3.1. Morphology of flux tubes

Figure 2 plots the temporal evolution of the kinetic energy $E_u$, magnetic energy $E_b$ and their time derivatives $\mathrm {d} E_u/\mathrm {d} t$ and $\mathrm {d} E_b/\mathrm {d} t$. In general, the same initial $E_u$ and $E_b$ decay with time owing to viscous and resistive dissipations. We observe that the profile of $E_u$ has a plateau at $1\leqslant t\leqslant 1.4$ in figure 2(a), and its negative time derivative reaches zero in this period. Meanwhile, $E_b$ keeps decaying at almost the same rate. Considering the small volume ratio of flux tubes to the computational domain, the changes of the decaying trends of $E_u$ and $E_b$ imply a remarkable conversion from $E_b$ to $E_u$ starting around $t=1$.

Figure 2.

Temporal evolution of the volume-averaged (a) kinetic energy and its time derivative and (b) magnetic energy and its time derivative.

We extract the MSF isosurface to study the morphology of flux tubes. figure 3 depicts the temporal evolution of isosurfaces of $\phi _b=0.01$ and $\phi _b=-0.01$, respectively, representing tubes $\mathcal {T}_1$ and $\mathcal {T}_2$. The field lines integrated from points on the isosurfaces almost lie on the surfaces, so the MSF isosurfaces effectively identify flux tubes with a very small averaged MSF deviation $\langle |\lambda _b| \rangle < 1\,\%$. The identification of flux tubes is further discussed in appendix A. The extracted flux tubes are colour coded by $-1\leqslant h^*\leqslant 1$ from blue to red, where $h^*\equiv (\boldsymbol {A}\boldsymbol {\cdot }\boldsymbol {b})/(|\boldsymbol {A}||\boldsymbol {b}|)$ denotes the normalized magnetic helicity density (see Moffatt & Tsinoher Reference Moffatt and Tsinoher1992). Compared with the helicity density, the overall magnitude of $h^*$ is not affected by the decay of $E_u$ and $E_b$.

Figure 3.

Temporal evolution of MSF isosurfaces of $\phi _b=\pm 0.01$ at (a) $t=1$, (b) $t=2$, (c) $t=3$ and (d) $t=4$. Some field lines are integrated and plotted on the isosurfaces colour coded by $-1\leqslant h^*\leqslant 1$ from blue to red.

From the MSF visualization, we observe that the initially orthogonal helical flux tubes first move towards each other and then merge. The colliding of $\mathcal {T}_1$ and $\mathcal {T}_2$ forms a convoluted plasmoid with knotted field lines (shown later in figure 7) at the centre of $\varOmega$. The plasmoid stretches outwards along the diagonal direction $(\boldsymbol {e}_z-\boldsymbol {e}_y)$ and gradually dissipates. Finally, two highly twisted U-shaped tubes are formed and ejected from the central region.

In addition, we observe that the evolution of the velocity–vorticity field shows no significant event of vortex dynamics at early times owing to the simple initial stagnation flow field in (2.9), but then intense local small-scale vortical structures are generated during the interaction of $\mathcal {T}_1$ and $\mathcal {T}_2$ near the plasmoid via the Lorentz force term in (2.3) with large $N_i$ (not shown).

From the topological features of MSF isosurfaces and field lines, the temporal evolution of $\mathcal {T}_1$ and $\mathcal {T}_2$ can be roughly divided into three stages: 1, incipient magnetic reconnection; 2, tying overhand knots; and 3, cascade of tying complex knots.

3.2. Stage 1: incipient magnetic reconnection

Driven by the straining velocity field (2.9), flux tubes $\mathcal {T}_1$ and $\mathcal {T}_2$ first collide and merge at the centre of $\varOmega$. We find that the minimum distance (Zhao et al. Reference Zhao, Yang and Chen2016) between MSF isosurfaces of $\phi _b=\pm 1\times 10^{-4}$ in the $x$-direction reaches zero around $t=1$ (not shown), which coincides with the time when notable energy conversion begins in figure 2(a). Thus, both the MSF results and flow statistics suggest that the incipient magnetic reconnection occurs around $t=1$.

Figure 4 plots the isosurfaces of $\phi _b=\pm 0.01$ in a subdomain $[{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]$ to provide a close-up view of the reconnection. We find that incipient reconnection occurs at the intersection between the MSF isosurfaces of $\phi _b=\pm 0.01$ and the isosurfaces of the current density magnitude $|\,\boldsymbol {j}|=30$, where the current sheet is shown later in figure 6 for clarity. Near this reconnection zone, we integrate two typical field lines before and after the reconnection in figures 4(a) and 4(b), respectively, which illustrates an X-type reconnection (see Priest & Forbes Reference Priest and Forbes2000; Pontin Reference Pontin2011).

Figure 4.

MSF isosurfaces of $\phi _b=\pm 0.01$ (translucent surfaces) at $t=1$ in the subdomain $[{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]$, with typical field lines (a) before and (b) after the magnetic reconnection. The reconnection zone is highlighted by the yellow dashed circle. The field lines are colour coded by $-1\leqslant h^*\leqslant 1$ from blue to red.

The field lines in figure 4 are colour coded by $h^*$. Before the reconnection, two field lines within $\mathcal {T}_1$ and $\mathcal {T}_2$, respectively marked by $\mathcal {A}_1$ and $\mathcal {A}_2$ in figure 4(a), have opposite $h^*$ owing to their opposite chirality, and they are nearly antiparallel in the reconnection region. After the reconnection of $\mathcal {A}_1$ and $\mathcal {A}_2$, two types of field lines, marked by $\mathcal {A}_1'$ and $\mathcal {A}_2'$, form in figure 4(b). Here, $\mathcal {A}_1'$, a U-shaped field line encircling the central region, moves along the direction of $(\boldsymbol {e}_y-\boldsymbol {e}_z)$ and further reconnects in the subsequent stage; $\mathcal {A}_2'$, a U-shaped field line, moves along the direction of $(\boldsymbol {e}_z-\boldsymbol {e}_y)$ away from the plasmoid.

Figure 5 sketches the reconnection process. Two field lines $\mathcal {A}_1$ and $\mathcal {A}_2$ with opposite $h^*$ are antiparallel in the reconnection region; then they form X-lines near magnetic null points. After the reconnection, the newly formed $\mathcal {A}_1'$ or $\mathcal {A}_2'$ consists of two pieces with opposite chirality.

Figure 5.

Schematic diagram for the magnetic reconnection in figure 4. The typical field lines are colour coded by red for $h^*>0$ and blue for $h^*<0$.

It is noted that this reconnection occurs symmetrically at two regions, as marked by yellow circles in figure 3(a). The reconnection regions can be roughly located by the closest antiparallel field lines between $\mathcal {T}_1$ and $\mathcal {T}_2$ with the initial magnetic field. Figures 1 and 3(a) illustrate that the flux tubes facing each other merge around $t=1$ with negligible deformation, and the field lines within $-{\rm \pi} /2\leqslant \theta _1\leqslant {\rm \pi}/2$ and ${\rm \pi} /2\leqslant \theta _2\leqslant 3{\rm \pi} /2$ first reconnect. Considering the symmetry with $r_1=r_2=r$ and a large twist parameter with $q^2r^2\gg 1$, we approximate the cosine of the alignment angle of $\boldsymbol {b}_1$ and $\boldsymbol {b}_2$ by

(3.1)\begin{equation} \cos\theta =\dfrac{q}{\sqrt{(1+q^2r_1^2)(1+q^2r_2^2)}} (qr_1r_2\sin\theta_1\sin\theta_2+r_1\cos\theta_1\!+\!r_2\cos\theta_2) \approx\sin\theta_1\sin\theta_2. \end{equation}

The reconnection of $\mathcal {T}_1$ and $\mathcal {T}_2$ implies antiparallel field lines with $\cos \theta \approx -1$. From (3.1), the reconnection region is close to the merged tubes in the central region with

(3.2a,b)\begin{equation} \theta_1=3{\rm \pi}/2,\ \theta_2={\rm \pi}/2\quad \textrm{or}\quad \theta_1={\rm \pi}/2,\ \theta_2=3{\rm \pi}/2. \end{equation}

These two locations are highlighted by yellow dashed circles in figure 3(a).

Figure 6(a) plots the contour of $|\,\boldsymbol {j}|$ on the $x$–$z$ plane at $y=7{\rm \pi} /8$. This slice cuts through the reconnection zone, which is marked by the green dashed line in figure 3(a). We observe that a current sheet forms in between two approaching flux tubes, as highlighted by the dashed box. Therefore, the reconnection X-points can be pinpointed by the intersection between the current sheet and the region with the maximum MSF gradient. In figure 6(b), the strong Lorentz force $\boldsymbol {F}_L=\boldsymbol {j}\times \boldsymbol {b}$ pointing to the current sheet implies that the merger of flux tubes is driven not only by the background straining flow, but also by the induced velocity from the deformation of the flux tubes themselves. The Lorentz force can further accelerate the local velocity and generate intense local vortical structures, leading to more complex magnetic reconnections in subsequent stages.

Figure 6.

Contour of $|\,\boldsymbol {j}|$ colour coded by $0\leqslant |\,\boldsymbol {j}|\leqslant 30$ from blue to red on the slice $y=7{\rm \pi} /8$ at $t=1$, with MSF isolines and the projected Lorentz force (arrows). (a) The $x$–$z$ slice $[3{\rm \pi} /4,5{\rm \pi} /4]\times [3{\rm \pi} /4,5{\rm \pi} /4]$ at $y=7{\rm \pi} /8$. (b) Close-up view of the region marked by the green dashed box in (a).

3.3. Stage 2: tying overhand knots

In stage 1, the helical field lines before and after the reconnection remain unknotted. Then in stage 2, some field lines in the plasmoid are tied into knots by the secondary reconnection around $t = 1.75$. The reconnection still occurs around the intersection between MSF isosurfaces of $\phi _b=\pm 0.01$ and isosurfaces of $|\,\boldsymbol {j}|=30$. From the reconnection region, we integrate a number of field lines to seek the X-type reconnection and knotted field lines.

The close-up view of MSF isosurfaces of $\phi _b=\pm 0.01$ at $t=1.75$ in figure 7 illustrates a delicate knotting process of field lines via the secondary magnetic reconnection. In figure 7(a), two typical field lines within $\mathcal {T}_1$ and $\mathcal {T}_2$, marked by $\mathcal {B}_1$ and $\mathcal {B}_2$, are almost antiparallel near the reconnection X-point before the secondary reconnection. Note that the $\Omega$-shaped field line $\mathcal {B}_1$ encircling the central region consists of two line segments with positive and negative $h^*$, indicating that it has undergone the first reconnection as $\mathcal {A}_1'$ and $\mathcal {A}_2'$ in figure 4(b); $\mathcal {B}_2$ is a helical line along the $y$-direction, similar to $\mathcal {A}_2$ in figure 4(a). The reconnection of $\mathcal {B}_1$ and $\mathcal {B}_2$ produces an overhand knot $\mathcal {B}_1'$ in figure 7(b). At the mean time, a U-shaped field line $\mathcal {B}_2'$ forms and moves along the direction of $(\boldsymbol {e}_z-\boldsymbol {e}_y)$ away from the central region.

Figure 7.

MSF isosurfaces of $\phi _b=\pm 0.01$ (translucent surfaces) at $t=1.75$ in the subdomain $[{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]$, with typical field lines (a) before and (b) after the magnetic reconnection. The reconnection zone is highlighted by the yellow dashed circle. The field lines are colour coded by $-1\leqslant h^*\leqslant 1$ from blue to red.

Figure 8 sketches the formation of the overhand knot via the secondary reconnection. It is interesting that the knotting process is via such a simple antiparallel reconnection in the present case, rather than more complex mechanisms such as bridging and threading in vortex reconnection (e.g. Melander & Hussain Reference Melander and Hussain1988; Kida & Takaoka Reference Kida and Takaoka1994; Yao & Hussain Reference Yao and Hussain2020). After this reconnection, $\mathcal {B}_1'$ consists of three pieces. The piece with positive $h^*$ is embedded into the other two with negative $h^*$. Thus, this topological change also makes the geometry of field lines more complex.

Figure 8.

Schematic diagram for tying the overhand knot via the magnetic reconnection in figure 7. The typical field lines are colour coded by red for $h^*>0$ and blue for $h^*<0$.

The reconnection location at $t=1.75$ is similar to that at $t=1$. Figure 9 plots the contour of $|\,\boldsymbol {j}|$ on the slice at $y=7{\rm \pi} /8$. Driven by the combination of the fluid velocity and the velocity induced by the Lorentz force, two MSF isosurfaces of $\phi _b=\pm 0.001$ collide at the reconnection point, along with the formation of the current sheet in between the two approaching flux tubes.

Figure 9.

Contour of $|\,\boldsymbol {j}|$ colour coded by $0\leqslant |\,\boldsymbol {j}|\leqslant 30$ from blue to red on the slice $y=7{\rm \pi} /8$ at $t=1.75$, with MSF isolines and the projected Lorentz force (arrows). (a) The $x$–$z$ slice $[3{\rm \pi} /4,5{\rm \pi} /4]\times [3{\rm \pi} /4,5{\rm \pi} /4]$ at $y=7{\rm \pi} /8$. (b) Close-up view of the region marked by the green dashed box in (a).

The knotting process can happen at a range of $\phi _b$. Since the MSF isosurfaces of different isocontour levels are coaxial tubes, we can distinguish inner and outer flux tubes by setting particular thresholds of $\phi _b$, and quantify their respective influences on the statistics and important events in the flow evolution. Considering the inner tubes within $\varOmega _{I}=\{\boldsymbol {x}\in \mathbb {R}^3\,|\,0.5<\phi _b(\boldsymbol {x})\leqslant 1\}$ and outer tubes within $\varOmega _{O}=\{\boldsymbol {x}\in \mathbb {R}^3\,|\,0.1<\phi _b(\boldsymbol {x})\leqslant 0.5\}$, the temporal evolution of the volume-averaged kinetic energy and normalized magnetic helicity density for the inner and outer flux tubes is shown in figure 10.

Figure 10.

Temporal evolution of volume-averaged (a) kinetic energy and (b) normalized magnetic helicity within inner or outer flux tubes.

At early times, both $\langle E_u \rangle _I$ and $\langle E_u \rangle _O$ decay due to the viscous dissipation in figure 10(a), where $\langle \cdot \rangle _I$ and $\langle \cdot \rangle _O$ denote the volume averages over $\varOmega _I$ and $\varOmega _O$, respectively. Around $t\approx 1$, $\langle E_u \rangle _I$ and $\langle E_u \rangle _O$ begin to rise with time, consistent with the incipient reconnection time $t=1$. Then the primary peaks of $\langle E_u \rangle _I$ and $\langle E_u \rangle _O$ occur at $1<t<2$. These volume averages effectively avoid the interference from the background flow outside the tube, i.e. in the region with $-0.1\leqslant \phi _b\leqslant 0.1$, on the flow statistics. The peak of $\langle E_u \rangle _I$ is higher than that of $\langle E_u \rangle _O$. This implies that the energy conversion from $E_b$ to $E_u$ within inner tubes is stronger than that within outer tubes through more frequent magnetic reconnections and stronger release of the magnetic energy.

The exact quantification of the helicity evolution is difficult in the present configuration owing to the periodic boundary condition with net fluxes (e.g. Berger Reference Berger1997) and the vanishing total helicity over the entire domain. As possible measures of knottedness or helical degree of field lines, $\langle h^* \rangle _I$ and $\langle h^* \rangle _O$ surge around $t=1.5$ in figure 10(b), and $\langle h^* \rangle _I$ peaks at $t=2$, consistent with the knotting of field lines in figure 7. We remark that the helicities $\langle h \rangle _I$ and $\langle h \rangle _O$ of inner and outer tubes generally decay with time due to the dissipation, so they cannot distinguish the knotting period and are omitted here. Moreover, the volume-averaged helicity density over the complement region $\{\boldsymbol {x}\in \mathbb {R}^3\,|\,0<\phi _b\leqslant 0.1\}$ is negligible.

Based on the MSF, the reconnection and tying of field lines can be characterized by the magnetic flux $F\equiv \int _{\mathcal {S}}\boldsymbol {b}\boldsymbol {\cdot }\boldsymbol {n}\, \mathrm {d}\mathcal {S}$ through a symmetric plane with the surface normal $\boldsymbol {n}$. Considering the flow symmetries, two typical magnetic fluxes

(3.3)\begin{equation} \left.\begin{gathered} F_1=\frac{1}{\sqrt{2}}\int_{\mathcal{S}_1}(b_z(\boldsymbol{x})- b_y(\boldsymbol{x}))\,\mathrm{d}\mathcal{S},\\ F_2=\frac{1}{\sqrt{2}}\int_{\mathcal{S}_2}(b_z(\boldsymbol{x})+ b_y(\boldsymbol{x}))\,\mathrm{d}\mathcal{S} \end{gathered}\right\} \end{equation}

of $\mathcal {T}_1$ through diagonal planes are calculated, where $\mathcal {S}_1$ denotes the region enclosed by $\phi _b=0$ and $\phi _b=1$ on the plane of $y+z=2{\rm \pi}$, and $\mathcal {S}_2$ denotes the region enclosed by $\phi _b=0$ and $\phi _b=1$ on the plane of $y-z=0$. We remark that the poloidal/toroidal flux (see Moffatt & Ricca Reference Moffatt and Ricca1992) is not calculated here, because it is difficult to accurately determine the centreline of the very distorted flux tubes during and after reconnection.

The temporal evolutions of $F_1$ and $F_2$ are shown in figure 11. The initial magnetic field in (2.6) and (2.10) of $\mathcal {T}_1$ implies $F_1(t=0)=F_2(t=0)=-\varGamma /\sqrt {2}$. We observe that both fluxes are nearly conserved before the first reconnection at $t=1$, consistent with the Alfvén theorem. Around $t=1$, the magnitude of the fluxes significantly decays owing to the direction change of field lines during the reconnection (see figure 4).

Figure 11.

Magnetic fluxes (3.3) through diagonal symmetric planes.

Figure 12 plots contours of the flux density $(b_z-b_y)/\sqrt {2}$ of $F_1$ on the diagonal symmetric plane $y+z=2{\rm \pi}$ in the subdomain $[3{\rm \pi} /4,5{\rm \pi} /4]\times [3{\rm \pi} /4,5{\rm \pi} /4]\times [3{\rm \pi} /4,5{\rm \pi} /4]$ with some typical field lines from two perspective views. The boundary of $\mathcal {T}_1$, the isosurface of $\phi _b=0.01$, is marked by the thick line on the plane. At $t=1$, the field lines integrated on the outer side of the central region in figure 12(a) only have slight deformation compared to the initial configuration. On the other hand, the directions of the field lines integrated on the inner side of the central region have significant changes in figure 12(b). Owing to the reconnection, some of these field lines of $\mathcal {T}_1$ originally through $\mathcal {S}_1$ no longer pass through $\mathcal {S}_1$, and the angles between the field lines and the symmetric plane are decreased. Hence, the strength of $F_1$ is weakened in figure 11 at $t=1$.

Figure 12.

Contours of the flux density of $F_1$ on the diagonal symmetric plane $y+z=2{\rm \pi}$ from two perspectives at (a,b) $t=1$ and (c,d) $t=1.75$ in the subdomain $[3{\rm \pi} /4,5{\rm \pi} /4]\times [3{\rm \pi} /4,5{\rm \pi} /4]\times [3{\rm \pi} /4,5{\rm \pi} /4]$, where the arrow denotes the normal $\boldsymbol {n}$ of the symmetric plane. Some field lines are integrated on the MSF isosurface of $\phi _b=\pm 0.01$ (translucent surface). Region $\mathcal {S}_1$ is enclosed by the intersection (yellow line) of the MSF isosurface and the diagonal plane.

From $t=1.5$ to $t=2$, the profile of $F_1$ shows a plateau in figure 11, signalling the secondary reconnection of field lines. At $t=1.75$, magnetic overhand knots are generated after the reconnection. The released kinetic energy shown in figure 2(a) accelerates the distortion of the field lines in the central region, and produces more positive flux with larger angle between field lines and the plane in figure 12(c). Meanwhile, the angle between field lines and the plane also increases in the central region, producing more negative flux in figure 12(d). Owing to the two competing effects, flux $F_1$ through $\mathcal {S}_1$ remains almost unchanged from $t=1.5$ to $t=2$.

3.4. Stage 3: cascade of tying complex knots

After the formation of overhand knots in the plasmoid, a sequence of reconnection events at the same regions as marked in figure 3(a) leads to a cascade of tying more complex knots. Figure 13 illustrates the formation of double overhand knots at $t=2$. Two typical field lines within $\mathcal {T}_1$ and $\mathcal {T}_2$ are marked by $\mathcal {C}_1$ and $\mathcal {C}_2$ before the tertiary reconnection. Here, $\mathcal {C}_1$ is an unknotted line encircling the plasmoid, and has highly convoluted geometry colour coded by $h^*$ with alternating signs; $\mathcal {C}_2$ is a helical line along the $y$-direction, with geometry similar to $\mathcal {A}_2$ and $\mathcal {B}_2$. The reconnection of $\mathcal {C}_1$ and $\mathcal {C}_2$ forms a double overhand knot $\mathcal {C}_1'$ along the $y$-direction and a U-shaped line $\mathcal {C}_2'$ moving away from the reconnection zone. We find that the positions of the three reconnections in figures 4, 7 and 13 are very close to the theoretical estimation in (3.2a,b). In addition, the initial magnetic condition has an impact on the knotting process, which is discussed in appendix B.

Figure 13.

MSF isosurfaces of $\phi _b=\pm 0.01$ (translucent surfaces) at $t=2$ in the subdomain $[{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]\times [{\rm \pi} /2,3{\rm \pi} /2]$, with typical field lines (a) before and (b) after the magnetic reconnection. The reconnection zone is highlighted by the yellow dashed circle. The field lines are colour coded by $-1\leqslant h^*\leqslant 1$ from blue to red.

The observations in figures 4, 7 and 13 imply that the major mechanism for the current magnetic reconnection is the X-type reconnection (see Priest & Forbes Reference Priest and Forbes2000; Pontin Reference Pontin2011), which is similar to viscous cancellation rather than bridging (see Melander & Hussain Reference Melander and Hussain1988; Kida & Takaoka Reference Kida and Takaoka1994) in vortex reconnection. It is noted that vortex lines can be tangled under the strong self-induced local velocity generated in the incipient reconnection of vortex lines, whereas the reconnection of field lines can only alter the local velocity via an implicit way at moderate and large interaction parameters (see Kivotides Reference Kivotides2018).

The influence of the magnetic knot cascade on the local flow motion is through the Lorentz force. figure 14 shows the knot motion projected onto the $y$–$z$ plane, along with the isoline of $\phi _b=0.01$ and the projected velocity field on the slice at $x=7{\rm \pi} /8$. Several knotted field lines are integrated from the reconnection regions. We observe that, in general, the local velocity is enhanced and small-scale vortical structures are generated during the magnetic reconnection. In figure 14(a), the velocity shows a vortex pattern, making the knots rotate around the $x$-axis and distort at $t=1.75$. In figure 14(b), the velocity components inclined to the $z$-direction gradually dominate. They drive the U-shaped field lines after the reconnection to separate, and stretch the knots along the $z$-direction. The knotting process is gradually spread from inner tubes to outer tubes, causing the growth of $\langle E_u \rangle _O$ and $\langle h^*_m \rangle _O$ in figure 10 around $t=1.75$. In figure 14(c), the knots move closer to the reconnection zone, triggering further reconnections to generate triple overhand knots and more complex knots at $t=2.25$. Furthermore, induced by the Lorentz force term in (2.3), the initially closed, unknotted vortex lines also become highly tangled and twisted, and some of them can be knotted (not shown), which is worth investigating in future work.

Figure 14.

The projected velocity (arrows) on the $y$–$z$ slice $[3{\rm \pi} /4,5{\rm \pi} /4]\times [3{\rm \pi} /4,5{\rm \pi} /4]$ at $x=7{\rm \pi} /8$, along with a set of knotted field lines integrated from the reconnection regions and the isoline of $\phi _b=0.01$: (a) $t=1.75$, (b) $t=2$ and (c) $t=2.25$.

We use the minimal crossing number, which is a knot invariant under Reidemeister moves (Reidemeister Reference Reidemeister1927; Alexander & Briggs Reference Alexander and Briggs1926), to quantify the knotting cascade via a sequence of magnetic reconnections. The typical knotted field lines in the plasmoid, shown in figures 4, 7 and 13, are projected from proper perspective views to depict a ‘knot diagram’ in figure 15(a–c). Reducible or nugatory crossings (Hoste, Thistlethwaite & Weeks Reference Hoste, Thistlethwaite and Weeks1998) can be removed in knot diagrams by proper Reidemeister moves. Meanwhile, non-reducible crossings, counted by the minimal crossing number $n_c$ in reduced knot diagrams, are marked by thick ‘$-$’ in figure 15(a–c). We find that $n_c$ grows with time, indicating the increasing complexity of knots in a cascade process.

Figure 15.

(a–c) Cascade of tying magnetic knots in the plasmoid, with non-reducible crossings marked by thick ‘$-$’: (a) overhand knot (trefoil knot) at $t=1.75$, (b) double overhand knot (cinquefoil knot) at $t=2.0$, and (c) triple overhand knot (septafoil knot) at $t=2.25$. (d–f) Corresponding reduced knot diagrams of torus knots (these knots are equivalent to those in the upper row): (d) trefoil knot, (e) cinquefoil knot, and (f) septafoil knot.

If the ends of the field lines at the boundary of $\varOmega$, e.g. points $\mathcal {P}_1$ and $\mathcal {P}_2$ of the overhand knot in figure 15(a), are closed at infinity without intersecting other field lines in $\varOmega$, the knots are equivalent to the torus knots, e.g. the trefoil knot with $n_c=3$ in figures 15(d). For more complex knots, we demonstrate how to deform the double overhand knot in figure 15(b) into the cinquefoil knot in figure 15(e) using Reidemeister moves. First, the two ends $\mathcal {P}_1$ and $\mathcal {P}_2$ of the field line are connected to form a loop. Second, part $\mathcal {L}_1$ marked by the red solid curve is moved completely over crossing ${O}_0$, from right to left as marked by the red arrow, by the type-III Reidemeister move, and part $\mathcal {L}_2$ marked by the blue dashed line is moved under crossing ${O}_0$ from left to right by the type-III Reidemeister move. Finally, $\mathcal {L}_1$ is moved over crossing ${O}_1$ and $\mathcal {L}_2$ is moved under crossing ${O}_2$ by type-III Reidemeister moves. Thus, the crossing number of the two knots in a minimal two-dimensional knot diagram is $n_c=5$. Similarly, the triple overhand knot in figure 15(c) is equivalent to the septafoil knot with $n_c=7$ in figure 15(f) after proper Reidemeister moves.

Figure 16 plots a highly coiled field line (blue) at $t=2.25$, surrounded by the triple overhand knot (red) shown in figure 15(c). The inner field line, in the global shape of a Mobius band, has thousands of turns around the plasmoid. The stretching effect shown in figure 14(b,c) appears to suppress the knotting process by merging neighbouring field lines, and both $\langle h^* \rangle _I$ and $\langle h^* \rangle _O$ begin to decay around $t=2.25$. In addition, we did not observe any clear sign of the relaxation of magnetic knots (e.g. Taylor Reference Taylor1974; Moffatt Reference Moffatt1990) as in Xiong & Yang (Reference Xiong and Yang2020). A possible reason is that each knotted field line in the present study is not isolated and it is distorted by fluctuating local fluid motion.

Figure 16.

Complex knots in the plasmoid at $t=2.25$. (a) MSF isosurfaces of $\phi _b=\pm 0.01$ (translucent surfaces) with typical field lines. (b) Close-up view of a triple overhand knot (red) and a highly coiled field line (blue).

The knot cascade slows down and then terminates at later times owing to the separation of the U-shaped tubes shown in figure 3(c,d). The plasmoid at the centre is stretched along the direction of $(\boldsymbol {e}_z-\boldsymbol {e}_y)$. The highly coiled field lines are severed by successive reconnections with the cancellation of opposite directions of the field lines near the centre of $\varOmega$. Finally, the untied magnetic knots gradually vanish in the central region at $t>4$ with resistive and viscous dissipations, which is similar to the observation in Linton et al. (Reference Linton, Dahlburg and Antiochos2001).

3.5. Quantification of the magnetic knot cascade

We have demonstrated a knot cascade of field lines through a sequence of reconnection events in the evolution of helical flux tubes. Starting from the unknotted state, the first overhand knot with $n_c=3$ forms around $t=1.75$, and then $n_c$ is incremented by two in each subsequent reconnection event at the same region near (3.2a,b) in the plasmoid until the separation of the U-shaped tubes and the vanishing of the plasmoid at late times.

In order to further quantify the knot cascade using the statistical distribution of knot types, we integrate 3700 field lines in both forward and backward directions from spatially uniform points sampled on the symmetric plane

(3.4)\begin{equation} \mathcal{S}^*=\left\{\boldsymbol{x} \,\Bigg|\,|\phi_b(\boldsymbol{x})|\geqslant0.1, \frac{3{\rm \pi}}{4}\leqslant x\leqslant \frac{5{\rm \pi}}{4},y={\rm \pi},{\rm \pi}\leqslant z\leqslant \frac{5{\rm \pi}}{4} \right\} \end{equation}

within the plasmoid at each of the three typical times $t=1.75$, 2 and 2.25. Each field line is extracted as a set of discretized points in three dimensions; then we artificially add an unknotted and unlinked line segment consisting of several ghost points outside $\varOmega$ to close the two ends of a field line to form a loop.

The knot type of the closed field lines, in terms of the Alexander–Briggs notation (Alexander & Briggs Reference Alexander and Briggs1926), is identified by a knot analysis toolkit ‘pyknotid’ (Taylor et al. Reference Taylor2017). In this toolkit, the Alexander polynomial is calculated via a standard matrix determinant algorithm (see Orlandini & Whittington Reference Orlandini and Whittington2007). Although the Alexander polynomial has some weaknesses to distinguish complex knots (see Kauffman Reference Kauffman2001; Liu & Ricca Reference Liu and Ricca2015), it is sufficient to identify simple torus knots (e.g. Kuei et al. Reference Kuei, Słowicka, Ekiel-Jeżewska, Wajnryb and Stone2015; Mesgarnezhad et al. Reference Mesgarnezhad, Cooper, Baggaley and Barenghi2017) in the present study.

The probability density function (p.d.f.) of the identified knot type at the three typical times is plotted in figure 17. Here, the p.d.f. of unknotted field lines is omitted for clarity. Additionally, $12.5\,\%$ of extracted field lines at $t=2.25$ are highly convoluted in the plasmoid as observed in figure 16. Their knot types, which have large crossing numbers or are not torus knots, may not be accurately identified owing to the finite spatial resolution and total integration steps for extracting the field lines, so the p.d.f.s for these field lines are also omitted.

Figure 17.

The p.d.f. of magnetic knot types in terms of the Alexander–Briggs notation at $t=1.75$, $t=2$ and $t=2.25$.

The knot statistics in figure 17 reveals that the ratio of knotted field lines to all the field lines in the central region surges with time, e.g. less than $4\,\%$ of field lines are knotted at $t=1.75$, and this ratio grows to $20\,\%$ at $t=2.25$. As illustrated in figure 15, the p.d.f. peak migrates from trefoil knots $3_1$ at $t=1.75$, through cinquefoil knots $5_1$ at $t=2$, to septafoil knots $7_1$ at $t=2.25$. Moreover, the p.d.f. is broadened with time, indicating that there are various types of complex knots at late times.

It is interesting that the knotting cascade of field lines in MHD flows is in contrast with the unknotting cascade of knotted vortex filaments found in HD flows (Kleckner & Irvine Reference Kleckner and Irvine2013; Xiong & Yang Reference Xiong and Yang2019a) and superfluids (Kleckner et al. Reference Kleckner, Kauffman and Irvine2016). The possible reason is that the field lines tend to be twisted by the Lorentz force. The resultant helical field lines have higher probabilities than straight lines to be tied into knots via the magnetic reconnection in MHD flows, as illustrated in figure 7. By contrast, the twist of vortex lines can decay rapidly without external forcing in HD flows (see Scheeler et al. Reference Scheeler, van Rees, Kedia, Kleckner and Irvine2017; Xiong & Yang Reference Xiong and Yang2020), so that vortex lines appear to have lower probabilities to be knotted than magnetic field lines.

In the characterization of the knotting cascade, the MSF plays an important role to determine the reconnection region and the starting points for integrating knotted field lines. On the other hand, there are several issues to hinder further quantifications of the knotting cascade in dissipative systems using topological diagnostic tools such as adapted knot polynomials (see Liu & Ricca Reference Liu and Ricca2015, Reference Liu and Ricca2016) and the pathway analysis in terms of the minimal crossing number and topological writhe (see Kleckner et al. Reference Kleckner, Kauffman and Irvine2016).

First, it is hard, or even not well defined in principle, to track a particular field line in a resistive MHD flow (see Hao et al. Reference Hao, Xiong and Yang2019). Second, there is a lack of a general algorithm to compute knot polynomials for complex knots in a periodic domain (see Liu & Ricca Reference Liu and Ricca2015; Cooper et al. Reference Cooper, Mesgarnezhad, Baggaley and Barenghi2019). Third, there are an infinite number of field/vortex lines in complex MHD/HD flows, so it is not clear how to characterize the influence of the knotting/unknotting cascade on important flow statistics. Therefore, the full topological diagram and the significance of knotting/unknotting cascade in MHD/HD turbulence remains an open problem.