Approach and separation of quantum vortices with balanced cores
Cecilia Rorai, Jack Skipper, Robert M. Kerr, Katepalli R. Sreenivasan
11 Approach and separation of quantumvortices with balanced cores
C. Rorai , J. Skipper , R. M. Kerr , K. R. Sreenivasan Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, UKCB2 1PZ [email protected] Department of Mathematics, University of Warwick, Coventry, UK CV4 7AL Departments of Physics, Mechanical and Aerospace Engineering, and the Courant Institute ofMathematical Sciences, New York University, Bobst Library, 70 Washington Square South,New York City, NY 100128 November 2018
The scaling laws of isolated quantum vortex reconnection are characterised by numer-ically integrating the three-dimensional Gross-Pitaevskii equations, the simplest mean-field equation for a quantum fluid. The primary result is the identification of distinctlydifferent temporal power laws for the pre- and post-reconnection separation distances δ ( t ) for two configurations. For the initially anti-parallel case, the scaling laws beforeand after the reconnection time t r obey the dimensional δ ∼ | t r − t | / prediction withtemporal symmetry about t r and physical space symmetry about the mid-point betweenthe vortices x r . The extensions of the vortex lines close to reconnection form the edgesof an equilateral pyramid. For all of the initially orthogonal cases, δ ∼ | t r − t | / be-fore reconnection and δ ∼ | t − t r | / after reconnection, which are respectively slowerand faster than the dimensional prediction. For both configurations, smooth scaling lawsare generated due to two innovations. One is to initialise with density profiles aboutthe vortex cores that suppress unwanted secondary temporal density fluctuations. Theother innovation is the accurate identification of the position of the vortex cores froma pseudo-vorticity constructed on the three-dimensional grid from the gradients of thewave function. These trajectories allow us to calculate the Frenet-Serret frames and thecurvature of the vortex lines, secondary results that might hold clues for the origin of thedifferences between the scaling laws of the two configurations. For the orthogonal cases,the reconnection takes place in a reconnection plane defined by the directions of the curva-ture and vorticity. To characterise the structure further, lines are drawn that connect thefour arms that extend from the reconnection plane, from which four angles θ i betweenthe lines are defined. Their sum is convex or hyperbolic, that is (cid:80) i =1 , θ i > ◦ , asopposed to the acute angles of the pyramid found for the anti-parallel initial conditions. Key words:
Gross-Pitaevskii equations, Bose-Einstein condensate, quantum fluids, vor-tex reconnection
1. Background
The term “quantum turbulence” refers to a tangle of quantum vortex lines, a tanglewhose formation and decay is determined by how these vortices collide, reconnect andseparate. Although superfluid tangles form in a variety of He or He experiments such ascounter-flow, moving grids, colliding vortex rings (Skrbek & Sreenivasan 2012; Walmsley& Golov 2008), until recently very little has been known directly about the underlying a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b microscopic interactions. Instead, the nature of the vortex interactions has been inferredfrom how rapidly the tangle decays.Theoretically, the observed decay has been linked to the conversion of the kinetic energyof the vortices into other forms of energy. This could be conversion into the kinetic en-ergy of the normal component in higher temperature experiments or into the interactionenergy and quantum waves in low temperature quantum fluids, including Bose-Einsteincondensates. Despite this, most of our current theoretical insight into quantum vortexreconnection has been through Lagrangian, Biot-Savart simulations of isolated vortexfilaments, a dynamical system that does not include the terms for the interaction energy.Why?Part of the reason is that the Lagrangian approach has experimental support, mostrecently by comparisons between experiments tracking quantum vortices with solid hy-drogen particles (Bewley et al. et al. δ , was interpreted in terms of the di-mensional analysis based upon the circulation Γ of the vortices. That is, if ddt δ ∼ v ∼ Γ /δ ,then one would expect that δ ( t ) ∼ (Γ | t r − t | ) / . (1.1)This will be called the dimensional scaling .Alternatively, one can simulate the underlying mean-field equations of quantum fluidsand visualise vortex reconnection by following the low density isosurfaces that surroundthe zero density cores. The problem with this approach is that tracking the motion ofthe vortices within these isosurfaces is difficult, even for single interactions.The aim of this paper is to begin to fill that gap using two innovations for solutionsof the mean-field, hard-sphere Gross-Pitaevskii equations. One innovation is an initialcondition that suppresses fluctuations in the temporal scaling of separations and thesecond is a method for identifying the position of the vortex cores. These innovationswill be used to determine scaling laws for two classes of initial configurations, orthogonalor anti-parallel vortices.The conclusion will be that the scaling laws for the minimum separation distancebetween the two vortices in the two configurations are distinctly different, even when thepairs are just several core radii apart. The anti-parallel case obeys the expectations from(1.1), but the orthogonal cases consistently obey a distinctly different type of scaling.The two sets of scaling laws will be associated with differences in the alignment of theirrespective Frenet-Serret coordinate frames, differences that form almost immediatelyThis paper is organised as follows. First, the equations and the initialisation of themodel are introduced, followed by overviews of the anti-parallel and orthogonal globalevolution in three-dimensions. Next, the methods used to identify the trajectories ofthe vortices and the local properties of the Frenet-Serret frame, including curvature,are explained. The numerical results, arranged by the type of simulation, orthogonaland anti-parallel, are then described. The results include the time dependence of theseparation of the vortices, the curvature along the vortices and the alignments in termsof the Frenet-Serret frames. Finally, the differences between the two classes of initialconditions are discussed and how these differences might affect the observed scaling lawsfor the approach and release of reconnecting vortices.
2. Equations, numerics and initial condition
Following Berloff (2004), the three-dimensional Gross-Piteavskii equations for the com-plex wave function or order-parameter ψ are1 i ∂∂t ψ = E v ∇ ψ + V ( | x − x (cid:48) | ) ψ (1 −| ψ | ) with E v = 0 . V ( | x − x (cid:48) | ) = 0 . δ ( x − x (cid:48) ) . (2.1)These are the mean-field equations of a microscopic, quantum system with (cid:126) and m non-dimensionalized to be 1, a chemical potential of E v = 0 . V ( | x − x (cid:48) | ). They are an example of a defocusing nonlinear Schr¨odingerequation. All calculations in this paper will use (2.1).These equations conserve mass: M = (cid:90) dV | ψ | (2.2)and a Hamiltonian H = 12 (cid:90) dV (cid:2) ∇ ψ · ∇ ψ † + 0 . − | ψ | ) (cid:3) (2.3)where ψ † is the complex conjugate of ψ . The local strength of the mass density, kineticor gradient energy K ψ and the interaction energy I are ρ = | ψ | , K ψ = 12 |∇ ψ | and I ( x ) = 14 (1 − | ψ | ) (2.4)Isosurfaces of ρ are used in all the three-dimensional visualisations and K ψ are includedin figures 2.1, 2.2, 2.3, 4.3 and 5.1.Gross-Pitaevskii calculations have previously identified the following features of quan-tum vortex reconnection. First, it has been demonstrated (Leadbeater et al. et al. et al. et al. K ψ is lost by these means during the initialreconnection (Kerr 2011). With added terms representing assumptions about the type ofenergy depletion at small-scales, these equations can also give us hints to why the vortextangle decays (Sasa et al. Quasi-classical approximations
But how can the continuum Gross-Pitaevskii equations provide us with details aboutthe Lagrangian dynamics and reconnections that underlie the vortex tangle of quantumfluids?This can be done by writing the wave function as ψ = √ ρe iφ , where ρ is the densityand φ is the complex phase, then defining the phase velocity v φ and quantised circulationΓ around the line defects: v φ = ∇ φ = Im( ψ † ∇ ψ ) /ρ Γ = (cid:90) v φ · s = 2 π (2.5)then identifing the ρ ≡ ω = ∇ × v φ = ∇ × ∇ · φ ≡
0. In this picture, vortex reconnection appearsnaturally as the instantaneous re-alignment of these lines and exchange of circulationwhen the line defects meet. If dimensions were added, the quantised circulation has theclassical units of circulation: Γ ∼ ρL /T . Note these two differences with classical vorticesgoverned by the Navier-Stokes equation: classical circulation is not quantised and viscousreconnection is never 100%.To extract the Lagrangian motion of quantum vortices from fields defined on three-dimensional meshes these issues must be addressed: • As the state relaxes from its initial form, it should not be dominated by either interalwaves (phonons) or strong fluctuations along the vortex trajectories (Kelvin waves). • Second, a method is needed for identifying the direction and positions that thevortices follow as they pass through the three-dimensional mesh.Two innovations introduced in this paper, (2.7) for the initial density and (3.1) fortracking vortices, resolve both problems and allow us to extract smooth motion forthe vortices from the calculated solutions of the Gross-Pitaevskii equations on Eulerianmeshes.To complete the discussion of the Gross-Pitaevskii equations (2.1), the full analogy toclassical hydrodynamic equations comes from inserting ψ = √ ρ exp( iφ ) into (2.1) to getthe standard equation for ρ and a Bernoulli equation for φ : ∂∂t ρ + ∇ · ( ρ v φ ) = 0 ∂∂t φ + ( ∇ φ ) = 0 . − ρ ) + ∇ √ ρ/ √ ρ . (2.6)The v φ velocity equation can then be formed by taking the gradient of the φ equation.As in Kerr (2011), the numerics are a standard semi-implicit spectral algorithm wherethe nonlinear terms are calculated in physical space, then transformed to Fourier spaceto calculate the linear terms. In Fourier space, the linear part of the complex equationis solved through integrating factors with the Fourier transformed nonlinearity addedas a 3rd-order Runge-Kutta explicit forcing. The domain is imposed by using no-stresscosine transforms in all three directions. For all of the calculations the domain size is L x × L y × L z = (16 π ) ≈ or (32 π ) . Both 128 and 256 grids were used, with the256 grid giving smoother temporal evolution. Most of the analysis and graphics will usethe δ = 3, 256 calculation.2.2. Choice of initial configurations and profiles
Configurations.
Two initial vortex configurations are used in this paper, anti-parallelvortices with a perturbation, and orthogonal vortices. Both configurations have been usedmany times for both classical (Navier-Stokes) and quantum fluids, including the firstcalculations using the three-dimensional Gross-Pitaevskii equations (Koplik & Levine1993).The advantage of focusing on these configurations is that the interactions leading toreconnection for most other configurations, for example colliding or initially linked, vortexrings, can be reduced to either anti-parallel or orthogonal dynamics, both of which canresolve the reconnection events in smaller global domains. This is because the initialreconnection events require, effectively, only half of each ring.These two configurations also represent the two extremes for the initial chirality orlinking number of the vortex lines in a quantum fluid. In classical fluids the correspond-ing global property is the helicity, h = (cid:82) ( ∇ × u ) · u dV . When h is large, it tends tosuppress nonlinear interactions. Anti-parallel initial conditions have zero net helicity andare sensitive to initial instabilities while orthogonal initial conditions have a large helicity,so reconnection can be delayed (Boratav et al. Density profile.
The density profiles for all the vortex cores in this paper are deter- (a) (b)
Figure 2.1.
Anti-parallel case: Density, kinetic energy K ψ = |∇ ψ | and pseudo-vorticity |∇ ψ r ×∇ ψ i | isosurfaces plus vortex lines at these times: (a) t = 0 . K ψ ) = 1 .
3, and(b) t = 4 with max( K ψ ) = 0 .
83. max | ω ψ | = 0 . ρ = 0 well, but the t = 4 frame also showsthat it can be used to follow isosurfaces of fixed ρ . The t = 0 .
125 frame shows that initially thekinetic energy is largest between the elbows of the two vortices, with large ω ψ at the elbows.The t = 4 frame shows the newly reconnected vortices as they are separating with undulationson the vortex lines that will form additional reconnections and vortex rings at later times. mined by the following Pad´e approximate: | ψ sb | = √ ρ sb = c r + c r d r + d r with c = 0 . , c = 0 . , d = 0 . , and d = 0 . . (2.7)Note that c (cid:46) d , which implies that as r → ∞ , the density approaches the usualbackground of ρ = 1 from below more slowly than the true Pad´e of this order does.The true Pad´e, derived by Berloff (2004), has c = d = 11 / ≈ . c (cid:46) d is designated as ψ sb because it is a sub-Berloff (2004) profile.Furthermore, because the calculations are in finite domains, to ensure that the Neumannboundary conditions are met, a set of up to 24 mirror images of the vortices are multipliedtogether. This multiplication process takes the slight, original ρ sb < r → ∞ and generates stronger differences. At the boundaries, this gives (1 − ρ ) ≈ . − . | ψ | (cid:46) Anti-parallel: Initial trajectory and global development.
Based upon past experience with classical vortices and Kerr (2011), the positions s ± ( y ) ofthe two nearly anti-parallel y -vortices, with a perturbation in the direction of propagation (a) t = 6(b) t = 14 Figure 2.2.
Overview of the evolution of the δ = 3 256 orthogonal calculation from a three--dimensional perspective for two times: t = 6 asreconnection is beginning and t = 14 after it hasended, where t r ≈ .
9. Green ρ = 0 .
05 isosur-faces encase the vortex cores, blue and red lines.The Frenet-Serret frames for t = 8 .
75 are givenin figure 4.3. A twist in the post-reconnection t = 14 right ( x >
0) vortex is visible. (a) t = 6(b) t = 14
Figure 2.3.
The same δ = 3 fields and timesas in figure 2.2 from the Nazarenko perspective,which looks down the 45 ◦ propagation plane inthe y − z -plane in figure 4.5. This perspectiveis useful because the pre- and post-reconnectionsymmetries can be seen most clearly, which iswhy it is the basis for comparison of reconnec-tion angles in figure 4.6 and section 4.6. Thearms are extending away from the reconnectionplane , either towards or away from the viewer,Also note that from this perspective, loops areclearly not forming. x , were s ± ( y ) = (cid:32) δ x (cid:0) ( y/δ y ) . (cid:1) − x c , y, ± z c (cid:33) . (2.8)The parameters used were δ x = − . δ y = 1 . x c = 21 .
04 and z c = 2 .
35. The powerof 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 (2011), two of the Neumann boundaries act as symmetry planes to increasethe effective domain size. These planes are the y = 0, x − z perturbation plane and z = 0, type mesh δ t r K ψ ( t = 0) I ⊥ (16 π ) ⊥ (16 π ) ⊥ (16 π ) ⊥ (16 π ) ⊥ (32 π ) × π ×
128 4 21 0.00178 0.00049 ⊥ (32 π ) ⊥ (32 π ) (cid:107) (16 π ) Table 1.
Cases: Type is orthogonal ⊥ or anti-parallel (cid:107) , initial separations, approximatereconnection times, initial kinetic and interaction energies. x − y dividing plane . Because the goal of this calculation was to focus upon the scalingaround the first reconnection, the long domain used in Kerr (2011) to generate a chainof vortices is unnecessary and L y is less. In addition, based upon recent experience withNavier-Stokes reconnection (Kerr 2013), L z was increased to ensure that the evolvingvortices do not see their mirror images across the upper z Neumann boundary condition.Figure 2.1 shows the state at t = 0 . t = 4, after the first reconnection event at t r ≈ .
4. Three isosurfaces are given. Lowdensity isosurfaces ( ρ = 0 . K ψ (2.4) and isosurfacesof | ω ψ | (3.1), a pseudo-vorticity that is introduced in the next section. The vortex linesdefined by ω ψ and Proposition 1 are shown using thickened curves. The structure atthe time of reconnection is discussed in section 5 using Figure 5.1 and how the flowwould develop later has already been documented by Kerr (2011), which shows severalreconnections forming a stack of vortex rings.Both K ψ and ω ψ are functions of the first-derivatives of the wave function, but showdifferent aspects of the flow. The K ψ isosurfaces show where the momentum is large.Initially, the momentum is dominated by forward motion between the perturbations, asshown for t = 0 . t = 4, the K ψ surfaces show that the primarymotion is around the vortices. | ω ψ | is large where the vortex cores bulge and have thegreatest curvature.2.4. Orthogonal: Initial separations and global development.
To place the orthogonal vortices one only needs to choose one line parallel to the y axis and another line parallel to the z axis through two points in x on either side of x = y = z = 0. The five separations and other details of the simulations are given inTable 1. Because all of the orthogonal cases with δ ≥ δ = 3 calculation, whose estimated reconnection is at time t r = 8 . y = z axis, which is also used forthe determination of three-dimensional angles in subsection 4.4. The two times chosenfor each are t = 6, pre-reconnection, and t = 19, post-reconnection.The two isosurfaces are for a low density of ρ = 0 .
05 and kinetic energy of K ψ = 0 . K ψ ) = 0 .
58. There are two pseudo-vortex lines (3.1) in each frame, one thatoriginates on the y = 0 plane and the other on the z = 0 plane. The t = 8 . Qualitative features are: • The initially orthogonal vortices are attracted towards each other at their points ofclosest approach, asymmetrically bending out towards each other. • During this stage there is a some loss of the kinetic energy between t = 8 and t = 11,∆ t K ψ < K ψ ( t = 0). This is converted into interaction energy I (2.4). There are nofurther noticeable changes in K ψ for t > • After reconnection, from one perspective there is a slight twist on one vortex, butit is not twisted enough for the vortices to loop back upon themselves and reconnectagain. Instead, the two new vortices pull back from one another, as shown by the t = 19frame in figure 2.2. Consistent with experimental observations of vortex interactionsusing solid hydrogen particles (Bewley et al. • Two sketches of the alignments and an addition isosurface perspective are used insubsection 4.3 to illustrate this evolution further.
3. Approach and separation of vortex lines: Methodology
The primary result in this paper will be the differences in the temporal scaling ofthe pre- and post-reconnection separation of the vortices for the two configurations. Thesecondary results are clues for the origin of these differences in the evolution of the localcurvature and Frenet-Serret frames on the vortices for the two configurations. To achievethis, one needs an initial condition for which the evolution of the vortices is smooth anda means to follow that evolution. The key ingredient of the initial condition is providedby the choice of coefficients in (2.7). This section will show what is needed to accuratelytrack the vortices. Two methods for detecting the vortices have been used.3.1.
Detecting lines by finding ρ = 0 mesh cells. The first approach is to estimate the locations of the ρ ≡ ρ (cid:38) ρ has to be smallenough so that there are only 3-6 points clustered in a plane perpendicular to the vortexlines. The method begins to fail around reconnection points because there is an extensive ρ ≈ ρ ≡ ω ψ = ∇ ρ × ∇ φ pseudo-vorticity method The second approach begins by recognising that the line of zero density should be perpen-dicular to the gradients of the real and imaginary parts of the wave function. Thereforeit is useful to define the following pseudo-vorticity : ω ψ = 0 . ∇ ψ r × ∇ ψ i (3.1)The inspiration for this approach comes from how to write the vorticity in terms ofa cross production of the scalars in Clebsch pairs, which is an alternative approach torepresenting the incompressible Euler equations. Proposition 1.
At points with ρ = 0 , the direction of the quantum vortex line isdefined by the direction of ω ψ = 0 . ∇ ψ r × ∇ ψ i .Proof. A quantum vortex line is defined by ρ ≡
0, which because ρ = ψ r + ψ i impliesthat ψ r = ψ i = 0 on this vortex line.Then define ˆ ω ρ the direction vector of the line at any arbitary point on the vortex line.By the definition of the line, the values of ψ r and ψ i in this direction must not changeand thus we know that ∇ ψ must satisfyˆ ω ρ · ∇ ψ r = ˆ ω ρ · ∇ ψ i = 0at these points. This is only possible if ˆ ω ρ = ( ∇ ψ r × ∇ ψ i ) / |∇ ψ r × ∇ ψ i | .By itself, this proposition does not tell us where the vortex lines lie because one stillneeds a method for identifying a point on the line. To find starting points for a streamlinefunction, the first method is used to identify points on the boundaries where ρ ≈ ρ ≈ ∇ ψ r,i are small, perhaps too small for the identifying the positionsof neigbouring lines with ρ ≈
0. In practice, this has not been a problem.Once the lines have been found, the derivatives along their trajectories of their three-dimensional positions can be determined, and from those derivatives the local curvature,Frenet-Serret coordinate frames and possibly the local motion of the lines can be found.Properties that could be compared to the predictions of vortex filament models.To analyse these properties, the following alternative definition of the pseudo-vorticityis useful.
Corollary 1. ˆ ω ρ = ˆ ω ψ where ˆ ω ρ = ∇ ρ × ∇ φ/ |∇ ρ × ∇ φ | Proof.
Start with ψ r = √ ρ cos φ and ψ i = √ ρ sin φ .Expand: 0 . ∇ ψ r × ∇ ψ i = [ ∇√ ρ cos φ − √ ρ sin φ ∇ φ ] × [ ∇√ ρ sin φ + √ ρ cos φ ∇ φ ].Remove all ψ r and ψ i terms sharing the same gradient to reduce this to2( ∇√ ρ × ∇ φ √ ρ cos φ − ∇ φ × ∇√ ρ √ ρ sin φ )Finally, use ∇√ ρ = ∇ ρ/ (2 √ ρ ) to get ω ψ = ∇ ρ × ∇ φ .Do these lines follow the cores of ρ ≡
0? One test is to interpolate the densities fromthe Cartesian mesh to the vortex lines. The result is that these densities are very small,but not exactly zero. Another test is simultaneously plot the pseudo-vorticity lines alongwith very low isosurfaces of density, examples of which is given in figure 2.1 and figure2.2. The centres of the isosurfaces and the lines are almost indistinguishable.Using the next proposition, the motion of the ρ = 0 lines given by the time derivative0of ψ can be written exactly using just the gradients and Laplacians of the wavefunction ψ . This will will be used in a later paper. Proposition 2.
The motion of the vortex line is given by the coupled set of equations ( ∇ ψ r × ∇ ψ i ) · ddt x ( s, t ) = 0 −∇ ψ r · ddt x ( s, t ) = − . ψ i −∇ ψ i · ddt x ( s, t ) = 0 . ψ r . The solution of which is ddt x ( s, t ) = − . ψ i ( ω ψ × ∇ ψ i ) + ∆ ψ r ( ω ψ × ∇ ψ i ) ω ψ (3.2) where pseudovorticity ω ψ := ∇ ψ r × ∇ ψ i Proof.
We already know that the trajectory of the vortex lines is defined by thepseudovorticity ω ψ = ∇ ψ r × ∇ ψ i from proposition above.Since the density remains zero along this line, the motion we are interested in isperpendicular to this direction.On the ρ = ψ r + ψ i ≡ ψ r,i are: ∂∂t ψ r = − . ψ i ∂∂t ψ i = 0 . ψ r Next we can Taylor expand to first order ψ r,i about the parameterised curve x ( s, t ). ψ r = ( ∇ ψ r )( x − x ( s, t )) and ψ i = ( ∇ ψ i )( x − x ( s, t ))and their time-derivatives again to first order are ∂∂t ψ r = ( ∇ ∂∂t ψ r )( x − x ( s, t )) − ∇ ψ r ddt x ( s, t ) ≈ −∇ ψ r ddt x ( s, t ) ∂∂t ψ i = ( ∇ ∂∂t ψ i )( x − x ( s, t )) − ∇ ψ i ddt x ( s, t ) ≈ −∇ ψ i ddt x ( s, t )By adding that the motion will be perpendicular to the vortex (i.e. the pseudovorticity ∇ ψ r × ∇ ψ i ) to the two time derivative equations, one gets the required three coupledequations. 3.3. Curvature obtained from the ω ψ lines The curvature of the lines identified by the pseudo-vorticity algorithm will be found byapplying the Frenet-Serret relations to derivatives of the trajectories r ( s ) of the vortexlines.Definition 3.1 The Frenet-Serret frame for any smooth curve r ( s ) : [0 , → R hasan orthonormal triple of unit vectors ( T , N , B ) at each point r ( s ) where T ( s ) is thetangent, N ( s ) is the normal and B ( s ) is the binormal. The following relations between T , N , B ) define the curvature κ and torsion τ . T ( s ) = ∂ s r ( s ) (3.3 a ) ∂ s T = κ N (3.3 b ) ∂ s N = τ B − κ T (3.3 c ) ∂ s B = − τ N (3.3 d )The numerical algorithm for calculating the curvature and normal uses the function gradient in Matlab twice. That is, first r ,s and then r ,ss are generated. Next, normalising r ,s gives the tangent vector T , the direction vector between points on the vortex lines.Finally, the derivative of T gives us both the curvature, κ = | ∂ s T | and the normal N = ∂ s T /κ . In practice it is better to calculate the curvature using: κ = | r ,s × r ,ss | / | r ,s | (3.4)
4. Orthogonal reconnection: New scaling laws and their geometry
The goals of this section are to to apply the pseudo-vorticity algorithm (3.1) to theevolution of the initially orthogonal vortex lines and use these positions to demonstratethat the separation scaling laws for the originally orthogal vortices deviate strongly fromthe mean-field prediction for all initial separations and for all times.The major points to be demonstrated for the orthogonal calculations are: • For strictly orthogonal initial vortices, there is just one reconnection and loops donot form out of the post-reconnection vortices in figure 2.2. • The sub-Berloff profiles are crucial for obtaining temporal evolution that is smoothenough to allow clear scaling laws for the pre- and post-reconnection separtations to bedetermined (Rorai 2012). • The separation scaling laws before and after reconnection are the same for each case,with these surprising results. Before reconnection δ in ∼ ( t r − t ) / , which is slower thanthe dimensional scaling (1.1). And after reconnection δ in ∼ ( t r − t ) / , faster than thedimensional scaling (1.1). • This non-dimensional scaling arises as soon as the vorticity tangent vectors at theirclosest points are anti-parallel and the alignment of the averaged Frenet-Serret frames atthese points with respect to the separation vector are respectively orthogonal, paralleland orthogonal for the averaged tangent, curvature and bi-normal. • Reconnection occurs in the reconnection or osculating plane defined by the vorticityand curvature vectors at t = t r and is for all times approximately the plane defined bythe average vorticity and curvature vectors of the two vortices at the points of closestapproach. • Angles taken between the reconnection event and the larger scale structure areconvex, not concave or acute, which could be the source of the non-dimensional separationscaling laws. 4.1.
Approach and separation
The steps used to determine the separation scaling laws are these: • First, identify the trajectories of the vortex lines with the pseudo-vorticity plusMatlab streamline algorithm. At any given time, both before and after reconnection: ◦ The vortex originating on the y = 0 plane will be the y -vortex. ◦ The vortex originating on the z = 0 plane will be the z -vortex.2 δ t p r e / , t po s t / y = 0.67*x + 0.064 Figure 4.1.
Reconnection times as afunction of the initial separation. Re-connection times are estimated by usingtimes immediately before and after re-connection plus the empirical 1/3 and2/3 scaling laws. −50 0 500100200 t−t r δ , δ −50 0 500100200 t−t r δ Figure 4.2.
Pre- and post-recon-nection separations for four cases: δ = 2 , , , , points. • : δ = 2, +: δ = 3, (cid:13) : δ = 4, (cid:62) : δ = 5, (cid:52) : δ = 6. Pre-recon-nection distances are raised to the power3, while post-reconnection distances areraised to the power 3/2. This scaling isvisibly better than the dimensional pre-diction ( δ versus time to reconnection)shown in the inset. The scaled separa-tions for t < t r fluctuate more stronglythan the t > t r scaled separations. • Identify the points, x y and x z , of minimum separation between the two vortex lines,defined as δ yz ( t ) = | x y − x z |◦ and identify approximate reconnection times ˜ t r ( δ ) when δ yz ( t ) was minimal. ◦ This generates δ yz versus t − t r curves such as those in the inset of figure4.2. • Once it was clear that neither the incoming nor outgoing separations obeyed thedimensional expectation (1.1), several alternative scaling laws were applied to the sepa-rations. Only the 1/3 incoming power law and outgoing 2/3 law working well for everycase. • By using these scaling laws to make the approach and separation linear, refinedestimates of ˜ t r ( δ ) can then be made. ◦ That is, cube the δ separations for t < ˜ t r ( δ ). ◦ And take the 3/2 power of δ for t > ˜ t r ( δ ). ◦ Then extrapolate these linear fits to the times when δ = 0. ◦ For all the initial δ , the t < t r and t > t r estimates of t r were nearly identical. • The combined results give the fit t r = (0 . δ + 0 . and are shown in figure 4.1. • Using these t r ( δ ), figure 4.2 compares the scaled pre- and post-reconnection sepa-rations δ ( t ) for all 5 cases to demonstrate as in Rorai (2012) that: ◦ δ in ∼ | t r − t | / for t < t r ( δ ) and δ out ∼ | t − t r | / for t > t r ( δ ). ◦ Note the inset which uses the dimensional scaling δ yz versus time to illustrate thedifferences between the new scaling laws and the dimensional prediction.4.2. Sub-Berloff profile.
Why was it necessary to use the sub-Berloff profile? That is, could a different profilegive similar separation collapse for all the δ cases and obtain clear scaling laws, as infigure 4.2? To show the benefits of the sub-Berloff profile, tests were done using all ofthe known Pad´e approximate profiles of steady-state two-dimensional quantum vortices3in an infinite domain, including tests with and without adding the mirror images. Thisincluded, the true Berloff profile, that is (2.7) with c = d = 0 . × d = 1 and c = d = 0, and a varietyof 3 × et al. (2013),solutions that are very close to the ideal diffusive solution. All gave roughly the sameoscillations in the approach and separation curves as in Zuccher et al. (2012) and only ahint of the clear scaling laws in figure 4.2. Only the sub-Berloff profile with at least someof the mirror images worked. Further work will be needed to identify why instabilitiesgenerated on the vortex lines are either suppressed by the sub-Berloff profile, or absorbedby it. 4.3. Evolution of the orthogonal geometry during reconnection
The three-dimensional evolution of the vortices is illustrated in figures 2.2, 2.3 and 4.3and the alignments at or near reconnection are illustrated with two sketches taken fromdifferent perspectives in figures 4.5 and 4.6. The purpose of the sketches is to emphasizethe strong qualitative differences between the orthogonal reconnection’s skew-symmetricalignment and the anti-parallel case with its planar symmetries. Quantitative alignmentsare then given in figures 4.7, 4.8 and 4.9. This initial discussion is divided into threeparts. First, the choice of three-dimensional images. Second, the role of the sketches.Third, how to use your fingers to put the pieces together into a mental picture.
Choice of 3D images.
Section 2.2 uses figures 2.2 and 2.3 to illustrate the globalchanges in structure from two perspectives. One is a general perspective and the otheris the Nazarenko perspective that shows the symmetries. Each figure has a t = 6 frame,long before the reconnection at t r = 8 .
9, and a frame at t = 14, long after reconnection.Figure 4.3 focuses upon the reconnection zone using three times: t = 6 is at thebeginning of reconnection, t = 8 .
75 is just before the reconnection time of t r ≈ . t = 12 shows the end of reconnection. Over this period the ρ = 0 .
05 isosurfaces changeslowly while the ρ ≡ Sketches: 2D and 3D.
Figures 4.5 and 4.6 provide two planar sketches at or near thereconnection time, with the best reference point for each being the mid-point betweenthe closest points on the two vortices: x r ( δ , t ) = 0 . x y ( t ) + x z ( t )) . (4.1)The sketch in figure 4.5 projects the vortices at the reconnection time t r onto the y − z plane around the point of reconnection: x r ( δ , t r ), along with projections of two planes ofinterest, the reconnection or osculating plane and the propagation/symmetry plane , whichare defined in terms of the average Frenet-Serret basis vectors (4.2) in subsection 4.5.The second sketch in figure 4.6 looks down the x = 0, y = z ◦ direction of propagation of x r ( δ , t ) onto the T av × N av (4.2) reconnection plane . Important features include: • Because figure 4.5 is at t = t r , the projections of the planes and vortices all crossat x = x r ( δ , t r ), which means that the red and blue curves trace both the pre- andpost-reconnection trajectories of the vortices, as follows: ◦ 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 bluelines in figure 4.6. ◦ The trajectories immediately after reconnection are indicated by the red and bluecurves coming out of the reconnection plane . • The two orthogonal lines through x r represent two planes:4 t = 6 a) t = 8 . ≈ t r b) t = 12 c) Figure 4.3.
Isosurfaces, quantum vortex linesand orientation vectors in three-dimensions forthe δ = 3, 256 calculation at t = 6 (4.3a), t = 8 .
75 (4.3b) and t = 12 (4.3c) where t r ≈ . D and Frenet-Serretvectors (3.3): T , N and B . a)b) Figure 4.4.
Curvature of the vortex lines isgiven only for the δ = 3 256 calculation with t r ≈ . t r ( δ ) are similar forall δ . (a) Curvatures against arclength s at fourtimes: t = 6, 8.75, 9.5 and 12. The profiles forthe other line are similar, with their maximumpeaks at t = 9 . t = t r . Figure 4.5.
Sketch of the orthogonal reconnectionaround the reconnection point x r at the reconnectiontime t r of the vortices projected onto the y − z plane.The pre-reconnection trajectories approximately followthe y − y mid = 0 and z − y mid = 0 axes and the post-re-connection trajectories are the red and blue lines. Twoprojected planes are indicated in black. The reconnec-tion plane ( z − z r ) = − ( y − y r ) contains the tangentsto the vortex lines T y,z and their curvature vectors N y,z both before and after reconnection, as well at the separa-tion vector between the closest points x y,z . The propaga-tion/symmetry plane ( z − z r ) = ( y − y r ) which containsthe bi-normals B y,z and the direction of propagation ofthe mid-point x r ( t ) between the closest points of the vor-tices: x y and x z . From this perspective the angles alongthe reconnection plane before reconnection are shallow,and those after reconnection are sharp. x r x
2, v2 x
1, v1 x
1, v2 x
2, v1 xzy
Figure 4.6.
Sketch using theNazarenko perspective, that islooking down on the reconnec-tion plane along the propaga-tion/symmetry plane , just beforereconnection. Four line segments(in three-dimensions) have beenadded that connect x r , the mid–point between the vortices at clos-est approach, to four points, eachlocated ∆ s units along the vor-tices from their respective closestpoints. Angles generated betweenadjacent segments as a function of∆ s are plotted in figure 4.9. ◦ The reconnection plane , defined by T av = 0 . T y − T z ) and N av = 0 . N y − N z )(4.2). Before or after reconnection, the separation vector D = ( x z − x y ) / | x z − x y | (cid:54) = 0is also in this plane. — It is shown below that T y,z and N y,z swap at reconnection, so all of thesebasis vectors stay in this plane after reconnection. ◦ And the propagation plane , which contains the velocity of x r ( δ , t ) and the averagebi-normal B av = 0 . B y + B z ) (4.2). • The x = 0, y = z projection in figure 4.6 is denoted the Nazarenko perspective orNP because it was used by linear model of Nazarenko & West (2003). ◦ That model tells us that x r = 0 . x y + x z ) translates in the ( − y, − z ) direc-tion, motion that implies that the reconnection does not occur at the centre of thecomputational box. ◦ Note that for the points on either side of x r , the tangents T y,z and curvaturevectors N y,z are anti-parallel. The components that are not anti-parallel are directedout of the Nazarenko perspective. This also holds for the lines across the central,green ρ = 0 .
05 isosurface in figure 4.3b. figure 4.7a shows how T y · T z , N y · N z and B y · B z converge to this state as t → t r .6 ◦ The lines drawn across the centre of figure 4.6 are used to determine the long-rangeangles discussed in section 4.6. •
3D by using fingers.
Cross your index fingers while pointing their knuckles towardsone another so they do not touch. ◦ Rotating this configuration reproduces figures 2.2a, 2.3a and 4.3a. ◦ Now move your fingers up, bending them as you do and bringing the knucklestogether. ◦ This is how, the alignments of the Frenet-Serret frames at the points of closestapproach in figure 4.7 form. ◦ Now hold that configuration and rotate it to get the configurations of the sketchesin figures 4.5 and 4.6 and the three-dimensional images in figures 2.2b, 2.3b and4.3b. 4.4.
Curvature
Curvature, has played a central role in our understanding of quantum turbulence due toits use in predicting velocities in the law of Biot-Savart and the local induction approx-imation. The connection between these approximations for the velocities and the truedynamics of quantum fluids, as modeled by the Gross-Pitaevskii equations, would be inhow the gradient of the phase φ of the wavefunction ψ is modified by the curvature ofthe vortex lines.In that context, could curvature profiles provide clues for the origins of the anomalousscaling exponents of the orthogonal separations? For example, if the local inductionapproximation is relevant, then a sudden increase in the maximum curvature of the linescould explain the change in scaling.To assess whether this is a possible explanation, figure 4.4 plots profiles of the curvaturealong the vortex lines at several times, and the curvature maxima κ max as a function oftime for the δ = 3 case. The other cases have similar behaviour, including one slightlyunder-resolved δ = 4 case in a large (32 π ) domain.Two primary features should be noted. First, in figure 4.4a there is very little asymme-try in s about the points of cloest approach, both before and after reconnection. Second,in figure 4.4b, there is some growth in κ max for t < t r , followed by a sharp jump in κ max at t = t r , which then relaxes rapidly to the pre-reconnection values of κ max .So there is some qualitative support for a local induction explanation change betweenthe δ in and δ out scaling, but there is also inconsistency with the following: • Even pre-reconnection, the δ in ∼ ( t r − t ) − / scaling would require a stronger growthin κ than is observed. • Post-reconnection, and after the curvature spike has relaxed, stronger curvaturesthan are observed would be needed to maintain the δ out ∼ ( t r − t ) − / .Therefore, one must conclude that a bigger picture is needed. Our proposal is to look atthe alignments of their respective Frenet-Serret frames as another reason for the changesin scaling. Looking first at the alignments of at the points of closest approach, then atdistances away from those points (Rorai 2012).Another possible explanation, if the local induction approximation is relevant, wouldbe if the curvature maxima are stronger post-reconnection. The inset in figure 4.4 doesshow that the maxima are slightly stronger post-reconnection, but this does not appearto be the whole story. To get a bigger picture we need to look beyond the curvatureprofiles and see how the vortices are aligned with each other for points along their entiretrajectories. Starting with the alignments of their respective Frenet-Serret frames at thepoints of closest approach, then for distances away from those points.74.5. Orthogonal: Frenet-Serret orientation.
Besides allowing us to calculate the curvature of the vortex lines, knowing their trajec-tories allows us to calculate the Frenet-Serret frame (3.3). This has been done for the256 , δ = 3 calculation for all times and provides quantitative support for the describingthe local frame at the reconnection point x r in terms of the reconnection plane and the propagation plane indicated in figure 4.5.Figures 4.7 and 4.8 show the following evolution of the Frenet-Serret frames at theclosest points x y,z : • For t < t r , pre-reconnection: ◦ figure 4.7a shows that the vorticity direction or tangent vectors T y,z , the curvaturevectors N y,z and the bi-normals B y,z converge to their opposites as t → t r , bothbefore and after reconnection. ◦ figure 4.7b shows that T y,z , N y,z and their unit separation vector D = ( x z − x y ) / | x z − x y | all lie in the same plane, the reconnection plane in figure 4.5. ◦ The bi-normals B y,z define the propagation plane . ◦ The useful averages of the Frenet-Serret frames between x y and x z are these: ◦ N av = 0 . N y − N z ) and is parallel to D ( t ) ◦ T av = 0 . T y − T z ) and is perpendicular to both N av and D . ◦ B av = 0 . B y + B z ) and is perpendicular to T av , N av and D . (4.2) ◦ That is: Subtract the tangent and curvature vectors because they become anti-parallel as t (cid:37) t r , and add the bi-normals because they are parallel as t (cid:37) t r . ◦ These alignments between T av , N av and B av with D form in the early stages,long before the reconnection at t r = 8 . • The post-reconnection Frenet-Serret flip: ◦ Figure 4.8 shows that the directions of T av and N av swap and D rotates by 90 ◦ so that all three are still in the reconnection plane with the same relations to oneanother. ◦ B av remains orthogonal to the reconnection plane .The alignments of the components of the Frenet-Serret frames at x y,z and the separa-tion of these points D is significantly different than their alignments for the anti-parallelcase in Sec. 5. Comparisons are discussed in the Summary in Sec. 6.4.6. Angles at reconnection
What additional dynamics might help us identify the differences between the orthogonaland anti-parallel separation scaling laws?One place to look is larger-scale alignments and long-range interactions. While theunderlying Gross-Pitaevskii equations are local, the existence of the vortex structuresmeans that such alignments should exist and should influence the local motion of the ρ ≡ i ) From the points of closest approach x y,z ( t ), define x r ( t ) = 0 . x y ( t ) + x z ( t )).( ii ) Move ± ∆ s along the arms of the vortices from x y,z ( t ) and identify four new points: x ,y , x ,y , x ,z and x ,z , illustrated in the Nazarenko perspective sketch in figure 2.28 a)b) Figure 4.7.
These two frames show how the basis vectors of the Frenet-Serret frames (3.3) at theclosest points x y,z of the δ = 3, 256 calculation are aligned and help define the reconnectionplane and the propagation plane used in figures 4.5 and 4.6. a): The alignments between theFrenet-Serret components at x y,z . For t < t r the tangent vectors T y,z and curvature vectors N y,z become increasingly anti-parallel as the reconnection time is approached while the bi-normalvectors B y,z become increasingly aligned. These trends are are reversed for t > t r . b): Thealignments between the averages over x y,z of the Frenet-Serret frames defined by (4.2) with theseparation vector D between x y and x z . Figure 4.8.
For the δ = 3, 256 calculation, the alignments over timeof D ( t − r ) = D ( t r − dt ) with T av and N av , the averages (4.2) over thetangent and curvature components ofthe Frenet-Serret frames at the closestpoints x y,z . B av · D ( t − r ) ≈ D ( t − r ) is the separation direc-tion at t = 8 .
9, just before reconnec-tion. ( iii ) To get a fully three-dimensional perspective, note that these points lie on out-stretched arms such as in figure 2.2b.( iv ) Connect the four points with x r to form an extended three-dimensional framethen calculate the angles θ i between these four vectors.( v ) Plots of θ i (∆ s ) show qualitatively similar variations independent of δ with theseproperties:9 ∆ s θ δ =6Pre−reconnection Post−reconnectionPre−reconnection Post−reconnection Figure 4.9.
Angles between the segments in 3D space displayed in figure 4.6 as a function of∆ s for δ = 6 and t = 63 (pre-reconnection) and t = 65 (post-reconnection). The angles betweenthe arms of the same vortex, triplets [ x ,v , x r , x ,v ] and [ x v , x r , x v ], are indicated bycircles (blue and red) before reconnection, and stars (blue and red) after reconnection. Theangles sum to 360 ◦ as ∆ s →
0, implying that the segments near the reconnection point x r lieon a plane, specifically, the reconnection plane in figure 4.6 The sum of the angles grows with∆ s and is about ∼ ◦ for ∆ s ∼ . x r , both before and after reconnection. The rapidly changing behavior for small ∆ s isa kinematic result of how the lines and angles indicated in figure 4.6 were chosen and is notsignificant. What could be more significant are the differences in the angles for intermediate ∆ s and their influence upon any Biot-Savart contributions to the velocities. ◦ The sum of the θ i grows as ∆ s increases, starting from (cid:80) i θ i (∆ s = 0) = 360 ◦ . ◦ This shows that the inner (∆ s ≈
0) structure is a plane. ◦ The vertical line in figure 4.9 represents the ∆ s p for which (cid:80) i θ i (∆ s ) = 362 ◦ > ◦ , indicating that the structure is mildly hyperbolic. ◦ (cid:80) i θ i (∆ s ) increases with ∆ s , implying that the global structure is convex orhyperbolic, the opposite of a structure with an acute angles such as a pyramid. ◦ The primary quantitative difference as δ decreases is that the θ = 90 ◦ pre-reconnection cross-over, at δs = 1 . δ = 6, decreases. ◦ In addition, the geometry becomes more hyperbolic. That is (cid:80) i θ i (∆ s ) increasesas δ decreases with (cid:80) i θ i (∆ s ) = 370 ◦ for δ = 3. ◦ The inner angles, that is the angles between the triplets [ x ,v , x r , x ,v ] and[ x v , x r , x v ], are larger than 90 ◦ before reconnection and smaller after. Thedifference is about 20 ◦ in all of the calculations and would be consistent with theobserved slower approach and faster separation. ◦ When looking down at the reconnection plane in either figures 4.6 or figure 2.3b,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 theextended arms as it moves, as in figure 2.2b.Understanding these features could provide us with some hints about the originsof the anomalous scaling laws. One hint could be the different angles at intermedi-ate scales, 0 . ≤ ∆ s ≤ .
5. Pre-reconnection, the this span has approximately re-tained its original θ ∼ ◦ orthogonal alignment. Post-reconnection the angles over0this span still sums to approximately 362 ◦ degrees, but the angles on either side ofthe reconnection have jumped by ± ◦ . The sudden jump is due to how the directionsof the tangent and curvature vectors swap at reconnection. Using the sketch in fig-ure 4.6, this means that prior to reconnection the angle in 3D space is between thetriplets [ x ,v , x r , x ,v ] and [ x v , x r , x v ] and afterwards between the triplets[ x v , x r , x v ] and [ x v , x r , x v ]. Similar behaviour is seen for all the δ cases.It is visualised for δ = 3, using two perspectives in figures 2.2 and 2.3.To goal is find a model that links the sudden swaps in local alignments in figures 4.7and 4.8 with the nonlocal changes in figure 4.9 and from that explains the anomalousorthogonal reconnection scaling. This model must also accommodate the scaling of anti-parallel reconnection, which obeys the expected dimensional scaling both before and afterreconnection. The final discussion in section 7 addresses what might be required.
5. Anti-parallel results: approach, separation, curvature
In contrast to the orthogonal vortices, whose scaling laws do not obey expectations, itwill now be shown that the scaling of initially anti-parallel vortices obeys those expecta-tions almost completely.Figure 2.1 illustrated the overall structure of our anti-parallel case before reconnectionat t = 0 .
125 and after reconnection at t = 4, where t r ≈ .
44. The very low density ρ = 0 .
05 isosurface is the primary diagnostic for the vortex lines and as in Kerr (2011),the post-reconnection density isosurfaces are developing a second set of reconnectionsnear y = ± y = 0 should form from the additional waves along theoriginal vortex lines. Large values of the gradient kinetic energy (2.4) and the magnitudeof the pseudovorticity (3.1) are shown using two additional isosurfaces and the trajectoriesof the ρ ≈ ρ ≈ y = 0, x − z perturbation plane and two blue curves seeded using ρ ≈ z = 0, x − y dividing plane , representing respectively the pre- and post-reconnectiontrajectories.As for the δ = 3 orthogonal case in figure 4.3b, the reconnection time isosurfaceshave an extended zone of very low density around the reconnection point and a strongisosurface of the gradient kinetic energy outside this zone. Large values of | ω ρ | are alsooutside the reconnection zone.Figure 5.2 presents several possible fits for the pre- and post-reconnection separationscaling laws, with both the pre-reconnection incoming and post-reconnection outgoingseparations following the predicted dimensional scaling of δ ∼ | t r − t | / (1.1), unlike theorthogonal cases just shown. Furthermore, for ∆ t = | t − t r | < .
5, the δ in ( t r − t ) and δ out ( t r − t ) are almost mirror images of each other. Although that is not the case forlarger ∆ t .This suggests that unlike the orthogonal case, where we have attempted to relate theasymmetric scaling laws to asymmetries in the underlying structure, for the anti-parallelcase we want to identify physical symmetries that would predispose the scaling laws tobe temporally symmetric and follow the dimensional prediction.The purpose of giving two perspectives near the reconnection time in figure 5.1 is toclarify the physical symmetries at this time. The overall structure in figure 5.1a shows howall four legs of vorticity converge on the y = z = 0 line where the x − z or x − y symmetry1 (a) (b) Figure 5.1.
Density, kinetic energy K ψ = 0 . |∇ ψ | (2.4) and pseudo-vorticity |∇ ψ r × ∇ ψ i | isosurfaces plus vortex lines a bit after the time of the first reconnection, t = 2 . (cid:38) t r = 2 . K ψ ) = 0 .
83. max | ω ρ | = 0 . y = 0 and z = 0symmetry planes respectively. (b) gives a second perspective that looks into a quadrant throughthe two symmetry planes and is designed to demonstrate that locally, around y = z = 0, theshape of the ρ = 0 .
05 isosurface is symmetric on the two symmetry planes. planes cross. Then figure 5.1b looks at the structure from the interior of the zone of nearlyzero density from the perspective of the ρ ≈ y − z reconnection plane upon which thevortices reconnect. In a manner analogous to when the orthogonal vortices reconnect infigure 4.3, the tangent and normal vectors swap directions, with pre-reconnection tangentin y becoming the post-reconnection normal, and the normal in z becoming the tangent.This results in the butterfly trajectories around the y = z = 0 line in figure 5.1a.A useful, but not perfect, way to characterise the resulting structure is to use the pro-posal by de Waele & Aarts (1994), based on their Biot-Savart calculation of anti-parallelquantum vortices, that near reconnection the vortices form an equilateral pyramid. Thepyramid for figure 5.1a can be formed by straightening the 2 red and 2 blue curved legsof vorticity surrounding the y = z = 0 line in x to ( x, y, z ) ≈ (18 , , ρ = 0 .
05 isosurface. These extensions would start from tangents to the red andblue lines. The angles θ yz of these tangents with respect to the symmetry planes woulddefine the sharpness of the tip of pyramid and depend on where the tangents are takenfrom. θ y,z = 45 ◦ is obtained for | y | = | z | ≤
2, which is where the vortices begin to bendback upon themselves.Why don’t the pseudo-vortex lines continue to bend, or kink, until a sharp tip with θ y,z = 45 ◦ is obtained? Figure 5.3 provides the clues by directly plotting the curvaturesand the inclinations of the tips of the y and z pseudo-vortex lines, which are the points ofclosest approach in figure 5.3a. The curvatures are determined by (3.4) and the inclina-tions come from N = ( n x , n y , n z ), the direction of the curvature from (3.3 b ). Independentof whether one considers the red y -vortices or blue z -vortices, their maximum curvatureshave nearly the same upper bounds and the same maxima as n y,z /n x , with n z /n x (cid:37) t (cid:37) t r for t > t r and n y /n x (cid:38) t (cid:38) t r for t r > t . (5.1)tan θ ± = 6 corresponds to 80 ◦ , not the 45 ◦ angles of a pyramid.This means that as reconnection is approached, the direction of the curvature N beginsto be parallel to their separation D , very reminiscent of what has been found for theorthogonal vortices as t → t r in figures 4.7 and 4.8. Which also means that the directions2 Figure 5.2.
Separations near the timeof the first reconnection of the anti-par-allel vortices. The positions are found us-ing the isosurface method on the sym-metry planes. Both pre- and post-re-connection curves ( in and out ) follow δ ∼ ( t r − t ) / most closely. of the curvature and tangents nearly swap during reconnection, which is also similar to,but not exactly the same as, the orthogonal cases. And not what a true pyramid with asharp tip would do.What is probably more important for getting the dimensional scaling for δ ( t ) is thatboth sets of curves are temporally symmetric. That is, κ ( t r − t ) ≈ κ ( t − t r ) and n z ( t r − t ) /n x ( t r − t ) ≈ n y ( t − t r ) /n x ( t − t r ). Figure 5.3b emphasises this further by showingthe dependence of κ y and κ z on their arclengths s at times just before and after theestimated reconnection time of t r = 2 . t < t r vortex lines in figure 2.1a evolve into the pinched red y -vortex lines in figure5.1a along a path in the x − z symmetry plane with ( n x , , n z )( t ) = (cos θ − , , sin θ − ) (cid:37) (cos 80 ◦ , , sin 80 ◦ ). Then note that the process is reversed for t > t r as the pinch in theblue z -vortex lines relaxes in z as δ + (cid:38) x − z symmetry plane fromits most extreme orientation: ( n x , n y , t ) = (cos θ + , sin θ + , (cid:38) (cos 80 ◦ , sin 80 ◦ , et al. (2012), which shows aslower approach and faster separation, similar to the scaling observed for a recent anti-parallel Navier-Stokes calculation (Hussain & Duraisamy 2011). There could be severalreasons for the differences. First, the initial perturbations to the anti-parallel trajectoriesin Zuccher et al. (2012) are pointed towards one another, and not in the direction ofpropagation as here. Another difference is that their periodic boundaries in y (the vorticaldirection) are relatively close, unlike that direction here. In Kerr (2011) and in a recentset of Navier-Stokes reconnection calculations Kerr (2013), the advantages of makingthat direction very long have already been discussed.3 (a) (b) Figure 5.3. a: Maximum curvatures, κ y , κ z , versus time. κ y and κ z are always on sym-metry planes. The approach before reconnection and separation after reconnection are similar.Reconnection time is where the curves cross, t r ≈ . z vortex taken abit before t r and the profile of the old y vortex a bit after t r . The curavatures near y = z = 0are similar, which drives similar approach and separation velocities.
6. Summary
The reconnection scalings of two configurations of paired vortices, orthogonal and anti-parallel, have been found to have different scaling exponents.
For the anti-parallel case, the temporal scaling of both the pre- and post-reconnectionseparations obey the dimensional prediction, δ ± ( t ) ∼ A (cid:112) Γ | t r − t | and the arms of thevortex pairs as the reconnection time is approached form an equilateral pyramid with asmooth tip, which is in most respects qualitatively similar to the prediction of a Biot-Savart model (de Waele & Aarts 1994). Around the smooth tip the curvatures N andseparation D are nearly parallel and as a result the directions of curvature N and thetangents T almost swap during reconnection. The orthogonal cases , in contrast, show asymmetric temporal scaling with respectto the reconnection time t r . For t < t r , δ − ( t ) ∼ A − (Γ | t r − t | ) / and for t > t r , δ + ( t ) ∼ A + (Γ | t r − t | ) / , where the coefficients A ± are independent of the initial separation δ .At t ≈ t r , the reconnecting vortices are anti-parallel, with the vortices interacting in a reconnection plane that contains the tangent and curvature vectors of both vortices aswell as their separation vector. This results in the directions of curvature and vorticityswapping during reconnection. Two innovations
There are two innovations that allow these calculations to generateclean scaling laws. One is an initial core profile that either minimises the formation ofsecondary waves by the interacting vortices, or absorbs these waves. The second innova-tion is a way to trace the vortex lines that minimises the need to identify computationalcells with small values of the density.6.1.
Contrasting geometries
While this paper has emphasised the differences between the reconnection scaling for theorthogonal and anti-parallel cases, a few similarites need to be noted when t ≈ t r and x ≈ x r . That is within the reconnection zone in both time and space. First, within thiszone the curvature vectors in both cases tend to align with the separation between thevortices and the opposing vortices are anti-parallel. The skew-symmetric alignment of the4reconnecting orthogonal vortices seems to be sufficient for imposing this local property.Post-reconnection, in both cases the curvature and tangent directions swap, or nearly soin the anti-parallel case.This also means that in neither case does a pyramid form in the zone immediatelyaround x r . Nor do any of the fixed point solutions identified in Meichle et al. (2012)form.However, further from x r , the situation is different. For the orthogonal cases, anglesbetween the vortices imply a convex or hyperbolic structure. In the anti-parallel case, apyramid forms with nearly acute angles.Let us summarise the additional key features of the orthogonal cases. Orthogonal
From an early time, the closest points of the originally orthogonal vorticesbecome locally anti-parallel and their respective curvature vectors become anti-alignedwith the line of separation. Combined, this implies that the local bi-normals for each lineare nearly parallel and do not point in the direction of separation. At reconnection, thedirections of the vorticity and curvature swap, and the sign of the bi-normal reverses. Allof this is in a reconnection plane defined by the averages of the curvature and vorticitydirections at the points of closest approach.Another useful perspective is the Nazarenko perspective in figure 2.3, which is alonga 45 ◦ angle in the y − z plane. From this perspective, the vortices are always distinct,without any loops, and one can see that the pre-reconnection vortices approach thereconnection from one direction, and post-reconnection vortices separate in another. Thisperspective is used for finding non-local alignments and angles as in figure 4.9, which showthat the global alignment of the initial orthogonal vortices is hyperbolic.While the three-dimensional graphics for our orthogonal cases are qualitatively similarto the equivalent Gross-Pitaevskii density isosurfaces in Zuccher et al. (2012), our inter-pretation of the underlying geometry is different. Zuccher et al. (2012) conclude that thedeviations of Gross-Pitaevskii separations from the dimensional prediction is only nearthe reconnection and probably due to the rarefaction waves they report. In contrast,our analysis shows that the derivations start much before that and continue until thereconnection time. Furthermore, this local scaling appears to be a result of the globalalignment that exists for almost all times.Our conclusion is that the new scaling laws appear at all times for initially orthog-onal vortices and these scaling laws are probably tied to the unique alignment of theFrenet-Serret frames that form early and continue through the reconnection period untilto the end of each calculation. That these anomalous scaling laws are identical, abouttheir respective reconnection times, for all initial separations, implies that the anomalousscaling laws could exist for initial vortices with macroscopic initial separations extendingto observable scales.
7. Discussion
These results leave us with several major questions. • First, could the scaling laws shown here be extended to the huge range of lengthscales in experiments? Because the new orthogonal scaling laws appear at all times forinitially orthogonal vortices and these scaling laws are tied to unique alignments that formearly and continue through the reconnection period, it is possible that the anomalousscaling could apply to vortices on the macroscopic, observable scales.However, what if the initial state is not strictly orthogonal? What seems to be true, basedupon several additional curved configurations considered in Rorai (2012) as well as casesfrom Zuccher et al. (2012), is that the scaling of all quantum reconnection events should5lie between the two extremes presented here. More work will be needed to determinewhen and for how long each type of scaling dominates. • Second, can these cases be compared with the experiments using solid hydrogenmarkers? Improvements in both the experimental and numerical data sets will be neededbefore that can be addressed properly. Currently, a few isolated events in some of theexperimental videos and the first experimental paper (Bewley et al. et al. • Finally, how can the alignments quantified here for the orthogonal cases be usedto explain the anomalous reconnection scaling laws? The local swaps in the alignmentof the Frenet-Serret frames for the orthogonal cases in figures 4.7 and 4.8 are probablytoo similar to the local swaps for the anti-parallel case in figure 5.3 to explain the non-dimensional scaling laws. So a better place to start might be to consider the large-scale alignments. However, to use these alignments together with Biot-Savart to predictvelocities could lead nowhere since all of full Biot-Savart calculations find the dimensional,temporally symmetric scaling laws. ◦ Nonetheless, Biot-Savart can be a useful place to start looking in the sense that (3.2)provides us with a means to exactly determine the Gross-Pitaevskii velocities, whichcould then be compared to the Biot-Savart predictions. Once the differences have beenidentified, and from there the sources of these differences, we should be on the road toexplaining this new behaviour.
Acknowledgements
CR acknowledges support from the National Science Foundation, NSF-DMR GrantNo. 0906109 and support of the Universit´a di Trieste. RMK acknowledges support fromthe EU COST Action program MP0806 Particles in Turbulence. Discussions with C.Barenghi and M.E. Fisher have been appreciated. Support with graphics from R. Hen-shaw is appreciated.
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