{'fraction_non_alphanumeric': 0.041269547515499216, 'fraction_numerical': 0.02627926344036273, 'mean_word_length': 4.60845874416191, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Separation scaling for viscous vortex reconnection

Reconnection plays a significant role in the dynamics of plasmas, polymers and macromolecules, as well as in numerous laminar and turbulent flow phenomena in both classical and quantum fluids. Extensive studies in quantum vortex reconnection show that the minimum separation distance {\delta} between interacting vortices follows a 1/2 scaling. Due to the complex nature of the dynamics (e.g., the formation of bridges and threads as well as successive reconnections and avalanche), such scaling has never been reported for (classical) viscous vortex reconnection. Using the direct numerical simulation of the Navier-Stokes equations, we study viscous reconnection of slender vortices, whose core size is much smaller than the radius of the vortex curvature. For separations that are large compared to the vortex core size, we discover that {\delta}(t) between the two interacting viscous vortices surprisingly also follows the 1/2 power scaling for both pre- and post-reconnection events. The prefactors in this 1/2-power law are found to depend not only on the initial configuration but also on the vortex Reynolds number (or viscosity). Our finding in the viscous reconnection, complementing numerous works on quantum vortex reconnection, suggests that there is indeed a universal route for reconnection -- an essential result for understanding the various facets of the viscous vortex reconnection phenomena and their potential modeling, as well as possibly explaining turbulence cascade physics.


Introduction
Reconnection, a fundamental topology-transforming event, has been a subject of intense recent study in both classical (Kida & Takaoka 1994;Pumir & Kerr 1987;Melander & Hussain 1989;Kleckner & Irvine 2013) and quantum (Koplik & Levine 1993;Barenghi et al. 2001;Bewley et al. 2008;Paoletti et al. 2010) fluids, as well as in many other fields, such as plasmas (Priest & Forbes 2000), polymers, and macromolecules (Vazquez & DE WITT 2004). In turbulent flows, vortex reconnection appears to be the main mechanism for energy cascade: i) in quantum fluids, reconnection excites a cascade of Kelvin waves leading to energy dissipation via emissions of phonons and rotons (Kivotides et al. 2001;Vinen et al. 2003); ii) in classical fluids, finer and finer scales and turbulence avalanche can occur through successive reconnections (Melander & Hussain 1989;Yao & Hussain 2020b). Reconnection is also believed to play an essential role in several other physical phenomena, such as fine-scale mixing (Hussain 1986), and noise generation (Leadbeater et al. 2001).
One simple but important question in reconnection is the time scaling of the minimum distance δ(t) between the two interacting vortices. Assuming that the reconnection is a local process in space and the circulation Γ is the only relevant dimensional quantity involved, dimensional analysis yields Note that the actual reconnection, which is intrinsically three-dimensional, is never complete in classical fluids, leaving unreconnected parts as threads.
where t 0 is the reconnection time, and A − and A + are dimensionless factors for pre-and post-reconnection, respectively. Such a 1/2-power scaling has been numerically observed for reconnection of line vortices using the Biot-Savart law (de Waele & Aarts 1994;Kimura & Moffatt 2017) and also for reconnection of quantized vortices by integrating the Gross-Pitaevksii equation (Nazarenko & West 2003;Villois et al. 2017). In addition, recent quantum experiments (Paoletti et al. 2010;Fonda et al. 2019) confirmed this scaling when the distances between two interacting vortices are large compared with the vortex diameter but small compared with those from other adjacent vortices. Note that deviations from this 1/2 scaling were also reported in several works (Zuccher et al. 2012;Allen et al. 2014;Rorai et al. 2016).
In contrast to the vast literature on the time scaling of δ(t) in quantum fluids, very limited results have been reported for reconnection in classical fluids, which are governed by the Navier-Stokes (N-S) equations (figure 1). By performing the direct numerical simulation (DNS) of two antiparallel vortex tubes reconnection, Hussain & Duraisamy (2011) found that the minimum distance δ between the vortex centroids scales asymmetrically as (t 0 − t) 3/4 and (t − t 0 ) 2 before and after the reconnection. Note that in this study, the vortex core size σ is comparable to the initial separation distance δ between these vortices (i.e., σ/δ ≈ 0.4) -which definitely breaks the local assumption required for the 1/2 scaling. Inspired by the recent works of Moffatt & Kimura (2019a,b) on the finite time singularity of Euler and Navier-Stokes equations, we studied reconnection of two colliding slender vortex rings (the ratio between the initial vortex core size σ and the radius of the ring R is approximately 0.01) and found that δ(t) before reconnection follows a 1/2 scaling when σ ≪ δ ≪ R (Yao & Hussain 2020a). The main objective of the present work is to further elucidate the time scaling of minimum separation distance for (classical) viscous vortex reconnection. In particular, we want to address the following questions: i) does the time scaling of the minimum distance follows δ ∼ t 1/2 scaling for both before and after reconnection? ii) what dictates the prefactors in the scaling? and iii) what are the similarities/differences between classical and quantum vortex reconnections?

Results
Previous studies of the dynamics of slender vortices are mainly based on the vortex filament (VF) method, which is based on the B-S law (Siggia 1985;de Waele & Aarts 1994;Kimura & Moffatt 2018). To regularize the singular kernel of the B-S integral, a cutoff needs to be employed. With such regularization, the B-S integration always diverges near the singular time of reconnection (Villois et al. 2017;Kimura & Moffatt 2018). An ad hoc "cut-and-paste" algorithm is typically required for studying post-reconnection scenario (Schwarz 1985;Baggaley 2012;Galantucci et al. 2019). However, as reconnection in classical fluids is very complex, such an algorithm is very difficult to implement. Hence, the VF method is mainly employed for studying the pre-reconnection event.
With the rapid development of supercomputers these days, DNSs for considerably large-scale flow problems are becoming feasible. Here, we aim to employ DNS of the N-S equations for studying viscous reconnection of slender vortices. The numerical method employed here is the same as those used in Yao & Hussain (2020b). To understand what is universal in reconnections, three different vortex configurations are considered. Case I is two colliding vortex rings, which is the same as that in Moffatt & Kimura (2019a,b); Yao & Hussain (2020a) for studying the possible formation of finite time singularity of Euler and N-S equations. Case II is the two initially rectilinear, orthogonal vortices, which corresponds to the limit where the radius of curvature κ of two vortices are extremely large. Finally, to study the interaction of vortices with significantly different curvatures, following Galantucci et al. (2019), we also consider a case of a vortex ring interacting with an isolated vortex tube (Case III). For all cases, the initial vorticity distribution in the cross-section is assumed to be Gaussian ω(r) = Γ 0 /(4πσ 2 0 ) exp[−r 2 /4σ 2 0 ] with the circulation Γ 0 = 1 and core scale σ 0 = 0.01. Compared with those in the past studies (Kida & Takaoka 1994;Boratav et al. 1992;Melander & Hussain 1989;Chatelain et al. 2003), the distinction of our simulations is the larger ratio of the radius of curvature to the core size (i.e., R 0 /σ 0 100). As the viscous effect is an essential issue in classical fluids, for each configuration, two different Reynolds numbers (Re Γ ≡ Γ 0 /ν = 2000 and 4000), achieved by changing the kinematic viscosity ν, are considered. More technical details are described in the supplementary material.

Colliding vortex rings
We first consider the interaction of two circular vortex rings, which are symmetrically placed with the initial inclination angle θ = π/4 (figure 2a). The initial radius of the ring is selected as R 0 = 1. In addition, the initial minimum distance between these two vortex rings is chosen as δ 0 = 0.2 so that the interaction between the vortices can be considered as localized (σ 0 ≪ δ 0 ≪ R 0 ). Note that this vortex setup represents the typical antiparallel configuration. The evolution of the flow structure for Re Γ = 2000 is shown in the insets of figure 2 and also in supplementary movie S1. The structures for Re Γ = 4000, which are quite similar, are not shown due to high computational cost for rendering. Several features that distinctly differ from quantum reconnection deserve to be noted. First, as the rings approach each other under self-induction, they also undergo significant core deformation and form two thin vortex sheets. Second, the reconnection process is not discrete as for quantized vortices; and circulation transfer rate and the reconnection time strongly depend on viscosity ν, hence on Re Γ (Hussain & Duraisamy 2011;Yao & Hussain 2020b). Finally, reconnection is never complete; as a consequence, the circulation in the reconnected bridges is relatively smaller than the initial circulation Γ 0 of the vortices.
The appropriate determination of δ(t) relies heavily on the accurate tracking of the location of the vortex axis (Fonda et al. 2014;Villois et al. 2016), which is rather challenging in classical fluids. First, unlike vortex filaments or quantized vortices, where the axis location is almost precise, the vorticity field in classical fluids is continuously distributed. Second, vortex cores are typically distributed in irregular shapes without any clear center: before reconnection, the vortices undergo significant core deformation; and after reconnection, the reconnected vortex lines take some time to collect together to form the bridge. Due to the two-fold symmetry of the initial condition considered, the minimum distance δ between these two interacting rings before and after reconnection should occur in the symmetry S s and collision S c planes, respectively -which makes the determination of δ(t) relatively easy. Following Hussain & Duraisamy (2011) and Yao & Hussain (2020a), we take the vorticity centroid (computed as the centroid of above 75% of its maximum) to be the center for vortices in these two planes. Figure 2(a) displays the evolution of δ 2 (t) for the pre-reconnection event, with the top inset showing δ as a function of t 0 − t on a log-log scale for Re Γ = 4000. The clear following of linear scaling for δ 2 (t) at the early time suggests that δ(t) ∼ a − (t − 0 − t) 1/2 , with a − the constant prefactors for prereconnection corresponding to A − Γ 1/2 in equation (1.1), and t − 0 the critical time when δ → 0. For both Re Γ cases, δ 2 (t) collapses initially and then slowly deviates from linear scaling when δ ∼ O(σ). The deviation happens earlier for the Re Γ = 2000 case, which is due to the more rapid increase of the core size caused by stronger viscous diffusion. A linear fit on δ 2 (t) between 0 < t < 0.15 for Re Γ = 4000 gives t − 0 = 0.26 and a − = 0.38. As the circulation remains constant at Γ = 1 during this time, the dimensionless prefactor A − = a − = 0.38, which is quite close to A = 0.4 reported in de Waele & Aarts (1994).
When two bridges move sufficiently apart from the interacting region, a clear linear scaling for δ 2 (t) can be observed for both Re Γ cases (figure 2b). Hence, δ ∼ a + (t − t + 0 ) 1/2 scaling also holds in the post-reconnection dynamics when the two bridges' vortices are mainly governed by the mutual interaction. The early evolution of δ 2 (t) deviates from  Table 1: Fitted values of the prefactors a ± and t ± 0 for the minimum distance scaling δ(t) ∼ a ± |t − t ± 0 | 1/2 . The superscript ± stands for before (−) and after (+) the reconnection, respectively. the linear scaling, presumably for two main reasons. First, when the bridges are too close, they are under the influence of other unreconnected structures, such as threads, and other parameters besides Γ may be relevant in determining δ. Second, the reconnected vortex lines, initially in a thin vortex sheet shape, take time to accumulate to form a circular shape, and the circulation Γ continuously increases during this phase. A fit in the linear region gives t + 0 ≈ 0.30 for both Re Γ cases, and a + = 2.19 and 2.27 for Re Γ = 2000 and 4000, respectively. Different from quantized vortices, where reconnection is discrete and t 0 is almost the same for pre-and post-reconnection, here reconnection is a continuous process, and hence t + 0 is slightly larger than t − 0 . Consistent with previous studies, a + is always larger than a − , indicating that the vortices separate much faster than their approach. Compared to the pre-reconnection process, the effect of Re Γ on δ(t) is more apparent for the post-reconnection. It is because, in classical fluids, the dynamics of reconnection, such as the reconnection time and the circulation transfer rate, strongly depends on the viscosity ν. In general, reconnection is faster at higher Re Γ , which explains why δ 2 (t) follows linear scaling earlier at Re Γ = 4000. In addition, as Re Γ increases, reconnection is more complete (Yao & Hussain 2020a). The variation of a + with respect to Re Γ is mainly attributed to different circulations Γ in the reconnected bridges -which is difficult to be precisely determined.

Orthogonal vortex tubes
As one of the simplest configurations, the reconnection of orthogonal vortex tubes has been extensively studied for both classical (Boratav et al. 1992;Beardsell et al. 2016;Jaque & Fuentes 2017) and quantum (Zuccher et al. 2012;Galantucci et al. 2019) fluids. Similar to Case I, here the initial distance between these two rectilinear vortices is chosen as δ 0 = 0.2 The insets in figures 3(a) and (b) and also supplementary movie S3 show the evolution of the flow structures for Re Γ = 2000. The evolution is quite similar to that in Boratav et al. (1992) for the thick vortex core case: the vortex tubes first develop into locally antiparallel configuration under mutual induction; then collide with each other due to self-induction; after reconnection, they recede away. Different from quantum cases (Villois et al. 2017;Galantucci et al. 2019), the unreconnected threads, which wrap around the bridges, are distinct after reconnection. In addition, a Kelvin wave is observed after reconnection. In quantum fluids, nonlinear interaction of Kelvin waves creates waves of shorter and shorter wavelength, which is considered as the main mechanism for energy cascade (Baggaley & Barenghi 2011); in classical fluids, however, the Kelvin wave would rapidly decay due to viscous effect. It would be interesting to compare the difference in the Kelvin wave evolution as well as its role on energy cascade between the quantum and classical reconnections.
To determine the minimum distance δ(t) between these two vortex tubes, the axis of the vortex tubes needs to be tracked. Here, we propose a vortex tracking method based on the vortex lines that go through the vortex center at the boundary. First, the centriod of the vortex tubes at the planes x = −π and y = −π is determined using same procedure as discussed above. Then, vortex lines that seeds from these two centers are integrated using the "stream3" function in Matlab. Figure 4 (and supplementary movie S4) shows the time evolution of the vortex axis for Re Γ = 2000; and the evolution at Re Γ = 4000 are quantitatively the same. It is clear that the axis of vortex tubes is unambiguously identified. Finally, δ is taken as the shortest distance between these two vortex lines.
For the pre-reconnection, δ 2 (t) initially varies slowly during the phase of the formation of anti-parallel configuration (figures 3a). Then, the perturbed vortex tubes approach each other rapidly with δ 2 (t) follows a clear linear scaling. A slight difference in the evolution of δ 2 (t) can be observed between Re Γ = 2000 and 4000 cases, indicating a weak Reynolds number effect on the pre-reconnection evolution. For both Re Γ cases, the linear fit shows that a − ≈ 0.29, which is slightly smaller than the case of the colliding rings. Figure 3(b) shows that δ 2 (t) also follows linear scaling after reconnection, and the δ ∼ t 1/2 scaling extend far beyond the initial separation distance δ 0 . Consistent with Case I, the prefactor increases with Re Γ , with a + = 0.93 and 0.99 for Re Γ = 2000 and 4000, respectively. Again, the vortices move faster after the reconnection than before it. Similar to the finding in Villois et al. (2017), the prefactors a + are smaller than those in Case I, which might be due to a smaller curvature of the cusps generated after reconnection in this case.

Vortex ring and tube interaction
The third case we considered is a vortex ring interacting with an isolated rectilinear vortex tube. The radius of the ring is chosen the same as the colliding vortex rings case, namely, R 0 = 1. To reveal the crossover from driven (δ ∼ t) to interaction (δ ∼ t 1/2 ) region observed in Galantucci et al. (2019), the initial distance is chosen as twice the previous cases, namely, , respectively (see also supplementary movies S5 and S6). Due to the self-induction, the vortex ring approaches the vortex tube; during this phase, both the vortex ring and tube are perturbed; at close approach, the vortex ring and tube are also deformed into locally antiparallel configuration (i.e., t = 1). It further confirms the argument that reconnection physics of two vortices should be independent of the initial spatial configuration (Siggia & Pumir 1985). After reconnection, parts of the vortex ring and tube exchange with each other; and due to the Kelvin wave, the newly formed vortex ring and tube become further perturbed with the threads connecting them. Figure 6(a) displays the evolution of δ(t) with figure 6(b) and (c) showing δ 2 (t) before and after reconnection, respectively. Initially, δ(t) scale almost linearly with t and the approaching velocity can be approximately determined by the initial self-induced velocity of the ring and the mutual-induced velocity between the ring and the tube. Consistent with the previous two cases, when the two vortices are close to each other, a clear t 1/2 scaling for δ is observed (inset in figure 6b). The transition between driven (δ ∼ t) and interaction (δ ∼ t 1/2 ) regions happens at δ ∼ 0.3. The prefactor for Re Γ = 4000 is a − = 0.40, which is very close to Case I.
From figure 6(c), it is clear that δ(t) ∼ t 1/2 scaling holds after reconnection, with the prefactor a + = 1.28 and 1.34 for Re Γ = 2000 and 4000, respectively. The values are between the colliding vortex rings and orthogonal tubes cases. The 1/2 scaling breaks down when the vortex ring moves sufficiently far away from the tube. Note that the crossover between the t 1/2 to t 1 scalings for δ(t) in the post-reconnection is not observed. Instead, for this case δ(t) remains almost constant at late times. The reason is that the traveling velocity of the perturbed vortex ring is roughly the same as that of the perturbed part of the tube. When the oscillations in the vortex tube and ring die out and the vortex ring regains its circular shape, we should expect δ ∼ t as suggested in Galantucci et al. (2019).

Conclusions
The question of whether there is a universal scaling/route for reconnection has been extensively studied and debated (Zuccher et al. 2012;Villois et al. 2017;Fonda et al. 2019). Prior works on quantum vortex reconnection have shown clear evidence for the existence of a universal δ ∼ t 1/2 scaling; however, due to the complex nature for reconnection in classical fluids (presumably due to viscosity), this scaling has never been confirmed previously. With the aid of recent advances in supercomputing, we performed direct numerical simulation of viscous reconnection for slender vortices at Re Γ = 2000 and 4000. Three different initial conditions are considered, namely, two colliding vortex rings; orthogonal and straight vortex tubes; and vortex ring interacting with a tube. For all these cases, the vortices evolve into locally antiparallel configuration -akin to the finding in Villois et al. (2017) for the reconnection of quantum vortices. When the distance between two interacting vortices is large compared with their core size, and the dynamics are predominately governed by their mutual induction, we observe, for the first time, that the approach and separation distances follow a symmetrical 1/2-power scaling, independent of the initial configuration. The discrepancies in previous studies (Hussain & Duraisamy 2011;Yao & Hussain 2020b) are due to the fact that the length scale of vortex core size σ is approximately the same order as the separation δ and should be incorporated when considering the scaling. Although the dynamics of the reconnection is substantially different from that in quantum fluids, the surprisingly similar results in classical fluids regarding δ(t) scaling suggest that there is indeed a universal route towards reconnection. Consistent with previous results (Zuccher et al. 2012;Boué et al. 2013;Villois et al. 2017), we find that the prefactors a ± in the square root law is not universal and depend on the initial configuration as well as the Reynolds number (or viscosity) -which is a distinct feature for classical vortex reconnection.