2.1. Reconnection phase scaling $(\sqrt {\nu }Z)^{-1/2}=(\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}})^{-1}$

In this subsection, figure 3(a) uses this viscous rescaling of $Z(t)$ (Kerr Reference Kerr2018b ), $B_\nu (t)=1/(\sqrt {\nu }Z(t))^{1/2}=(\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}})^{-1}$ (1.15), while figure 3(b) introduces scaling of the dissipation rate $\epsilon =\nu Z$ (1.12) that is covered more throughly in figure 6 when enstrophy spectra are discussed.

Figure 6. Dissipation rate $\epsilon =\nu Z$ for seven cases, $\nu =5\times 10^{-4}$ to $4\times 10^{-6}$ , showing approximate convergence of the dissipation rates beginning at $t\sim 70$ for $\nu \leqslant 1.25\times ^{-5}$ , with peaks at $t_\epsilon \approx 93\approx 2t_x$ that continue for an extended period. These trends are consistent with the formation of a dissipation anomaly, that is, finite energy dissipation $\Delta E_\epsilon$ (1.2) in a finite time as $\nu$ decreases. Inset: $\epsilon$ rescaled as the dissipation coefficient $C_\epsilon$ (1.3) using $\mathcal{L}=2r_{\!f}=4$ and $\mathcal{U}=\|u\|_\infty (t_\epsilon =93)\approx 0.34$ for calculations whose maximum Taylor microscale Reynolds number at $t=93$ is $R_\lambda =218$ .

Since originally introduced by Kerr (Reference Kerr2018a , Reference Kerr2018b ), convergent, inverse linear $B_\nu (t)$ regimes have been found for the interaction of coiled vortex rings (Kerr Reference Kerr2018c ), three-fold symmetric trefoils (Kerr Reference Kerr2023) and additional configurations discussed in § 5.1.

The extensions in figure 3(a) come from linearising $B_\nu (t)$ about $\nu$ -independent $B_x=B_\nu (t_x)\approx 2.67$ for $t\lesssim t_x$ ( $30\leqslant t\leqslant 40$ ) and extending each linear $B_\nu (t)$ to $T_c(\nu )\gt t_x$ , when $t\gt t_x$ inverse linear ends with $B_\nu (T_c(\nu ))=0$ .

Specifically, for $t_{30}=30$ and $t_x=40$ , one gets the $\Delta t(\nu )$ and $T_{c1}(\nu )$ in figure 3(a,c) using

(2.2) \begin{equation} \Delta t(\nu )=B_x\frac {t_x-t_{30}}{B_\nu (t_{30})-B_x}\quad \text{and}\quad T_{c1}(\nu )=t_x+\Delta t(\nu ). \end{equation}

The purpose of identifying these $t\gt t_x$ linear extensions is to emphasise the inverse linearity of the pre- $t_x$ $B_\nu (t)$ , which suggests some type of self-similar process might be found. Figure 3(c) shows that $\Delta t(\nu )$ decreases significantly as $\nu$ decreases.

These observations suggest that an underlying scaling process exists with a role for the double square-root on the viscosity, that is, when scaled using $\nu ^{1/4}$ (1.15). The next section will follow this up by extending the $\nu ^{-1/4}\mathcal{O}_{\textit{V}\text{1}}^{-1}$ scaling to higher order $\mathcal{O}_{\textit{Vm}}$ and § 6 will begin by mentioning other configurations with this scaling.

2.2. Additional $\nu ^{1/4}$ scaling examples

Figure 4 shows inverse $\nu ^{1/4}$ scaling of $\varOmega _\infty (t)$ and $\mathcal{O}_{V9}(t)$ (1.1): $(\nu ^{1/4}\varOmega _\infty (t))^{-1}$ and $\bigl (\nu ^{1/4}\mathcal{O}_{V9}(t)\bigr )^{-1}$ , recalling that at $t=0$ , $\varOmega _\infty =\|\omega \|_\infty =1$ is imposed. For all $m$ , the $\nu \leqslant 6.25\times 10^{-5}$ cases have the best temporal convergence, with the larger $\nu$ cases gradually joining those convergences as $m$ decreases.

Figure 4(a) shows that the $\nu$ -independent temporal convergence of $(\nu ^{1/4}\varOmega _\infty (t))^{-1}$ is at $t_\infty =18.4$ and figure 4(b) shows this for $\bigl (\nu ^{1/4}\mathcal{O}_{V9}(t)\bigr )^{-1}$ at $t_9=19.1$ . For both, there are temporal convergences at $t_\infty$ and $t_9$ for $\nu =6.25\times 10^{-5}$ to $4\times 10^{-6}$ . For the dash-dotted $\nu \lt 1.56\times 10^{-5}$ cases, the resolution is adequate only up to $t\sim 19$ , but inadequate after that for the highest-order moments. The lower Reynolds number $\nu =1.25\times 10^{-4}$ curves at $t_\infty$ and $t_9$ have smaller $(\nu ^{1/4}\varOmega _m(t))^{-1}$ , larger $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}(t)$ , consistent with their temporal convergences tending to be slightly later, as seen for $m=4$ in the next figure.

The convergences of these large $m$ , $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}$ , provide the earliest quantitative evidence that the size of the domain is affecting vorticity growth. That is, a sign that the local growth of $\varOmega _\infty =\|\omega \|_\infty$ (and $\mathcal{O}_{V9}$ ) is being affected by interactions across the periodic boundaries, interactions that can be suppressed if the periodic domains are increased as $\ell \sim \nu ^{-1/4}$ as $\nu$ decreases. The $t\lesssim t_m$ evolution of all the $\bigl (\nu ^{1/4}\mathcal{O}_{\textit{Vm}}(t)\bigr )^{-1}$ are approximately inverse-linear, similar to how $(\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}}(t))^{-1}$ behaves in figure 3. Note that both $t_\infty$ and $t_9$ are at $\approx t_x/2$ .

Can this early time evolution of $\nu ^{1/4}\varOmega _\infty (t)$ and $\nu ^{1/4}\mathcal{O}_{V9}(t)$ tell us anything about the origins of the $\nu ^{1/4}$ scaling? For that, a physical model will be needed that takes into account the origin, extent and scaling of vortex sheets like those in figures 11 and 12. However, first, let us consider the evolution of the intermediate vorticity moments, $\mathcal{O}_{\textit{Vm}}(t)$ .

For all orders $m$ , there is convergence of almost all the $\nu ^{-1/4}\mathcal{O}_{\textit{Vm}}^{-1}$ at their respective $t_m$ , ordered as $t_\infty \lt \ldots \lt t_m\ldots t_1=t_x$ , with $\nu$ -independent $\sqrt {\nu }Z(t_x)$ above all others. Figure 5 shows two of these intermediate scaled, inverse moments. Here, $\nu ^{-1/4}\mathcal{O}_{V4}^{-1}$ and $\nu ^{-1/4}\mathcal{O}_{V2}^{-1}$ . For $\nu ^{-1/4}\mathcal{O}_{V4}^{-1}$ from $\nu =6.25\times 10^{-5}$ to $\nu =1.56\times 10^{-5}$ , there is convergence at $t_4=22$ and for $\nu ^{-1/4}\mathcal{O}_{V2}^{-1}$ for the same $\nu$ at $t_2=29.3$ .

One way to understand the relationship between $\nu ^{1/4}\varOmega _\infty$ and the $m\lt \infty$ volume-integrated moments $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}(t)$ , including the $(\sqrt {\nu }Z)^{1/2}=(\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}})^2$ , is to realise that $\|\omega \|_\infty =\varOmega _\infty$ is point-like. Because the $t_\infty =18.4$ convergence comes first, as $\omega$ near $\varOmega _\infty (t)$ continues to grow, the region with $\omega \sim \|\omega \|_\infty$ begins to spread out, resulting in increases of the other $\mathcal{O}_{\textit{Vm}}$ , until one gets vortex sheets like those in figure 11.

To determine whether the resolution is adequate, one common diagnostic is to follow and compare values of $\|\omega \|_\infty =\varOmega _\infty$ , with figure 4 showing both $\nu ^{1/4}\varOmega _\infty (t)$ and $\nu ^{1/4}\mathcal{O}_{V9}(t)$ converging at $t_\infty =18.9$ and $t_9=19.1$ , respectively, for all $\nu$ . Furthermore, for $\nu \gtrsim 2.2\times 10^{-5}$ and $t\geqslant 20$ , as $\nu$ decreases, the inverse $\|\omega (t)\|_\infty$ and $\mathcal{O}_{V9}(t)$ continue to decrease as expected.

However, for $t\gt 20$ , the smaller $\nu$ , $\nu \lt 2.2\times 10^{-5}$ , the highest-order $\varOmega _m$ do not decrease as would be expected, indicating that they are no longer fully resolved and are not shown, which leads to the conclusion that the only calculations that are adequately resolved for all times are those with $\nu \geqslant 2.21\times 10^{-5}$ , which is 1/25 of $\max (\nu )=5\times 10^{-4}$ .

3.1. Enstrophy spectra and production

One set of tools that can demonstrate how the energy and enstrophy go from large to small scales is their spectra, complemented by their spectral transfer and production spectra.

Energy spectra $E_V(k,t)$ are formed by taking Fourier transforms of the velocity (1.17) and enstrophy spectra $Z_V(k,t)$ are formed with the vorticity $\boldsymbol{\omega }$ replacing the velocity $\boldsymbol{u}$ in the Fourier transforms. Using (1.20), these are

(3.1) \begin{equation} E_V(k,t)=\frac {1}{2}\sum _{\boldsymbol{k}}|\hat {\boldsymbol{u}}_{\boldsymbol{k}}(t)|^2\ \quad \text{and}\ \quad Z_V(k,t)=\sum _{\boldsymbol{k}}|\hat {\boldsymbol{\omega }}_{\boldsymbol{k}}(t)|^2. \end{equation}

The types of sums taken depend upon the definition of $k$ . If $k$ is in one of the three wavenumber directions $k_x$ , $k_y$ and $k_z$ , the sums are taken over the two perpendicular directions. In this paper, all spectra use three-dimensional wavenumbers

(3.2) \begin{equation} k=|\boldsymbol{k}|=\sqrt {k_x^2+k_y^2+k_z^2}\,, \end{equation}

with $E_V(k,t)$ and $Z_V(k,t)$ collected in wavenumber shells $\mathbb{Z}/\ell \leqslant k\lt \mathbb{Z}+1$ whose element-numbers for small $k$ are discontinuous, which can lead to ambiguities (Kerr Reference Kerr1990). The wavenumbers of the shells are the mean-square averages of the wavenumbers within them. For the variable $(2\ell \pi )^3$ domains in this paper, there are minimum wavenumbers $k_{{\textit{min}}}$ such that all $\boldsymbol{k}$ obey

(3.3) \begin{equation} |\boldsymbol{k}|\geqslant k_{{\textit{min}}}=\min _{k\neq 0}(k)=1/\ell =1/3. \end{equation}

The spectral equation for $E_V(k,t)$ is

(3.4) \begin{equation} \frac {\rm d}{{\rm d}t} E_V(k,t)=T(k)-\nu Z_V(k,t), \end{equation}

with the integral of the energy-transfer identically zero: $\int T(k)\,{\rm d}k\equiv 0$ . The spectral equation for the enstrophy is similar, with non-zero enstrophy production $\zeta _p$ replacing the energy transfer,

(3.5) \begin{equation} \frac {\rm d}{{\rm d}t} Z_V(k,t)=\zeta _p(k)-2\nu P_V(k,t), \end{equation}

with $\zeta _p(k)$ being the Fourier transform of the physical space enstrophy density production $\zeta _p(\boldsymbol{x},t)=2\boldsymbol{\omega }\boldsymbol{S}\boldsymbol{\omega }$ (1.13) and $P_V(k,t)$ being the palinstrophy, which at individual wavenumbers $\boldsymbol{k}$ is $P_V(\boldsymbol{k},t)=\boldsymbol{k}^2 Z_V(\boldsymbol{k},t)=2\boldsymbol{k}^4 E_V(\boldsymbol{k},t)$ with $P_V(k,t)\approx 2k^4 E_V(k,t)$ . The $\zeta _p(\boldsymbol{k})$ are determined numerically at run-time by Fourier transforming the vortex stretching term $\boldsymbol{S}\boldsymbol{\omega }$ and the enstrophy $\boldsymbol{\omega }$ simultaneously, then multiplying $\widetilde {\{\boldsymbol{S}\boldsymbol{\cdot }\boldsymbol{\omega }\}}(\boldsymbol{k})$ and $\boldsymbol{\omega }(\boldsymbol{k})$ in Fourier space, with $\widetilde {\{\ \}}$ indicating a Fourier transform. This gives the following enstrophy production spectra and global enstrophy production in the enstrophy equation (1.13):

(3.6) \begin{align} \zeta _p(k,t)&= 2\boldsymbol{\omega }(\boldsymbol{k})\boldsymbol{\cdot }\widetilde {\{\boldsymbol{S}\boldsymbol{\cdot }\boldsymbol{\omega }\}}(\boldsymbol{k}) \approx 2k^2 T(k) \nonumber\\ \text{and} \quad Z_p(t)&= \int {\rm d}k\,\zeta _p(k,t) =\int _{\mathcal{V}_\ell } 2\boldsymbol{\omega }\boldsymbol{S}\boldsymbol{\omega }\,{\rm d}V. \end{align}

The evolution of the dissipation rate $\epsilon (t)=\nu Z_p(t)$ and its nonlinear production $\nu Z(t)$ are given in figure 10 after the $Z_V(k,t)$ enstrophy spectra. The $T(k,t)$ and $\zeta _p(k,t)$ spectra are mentioned, but a full discussion will require another paper.

One numerical paradigm for representing turbulent states is homogeneous, isotropic turbulence generated by forcing the flow at small wavenumbers (large length scales). With statistically steady data from calculations run in large enough domains, these can generate $k^{-5/3}$ Kolmogorov spectra, starting with Kerr (Reference Kerr1985) and since then, much higher Reynolds numbers (Ishihara, Gotoh & Kaneda Reference Ishihara, Gotoh and Kaneda2009). One advantage that these statistically steady data sets have is that time averages can be taken that smooth out transients.

Taking steady-state averages is less of an option for the transient flows generated from configurations of reconnecting vortices. Without statistically steady data that allow averaging over long times to get wavenumber scaling, transient energy spectra $E_V(k,t)$ and enstrophy spectra $Z_V(k,t)$ (3.1) can be followed through the wavenumber scales to indicate how the energy and enstrophy move, grow and decay through the wavenumbers and how energy dissipation rates, $\epsilon = \nu Z$ , evolve through scales to generate finite-time dissipation. If a flow is believed to generate finite energy dissipation, or other signatures of turbulence, some transient $E_V(k,t)\sim k^{-5/3}$ spectra might develop, which for the enstrophy would be $Z_V(k,t)\sim k^{1/3}$ .

The only Kolmogorov-like $E_V(k)\sim k^{-5/3}$ spectra from reconnection simulations are from anti-parallel cases. With a Gaussian profile, Yao & Hussain (Reference Yao and Hussain2020) obtained $k^{-5/3}$ for a short time-span (see also Yao & Hussain Reference Yao and Hussain2022). Additionally, with an algebraic profile, Kerr (Reference Kerr2013a ) obtained $k_y^{-5/3}$ over a sustained time-span when normalised by time-varying $\epsilon ^{2/3}$ as the the energy $E(t)$ decreases by 50 %. These are mentioned again at the end of § 5.2.

Only one Gaussian profile trefoil calculation reports spectra, with Yao et al. (Reference Yao, Yang and Hussain2021) finding a $E_V(k)\sim k^{-7/3}$ regime that, like the Yao & Hussain (Reference Yao and Hussain2020) inertial range, begins as high wavenumbers with very little associated energy dissipation.

3.2. Algebraic trefoil spectra

Could a trefoil vortex knot with an algebraic initial profile overcome the limitations of Yao et al. (Reference Yao, Yang and Hussain2021) without taking ensemble averages? That is, can it generate finite energy dissipation and with Kolmogorov-like spectral scaling $?$

To facilitate comparisons over several viscosities, only an $\ell =3$ , $(6\pi )^3=(2\ell \pi )^3$ domain is used for plotting three-dimensional enstrophy spectra $Z_V(k)$ . The enstrophy spectra are used because under Kolmogorov scaling, the $Z_V(k)$ scale as $k^{1/3}$ ( $=k^2k^{-5/3}$ ), which can be easily recognised, unlike $E_V(k)\sim k^{-5/3}$ scaling, which requires plotting $k^{5/3}E_V(k)$ (Kerr Reference Kerr2013a ).

Figure 7. Enstrophy spectrum $Z_V(k,t)$ (3.1) for seven times from the $\nu =3.125\times 10^{-5}$ perturbed trefoil case run in a $(6\pi )^3$ domain. Times are $t=6$ , 18, 36, 48, 72, 96 and 120, plus three power laws, $k^{-5/3+2}=k^{1/3}$ , $k^{-3+2}=k^{-1}$ and $k^{-4+2}=k^{-2}$ . Due to the logarithmic $k$ -scale, $k\lt 2$ values are over-emphasised compared with $k\gt 2$ values. The $Z_V(k,t)\lt 2$ spectra generally decrease with time as small-wavenumber energy begins to transfer to higher wavenumbers, with the exception of $k=2$ at $t=48$ as noted for figure 9(b). The $k\geqslant 6$ spectra gradually grow over time until all approximately obey $k^{-1}$ for all $10\leqslant k\leqslant 30$ .

Figure 8. Comparisons of enstrophy spectra $Z_V(k,t)$ (3.1) for two intermediate resolved Reynolds number ( $Re$ ) calculations at $t=0$ and intermediate times ( $t=36$ to 92 or 93) for the wavenumber span $2\lt k\lt 10$ to show the first step in how the enstrophy’s form of Lundgren scaling, $Z_V(k)\sim k^{1/3}$ , develops. For $\nu =3.125\times 10^{-5}$ , panel (a) shows that at $t=48$ for $k\sim 3$ , the $Z_V(k,t)$ spectrum briefly overshoots $k^{1/3}$ , before becoming $Z_V(k,t)\sim k^{1/3}$ for $t=72$ to 80. For $\nu =6.25\times 10^{-5}$ , panel (b) shows that for smaller $Re$ , the spectrum also overshoots $k^{1/3}$ , but a clear $Z_V(k,t)\sim k^{1/3}$ regime does not form afterwards.

Figure 9. Enstrophy spectra $Z_V(k,t)$ (3.1) for early, intermediate and late times ( $t=24$ to 111) for the highest resolved Reynolds number ( $Re$ ) calculation ( $\nu =2.2\times 10^{-5}$ ) run on a $2048^3$ mesh. (a–c) Wavenumber span $2\lt k\lt 6$ with $t=48$ shown in both panels (a) and (b). For $t\leqslant 36$ in panel (a), the spectra briefly overshoot $k^{1/3}$ . Panel (b) provides the best evidence for persistant, approximately $Z(k,t)\sim k^{1/3}$ for $2.67\leqslant k\leqslant 4.33$ over a finite time span of $t=66$ to 75, and less to $t=78$ . Recall that $t\sim 70$ is roughly the time when convergence of the dissipation rates $\epsilon (t)$ begins in figure 6. Note that the leading high-wavenumber of that $k^{1/3}$ span increases slightly with time to $t=78$ . Panels (c) and (d) show the same times with the $k^{1/3}$ span continuing to retreat for $t\gt 78$ , then decaying for $t\gt 93$ . Panel (d) shows both very high and very low wavenumbers. At high wavenumbers, there is a continuation of the $k\geqslant 5$ formation of a $k^{-1}$ enstrophy spectral regime from panel (c), with the Kolmogorov-dissipation wavenumber $k_\eta =183$ noted. The $k^{-1}$ decay persists until $k\sim 50\sim k_\eta /4$ , with approximately exponential decay for $k\gt 50$ , indicated by the rough red-dashed fit. The inset of panel (d) shows very low wavenumbers of $k\leqslant 2$ with those $Z_V(k,t)$ decreasing with $k$ , resuming the trend shown in figure 7.

Figure 10. Time evolution of the dissipation rate $\epsilon (t)=\nu Z(t)$ and its scaled production $C_{p}\nu Z_p(t)$ , where $Z_p=\int \zeta _p\, {\rm d}V$ and the $C_{p}$ are chosen to allow comparisons. For these two resolved Reynolds numbers, $Z(t)$ and $C_{p}\nu Z_p(t)$ follow one another for $t_x\leqslant t\lesssim t_\epsilon$ .

To identify the significant temporal regimes, several plausible comparison power laws, $k^{1/3}$ , $k^{-1}$ and $k^{-2}$ , are shown. Comparisons to these power laws are meant less for determining the particular subranges and more for showing what the spectra are not.

If $Z_V(k)\sim k^{1/3}$ is found, further questions might include whether $Z_V(k)$ overshoots $k^{1/3}$ , whether there are brief $Z_V(k,t)$ , sub- $k^{1/3}$ periods, how long $k^{1/3}$ scaling is maintained and then how $Z_V(k)$ dissipates. The extent of any brief Lundgren-like inertial subranges is given in terms of both $k$ and $k_{{\textit{min}}}$ (3.3). Figure 7 shows the progression over $t=0$ to $t= 120$ of $Z_V(k,t)$ for $2/3\leqslant k\leqslant 33$ from the $\nu =3.125\times 10^{-5}$ calculation, with the following points.

1. (i) The spectra for $k\lt 2$ generally decrease with time.

2. (ii) At higher $k\geqslant 6=18k_{{\textit{min}}}$ , there is a steady progression from the very steep $t=0$ spectrum to $Z_V(k)\sim k^{-1}$ spectra (like $E_V(k)\sim k^{-3}$ ) that are discussed further using figure 9(d).

3. (iii) The $t=36$ and $t=48$ spectra overshoot the $k^{1/3}$ scaling for $k\gt 2$ .

4. (iv) $t=48$ is marked using a dash-dotted line because this represents the transition between pre- and post-reconnection scaling, after which $Z_V(k,t)$ spreads out in $k$ until $t=72$ .

5. (v) $k^{1/3}$ scaling goes through both the $t=72$ and $t=80$ curves for $3\leqslant k\leqslant 4$ ( $9\leqslant k/k_{{\textit{min}}}\leqslant 12$ ) in both figures 7 and 8(a), with the full $2\leqslant k \leqslant 10$ behaviour in figure 8(a).

Figure 8 shows the progression of $Z_V(k,t)$ for two viscosities from $t=0$ to $t=92$ or 93 for $\nu =3.125\times 10^{-5}$ and $\nu =6.25\times 10^{-5}$ , respectively. For both viscosities, for $2\lt k\lt 4$ , there is growth of $Z_V(k,t)$ for $t\lt 48$ ; for $k\gt 6$ over all times, as $t$ increases, $Z_V(k,t)\sim k^{-1}$ regimes are forming. At intermediate $k\sim 2.5{-}3.5$ , at both $t=36$ and 48, there is overshooting of any developing $k^{1/3}$ scaling.

There are differences in whether $Z_V\sim k^{1/3}$ regimes are forming for the two Reynolds numbers in figure 8. For panel (a), $\nu =3.125\times 10^{-5}$ and $t=72$ , there is a $Z_V\sim k^{1/3}$ regime for $2\leqslant k\leqslant 4$ , which at $t=80$ , moves to $3\leqslant k \leqslant 4.5$ ( $9\leqslant k/k_{{\textit{min}}} \leqslant 13.5$ ).

The lower Reynolds number in figure 8(b) with $\nu =6.25\times 10^{-5}$ shows the $k\geqslant 2$ spectra increasing until $t=48$ , including a growing high-wavenumber $Z_V(k,t)\sim k^{-1}$ regime. However, the $k\gtrsim 3$ spectra never develop a convincing $Z_V\sim k^{1/3}$ and instead tend to decay, but not uniformly.

These observations raise the following questions.

1. (i) How does the approximately $Z_V(k)\sim k^{1/3}$ spectra scaling develop $?$

2. (ii) Does this spectrum persist long enough to attain finite energy dissipation $\Delta E_\epsilon$ (1.2) $?$

To help answer those questions, figures 9 and 10 show enstrophy and enstrophy production spectra from the highest Reynolds number $\nu =2.2\times 10^{-5}$ resolved simulation. Figure 9 uses four time frames to show how a transient $Z_V(k,t)\sim k^{1/3}$ regime forms and how long it persists.

1. (i) Figure 9(a–c) all cover $2\leqslant k\leqslant 6$ ( $6\leqslant k/k_{{\textit{min}}}\leqslant 18$ ) as follows.

2. (ii) Figure 9(a) highlights the overshoot phase that ends between $t=36$ and 48.

3. (iii) Figure 9(b) shows a settling phase with the best evidence for Lundgren-like $Z_V(k)\sim k^{1/3}$ scaling over a brief time span of $t=66$ to $t=78$ as the enstrophy spectrum settles over $2.67\leqslant k\leqslant 4.33$ , ( $8\leqslant k/k_{{\textit{min}}}\leqslant 13)$ , with the peak of $Z_V(k\gt 3)$ moving to larger $k$ as $t$ increases. This span of times and wavenumbers provides the best evidence for how energy and enstrophy are moving to large $k$ coherently.

4. (iv) Figure 9(c) covers times $t\sim 75{-}111$ within the temporal span of figure 6 with temporal convergence of the energy dissipation rates $\epsilon (t)$ (1.12) for several viscosities, with the maxima of the $\epsilon (t)$ at $t_\epsilon \approx 93$ .

5. (v) Over this time span, the higher wavenumbers gradually decay from $k^{1/3}$ to $k^{-1}$ due to high wavenumber dissipation.

6. (vi) Figure 9(d) shows both (1) very low $k\leqslant 2$ in the inset and (2) the highest $k$ for the later times.

7. (vii) For small $k$ , the $t=75$ to 111 evolution is similar to the $t=72$ to 120 evolution for $\nu =3.125\times 10^{-5}$ in figure 7, with the very largest scales (smallest $k$ ) still influenced by the original, now decaying, trefoil structure.

8. (viii) For $k\gt 10$ , $Z_V\sim k^{-1}$ from the $k\sim 6$ span continues to dominate for $k\leqslant 50$ , followed by approximately exponential $k$ -spectra for $k\gt 50$ . Is this exponential regime consistent with the type of exponential regimes reported for forced turbulence $?$ To determine that requires that a characteristic high, dissipation range wavenumber be defined. Using $\epsilon =\nu Z$ , the enstrophy $Z$ is modified within the formula for the effective Kolmogorov wavenumber, $k_\eta =(\epsilon /\nu ^3)^{1/4}$ , is as follows.

9. (ix) If turbulence fills the entire domain, using the volume-integrated enstrophy $Z$ (1.13) would be the appropriate choice for determining $\epsilon$ and $k_\eta$ . However, that choice is inappropriate for these calculations because significant vorticity occupies only a fraction of the compuational domain.

10. (x) Since most of the enstrophy remains within the $t=0$ inner $(2\pi )^3$ domain, that inner domain has been used to determine $k_\eta$ , with the relevant volume-averaged enstrophy, dissipation rate and Kolmogorov-dissipation wavenumber being

    (3.7) \begin{equation} Z_{\varOmega 2\pi }\sim Z/(2\pi )^3\,,\quad \epsilon _{\varOmega 2\pi }=\nu Z_{\varOmega 2\pi }\quad \text{and}\quad k_\eta =(\epsilon _{\varOmega 2\pi }/\nu ^3)^{0.25}. \end{equation}
    Based upon this, $k_\eta =183$ and is marked within the exponential regime in figure 9(d). Additionally, based on having an exponential wavenumber regime after the power-law regimes, these simulations are resolved.

11. (xi) Is this $k_\eta$ -based exponential regime consistent with what has been observed previously for numerical turbulence $?$ For those comparisons, $k_\epsilon$ is taken to be where the $Z_V(k,t)\sim k^{1/3}$ regime ends. For both forced turbulence (Kerr Reference Kerr1990; Ishihara et al. Reference Ishihara, Gotoh and Kaneda2009) and for the algebraic, anti-parallel case (Kerr Reference Kerr2013b ), $k_\epsilon \sim k_\eta /6$ . (Note: $k_\eta$ was not given for the Gausian profile anti-parallel case (Yao & Hussain Reference Yao and Hussain2020).)

12. (xii) For this case, $k_\epsilon \sim 4{-}5\approx 0.02 k_\eta$ , which is inconsistent with those cases. However, where the $Z_V(k,t)\sim k^{-1}$ regime in figure 9(d) ends is at $k\sim k_\eta /6$ .

13. (xiii) Is there another diagnostic that could support how $Z_V(k,t)$ moves to large $k$ in figure 9(b) $?$ Two options are the energy transfer spectra (3.4) and the enstrophy production spectra $\zeta (k,t)$ (3.6). The drawback to both is that there are compensating negative and positive values, making it impossible to get useful averages as a pulse moves to large wavenumbers. Introducing time-lags might be applied later (Kerr Reference Kerr1990).

As an alternative, figure 10 follows the global time evolution of the dissipation rate $\epsilon =\nu Z(t)$ and its scaled production $C_{p}\nu Z_p(t)=C_{p}\nu \int \zeta _p \,{\rm d}V$ , ((1.13), (3.6)). Note that $5\nu Z_p(t)$ and $4\nu Z_p(t)$ follow their respective $\epsilon =\nu Z(t)$ for $t_x=40\leqslant t\lesssim 70 \lt t_\epsilon$ , suggesting (if dissipation is ignored) that over this period, there is an effective stretching that is maintained within the coiled structures. This would allow for the accelerated, exponential growth of $Z(t)$ and $\epsilon (t)$ .

5.1. Supporting numerical analysis and mathematical scaling for $(2\ell \pi )^3$ domains

The second aspect is providing new quantitative numerical analysis and mathematical scaling to explain why the computational domains need to be increased. Because they are compact, trefoil vortex knots have been ideal for showing that the $B_{\nu }(t)$ (1.15) scaling can be retained as $\nu$ decreases by increasing the domain size, leading to the following question. What controls the domain dependence of the enstrophy suppressing critical viscosities $?$

Previously, to maintain the temporal convergence at $t_x$ , the domain size $\mathcal{L}=2\ell \pi$ had to be empirically increased as $\nu$ decreased (Kerr Reference Kerr2018a ,Reference Kerr b ), although an estimate of that dependence was not given. Here, with a wider range of $\nu$ , especially for the convergence of $(\nu ^{1/4}\varOmega _\infty )^{-1}$ at $t_\infty \approx 18.4$ , combined with the consistency of the convergence of each ( $t\lt t_x$ ) $(\nu ^{1/4}\mathcal{O}_{\textit{Vm}}(t))^{-1}$ , it is clear that the increase is $\ell \sim \nu ^{-1/4}$ . The structural evidence points to the growth of vortex sheets as in figures 11 and 12(a) as the source of the limiting factor. Overall, combining the requirements at large scales with $\ell \sim \nu ^{-1/4}$ , with those at small scales going as $\eta \sim \nu ^{3/4}$ , the range of scales should increase as $\ell /\eta \sim \nu ^{-1/4}/\nu ^{3/4}\sim \nu ^{-1}$ .

A more rigorous supporting result is given in Appendix A. This provides an extremal mathematical justification for why increasing $\ell$ is needed. The method consists of determining domain size dependent mathematical critical viscosities $\nu _s(\ell )$ as $\ell$ increases. These analytic/Sobolev $\nu _s(\ell )$ are found by rescaling the high-order $s\gt 5/2$ Sobolev analysis for $\ell =1$ , $V=(2\pi )^3$ into larger $V_\ell =(2\ell \pi )^3$ domains (1.16). With some $(2\pi )^3$ estimates of the $\nu _s$ made from the Euler norms in Appendix A.5, before they are transformed into the original domain in Appendix A.6.

Is there mathematics that could bring the two determinations of the critical viscosities more in line with one another $?$ Currently, we know that the $\nu _s(\ell )$ should be, and are, orders of magnitude smaller than the empirical $\nu _c$ . However, it is believed by many that $\ell \to \infty$ limiting behaviour can be determined by a more direct extension of the $(2\pi )^3$ mathematics (Constantin Reference Constantin1986) to whole space, infinite $\mathbb{R}^3$ . Part of that extension has been done (Robinson Reference Robinson2021) and having a further extension might bridge that gap by showing explicitly how the $\nu _s(\ell )$ depend upon $\ell$ , the domain size.

The conclusion of both approaches is that getting a dissipation anomaly that is finite-time finite- $\Delta E_\epsilon$ , without finite-time singularities, is possible only when the physical problem is done in whole space, that is, infinite $\mathbb{R}^3$ .

5.2. Spectral summary

In addition to generating convergence of $\epsilon (t)=\nu Z(t)$ , then finite dissipation $\Delta E_\epsilon$ (1.2) during the period $t\gtrsim 72$ , precursors to $k^{1/3}$ spectral scaling also develop, from short $Z_V(k)\sim k^{1/3}$ regimes for $3\leqslant k\leqslant 4$ at $t=72$ and 80 for $\nu =3.12\times 10^{-5}$ to slightly broader $2.67\leqslant k\leqslant 4.33$ regimes for $t=66$ to 78 for $\nu =2.2\times 10^{-5}$ . These are distinctly less steep than the $E(k)\sim k^{-7/3}$ spectra, or equivalent $Z(k)\sim k^{-1/3}$ , reported for a Gaussian-profile trefoil (Yao et al. Reference Yao, Yang and Hussain2021), whose decrease in energy is approximately 25 % over that span. By integrating $\epsilon (t)$ from figure 6, the algebraic trefoil’s energy decays by approxomately 50 % as the coiled vortices shown in figures 13 and 14 form.

1. (i) These indications of enstrophy and energy flow through wavenumbers are usually associated with turbulent flows, showing that for these and the earlier Kerr (Reference Kerr2023) calculations, an energy cascade to small scales does not require the formation of a statistically steady state and instead can form as the enstrophy moves through a brief $Z_V(k)\sim k^{1/3}$ regime, as in figure 9(b).

2. (ii) However, are these transient quasi- $k^{1/3}$ regimes consistent with the experimental observations of Kolmogorov spectra from statistically steady data $?$ They are not, as shown by how the $Z_V(k)\sim k^{-1}$ spectrum persists after the convergence of the dissipation rates after $t_\epsilon \sim 93$ . Specifically ,to $t=120$ for $\nu =3.12\times 10^{-5}$ in figure 7 and $t=111$ for $\nu =2.21$ in figure 9(d), along with how the Kolmogorov wavenumber (3.7) is unexpectedly large.

3. (iii) Perhaps this should be expected, because unlike these calculations, having a statistically steady state implies the existence of a steady energy source, such as a constant flow, forcing or in the algebraic anti-parallel case (Kerr Reference Kerr2013b ), very long initial vortices, which allows for the formation of multiple reconnection sites with coiled or twisted vortices. Additionally, with multiple reconnection sites, off-set in time, it is likely that the contributions to the spectra from these sites overlap, allowing for smoother-in- $k$ evolution of the spectra.

5.3. Does $(2\ell \pi )^3$ rescaling lead to turbulence?

Is this evidence for a dissipation anomaly and Lundgren-like spectra consistent with the seminal proposal of Onsager (Reference Onsager1949) that there might be singularities, or near singularities, around which spatial velocity increments have a 1/3 Hölder exponent (Eyink Reference Eyink2003), from which $k^{-5/3}$ Kolmogorov spectra and finite dissipation might develop.

In a strict sense, the existing numerics does not support a role for singularities. Singularities of either the Navier–Stokes or Euler equations away from boundaries have not been identified, with the strongest growth for the Euler equations being doubly exponential (Kerr Reference Kerr2013a ; Meng & Yang Reference Meng and Yang2024). The latest forced, very-high-Reynolds-numbers calculations are not providing convincing numerical evidence because the dissipation rate $\epsilon$ (1.12) increases slowly when the Reynolds numbers are increased (Iyer, Sreenivasan & Yeung Reference Iyer, Sreenivasan and Yeung2022). Given these difficulties, a possible role for sub-Kolmogorov fluctuations has been floated (Eyink & Jafari Reference Eyink and Jafari2022).

However, if one replaces the assumption of singularities with how vortex sheets are shed, the calculations here present an alternative approach to generating numerical turbulence with finite dissipation where we have the following.

1. (i) The large-scale forcing is replaced by large-scales that grow without bounds.

2. (ii) The first vorticity that is shed from the original configuration is in sheets, not thin bridges or braids, with the finite-time energy dissipation resulting from the interactions between those sheets.

3. (iii) When there is only one self-interaction and reconnection site, and the energy flow through the wavenumbers is not continuous, but occurs as a pulse. A pulse whose slope is consistent with $1/3$ scaling as the time convergent dissipation rates are reached.

These observations demonstrate that the incompressible Navier–Stokes are capable of generating many, but not all, of the properties of fully developed Navier–Stokes turbulence, without the types of Navier–Stokes modifications referenced previously.