2.1. Introduction to the spin Euler equation

The 3-D incompressible Euler equation is

(2.1)\begin{equation} \partial_t\boldsymbol{u} + \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u} ={-} \boldsymbol{\nabla} p, \end{equation}

with $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u} = 0$, where $\boldsymbol {u}$ is the velocity, and $p$ is the pressure.

By applying the spherical Clebsch mapping with a Hamiltonian structure (Kuznetsov & Mikhailov Reference Kuznetsov and Mikhailov1980), (2.1) is transformed into a Lagrangian form

(2.2)\begin{equation} \partial_t\boldsymbol{s} + \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{s} = \boldsymbol{0}, \end{equation}

where $\boldsymbol {s}$ is of class $C^k$, $k\ge 1$. Note that $\boldsymbol {u}$ can be obtained from $\boldsymbol {s}$, which is discussed further in (3.1). Here, the Hopf fibration (Hopf Reference Hopf1931)

(2.3)\begin{equation} \boldsymbol{s} = (a^2+b^2-c^2-d^2, 2(bc-ad), 2(ac+bd)) \end{equation}

with $|\boldsymbol {s}|=1$ establishes a correspondence between the unit spin vector $\boldsymbol {s}\in \mathbb {S}^2$ and a two-component wave function $\boldsymbol {\psi }=[\psi _1,\psi _2]^{\mathrm {T}}\in \mathbb {S}^3$ (Chern et al. Reference Chern, Knöppel, Pinkall, Schröder and Weißmann2016; Chern Reference Chern2017), where $\psi _1=a+\mathrm {i} b$ and $\psi _2=c+\mathrm {i} d$ are complex functions of real-valued potentials $a,b,c, d$, and $\mathrm {i}$ denotes the imaginary unit. The Clebsch potentials $a,b,c,d$ are subject to the constraint $a\,\Delta b-b\,\Delta a+c\,\Delta d-d\,\Delta c=0$, which ensures the incompressibility $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u}=0$.

Then the velocity and vorticity $\boldsymbol {\omega }\equiv \boldsymbol {\nabla }\times \boldsymbol {u}$ can be re-expressed by $\boldsymbol {u} = a\,\boldsymbol {\nabla } b - b\,\boldsymbol {\nabla } a\, +$ $c\,\boldsymbol {\nabla } d - d\,\boldsymbol {\nabla } c$ and

(2.4)\begin{align} \boldsymbol{\omega} &= \tfrac{1}{4}\varepsilon_{ijk}s_i\,\boldsymbol{\nabla} s_j\times \boldsymbol{\nabla} s_k \nonumber\\ &= \tfrac{1}{2}\left(s_1\,\boldsymbol{\nabla} s_2\times\boldsymbol{\nabla} s_3 + s_2\,\boldsymbol{\nabla} s_3\times\boldsymbol{\nabla} s_1 + s_3\,\boldsymbol{\nabla} s_1\times\boldsymbol{\nabla} s_2 \right), \end{align}

respectively, where $\varepsilon _{ijk}$ is the Levi–Civita symbol. The derivation of (2.4) is detailed in Appendix A. Note that $\boldsymbol {s}$ remains a gauge invariant in the transformation from the velocity to ‘magnetization’ (Chorin Reference Chorin1994).

From the perspective of vortex dynamics, the spin vector encodes vortex lines and surfaces (Chern et al. Reference Chern, Knöppel, Pinkall, Schröder and Weißmann2016; Chern Reference Chern2017; Yang et al. Reference Yang, Xiong and Lu2023). Isosurfaces of one component of $\boldsymbol {s}$ are a family of vortex surfaces. Correspondingly, intersections of isosurfaces of two components of $\boldsymbol {s}$ are a family of vortex lines.

We consider the quaternion form (Gibbon & Holm Reference Gibbon and Holm2007) of the two-component wave function $\boldsymbol {\psi }=a+\boldsymbol {i}b+\boldsymbol {j}c+\boldsymbol {k}d$, where $\{\boldsymbol {i},\boldsymbol {j},\boldsymbol {k}\}$ are the basis vectors of the imaginary part of the quaternion. The velocity and spin vector are then given by $\boldsymbol {u}=(\boldsymbol {\nabla }\bar {\boldsymbol {\psi }}\boldsymbol {i\psi } - \bar {\boldsymbol {\psi }}\boldsymbol {i}\,\boldsymbol {\nabla }\boldsymbol {\psi })/2$ and $\boldsymbol {s}=\bar {\boldsymbol {\psi }}\boldsymbol {i\psi }$, respectively, where $\bar {\boldsymbol {\psi }}$ denotes the quaternion conjugate of $\boldsymbol {\psi }$. Then we derive

(2.5)\begin{align} \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{s} &= \tfrac{1}{2}\left(\boldsymbol{\nabla}\bar{\boldsymbol{\psi}}\boldsymbol{i\psi}\boldsymbol{\cdot} \boldsymbol{\nabla}\bar{\boldsymbol{\psi}}\boldsymbol{i\psi} - |\boldsymbol{\nabla}\boldsymbol{\psi}|^2 + |\boldsymbol{\nabla}\boldsymbol{\psi}|^2 - \bar{\boldsymbol{\psi}}\boldsymbol{i}\,\boldsymbol{\nabla}\boldsymbol{\psi}\boldsymbol{\cdot}\bar{\boldsymbol{\psi}}\boldsymbol{i}\,\boldsymbol{\nabla}\boldsymbol{\psi}\right) \nonumber\\ &= \tfrac{1}{2}\left(\boldsymbol{s}\,\boldsymbol{\nabla}\bar{\boldsymbol{\psi}}\boldsymbol{\cdot}\boldsymbol{i}\,\boldsymbol{\nabla}\boldsymbol{\psi} -\boldsymbol{\nabla}\bar{\boldsymbol{\psi}}\boldsymbol{\cdot}\boldsymbol{i}\,\boldsymbol{\nabla}\boldsymbol{\psi}\boldsymbol{s}\right) = \boldsymbol{s}\times\boldsymbol{m}, \end{align}

where $\boldsymbol {m} \equiv \boldsymbol {\nabla }\bar {\boldsymbol {\psi }}\boldsymbol {\cdot }\boldsymbol {i}\,\boldsymbol {\nabla }\boldsymbol {\psi }$ is a pure quaternion (i.e. a vector in $\mathbb {R}^3$) and can be expanded as $\boldsymbol {m}=\boldsymbol {i}m_1+\boldsymbol {j}m_2+\boldsymbol {k}m_3$, with

(2.6) \begin{equation} \left.\begin{aligned} m_1 &= |\boldsymbol{\nabla} a|^2 + |\boldsymbol{\nabla} b|^2 - |\boldsymbol{\nabla} c|^2 - |\boldsymbol{\nabla} d|^2, \\ m_2 &= 2(\boldsymbol{\nabla} b\boldsymbol{\cdot}\boldsymbol{\nabla} c - \boldsymbol{\nabla} a\boldsymbol{\cdot}\boldsymbol{\nabla} d), \\ m_3 &= 2(\boldsymbol{\nabla} a\boldsymbol{\cdot}\boldsymbol{\nabla} c + \boldsymbol{\nabla} b\boldsymbol{\cdot}\boldsymbol{\nabla} d). \end{aligned}\right\} \end{equation}

Thus we rewrite $\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {s}= \boldsymbol {s}\times \boldsymbol {m}$ with an effective field $\boldsymbol {m}=(m_1,m_2,m_3)$.

In general, $\boldsymbol {m}$ cannot be represented solely in terms of $\boldsymbol {s}$, because the Hopf mapping (2.3) is non-invertible. However, given $\boldsymbol {s}$ with a boundary condition, we can obtain $\boldsymbol {u}$ by calculating $\boldsymbol {\omega }$ with (2.4) and applying the generalized Biot–Savart law, and then obtain $\boldsymbol {s}\times \boldsymbol {m}$ at any point in $\mathbb {R}^3$ by (2.5).

Substituting (2.5) into (2.2), we obtain the spin Euler equation

(2.7)\begin{equation} \partial_t\boldsymbol{s} + \boldsymbol{s}\times\boldsymbol{m} = \boldsymbol{0}. \end{equation}

This is equivalent to the original incompressible Euler equation (2.1). In contrast to (2.2), (2.7) characterizes the evolution of $\boldsymbol {s}$ by its precession about $\boldsymbol {m}$ rather than the convection with $\boldsymbol {u}$. The spin Euler equation (2.7) can be more suitable to study fluid dynamics from a Lagrangian perspective than its original form (2.1), because the isosurfaces of $s_i$, $i=1,2,3$, are vortex surfaces consisting of vortex lines (Yang & Pullin Reference Yang and Pullin2010,  Reference Yang and Pullin2011; Yang et al. Reference Yang, Xiong and Lu2023). From the Helmholtz vorticity laws, the surfaces are material surfaces for all $t\ge 0$ in Euler flows.

Therefore, solving the spin Euler equation (2.7) is similar to a vortex method (Yang et al. Reference Yang, Xiong, Zhang, Feng, Liu and Zhu2021; Nabizadeh et al. Reference Nabizadeh, Wang, Ramamoorthi and Chern2022; Xiong et al. Reference Xiong, Wang, Wang and Zhu2022) for simulating ideal flows. Since the primary variable $\boldsymbol {s}$ of (2.7) has unit length, the fixed magnitude of $\boldsymbol {s}$ can avoid the numerical blowup arising from numerical instabilities.

In particular, the spin Euler equation contains the inherent Lagrangian vortex dynamics via level sets of $s_i$ (i.e. vortex surfaces). This can facilitate the regularity analysis of the Euler equation, similar to the level set representation of $\boldsymbol {\omega }$ (Constantin Reference Constantin2001a,Reference Constantinb; Deng et al. Reference Deng, Hou and Yu2005).

Note that the initial conditions $\boldsymbol {s}_0$ of (2.7) for simulating ideal flows are a subset of the classical Euler equations, because finding a globally smooth $\boldsymbol {s}_0$ for a given vorticity field $\boldsymbol {\omega }_0$ remains an open problem. Currently, a useful approximation of $\boldsymbol {s}_0$ can be obtained using the numerical optimization (Chern et al. Reference Chern, Knöppel, Pinkall and Schröder2017) and the Poincaré recurrence theorem (Poincaré Reference Poincaré1890).

The spin Euler equation is also equivalent to a special case of the Landau–Lifshitz equation with a specific effective magnetic field $\boldsymbol {m}$, revealing a possible connection between ideal flow and magnetic crystal. As sketched in figure 1, the spin vector (or magnetization) $\boldsymbol {s}$ in (2.3) at each point in space precesses around the effective magnetic field $\boldsymbol {m}$ in (2.6). More discussion is provided in Appendix B.

Figure 1.

The domain is discretized on a uniform grid. The smallest lattice configuration, consisting of $2^3$ grid points, is sketched. (a) Schematic of a $2^3$ lattice of spin vectors, where the orientation of $\boldsymbol {s}$ fully characterizes the ideal flow by (2.4). (b) At each grid point, the spin vector $\boldsymbol {s}$ in (2.3) precesses around an ‘effective magnetic field’ $\boldsymbol {m}$ in (2.6).

2.2. Non-blowup condition of the spin Euler equation

Next, we discuss the regularity of the spin Euler equation. The derivation for the non-blowup condition is outlined below and detailed in Appendix C. First, we estimate the upper bound of $\| \boldsymbol {\omega } \|_p$ in terms of $\| \boldsymbol {\nabla }\boldsymbol {s} \|_{2p}$, $p\ge 1$, as

(2.8)\begin{equation} \|\boldsymbol{\omega}\|_{p} \le \tfrac{1}{2}\,\|\boldsymbol{\nabla}\boldsymbol{s}\|_{{2p}}^2, \end{equation}

where the $L^p$-norm is defined in Appendix C. Then we estimate the growth of $\| \boldsymbol {\nabla }\boldsymbol {s} \|_{2p}$ as

(2.9)\begin{equation} \partial_t\| \boldsymbol{\nabla}\boldsymbol{s} \|_{{2p}}^{2p} \le 2p\,\| \boldsymbol{m} \|_{1}\,\| \Delta\boldsymbol{s} \|_{\infty}^p, \end{equation}

and the upper bound of $\| \boldsymbol {m} \|_1$ in terms of $\| \boldsymbol {\nabla }\boldsymbol {s} \|_2$ and $\| \Delta \boldsymbol {s} \|_\infty$ as

(2.10)\begin{equation} \| \boldsymbol{m} \|_{1}\le \tfrac{1}{4}\,\| \boldsymbol{\nabla}\boldsymbol{s} \|_{2}^2\left(\sqrt{6} + \sqrt{2}C_\omega\,\| \Delta\boldsymbol{s} \|_{\infty} \right), \end{equation}

where $C_\omega$ is a positive constant. Finally, substituting (2.9) and (2.10) into (2.8), we obtain a sufficient condition for bounded $\| \boldsymbol {\omega } \|_p$ as

(2.11)\begin{equation} \int_0^t\| \Delta\boldsymbol{s}(\,\cdot\,,\tau) \|_{\infty}^{p+1}\,{\rm d}\tau < \infty, \quad \forall p\ge 1. \end{equation}

In summary, we obtain a non-blowup condition (2.11) of the spin Euler equation (equivalent to the original Euler equation with well-posed initial conditions), which guarantees a bounded $\| \boldsymbol {\omega } \|_{p}$. It implies that if the solution loses regularity beyond a certain time, then the Laplacian of the spin vector must grow unboundedly. The transport equation and the estimation of the norm of $\Delta \boldsymbol {s}$ are discussed further in Appendix D.

4.1. Evolution of vortex surfaces

The DNS of the spin Euler equation (2.7) are carried out to investigate Lagrangian dynamics of ideal flows listed in table 1, and to validate the non-blowup criterion (2.11). To illustrate the Lagrangian vortex dynamics, figure 3 shows the top view of the isosurface of $s_1=0.5$ (i.e. vortex surface) for the trefoil knot at $t^*=0$, 0.9 and 1.8. Note that isosurfaces of $s_2$ and $s_3$ can show similar structures (Tao et al. Reference Tao, Ren, Tong and Xiong2021), and the isosurfaces of $|\boldsymbol {\omega }|$ (not shown) fail to capture the complete vortex tube as visualized by $s_1$ (as discussed in Xiong & Yang Reference Xiong and Yang2019; Shen et al. Reference Shen, Yao, Hussain and Yang2023).

Figure 3.

(a–c) Evolution of the isosurface of $s_1=0.5$ (vortex surfaces) colour-coded by $|\boldsymbol {\omega }|$ for the trefoil knot at $t^*=0$, 0.9 and 1.8 in the top view. Some vortex lines (cyan) are integrated and plotted on the isosurfaces. (d) The vortex surface for $s_1=0.5$, the region enclosed by this vortex surface, and the three cyan vortex lines are mapped to the red circle, the purple spherical cap within the red circle, and the three cyan points on the Bloch sphere $\mathbb {S}^2$, respectively.

Near the three crossings of the initial vortex knot, adjacent parts of the vortex tube are nearly orthogonal. Driven by the self-induced velocity with the Biot–Savart law, the vortex tube and vortex lines are stretched and twisted. The adjacent parts of the vortex knot approach each other, and they are progressively flattened and rolled up, instead of undergoing the vortex reconnection in viscous flows (Yao & Hussain Reference Yao and Hussain2022). The regions with large vorticity magnitude $|\boldsymbol {\omega }|$ are rapidly stretched into spiral sheets with strong twist.

In figure 3, the evolving vortex surfaces and lines preserve their initial mapping to the red circle and cyan points on $\mathbb {S}^2$, respectively, due to the Lagrangian nature of the spin Euler equation. Namely, the vortex topology is preserved in ideal flows. In addition, figure 4 shows the top view of the isosurface of $s_1=0.5$ for the Hopf link at $t^*=0$, 0.88 and 2.19. The structural evolution is similar to that of the trefoil knot.

Figure 4.

Evolution of the isosurface of $s_1=0.5$ (vortex surface) colour-coded by $|\boldsymbol {\omega }|$ for the Hopf link at $t^*=0$, 0.88 and 2.19 in the top view. Some vortex lines (cyan) are integrated and plotted on the isosurfaces.

Figure 5 plots the contour of $|\boldsymbol {\omega }|$ on the $x$–$y$ plane at $z=2.55$, and on the $y$–$z$ plane at $x=3.24$ for the trefoil knot, along with the contour lines of $s_1$. These planes intersect the point with the largest $|\boldsymbol {\omega }|$, so their contours show the most intense swirling motion. In figure 5(b), ‘vorticity pancakes’ (Brachet et al. Reference Brachet, Meneguzzi, Vincent, Politano and Sulem1992) form in the regions of large $|\boldsymbol {\omega }|$ among highly stretched and curved vortex surfaces. These structures appear when the vortex surfaces approach each other and undergo strong deformation. The formation of the high-vorticity region within sheet-like structures was observed in the collapse of vortex pairs (e.g. Pumir & Siggia Reference Pumir and Siggia1990; Kerr Reference Kerr1993) and Taylor–Green and Kida–Pelz flows (Yang & Pullin Reference Yang and Pullin2010). Furthermore, we observe the energy spectra with the $k^{-3}$ scaling (not shown) in the evolution of the trefoil knot and Hopf link, consistent with the result for the collision of two Lamb dipoles in Orlandi et al. (Reference Orlandi, Pirozzoli and Carnevale2012).

Figure 5.

Contour of the vorticity magnitude and contour lines (white) of $s_1$ for the trefoil knot on (a) the $x$–$y$ plane at $z=2.55$, and (b) the $y$–$z$ plane at $x=3.24$, at $t^*=1.8$. The planes intersecting the point with the largest $|\boldsymbol {\omega }|$ contain the most intense swirling motion in the flow.

In the highly symmetric MTG flow, a finite-time singularity may not occur according to the theoretical analysis (Constantin et al. Reference Constantin, Fefferman and Majda1996). Figure 6 plots the evolution of the isosurfaces of $s_1=0.8$ (red) and $s_1=-0.8$ (blue) for the MTG flow. A pair of vortex blobs are compressed and flattened into pancakes. Since the vortex surface is compressed in a quasi-2-D configuration, preserving the smoothness $\boldsymbol {\nabla }(\boldsymbol {\omega }/|\boldsymbol {\omega }|)$ of the vorticity direction (Constantin et al. Reference Constantin, Fefferman and Majda1996), the MTG flow does not exhibit a finite-time singularity, even though the vorticity grows rapidly (Brachet et al. Reference Brachet, Meneguzzi, Vincent, Politano and Sulem1992).

Figure 6.

Evolution of the isosurfaces of $s_1=0.8$ (red) and $s_1=-0.8$ (blue) in the MTG flow: (a) $t^{*}=0$, (b) $t^{*}=2$, (c) $t^{*}=4$.

The comparison of the two types of ideal flows implies that the Euler equation cannot form a singularity in a 2-D process. Constantin et al. (Reference Constantin, Fefferman and Majda1996) proved that if $\boldsymbol {u}$ remains uniformly bounded and $\boldsymbol {\omega }/|\boldsymbol {\omega }|$ stays $C^1$, then no singularity can occur. In other words, the vorticity must change its direction very rapidly to form a potential singularity. Note that if a singularity occurs at a vorticity null for all $t^*< t_b^*$ (e.g. Elgindi Reference Elgindi2021), then the vorticity direction becomes discontinuous at the time of singularity. The MTG flow vortex lines near the vorticity nulls maintain a quasi-2-D smooth shape, which contradicts the necessary blowup conditions. Hence the MTG flow does not exhibit finite-time singularities. By contrast, the trefoil knot and Hopf link have large vortex-line curvature in figures 3(c) and 4(c), with a rapid change in the vorticity direction.

4.2. Assessment of the non-blowup criterion

We apply (2.11) to the three ideal flows to test the non-blowup criterion based on the spin Euler equation, and compare our criterion to the BKM criterion (Beale et al. Reference Beale, Kato and Majda1984) by examining growth rates of the maximum vorticity and Laplacian spin vector. Before $t^* = T_\mathcal {R}$, $\| \boldsymbol {\omega } \|_{\infty }$ increases by a factor of approximately 16 for the trefoil knot and the Hopf link in figure 7. Both $\| \boldsymbol {\omega } \|_{\infty }$ and $\| \Delta \boldsymbol {s} \|_{\infty }$ exhibit the nearly double-exponential growth for the trefoil knot and Hopf link. The double-exponential growth of $\| \boldsymbol {\omega } \|_{\infty }$ is consistent with the results in Hou & Li (Reference Hou and Li2007) and Kerr (Reference Kerr2013). As the number of grid points increases (up to $1536^3$), the growth rate of $\| \Delta \boldsymbol {s} \|_{\infty }$ appears to remain constant for the trefoil knot and Hopf link.

Figure 7.

Evolution of (a–c) the maximum vorticity and (d–f) the maximum Laplacian spin vector, for (a,d) the trefoil knot, (b,e) the Hopf link and (c,f) MTG flow, with different grid resolutions, respectively. Note that we plot $\ln \ln (50\| \boldsymbol {\omega } \|_{\infty })$ for the MTG flow, where the factor 50 is used to avoid complex values of the logarithm. The vertical dashed lines with different colours mark the time with $\mathcal {R}=2$ for each resolution. The simulations are well-resolved on the left of the dashed lines (shaded in corresponding colours).

The present criterion has some advantages that $\ln \ln \| \Delta \boldsymbol {s} \|_{\infty }$ grows more slowly than $\ln \ln \| \boldsymbol {\omega } \|_{\infty }$ (4–6 times slower), and exhibits better convergence with the mesh resolution. Therefore, $\Delta \boldsymbol {s}$ appears to be resolved more easily than $\boldsymbol {\omega }$ with the same numerical accuracy. Moreover, the duration of the linear stage for $\ln \ln \| \Delta \boldsymbol {s} \|_\infty$ exceeds that for $\ln \ln \| \boldsymbol {\omega } \|_\infty$ more than five-fold.

The highly symmetric MTG flow shows no evidence of a finite-time singularity. The profile of $\ln \ln \| \Delta \boldsymbol {s} \|_\infty$ in figure 7(f) clearly bends downwards before $t^* = T_\mathcal {R}$ with $N^3=1024^3$. The growth rate of $\| \Delta \boldsymbol {s} \|_\infty$ is weaker than double-exponential, whereas the growth of $\| \boldsymbol {\omega } \|_\infty$ remains double-exponential in figure 7(c) when $t^*< T_{\mathcal {R}}$. Therefore, the criterion based on $\Delta \boldsymbol {s}$ can effectively identify the flows that are unlikely to develop a finite-time singularity.

Assuming that the double-exponential growth is bounded in a finite time $t$, the time integral of $\| \Delta \boldsymbol {s} \|_\infty ^{p+1}$ becomes

(4.1)\begin{equation} \int_0^t\| \Delta \boldsymbol{s}(\,\cdot\,,\tau) \|_{\infty}^{p+1}\,{\rm d}\tau \le \frac{(\mathrm{e}^{c_1t}-1)\exp[(p+1)\,\mathrm{e}^{c_1t+c_2}]}{c_1\mathrm{e}^{c_1t}}, \end{equation}

with constants $c_1$ and $c_2$. According to the non-blowup condition (2.11), the Euler equation can avoid singularity formation in finite time for the double-exponential growth of $\| \Delta \boldsymbol {s} \|_\infty$.

4.3. Difference of non-blowup criteria

We highlight the major difference between (2.11) and the BKM criterion, and explain why $\| \Delta \boldsymbol {s} \|_{\infty }$ grows more slowly than $\| \boldsymbol {\omega } \|_{\infty }$. Figure 8 plots the trajectories of $\arg \max |\boldsymbol {\omega }|$ and $\arg \max |\Delta \boldsymbol {s}|$, colour-coded by $t^*$, and their projections on the $x$–$y$ plane for the trefoil knot and Hopf link. The trajectories of $\arg \max |\boldsymbol {\omega }|$ and $\arg \max |\Delta \boldsymbol {s}|$ starting from the same locations do not collapse, implying that the present criterion is distinct from the BKM criterion.

Figure 8.

Three groups of trajectories of the maximum points of $|\boldsymbol {\omega }|$ (from blue to green) and $|\Delta \boldsymbol {s}|$ (from red to yellow) during $t^*\in [0,2.2]$ for (a) the trefoil knot and (b) the Hopf link, along with their projections on the $x$–$y$ plane (shaded in grey). The two trajectories of maximum $|\boldsymbol {\omega }|$ and $|\Delta \boldsymbol {s}|$ in each group start at the same location.

The continuous trajectory of $\arg \max |\Delta \boldsymbol {s}|$ is more tractable than the discontinuous one of $\arg \max |\boldsymbol {\omega }|$ in figure 8. Figure 9 illustrates the $z$-coordinates of the maximum $|\boldsymbol {\omega }|$ and $|\Delta \boldsymbol {s}|$, the Lagrangian trajectory of particles that locate at the position of the maximum values at $t^*=0$, and the evolution of the centroid positions $z_c = \int _{s_1\ge 0}z\,{\rm d}V /\int _{s_1\ge 0}\,{\rm d}V$ of the trefoil knot and Hopf link. We find that $\arg \max |\Delta \boldsymbol {s}|$ remains continuous over time and moves at a constant speed in the $z$-direction in both flows, which is close to the Lagrangian velocity of the particle at the location of $\arg \max |\Delta \boldsymbol {s}|$ (or $\arg \max |\boldsymbol {\omega }|$) at $t^*=0$. This implies that the maximum $|\Delta \boldsymbol {s}|$ could have some Lagrangian nature.

Figure 9.

Evolution of the $z$-direction coordinates of the maximum points of $|\boldsymbol {\omega }|$ and $|\Delta \boldsymbol {s}|$, the Lagrangian tracing particle located at the maximum point at $t^*=0$, and the centroid position for (a) the trefoil knot and (b) the Hopf link.

By contrast, $\arg \max |\boldsymbol {\omega }|$ exhibits a sharp jump at $t^*=1.26$ for the trefoil knot and $t^*=1.98$ for the Hopf link. The speed of $\arg \max |\boldsymbol {\omega }|$ in the $z$-direction is close to that of the centroid of the vortex at early times. During the growth of $|\boldsymbol {\omega }|$, the locus of the peak vorticity exhibits a notable displacement, converging to $\arg \max |\Delta \boldsymbol {s}|$ (marked in a dashed box in figure 9).

We examine the correlation and distribution of the values of $|\Delta \boldsymbol {s}|$ and $|\boldsymbol {\omega }|$, normalized by their respective maxima, for the trefoil knot at $t^*=1.8$ and the Hopf link at $t^*=2.19$. The scatter plots in figure 10 show a low positive correlation between $|\Delta \boldsymbol {s}|/\| \Delta \boldsymbol {s} \|_{\infty }$ and $|\boldsymbol {\omega }|/\| \boldsymbol {\omega } \|_{\infty }$, with correlation coefficients $\rho =0.675$ for the trefoil knot, and $\rho =0.627$ for the Hopf link. Therefore, the criterion based on $\Delta \boldsymbol {s}$ in (2.11) has a notable statistical difference from that based on $\boldsymbol {\omega }$.

Figure 10.

Scatter plots of $|\Delta \boldsymbol {s}|$ and $|\boldsymbol {\omega }|$ normalized with their maximum values for (a) the trefoil knot at $t^*=1.8$, and (b) the Hopf link at $t^*=2.19$, along with the correlation coefficients $\rho$ and linear fits (dashed lines) of data points. The scattered points are coloured from purple to red by the density of data points.

The probability density functions (p.d.f.s) of the normalized values of $|\Delta \boldsymbol {s}|$ and $|\boldsymbol {\omega }|$ for the trefoil knot at $t^*=1.8$ and the Hopf link at $t^*=2.19$ are shown in figure 11. For both configurations, the p.d.f. profiles of $|\Delta \boldsymbol {s}|/\| \Delta \boldsymbol {s} \|_{\infty }$ are smoother than those of $|\boldsymbol {\omega }|/\| \boldsymbol {\omega } \|_{\infty }$, and they obey a Pareto distribution (Arnold Reference Arnold2015) with the $-2$ power law except for very large values, indicating that the extreme values at a few locations can dominate norms of $\Delta \boldsymbol {s}$ and $\boldsymbol {\omega }$.

Figure 11.

P.d.f.s of normalized $|\Delta \boldsymbol {s}|$ and $|\boldsymbol {\omega }|$ for (a) the trefoil knot at $t^*=1.8$, and (b) the Hopf link at $t^*=2.19$.