Sensitivity of trefoil vortex knot reconnection to the initial vorticity profile

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Sensitivity of trefoil vortex knot reconnection to the initial vorticity profile

Robert M. Kerr

Phys. Rev. Fluids 8, 074701 – Published 5 July 2023

Abstract

Five sets of Navier-Stokes trefoil vortex knots in ${(2 π)}^{3}$ domains show how the shape of their initial profiles, Gaussian/Lamb-Oseen or algebraic, and their widths influence their evolution, as defined by their enstrophy $Z (t)$ , helicity $H (t)$ , and changes in their dissipation-scale structures. Significant differences develop even when all have the same three-fold symmetric trajectory, the same initial circulation and the same range of the viscosities $ν$ . The focus is upon how the dynamics of helicity density $h = u \cdot ω$ affects reconnection and the evolution of enstrophy. $h ≲ 0$ patches on the vorticity isosurfaces show where and how reconnection forms. For the Lamb-Oseen profile, the tightest and most linearly unstable, there is only a brief spurt of enstrophy growth as thin braids form at these positions; before being dissipated as the post-reconnection helicity $H$ grows significantly. For the algebraic cases: as $h < 0$ vortex sheets form prior to reconnection, there is $ν$ -independent convergence of $\sqrt{ν} Z (t)$ at a common $t_{x}$ . For those with the broadest wings, enstrophy growth accelerates after reconnection, leading to approximately convergent dissipation rates $ε = ν Z (t)$ . Maps of terms from the budget equations onto centerlines illustrate the divergent behavior. Lamb-Oseen briefly forms six locations of centerline convergence with local negative dips in the helicity dissipation $ε_{h}$ and vortical-helicity flux $h_{f}$ . These are the source of the following positive increase in the global $H$ and suppression of enstrophy production. For the algebraic profiles there are only three locations of centerline convergence, each with spans of less localized $ε_{h} < 0$ that could be the seeds for the $h < 0$ vortex sheets and whose interactions can explain the later accelerated growth of the enstrophy, approximate $ν$ -independent convergence of the energy dissipation rates $ε$ , and evidence for finite-time energy dissipation $Δ E_{ε}$ , despite the initial symmetries.

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Received 14 November 2022
Accepted 18 April 2023

DOI:https://doi.org/10.1103/PhysRevFluids.8.074701

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Vortex dynamics Vortex flows Vortex interactions

Direct numerical simulations

Fluid Dynamics

Authors & Affiliations

Robert M. Kerr^*

Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom

^*Robert.Kerr@warwick.ac.uk

Article Text

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Issue

Vol. 8, Iss. 7 — July 2023

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Figure 1
Global diagnostics of a three-fold symmetric trefoil for case Gd05 with a Gaussian/Lamb-Oseen profile (10). The three Reynolds numbers are [2000, 6000, 12 000], with their viscosities in the legend. The diagnostics are the evolution of: (a) the enstrophy $Z (t)$ , (b) the dissipation rate $ε (t) = ν Z$ , and (c) the global helicity $H (t)$ . These are similar to Fig. 3 of Ref. [3]. All calculations are in ${(2 π)}^{3}$ periodic boxes.
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Figure 2
Global diagnostics for algebraic case r1d015 whose profile (9) uses $p_{r} = 1, r_{o} = 0.015$ and $r_{e} = 0.08$ . The Reynolds numbers are [24 000, 12 000, 6000, 3000], with their viscosities in the legend. Diagnostics are the evolution of: (a) the enstrophy $Z (t)$ and (b) the global helicity $H (t)$ .
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Figure 3
For the case and viscosities from Fig. 2, two rescaled enstrophy diagnostics. (a) The time dependence of the reconnection-enstrophy $\sqrt{ν} Z (t)$ , with convergence at $t_{x} = 6.1$ indicated. This is used later to define the end of the first reconnection. (b) The dissipation rate $ε (t) = ν Z$ , whose convergence at $t \approx 11$ might be preliminary evidence for the formation of a dissipation anomaly $Δ E_{ε}$ . (1).
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Figure 4
Three-dimensional vorticity isosurfaces with mapped helicity at $t = 1.2$ for the two primary three-fold symmetric trefoils, both for $ν \sim 8.4 \times 10^{- 5}$ . (a) From algebraic case r1d015 with $p_{r} = 1$ and $r_{o} = 0.015$ (9). (b) Lamb-Oseen profile (10) (Gd05). The primary extrema of interest are the maximum vorticity $ω_{m}$ , the helicity minima $h_{m n}$ , its maxima $h_{m x}$ , and the maximum velocity $u_{m}$ . All are indicated in both panels by the symbols in the legends. In addition, each panel indicates the three-dimensional $s_{f}$ positions of the local centerline minima of the vortical helicity flux $min (h_{f})$ (6), as defined on budget profiles such as Fig. 9. Also shown are the $s_{f}$ 's opposing $s_{o}$ points, the closest points in three dimensions on their opposite loops. Also indicated in panel (a) are the $s_{d}$ positions of the local $min (ε_{h})$ , the minima of the centerline helicity dissipation. These positions are also marked on the $t = 1.2$ centerline budget profiles in Figs. 9 and 11 and will be used for reference later.
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Figure 5
Comparisons of $ω_{y} (z)$ at $t$ = 0 between vortices initialized with algebraic and Lamb-Oseen core profiles. The $ω_{y} (z)$ profiles are taken through the $min (ω_{y})$ of the $y = 0 x − z$ planes as in Figs. 7 and 8. Three cases from Table 1 are used. All except one curve are taken after the nonsolenoidal Fourier components have been removed. The profiles are for the $r_{o} = 0.05$ Lamb-Oseen case (10) (Gd05) and two of the algebraic profiles that use the Rosenhead regularization (9). r2d05: $p_{r} = 2, r_{o} = 0.05$ , referred to as K-S-R, and r1d015: $p_{r} = 1, r_{o} = 0.015$ . The other curve is the “raw” $p_{r} = 1, r_{o} = 0.015$ curve, taken through its pre-Fourier-projected $ω_{y}$ field. (a) The primary figure shows the full profiles in $z$ . (b) The lower-left inset focuses upon $z < 0.4$ wings with small $ω_{y}$ . Note the slight $ω_{y} > 0$ overshoot at the boundaries of the Lamb-Oseen profile. This is the likely seed for the oscillations about $ω_{y} = 0$ in Fig. 8.
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Figure 6
Stability at $t$ = 0 of the Lamb-Oseen (L-O) (10) and K-S-R (9) profiles using the Richardson functions $J (ρ)$ (12) of Howard-Gupta [12] in two steps. (a) First, because each profile uses $r_{o} = 1$ for their width, their $Ω (ρ)$ and $Ω^{'} (ρ)$ profiles are similar. (b) However, what determines stability is how their $J (ρ)$ asymptote as $ρ \to \infty$ . For Lamb-Oseen its $J (ρ) \to 0$ (15), suggesting instability, while K-S-R is almost always stable because, by (14), $J (ρ) \to r_{o}^{2}$ is finite.
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Figure 7
From algebraic case r1d015 at $t = 2.4$ , a $y = 0 ω_{y} x - z$ cross section with $ω_{y} (z)$ profiles through the local $min (ω_{y})$ indicated. (a) Contour plot whose primary $min (ω_{y})$ is at $x_{1} = 1.58$ with a secondary minima at $x_{2} = 0.81$ . $| ω_{y} | \sim 0$ contours do not appear. (b) $ω_{y} (z)$ profiles through those local minima. First, the full $ω_{y} (z)$ , then focusing on small $ω_{y}$ . Contours and profiles at $t = 1.2$ are similar.
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Figure 8
From the Lamb-Oseen (10) case Gd05 at $t = 1.2$ , a $y = 0 ω_{y} x - z$ cross section with $ω_{y} (z)$ profiles through the local $min (ω_{y})$ indicated. (a) Contour plot whose primary $min (ω_{y})$ is at $x_{1} = 1.52$ with a secondary minima at $x_{2} = 0.65$ . A few $| ω_{y} | \sim 0.001$ contours are included to show the fluctuations about $ω_{y} = 0$ . (b) $ω_{y} (z)$ profiles through those local minima. Values of the negative $min (ω_{y} (z))$ are indicated. The positive overshoots of $ω_{y} (z)$ show the magnitude of the $| ω_{y} | \sim 0$ contours on the left. Note that the $y = 0, x − z$ plane negative $ω_{y}$ extrema are not at the positions of the global $max (| ω |)$ for these fields.
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Figure 9
Centerline budget profiles at $t = 1.2$ for algebraic case r1d015 $ν = 1.6 \times 10^{- 4}$ of $h, ε_{h}, | ω |, h_{f}, ε_{ζ}$ and $ζ_{p}$ . (a) $h$ and $ε_{h}$ (6). (b) $| ω |$ . (c) $h_{f}$ . (d) Enstrophy density production $ζ_{p}$ and its dissipation rate $ε_{ζ}$ (5). Each panel has three vertical maroon lines at the $s_{f}$ positions of the local $min (h_{f})$ . Panel (a) has two additional sets: $s_{d}$ positions of the local $min (ε_{h})$ ; $s_{o}$ positions that oppose the $s_{f}$ . All of the $0.4 < t ≲ 2.4$ algebraic budget profiles are similar to these.
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Figure 10
Early $t = 0.4$ Lamb-Oseen centerline budget profiles for $ν = 8.35 \times 10^{- 5}, r_{o} = 0.05$ of $h, ε_{h}, | ω |, h_{f}, ε_{ζ}$ , and $ζ_{p}$ . These are similar, but not identical, to the algebraic profiles at $t$ = 1.2 in Fig. 9. From panel (c), the three $s_{f}$ positions of local $min (h_{f})$ are colocated with: (a) local $max (h)$ and $min (ε_{h})$ ; (b) secondary local $max | ω |$ ; (d) local $min (ζ_{p})$ , meaning at points of maximum centerline compression.
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Figure 11
Lamb-Oseen centerline budget profiles at $t = 1.2$ for $r_{o} = 0.05, ν = 8.35 \times 10^{- 5}$ . These are very different than the $t = 1.2$ algebraic budget profiles in Fig. 9. In panel (a) there are six positions with strong negative helicity dissipation, local $min (ε_{h})$ and local $min (h)$ . The positions are separated into two sets of three. The $s_{f}$ in maroon are at the strongest $min (ε_{h})$ , adjacent to the local $min (h_{f})$ ( $h_{f}$ panel is not shown). The $s_{o}$ in turquoise are the points that oppose the $s_{f}$ in three dimensions (Fig. 4). In panel (b), all six positions are at very large positive gradients of $ζ_{p}$ between local $min (ζ_{p})$ and $max (ζ_{p})$ . Strong local $min (ζ_{p})$ means strong local centerline compression. The $s_{f}$ are also at $max (ε_{ζ})$ positions, maxima of the enstrophy dissipation.
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Figure 12
Lamb-Oseen centerline budget profiles at $t = 2.4$ for $r_{o} = 0.05, ν = 8.35 \times 10^{- 5}$ . (a) $h (s), ε_{h} (s), s_{f}$ (maroon) for local $min (h_{f})$ and the $s_{f}$ 's opposing $s_{o}$ (turquoise) are marked. The $ε_{h} (s)$ profiles are three-fold symmetric again and more like the algebraic profiles at $t = 1.2$ and $t = 2.4$ and Lamb-Oseen at $t = 0.4$ . (b) However, there are still six positions of local $min (ζ_{p}) < 0$ compression: The three $s_{f}$ and three $s_{o}$ . Having this many local compression locations is why the post-reconnection Lamb-Oseen vortex structures in Sec. 3c are braids, not the sheets generated by the algebraic profiles.
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Figure 13
Algebraic mapped-helicity $ω$ -isosurface at $t = 2.4$ from the r1d015 $p_{r} = 1$ (9), $ν = 8.4 \times 10^{- 5}$ calculation at the beginning of the initial phase of reconnection. Symbols (from legend) show the three-dimensional positions of the basic $u, ω$ , and $h$ extrema as well as the extrema of terms taken from the enstrophy and helicity budget equations (5) and (6). Also shown, taken from the next figure, are the three-dimensional locations of these centerline positions: the $s_{f}$ (maroon) positions of local $min (h_{f})$ ; the $s_{o}$ (turquoise) positions that oppose the $s_{f}$ ; and the $s_{d}$ (yellow) positions of the local $min (ε_{h})$ . The $s_{f}$ points are on one side of each reconnection, with the $s_{d} − s_{o}$ zones representing the other side of those reconnections. The best diagnostic for following the Biot-Savart evolution of the vortex centerlines from $t = 1.2$ to 3.6 is the convergence within each of the three color-coded clusters of $\circ$ 's, $★$ 's and $⋄$ 's. These clusters, plus the extrema appearing with them, are: a $\circ$ cluster on the left with $ω_{m}$ (X), $max (ε_{ζ})$ ; a $★$ cluster above $u_{m}$ (green $+$ ) near the global $min (h_{f})$ (maroon square) and $min (ζ_{p})$ ; and a $⋄$ cluster at the bottom with $min (h)$ and $max (ζ_{p})$ . Over time, the $s_{d}$ and $s_{o}$ with the same symbols are approaching one another on the same centerline spans of the trefoil and together they approach the $s_{f}$ in three-dimensions.
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Figure 14
Algebraic centerline budget profiles at $t = 2.4$ of $h, ε_{h}, | ω |, h_{f}, ε_{ζ}$ , and $ζ_{p}$ , for case r1d015, $ν = 8.4 \times 10^{- 5}$ . Added to each panel are three sets of three dashed vertical lines. Maroon lines are at the $s_{f}$ of local $min (h_{f})$ . Yellow is for the $s_{d}$ at local $min (ε_{h})$ . And turquoise is for the $s_{o}$ , the points opposing the $s_{f}$ in three-dimensions. The $s_{f}$ points are on one side of each reconnection, with of the $s_{d} − s_{o}$ zones representing the other side of those reconnections. Over time, the $s_{d}$ and $s_{o}$ with the same symbols are approaching one another on the same centerline spans of the trefoil.
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Figure 15
Algebraic centerline budget profiles and an isosurface at $t = 3.6$ for case r1d015 for $ν = 8.4 \times 10^{- 5}$ . Budget profiles: $h, ε_{h}, h_{f}, ε_{ζ}$ , and $ζ_{p}$ , with added vertical dashed lines in each panel for these local positions: $s_{f}$ [maroon, $min (h_{f})], s_{d}$ [yellow, $min (ε_{h})]$ , and in the upper-left panel with $s_{g}$ (green) for the $s_{d}$ opposing points. The $s_{f}$ are also at $min (ζ_{p})$ and are at two of the $max (ε_{ζ})$ positions, local enstrophy dissipation peaks. The $s_{d}$ are also at the local minima of the helicity $min (h) < 0$ , at crossovers between secondary local $min (ζ_{p})$ to $max (ζ_{p})$ and at two of the local $max (ε_{ζ})$ positions. They are colocated with the opposing positions to the $s_{f}$ . The $s_{g}$ oppose the $s_{d}$ and are nearly coincide with the $s_{f}$ . Where might reconnection form? The positioning of the $s_{f}$ and $s_{d}$ , plus their opposing points, suggests that reconnection is forming between the $s_{f}$ and $s_{d}$ . Panel (d) shows that at all six points there is local $ζ_{p} < 0$ , that is $d u_{s} / d s < 0$ . Due to incompressibility this implies the existence of stretching perpendicular to the vorticity at these points, the stretching needed to needed to create the $h < 0$ vortex sheets. The large vorticity ( $ω = 0.2 ω_{m}$ ) isosurface in the upper-right panel encases the centerline and includes centerline symbols from panel (a), with the labels for the auxiliary symbols given in Fig. 16. This isosurface also shows continuity with the earlier evolution of its inner isosurfaces.
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Figure 16
Algebraic $t = 3.6$ mapped helicity, $ω$ -isosurface from three-perspectives for case r1d015 with $ν = 8.35 \times 10^{- 5}$ , with a color-coded centerline. Symbols show the three-dimensional positions of the basic $u, ω$ , and $h$ extrema as well as extrema from the enstrophy and helicity budget equations (5) and (6). Also, the $s_{f}$ (maroon) positions of local $min (h_{f})$ and the $s_{d}$ (yellow) positions of the local $min (ε_{h})$ , which also oppose the $s_{f}$ (the $s_{o}$ in Fig. 15). Panel (a) is a plan view perspective with faint $h ≲ 0$ yellow sheets extending out from the lower reddish ring and panels (b) and (c) are side views from the same perspective. Panel (b) shows the entire domain. Panel (c) shows only $z < - 0.8$ with the yellow lower ring emerging below the blue, largely $h > 0$ , upper portion. In panels (b) and (c) the $★ s_{d} = 9.2$ mark is enlarged. The centerline vortex has mapped helicity ranging from red ( $h = - 13$ ) to blue ( $h = 26$ ). By using a small $ω \sim 1.4 \sim 0.03 ω_{m}$ vorticity isosurface, a gradation can be seen in the lower $h < 0$ zone from a red $h \sim - 0.4$ inward facing half to the yellow-green $h ≲ 0$ outward half. This is the first step in the formation of the yellow negative helicity $h ≲ 0$ vortex sheets at later times.
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Figure 17
Two $t = 3.6$ Lamb-Oseen isosurfaces with different vorticity thresholds. (a) The primary $ω = 19$ isosurface is similar to the higher- $ω$ algebraic isosurface in Fig. 15. Additional markers indicate the three-dimensional locations of the $s_{d}$ (yellow), local $min (ε_{h}), s_{f}^{+}$ (cobalt) for the local $max (h_{f})$ points, and $s_{o}^{+}$ (turquoise), points opposing the $s_{f}^{+}$ that are also $min (h) < 0$ and $min (ζ_{p})$ points. Reconnection is commencing between the $s_{f}^{+}$ and $s_{o}^{+}$ points. The local $s_{d}$ (yellow), $min (ε_{h})$ sit in strongly positive $h > 0$ zones, not $h < 0$ as for the algebraic calculations or Lamb-Oseen for $t ⩽ 2.4$ . (b) The vorticity of the second isosurface uses very small $ω = 1.7$ to show that the outer edges of the isosurface are shedding sheets with slightly negative helicity.
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Figure 18
Lamb-Oseen centerline budget profiles at $t = 3.6$ for $ν = 8.35 \times 10^{- 5}$ . The $s_{d}$ (yellow/maroon) at local $min (ε_{h})$ and colocated with local $max (ε_{ζ})$ and $max (| ω |)$ , are in large $h > 0$ zones far from the reconnections. The $s_{f}^{+}$ (cobalt) are at local $max (h_{f})$ points and colocated with local $max (ζ_{p})$ and secondary velocity minima. The $s_{o}^{+}$ (turquoise) points oppose the $s_{f}^{+}$ and are colocated with $min (h) < 0$ and $min (ζ_{p})$ points. Reconnection is commencing between the $s_{f}^{+}$ and their opposing $s_{o}^{+}$ points.
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Figure 19
Two Lamb-Oseen isosurfaces at $t = 4.0$ . The panels are related to one another by an boxed-outline they share that contains a bridge. (a) The primary isosurface shows the overall structure using a small vorticity of $ω = 9.3 = 0.014 ω_{m}$ . There are the three symmetric locations of yellow-green $h ≲ 0$ where the loops are interacting and forming bridges between the rings. Panel (b) shows a $ω = 37$ isosurface that focuses upon the lower-left reconnection site to highlight one of the reconnection bridges. There is a noticeable $h < 0$ string within the bridge, but no $h < 0$ vortex sheets.
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Figure 20
Three views of a $t = 4.4$ Lamb-Oseen isosurface with $ω = 49 = 0.015 ω_{m}$ (=312), with the bottom two focusing upon the smallest structures. (a) The primary $t = 4.4$ isosurface shows the overall structure to indicate how braids are forming from bridges, as seen for previous Lamb-Oseen calculations. Two vortex rings have formed whose centerline vortices are an upper ring with a blue centerline that was seeded by $u_{m} = max (u)$ and a lower, red centerline that was seeded by $h_{m} = max (h)$ . (b) The full length of one of the double braids is shown, including where it attaches to the new upper and lower vortex rings. This is similar to $t = 4.29$ of Fig. 18 from Ref. [3]. (c) The small box at one end of the double braid shows it winding around the primary vortex.
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Figure 21
Two views of the same $t = 4.8$ isosurface from different elevation angles from the $p_{r} = 1, r_{o} = 0.015$ (r1d015) calculation: (a) (planar view) and (b) (side view). The time of $t = 4.8 > t_{r} \sim 4$ represents the middle of the reconnection phase that ends with the first reconnection at $t_{x}$ =6. The $ω$ -isosurfaces are: a blue inner $ω = 14$ surface and a small $ω = 0.65 = 0.02 ω_{m}$ isosurface with mapped helicity. The positions of $ω_{m}, max (h), min (h)$ , and $u_{m}$ are given along with extrema of terms from the enstrophy and helicity budgets. The red hashes indicate where sheets arise from the marked centerline spans of $ε_{h} < 0$ in budget [Fig. 23], plus three triplets of the local positions $s_{f}, s_{d}$ , and $s_{g}$ at local $min (h_{f}), min (ε_{h})$ , and the $s_{d}$ opposing points. The symbols given in the legend are also used in Fig. 23. In panel (a) the overall structure of the lobes is emphasized. Panel (b) shows that the red hashes are all in the lower portion and represent where a separate lower vortex ring is forming. The origins and location of the yellow regions are given in the next figure.
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Figure 22
Two additional views of the lower region of the same $t = 4.8$ algebraic isosurface as the last figure. The views show $z < - 1.1$ and $- 0.65$ , respectively. Each perspective is dominated by yellow $h ≲ 0$ with (a) looking down and (b) looking up with the domain flipped across a line from $[x y] = [- 1 1.5]$ (green triangle) to the $[x y] = [1 - 1]$ corner, with some of the upper $h > 0$ zone included. It is also rotated a bit about the $z$ axis to give a flavor of how the legs of the lower ring are connecting with the bridge. Gray is where we are looking through both the lower yellow and upper blue. Some of the $h > 0$ zone is included to show the while the $h ≲ 0$ sheets are being shed from the lower $h < 0$ centerline, they extend up to the upper $h > 0$ blue-marked centerline. The orange $s_{d}$ and the opposing green $s_{g}$ , both marked with $⋄$ 's, are highlighted to show how the legs might be starting to wind around each other.
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Figure 23
Algebraic centerline budget profiles at $t = 4.8$ for case r1d015 with $ν = 1.67 \times 10^{- 4}$ . Profiles: $h, ε_{h}, h_{f}, ε_{ζ}$ , and $ζ_{p}$ , with added vertical dashed lines for these local positions: $s_{f}$ [maroon, $min (h_{f})], s_{d}$ [yellow, $min (ε_{h})]$ , and $s_{g}$ (green) for the $s_{d}$ opposing points. The $s_{f}$ are also at $min (ζ_{p})$ and at large local enstrophy dissipation $ε_{ζ}$ positions. The $s_{d}$ are at secondary $min (ζ_{p})$ and at local $max (ε_{ζ})$ positions. The $ε_{h} (s) ≲ 0$ spans over which the $h < 0$ sheets are being shed are indicated by thick, dashed red lines that are to the right of each $s_{g}$ . Reconnection is forming between spans near each $s_{d}$ and the red hashed patches on the opposing loops with green $s_{g}$ symbols at one end, for example, at the $s_{d} = 4.7$ yellow diamond at the other end and the span next to the green diamond at $s_{g} = 16$ .
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Figure 24
Two isosurface perspectives ([a e]) at $t = t_{x} = 6$ for algebraic r1d015, $ν = 1.6 \times 10^{- 4}$ , at the end of the first reconnection, as defined by Fig. 3. This is when the dissipation in Fig. 3 begins to accelerate, with convergence of $ε = ν Z$ at $t \approx 10$ . The two isosurfaces are an inner $ω = 12$ blue that encases the centerline and an outer $ω = 15$ with helicity-mapping. The perspective angles [a e]= $Ω_{z}$ [rotation elevation] are in the figures. The two perspectives are similar to those at $t = 4.8$ : (a) is a side view similar to that in Fig. 21; (b) is a cropped plan view, similar to Fig. 22 but with the helicity brightened. A box is drawn on both panels to show where the subdomain in panel (b) has been taken from the full domain in panel (a). In panel (a) the dominant structure is the pure blue $ω = 12$ centerline isosurface with three bridges connecting the separating upper and lower vortex rings. This illustrates what direct experimental visualizations of cores are probably observing [14]. The plan view shows what those experiments cannot see: lower $ω$ magnitude $h ≲ 0$ vortex sheets. Two differences with Fig. 22 are that the sheets shed from the legs change pigmentation along their length, and they are wrapping around one another at the bridges. The “left” bridge has the $min (h)$ (red $▿$ ) mark. The “right” bridge has the $ω_{m}$ (X) and $u_{m}$ (green $+$ ) marks. There is a color change on the bottom leg from orange $h < 0$ at the (X, $+$ ) “right” bridge to green at the “left” bridge. With the “left” green wrapping around the “left” bridge in the upper left and green from the leg on the right wrapping about “right” bridge and some of the $y$ axis leg.
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Figure 25
Global diagnostics for case r2d1, a K-S-R algebraic profile (9) with $p_{r} = 2$ and $r_{o} = 0.1$ . The figure shows the evolution of (a) helicity $H$ for different viscosities and (b) two ways of scaling the enstrophy $Z$ . The dissipation rate $ε (t) = ν Z$ with approximate convergence at $t_{e} = 10.75$ , and in the inset convergence of the reconnection enstrophy $\sqrt{ν} Z (t)$ at $t_{x} = 5.45$ . These curves are similar to those for case r1d015 in Figs. 2 and 3.
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Figure 26
Global diagnostics for case r2d05, algebraic with $p_{r} = 2$ and $r_{o} = 0.05$ , for different viscosities, showing (a) convergence of $\sqrt{ν} Z (t)$ at $t = 4.45$ , but without convergence of the dissipation rate $ε (t) = ν Z$ as shown in the inset; (b) strong post-reconnection growth of the helicity $H$ . Case r1d006 ( $p_{r} = 1, r_{o} = 0.006$ ) has similar $Z (t), \sqrt{ν} Z (t)$ and $H (t)$ evolution. In the next figure, r2d05 does not develop strong $h < 0$ vortex sheets, nor does r1d006.
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Figure 27
Side-views of isosurfaces for algebraic case r2d05, $p_{r} = 2$ (9) at $t = 4.4$ and 5.2. Perspective angles and the range of the helicity density $h$ are given at the bottom of each panel. High $ω$ isosurfaces are used instead of vortex lines to indicate the centerlines. (a) Time $t = t_{x} = 4.4$ is when the $\sqrt{ν} Z (t)$ cross in Fig. 26 and a vortex sheet is starting to form. However, $t_{x} = 4.4$ is earlier ( $t < 5$ ) than in Figs. 3 and 25. (b) And at $t = 5.2$ , instead of sheets continuing to form, there are connecting bridges at $t = 5.2$ . The formation of bridges instead of vortex sheets is similar to Lamb-Oseen in Fig. 19, possibly because for both the outer gradients are steep. For case r1d05 this is due to the thin core radius and the influence of the restrictive ${(2 π)}^{3}$ domain.
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Figure 28
Six views of the three-dimensional vorticity isosurface with mapped helicity at $t = 3.6$ for algebraic case r1d015 with $ν = 8.4 \times 10^{- 4}$ . The legend on the left of (a) and the color bar to its right apply to all panels, with [a e] = $Ω_{z}$ [rotation elevation]. In these panels, the dominant features are on or around the centerline. Starting with panel (a) where the reddish regions hugging the centerlines show where $h < 0$ is being shed. The sequence starts in upper-left with: (a) $[a e] = [- 18, 85]$ , as in Fig. 16; next is panel (b), which is rotated to $[a e] = [- 44 43]$ to be similar to the high- $ω$ isosurface in Fig. 15. Panels (c), (d) are rotated gradually into a side view, indicating where slightly negative helicity $h ≲ 0$ sheets are appearing underneath the centerline. Panels (d)–(f) shave off, then lop off, the blue top of the trefoil, leaving its bottom half, which is dominated by shed, yellow $h ≲ 0$ vortex sheets. These sheets have the appearance of the ruffles of a yellow skirt sticking out below the main blue $h > 0$ trefoil. To follow the rotation between panels (d), (e), use the lime bulge with the maroon $□$ of $min (h_{f})$ . Then panel (f) shows the bottom half of panel (a), providing a view of the $h < 0$ vortex sheets on the periphery, including three very faint $h ≲ 0$ yellow lobes. These become the more extensive $h ≲ 0$ sheets at $t = 4.8$ in Fig. 22.
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