3.1 Decay controlled by mean and fluctuating magnetic helicities

In the helical case with $I_{M}\neq 0$, we have $\xi _{M}\propto t^{2/3}$ and ${\mathcal {E}}_{M}\propto t^{-2/3}$ (Hatori Reference Hatori1984; Biskamp & Müller Reference Biskamp and Müller1999; Brandenburg & Kahniashvili Reference Brandenburg and Kahniashvili2017). In the present context, the pre-factors are important. Using the data from figures 1(c) and 2(c) of Brandenburg & Kahniashvili (Reference Brandenburg and Kahniashvili2017), here referred to as Run A, we find

(3.2a–c)\begin{equation} \xi_{M}(t)\approx0.12\,I_{M}^{1/3}\,t^{2/3},\quad {\mathcal{E}}_{M}(t)\approx4.3\,I_{M}^{2/3}\,t^{{-}2/3},\quad E_{M}(k,t)\lesssim 0.7\,I_{M}. \end{equation}

Generally, we can write

(3.3a–c)\begin{equation} \xi_{M}(t)=C_{i}^{(\xi)}\,I_{i}^{\sigma}t^q,\quad {\mathcal{E}}_{M}(t)=C_{i}^{({\mathcal{E}})}\,I_{i}^{2\sigma}t^{{-}p},\quad E_{M}(k)=C_{i}^{(E)}\,I_{i}^{(3+\beta)\sigma}(k/k_0)^{\beta}, \end{equation}

where the index $i$ in the integrals $I_i$ and the coefficients $C_{i}^{(\xi )}$, $C_{i}^{({\mathcal {E}})}$ and $C_{i}^{({E})}$ stands for $M$ or $H$ for magnetic helicity and Hosking scalings, respectively, and $\sigma$ is the exponent with which length enters in $I_{i}$: $\sigma =1/3$ for the magnetic helicity density ($i={M}$) and $\sigma =1/9$ for the Hosking integral ($i={H}$); see table 2 for a summary of the coefficients.Footnote ²

Table 2. Summary of the coefficients characterizing the decays governed by the conservation of magnetic helicity ($i={M}$) and the Hosking integral ($i={H}$).

In figure 1, we show the magnetic energy spectra, as well as compensated evolutions of $\xi _{M}(t)$ and ${\mathcal {E}}_{M}(t)$ for the maximally helical run (Run A). We see that the peak of ${\mathcal {E}}_{M}(t)$ remains underneath a nearly flat envelope (its slope is $\beta =0$), as is expected for a fully helical turbulent decay at late times. The compensated evolutions of $\xi _{M}(t)$ and ${\mathcal {E}}_{M}(t)$ are not yet fully converged towards the end of that run (the lines are not yet flat). This is partially caused by the insufficient scale separation between the box wavenumber $k_1$ and that of the spectral peak by the end of the run. Nevertheless, we can read off the approximate values $C_{M}^{(\xi )}\approx 0.12$ and $C_{M}^{({\mathcal {E}})}\approx 4.3$ towards the end of the run. The values of these coefficients are revisited later in this paper.

Figure 1. (a) Magnetic energy spectra, as well as compensated evolutions of (b) $\xi _{M}(t)$ and (c) ${\mathcal {E}}_{M}(t)$ for the maximally helical run of figure 2(c) of Brandenburg & Kahniashvili (Reference Brandenburg and Kahniashvili2017), here referred to as Run A. In (a), the red symbols denote the spectral peaks.

In figure 2, we again show the compensated evolutions of $\xi _{M}(t)$ and ${\mathcal {E}}_{M}(t)$, but now for Run B, which is nearly perfectly non-helical ($\varsigma =0.003$) and has $k_0/k_1=30$. The resulting coefficients are close to those estimated previously, namely $C_{H}^{(\xi )}\approx 0.12$ and $C_{H}^{({\mathcal {E}})}\approx 4.7$. This supports the previous hypotheses of Brandenburg & Larsson (Reference Brandenburg and Larsson2023) and Brandenburg et al. (Reference Brandenburg, Sharma and Vachaspati2023c) that these coefficients may indeed be universal.

Figure 2. Evolution of $\xi _{M}(t)$ (a,b) and ${\mathcal {E}}_{M}(t)$ (c,d) for Run B with $k_0/k_1=30$ and $\varsigma =0.003$, compensated by the expected evolution if the decay is controlled either by $I_{H}$ (a,c) or by $I_{M}$ (b,d). The dashed line denotes the use of $I_{M}$ at the end of the run, while for the solid line, the time-dependent value was taken.

To determine the value of $I_{H}(t)$, we plot in figure 3 the evolutions of $(2{\rm \pi} ^2/k^2)\, {\rm Sp}(h)$ (normalized by $v_\mathrm {Ae}^4/k_{e}^5$) for $k/k_1=1$, 2 and 3 for Run B. We see that for $k/k_1=1$, the result shows a nearly negligible decline proportional to $t^{-0.07}$. Note that, in units of $v_\mathrm {Ae}^4/k_{e}^5$, the value of $I_{H}$ is about 500.

Figure 3. Evolutions of $(2{\rm \pi} ^2/k^2)\, {\rm Sp}(h)$, normalized by $v_\mathrm {Ae}^4/k_{e}^5$, for $k/k_1=1$ (solid line), 2 (dashed-dotted line) and 3 (dashed line), for the nearly non-helical Run B with $\varsigma =0.003$ and $k_0/k_1=30$.

3.2 Decay controlled by $I_{M}$ and $I_{H}$

If both $I_{M}$ and $I_{H}$ control the decay, we have a combination of the two decay laws such that the late times are always controlled by the more strongly conserved quantity, i.e. by $I_{M}$. One might expect that the resulting expression for the combination of the decay laws (3.1a–c) and (3.1a–c) is given by the sum of both expressions. This would be analogous to the way how in radiation transport the cooling time is given by the sum of the cooling times for the optically thick and thin cases; see (7) of Brandenburg & Das (Reference Brandenburg and Das2020). In the present case, this would translate to

(3.4)\begin{gather} \xi_{M}\approx 0.12 \, I_{M}^{1/3} \, t^{2/3} +0.12 \, I_{H}^{1/9} \, t^{4/9}, \end{gather}

(3.5)\begin{gather}{\mathcal{E}}_{M}\approx 4.3 \, I_{M}^{2/3} \, t^{{-}2/3} +3.7 \, I_{H}^{2/9} \, t^{{-}10/9}. \end{gather}

Since the second terms involving $I_{H}$ are initially larger, but their contributions to $\xi _{M}$ grow more slowly and that to ${\mathcal {E}}_{M}$ decay faster than the first terms, one expects their contributions to become subdominant after some time. Thus, the magnetic helicity will always survive and be the dominant contribution to explaining the decay.

To examine now a run where the decay is controlled by both $I_{M}$ and $I_{H}$, we now increase the initial fractional helicity slightly from 0.003 to 0.01; see figure 4 for magnetic energy spectra at different times for Run C. Note that the peaks of the spectra evolve at first underneath an envelope with the slope $\beta =3/2$, as expected for a decay controlled by $I_{H}$. At later times, however, the envelope becomes flat (slope $\beta =0$), as expected for a decay controlled by $I_{M}$.

Figure 4. Magnetic energy spectra for Run C with $k_0/k_1=60$ and $\varsigma =0.01$ at times $v_\mathrm {Ae} k_{e}\,t=0.07$, 0.18, 0.40, 0.82, 1.65, 3.3, 6.1, 11.1 and 20.7.

In figure 5, we show the evolutions of $\xi _{M}(t)$ and ${\mathcal {E}}_{M}(t)$ for Run C, compensated by $t^{-2/3}$ and $t^{2/3}$, respectively, as well as $t^{-4/9}$ and $t^{10/9}$, respectively. We see that now the curves compensated by $t^{-2/3}$ and $t^{2/3}$, respectively, become nearly constant, as expected for a decay that is governed by magnetic helicity conservation. Specifically, we find $\xi _{M}/(I_{M}^{1/3}\,t^{2/3})\approx 0.14$ and ${\mathcal {E}}_{M}/(I_{M}^{2/3}\,t^{-2/3})\approx 4.0$. During a short intermediate interval, however, we see that the curves compensated by $t^{-4/9}$ and $t^{10/9}$, respectively, show brief plateaus around $v_\mathrm {Ae}k_{e}t=1$ with $\xi _{M}/(I_{H}^{1/9}\,t^{4/9})\approx 0.12$ and ${\mathcal {E}}_{M}/(I_{H}^{2/9}\,t^{-10/9})\approx 4.0$.

Figure 5. Similar to figure 2, but for Run C with $k_0/k_1=60$ and $\varsigma =0.01$.

3.3 Improved fits with $I_{M}$ and $I_{H}$

We have seen that the limiting cases where the decay is controlled either by $I_{M}$ or by $I_{H}$ are well reproduced by (3.1a–c). It turns out, however, that the combined fits given by (3.4) and (3.5) are not very accurate. Improved fits can be obtained by using large weighting exponents for both contributions, i.e.

(3.6)\begin{gather} \xi_{M}\approx \left[ \left(0.12 \, I_{M}^{1/3} \, t^{2/3}\right)^s + \left(0.14 \, I_{H}^{1/9} \, t^{4/9}\right)^s\right]^{1/s}, \end{gather}

(3.7)\begin{gather}{\mathcal{E}}_{M}\approx \left[ \left(4.0 \, I_{M}^{2/3} \, t^{{-}2/3}\right)^s + \left(4.0 \, I_{H}^{2/9} \, t^{{-}10/9}\right)^s\right]^{1/s}. \end{gather}

The result is shown in figure 6 for Run C, where we show that $s=10$ yields satisfactory fits, while $s=2$ and $s=1$ (our original hypothesis) are poor. The fact that the coefficients for both parts are different from those of the individual fits and that they happen to be $4.0$ in (3.7), but different from each other in (3.6), is probably just by chance and reflect that degree of uncertainty of these values.

Figure 6. Decay of magnetic energy (black line) and the fit given by (3.5) (dotted blue line, denoted by $s=1$) as well as (3.7) with $s=2$ (dashed orange line) and $s=10$ (solid red line). The dotted red line corresponds to the limit $s\to \infty$, as realized by (3.8) and (3.9).

It is important to emphasize that the limit $s\to \infty$ corresponds to

(3.8)\begin{gather} \xi_{M}\approx \max\left( 0.12 \, I_{M}^{1/3} \, t^{2/3}, 0.14 \, I_{H}^{1/9} \, t^{4/9}\right), \end{gather}

(3.9)\begin{gather}{\mathcal{E}}_{M}\approx \max\left( 4.0 \, I_{M}^{2/3} \, t^{{-}2/3},\; 4.0 \, I_{H}^{2/9} \, t^{{-}10/9}\right). \end{gather}

These expressions yield discontinuities in the derivative. An advantage of such expressions is that one can clearly see the regimes of validity of both expressions. The critical times when characterizing the crossover from Hosking scaling to magnetic helicity scaling are given by

(3.10)\begin{gather} t_\xi\approx(0.12/0.14)^{9/2}\left(I_{H}/I_{M}^3\right)^{1/2} \approx0.50\,I_{H}^{1/2}I_{M}^{{-}3/2}, \end{gather}

(3.11)\begin{gather}t_{\mathcal{E}}\approx I_{H}^{1/2}I_{M}^{{-}3/2}. \end{gather}

It would be plausible to assume that both times should equal each other. The fact that they are not equal to each other might hint, again, at the possibility that the precise values of these coefficients are still uncertain. On the other hand, looking at figure 5, it is actually true that $\xi _{M}$ approaches the $I_{M}$-dominated scaling by a factor two earlier than ${\mathcal {E}}_{M}$. A possible explanation for this behaviour could lie in the fact that the spectral shapes change as the system becomes fully helical. We return to this in § 3.4, where we discuss the spectral shapes in more detail.

We recall that an essential assumption in our dimensional argument was the fact that the magnetic field is understood to be in Alfvén units and thus has dimensions of ${\rm cm}\ {\rm s}^{-1}$. In neutron star crusts, by contrast, where the magnetic field is governed by the Hall effect (Goldreich & Reisenegger Reference Goldreich and Reisenegger1992), it has units of ${\rm cm}^2\ {\rm s}^{-1}$, so $[I_{M}]={\rm cm}^5\ {\rm s}^{-2}$ and $[I_{H}]={\rm cm}^{13}\ {\rm s}^{-4}$ (Brandenburg Reference Brandenburg2023), so $t_\xi$ and $t_{\mathcal {E}}$ are now proportional to $I_{H}^{5/6}I_{M}^{-13/6}$, so both exponents are larger than in the ordinary magnetohydrodynamic case; cf. (3.10) and (3.11). An example of the corresponding switch between the two regimes was presented in figure 10(b) of Brandenburg (Reference Brandenburg2020).

3.4 Collapsed spectra

The quality of the fits of (3.6) and (3.7) for $s=1$ and $s\to \infty$ can be examined further by computing compensated spectra. This is shown in figure 7, where the abscissa is scaled with $\xi _{M}(t)$ and the ordinate with $[{\mathcal {E}}_{M}(t)\xi _{M}(t)]^{-1}$. We see that the collapse in figure 7(b), where $s\to \infty$, is much better than that in figure 7(a), where $s=1$, and it is almost as good as that in figure 7(c), where the actual values of $\xi _{M}(t)$ and ${\mathcal {E}}_{M}(t)$ are used. This supports our finding that in the fractionally helical case, the magnetic energy and correlation length are approximately given by the maximum of the values for the purely helical and purely non-helical cases, and not by their sum, as might naively have been expected.

Figure 7. Magnetic energy spectra similar to figure 4, but the abscissa is scaled with $\xi _{M}(t)$ and the ordinate with $[{\mathcal {E}}_{M}(t)\xi _{M}(t)]^{-1}$, where (3.6) and (3.7) are used with $s=1$ in (a) and with $s\to \infty$ in (b). In (c), the actual values of $\xi _{M}(t)$ and ${\mathcal {E}}_{M}(t)$ are used. The last time is shown as a thick red line.

As alluded to in § 3.3, there is a change in the shape of the spectrum as the system becomes fully helical. In particular, the position of the peak appears for slightly larger values of $k\xi _{M}(t)$ at later times; see figure 7(c). Thus, the value of $\xi _{M}(t)$ is slightly overestimated, which would explain the smaller value of $t_\xi$ compared with $t_{\mathcal {E}}$.

3.5 Comparison with earlier work

As mentioned in the introduction, the switchover time from non-helically to helically dominated decay has been studied by Tevzadze et al. (Reference Tevzadze, Kisslinger, Brandenburg and Kahniashvili2012) under the assumption that $p=1$ and $q=1/2$ (Christensson et al. Reference Christensson, Hindmarsh and Brandenburg2001) instead of $p=10/9$ and $q=4/9$, as now motivated by the conservation of the Hosking integral. The basic idea is to assume that at the time of switchover, $t_\ast$, the real-space realizability condition (Biskamp Reference Biskamp2003; Kahniashvili et al. Reference Kahniashvili, Brandenburg, Tevzadze and Ratra2010) is saturated, i.e. $2\xi _{M}(t_\ast ){\mathcal {E}}_{M}(t_\ast )/\rho _0=I_{M}$. Next, inserting $\xi _{M}(t_\ast )=\xi _{M}(t_0)\,(t_\ast /t_0)^q$ and ${\mathcal {E}}_{M}(t_\ast )={\mathcal {E}}_{M}(t_0)\,(t_\ast /t_0)^{-p}$, we find $2\xi _{M}(t_0){\mathcal {E}}_{M}(t_0)/\rho _0\,(t_\ast /t_0)^{-(p-q)}=I_{M}$, and therefore

(3.12)\begin{equation} t_\ast{=}t_0\,\left[2\xi_{M}(t_0){\mathcal{E}}_{M}(t_0)/I_{M}\rho_0\right]^{1/(p-q)}. \end{equation}

For $p=1$ and $q=1/2$, we have $1/(p-q)=2$ and recover the result of Tevzadze et al. (Reference Tevzadze, Kisslinger, Brandenburg and Kahniashvili2012), while for $p=10/9$ and $q=4/9$, we have $1/(p-q)=3/2$. Comparing with (3.11), we see that $I_{M}$ enters with the same exponent $3/2$, and that the remainder can be identified with

(3.13)\begin{equation} I_{H}=\left[2\xi_{M}(t_0){\mathcal{E}}_{M}(t_0)\right]^3 t_0^2. \end{equation}

This suggests that $I_{H}$ is related to $\xi _{M}$ and ${\mathcal {E}}_{M}$, but the problem is that $t_0$ is not straightforwardly related to the Alfvén time $\xi _{M}/v_{A}$, where $v_{A}^2=2{\mathcal {E}}_{M}/\rho _0$. Indeed, Brandenburg, Neronov & Vazza (Reference Brandenburg, Neronov and Vazza2024) found that there is a pre-factor $C_{M}$ that increases with increasing magnetic Reynolds number. Such a factor has been motivated based on magnetic reconnection arguments (Hosking & Schekochihin Reference Hosking and Schekochihin2023a). Thus, inserting $t=C_{M}\xi _{M}/v_{A}$, we find

(3.14)\begin{equation} I_{H}=C_{M}^2\xi_{M}^5v_{A}^4. \end{equation}

The fact that $C_{M}$ enters quadratically and approaches values in the range 20–50 for large magnetic Reynolds numbers explains why $I_{H}$ strongly exceeds the naive estimate $\xi _{M}^5v_{A}^4$. Interestingly, Zhou et al. (Reference Zhou, Sharma and Brandenburg2022) found that part of the large excess over the naive estimate is related to non-Gaussianity. Another smaller part has to do with the spectral shape. Linking the value of $C_{M}$ to non-Gaussianity of the magnetic field provides a new clue to the question of why there is a resistivity-dependent relation between decay and Alfvén times in hydromagnetic turbulence.

3.6 Can the switchover time be resistively limited?

We know that the decay time, $\tau (t)=(-{\rm d} {}\ln {\mathcal {E}}_{M}/{\rm d} {} t)^{-1}=t/p(t)$, can be regarded as resistively limited when relating it to the Alfvén time, $\tau _{A}=\xi _{M}/v_{A}$. In particular, as alluded to in § 3.5, it turns out that $\tau /\tau _{A}=C_{M}( {\textit {Lu}})$, which is a monotonically increasing function of ${\textit {Lu}}$ that saturates near ${\textit {Lu}}_\ast$ at $C_{M}^\ast \approx 50$ (Brandenburg et al. Reference Brandenburg, Neronov and Vazza2024). Such a relation was theoretically expected and has been associated with magnetic reconnection (Hosking & Schekochihin Reference Hosking and Schekochihin2023a). However $C_{M}( {\textit {Lu}})$ was found to be independent of the value of the magnetic Prandtl number, which raised doubts about this interpretation.

The question now is whether the switchover time might also depend on the value of ${\textit {Lu}}$. Differentiating (3.7) or (3.9), we see that the decay time depends on whether $t< t_\ast$ or $t>t_\ast$ and is equal to $9t/10$ or $3t/2$, respectively, but the switchover time itself is unaffected by resistivity effects. In other words, the decay time is always $t/p$, where the value of $p$ depends on the time.

There is also the possibility that for ${\textit {Lu}}< {\textit {Lu}}_\ast$, the exponent $p$ might depend on the order of the diffusion operator, i.e. on whether it is proportional to $\nabla ^2$ or some higher power; see Zhou et al. (Reference Zhou, Sharma and Brandenburg2022) for details. However, this consideration only applies to the regime of low enough values of ${\textit {Lu}}$ and it would still not affect the actual value of $t_\ast$.

3.7 Hosking scaling at intermediate length scales

Hosking & Schekochihin (Reference Hosking and Schekochihin2021) presented arguments that for finite magnetic helicity, the Hosking scaling should only be obeyed at intermediate length scales. To check this, we now use the box-counting method as described by (2.7) to plot $\mathcal {I}_{H}(t,R)$ at different times $t$. The result is shown in figure 8 and resembles the sketch provided in figure 10 of Hosking & Schekochihin (Reference Hosking and Schekochihin2021). We do indeed see a short plateau where the Hosking scaling can be discerned for intermediate times. Furthermore, at later times, we see the expected $R^3$ scaling over the whole range of $R$.

Figure 8. Box-counting result for $\mathcal {I}_{H}(R)$ for runs B and C in the left- and right-hand panels. Note the plateau for intermediate values of $R$ at early times.

3.8 Comment on the case of a finite mean field

Our simulations presented above had zero mean field. It is well known that in the presence of a mean field across the entire domain, the magnetic helicity is no longer conserved (Berger Reference Berger1997; Brandenburg & Matthaeus Reference Brandenburg and Matthaeus2004); see Brandenburg et al. (Reference Brandenburg, Durrer, Huang, Kahniashvili, Mandal and Mukohyama2020) for corresponding decay simulations in the presence of a mean field. To check whether the Hosking integral could still be meaningful in such a case, we now present the time dependence of $I_{H}(t)$ for Run D with different magnetic field strength, where a mean field $\boldsymbol {B}_{m}=B_{m}\hat {\boldsymbol {x}} {}$ is now imposed, so the magnetic field is given by $\boldsymbol {B}=\boldsymbol {B}_{m}+{\boldsymbol {\nabla }}\times \boldsymbol {A}$. As before, we evaluate $I_{H}(t)=2{\rm \pi} ^2 {\rm Sp}(h)/k^2$ at $k=k_1$. The result is shown in figure 9 for four values of $v_{\rm Am}$. We see that $I_{H}(t)$ is now decaying $\propto t^{-2}$, i.e. the Hosking integral is not conserved. Thus, with periodic boundary conditions, not only is $I_{M}$ not conserved, but $I_{H}$ is also not conserved.

Figure 9. Time dependence of ${\rm Sp}(h)$ for Run D (solid curve, $v_{\rm Am}/c_{s}=0.1$) and several cases with weaker mean field ($v_{\rm Am}/c_{s}=0.05$ for the dashed-dotted line, 0.02 for the dashed line and 0.01 for the dotted line).

3.9 Comment on the case with chiral fermions

As we have mentioned in the introduction, the Hosking integral also describes the decay of helical turbulence in the presence of chiral fermions if their chirality exactly balances the magnetic helicity. One may therefore ask whether a switchover to helical scaling could also occur in this case of the initial balance being not perfect. In Brandenburg et al. (Reference Brandenburg, Kamada, Mukaida, Schmitz and Schober2023a), two cases of imbalanced chirality were presented. In the case where the magnetic helicity exceeds the negative contribution to the chirality, a helical decay scaling $\propto t^{-2/3}$ of the magnetic energy was found, thus supporting our expectation. In the opposite case of an excess of fermion chirality, the time evolution of the magnetic energy was more complicated and no clear scaling suggestive of a helical decay was found.