2.1. Far-field velocity due to the non-local pressure force

The MHD momentum equation for an incompressible fluid is

(2.1) \begin{align} \frac {\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u}\boldsymbol{\cdot } \boldsymbol{\nabla } \boldsymbol{u} = - \boldsymbol{\nabla } P + \boldsymbol{B} \boldsymbol{\cdot } \boldsymbol{\nabla } \boldsymbol{B}+\nu {\nabla} ^2 \boldsymbol{u}, \end{align}

where $\boldsymbol{B}$ is the magnetic field measured in units of the Alfvén speed, $P = (p+B^2/2)/\rho _0$ the total pressure scaled by the constant density $\rho _0$ , $p$ the thermal pressure and $B=|\boldsymbol{B}|$ . The pressure-gradient force is determined non-locally from the condition $\boldsymbol{\nabla } \boldsymbol{\cdot } \boldsymbol{u} = 0$ , which requires

(2.2) \begin{align} {\nabla} ^2 P = -\boldsymbol{\nabla } \boldsymbol{\cdot } \left [(\boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }) \boldsymbol{u} - (\boldsymbol{B} \boldsymbol{\cdot } \boldsymbol{\nabla }) \boldsymbol{B} \right ]\!. \end{align}

The Green’s function solution of (2.2) is

(2.3) \begin{align} P(\boldsymbol{x}) = \frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime }}{| \boldsymbol{x}^{\prime } - \boldsymbol{x} |}\frac {\partial }{\partial x_i^{\prime }}\frac {\partial }{\partial x_j^{\prime }} \left [ u_i(\boldsymbol{x}^{\prime } )u_j(\boldsymbol{x}^{\prime }) - B_i(\boldsymbol{x}^{\prime }) B_j (\boldsymbol{x}^{\prime })\right ]\!, \end{align}

which, after employing the expansion

(2.4) \begin{align} \frac {1}{| \boldsymbol{x}^{\prime } - \boldsymbol{x} |} = \frac {1}{x} - x_i^\prime \frac {\partial }{\partial x_i} \left (\frac {1}{x}\right ) + \frac {1}{2} x_i^\prime x_j^\prime \frac {\partial ^2}{\partial x_i \partial x_j} \left (\frac {1}{x}\right ) + \cdots \end{align}

to find the pressure field far from the origin, becomes

(2.5) \begin{align} P(\boldsymbol{x}) = \frac {1}{4\pi } \frac {\partial ^2}{\partial x_i \partial x_j}\biggl (\frac {1}{x}\biggl ) \int _{\mathbb{R}^3} \textrm{d}^3 \boldsymbol{x}^{\prime } \left [ u_i(\boldsymbol{x}^{\prime } )u_j(\boldsymbol{x}^{\prime }) - B_i(\boldsymbol{x}^{\prime }) B_j (\boldsymbol{x}^{\prime })\right ] + \mathcal{O}(x^{-4}) \end{align}

for $x=|\boldsymbol{x}|\to \infty$ (note that the first two terms in the Taylor expansion vanish after substitution). Thus, even though $\boldsymbol{u}$ vanishes outside the patch at the origin at $t=0$ , the pressure field (2.5) sources it throughout space for all $t\gt 0$ .

2.2. Far-field vector potential due to a non-local gauge

A second source of non-local interaction arises if $\zeta$ in (1.5) is a non-local function. In that case, the non-solenoidal part of $\boldsymbol{A}$ (i.e. the part that can be written as a gradient) may become correlated between distant points. Consider, for example, the class of gauges for which $\zeta$ is determined by the Poisson equation

(2.6) \begin{align} {\nabla} ^2 \zeta = \phi , \end{align}

where $\phi$ is a local function of $\boldsymbol{u}$ and $\boldsymbol{B}$ (i.e. depends on these fields and their spatial derivatives, but not their integrals). A prominent member of this class of gauges is the Coulomb gauge $\boldsymbol{\nabla } \boldsymbol{\cdot } \boldsymbol{A} = 0$ , for which $\phi = - \boldsymbol{\nabla } \boldsymbol{\cdot } (\boldsymbol{u} \times \boldsymbol{B})$ . Gauges that satisfy (2.6) are non-local because $\zeta$ is determined by an integral over space analogous to (2.3). The Green’s function solution of (2.6) is

(2.7) \begin{align} \zeta (\boldsymbol{x}) = -\frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime }}{| \boldsymbol{x}^{\prime } - \boldsymbol{x} |}\phi (\boldsymbol{x}'). \end{align}

Applying (2.7) for the compact patch of velocity and magnetic field considered in § 2.1, we find that, for all $t\gt 0$ , a finite non-solenoidal vector potential is generated throughout space. At large $x$ , the asymptotic form of $\zeta$ is

(2.8) \begin{align} \zeta (\boldsymbol{x}) = -\frac {1}{4\pi x} \int _{\mathbb{R}^3} \textrm{d}^3 \boldsymbol{x}^{\prime } \phi (\boldsymbol{x}^{\prime }) + \mathcal{O}(x^{-2}). \end{align}

We note that, if $\phi$ is a gradient, the leading-order term in (2.8) vanishes; for example, in the case of the Coulomb gauge, (2.8) becomes

(2.9) \begin{align} \zeta _{\mathrm{Coulomb}}(\boldsymbol{x}) = -\frac {1}{4\pi }\frac {\partial }{\partial \boldsymbol{x}}\left (\frac {1}{x}\right ) \boldsymbol{\cdot } \int _{\mathbb{R}^3} \textrm{d}^3 \boldsymbol{x}^{\prime }\, \boldsymbol{u}(\boldsymbol{x}^\prime ) \times \boldsymbol{B}(\boldsymbol{x}^\prime ) + \mathcal{O}(x^{-3}). \end{align}

Naturally, only the part of $\boldsymbol{A}$ that can be expressed as a gradient, which does not contribute to $\boldsymbol{B}$ , is affected by the non-locality (2.7) – gauge effects cannot influence physical quantities. Nonetheless, this part of $\boldsymbol{A}$ does contribute to the helicity density, so contributes to $C_{\infty }$ in homogeneous turbulence.

3.1. Batchelor & Proudman (Reference Batchelor and Proudman1956) theory of the large- $r$ asymptotics of correlation functions

The theory of the asymptotic tails of correlation functions for hydrodynamic turbulence is due to Batchelor & Proudman (Reference Batchelor and Proudman1956). We review this theory briefly in this section, before applying a similar methodology to determine $C_{\infty }$ in MHD turbulence in §§ 3.2 and 3.3.

Batchelor & Proudman (Reference Batchelor and Proudman1956) considered long-range correlations that arise dynamically owing to pressure-mediated interactions (§ 2.1) from an initial condition for which distant points are statistically independent. Stated precisely, at $t=0$ , all cumulants of the velocity field decay more quickly with separation than any power law.Footnote ⁴ To determine correlation functions at later times, Batchelor & Proudman (Reference Batchelor and Proudman1956) assumed that they could be written as convergent Taylor series in $t$ , with derivatives evaluated at $t=0$ . The terms in the Taylor series that decay most slowly with the separation $r$ give the large- $r$ asymptotic of the correlation function at $t\gt 0$ .

To illustrate the method, consider the velocity triple correlation $\langle u_i u_j u_k^\prime \rangle$ , where primed variables are evaluated at $\boldsymbol{x}^{\prime } = \boldsymbol{x} + \boldsymbol{r}$ and unprimed variables are evaluated at $\boldsymbol{x}$ . Its first time derivative involves the non-locally determined pressure:

(3.1) \begin{align} \frac {\partial }{\partial t}\langle u_i u_j u_k^{\prime} \rangle & = \cdots - \left\langle u_i u_j \frac {\partial P^\prime }{\partial x_k^\prime } \right\rangle + \cdots \end{align}

(3.2) \begin{align} & = \cdots - \frac {\partial }{\partial r_k}\left\langle u_i u_j \frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime \prime }}{| \boldsymbol{x}^{\prime \prime } - \boldsymbol{x} ^\prime |}\frac {\partial }{\partial x_l^{\prime \prime }}\frac {\partial }{\partial x_m^{\prime \prime }} u_l^{\prime \prime } u_m^{\prime \prime }\right\rangle + \cdots ,\end{align}

where we have used (2.3) (dropping the terms involving the magnetic field, which are not present in Batchelor & Proudman’s analysis, but behave in the same way as the velocity terms and do not change the result). We define $\boldsymbol{s} = \boldsymbol{x} - \boldsymbol{x}^{\prime \prime }$ , so that

(3.3) \begin{align} \frac {1}{|\boldsymbol{x}^{\prime \prime }-\boldsymbol{x}^{\prime }|}=\frac {1}{|\boldsymbol{r}+\boldsymbol{s}|}= \frac {1}{r}+s_i\frac {\partial }{\partial r_i}\frac {1}{r}+\frac {1}{2}s_i s_j \frac {\partial }{\partial r_i}\frac {\partial }{\partial r_j}\frac {1}{r}+\mathcal{O}\left (\frac {1}{r^4}\right )\!. \end{align}

Substituting (3.3) into (3.2) yields

(3.4) \begin{align} \frac {\partial }{\partial t}\langle u_i u_j u_k^{\prime} \rangle = \cdots - \frac {1}{4\pi } \frac {\partial ^3}{\partial r_k \partial r_l \partial r_m} \left (\frac {1}{r}\right )\int _{\mathbb{R}^3} \textrm{d}^3 \boldsymbol{s} \left (\langle u_i u_j u_l^{\prime \prime } u_m^{\prime \prime }\rangle - \langle u_i u_j \rangle \langle u_l^{\prime \prime } u_m^{\prime \prime } \rangle \right )\ldots , \end{align}

where, as in (2.5), the first two terms in the series (3.3) vanish. We now evaluate (3.4) at $t=0$ , at which time, the decay of all cumulants being faster than any power law, all their integral moments converge, so all the higher-order terms arising from the expansion (3.3) converge. No stronger dependence on $r$ exists in any term in the Taylor expansion in $t$ , and hence

(3.5) \begin{align} \langle u_i u_j u_k^{\prime} \rangle = \mathcal{O}\left (\frac {1}{r^4}\right )\!, \quad t\gt 0. \end{align}

We note that the $r^{-4}$ scaling of the velocity triple-correlation function (3.5) is responsible for the non-conservation of the Loitsyansky (Reference Loitsyansky1939) integral in decaying hydrodynamic turbulence – see Davidson (Reference Davidson2015) for a review.

3.2. The case of a local gauge function $\zeta$

We now apply a similar analysis to the one that led to (3.5) to the problem of determining $C_{\infty }$ (1.7). We take the statistical independence of distant points at $t=0$ to mean that all cumulants involving $\boldsymbol{u}$ , $\boldsymbol{A}$ and $\boldsymbol{B}$ decay with separation faster than any power law at $t=0$ . Under this assumption, we seek the large- $r$ tails of the correlation functions appearing in (1.7). We do not make any assumption about the relative sizes of the fields $\boldsymbol{u}$ and $\boldsymbol{B}$ (i.e. about the Alfvénic Mach number) in the initial state.Footnote ⁵ We first consider a local gauge – i.e. one for which $\zeta$ is a local function of $\boldsymbol{A}$ , $\boldsymbol{B}$ and $\boldsymbol{u}$ – and examine pressure-induced correlations (§ 2.1) only, returning to the issue of gauge-induced correlations (§ 2.2) in § 3.3. We find that all terms in the Taylor expansion in which the total pressure $P$ appears exactly once, which might lead to the correlation functions decaying as $r^{-4}$ , as in (3.1), vanish due to symmetry considerations. This means that the only surviving terms are ones in which $P$ appears more than once, which decay with separation as $r^{-8}$ or faster. Thus, $C_{\infty }=0$ . We first present a specific example of a contribution that vanishes, then give a general argument that all similar contributions also vanish.

3.2.1. Explicit example of a vanishing term

The two fifth-order correlation functions that appear in (1.7) for $C_{\infty }$ have general form $\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle$ , with contractions over different pairs of the free indices $ijk$ . Let us consider their Taylor expansion in time, evaluated at $t=0$ . We first present a specific example of a contribution to the Taylor series that vanishes, then give a general argument that all similar contributions (i.e. those with one appearance of $P$ ) also vanish.

Differentiating $\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle$ twice with respect to time yields terms involving the non-locally determined pressure:

(3.6) \begin{align} & \frac {\partial ^2 }{\partial t^2}\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle \nonumber \\ & \quad =\cdots - \bigg \langle \frac {\partial u_m}{\partial t} \frac {\partial u_i}{\partial x_m}A_jB_kA_l^{\prime}B_l^{\prime}\bigg \rangle \cdots \nonumber \\ & \quad = \cdots +\bigg \langle \left (\frac {\partial }{\partial x_m}\frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime \prime }}{| \boldsymbol{x}^{\prime \prime } - \boldsymbol{x} |}\frac {\partial }{\partial x_r^{\prime \prime }}\frac {\partial }{\partial x_s^{\prime \prime }} \left ( u_r^{\prime \prime }u_s^{\prime \prime } - B_r^{\prime \prime } B_s^{\prime \prime }\right )\right ) \frac {\partial u_i}{\partial x_m} A_j B_k A^{\prime }_l B^\prime _l\bigg \rangle \cdots \nonumber \\ & \quad = \cdots +\frac {\partial }{\partial r_m}\frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime \prime }}{| \boldsymbol{x}^{\prime \prime } - \boldsymbol{x} |}\frac {\partial }{\partial x_r^{\prime \prime }}\frac {\partial }{\partial x_s^{\prime \prime }}\bigg \langle B_r^{\prime \prime } B_s^{\prime \prime } \frac {\partial u_i}{\partial x_m} A_j B_k A^{\prime }_l B^\prime _l\bigg \rangle \nonumber \\ &\qquad \,\,\,\quad + \frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime \prime }}{| \boldsymbol{x}^{\prime \prime } - \boldsymbol{x} |}\frac {\partial }{\partial x_r^{\prime \prime }}\frac {\partial }{\partial x_s^{\prime \prime }} \bigg \langle B_r^{\prime \prime } B_s^{\prime \prime } \frac {\partial }{\partial x_m}\left ( \frac {\partial u_i}{\partial x_m} A_j B_k\right ) A^{\prime }_l B^\prime _l\bigg \rangle +\cdots , \end{align}

where in the second equality we have substituted (2.3), and in the third equality we have restricted attention to the terms inside the integral that involve $\boldsymbol{B}$ , the ones that involve $\boldsymbol{u}$ being analogous.

We now evaluate (3.6) at the initial time in the limit of large $r=|\boldsymbol{x}^\prime -\boldsymbol{x}|$ , seeking a result analogous to (3.4). Unlike in (3.2), the correlation functions in the final line of (3.6) contain fields evaluated at three different points, i.e. $\boldsymbol{x}''$ , $\boldsymbol{x}'$ and $\boldsymbol{x}$ . Let us define $\boldsymbol{s}=\boldsymbol{x}''-\boldsymbol{x}$ and $\boldsymbol{s}'=\boldsymbol{x}''-\boldsymbol{x}'$ to be the two independent displacement vectors between these points. Because $\boldsymbol{r} = \boldsymbol{x}^\prime - \boldsymbol{x} = \boldsymbol{s} - \boldsymbol{s}^\prime$ , it must be the case that $|\boldsymbol{s}|\sim r$ or $ |\boldsymbol{s}'|\sim r$ (or both) as $r\to \infty$ . In the former case, our assumption about the vanishing of (second-order) cumulants (see start of § 3.2) means that correlation functions of the form $\langle X Y^\prime Z^{\prime \prime }\rangle \to \langle X \rangle \langle Y^\prime Z^{\prime \prime }\rangle + o(r^{-n})$ for all $n\gt 0$ as $r\to \infty$ , while in the latter case, $\langle X Y^\prime Z^{\prime \prime }\rangle \to \langle Y^\prime \rangle \langle X Z^{\prime \prime }\rangle + o(r^{-n})$ . Applied to the first of the two integrals appearing after the final equality in (3.6), these formulae become

(3.7) \begin{align} \bigg \langle B_r^{\prime \prime } B_s^{\prime \prime } \frac {\partial u_i}{\partial x_m} A_j B_k A^{\prime }_l B^\prime _l\bigg \rangle \to \begin{cases} \displaystyle \bigg \langle B_r^{\prime \prime } B_s^{\prime \prime } \frac {\partial u_i}{\partial x_m} A_j B_k \bigg \rangle \langle A^{\prime }_l B^\prime _l\rangle + o(r^{-n})& \text{if }|\boldsymbol{s}'|\sim r, \\[10pt] \displaystyle \bigg \langle \frac {\partial u_i}{\partial x_m} A_j B_k \bigg \rangle \langle B_r^{\prime \prime } B_s^{\prime \prime } A^{\prime }_l B^\prime _l\rangle + o(r^{-n}) & \text{if } |\boldsymbol{s}|\sim r, \end{cases} \end{align}

for all $n\gt 0$ as $r\to \infty$ . In both cases, the leading-order term vanishes. In the first case, this is because $\langle A_l^{\prime} B_l^{\prime}\rangle = 0$ . In the second case, the correlation function $\langle B_r^{\prime \prime } B_s^{\prime \prime } A^{\prime }_l B^\prime _l\rangle$ must change sign under reflection, because $\boldsymbol{B}$ is an axial (pseudo-)vector and $\boldsymbol{A}$ is a polar (true) vector.Footnote ⁶ In reflection-symmetric turbulence, the most general decomposition of this correlation function consistent with this fact is (see e.g. Robertson Reference Robertson1940 or Batchelor Reference Batchelor1953)

(3.8) \begin{align} \langle B_r^{\prime \prime } B_s^{\prime \prime } A^{\prime }_l B^\prime _l\rangle = a(|\boldsymbol{s}^\prime |) \epsilon _{rsn} s_n^\prime , \end{align}

for which $a(|\boldsymbol{s}^\prime |)$ must be zero given that the left-hand side is symmetric under exchanging the indices $r$ and $s$ .

In the equivalent expression to (3.7) for the second of the two integrals appearing after the final equality in (3.6), both cases are zero: $\langle A_l^{\prime} B_l^{\prime}\rangle = 0$ by assumption and $\langle \partial _m[(\partial _m u_i)A_jB_k]\rangle = 0$ because $\langle \partial _m (\ldots )\rangle = 0$ for homogeneous turbulence.

3.2.2. Generalisation to other terms in the Taylor expansion

Let us now consider whether there are any terms in the Taylor expansion of $\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle$ that do not vanish after contraction of $i$ with $j$ or of $j$ with $k$ , as in $C_{\infty }$ (1.7). Consider terms in which the total pressure $P$ appears once. Such terms have the general form

(3.9) \begin{align} C_{i_1 i_2 \ldots i_n} \frac {\partial ^{m}}{\partial r_{j_1} \partial r_{j_2} \ldots \partial r_{j_m}} \left (\frac {1}{r}\right )\int _{\mathbb{R}^3} \textrm{d}^3 \boldsymbol{s} \, D_{k_1 k_2 \ldots k_p}(\boldsymbol{s}), \end{align}

where $C_{i_1 i_2 \ldots i_n}$ and $D_{k_1 k_2 \ldots k_m}$ are pseudo-tensors. Since the quantity (3.9) must be a vector, all but one of its indices are contracted. However, none of the indices $i_1 \ldots i_n$ can be contracted with the indices $k_1 \ldots k_m$ , because independent fields (for which the correlation functions can be split) cannot appear in contraction in the Taylor expansion. Thus, all of these indices, with the possible exception of one (the single free index), must be contracted with the indices $j_1 \ldots j_m$ . However, the indices $j_1 \ldots j_m$ are symmetric under interchange, while $C_{i_1 i_2 \ldots i_n}$ and $D_{k_1 k_2 \ldots k_m}$ are each antisymmetric under the exchange of at least two of their indices, by virtue of being pseudo-tensors. It follows that all terms having the form (3.9) – i.e. having one appearance of pressure – are zero.

We conclude that the only terms in the Taylor expansion that are non-vanishing are those for which pressure appears twice. Because these involve two appearances of $\partial _i P$ , they decay as $r^{-8}$ or faster, so we have that

(3.10) \begin{align} \langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle = \mathcal{O}\left (\frac {1}{r^8}\right )\!. \end{align}

Precisely analogous arguments to those just given imply that $\langle \zeta B_iA_l^{\prime}B_l^{\prime}\rangle = \mathcal{O}(r^{-8})$ .Footnote ⁷

Because the tails of all the correlation functions that appear in (1.7) decay faster than $r^{-2}$ as $r\to \infty$ , we conclude that $C_{\infty }=0$ if $\zeta$ is a local function.

3.3. The case of non-local $\zeta$

We turn now to long-range interactions arising from the use of a non-local gauge function $\zeta$ , which can cause the non-solenoidal part of $\boldsymbol{A}$ to become correlated between distant points, as explained in § 2.

3.3.1. Poisson-type non-locality

We prove in this section that sufficiently strong correlations for $C_{\infty }\neq 0$ never arise for any gauge for which $\zeta$ is determined by Poisson’s equation (2.6), with the source $\phi$ a local function of $\boldsymbol{u}$ and $\boldsymbol{B}$ . For such gauges, we can evaluate all the relevant contributions to $C_{\infty }$ explicitly: because $\zeta$ only appears in terms for which $\boldsymbol{A}$ is differentiated with respect to time, the number of such terms is limited. We address them in turn below.

As we utilised in § 3.2, the first two correlation functions in (1.7) have general form $\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle$ . Differentiating this once with respect to time and isolating the gauge terms, we have

(3.11) \begin{align} \frac {\partial }{\partial t}\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle &= \cdots \bigg \langle u_i \frac {\partial \zeta }{\partial x_j} B_kA_l^{\prime}B_l^{\prime}\bigg \rangle + \frac {\partial }{\partial r_l}\langle u_iA_jB_k \zeta ' B_l^{\prime}\rangle + \cdots \nonumber \\ & = \cdots \frac {\partial }{\partial r_j}\frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3\boldsymbol{x}''}{|\boldsymbol{x}'' - \boldsymbol{x}|} \langle u_i \phi ''B_kA_l^{\prime}B_l^{\prime}\rangle \nonumber \\ & \phantom {= \cdots } + \frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3\boldsymbol{x}''}{|\boldsymbol{x}'' - \boldsymbol{x}|} \bigg \langle \frac {\partial }{\partial x_j}(u_iB_k)\phi ''A_l^{\prime}B_l^{\prime}\bigg \rangle \nonumber \\ & \phantom {= \cdots } - \frac {\partial }{\partial r_l}\,\frac {1}{4\pi } \int _{\mathbb{R}^3} \frac {\textrm{d}^3\boldsymbol{x}''}{|\boldsymbol{x}'' - \boldsymbol{x}'|} \langle u_iA_jB_k \phi '' B_l^{\prime}\rangle + \cdots .\end{align}

Taking the limit $r=|\boldsymbol{x}' - \boldsymbol{x}|\to \infty$ , the correlation functions appearing inside the three integrals in (3.11) split, as in (3.7). In the first integral,

(3.12) \begin{align} \langle u_i \phi ''B_kA_l^{\prime}B_l^{\prime}\rangle \to \begin{cases} \langle u_i \phi ''B_k\rangle \langle A_l^{\prime}B_l^{\prime}\rangle + o(r^{-n}) & \text{if }|\boldsymbol{s}'|\sim r, \\ \langle u_i B_k \rangle \langle \phi '' A_l^{\prime}B_l^{\prime}\rangle + o(r^{-n}) & \text{if }|\boldsymbol{s}|\sim r. \end{cases} \end{align}

In the second,

(3.13) \begin{align} \langle \partial _j(u_iB_k)\phi ''A_l^{\prime}B_l^{\prime}\rangle \to \begin{cases} \langle \partial _j(u_iB_k)\phi ''\rangle \langle A_l^{\prime}B_l^{\prime}\rangle + o(r^{-n}) & \text{if }|\boldsymbol{s}'|\sim r, \\ \langle \partial _j(u_iB_k)\rangle \langle \phi ''A_l^{\prime}B_l^{\prime}\rangle + o(r^{-n}) & \text{if }|\boldsymbol{s}|\sim r. \end{cases} \end{align}

In the third,

(3.14) \begin{align} \langle u_iA_jB_k \phi '' B_l^{\prime}\rangle \to \begin{cases} \langle u_iA_jB_k \phi '' \rangle \langle B_l^{\prime}\rangle + o(r^{-n}) & \text{if }|\boldsymbol{s}'|\sim r,\\ \langle u_iA_jB_k \rangle \langle \phi '' B_l^{\prime}\rangle + o(r^{-n}) & \text{if }|\boldsymbol{s}|\sim r. \end{cases} \end{align}

In all of these cases, the product of the split correlation function is zero, as follows. In (3.12) and (3.13), the cases with $|\boldsymbol{s}'|\sim r$ vanish because $\langle A_l B_l\rangle = 0$ in reflection-symmetric turbulence. In (3.14), the case with $|\boldsymbol{s}'|\sim r$ finite vanishes because $\langle B_l\rangle = 0$ , by isotropy. Of the cases for which $|\boldsymbol{s}|\sim r$ : in (3.12), $\langle u_i B_k \rangle = \delta _{ij}\langle \boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{B}\rangle /3=0$ in reflection-symmetric turbulence; in (3.13), $\langle \partial _j(u_iB_k)\rangle =0$ in homogeneous turbulence; and in (3.14), $\langle \phi '' B_l^{\prime}\rangle = 0$ in homogeneous turbulence because $B_l$ is solenoidal. It follows that there is no contribution to $\partial _t\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle$ from the gauge at $t=0$ .

Let us now consider the second time derivative of $\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle$ . Its only non-trivial term that is qualitatively distinct from those appearing in (3.11) is that for which the two appearances of the vector potential are each differentiated with respect to time:

(3.15) \begin{align} \frac {\partial ^2}{\partial t^2 }\langle u_iA_jB_kA_l^{\prime}B_l^{\prime}\rangle &= \cdots \frac {\partial }{\partial r_l}\left\langle u_i \frac {\partial \zeta }{\partial x_j} B_k \zeta ' B_l^{\prime}\right\rangle + \cdots \nonumber \\& = \cdots - \frac {\partial ^2}{\partial r_l r_j}\frac {1}{(4\pi )^2} \int _{\mathbb{R}^3} \frac {{d}^3\boldsymbol{x}''}{|\boldsymbol{x}'' - \boldsymbol{x}|}\int _{\mathbb{R}^3} \frac {{d}^3\boldsymbol{x}'''}{|\boldsymbol{x}''' - \boldsymbol{x}'|} \langle u_i \phi ''B_k\phi '''B_l^{\prime}\rangle \nonumber \\ & \phantom {= \cdots } - \frac {\partial }{\partial r_l}\frac {1}{(4\pi )^2} \int _{\mathbb{R}^3} \frac {{d}^3\boldsymbol{x}''}{|\boldsymbol{x}'' - \boldsymbol{x}|}\int _{\mathbb{R}^3} \frac {{d}^3\boldsymbol{x}'''}{|\boldsymbol{x}''' - \boldsymbol{x}'|} \bigg \langle \frac {\partial }{\partial x_j}(u_iB_k)\phi ''\phi '''B_l^{\prime}\bigg \rangle \nonumber \\ & \phantom {= \cdots } +\cdots . \end{align}

The correlation functions appearing in the final line of (3.15) involve fields evaluated at four points, $\boldsymbol{x}$ , $\boldsymbol{x}^\prime$ , $\boldsymbol{x}^{\prime \prime }$ and $\boldsymbol{x}^{\prime \prime \prime }$ . In the limit $r\to \infty$ , there are four qualitatively distinct possibilities for how these points can be arranged. The first is that all of their separations are ${\sim}r$ , in which case the correlation functions vanish with $r$ faster than any power law at $t=0$ . The remaining three cases involve some of the separations being held finite as the limit is taken. As concerns the correlation function in the first integral, the possibilities are

(3.16) \begin{align} \langle u_i \phi ''B_k\phi '''B_l^{\prime}\rangle \to \begin{cases} \langle u_i \phi ''B_k\phi '''\rangle \langle B_l^{\prime}\rangle + o(r^{-n}) & \text{for } |\boldsymbol{x}^{\prime \prime }-\boldsymbol{x}|, |\boldsymbol{x}^{\prime \prime \prime }-\boldsymbol{x}| \text{ finite,}\\ \langle u_i B_k\phi '''\rangle \langle \phi ''B_l^{\prime}\rangle + o(r^{-n}) & \text{for } |\boldsymbol{x}^{\prime \prime }-\boldsymbol{x}^{\prime }|, |\boldsymbol{x}^{\prime \prime \prime }-\boldsymbol{x}| \text{ finite,}\\ \langle u_i B_k\rangle \langle \phi ''\phi '''B_l^{\prime}\rangle + o(r^{-n}) & \text{for } |\boldsymbol{x}^{\prime \prime }-\boldsymbol{x}^{\prime }|, |\boldsymbol{x}^{\prime \prime \prime }-\boldsymbol{x}^\prime | \text{ finite.} \end{cases} \end{align}

Because $\langle B_l^{\prime}\rangle$ , $\langle \phi '' B_l^{\prime}\rangle$ and $ \langle u_i B_k \rangle$ are all zero, each of the cases in (3.16) are zero. As concerns the second line of (3.15), it is readily verified that replacing $u_i B_k$ in (3.16) by $\partial _j(u_i B_k)$ yields terms that still vanish. We conclude that the terms presented explicitly in (3.15) vanish. By precisely analogous reasoning, there are also no terms in the Taylor expansion in time of $\langle \zeta u_iA_l^{\prime}B_l^{\prime}\rangle$ that have power-law tails in $r$ induced by substituting (2.7).

In the above analysis, we have not included terms for which pressure and gauge enter together – such terms decay more quickly with $r$ than would pressure-only terms, i.e. as $\mathcal{O}(r^{-5})$ , so cannot lead to $C_{\infty }\neq 0$ . We have also not treated the case where $\phi$ is a function of $\boldsymbol{A}$ as well as of $\boldsymbol{B}$ and $\boldsymbol{u}$ ; we anticipate that, for a large class of such gauges, $C_{\infty }$ does vanish, but we have not proven this in general.

3.3.2. An $I_{\!H}$ -non-conserving gauge

We have so far shown that $C_{\infty }$ vanishes under a wide class of gauge choices, including all those for which $\zeta$ is determined by Poisson’s equation (2.6), with $\phi$ a local function of $\boldsymbol{u}$ and $\boldsymbol{B}$ . Let us now ask whether there exist more exotic gauges for which $C_{\infty }\neq 0$ . Interestingly, the answer appears to be yes. An explicit example is the gauge defined by

(3.17) \begin{align} \zeta = B_i \mathcal{Z}_i,\quad {\nabla} ^2 \mathcal{Z}_i = \phi B_i, \end{align}

where $\phi$ is any local function of $\boldsymbol{u}$ , $\boldsymbol{B}$ and $\boldsymbol{A}$ . The Green’s function solution for $\zeta$ is

(3.18) \begin{align} \zeta (\boldsymbol{x}) = -\frac {B_i}{4\pi } \int _{\mathbb{R}^3} \frac {{d}^3 \boldsymbol{x}^{\prime }}{| \boldsymbol{x}^{\prime } - \boldsymbol{x} |}B_i'\phi '. \end{align}

It follows that, at $t=0$ , the third correlation function that appears in the definition of $C_{\infty }$ (1.7) is

(3.19) \begin{align} \langle \zeta B_iA_l^{\prime}B_l^{\prime}\rangle & = - \frac {1}{4\pi }\int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime \prime }}{| \boldsymbol{x}^{\prime \prime } - \boldsymbol{x} |}\langle B_k B_i B_k^{\prime\prime}\phi^{\prime\prime} A_l^{\prime}B_l^{\prime}\rangle \nonumber \\ & = -\frac {1}{4\pi }\langle B_k B_i \rangle \int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{x}^{\prime \prime }}{| \boldsymbol{x}^{\prime\prime} - \boldsymbol{x} |} \langle B_k^{\prime\prime}\phi^{\prime\prime} A_l^{\prime}B_l^{\prime}\rangle \nonumber \\ & = -\frac {\langle B^2 \rangle }{12\pi }\int _{\mathbb{R}^3} \frac {\textrm{d}^3 \boldsymbol{s}^\prime }{| \boldsymbol{s}^\prime + \boldsymbol{r} |} \langle B_i^{\prime\prime}\phi^{\prime\prime} A_l^{\prime}B_l^{\prime}\rangle \nonumber \\ & = -\frac {\langle B^2 \rangle }{12\pi } \frac {1}{r}\int _{\mathbb{R}^3} \textrm{d}^3 \boldsymbol{s}^\prime \langle B_i^{\prime\prime}\phi^{\prime\prime} A_l^{\prime}B_l^{\prime}\rangle - \frac {\langle B^2 \rangle }{12\pi } \frac {\partial }{\partial r_j}\left (\frac {1}{r}\right )\int _{\mathbb{R}^3} \textrm{d}^3 \boldsymbol{s}^\prime s_j^{\prime} \langle B_i^{\prime\prime}\phi^{\prime\prime} A_l^{\prime}B_l^{\prime}\rangle \nonumber \\ &\quad + \mathcal{O}(r^{-3}). \end{align}

Provided that $\phi$ is not a true scalar (the natural choice for $\phi$ would, presumably, be the pseudo-scalar $\phi \,\propto \,h = \boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}$ ), the correlation function appearing in the above is

(3.20) \begin{align} \langle B_i''\phi '' A_l^{\prime}B_l^{\prime}\rangle = a(|\boldsymbol{s}^\prime |)s_i^\prime , \end{align}

where $a(|\boldsymbol{s}^\prime |)$ is an undetermined function. This does not, in general, vanish, although the first integral in (3.19) vanishes by symmetry. The second integral in (3.19) does not obviously vanish, so we expect that $\langle \zeta B_iA_l^{\prime}B_l^{\prime}\rangle = \mathcal{O}(r^{-2})$ , indicating that $C_{\infty }$ is finite for this gauge (see (1.7)). We suggest a possible interpretation of this phenomenon in § 5, but defer detailed investigation of this interesting gauge to future work.

4.1. Numerical method and initial conditions

We employ a modified version of the flash code (Fryxell et al. Reference Fryxell, Olson, Ricker, Timmes, Zingale, Lamb, MacNeice, Rosner, Truran and Tufo2000; Dubey et al. Reference Dubey, Fisher, Graziani, Jordan, Lamb, Reid, Rich, Sheeler, Townsley and Weide2008; Federrath et al. Reference Federrath, Klessen, Iapichino and Beattie2021) with the 5-wave HLL5R approximate Riemann solver (Waagan, Federrath & Klingenberg Reference Waagan, Federrath and Klingenberg2011), which uses a positivity-preserving MUSCL-Hancock scheme (Bouchut, Klingenberg & Waagan et al. Reference Bouchut, Klingenberg and Waagan2007, Reference Bouchut, Klingenberg and Waagan2010; Waagan Reference Waagan2009) to solve the equations of three-dimensional compressible isothermal MHD – i.e.

(4.1) \begin{align} \rho \left (\frac {\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla } \boldsymbol{u}\right ) &= -\boldsymbol{\nabla } p + \frac {1}{4\pi }(\boldsymbol{\nabla }\times \boldsymbol{B})\times \boldsymbol{B}+\rho \nu {\nabla} ^2 \boldsymbol{u}, \end{align}

(4.2) \begin{align} \frac {\partial \boldsymbol{B}}{\partial t} &= \boldsymbol{\nabla }\times (\boldsymbol{u}\times \boldsymbol{B}) + \eta {\nabla} ^2 \boldsymbol{B}, \end{align}

and $p = \rho c_{ s}^2$ , with $c_{ s}$ the constant speed of sound ( $c_{ s} = 1$ in our code units) – in a periodic box of volume $L^3$ , where $L=1$ in our code units. Because our simulations evolve a compressible fluid, we shall in what follows measure the magnetic field $\boldsymbol{B}$ in full CGS (Gaussian) units, rather than in units of the Alfvén speed, as was convenient in the preceding sections. We use a uniform grid with $2304^3$ cells (see Appendix D for comparison with a simulation at lower resolution). The simulation uses explicit kinematic viscosity $\nu =10^{-7}\,Lc_{s}$ and magnetic diffusivity $\eta =10^{-7}\,Lc_{s}$ , so the magnetic Prandtl number is $Pm\equiv \nu /\eta =1$ .

The simulation is initialised with a uniform density $\rho _0$ , with $\rho _0=1$ in our code units, and zero velocity.Footnote ⁸ As in Hosking & Schekochihin (Reference Hosking and Schekochihin2021) and Zhou et al. (Reference Zhou, Sharma and Brandenburg2022), the initial condition for the magnetic field is a non-helical Gaussian random field. We choose a magnetic-energy spectrum $\mathcal{E}_{M}(k)\propto k^4$ for $1\leqslant kL/2\pi \leqslant 60 \equiv k_{\mathrm{peak},0} L /2\pi$ , and zero elsewhere, generated with TurbGen (Federrath et al. Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010, Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2022). We choose the initial amplitude of the magnetic field such that $\langle B_0^2\rangle = \rho _0 c_{ s}^2$ , i.e. the initial (root-mean-square) Alfvén speed is $v_{ A0}\equiv \langle B_0^2\rangle ^{1/2}/\sqrt {4\pi \rho _0} = c_{ s}/\sqrt {4\pi }\simeq 0.3c_{ s}$ . The mean magnetic field in the periodic domain (i.e. its $\boldsymbol{k}=0$ component) is zero. Thus, the magnetic vector potential is well defined, as is the net magnetic helicity in the periodic box, which is equal to zero. Using $\mathcal{E}_{M}$ we define the magnetic-field correlation scale (integral scale) by

(4.3) \begin{align} \xi _{M} = \frac {2\pi }{E_{ M}}\int _0^{\infty } \text{d}k\,\frac {\mathcal{E}_{M}(k)}{k}, \end{align}

where $E_{ M} = \int _0^{\infty } {\rm d}k\,\mathcal{E}_{M}(k)$ is the magnetic energy. The initial Lundquist number is $S_0 \equiv v_{A0} \xi _{M0}/\eta \simeq 6\times 10^4$ , where $\xi _{M0}=2\pi \,{\times }\,5/(4k_{\mathrm{peak},0})$ is the initial $\xi _{M}$ . In the results presented below, we restrict attention to times for which $\xi _{ M} \ll L$ ; as far as any individual magnetic structure is concerned, box-scale topological effects (Berger Reference Berger1997) can then be neglected, and the periodic box serves as a proxy for an infinite open domain, as considered in §§ 2 and 3.

4.2. Evolution of the simulation

The out-of-equilibrium initial condition (see § 4.1) undergoes turbulent decay as visualised in figure 1, which shows slices of the squared current density at different times. We observe that the coherence scale of the magnetic field increases with time, as expected from the decay laws (1.8).

Figure 1.

Two-dimensional slices of the current density squared $J^2 = |\boldsymbol{\nabla } \times \boldsymbol{B}|^2$ , normalised to its root-mean-square value, at various times during the decay. After $t \sim t_{ A0}$ , we see growth of magnetic structures resulting from the merging of smaller structures.

Figure 2 shows the evolution of the magnetic and kinetic energies and their spectra, as well as that of the variance of magnetic helicity. Both energies decay as a power law that is close to $E_{ M}\propto t^{-1}$ (figure 2 a). In order to diagnose the decay laws more precisely, we follow Brandenburg & Kahniashvili (Reference Brandenburg and Kahniashvili2017) in plotting in figure 3 the instantaneous scaling exponents of the magnetic energy and correlation length:

(4.4) \begin{align} p = -\frac{\text{d} \ln {E_{ M}}}{\text{d} \ln {t}}, \quad q = \frac{\text{d} \ln {\xi _{ M}}}{\text{d}\ln {t}}. \end{align}

Both $p$ and $q$ increase with time somewhat, reaching $p\simeq 1.02$ and $q \simeq 0.43$ at the last time shown. These values are close to the expected $p = 10/9 \simeq 1.11$ and $q = 4/9 \simeq 0.44$ (1.8). In particular, their evolution always satisfies $\beta \equiv p/q - 1 \simeq 3/2$ , as is consistent with self-similar decay that conserves $I_{\!H} \sim E_{M}^2\xi _M^5$ (and inconsistent with self-similar decay conserving the mean squared vector potential, an alternative possibility mooted by Bhat et al. (Reference Bhat, Zhou and Loureiro2021) and Dwivedi, Anandavijayan & Bhat (Reference Dwivedi, Anandavijayan and Bhat2024), which corresponds to $\beta = 1$ ).Footnote ⁹

Figure 2.

Temporal and spectral statistics of decaying MHD turbulence in the simulations with $2304^3$ grid cells. (a) Magnetic and kinetic energy, $E_M$ and $E_K$ , as functions of time in units of the Alfvén box-crossing time $t_{A0}$ . The dotted vertical lines indicate times in which the correlation functions are plotted in figures 5 and 6. (b) Magnetic- and kinetic-energy spectra. The darkest lines correspond to the initial condition, and the other three to the times indicated in (a), with lighter lines indicating later times. We observe a $k^4$ scaling until the peak scale, after which the spectrum is proportional to $k^{-2}$ , which may be an artefact of current-sheet discontinuities. (c) Magnetic-helicity-variance spectrum, $\mathcal{E}_H$ . The vertical axis of each panel uses the code units defined in § 4.1.

Figure 3.

The evolution of $p(t)$ and $q(t)$ (4.4) as a function of time in the simulation. Different values of $\beta$ correspond to different scaling relations between $E_{M}$ and $\xi _M$ . The simulation evolves somewhat along the line $\beta = 3/2$ , which corresponds to self-similar decay that conserves $I_{\!H}$ (see (1.8)).

Figure 2(a) shows that the velocity field remains energetically subdominant to the magnetic field throughout the evolution, which may be a consequence of the velocity field being more intermittent than the volume-filling magnetic field, being concentrated in Alfvénic reconnection outflows (see Hosking & Schekochihin (Reference Hosking and Schekochihin2021) for further discussion). Figure 2(b) shows the evolution of the energy spectra. The magnetic spectrum $\mathcal{E}_M(k)$ exhibits a near- $k^{-2}$ power law between its peak and the dissipation scale, which may be a signature of current-sheet discontinuities. The kinetic-energy spectrum $\mathcal{E}_K(k)$ is nearly flat over the same range, which may be associated with the sheet-like structure of the reconnection outflows (Hosking & Schekochihin Reference Hosking and Schekochihin2021). At large scales, the magnetic spectrum exhibits a $k^4$ tail that grows in amplitude with time (this is the ‘inverse transfer’ effect discovered by Zrake (Reference Zrake2014) and Brandenburg, Kahniashvili & Tevzadze (Reference Brandenburg, Kahniashvili and Tevzadze2015), which was explained as a kinematic consequence of the conservation of $I_{\!H}$ by Hosking & Schekochihin (Reference Hosking and Schekochihin2021)); the amplitude of the $k^4$ tail of the kinetic spectrum also grows somewhat over time. We show in figure 2(c ) the spectrum of the variance of the magnetic-helicity density at different times. The conservation of its $k^2$ tail indicates the conservation of $I_{\!H}$ (Hosking & Schekochihin Reference Hosking and Schekochihin2021).

Figure 4.

Evolution of the Mach number $Ma$ and Lundquist number $S$ . The vertical lines indicate the times at which the correlation functions are shown in figures 5 and 6.

As we show in figure 4, the Lundquist number $S = v_A \xi _M/\eta$ decreases slowly with time (also as predicted by the decay laws (1.8)) but remains ${\sim}10^{4}$ . Our simulations are therefore marginal with respect to plasmoid-mediated fast reconnection of magnetic structures at the integral scale (see Uzdensky, Loureiro & Schekochihin Reference Uzdensky, Loureiro and Schekochihin2010 Footnote ¹⁰ ). The velocity Reynolds number (not shown) based on the integral scale of the velocity field (i.e. (4.3), but evaluated for the velocity spectrum) peaks at $\simeq 2\times 10^4$ at $t/t_{A0}\simeq 10^{-2}$ , and then decays by a factor of $\simeq 3$ by $t/t_{A0}\simeq 3$ . Figure 4 shows that the Mach number $Ma \equiv \langle v^2\rangle ^{1/2}/c_s \lesssim 0.1$ at all times, making the turbulence nearly incompressible throughout.

4.3. Measurements of correlation functions

Figure 5.

The three correlation functions that contribute to $C_{\infty }$ (1.7) measured at $t/t_{A0} = 0.112$ (blue), $t/t_{A0} = 0.447$ (red) and $t/t_{A0} = 1.780$ (green) as a function of $r/L$ . Each vertical axis is measured in the code units defined in § 4.1. Positive values are plotted with solid circles, negative ones with crosses. Dashed vertical lines indicate the integral scale $\xi _M$ (4.3) at each time. The correlation functions decay in amplitude and shift to larger spatial scales over time. We indicate a number of power laws to guide the eye, but observe that none of these fit any given curve over more than half a decade in $r/L$ .

We now describe our measurements of the correlation functions that appear in (1.7). We calculate these using an analogous method to the calculation of higher-order structure functions by Federrath et al. (Reference Federrath, Klessen, Iapichino and Beattie2021), using $10^{11}$ sampling points. We plot the correlation functions in figure 5 for $t/t_{A0}$ of $0.112$ , $0.447$ and $1.780$ , at which times the magnetic energy has decayed by factors of roughly $10$ , $30$ and $100$ , respectively (see figure 2 a). In each case, the correlation functions vanish as $r \to 0$ (where they become dominated by noise). This is as expected, since the expectation value of a vector is zero in isotropic turbulence. The correlation functions peak in magnitude at around $r\sim \xi _M$ . At $r/\xi _M \gtrsim 1$ , their amplitudes decay with $r$ in what we term the ‘decorrelation range’, ultimately becoming dominated by numerical noise. We observe that they decrease in amplitude and shift to larger $r$ over time, consistent with the decay of the turbulence and transfer of energy to larger scales (see the decay laws (1.8) and figure 2).

In the decorrelation range $r/\xi _M \gtrsim 1$ , each correlation function ultimately decays faster than $r^{-2}$ , which is the condition for the conservation of $I_{\!H}$ ( $C_{\infty }=0$ in (1.6)) (we note that the top panel of figure 5 appears to show $r^{-1}$ over a short intermediate range, but this steepens at larger $r$ ). It is difficult to judge whether any of these plots show a power-law decay at $r \gg \xi _M$ : the $r^{-3}$ and $r^{-4}$ power laws that we plot for reference match our measurements reasonably well locally, although this is true only for around half a decade in $r$ . After this range, the behaviour transitions to steeper decay in most cases. Because the Coulomb gauge is of the Poisson type (see (2.6) and (2.9)), the analysis in § 3 rules out decay of these correlation functions slower than $r^{-6}$ (which corresponds to terms in the Taylor expansion that involve both pressure and gauge, i.e. counting two inverse powers of $r$ from (2.9) and four from the gradient of (2.5)). Supposing that the $r^{-3}$ and $r^{-4}$ power laws identified in figure 5 are real, therefore, they are somewhat less steep than our theoretical analysis predicts. One way to rationalise this behaviour is by noting that our analysis involved frequent use of the vanishing of terms like $\langle \boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{B} \rangle$ and $\langle \boldsymbol{A} \boldsymbol{\cdot } \boldsymbol{B} \rangle$ due to parity symmetry. While these quantities would be zero in an infinite domain, the box-averaged $\boldsymbol{u}\boldsymbol{\cdot } \boldsymbol{B}$ and $\boldsymbol{A}\boldsymbol{\cdot } \boldsymbol{B}$ in a finite periodic box can be non-zero, owing to fluctuations sourced by compressibility and resistivity. Thus, parity-dependent terms that would, if non-zero, contribute shallower power laws to the various correlation functions (such as (3.12), (3.13) and (3.14)), might, at $r\gg \xi _M$ , overwhelm the faster-decaying parity-invariant contributions.

In figure 6, we show the same correlation functions as in figure 5, but normalised to the dimensionally appropriate combinations of $\langle u^2\rangle ^{1/2}$ , $\langle B^2\rangle ^{1/2}$ and $\xi _M$ . We find, under this rescaling, that the functions collapse onto one another and become nearly time-independent, as is consistent with self-similar decay.

Figure 6.

The same as figure 5, but with all quantities normalised to the dimensionally appropriate combinations of $\langle u^2\rangle ^{1/2}$ , $\langle B^2\rangle ^{1/2}$ and $\xi _M$ .