2.1. Roberts fields

To fix our geometry, we assume magnetic flux tubes to extend in the $z$ -direction and being perpendicular to the $xy$ -plane. Such a field can be realised by the so-called Roberts field I, i.e. the magnetic field $\boldsymbol{B}$ is given by

(2.1) \begin{equation} {\boldsymbol{B}}={\boldsymbol{B}}_{\mathrm{I}}\equiv {\boldsymbol{\nabla }}\times \phi \hat {{\boldsymbol{z}}} {}+\sqrt {2}k_0\phi \hat {{\boldsymbol{z}}} {}, \quad \textrm{where}\; \phi =k_0^{-1} B_0 \sin k_0 x\sin k_0 y \end{equation}

is an $xy$ periodic field. Such a magnetic field has a component in the $z$ -direction, but no variation along that direction, so it is highly anisotropic. This may change with time as the magnetic field undergoes a turbulent decay. The Roberts field I is maximally helical with $\boldsymbol{A}\boldsymbol{\cdot} {\boldsymbol{B}}=\sqrt {2} k_0^{-1} B_0^2 (\sin ^2 k_0x+\sin ^2 k_0y)$ , so $\langle \boldsymbol{A}\boldsymbol{\cdot} {\boldsymbol{B}}\rangle =\sqrt {2} k_0^{-1} B_0^2$ . Here, $\boldsymbol{A}$ is the magnetic vector potential and ${\boldsymbol{B}}={\boldsymbol{\nabla }}\times \boldsymbol{A}$ . The Roberts field II, by contrast, is given by

(2.2) \begin{equation} {\boldsymbol{B}}={\boldsymbol{B}}_{\mathrm{II}}\equiv {\boldsymbol{\nabla }}\times \phi \hat {{\boldsymbol{z}}} {}+k_{\textrm{f}}\tilde {\phi }\hat {{\boldsymbol{z}}} {}, \quad \textrm{where}\ \tilde {\phi }=k_0^{-1} B_0 \cos k_0 x\cos k_0 y, \end{equation}

where $\tilde {\phi }$ is $90^\circ$ phase shifted in the $x$ - and $y$ -directions relative to $\phi (x,y)$ , and $k_{\textrm{f}}=\sqrt {2}k_0$ is the eigenvalue of the curl operator for field I, i.e. ${\boldsymbol{\nabla }}\times {\boldsymbol{B}}_{\mathrm{I}}=k_{\textrm{f}}{\boldsymbol{B}}_{\mathrm{I}}$ , so ${\boldsymbol{B}}_{\mathrm{I}}\boldsymbol{\cdot} {\boldsymbol{\nabla }}\times {\boldsymbol{B}}_{\mathrm{I}}=k_{\textrm{f}}{\boldsymbol{B}}_{\mathrm{I}}^2$ , while ${\boldsymbol{B}}_{\mathrm{II}}\boldsymbol{\cdot} {\boldsymbol{\nabla }}\times {\boldsymbol{B}}_{\mathrm{II}}=0$ pointwise. In the Coulomb gauge, we have, for field II, ${\boldsymbol{B}}_{\mathrm{II}}={\boldsymbol{\nabla }}\times \boldsymbol{A}_{\mathrm{II}}$ , where

(2.3) \begin{equation} \boldsymbol{A}_{\mathrm{II}}=k_{\textrm{f}}^{-1}\big({\boldsymbol{\nabla }}\times \tilde {\phi }\hat {{\boldsymbol{z}}} {}+k_{\textrm{f}}\phi \hat {{\boldsymbol{z}}} {}\big), \end{equation}

and therefore also $\boldsymbol{A}_{\mathrm{II}}\boldsymbol{\cdot} {\boldsymbol{B}}_{\mathrm{II}}=0$ . Thus, for field II, not just the current helicity density vanishes pointwise, but in the Coulomb gauge, also the magnetic helicity density vanishes pointwise. Both for fields I and II, we have $\langle {\boldsymbol{B}}^2\rangle =2B_0^2$ .

2.2. Quantifying the emerging anisotropy

To quantify the degree of anisotropy, we must separate the derivatives of the magnetic field along the $z$ -direction ( ${\boldsymbol{\nabla }}_\|$ ) from those perpendicular to it ( ${\boldsymbol{\nabla }}_\perp$ ), so ${\boldsymbol{\nabla }}={\boldsymbol{\nabla }}_\|+{\boldsymbol{\nabla }}_\perp$ . We also decompose the magnetic field analogously, i.e. ${\boldsymbol{B}}={\boldsymbol{B}}_\|+{\boldsymbol{B}}_\perp$ . The mean current density can be decomposed similarly, i.e. $\boldsymbol{J}=\boldsymbol{J}_\|+\boldsymbol{J}_\perp$ , but this decomposition mixes the underlying derivatives. We see this by computing $\boldsymbol{J}\equiv {\boldsymbol{\nabla }}\times {\boldsymbol{B}}$ (where the permeability has been set to unity). Using this decomposition, we find

(2.4) \begin{equation} \boldsymbol{J}={\boldsymbol{\nabla }}_\|\times {\boldsymbol{B}}_\perp +{\boldsymbol{\nabla }}_\perp \times {\boldsymbol{B}}_\|+{\boldsymbol{\nabla }}_\perp \times {\boldsymbol{B}}_\perp , \end{equation}

noting that ${\boldsymbol{\nabla }}_\|\times {\boldsymbol{B}}_\|=0$ . The term of interest for characterising the emergent isotropisation is the first one, ${\boldsymbol{\nabla }}_\|\times {\boldsymbol{B}}_\perp$ , because it involves only parallel derivatives ( $z$ -derivatives), which vanish initially. We monitor the ratio of its mean squared value to $\langle \boldsymbol{J}^2\rangle$ .

The last term in (2.4) is just $\boldsymbol{J}_\|={\boldsymbol{\nabla }}_\perp \times {\boldsymbol{B}}_\perp$ , but the first and second terms cannot simply be expressed in terms of $\boldsymbol{J}_\perp$ , although ${\boldsymbol{\nabla }}_\|\times {\boldsymbol{B}}_\perp$ would be $\boldsymbol{J}_\perp$ if the magnetic field only had a component in the plane and ${\boldsymbol{\nabla }}_\perp \times {\boldsymbol{B}}_\|$ would be $\boldsymbol{J}_\perp$ if the magnetic field only had a component out of the plane. We therefore denote those two contributions in what follows by $\boldsymbol{J}_{\perp \perp }$ and $\boldsymbol{J}_{\perp \|}$ , respectively, so that $\boldsymbol{J}_{\perp \perp }+\boldsymbol{J}_{\perp \|}=\boldsymbol{J}_\perp$ .

Thus, with the abovementioned motivation, to monitor the emergent isotropisation, we determine $\langle \boldsymbol{J}_{\perp \perp }^2\rangle /\langle \boldsymbol{J}^2\rangle$ . For isotropic turbulence, we find that this ratio is approximately $4/15\approx 0.27$ , and this is also true for $\langle \boldsymbol{J}_{\perp \|}^2\rangle /\langle \boldsymbol{J}^2\rangle$ ; see Appendix A for an empirical demonstration. In the expression for $\langle \boldsymbol{J}^2\rangle$ , there is also a mixed term, $\boldsymbol{J}_{\perp {m}}^2=-2\langle B_{x,z}B_{z,x}+B_{y,z}B_{z,y}\rangle$ , which turns out to be positive in practice. Here, commas denote partial differentiation. Thus, we have

(2.5) \begin{equation} \langle \boldsymbol{J}^2\rangle = \big\langle \boldsymbol{J}_{\perp \perp }^2\big\rangle + \big\langle \boldsymbol{J}_{\perp \|}^2\big\rangle + \big\langle \boldsymbol{J}_{\perp \textrm{m}}^2\big\rangle + \big\langle \boldsymbol{J}_{\|}^2\big\rangle . \end{equation}

In the isotropic case, we find $\langle \boldsymbol{J}_{\|}^2\rangle /\langle \boldsymbol{J}^2\rangle =1/3$ and for the mixed term, we then have $\langle \boldsymbol{J}_{\perp {\mathrm{m}}}^2\rangle /\langle \boldsymbol{J}^2\rangle =2/15\approx 0.13$ .

2.3. Evolution equations

To study the decay of the magnetic field, we solve the evolution equations of magnetohydrodynamics (MHD) for an isotropic compressible gas with constant sound speed $c_{\mathrm{s}}$ , so the gas density $\rho$ is proportional to the pressure $p=\rho c_s^2$ . In that case, $\ln \rho$ and the velocity $\boldsymbol{u}$ obey

(2.6) \begin{equation} \frac {{\textrm{D} {}}\ln \rho }{{\textrm{D} {}} t}=-{\boldsymbol{\nabla }}\boldsymbol{\cdot} {\boldsymbol{u}}, \end{equation}

(2.7) \begin{equation} \frac {{\textrm{D} {}}{\boldsymbol{u}}}{{\textrm{D} {}} t}=-c_{\mathrm{s}}^2{\boldsymbol{\nabla }}\ln \rho +\frac {1}{\rho }\left [\boldsymbol{J}\times {\boldsymbol{B}}+{\boldsymbol{\nabla }}\boldsymbol{\cdot} (2\rho \nu {\boldsymbol{\sf S}} {})\right ],\\[-12pt] \end{equation}

where ${\textrm{D} {}}/{\textrm{D} {}} t=\partial /\partial t+{\boldsymbol{u}}\boldsymbol{\cdot} {\boldsymbol{\nabla }}$ is the advective derivative, $\nu$ is the kinematic viscosity and ${\boldsymbol{\sf S}} {}$ is the rate-of-strain tensor with components $\mathsf{S}_{ij}=(u_{i,j}+u_{j,i})/2-\delta _{ij}{\boldsymbol{\nabla }}\boldsymbol{\cdot} {\boldsymbol{u}}/3$ . To ensure that the condition ${\boldsymbol{\nabla }}\boldsymbol{\cdot} {\boldsymbol{B}}=0$ is obeyed at all times, we also solve the uncurled induction equation for $\boldsymbol{A}$ , i.e.

(2.8) \begin{equation} \frac {\partial \boldsymbol{A}}{\partial t}={\boldsymbol{u}}\times {\boldsymbol{B}}-\eta \boldsymbol{J}. \end{equation}

As before, the permeability is set to unity, so $\boldsymbol{J}={\boldsymbol{\nabla }}\times {\boldsymbol{B}}$ is the current density.

We use the Pencil Code (Pencil Code Collaboration et al. 2021), which is well suited for our MHD simulations. It uses sixth-order accurate spatial discretisations and a third-order time-stepping scheme. We adopt periodic boundary conditions in all three directions, so the mass is conserved and the mean density $\langle \rho \rangle \equiv \rho _0$ is constant. The size of the domain is $L_\perp \times L_\perp \times L_\|$ and the lowest wavenumber in the plane is $k_1=2\pi /L_\perp$ . By default, we choose $\rho _0=k_1={c_{\mathrm{s}}}=\mu _0=1$ , which fixes all dimensions in the code.

2.4. Input and output parameters

In the following, we study cases with different values of $k_0$ . We specify the amplitude of the vector potential to be $A_0=0.02$ for most of the runs with Roberts field I and $A_0=0.05$ for Roberts field II. We use $k_0=16$ , so $B_0=k_0A_0=0.32$ for field I and $0.8$ for field II. For other values of $k_0$ , we adjust $A_0$ such that $B_0$ is unchanged in all cases. This implies $\langle {\boldsymbol{B}}^2\rangle =2B_0^2=0.2$ and $1.28$ , and therefore $B_{\mathrm{rms}}=0.45$ and $1.13$ , respectively. The initial values of the Alfvén speed, $v_{\mathrm{A0}}=B_{\mathrm{rms}}/\sqrt {\mu _0\rho _0}$ , are therefore transonic. We often give the time in code units, $({c_{\mathrm{s}}} k_1)^{-1}$ , but sometimes we also give it in units of $(v_{\mathrm{A0}} k_0)^{-1}$ , which is physically more meaningful. However, we must remember that the actual magnetic field and therefore the actual Alfvén speed are of course decaying.

In addition to the Roberts field, we add to the initial condition Gaussian-distributed noise of a relative amplitude of $10^{-6}$ . This allows us to study the stability of the field to small perturbations. To measure the growth rate, we compute the semilogarithmic derivative of $\langle \boldsymbol{J}_{\perp \perp }^2\rangle /\langle \boldsymbol{J}^2\rangle$ for a suitable time interval.

The number of eddies in the plane is characterised by the ratio $k_0/k_1$ . The aspect ratio of the domain is quantified by $L_\|/L_\perp$ . The electric conductivity is quantified by the Lundquist number ${\textrm{Lu}}=v_{\mathrm{A0}}/\eta k_0$ , and the kinematic viscosity is related to $\eta$ through the magnetic Prandtl number, ${\textrm{Pr}_{\mathrm{M}}}=\nu /\eta$ . In all our cases, we take ${\textrm{Pr}_{\mathrm{M}}}=5$ . This is an arbitrary choice, just like ${\textrm{Pr}_{\mathrm{M}}}=1$ would be arbitrary. The value of $\textrm{Pr}_{\mathrm{M}}$ affects the ratio of kinetic to magnetic energy dissipation (Brandenburg Reference Brandenburg2014; Brandenburg & Rempel Reference Brandenburg and Rempel2019). While this topic is interesting and important, it is not the focus of our present study. Laboratory plasmas tend to have large values of $\textrm{Pr}_{\mathrm{M}}$ , so the choice ${\textrm{Pr}_{\mathrm{M}}}=5$ instead of unity is at least qualitatively appropriate. Much larger values of $\textrm{Pr}_{\mathrm{M}}$ would become computationally prohibitive. Furthermore, the choice ${\textrm{Pr}_{\mathrm{M}}}=1$ can lead to exceptional behaviour, particularly when the cross-helicity is finite; see figure 1 of Rädler & Brandenburg (Reference Rädler and Brandenburg2010).

Figure 1. Evolution of $\langle \boldsymbol{J}_{\perp \perp }^2\rangle /\langle \boldsymbol{J}^2\rangle$ for (a) Roberts field I with $k_0=4$ (blue), $8$ (green), $16$ (orange), $32$ (red) and $64$ (black dashed), and for (b) Roberts field II with $k_0=2$ (black), $4$ (blue), $8$ (green), $16$ (orange), $32$ (red) and $64$ (black dashed). The short thick line on the upper right indicates the value of 4/15, which is reached only at much later times outside this plot. The insets demonstrate that $\langle \boldsymbol{J}_{\perp \perp }^2\rangle /\langle \boldsymbol{J}^2\rangle \to 4/15$ much later.

Important output parameters are the growth rate $\lambda ={\textrm{d} {}}\ln (\langle \boldsymbol{J}_{\perp \perp }^2\rangle /\langle \boldsymbol{J}^2\rangle )/{\textrm{d} {}} t$ , evaluated in the regime where it is non-vanishing and approximately constant. It is made non-dimensional through the combination $\lambda /v_{\mathrm{A0}} k_0$ . We also present magnetic energy and magnetic helicity variance spectra, $\textrm{Sp}({\boldsymbol{B}})$ and $\textrm{Sp}(h)$ , respectively. These spectra depend on $k$ and $t$ , so we denote the spectra sometimes also as $\textrm{Sp}({\boldsymbol{B}};k,t)$ and $\textrm{Sp}(h;k,t)$ , respectively.

Since $\rho \approx \rho _0=1$ , the value of $B_{\mathrm{rms}}$ is also equal to the instantaneous Alfvén speed, $v_{\mathrm{A}}$ , and its square is the mean magnetic energy density, $\mathcal{E}_{\mathrm{M}}=\langle {\boldsymbol{B}}^2\rangle /2$ . The latter can also be computed from the magnetic energy spectrum ${E_{\mathrm{M}}}(k,t)=\textrm{Sp}({\boldsymbol{B}})$ through $\mathcal{E}_{\mathrm{M}}=\int\!\!{E_{\textrm{M}}}(k,t)\,{\textrm{d} {}} k$ . The integral scale of the magnetic field is given by

(2.9) \begin{equation} \xi _{\mathrm{M}}(t)=\int k^{-1}{E_{\mathrm{M}}}(k,t)\,{\textrm{d} {}} k/\mathcal{E}_{\mathrm{M}}. \end{equation}

It is of interest to compare its evolution with the magnetic Taylor microscale, $\xi _{\mathrm{T}}=B_{\mathrm{rms}}/J_{\mathrm{rms}}$ , where $J_{\mathrm{rms}}$ is the root-mean-squared current density, i.e. $({\boldsymbol{\nabla }}\times {\boldsymbol{B}})_{\mathrm{rms}}$ . (We recall that the permeability was set to unity; otherwise, there would have been an extra $\mu _0$ factor in front of $J_{\mathrm{rms}}$ .) Both in experiments and in simulations, $\xi _{\mathrm{T}}$ may be more easily accessible than $\xi _{\mathrm{M}}$ , so it is important to find out whether the two obey similar scaling relations.

During the decay, $\mathcal{E}_{\mathrm{M}}=v_{\mathrm{A}}^2/2$ decreases and $\xi _{\mathrm{M}}$ increases. The Alfvén time, i.e. the ratio $\tau _{\mathrm{A}}\equiv \xi _{\mathrm{M}}/v_{\mathrm{A}}$ , therefore also increases; see Banerjee & Jedamzik (Reference Banerjee and Jedamzik2004) and Hosking & Schekochihin (Reference Hosking and Schekochihin2023a) for early considerations of this point. Both for standard (isotropic) helical decay with $v_{\mathrm{A}}\propto t^{-1/3}$ and $\xi _{\mathrm{M}}\propto t^{2/3}$ , as well as for non-helical decay with $v_{\mathrm{A}}\propto t^{-5/9}$ and $\xi _{\mathrm{M}}\propto t^{4/9}$ , the value of $\tau _{\mathrm{A}}$ increases linearly with $t$ , i.e.

(2.10) \begin{equation} t\propto \tau _{\mathrm{A}}(t). \end{equation}

This is also consistent with the idea that the turbulent decay is self-similar (Brandenburg & Kahniashvili Reference Brandenburg and Kahniashvili2017). It was found that the ratio $t/\tau _{\mathrm{A}}(t)$ approaches a constant that increases with the Lundquist number (Brandenburg et al. Reference Brandenburg, Neronov and Vazza2024). The difference between the quantity $t/\tau _{\mathrm{A}}(t)$ and the factor $C_{\mathrm{M}}$ defined by Brandenburg et al. Reference Brandenburg, Neronov and Vazza(2024) is the exponent $p=10/9$ in the relation $\mathcal{E}_{\mathrm{M}}\propto t^{-p}$ for non-helical and $p=2/3$ for helical turbulence with $t/\tau _{\mathrm{A}}=C_{\mathrm{M}}/p$ .

To compute the Hosking integral, we need the function $\mathcal{I}_{\mathrm{H}}(R,t)$ , which is a weighted integral over $\textrm{Sp}(h)$ , given by

(2.11) \begin{align} \mathcal{I}_{\mathrm{H}}(R,t)&=\int _0^\infty w(k,R)\, \textrm{Sp}(h;k,t) \, \mathrm{d}k, \quad \textrm{where} \; w(k,R)=\frac {4\pi R^3}{3} \left [\frac {6j_1(kR)}{kR}\right ]^2,\end{align}

and $j_1(x)=(\sin x-x\cos x)/x^2$ is the spherical Bessel function of order one. As shown by Zhou et al. (Reference Zhou, Sharma and Brandenburg2022), the function $\mathcal{I}_{\mathrm{H}}(R,t)$ yields the Hosking integral in the limit of large radii $R$ , although $R$ must still be small compared with the size of the domain. They referred to this as the box-counting method for a spherical volume with radius $R$ .

4.1. Evolution in the diagnostic diagram

In view of the cosmological applications of decaying MHD turbulence, it is of interest to consider the evolution of the actual Alfvén speed $v_{\mathrm{A}}(t)=\sqrt {2\mathcal{E}_{\mathrm{M}}/\rho }$ in an evolutionary diagram as a parametric representation versus $\xi _{\mathrm{M}}(t)$ ; see figure 7(a). With $v_{\mathrm{A}}\propto t^{-p/2}$ and $\xi _{\mathrm{M}}\propto t^{q}$ , we expect that $v_{\mathrm{A}}\propto \xi _{\mathrm{M}}^{-\kappa }$ with $\kappa =p/2q=1/2$ for the fully helical case of Roberts field I. This is in agreement with early work showing that $v_{\mathrm{A}}\propto t^{1/3}$ and $\xi _{\mathrm{M}}\propto t^{2/3}$ (Hatori Reference Hatori1984; Biskamp & Müller Reference Biskamp and Müller1999).

Figure 7. (a) Parametric representation of $v_{\mathrm{A}}$ versus $\xi _{\mathrm{M}}$ for Roberts fields I (red) and II (blue). The solid (dotted) curves are for $\eta =2\times 10^{-7}$ ( $\eta =4\times 10^{-6}$ ). Note that the red dotted line for $\eta =4\times 10^{-6}$ starts at the same value $v_{\mathrm{A}}=\sqrt {1.28}$ as the non-helical runs (blue lines). The similarity between the dotted and solid red lines shows that the initial amplitude does not matter much. The open (filled) symbols indicate the times $t=10$ ( $t=100$ ). The dash-dotted lines give the slopes $\kappa =1/2$ and 5/4 for Roberts fields I (red) and II (blue), respectively. (b) $pq$ diagram field fields I (red) and II (blue) with $\eta =2\times 10^{-7}$ . Larger symbols indicate later times.

In figure 7(a), we have also marked the times $t=10$ (open symbols) and $t=100$ (filled symbols). The points of constant times depart significantly from the lines of constant Alfvén time, $\tau _{\mathrm{A}}$ , for which $v_{\mathrm{A}}=\xi _{\mathrm{M}}/\tau _{\mathrm{A}}$ grows linearly with $\xi _{\mathrm{M}}$ . We expect the times to be larger by a factor $C_{\mathrm{M}}$ than the corresponding values of $\tau _{\mathrm{A}}(t)$ . This is indeed the case: the point $t=100$ lies on the line $\tau _{\mathrm{A}}=1$ , i.e. $t/\tau _{\mathrm{A}}=100$ . This is twice as much as our nominal value of approximately 50.

There is an interesting difference between the cases of Roberts fields I and II in that for field II, there is an extended period during which $\xi _{\mathrm{M}}$ shows a rapid decrease before the expected increase emerges. The fact that such an initial decrease of the characteristic length scale is not seen for Roberts field I is remarkable. The rapid development of smaller length scales is probably related to the breakup of the initially organised tube-like structures into smaller scales. In the helical case, however, the nonlinear interaction among helical modes can only result in the production of modes with smaller wavenumbers, i.e. larger length scales; see Frisch et al. Reference Frisch, Pouquet, Leorat and Mazure(1975) and Brandenburg & Subramanian (Reference Brandenburg and Subramanian2005) for a review. Such a constraint does not exist for the non-helical modes, where this can then reduce the effective starting values of $\xi _{\mathrm{M}}$ and therefore also of the effective Alfvén time, $\tau _{\mathrm{A}}=\xi _{\mathrm{M}}/v_{\mathrm{A}}$ , early in the evolution. In Appendix B, we present similar diagrams for different values of $k_0$ , but with a drag term included that could be motivated by cosmological applications.

We inspect the time-dependences of $t/\tau _{\mathrm{A}}=v_{\mathrm{A}} t/\xi _{\mathrm{M}}$ and ${\textrm{Lu}}=v_{\mathrm{A}}\xi _{\mathrm{M}}/\eta$ for Roberts fields I and II in figure 8. We see that $t/\tau _{\mathrm{A}}(t)$ reaches values in excess of 100 for $t=100$ in both cases. This is more than what has been seen before, but it also shows significant temporal variations.

Figure 8. (a) $t/\tau _{\textrm{A}}$ and (b) $\textrm{Lu}$ versus time for Roberts fields I (red) and II (blue).

4.2. Universality of prefactors in the decay laws?

The decay of a turbulent magnetic field is constrained by certain conservation laws: the conservation of mean magnetic helicity density $I_{\mathrm{M}}=\langle h\rangle$ , where $h=\boldsymbol{A}\boldsymbol{\cdot} {\boldsymbol{B}}$ is the local magnetic helicity density, and the Hosking integral, $I_{\mathrm{H}}=\int h(\boldsymbol{x})h(\boldsymbol{x}+{\boldsymbol{r}})\,{\textrm{d} {}}^3r$ . When the magnetic field is fully helical, the decay is governed by the conservation of $I_{\mathrm{M}}$ , and when it is non-helical, it is governed by the conservation of $I_{\mathrm{H}}$ . The time of cross-over depends on the ratio $t_\ast \equiv I_{\mathrm{H}}^{1/2}/I_{\mathrm{M}}^{3/2}$ (Brandenburg & Banerjee Reference Brandenburg and Banerjee2025). Specifically, the correlation length $\xi _{\mathrm{M}}(t)$ , the mean magnetic energy density $\mathcal{E}_{\mathrm{M}}(t)$ and the envelope of the peaks of the magnetic energy spectrum ${E_{\mathrm{M}}}(k,t)$ depend on the values of the conserved quantities with (Brandenburg & Larsson Reference Brandenburg and Larsson2023)

(4.1) \begin{equation} \xi _{\mathrm{M}}(t)=C_i^{(\xi )} I_i^\sigma t^q,\quad \mathcal{E}_{\mathrm{M}}(t)=C_i^{(\mathcal{E})} I_i^{2\sigma } t^{-p},\quad {E_{\mathrm{M}}}(k)\leqslant C_i^{(E)} I_i^{(3+\beta )\,q} k^\beta , \end{equation}

where $\sigma$ is the exponent with which the length enters in $I_{i}$ : $\sigma =1/3$ when the mean magnetic helicity density governs the decay ( $i=\textrm{M}$ ) and $\sigma =1/9$ for the Hosking integral ( $i=\textrm{H}$ ). In figure 9, we show the appropriately compensated evolutions of $\xi _{\mathrm{M}}$ and $\mathcal{E}_{\mathrm{M}}$ such that we can read off the values of $C_i^{(\xi )}$ and $C_i^{(\mathcal{E})}$ for the helical and non-helical cases.

Figure 9. Compensated evolutions of $\xi _{\mathrm{M}}$ and $\mathcal{E}_{\mathrm{M}}$ allowing the non-dimensional prefactors in (4.1) to be estimated.

In table 2, we summarise the values for the six coefficients reported previously in the literature and compare with those determined here. The fact that the coefficients are now somewhat different under the present circumstances suggests that they might not be universal, although the anisotropy of the present set-up as well as the limited scale separation may have contributed to the new results. For the purpose of providing relevant information for future studies of anisotropic magnetic decay, we present in Appendix C the temporal evolution of the length scales and field strengths in the parallel and perpendicular directions.

Table 2. Comparison of the dimensionless prefactors with values in earlier papers.

The question of universality is significant, however, because universality would mean that the decay laws of the form (e.g. Vachaspati Reference Vachaspati2021)

(4.2) \begin{equation} \xi _{\mathrm{M}}(t)=\xi _{\mathrm{M}}(t_0)\,\left (t/t_0\right )^q,\quad \mathcal{E}_{\mathrm{M}}(t)=\mathcal{E}_{\mathrm{M}}(t_0)\,\left (t/t_0\right )^{-p} \end{equation}

could be misleading in that they suggest some freedom in the choice of the values of $\xi _{\mathrm{M}}(t_0)$ and $\mathcal{E}_{\mathrm{M}}(t_0)$ at the time $t_0$ . Comparing with (4.1), we see that

(4.3) \begin{equation} \xi _{\mathrm{M}}(t_0)/t_0^q=C_i^{(\xi )} I_i^\sigma \quad \textrm{and}\quad \mathcal{E}_{\mathrm{M}}(t_0)\,t_0^p=C_i^{(\mathcal{E})} I_i^{2\sigma }, \end{equation}

so they cannot be chosen arbitrarily, but they must obey a constraint that depends on the relevant conservation law.