldsymbol{x}_{2v2}$ ], are all larger than $90^{\circ }$ before reconnection, and smaller after.

      The difference is about $20^{\circ }$ in all of the calculations and would be consistent with the observed slower approach and faster separation.

   4. (4) When looking down at the reconnection plane in either figure 8(b) or figure 3(b), do not forget that the centre of the reconnection plane is moving along the $y=z$ direction of the propagation plane and simultaneously dragging or pushing the extended arms as it moves, as in figure 2(b).

6. (vi) $\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x0394}s)$ depends upon $\unicode[STIX]{x1D6FF}_{0}$ as follows:

   1. (1) As $\unicode[STIX]{x1D6FF}_{0}$ decreases, the $\unicode[STIX]{x1D703}=90^{\circ }$ pre-reconnection cross-over, at $\unicode[STIX]{x1D6FF}s=0.8$ for $\unicode[STIX]{x1D6FF}_{0}=5$ , decreases.

   2. (2) The geometry becomes more hyperbolic. That is $\sum _{i}\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x0394}s)$ increases as $\unicode[STIX]{x1D6FF}_{0}$ decreases with $\sum _{i}\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x0394}s)=370^{\circ }$ for $\unicode[STIX]{x1D6FF}_{0}=3$ .

Understanding these features could provide us with some hints about the origins of the anomalous scaling laws. One hint could be the different angles at intermediate scales, $0.5\leqslant \unicode[STIX]{x0394}s\leqslant 2.5$ . Pre-reconnection, this span has approximately retained its original $\unicode[STIX]{x1D703}\sim 90^{\circ }$ orthogonal alignment. Post-reconnection the angles over this span still sum to approximately $362^{\circ }$ , but the angles on either side of the reconnection have jumped by $\pm 20^{\circ }$ . The sudden jump is due to how the directions of the tangent and curvature vectors swap at reconnection. Using the sketch in figure 8(b), this means that prior to reconnection the angle in three-dimensional space is between the triplets [ $\boldsymbol{x}_{1,v1},\boldsymbol{x}_{r},\boldsymbol{x}_{2,v1}$ ] and [ $\boldsymbol{x}_{1v2},\boldsymbol{x}_{r},\boldsymbol{x}_{2v2}$ ] and afterwards between the triplets [ $\boldsymbol{x}_{1v1},\boldsymbol{x}_{r},\boldsymbol{x}_{1v2}$ ] and [ $\boldsymbol{x}_{2v1},\boldsymbol{x}_{r},\boldsymbol{x}_{2v2}$ ]. Similar behaviour is seen for all the $\unicode[STIX]{x1D6FF}_{0}$ cases. It is visualised for $\unicode[STIX]{x1D6FF}_{0}=3$ , using two perspectives in figures 2 and 3.

The goal is find a model that links the sudden swaps in local alignments in figures 9 and 10 with the non-local changes in figure 11 and from that explains the anomalous orthogonal reconnection scaling. This model must also accommodate the scaling of anti-parallel reconnection, which obeys the expected dimensional scaling both before and after reconnection. The discussion in § 6 addresses what might be required.