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Helicity Conservation in Topology-Changing Reconnections: the Flow of Linking and Coiling Across Scales

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Helicity Conservation in Topology-Changing Reconnections: the Flow of Linking and Coiling Across Scales Helicity conservation in topology-changing reconnections: the flow of linking and coiling across scales Martin W. Scheeler,1, ∗ Dustin Kleckner∗,1, y Davide Proment,2 Gordon L. Kindlmann,3 and William T.M. Irvine1, z 1James Franck Institute, Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA. 2School of Mathematics, The University of East Anglia, NR4 7TJ Norwich, Norfolk, UK. 3Computation Institute, Department of Computer Science, The University of Chicago, Chicago, Illinois 60637, USA. The conjecture that knottedness is a fundamental conserved physical quantity has a long history in fluid mechanics. In ideal flows, the conservation of helicity arises because the topology of vortex lines is invariant. In real flows (including superfluids), the large scale topology of vortex tubes changes through `reconnection' events, so helicity can only be conserved by transferring to different spatial scales. By measuring the behavior of vortex knots and links in viscous fluid experiments and quantum fluid simulations, we identify a mechanism for helicity scale transfer through reconnections, allowing helicity to be conserved even when the topology is not. We also describe a new method for quantifying helicity across scales, and discuss the transfer of helicity to sub-core local twisting where it may ultimately be dissipated. Remarkably, we find that topology-changing reconnections proceed in a manner than tends to conserve helicity, suggesting that it plays a fundamental role in real fluids, from turbulent viscous flows to plasmas. In addition to energy, momentum and angular momen- have been hindered by the lack of techniques to create tum, ideal (inviscid) fluids have an additional conserved vortices with topological structure. Thanks to a recent quantity { helicity (eqn. 1) { which measures the linking advance [19], this is finally possible. and knotting of the vortex lines composing a flow [1]. For While on the one hand the diffusion of vorticity en- an ideal fluid, the conservation of helicity is a direct con- ables reconnections that remove topology, it also guaran- sequence of the Helmholtz laws of vortex motion, which tees that vortex tubes have finite thickness. Finite thick- both forbid vortex lines from ever crossing and preserve ness allows for internal structure { like the filaments that the flux of vorticity, making it impossible for linked or make up a strand of yarn { which may also store helic- knotted vortices to ever untie [1, 2]. Since conservation ity [20{22]. When topology changes through a reconnec- laws are of fundamental importance in understanding tion event, helicity conservation is therefore dependent flows, the question of whether this topological conser- on whether and how global linking is transferred to this vation law extends to real, dissipative systems is of clear interior structure. and considerable interest. The general importance of this By performing experiments on linked and knotted vor- question is further underscored by the recent and growing tices in water, as well as numerical simulations of Bose- impact knots and links are having across a range of fields, Einstein condensates (a compressible superfluid [23]) and including plasmas [3, 4], liquid crystals [5, 6], optical [7], Biot-Savart vortex evolution, we investigate the conser- electromagnetic [8] and biological structures [9{11], cos- vation of helicity, in so far as it can be inferred from the mic strings [12, 13] and beyond [14]. Determining if and shape of the reconnecting flux tubes. We report a system- how helicity is conserved in the presence of dissipation independent geometric mechanism for helicity conserva- is therefore paramount in understanding the fundamen- tion through a reconnection. We also describe a new tal dynamics of real fluids and the connections between method for quantifying the storage of helicity on differ- tangled fields across systems. ent spatial scales of a thin-core vortex: a `helistogram'. Using this analysis technique, we find that self-induced The robustness of helicity conservation in real fluids is vortex stretching generates a rich structure in the flow of unclear because dissipation allows the topology of field helicity, driving both vortex reconnections and geometric lines to change. For example, in viscous flows vortic- deformations that transport helicity between scales. ity will diffuse, allowing nearby vortex tubes to `recon- The hydrodynamic helicity is defined in terms of the nect' (Fig. 1), creating or destroying the topological link- fluid flow field, u(r): ing of vortices. This behavior is not unique to classi- cal fluids: analogous reconnection events have also been Z experimentally observed in superfluids [15] and coronal H = u · ! dV; (1) loops of plasma on the surface of the sun [16]. In gen- eral, these observed reconnection events exhibit diver- where the vorticity is !(r) = r × u(r). The connection gent, non-linear dynamics which makes it difficult to re- between this volumetric representation and the linking solve these questions theoretically [4, 17, 18]. On the between vortex tubes was first noted by Moffatt [1], who other hand, experimental tests of helicity conservation showed that for flows consisting of thin, closed vortex 2 A ideal viscous B C t = 832 ms t = 850 ms flow flow 116 mm 186 mm 151 mm E F D FIG. 1: (A) A sketch of the evolution of vortex tube topology in ideal (Euler) and viscous (Navier-Stokes) flow. Dissipative flows allow for reconnections of vortex tubes (highlighted in blue), and so topology is not conserved. (B) A diagram of a single reconnection event. (C) Two frames of a 3D reconstruction of a vortex reconnection in experiment, which turns an initially linked pair of rings into a single twisted ring. (D) A close-up view of the reconnection in C.(E) Vortex `tubes' may contain internal structure even when their centerline is straight, leading to the storage of helicity as twist. (F) If the vortex tube `writhes', it may store helicity even when the topology of the centerline is trivial. tubes Cn, with compact internal structure, the helicity is writhes in space (Fig. 1F). In this sense, the writhe is a equivalently given by: measure for the internal linking of vortex filaments for a locally untwisted bundle, providing valuable information 1 I I x − x X i j about the internal topology even when the fine structure H = ΓiΓj 2 · (dxi × dxj) (2) 4π jxi − xjj can not be directly resolved. Though total helicity is i;j C C i j agnostic to how it is encoded in the geometry of the vor- X X 2 = ΓiΓj Lij + Γi W ri: (3) tex lines, it is useful to distinguish these three geometric i6=j i contributions: centerline linking L, centerline coiling (or writhing) Wr and local twist T w. The latter cannot be where Γ and x correspond to the circulation (vorticity i i resolved without looking in the vortex core, so we define flux) and center-line path for vortex tube C , and we have i the centerline helicity, H =Γ2 = P L + P W r , which assumed that each vortex-tube is locally untwisted and c ij i we measure directly from the vortex centerline geometry. non-intersecting. The linking number, Lij, is a topolog- ical invariant which measures the signed integer number Recently, elementary vortex knots and links have been of times a pair of closed tubes is wrapped around one found to be intrinsically unstable in experimental vis- another. The writhe, W ri, is an analogous quantity for a cous fluids [19] and simulations of Bose-Einstein con- single path, which includes contributions from both knot- densates (BECs) [24]. In both cases, thin-core vortices ting and helix-like coiling, and in general is not integer. are seen to spontaneously deform towards a series of lo- If a vortex tube has finite thickness, the vortex `bundle' cal reconnections that untie knots and disconnect linked (with total circulation Γ) can be mathematically decom- rings. Thin-core vortices of this type are ideal model posed into ‘filaments' (each with Γfilament = Γ=N) that systems for studying the conservation of helicity; they describe its internal structure. As the number of fila- concentrate the region of interest to a small subset of the ments increases, N ! 1, the writhe contribution to the broader three-dimensional flow, while efficiently encoding helicity is seen to vanish, replaced by the linking between the fundamental physics of the problem in their three- individual filaments inside the bundle [21]. Thus a vortex dimensional shape. Indeed, from their shape, physical tube whose internal filaments are linked stores topology quantities such as energy, momentum and helicity can even though its centerline is topologically trivial. For ex- be determined [25], and the geometric nature of the de- ample, in a vortex loop whose centerline has no torsion, scription allows natural generalizations to other knotted such as a circle, internal topology can be achieved by uni- physical systems. Here, we generate knots, links, and formly twisting the vortex sub-tubes around each other unknots (distorted rings) in vortices in water, simula- (Fig. 1E). Conversely, the same topology can be achieved tions of Gross-Pitaevski equation (GPE, a simple model without internal twisting if instead the vortex bundle of a BEC [23]), and Biot-Savart simulations, and recon- 3 4 4 4 A C E ) 3 ) 3 ) 3 2 2 2 Γ Γ Γ / / / c c c H H H ( ( ( 2 2 2 y y y t t t i i i c c c i i i l l l e e e h 1 h 1 h 1 0 0 0 1.6 1.6 1.6 B D F ¯r/ξ = 48 ) ) ) 36 0 0 0 24 L L L / 1.4 / 1.4 / 1.4 18 L L L ( ( ( 12 raw smooth raw smooth 9 h h h t t t 6 g 1.2 g 1.2 g 1.2 n n n e e e l l l 1.0 1.0 1.0 1 2 3 1 2 3 0 1 2 3 4 Γ Γ Γ time (t0 =t ) time (t0 =t ) time (t0 =t ) × ¯r2 × ¯r2 × ¯r2 G experiment (linked rings, side view) H GPE (trefoil knot, front view) ) 1.0 2 ) I Γ Γ / / 10 L H × ∆ t 5 1 3 u ( ( p y 0 -1 o t 0.1 i r s d n -5 e H d -10 10 50 H t0 =2.93 t0 =3.04 t0 =0.00 t0 =1.92 t0 =3.84 scale (¯r/ξ) FIG.
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