2.1 General formalism

As per standard MFE practice, we separate any vector field $\boldsymbol{A}$ into a mean part $\overline{\boldsymbol{A}}$ and a fluctuation part $\boldsymbol{a}$ ,

(2.1) $$\begin{eqnarray}\boldsymbol{A}(\boldsymbol{x})=\overline{\boldsymbol{A}}(\boldsymbol{x})+\boldsymbol{a}(\boldsymbol{x}).\end{eqnarray}$$

The mean part is defined via

(2.2) $$\begin{eqnarray}\overline{\boldsymbol{A}}(\boldsymbol{x})=G_{l}(\boldsymbol{x})\ast \boldsymbol{A}(\boldsymbol{x})=\int \text{d}^{3}x^{\prime }G_{l}(\boldsymbol{x}-\boldsymbol{x}^{\prime })\boldsymbol{A}(\boldsymbol{x}^{\prime }),\end{eqnarray}$$

where ‘ $\ast$ ’ denotes a convolution. The filtering kernel $G_{l}(\boldsymbol{x})$ is a prescribed function with a characteristic scale of averaging $l$ , satisfying $l_{s}<l_{\text{eff}}(l)<l_{L}$ , where $l_{\text{eff}}$ can be viewed as the configuration space dividing scale between large and small scale fields. We define $l$ such that $l=l_{\text{eff}}$ for our analytic derivations.Footnote ² We have assumed that the system under consideration is statistically homogeneous and isotropic on scales ${\leqslant}l$ , so that $l$ is independent of location and $G_{l}(\boldsymbol{x})$ is isotropic. For anisotropic or inhomogeneous systems, $G_{l}(\boldsymbol{x})$ could be anisotropic and $l$ could be a function of spatial coordinates.

We use the following definition of the Fourier transform:

(2.3) $$\begin{eqnarray}{\mathcal{F}}[f(\boldsymbol{x})](\boldsymbol{k})={\displaystyle\mathop{f}\limits_{{\sim}}}(\boldsymbol{k})=\int \text{d}^{3}x\,f(\boldsymbol{x})\text{e}^{-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}},\end{eqnarray}$$

and

(2.4) $$\begin{eqnarray}{\mathcal{F}}^{-1}[{\displaystyle\mathop{f}\limits_{{\sim}}}(\boldsymbol{k})](\boldsymbol{x})=f(\boldsymbol{x})=\frac{1}{(2\unicode[STIX]{x03C0})^{3}}\int \text{d}^{3}k{\displaystyle\mathop{f}\limits_{{\sim}}}(\boldsymbol{k})\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}},\end{eqnarray}$$

and therefore the Fourier transform of $\overline{\boldsymbol{A}}$ is given by

(2.5) $$\begin{eqnarray}\overline{{\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}}(\boldsymbol{k})={\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k}){\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}(\boldsymbol{k}).\end{eqnarray}$$

Unlike idealized ensemble averages, equation (2.5) implies $\overline{\overline{\boldsymbol{A}}}\neq \overline{\boldsymbol{A}}$ since ${\displaystyle\mathop{G}\limits_{{\sim}}}_{l}^{2}(\boldsymbol{k})\neq {\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})$ unless ${\displaystyle\mathop{G}\limits_{{\sim}}}(\boldsymbol{k})=0$ or 1, such as for a step function in Fourier space. However, interchangeability of differential and average operations, as commonly invoked, is manifest in Fourier space since $k_{i}[{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k}){\displaystyle\mathop{A}\limits_{{\sim}}}_{j}(\boldsymbol{k})]={\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})[k_{i}{\displaystyle\mathop{A}\limits_{{\sim}}}_{j}(\boldsymbol{k})]$ .

The kernel $G_{l}(\boldsymbol{x})$ must meet several requirements for a practical mean field theory. First, it should be a spatially local function that decreases rapidly for $|\boldsymbol{x}|\gtrsim l$ , being that it is used to extract a filtered value at a scale  $l$ . Complementarily, its Fourier transform ${\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})$ should also monotonically decrease and vanish for large $|\boldsymbol{k}|$ . Furthermore, in the limit $l\rightarrow 0$ , ${\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})$ approaches unity, since no filtering is needed for large scales. Thus, ${\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})$ can be expanded around $|\boldsymbol{k}|=k=0$ when $|\boldsymbol{k}l|/2\unicode[STIX]{x03C0}=|\boldsymbol{k}|/k_{l}$ is small compared to unity, yielding

(2.6) $$\begin{eqnarray}{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})=1-{\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}+O({\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}^{2}),\end{eqnarray}$$

where ${\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}$ is a small parameter related to $|\boldsymbol{k}|/k_{l}$ , and the minus sign is for future convenience. Note that ${\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}$ is independent of the direction of $\boldsymbol{k}$ due to isotropy.

The inverse Fourier transform of ${\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}$ is an operator $\hat{\unicode[STIX]{x1D6FE}}$ which is determined by

(2.7) $$\begin{eqnarray}(\hat{\unicode[STIX]{x1D6FE}}f)(\boldsymbol{x})={\mathcal{F}}^{-1}[{\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}(\boldsymbol{k}){\displaystyle\mathop{f}\limits_{{\sim}}}(\boldsymbol{k})](\boldsymbol{x}).\end{eqnarray}$$

Hereafter we assume that these fields are either vanishing or periodic at the spatial boundaries, and therefore any ${\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}(\boldsymbol{k})$ proportional to a power of $i\boldsymbol{k}$ is simply translated to a $\hat{\unicode[STIX]{x1D6FE}}$ which is a spatial derivative raised to the corresponding power. When applied to a quantity $Q$ with smallest characteristic scale $l_{\text{ch}}>l$ , the order-of-magnitude estimate yields

(2.8) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FE}}Q\sim \left(\frac{l}{l_{\text{ch}}}\right)^{c}Q,\end{eqnarray}$$

with $c$ being a positive number that depends on the specific choice of kernel.

2.2 Expressions for averages of fluctuations and double averages

Here we obtain formulae for averages of fluctuations and double averages, both of which do not strictly obey Reynolds rules in the presence of mesoscale fluctuations. In particular, the averages of fluctuations do not vanish and the double averages will not agree with single-averaged values. We will use the expressions in subsequent sections.

We first derive an expression for the mean of fluctuations, namely $\overline{\boldsymbol{a}}=\overline{\boldsymbol{A}}-\overline{\overline{\boldsymbol{A}}}$ . This vanishes in conventional MFE using the ensemble average, but not for spatial averages. In Fourier space, by definition,

(2.9) $$\begin{eqnarray}\overline{{\displaystyle\mathop{\boldsymbol{a}}\limits_{{\sim}}}}(\boldsymbol{k})={\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k}){\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}(\boldsymbol{k})-{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}^{2}(\boldsymbol{k}){\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}(\boldsymbol{k})=(1-{\displaystyle\mathop{G}\limits_{{\sim}}}_{l})\overline{{\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}}.\end{eqnarray}$$

Since $\overline{{\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}}$ is a large scale quantity, it decays rapidly when $|\boldsymbol{k}|/k_{l}\gg 1$ . Therefore we can expand the right-hand side of (2.9) for $|\boldsymbol{k}|/k_{l}\ll 1$ using (2.6) to obtain

(2.10) $$\begin{eqnarray}\overline{{\displaystyle\mathop{\boldsymbol{a}}\limits_{{\sim}}}}(\boldsymbol{k})=[{\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}+O({\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}^{2})]\,\overline{{\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}}.\end{eqnarray}$$

In configuration space, this implies

(2.11) $$\begin{eqnarray}\overline{\boldsymbol{a}}(\boldsymbol{x})=\overline{\boldsymbol{A}}-\overline{\overline{\boldsymbol{A}}}=[\hat{\unicode[STIX]{x1D6FE}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})]\,\overline{\boldsymbol{A}}\sim \left(\frac{l}{l_{L}}\right)^{c}\overline{\boldsymbol{A}},\end{eqnarray}$$

if $\overline{\boldsymbol{A}}$ has a characteristic variation scale of $l_{L}$ . Equivalently

(2.12) $$\begin{eqnarray}\overline{\overline{\boldsymbol{A}}}=[1-\hat{\unicode[STIX]{x1D6FE}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})]\,\overline{\boldsymbol{A}}.\end{eqnarray}$$

To recover conventional MFE, we simply take the limit $l/l_{L}\rightarrow 0$ and get $\overline{\boldsymbol{a}}=\mathbf{0}$ .

Next, we obtain an expression for the mean of the product of two fields, $\overline{AB}$ , in terms of the mean fields. Here $A$ and $B$ can be either two scalar fields or the components of some vector fields. We adopt a two-scale approach, assuming that the fields have double-peaked spectra and scale separations are large but finite, i.e. we relax the assumption of infinite scale separation in conventional approaches (for details see appendix A where the valid range of scale separation is quantified). Other closures may include a test filtering process like that used in the Smagorinsky model (Smagorinsky Reference Smagorinsky1963; Germano et al. Reference Germano, Piomelli, Moin and Cabot1991; Lilly Reference Lilly1992).

By straightforward expansion we have

(2.13) $$\begin{eqnarray}\overline{AB}=\overline{\overline{A}~\overline{B}}+\overline{a\overline{B}}+\overline{\overline{A}b}+\overline{ab}.\end{eqnarray}$$

We refer to the terms on the right-hand side of (2.13) as $T_{1}$ , $T_{2}$ , $T_{3}$ and $T_{4}$ , respectively. The calculation of $T_{1}$ involves only mean quantities, but for practical purposes, it is convenient to make some further approximations to avoid integro-differential equations. If $\overline{A}$ and $\overline{B}$ both have a characteristic scale of variation $l_{L}$ , then the spectrum of the Fourier transform of their product will roughly extend to $k=2k_{L}$ . If the scale of average satisfies $2k_{L}l/2\unicode[STIX]{x03C0}=k_{L}l/\unicode[STIX]{x03C0}\ll 1$ , we can use (2.12) to calculate $T_{1}$ , namely,

(2.14) $$\begin{eqnarray}\overline{\overline{A}~\overline{B}}=[1-\hat{\unicode[STIX]{x1D6FE}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})](\overline{A}~\overline{B}).\end{eqnarray}$$

The Fourier transform of $T_{2}$ is

(2.15) $$\begin{eqnarray}{\displaystyle\mathop{T}\limits_{{\sim}}}_{2}(\boldsymbol{k})={\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})[{\displaystyle\mathop{a}\limits_{{\sim}}}(\boldsymbol{k})\ast \overline{{\displaystyle\mathop{B}\limits_{{\sim}}}}(\boldsymbol{k})]={\displaystyle\mathop{G}\limits_{{\sim}}}_{l}\{[({\displaystyle\mathop{G}\limits_{{\sim}}}_{l}^{-1}-1)\overline{{\displaystyle\mathop{A}\limits_{{\sim}}}}]\ast \overline{{\displaystyle\mathop{B}\limits_{{\sim}}}}\},\end{eqnarray}$$

where we have used the definition ${\displaystyle\mathop{\boldsymbol{a}}\limits_{{\sim}}}=(1-{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}){\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}$ . The convolution of two quantities with characteristic wavenumbers $k_{1}$ and $k_{2}$ will yield wavenumbers $k_{1}\pm k_{2}$ . Note that $G_{l}$ is outside of the square brackets and so with periodic or vanishing boundary conditions, only the low wavenumber part of the factor $(G^{-1}-1)\overline{A}$ survives on the right-hand side of (2.15). (The validity of this approximation is discussed in more detail in appendix A.) We therefore expand $G_{l}^{-1}$ in a Taylor series, yielding

(2.16) $$\begin{eqnarray}({\displaystyle\mathop{G}\limits_{{\sim}}}_{l}^{-1}-1)\overline{{\displaystyle\mathop{A}\limits_{{\sim}}}}=[{\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}+O({\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}^{2})]\,\overline{{\displaystyle\mathop{A}\limits_{{\sim}}}},\end{eqnarray}$$

which upon Fourier inversion then implies

(2.17) $$\begin{eqnarray}T_{2}=\overline{a\overline{B}}=\overline{\overline{B}[\hat{\unicode[STIX]{x1D6FE}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})]\,\overline{A}}.\end{eqnarray}$$

Similarly, for $T_{3}$ we obtain

(2.18) $$\begin{eqnarray}T_{3}=\overline{b\overline{A}}=\overline{\overline{A}[\hat{\unicode[STIX]{x1D6FE}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})]\overline{B}}.\end{eqnarray}$$

The sum $T_{2}+T_{3}$ , using (2.14), is then

(2.19) $$\begin{eqnarray}\overline{a\overline{B}}+\overline{b\overline{A}}=[\hat{\unicode[STIX]{x1D6FE}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})](\overline{\overline{A}~\overline{B}})-\overline{\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{A},\overline{B})}=[\hat{\unicode[STIX]{x1D6FE}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})](\overline{A}~\overline{B})-\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{A},\overline{B}),\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D6FE}}^{\prime }$ , a binary operator, is introduced to account for the violation of the distribution rule of $\hat{\unicode[STIX]{x1D6FE}}$ ; that is,

(2.20) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FE}}^{\prime }(A,B)=\hat{\unicode[STIX]{x1D6FE}}(AB)-(A\hat{\unicode[STIX]{x1D6FE}}B+B\hat{\unicode[STIX]{x1D6FE}}A).\end{eqnarray}$$

Note that $\hat{\unicode[STIX]{x1D6FE}}^{\prime }(A,B)$ has the same order of magnitude as $B\hat{\unicode[STIX]{x1D6FE}}A$ or $A\hat{\unicode[STIX]{x1D6FE}}B$ if $A$ and $B$ have the same characteristic length scale.

Combining (2.13), (2.14) and (2.19) we obtain

(2.21) $$\begin{eqnarray}\overline{AB}=[1+O(\hat{\unicode[STIX]{x1D6FE}}^{2})](\overline{A}~\overline{B})-\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{A},\overline{B})+\overline{ab}.\end{eqnarray}$$

Furthermore, it can be verified using (2.12), (2.17) and (2.21) together that

(2.22) $$\begin{eqnarray}\displaystyle \overline{A\overline{B}} & = & \displaystyle \overline{A}~\overline{\overline{B}}-\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{A},\overline{\overline{B}})+\overline{a\overline{b}}\nonumber\\ \displaystyle & = & \displaystyle \overline{A}(1-\hat{\unicode[STIX]{x1D6FE}})\overline{B}-\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{A},(1-\hat{\unicode[STIX]{x1D6FE}})\overline{B})+\overline{\overline{b}\hat{\unicode[STIX]{x1D6FE}}\overline{A}}+O(\hat{\unicode[STIX]{x1D6FE}}^{2})\nonumber\\ \displaystyle & = & \displaystyle \overline{A}(1-\hat{\unicode[STIX]{x1D6FE}})\overline{B}-\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{A},\overline{B})+O(\hat{\unicode[STIX]{x1D6FE}}^{2})\nonumber\\ \displaystyle & = & \displaystyle (1-\hat{\unicode[STIX]{x1D6FE}})(\overline{A}~\overline{B})+\overline{B}\hat{\unicode[STIX]{x1D6FE}}\overline{A}+O(\hat{\unicode[STIX]{x1D6FE}}^{2}).\end{eqnarray}$$

2.3 Comparison to previous work

Expressing a turbulent field as ${\displaystyle\mathop{a}\limits_{{\sim}}}=(1-{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}){\displaystyle\mathop{A}\limits_{{\sim}}}$ is equivalent to applying a high-pass filter on  $A$ , as has been discussed in Yeo (Reference Yeo1987). In Yeo (Reference Yeo1987) all fields are expressed in terms of mean fields and their derivatives (Yeo–Bedford expansion), including small scale fields, so the approach facilitates a closure in their context of the inertial range for large eddy simulations (LES). In our approach, we focus on the large scale mean fields, not the inertial range. We keep two-point correlations of turbulent fields ( $\overline{ab}$ -like terms in (2.21)) but use a separate closure for triple correlations (compare (5.8) to (5.11) of Yeo (Reference Yeo1987) to our (2.21)). In our formalism, $\hat{\unicode[STIX]{x1D6FE}}$ terms enter as corrections to capture finite scale separation effects, facilitating comparisons to conventional approaches (e.g. ensemble averages), while allowing different closures. Also, Yeo (Reference Yeo1987) use a Gaussian kernel, whereas our discussions in the previous sections apply to any kernel meeting the requirements in § 2.1.

2.4 Unifying different averaging methods using kernels

Here we discuss commonly used averages and their kernel forms (if possible). Recall from above that in order to accurately capture large scale features, a suitable kernel for mean field theories should at least be monotonically decreasing in Fourier space.

2.4.1 Gaussian average

For isotropic and homogeneous turbulence, the Gaussian kernel is defined as

(2.23) $$\begin{eqnarray}G_{l}(\boldsymbol{x})=\left(\frac{k_{l}^{2}}{2\unicode[STIX]{x03C0}}\right)^{3/2}\text{e}^{-k_{l}^{2}|\boldsymbol{x}|^{2}/2},\end{eqnarray}$$

where $k_{l}=2\unicode[STIX]{x03C0}/l$ . It is then evident that $\overline{\boldsymbol{A}}$ represents the large scale part of $\boldsymbol{A}$ by rewriting it in Fourier space. This gives

(2.24) $$\begin{eqnarray}\overline{{\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}}(\boldsymbol{k})={\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k}){\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}(\boldsymbol{k})=\text{e}^{-k^{2}/2k_{l}^{2}}{\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}(\boldsymbol{k}).\end{eqnarray}$$

Since the kernel decreases rapidly for large $k$ , as long as the spectrum of ${\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}(\boldsymbol{k})$ does not increase exponentially at large $k$ , the spectrum of $\overline{{\displaystyle\mathop{\boldsymbol{A}}\limits_{{\sim}}}}(\boldsymbol{k})$ has little power for $k>k_{l}$ . For $k<k_{l}$ we can then write

(2.25) $$\begin{eqnarray}{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})=1-\frac{k^{2}}{2k_{l}^{2}}+O\left(\frac{k^{4}}{k_{l}^{4}}\right)+\cdots ,\end{eqnarray}$$

so that in configuration space $\hat{\unicode[STIX]{x1D6FE}}=-\unicode[STIX]{x1D6FB}^{2}/2k_{l}^{2}$ (recall the ‘ $-$ ’ sign in the definition of ${\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}$ from (2.6)) and $\hat{\unicode[STIX]{x1D6FE}}^{\prime }(A,B)=-\unicode[STIX]{x1D735}A\boldsymbol{\cdot }\unicode[STIX]{x1D735}B/k_{l}^{2}$ for any $A$ and  $B$ . Overall, $\hat{\unicode[STIX]{x1D6FE}}$ operating on quantity $Q$ gives $\hat{\unicode[STIX]{x1D6FE}}Q\sim (l/l_{\text{ch}})^{2}Q$ where $l_{\text{ch}}$ is the characteristic variation scale of  $Q$ .

2.4.2 Moving box average

Here fields at a point $\boldsymbol{x}$ are averaged in a finite box with sides of length  $l$ . Expressing the average using a kernel allows the integral bounds to be taken to infinity, that is

(2.26) $$\begin{eqnarray}\overline{\boldsymbol{A}}(\boldsymbol{x})=\frac{1}{l^{3}}\int _{-l/2}^{l/2}\,\text{d}x^{\prime }\int _{-l/2}^{l/2}\,\text{d}y^{\prime }\int _{-l/2}^{l/2}\,\text{d}z^{\prime }\boldsymbol{A}(\boldsymbol{x}-\boldsymbol{x}^{\prime })=\int \text{d}^{3}x^{\prime }G_{l}(\boldsymbol{x}^{\prime })\boldsymbol{A}(\boldsymbol{x}-\boldsymbol{x}^{\prime }),\end{eqnarray}$$

where $G_{l}(\boldsymbol{x})=\unicode[STIX]{x1D703}_{l}(x)\unicode[STIX]{x1D703}_{l}(y)\unicode[STIX]{x1D703}_{l}(z)$ is the product of three rectangular functions defined by

(2.27) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{l}(x)=\left\{\begin{array}{@{}ll@{}}1/l & -l/2\leqslant x\leqslant l/2\\ 0 & \text{otherwise}\end{array}\right\}.\end{eqnarray}$$

We call this a ‘moving’ average because it is not taken on a fixed grid, but centred around each point  $\boldsymbol{x}$ . Although suitable for numerical simulation analyses and seemingly benign, this has limitations for applicability to realistic contexts. The reason is evident from the Fourier transform of the kernel of a one-dimensional running box, namely

(2.28) $$\begin{eqnarray}{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(k)=\text{sinc}\left(\frac{kl}{2}\right).\end{eqnarray}$$

Here $|{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}|$ is a non-monotonic function of $k$ with zero points at $kl=n\unicode[STIX]{x03C0}$ , where $n\in \mathbb{Z}$ . As a result, some modes with large wave numbers may contribute more to the mean than those with small wavenumbers. This contradicts our basic notion of mean field theory and highlights why monotonicity of the kernel is a requirement for a physically motivated kernel. A secondary pathology is that modes with wavelengths $2l/n$ are completely absent from the mean fields calculated using this kernel, although this problem is lessened for large scale separation $l\ll l_{L}$ , since then only a few modes lie near $k=n\unicode[STIX]{x03C0}/l$ .

2.4.3 Moving line segment average

A one-dimensional, or line average over a segment of length  $l$ , is a variant of the moving box average but with the averages taken along a single direction  $\hat{\boldsymbol{r}}_{0}$ :

(2.29) $$\begin{eqnarray}\overline{\boldsymbol{A}}(\boldsymbol{x},\hat{\boldsymbol{r}}_{0})=\frac{1}{l}\int _{0}^{l}\text{d}s\,\boldsymbol{A}(\boldsymbol{x}+s\hat{\boldsymbol{r}}_{0})=\int \text{d}s\,\unicode[STIX]{x1D703}_{l}\left(s-\frac{l}{2}\right)\boldsymbol{A}(\boldsymbol{x}+s\hat{\boldsymbol{r}}_{0}).\end{eqnarray}$$

Note that the argument of $\unicode[STIX]{x1D703}_{l}$ is shifted here because the line segment over which the average is taken starts from  $\boldsymbol{x}$ , rather than being centred at  $\boldsymbol{x}$ . The Fourier transform of the kernel is obtained by directly calculating the Fourier transform of  $\overline{\boldsymbol{A}}$ , which gives

(2.30) $$\begin{eqnarray}{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k},\hat{\boldsymbol{r}}_{0})=\frac{\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\hat{\boldsymbol{r}}_{0}l}-1}{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\hat{\boldsymbol{r}}_{0}l}.\end{eqnarray}$$

When $|\boldsymbol{k}l|\ll 1$ , the expansion of ${\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k},\hat{\boldsymbol{r}}_{0})$ gives ${\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}=-(\text{i}/2)\boldsymbol{k}\boldsymbol{\cdot }\hat{\boldsymbol{r}}_{0}l$ and thus $\hat{\unicode[STIX]{x1D6FE}}=(1/2)l\hat{\boldsymbol{r}}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ and $\hat{\unicode[STIX]{x1D6FE}}^{\prime }=0$ . The moving line segment average has the same problems of physical applicability as the moving box average for MFE.

2.4.4 Fixed grid averages

We may also average fields inside a set of fixed boxes, i.e. fixed-grid average. For this case, the mean of $\boldsymbol{A}(\boldsymbol{x})$ is given by

(2.31) $$\begin{eqnarray}\overline{\boldsymbol{A}}(\boldsymbol{x})=\mathop{\sum }_{i=1}^{N_{\text{box}}}\unicode[STIX]{x1D703}_{l}(\boldsymbol{x}-\boldsymbol{x}_{i})\overline{\boldsymbol{A}}_{\text{m.b.}}(\boldsymbol{x}_{i}),\end{eqnarray}$$

where $\{\boldsymbol{x}_{i}\}$ , $i=1,2,\ldots ,N_{\text{box}}$ is the set of points of a grid with side length $l$ and the subscript m.b. denotes that $\overline{\boldsymbol{A}}_{\text{m.b.}}$ is a moving box average. Note that since in each grid cell $\overline{\boldsymbol{A}}(\boldsymbol{x})$ is a constant, $\overline{\overline{\boldsymbol{A}}}=\overline{\boldsymbol{A}}$ , this fixed-grid average results in a mean field valued discretely in space. To recover a mean field which is smooth in space, one may apply a second average using a proper (e.g. Gaussian) kernel. Nevertheless, the averaged field still misses those modes which are resonant to the side length of the grid, and thus again is physically problematic for observational applications of MFE.

2.4.5 Planar average

The planar average is widely used in simulations (e.g. Brandenburg Reference Brandenburg2009; Hubbard & Brandenburg Reference Hubbard and Brandenburg2011; Bhat, Ebrahimi & Blackman Reference Bhat, Ebrahimi and Blackman2016). It is manifestly anisotropic since it integrates out, say, $x$ and $y$ but leaves the full $z$ dependence. We write

(2.32) $$\begin{eqnarray}\overline{\boldsymbol{A}}(z)=\frac{1}{L^{2}}\int \text{d}x\,\text{d}y\,\boldsymbol{A}(\boldsymbol{x}),\end{eqnarray}$$

if $L$ is the side length of the simulation box. The full Reynolds rules, especially the interchangeability of differential and average operations, are respected if boundary conditions are periodic. That is

(2.33) $$\begin{eqnarray}\overline{\unicode[STIX]{x2202}_{x}\boldsymbol{A}(\boldsymbol{x})}=\frac{1}{L^{2}}\int \text{d}x\,\text{d}y\unicode[STIX]{x2202}_{x}\boldsymbol{A}(\boldsymbol{x})=\frac{1}{L^{2}}\int \text{d}y\,\boldsymbol{A}(\boldsymbol{x})|_{x=0}^{x=L}=0=\unicode[STIX]{x2202}_{x}\overline{\boldsymbol{A}}(\boldsymbol{x}).\end{eqnarray}$$

The planar average, although fine for simulation boxes, does not remove large $k$ modes in the $z$ direction from the mean and so does not fully filter small scale fields from large scale fields for a real system.

2.5 On averages in simulations versus observations

Planar or box averages used in simulations yield reliable results if compared to a theory based on corresponding averages, and interpreted appropriately. However, it is a different question as to how well lessons learned from box averages apply to observations, for which a differently defined mean is more appropriate.

3.1 Derivation

We average the MHD magnetic induction equation and use (2.21), which yields

(3.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\overline{\boldsymbol{B}}=\unicode[STIX]{x1D735}\times \overline{\boldsymbol{U}\times \boldsymbol{B}}+\unicode[STIX]{x1D708}_{m}\unicode[STIX]{x1D6FB}^{2}\overline{\boldsymbol{B}}=\unicode[STIX]{x1D735}\times [(1+O(\hat{\unicode[STIX]{x1D6FE}}^{2}))\overline{\boldsymbol{U}}\times \overline{\boldsymbol{B}}-\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{\boldsymbol{U}},\overline{\boldsymbol{B}})+\boldsymbol{{\mathcal{E}}}]+\unicode[STIX]{x1D708}_{m}\unicode[STIX]{x1D6FB}^{2}\overline{\boldsymbol{B}},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}_{m}$ is the magnetic diffusivity assumed to be a constant, and $\boldsymbol{{\mathcal{E}}}=\overline{\boldsymbol{u}\times \boldsymbol{b}}$ is the turbulent EMF. The relative magnitude of the correction terms to the standard terms that we arrive at in this section are unchanged if $\overline{\boldsymbol{U}}$ is included.

To express $\boldsymbol{{\mathcal{E}}}$ in terms of large scale quantities we adopt the minimal- $\unicode[STIX]{x1D70F}$ approach (MTA Blackman & Field Reference Blackman and Field2002). In deriving $\boldsymbol{{\mathcal{E}}}$ , terms involving $\overline{\boldsymbol{U}}$ come proportional to scalar or pseudoscalar cross-correlations between functions of $\boldsymbol{u}$ and $\boldsymbol{b}$ (e.g. Yoshizawa & Yokoi Reference Yoshizawa and Yokoi1993; Blackman Reference Blackman2000) and for present purposes we ignore these, i.e. we ignore terms linear in $\overline{\boldsymbol{U}}$ or $\overline{\boldsymbol{u}}$ ( $\simeq \hat{\unicode[STIX]{x1D6FE}}\overline{\boldsymbol{U}}=0$ ) in the evolution equations for $\boldsymbol{u}$ and $\boldsymbol{b}$ (but not in that for $\overline{\boldsymbol{B}}$ ). The incompressible momentum equation for velocity fluctuations then reads

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}u_{l} & = & \displaystyle \hat{P}_{ml} [\overline{B}_{n}\unicode[STIX]{x2202}_{n}b_{m}+b_{n}\unicode[STIX]{x2202}_{n}\overline{B}_{m}+b_{n}\unicode[STIX]{x2202}_{n}b_{m}-\overline{b_{n}\unicode[STIX]{x2202}_{n}b_{m}}\nonumber\\ \displaystyle & & \displaystyle -\,u_{n}\unicode[STIX]{x2202}_{n}u_{m}+\overline{u_{n}\unicode[STIX]{x2202}_{n}u_{m}}+\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{B}_{n},\unicode[STIX]{x2202}_{n}\overline{B}_{m})]+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}u_{l},\end{eqnarray}$$

where $\hat{P}_{ml}=\unicode[STIX]{x1D6FF}_{ml}-\unicode[STIX]{x2202}_{m}\unicode[STIX]{x2202}_{l}\unicode[STIX]{x1D6FB}^{-2}$ is the projection operator used to eliminate the sum of thermal and magnetic pressures, and $\unicode[STIX]{x1D708}$ is the viscosity. The units are such that the mass density $\unicode[STIX]{x1D70C}_{f}=1$ and the magnetic permeability $\unicode[STIX]{x1D707}=1$ . The induction equation for $\boldsymbol{b}$ is

(3.3) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\boldsymbol{b}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \overline{\boldsymbol{B}}+\boldsymbol{u}\times \boldsymbol{b}-\overline{\boldsymbol{u}\times \boldsymbol{b}})+\unicode[STIX]{x1D708}_{m}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{b}.\end{eqnarray}$$

Using (2.12), and carrying through the algebra keeping only first-order terms in $\hat{\unicode[STIX]{x1D6FE}}$ or $\hat{\unicode[STIX]{x1D6FE}}^{\prime }$ , we have

(3.4) $$\begin{eqnarray}\displaystyle \{\overline{\boldsymbol{u}\times \unicode[STIX]{x2202}_{t}\boldsymbol{b}}\}_{i} & = & \displaystyle \unicode[STIX]{x1D716}_{ijk} [\hspace{2.39996pt}(1-\hat{\unicode[STIX]{x1D6FE}})(\overline{u_{j}\unicode[STIX]{x2202}_{n}u_{k}}\,\overline{B_{n}})-(1-\hat{\unicode[STIX]{x1D6FE}})(\overline{u_{j}u_{n}}\,\unicode[STIX]{x2202}_{n}\overline{B_{k}})\nonumber\\ \displaystyle & & \displaystyle +\,\overline{B_{n}}\hat{\unicode[STIX]{x1D6FE}}\overline{u_{j}\unicode[STIX]{x2202}_{n}u_{k}}-\unicode[STIX]{x2202}_{n}\overline{B_{k}}\hat{\unicode[STIX]{x1D6FE}}\overline{u_{j}u_{n}}+\overline{u_{j}b_{n}\unicode[STIX]{x2202}_{n}u_{k}}-\overline{u_{j}u_{n}\unicode[STIX]{x2202}_{n}b_{k}} ]+\unicode[STIX]{x1D708}_{m}\unicode[STIX]{x1D716}_{ijk}\overline{u_{j}\unicode[STIX]{x2202}_{nn}b_{k}},\qquad\end{eqnarray}$$

and

(3.5) $$\begin{eqnarray}\displaystyle \{\overline{\unicode[STIX]{x2202}_{t}\boldsymbol{u}\times \boldsymbol{b}}\}_{i} & = & \displaystyle \unicode[STIX]{x1D716}_{ijk}\overline{b_{k}\hat{P}_{lj}(\overline{B}_{n}\unicode[STIX]{x2202}_{n}b_{l}+b_{n}\unicode[STIX]{x2202}_{n}\overline{B}_{l})}+\unicode[STIX]{x1D716}_{ijk}\overline{b_{k}\hat{P}_{lj}(b_{n}\unicode[STIX]{x2202}_{n}b_{l}-u_{n}\unicode[STIX]{x2202}_{n}u_{l})}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}_{ijk}[(1-\hat{\unicode[STIX]{x1D6FE}})(\overline{b}_{k}\overline{C}_{j})+\overline{C}_{j}\hat{\unicode[STIX]{x1D6FE}}\overline{b}_{k}]+\unicode[STIX]{x1D716}_{ijk}\overline{b_{k}\hat{P}_{lj}\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{B}_{n},\unicode[STIX]{x2202}_{n}\overline{B}_{l})}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D716}_{ijk}\overline{b_{k}\unicode[STIX]{x2202}_{nn}u_{j}},\qquad\end{eqnarray}$$

where $C_{j}=\hat{P}_{lj}(u_{n}\unicode[STIX]{x2202}_{n}u_{l}-b_{n}\unicode[STIX]{x2202}_{n}b_{l})$ . Assuming all small scale quantities are isotropic and homogeneous below scale $l$ but could vary on large scales ( ${\sim}l_{L}$ ), the two previous equations become

(3.6) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{u}\times \unicode[STIX]{x2202}_{t}\boldsymbol{b}} & = & \displaystyle (1-\hat{\unicode[STIX]{x1D6FE}})\big(-{\textstyle \frac{1}{3}}\overline{\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}}\,\overline{\boldsymbol{B}}+{\textstyle \frac{1}{3}}\overline{u^{2}}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}\big)+\unicode[STIX]{x1D708}_{m}\overline{\boldsymbol{u}\times \unicode[STIX]{x1D6FB}^{2}\boldsymbol{b}}+\boldsymbol{T}^{M}\nonumber\\ \displaystyle & & \displaystyle -\,\hat{\unicode[STIX]{x1D6FE}}\left({\textstyle \frac{1}{3}}\overline{\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}}\right)\overline{\boldsymbol{B}}+\hat{\unicode[STIX]{x1D6FE}}\big({\textstyle \frac{1}{3}}\overline{u^{2}}\big)\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}},\end{eqnarray}$$

where $\boldsymbol{T}^{M}=\overline{\boldsymbol{u}\times \unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{b}-\overline{\boldsymbol{u}\times \boldsymbol{b}})}$ , and

(3.7) $$\begin{eqnarray}\overline{\unicode[STIX]{x2202}_{t}\boldsymbol{u}\times \boldsymbol{b}}=(1-\hat{\unicode[STIX]{x1D6FE}})\left({\textstyle \frac{1}{3}}\overline{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{b}}\,\overline{\boldsymbol{B}}\right)+\hat{\unicode[STIX]{x1D6FE}}\left({\textstyle \frac{1}{3}}\overline{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{b}}\right)\overline{\boldsymbol{B}}+\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}\times \boldsymbol{b}}+\boldsymbol{T}^{U},\end{eqnarray}$$

where $T_{i}^{U}=\unicode[STIX]{x1D716}_{ijk}\overline{b_{k}\hat{P}_{lj}(b_{n}\unicode[STIX]{x2202}_{n}b_{l}-u_{n}\unicode[STIX]{x2202}_{n}u_{l})}$ . The derivation of the first and second terms in (3.7) is given in appendix B. Also note that the small scale part in the $\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{B}_{n},\unicode[STIX]{x2202}_{n}\overline{B}_{l})$ term will have its maximum wavenumber at ${\sim}2k_{L}$ . Therefore if the scale separation is large enough such that $2k_{L}\ll k_{s}$ , the $\hat{\unicode[STIX]{x1D6FE}}^{\prime }(\overline{B}_{n},\unicode[STIX]{x2202}_{n}\overline{B}_{l})$ term can be roughly treated as a large scale quantity in (3.5).

Adding (3.6) and (3.7) gives

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{{\mathcal{E}}} & = & \displaystyle (1-\hat{\unicode[STIX]{x1D6FE}})(\tilde{\unicode[STIX]{x1D6FC}}\overline{\boldsymbol{B}}-\tilde{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}})+(\hat{\unicode[STIX]{x1D6FE}}\tilde{\unicode[STIX]{x1D6FC}})\overline{\boldsymbol{B}}-(\hat{\unicode[STIX]{x1D6FE}}\tilde{\unicode[STIX]{x1D6FD}})\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D708}_{m}\overline{\boldsymbol{u}\times \unicode[STIX]{x1D6FB}^{2}\boldsymbol{b}}+\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}\times \boldsymbol{b}}+\boldsymbol{T}^{M}+\boldsymbol{T}^{U},\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D6FC}}=(\overline{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{b}}-\overline{\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}})/3$ and $\tilde{\unicode[STIX]{x1D6FD}}=\overline{u^{2}}/3$ . In the spirit of the MTA, the sum of the triple correlation terms in (3.8) is equated to a damping term $-\boldsymbol{{\mathcal{E}}}/\unicode[STIX]{x1D70F}$ . For $|\unicode[STIX]{x1D70F}\unicode[STIX]{x2202}_{t}\boldsymbol{{\mathcal{E}}}|\ll |\boldsymbol{{\mathcal{E}}}|$ , equation (3.8) then gives

(3.9) $$\begin{eqnarray}\boldsymbol{{\mathcal{E}}}=(1-\hat{\unicode[STIX]{x1D6FE}})(\unicode[STIX]{x1D6FC}\overline{\boldsymbol{B}}-\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}})+(\hat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FC})\overline{\boldsymbol{B}}-(\hat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}\end{eqnarray}$$

in the ideal MHD limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D708}_{m}\rightarrow 0$ , where $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70F}\tilde{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D70F}\tilde{\unicode[STIX]{x1D6FD}}$ are the helical and diffusion dynamo coefficients and $\unicode[STIX]{x1D70F}$ is the damping time for the EMF when mean fields are removed. Empirically, this is approximately equal to the turnover time at the turbulent driving scales in forced isotropic simulations (Brandenburg & Subramanian Reference Brandenburg and Subramanian2005b ). We also define $\unicode[STIX]{x1D6FC}_{k}=-\unicode[STIX]{x1D70F}\overline{\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}}/3$ and $\unicode[STIX]{x1D6FC}_{m}=\unicode[STIX]{x1D70F}\overline{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{b}}/3$ being the kinetic and magnetic contributions to the $\unicode[STIX]{x1D6FC}$ -effect, respectively.

When there is large scale separation $\hat{\unicode[STIX]{x1D6FE}},\hat{\unicode[STIX]{x1D6FE}}^{\prime }\rightarrow 0$ , equations (3.1) and (3.9) reduce exactly to the standard dynamo equations derived with ensemble average. This important feature indicates that different kinds of suitable averaging – like local Gaussian average or ensemble average – converge to the same set of equations when scale separation is large.

The turbulent EMF now has routes of expansion: (i) higher gradients of $\overline{\boldsymbol{B}}$ ; (ii)  $\hat{\unicode[STIX]{x1D6FE}}$ due to the violation of Reynolds rules. Expanding to every higher-order results in (using order-of-magnitude estimates) an extra factor of $l_{s}/l_{L}$ for the former, and $(l/l_{L})^{c}$ for the later. Interestingly, both of these two ratios are related to the scale separation, and the question of which dominates higher-order terms in $\boldsymbol{{\mathcal{E}}}$ varies for different models. In this work we assume the $\hat{\unicode[STIX]{x1D6FE}}$ corrections dominate.

3.2 Comparison to previous work on non-local EMF kernels

We see from (3.9) that the violation of the Reynolds rules from mesoscale fluctuations is a direct source of contributions to the EMF from terms with higher than linear order in derivatives of $\overline{\boldsymbol{B}}$ . In Fourier space these terms imply that ${\displaystyle\mathop{{\mathcal{E}}}\limits_{{\sim}}}_{i}(\boldsymbol{k})={\displaystyle\mathop{K}\limits_{{\sim}}}_{ij}(\boldsymbol{k}){\displaystyle\mathop{\{}\limits_{{\sim}}}_{j}(\boldsymbol{k})$ where ${\displaystyle\mathop{K}\limits_{{\sim}}}_{ij}$ could contain terms of order higher than linear in $\boldsymbol{k}$ , in contrast to the conventional mean field dynamo theory where ${\displaystyle\mathop{K}\limits_{{\sim}}}_{ij}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}_{ij}-i\unicode[STIX]{x1D716}_{imj}\unicode[STIX]{x1D6FD}k_{m}$ .

Consequently, in configuration space we have $\boldsymbol{{\mathcal{E}}}(\boldsymbol{x})=\mathbf{K}\ast \overline{\boldsymbol{B}}$ , and the turbulent EMF depends on $\overline{\boldsymbol{B}}$ through its weighted average in the vicinity of $\boldsymbol{x}$ , i.e. non-locally. More generally, if we have used a time average in (3.9), $\boldsymbol{K}$ could also be time dependent, and correspondingly $\boldsymbol{{\mathcal{E}}}$ becomes non-local in both space and time.

The EMF kernel $\boldsymbol{K}$ that we derive includes terms caused by violation of Reynolds rules, and varies depending on the choice of our (potentially anisotropic) averaging kernel  $\boldsymbol{G}$ . Although previous work has identified the need for an EMF kernel to capture non-locality (Krause & Rädler Reference Krause and Rädler1980; Rädler Reference Rädler, Page and Hirsch2000; Rädler & Rheinhardt Reference Rädler and Rheinhardt2007; Brandenburg, Rädler & Schrinner Reference Brandenburg, Rädler and Schrinner2008; Hubbard & Brandenburg Reference Hubbard and Brandenburg2009; Rheinhardt & Brandenburg Reference Rheinhardt and Brandenburg2012), this previous work did not address the contribution to this kernel from the violation of the Reynolds rules. For example, Rheinhardt & Brandenburg (Reference Rheinhardt and Brandenburg2012) used numerical simulations (DNS) to test an ansatz for the EMF kernel in the case of homogeneous isotropic turbulence with mean fields defined by an average over the $x{-}y$ plane. They assessed whether the EMF kernel takes the form

(3.10) $$\begin{eqnarray}{\displaystyle\mathop{K}\limits_{{\sim}}}_{ij}(\unicode[STIX]{x1D714},\boldsymbol{k})=\frac{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}_{ij}-\text{i}\unicode[STIX]{x1D716}_{imj}\unicode[STIX]{x1D6FD}k_{m}}{1-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{\text{RB}}+l_{\text{RB}}^{2}k^{2}}\end{eqnarray}$$

at low wavenumbers, where $\unicode[STIX]{x1D70F}_{\text{RB}}$ is approximately equal to the eddy turnover time $\unicode[STIX]{x1D70F}$ , and $l_{\text{RB}}$ is a parameter whose value is to be extracted from fits to simulation data. Their resulting evolution equation for the EMF reads

(3.11) $$\begin{eqnarray}(1+\unicode[STIX]{x1D70F}_{\text{RB}}\unicode[STIX]{x2202}_{t}-l_{\text{RB}}^{2}\unicode[STIX]{x2202}_{z}^{2})\boldsymbol{{\mathcal{E}}}=\unicode[STIX]{x1D6FC}\overline{\boldsymbol{B}}^{(p)}-\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}^{(p)},\end{eqnarray}$$

where the superscript ‘ $(p)$ ’ distinguishes their planar average from our kernel averages. Rheinhardt & Brandenburg (Reference Rheinhardt and Brandenburg2012) found that (3.10) was at least consistent with simulation data up to $k/k_{1}\approx 3$ where $k_{1}$ is the simulation box wavenumber, for $\unicode[STIX]{x1D70F}_{\text{RB}}\sim \unicode[STIX]{x1D70F}$ and $l_{\text{RB}}\sim l_{s}$ , the energy-dominating eddy scale.

To compare (3.11) with our result equation (3.8), we ignore the second to fifth terms on the right-hand side of the latter (i.e. assuming $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are constants and taking the ideal MHD limit), and identify the sixth and seventh terms (triple correlations) with $\hat{T}\boldsymbol{{\mathcal{E}}}$ where $\hat{T}$ is an operator. This gives

(3.12) $$\begin{eqnarray}(-\unicode[STIX]{x1D70F}\hat{T}+\unicode[STIX]{x1D70F}\unicode[STIX]{x2202}_{t})\boldsymbol{{\mathcal{E}}}=(1-\hat{\unicode[STIX]{x1D6FE}})(\unicode[STIX]{x1D6FC}\overline{\boldsymbol{B}}-\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D735}\times \overline{\boldsymbol{B}}).\end{eqnarray}$$

The identification of the triple correlations with a damping term, $\hat{T}=-1/\unicode[STIX]{x1D70F}$ , serves as the closure in MTA. Comparison to (3.11) shows that the left sides of the two equations can be made to mutually correspond if we replace the triple correlations by the sum of a damping term and a diffusion term, that is $\hat{T}\boldsymbol{{\mathcal{E}}}=-\boldsymbol{{\mathcal{E}}}/\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D702}_{\text{t.c.}}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{{\mathcal{E}}}$ , where $\unicode[STIX]{x1D702}_{\text{t.c.}}=l_{\text{RB}}^{2}/\unicode[STIX]{x1D70F}$ is a diffusion coefficient determined by statistical properties of turbulent fields. The spatially non-local term in (3.11), $-l_{\text{RB}}^{2}\unicode[STIX]{x2202}_{z}^{2}\boldsymbol{{\mathcal{E}}}$ , can thus be understood as textured specification of the form of terms for which the crude MTA approximates. This additional term plays a similar role to that of the standard MTA term, namely that it depletes the turbulent EMF in the absence of any other mean fields.

We emphasize that the derivation of (3.12) differs from that of (3.11) in that the correction terms appearing on the right of (3.12) are derived from the averaging procedure itself, and represent the lowest-order corrections when Reynolds rules are violated. Higher-order terms can also be derived. The form of $\hat{\unicode[STIX]{x1D6FE}}$ is determined by the scale $l$ and the kernel of average. These terms are not included in the semi-empirical approach of Rheinhardt & Brandenburg (Reference Rheinhardt and Brandenburg2012) that produced (3.11), because they vanish identically due to the planar average.

Finally, we note that in deriving (3.12), we averaged the MHD equations using a kernel that retains a spatial dependence, so that $\overline{\boldsymbol{B}}$ can depict large scale magnetic fields and retain large scale gradients in all directions. In contrast, the $x{-}y$ planar average used in Rheinhardt & Brandenburg (Reference Rheinhardt and Brandenburg2012) does not retain large scale field gradients in $x$ and $y$ directions, which is self-consistent for the simulation boxes but not sufficiently general for investigating mean fields. In addition planar averages do not remove large $k_{z}$ modes from $\overline{\boldsymbol{B}}^{(p)}$ , and hence (3.11) might not be complete even for the simulations in the absence of including higher-order terms since the EMF kernel (3.10) is valid only for small $|\boldsymbol{k}|$ .

4.1 Intrinsic error from statistical fluctuations in inputs to mean field equations

The dynamo input parameters in the mean field equations (e.g. $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ in (3.9) along with boundary and initial conditions) are themselves random variables (in an ensemble) and so is $\overline{\boldsymbol{B}}=\overline{\boldsymbol{B}}(\boldsymbol{x};\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\ldots )$ , because the small scale fields $\boldsymbol{u}$ and $\boldsymbol{b}$ are statistically fluctuating. The intrinsic error is thus defined as the variation of statistical fluctuations of $\overline{\boldsymbol{B}}$ (about its ensemble mean) due to these small scale fluctuations, which we denote by $\unicode[STIX]{x1D70E}_{\text{IE}}^{2}$ :

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{IE},B_{i}}^{2}=\langle (\overline{B}_{i}-\langle \overline{B}_{i}\rangle )^{2}\rangle ,\quad \text{for}~i=1,2,3.\end{eqnarray}$$

With this definition the IE vanishes if the mean field theory is defined using ensemble average, i.e. $\unicode[STIX]{x1D70E}_{\text{IE},B_{i}}^{2}=\langle (\langle B_{i}\rangle -\langle \langle B_{i}\rangle \rangle )^{2}\rangle =0$ .

The IE can be calculated by propagating the statistical variations of input parameters to the solutions of mean field equations. We consider the IE of the steady-state solutions of MFE dynamo equations for a minimalist model where $\unicode[STIX]{x1D6FC}_{k}$ and $\unicode[STIX]{x1D6FD}$ are the only input parameters: $\overline{\boldsymbol{B}}=\overline{\boldsymbol{B}}(\boldsymbol{x};\unicode[STIX]{x1D6FC}_{k},\unicode[STIX]{x1D6FD})$ . The magnetic $\unicode[STIX]{x1D6FC}$ -effect, $\unicode[STIX]{x1D6FC}_{m}$ , is dynamical in our model, and not an input parameter, since it is governed by the transport equation of the helicity density, equation (6.2).

Let us consider a minimalist model where all turbulent transport coefficients are statistically homogeneous over the whole space. In one such dynamo model that we discuss later, turbulent transport coefficients depend on the radial coordinate, but since its variation scale is greater than  $l$ , they remain locally approximately homogeneous.

The deviation of kernel-filtered values from the ensemble averages of turbulent coefficients contributes to the IE of the mean fields. The resulting average (in the sense of an ensemble average) imprecision in $\overline{\boldsymbol{B}}$ can be calculated by propagating the imprecision to turbulent coefficients. For $\unicode[STIX]{x1D6FC}_{k}$ , this is

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}}^{2}=\langle (\unicode[STIX]{x1D6FC}_{k}-\langle \unicode[STIX]{x1D6FC}_{k}\rangle )^{2}\rangle .\end{eqnarray}$$

Similarly, we have

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2}=\langle (\unicode[STIX]{x1D6FD}-\langle \unicode[STIX]{x1D6FD}\rangle )^{2}\rangle ,\end{eqnarray}$$

and

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}\unicode[STIX]{x1D6FD}}^{2}=\langle (\unicode[STIX]{x1D6FC}_{k}-\langle \unicode[STIX]{x1D6FC}_{k}\rangle )(\unicode[STIX]{x1D6FD}-\langle \unicode[STIX]{x1D6FD}\rangle )\rangle .\end{eqnarray}$$

The uncertainty in $\overline{\boldsymbol{B}}$ derives from the uncertainties from $\unicode[STIX]{x1D6FC}_{k}$ and $\unicode[STIX]{x1D6FD}$ as follows:

(4.5) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{IE},\overline{B}_{i}}^{2}=(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}_{k}}\overline{B}_{i})^{2}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}}^{2}+(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}\overline{B}_{i})^{2}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2}+2(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}_{k}}\overline{B}_{i})(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}\overline{B}_{i})\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

To estimate magnitudes of (4.2) to (4.4), we decompose filtered quantities into (ensemble averaged) means and random parts. Consequently we have

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{k,r}=\unicode[STIX]{x1D6FC}_{k}-\langle \unicode[STIX]{x1D6FC}_{k}\rangle =\frac{\unicode[STIX]{x1D70F}}{3}\overline{\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}-\langle \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}\rangle },\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{k,r}$ is the random part. Combining (4.2) and (4.6) we have

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}}^{2}=\langle \unicode[STIX]{x1D6FC}_{k,r}^{2}\rangle .\end{eqnarray}$$

Similarly,

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2}=\langle \unicode[STIX]{x1D6FD}_{r}^{2}\rangle ,\end{eqnarray}$$

where

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{r}=\unicode[STIX]{x1D6FD}-\langle \unicode[STIX]{x1D6FD}\rangle =\frac{\unicode[STIX]{x1D70F}}{3}\overline{u^{2}-\langle u^{2}\rangle }\end{eqnarray}$$

is the random part of $\unicode[STIX]{x1D6FD}$ .

To estimate the quantities in (4.7) and (4.8), we consider the system of study to be divided into cells of typical length $l_{s}$ and crudely assume that in each cell, $\boldsymbol{u}$ is nearly uniform with components drawn from independent Gaussian distributions,

(4.10) $$\begin{eqnarray}f(u_{i})=\frac{1}{\sqrt{2\unicode[STIX]{x03C0}}u_{0}}\text{e}^{-u_{i}^{2}/2u_{0}^{2}},\quad i=1,2,3.\end{eqnarray}$$

Then for each cell,

(4.11a-c ) $$\begin{eqnarray}\langle u_{i}\rangle =0,\quad \langle u_{i}^{2}\rangle =u_{0}^{2},\quad \langle u_{i}^{4}\rangle =3u_{0}^{4},\end{eqnarray}$$

and

(4.12a,b ) $$\begin{eqnarray}\langle u^{2}\rangle =\mathop{\sum }_{i=1}^{3}\langle u_{i}^{2}\rangle =3u_{0}^{2},\quad \langle u^{4}\rangle =\left\langle \left(\mathop{\sum }_{i=1}^{3}u_{i}^{2}\right)^{2}\right\rangle =15u_{0}^{4}=\frac{5}{3}\langle u^{2}\rangle ^{2},\end{eqnarray}$$

so that

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{u^{2}}^{2}=\langle u^{4}\rangle -\langle u^{2}\rangle ^{2}={\textstyle \frac{2}{3}}\langle u^{2}\rangle ^{2},\end{eqnarray}$$

which links fluctuations to mean quantities.

The filtering $\overline{(\cdot )}$ in (4.7) and (4.8) can be roughly seen as the algebraic average of the quantity $(\cdot )$ of $N=(2l/l_{s})^{3}$ cells, with the factor of two accounting for the fact that the variation scale of $u^{2}$ will be $l_{s}/2$ if that of $\boldsymbol{u}$ is  $l_{s}$ . The central limit theorem (CLT) then yields

(4.14a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}}^{2}\simeq \frac{\langle \unicode[STIX]{x1D6FC}_{k,r}^{2}\rangle }{N}=\frac{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k,r}}^{2}}{N},\quad \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2}\simeq \frac{\langle \unicode[STIX]{x1D6FD}_{r}^{2}\rangle }{N}=\frac{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}_{r}}^{2}}{N},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k,r}}^{2}$ and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}_{r}}^{2}$ are the variances of the random parts in each cell.

Since both $\unicode[STIX]{x1D6FC}_{k}$ and $\unicode[STIX]{x1D6FD}$ are quadratic in $\boldsymbol{u}$ , equations (4.13) and (4.14) then yield

(4.15a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}}^{2}\simeq \frac{2\unicode[STIX]{x1D6FC}_{k}^{2}/3}{(2l/l_{s})^{3}},\quad \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2}\simeq \frac{2\unicode[STIX]{x1D6FD}^{2}/3}{(2l/l_{s})^{3}}.\end{eqnarray}$$

It then follows from (4.5) that

(4.16) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{IE},\overline{B}_{i}}^{2}\simeq \frac{1}{12(l/l_{s})^{3}}[(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}_{k}}\overline{B}_{i})^{2}\unicode[STIX]{x1D6FC}_{k}^{2}+(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}\overline{B}_{i})^{2}\unicode[STIX]{x1D6FD}^{2}]+2(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}_{k}}\overline{B}_{i})(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}\overline{B}_{i})\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D70E}_{\text{IE},\overline{B}_{i}}^{2}$ depends on spatial coordinates $\boldsymbol{x}$ , just as $\overline{\boldsymbol{B}}$ does. In galaxies, a typical value of the variation scale of turbulent fields satisfies $l_{s}\lesssim 0.1~\text{kpc}$ . Hence $(l/l_{s})^{3}\gtrsim 8$ for $l=0.2~\text{kpc}$ and ${\gtrsim}64$ for $l=0.4~\text{kpc}$ .

From the CLT, this IE decreases with increasing $l$ because the average variations from turbulence are inversely proportional to the number of eddy cells in the region being averaged, $(l/l_{s})^{3}$ , provided that $l_{s}$ is rather insensitive to the choice of  $l$ .

4.2 Filtering error from mismatch between measurement and theoretical kernels

Measuring physical quantities always results in measuring mean quantities to a certain extent. Detectors have limited sensitivity so measurements represent a convolution between true physical quantities and an instrument kernel. Furthermore, the physical quantity being measured typically involves a superposition of microphysical contributions and an average over many local macroscopic contributions. In particular, the predicted values of observed FR and synchrotron polarization are limited in precision when these predictions are made using MFE.

For a given physical quantity of a real system $Q^{A}$ (e.g. the actual magnetic field of a galaxy) we define the measured value as $(Q^{A})_{{\mathcal{M}}}$ , where the subscript ${\mathcal{M}}$ indicates that quantity subjected to a measuring kernel that the instrument uses to project out the actual measured value. Complementarily, we write $\overline{Q^{A}}$ to indicate the value of $Q^{A}$ subjected to a theoretically chosen mean field theory filter. We will assume these two filters commute, i.e. $\overline{(Q^{A})_{{\mathcal{M}}}}=(\overline{Q^{A}})_{{\mathcal{M}}}$ . We use $Q$ to indicate a theoretically predicted value of $Q^{A}$ . Like $Q^{A}$ we can subject $Q$ mathematically to a theoretical mean field filtering and obtain $\overline{Q}$ or to measurement filtering to obtain $(Q)_{{\mathcal{M}}}$ , or both $(\overline{Q})_{{\mathcal{M}}}$ ( $=\overline{(Q)_{{\mathcal{M}}}}$ by assumption). For the common practice in which observations are not subjected to the theoretical mean filtering but the theory is subjected to the instrument filtering, the difference the measured value and the theoretically predicted mean can be written

(4.17) $$\begin{eqnarray}\displaystyle (Q^{A})_{{\mathcal{M}}}-(\overline{Q})_{{\mathcal{M}}} & = & \displaystyle [\overline{(Q^{A})}_{{\mathcal{M}}}-(\overline{Q})_{{\mathcal{M}}}+(q^{A})_{{\mathcal{M}}}-(q)_{{\mathcal{M}}}]+(q)_{{\mathcal{M}}}\nonumber\\ \displaystyle & = & \displaystyle [(\overline{Q^{A}}-\overline{Q})_{{\mathcal{M}}}+(q^{A}-q)_{{\mathcal{M}}}]+(q)_{{\mathcal{M}}},\end{eqnarray}$$

where $(q^{A})_{{\mathcal{M}}}=(Q^{A})_{{\mathcal{M}}}-\overline{(Q^{A})}_{{\mathcal{M}}}$ is the difference between the actual quantity and its value using the theoretical mean filter, then both filtered through the measuring kernel. Analogously, $(q)_{{\mathcal{M}}}=(Q)_{{\mathcal{M}}}-(\overline{Q})_{{\mathcal{M}}}$ is the difference between the theoretical quantity and its theoretical mean filter then both filtered through the measuring kernel. The terms in the square brackets on the right of (4.17), measure accuracy of the theoretical model and these terms will be small if the theory provides a good match to the real system. We focus on $(q)_{{\mathcal{M}}}$ , the last term in (4.17), which is a precision error of the theory and the FE that we will quantify. The smaller its magnitude, the more precise the theory.

In principle, one would like to subject $(Q^{A})_{{\mathcal{M}}}$ to the same filtering which corresponds to that of the mean field model, that is, compute $\overline{(Q^{A})_{{\mathcal{M}}}}$ , and compare it to $(\overline{Q})_{{\mathcal{M}}}$ . This would obviate computation of the FE. For simulations this may be possible, but for observations one cannot always compute $\overline{(Q^{A})_{{\mathcal{M}}}}$ , due to limited resolution. Moreover, it is typically not done in practice, and cannot be done if $\overline{(\cdot )}$ represents the ensemble average and the system has finite scale separation. If both $\overline{(\cdot )}$ and $(\cdot )_{{\mathcal{M}}}$ averages were equivalent to ensemble averages due to infinite scale separation, then $(q)_{{\mathcal{M}}}=\langle Q-\langle Q\rangle \rangle \rightarrow 0$ ; but this is not the case with finite scale separation and local spatial averages.

Unlike the IE of the previous subsection, the FE increases with increasing $l$ , since smaller $l$ means including a greater fraction of modes into what comprises the mean field, and so the theoretical predictions from mean field theory would be less coarse grained and thus more capable of characterizing the actual field. If the presumption is made that IE and FE are statistically independent and uncorrelated for a given $l$ , then the total uncertainty of the mean field theory is given by $\unicode[STIX]{x1D70E}^{2}=\unicode[STIX]{x1D70E}_{\text{IE}}^{2}+\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ . Due to their competitive behaviours when changing $l$ , an optimal scale of average $l_{\text{opt}}$ which minimizes either $\unicode[STIX]{x1D70E}^{2}$ or the relative uncertainty $\unicode[STIX]{x1D70E}^{2}/\overline{B}^{2}$ can arise, satisfying $l_{s}<l_{\text{opt}}<l_{L}$ .

In the next section we combine all of the formalism of this section into a specific example. We discuss implications of a finite precision when comparing observations and MFE theory for measuring galactic fields by FR from extragalactic sources. Our formalism is not restricted to that particular example and the precision of theoretical predictions for other kinds of observations, such as pulsar FRs, or polarized synchrotron emission, can also similarly be worked out.

6.1 Galactic dynamo model

We augment the simplified galactic dynamo model from § 4.5 of Zhou & Blackman (Reference Zhou and Blackman2017),Footnote ⁴ where the ‘no- $z$ ’ approximation (Subramanian & Mestel Reference Subramanian and Mestel1993; Moss Reference Moss1995; Phillips Reference Phillips2001; Sur, Shukurov & Subramanian Reference Sur, Shukurov and Subramanian2007; Chamandy et al. Reference Chamandy, Shukurov, Subramanian and Stoker2014) is used. The resulting $\boldsymbol{B}$ is $r$ -dependent and cylindrically symmetric (i.e. azimuthally averaged). We include the correction terms of § 3 employing a Gaussian kernel (and thus $\hat{\unicode[STIX]{x1D6FE}}=-l^{2}\unicode[STIX]{x1D6FB}^{2}/8\unicode[STIX]{x03C0}^{2}$ ), which gives

(6.1) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FE}}=-\frac{1}{2}\frac{l^{2}}{4\unicode[STIX]{x03C0}^{2}}\unicode[STIX]{x2202}_{z}^{2}\rightarrow \frac{l^{2}}{32h^{2}},\end{eqnarray}$$

where derivatives in the radial direction are dropped assuming the disk is thin, $h/R\ll 1$ . The last relation in (6.1) follows from the ‘no- $z$ ’ approximation, $\unicode[STIX]{x2202}_{z}^{2}\rightarrow -(k_{h}/4)^{2}$ where $k_{h}=2\unicode[STIX]{x03C0}/h$ (Phillips Reference Phillips2001; Sur et al. Reference Sur, Shukurov and Subramanian2007). To the helicity density evolution equation with flux terms (Brandenburg & Subramanian Reference Brandenburg and Subramanian2005a ; Subramanian & Brandenburg Reference Subramanian and Brandenburg2006; Sur et al. Reference Sur, Shukurov and Subramanian2007) we also add the correction terms resulting from violation of the Reynolds rules and obtain

(6.2) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FC}_{m}=-\frac{2\unicode[STIX]{x1D6FD}}{l_{s}^{2}}\left[\frac{(1-\hat{\unicode[STIX]{x1D6FE}})(\boldsymbol{{\mathcal{E}}}\boldsymbol{\cdot }\overline{\boldsymbol{B}})+\overline{\boldsymbol{B}}\hat{\unicode[STIX]{x1D6FE}}\boldsymbol{{\mathcal{E}}}}{B_{eq}^{2}}+\frac{\unicode[STIX]{x1D6FC}_{m}}{R_{m}}\right]-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D6FC}_{m}\overline{\boldsymbol{U}})-\unicode[STIX]{x1D6FD}_{d}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6FC}_{m}.\end{eqnarray}$$

The last term of (6.2) governs the diffusive flux and we adopt $\unicode[STIX]{x1D6FD}_{d}=\unicode[STIX]{x1D6FD}$ .

With the requirement that $l<h$ , we find that the $\hat{\unicode[STIX]{x1D6FE}}$ correction terms produce only small changes in the dynamo model solutions and we can omit them in the later discussion of the precision error. However, the smallness of the effect on the solutions is a feature of our particular dynamo model that is exacerbated by the aforementioned ‘no- $z$ ’ approximation. To see this note that for our choice of $l$ , ${\displaystyle\mathop{\unicode[STIX]{x1D6FE}}\limits_{{\sim}}}(k_{h})<1$ and the magnitude of $\hat{\unicode[STIX]{x1D6FE}}$ is always less than $1/16$ . The maximum value of $(\overline{B})_{L}(r)$ when $l$ is increased from $0.1h$ to $0.9h$ from the solutions changes by just ${\sim}1\,\%$ . If instead we had used the approximation that $\unicode[STIX]{x2202}_{z}^{2}\sim -k_{h}^{2}$ , there would be a ${\sim}40\,\%$ decrease in the maximum value of $(\overline{B})_{L}(r)$ when $l$ is increased from $0.1h$ to $0.9h$ from the solutions with the correction terms. This highlights that the correction terms are not necessarily small for every model. Moreover, in the absence of any significant scale separation between large and small scale parts of the magnetic energy spectrum, the expansion of ${\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(\boldsymbol{k})$ in (2.6) would itself be invalid, and corrections to the MFE would be non-perturbative.

Numerically, Shapovalov & Vishniac (Reference Shapovalov and Vishniac2011) found, from the (uncorrected) evolution equation of small scale helicity, that the resultant spectra of large scale quantities are insensitive to different filtering methods, for reasonable spectra of relevant total quantities.

The steady state,Footnote ⁵ non-dimensionalized dynamo equations read

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\unicode[STIX]{x2202}_{t}B_{r}=-\frac{2}{\unicode[STIX]{x03C0}}R_{\unicode[STIX]{x1D6FC}}(1+\unicode[STIX]{x1D6FC}_{m})B_{\unicode[STIX]{x1D719}}-\left(R_{U}+\frac{\unicode[STIX]{x03C0}^{2}}{4}\right)B_{r} & \displaystyle\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\unicode[STIX]{x2202}_{t}B_{\unicode[STIX]{x1D719}}=R_{\unicode[STIX]{x1D714}}B_{r}-\left(R_{U}+\frac{\unicode[STIX]{x03C0}^{2}}{4}\right)B_{\unicode[STIX]{x1D719}} & \displaystyle\end{eqnarray}$$

(6.5) $$\begin{eqnarray}\displaystyle \displaystyle 0 & = & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FC}_{m}=-R_{U}\unicode[STIX]{x1D6FC}_{m}-\frac{\unicode[STIX]{x1D6FD}_{d}}{\unicode[STIX]{x1D6FD}}\frac{\unicode[STIX]{x03C0}}{2}\unicode[STIX]{x1D6FC}_{m}\nonumber\\ \displaystyle & & \displaystyle -\,C\left[(1+\unicode[STIX]{x1D6FC}_{m})(B_{r}^{2}+B_{\unicode[STIX]{x1D719}}^{2})+\frac{3}{8}\sqrt{\frac{-\unicode[STIX]{x03C0}(1+\unicode[STIX]{x1D6FC}_{m})R_{\unicode[STIX]{x1D714}}}{R_{\unicode[STIX]{x1D6FC}}}}B_{r}B_{\unicode[STIX]{x1D719}}+\frac{\unicode[STIX]{x1D6FC}_{m}}{R_{m}}\right],\end{eqnarray}$$

where

(6.6a-d ) $$\begin{eqnarray}R_{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D6FC}_{k}h}{\unicode[STIX]{x1D6FD}},\quad R_{U}=\frac{|\overline{\boldsymbol{U}}|h}{\unicode[STIX]{x1D6FD}},\quad R_{\unicode[STIX]{x1D714}}=-\frac{h^{2}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x1D6FD}},\quad C=2\left(\frac{h}{l_{s}}\right)^{2}\end{eqnarray}$$

are dimensionless parameters with a flat rotation curve $\unicode[STIX]{x1D6FA}\propto 1/r$ adopted, and magnetic fields are normalized by the equipartition field strength $B_{\text{eq}}=\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{f}u^{2}}$ with $\unicode[STIX]{x1D70C}_{f}$ being the fluid density. The $\unicode[STIX]{x1D6FC}$ -coefficients are normalized by $\unicode[STIX]{x1D6FC}_{k}$ . The $r$ -dependence of (6.6) is described in detail in § 2.4 and (41) in Zhou & Blackman (Reference Zhou and Blackman2017). The approximation for the $\boldsymbol{{\mathcal{E}}}\boldsymbol{\cdot }\overline{\boldsymbol{B}}$ term can be found in the appendix of Sur et al. (Reference Sur, Shukurov and Subramanian2007) or that of Chamandy, Subramanian & Shukurov (Reference Chamandy, Subramanian and Shukurov2013).

Analytical expressions of $\overline{B}_{\unicode[STIX]{x1D719}}$ and $\overline{B}_{r}$ are obtainable from (6.3) to (6.5). The intrinsic error of $\overline{\boldsymbol{B}}(\boldsymbol{x})$ is then given by (4.16) in terms of $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}}^{2}$ , $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2}$ and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}\unicode[STIX]{x1D6FD}}$ . The first two are given in (4.15), whereas for $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}\unicode[STIX]{x1D6FD}}$ we assume that fluctuations of $\unicode[STIX]{x1D6FC}_{k}$ and $\unicode[STIX]{x1D6FD}$ are uncorrelated, and

(6.7) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}\unicode[STIX]{x1D6FD}}\simeq (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}_{k}}^{2}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2})^{1/2}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}}^{2}R_{\unicode[STIX]{x1D6FC}}/h.\end{eqnarray}$$

(For galaxies, $R_{\unicode[STIX]{x1D6FC}}\simeq 1$ .) At a fixed location, $\overline{\boldsymbol{B}}$ is a function of $R_{\unicode[STIX]{x1D6FC}}$ , $R_{U}$ and $R_{\unicode[STIX]{x1D714}}$ . Therefore the partial derivatives with respect to $\unicode[STIX]{x1D6FC}_{k}$ and $\unicode[STIX]{x1D6FD}$ can be evaluated using the chain rule,

(6.8a,b ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}_{k}}=\frac{h}{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D6FC}}},\quad \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}=-\frac{1}{\unicode[STIX]{x1D6FD}}(R_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D6FC}}}+R_{U}\unicode[STIX]{x2202}_{R_{U}}+R_{\unicode[STIX]{x1D714}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D714}}}).\end{eqnarray}$$

Combining (4.16), (6.7) and (6.8), we have for the intrinsic error of  $\overline{B}_{i}$ ,

(6.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{\text{IE},\overline{B}_{i}}^{2} & = & \displaystyle \frac{1}{12(l/l_{s})^{3}}\{(\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D6FC}}}\overline{B}_{i})^{2}R_{\unicode[STIX]{x1D6FC}}^{2}+[(R_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D6FC}}}+R_{U}\unicode[STIX]{x2202}_{R_{U}}+R_{\unicode[STIX]{x1D714}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D714}}})\overline{B}_{i}]^{2}\nonumber\\ \displaystyle & & \displaystyle -\,2(R_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D6FC}}}\overline{B}_{i})[(R_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D6FC}}}+R_{U}\unicode[STIX]{x2202}_{R_{U}}+R_{\unicode[STIX]{x1D714}}\unicode[STIX]{x2202}_{R_{\unicode[STIX]{x1D714}}})\overline{B}_{i}]\!\},\end{eqnarray}$$

and that of $(\overline{B})_{L}$ is given by substituting (6.9) into (5.4), given the solutions of (6.3) to (6.5).

6.2 Edge-on view

We first consider a special case representing the measurement of FR of a perfectly edge-on disc galaxy with radius $R=12~\text{kpc}$ (see the schematic diagrams of figure 1). Note that the integration path segments along the line of sight within the galaxy form chords with lengths $L(\unicode[STIX]{x1D71B})=2\sqrt{R^{2}-\unicode[STIX]{x1D71B}^{2}}$ , where $\unicode[STIX]{x1D71B}$ is the distance from the galactic centre to the closest point on the chord. From the geometry of the configuration, the line of sight average is

(6.10) $$\begin{eqnarray}\overline{B}_{L}(\unicode[STIX]{x1D71B})=\frac{2\unicode[STIX]{x1D71B}}{L(\unicode[STIX]{x1D71B})}\int _{0}^{L/2}\,\text{d}y\frac{\overline{B}_{\unicode[STIX]{x1D719}}(r)}{r},\end{eqnarray}$$

and

(6.11) $$\begin{eqnarray}(\overline{B}^{2})_{L}(\unicode[STIX]{x1D71B})=\frac{2}{L(\unicode[STIX]{x1D71B})}\int _{0}^{L/2}\,\text{d}y\overline{B}^{2}(r),\end{eqnarray}$$

where $r=\sqrt{\unicode[STIX]{x1D71B}^{2}+y^{2}}$ is the radial coordinate from the galactic centre. Only $\overline{B}_{\unicode[STIX]{x1D719}}$ contributes to $(\overline{B})_{L}$ for the edge-on view because $\overline{B}_{r}$ is mirror symmetric about the $x$ -axis and its contributions from the $y>0$ and $y<0$ regions cancel each other. The intrinsic error is given by

(6.12) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{IE}}^{2}=\frac{2\unicode[STIX]{x1D71B}^{2}}{L(\unicode[STIX]{x1D71B})}\int _{0}^{L/2}\,\text{d}y\frac{\unicode[STIX]{x1D70E}_{\text{int},\overline{B}_{\unicode[STIX]{x1D719}}}^{2}}{r^{2}}.\end{eqnarray}$$

The imprecision associated with the observation is given by (5.15) and is

(6.13) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{FE}}^{2}=\frac{2l_{s}}{L^{2}}\frac{(1-\text{e}^{-l^{2}/2l_{L}^{2}})^{2}+(1-\text{e}^{-l^{2}/2l_{s}^{2}})^{2}q}{\text{e}^{-l^{2}/l_{L}^{2}}+\text{e}^{-l^{2}/l_{s}^{2}}q}\int _{0}^{L/2}\,\text{d}y\overline{B}^{2}(r),\end{eqnarray}$$

where we take $l_{L}\simeq h=0.5~\text{kpc}$ , for galactic disk semi-thickness $h$ , and the variation scale of turbulent fields $l_{s}\simeq 0.1~\text{kpc}$ is assumed to be the same for velocity and magnetic fields. Here $l_{L}\simeq h$ because $\unicode[STIX]{x2202}_{r}\ll \unicode[STIX]{x2202}_{z}$ in a thin disk and $h$ is the smallest natural scale of variation for the mean field. Correspondingly we take an averaging scale $0.12\leqslant l\leqslant 0.48~\text{kpc}$ .

Figure 2. (a) Theoretical predictions of the line-of-sight-averaged magnetic field $\overline{B}_{L}$ with the filtering error $\unicode[STIX]{x1D70E}_{\text{df}}$ shown as error bars in an edge-on view of a disc galaxy, assuming that the mean field has the form $\overline{\boldsymbol{B}}=B_{0}\hat{\unicode[STIX]{x1D753}}$ with $B_{0}=1$ . Lengths are normalized by the galactic radius $R=12~\text{kpc}$ . Two sets of error bars are shown for different choices of  $l$ . (b) Fractional error bar values at different radii as a function of the averaging scale  $l$ .

Figure 3. Similar to figure 2(a) but using analytic dynamo solutions for $\overline{\boldsymbol{B}}$ from § 4.5 of Zhou & Blackman (Reference Zhou and Blackman2017) by solving equations (6.3) to (6.5). (a) Intrinsic error and (b) filtering error.

The predicted line-of-sight average of the magnetic field, together with the error bars are shown in figures 2 and 3, where two different profiles of $\boldsymbol{B}$ are separately considered: (i) in figure 2(a)  $\overline{\boldsymbol{B}}(\boldsymbol{x})=B_{0}\hat{\unicode[STIX]{x1D753}}$ where $B_{0}=1$ is a constant, and (ii) in figure 3 the analytic solution of the mean field dynamo model from § 6.1, normalized by the equipartition field strength $B_{\text{eq}}=\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{f}u^{2}}$ . The dimensionless parameters we have used for the analytic solution are the same as those in Zhou & Blackman (Reference Zhou and Blackman2017):

(6.14a-d ) $$\begin{eqnarray}R_{\unicode[STIX]{x1D6FC}}=R_{\unicode[STIX]{x1D6FC}0}/2,\quad R_{U}=2R_{U0}/(r/r_{\odot })^{2}F^{5/2},\quad R_{\unicode[STIX]{x1D714}}=2R_{\unicode[STIX]{x1D714}0}/(r/r_{\odot })^{2}F^{3},\quad C=4C_{0}/(r/r_{\odot })^{2}F^{3},\end{eqnarray}$$

where quantities with subscripts 0 are computed using

(6.15) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D70F}_{\text{ed}}=10^{15}~\text{s},\quad u=10~\text{km s}^{-1},\quad r\unicode[STIX]{x1D6FA}=200~\text{km s}^{-1},\\ l_{s}=0.1~\text{kpc},\quad h=0.5~\text{kpc},\quad U_{0}=1~\text{km s}^{-1},\end{array}\right\}\end{eqnarray}$$

which yields

(6.16a-c ) $$\begin{eqnarray}R_{\unicode[STIX]{x1D6FC}0}=1,\quad R_{U0}=0.3,\quad R_{\unicode[STIX]{x1D714}0}=-15,\end{eqnarray}$$

and we use $R_{m}=10^{5}$ . Above $r_{\odot }\equiv 8~\text{kpc}$ is the location of the Sun, and the function $F$ determines the $r$ -dependence of the dimensionless parameters and is described in detail in the appendix in Zhou & Blackman (Reference Zhou and Blackman2017).

The line-of-sight averages of the mean magnetic fields are shown as black solid curves, along with different types of error bars $=\pm \unicode[STIX]{x1D70E}$ about the mean computed from (5.4) and (6.13) for the cases associated with two different choices of the scale of average,  $l$ . The blue dashed lines with circular markers give error bars with $l=0.2~\text{kpc}$ , and the yellow solid lines with triangular markers give those with $l=0.4~\text{kpc}$ . In the constant magnetic field case, the intrinsic error does not exist because here $\overline{\boldsymbol{B}}$ is presumed, rather than derived from MFE equations.

Different choices of $l$ conspicuously show different levels of precision in the predictions for measurements, as evidenced by a comparison of the blue versus yellow IE bars in the $r$ -dependent model (figure 3 a). Variations in a data curve beneath the level of the error bars cannot be deemed a disagreement with the MFE theory. That is, whether uncorrelated or weakly correlated deviations with amplitudes below the error bars are systematic (Chamandy, Shukurov & Taylor Reference Chamandy, Shukurov and Taylor2016) or stochastic is beyond the resolution of the theory.

Comparing figure 3(a,b) highlights competing dependences of $\unicode[STIX]{x1D70E}_{\text{IE}}^{2}$ and $\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ on $l$ , as discussed in § 4: $\unicode[STIX]{x1D70E}_{\text{IE}}^{2}$ grows with $l$ but $\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ decreases with  $l$ . Assuming $\unicode[STIX]{x1D70E}_{\text{IE}}^{2}$ and $\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ are independent and uncorrelated, adding them in quadrature gives the total uncertainty  $\unicode[STIX]{x1D70E}^{2}$ .

In figure 2(b) and in figure 4, we show the relative total errors, $\unicode[STIX]{x1D70E}^{2}/(\overline{B})_{L}^{2}$ , as a function of $0.12~\text{kpc}\leqslant l\leqslant 0.48~\text{kpc}$ at different galactic radii. For figure 2 there is only one uncertainty, namely $\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ which is a monotonic function of $l$ for all radii shown. More interesting case is figure 4 where both $\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ and $\unicode[STIX]{x1D70E}_{\text{IE}}^{2}$ are competitive. There is an optimal scale of average, located at $0.15\leqslant r/R\leqslant 0.20$ for all four chosen radii, that minimizes the total error, and thus maximizes the precision of comparing theory and observation. In general. the existence and location of such a ‘sweet spot’ depends on the solution to a given dynamo model, and the observational method used.

Figure 4. The total relative error using the analytic dynamo solutions at different radii as a function of averaging scale  $l$ . An optimal scale arises at 0.15–0.20 kpc which minimizes the relative error, and therefore provides the best precision of theoretical predictions.

6.3 Face-on view

A complementary extreme to the edge-on case is a face-on view. Here every line of sight is perpendicular to the galactic disk, taken along the $z$ direction. In this orientation, $B_{\unicode[STIX]{x1D719}}$ and $B_{r}$ do not contribute to $(\overline{B})_{L}$ , and for a weak $\overline{B}_{z}$ , the dominant non-vanishing RM would come from small scale fluctuations. If we assume quasi-equipartition between the total mean and fluctuating small scale magnetic energies, the FR measurements still predict a precision error about which the mean field is indeterminate.

Taking $L=2h$ , the thickness of the galactic disk, and noting that $\overline{\boldsymbol{B}}$ is solely a function of $r$ in (5.15), we have

(6.17) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{FE}}^{2}(r)=\frac{l_{s}}{2h}\frac{|1-{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(k_{L})|^{2}+|1-{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(k_{s})|^{2}q}{|{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(k_{L})|^{2}+|{\displaystyle\mathop{G}\limits_{{\sim}}}_{l}(k_{s})|^{2}q}(\overline{B}_{\unicode[STIX]{x1D719}}^{2}+\overline{B}_{r}^{2})_{L}.\end{eqnarray}$$

Figure 5 shows $\overline{B}_{L}$ as a function of the galactic radial coordinate $r$ (normalized by the galactic radius  $R$ ) from a face-on view of the same $r$ -dependent dynamo model used in the last subsection (Zhou & Blackman Reference Zhou and Blackman2017). The predicted RM is now zero and its filtering error is given in blue dashed lines with circular markers for $l=0.2~\text{kpc}$ , and in yellow solid lines with triangular markers for $l=0.4~\text{kpc}$ . These emerge purely from stochastic fluctuations. The intrinsic error is zero because $\overline{B}_{z}=0$ everywhere.

Figure 5. Similar to figure 3 but for a face-on view of a disc galaxy, using the analytic dynamo solution.

6.4 Views at intermediate inclinations

The formulation becomes a bit more complicated when the line of sight is at an intermediate inclination. We adopt Cartesian coordinates in this subsection, where the $z$ -axis coincides with the galactic rotation axis, $x{-}y$ plane coincides with the galactic mid-plane, the $y$ -axis is parallel to the line of sight. Figure 1 shows a schematic plot. Let the angle between the $z$ axis and the line of sight be $\unicode[STIX]{x1D703}$ , and $0<\unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}/2$ . The line-of-sight averages depend on the location of the intersection point of the line of sight and the galactic mid-plane, $(x,y)$ , and are given by

(6.18) $$\begin{eqnarray}(\overline{B})_{L}(x,y)=\frac{\sin \unicode[STIX]{x1D703}}{2h\sqrt{x^{2}+y^{2}}}\int _{-h}^{h}\,\text{d}z\,[x\overline{B}_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D70C})+y\overline{B}_{r}(\unicode[STIX]{x1D70C})],\end{eqnarray}$$

and

(6.19) $$\begin{eqnarray}(\overline{B}^{2})_{L}(x,y)=\frac{1}{2h}\int _{-h}^{h}\,\text{d}z\,\overline{B}^{2}(\unicode[STIX]{x1D70C}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}=\sqrt{x^{2}+(z\tan \unicode[STIX]{x1D703}+y)^{2}}$ . We include only the region $\{(x,y)|\unicode[STIX]{x1D70C}\leqslant R\}$ . Equation (6.19) can then be used in (5.15) to compute the precision error associated with FR measures, and the intrinsic error is given by

(6.20) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{IE}}^{2}(x,y)=\frac{\sin ^{2}\unicode[STIX]{x1D703}}{2h(x^{2}+y^{2})}\int _{-h}^{h}\,\text{d}z\,[x^{2}\unicode[STIX]{x1D70E}_{\text{int},\overline{B}_{\unicode[STIX]{x1D719}}}^{2}(\unicode[STIX]{x1D70C})+y^{2}\unicode[STIX]{x1D70E}_{\text{int},\overline{B}_{r}}^{2}(\unicode[STIX]{x1D70C})],\end{eqnarray}$$

which can be determined once the intrinsic error of $\overline{\boldsymbol{B}}$ is calculated.

6.5 View from within our Galaxy

Finally, we discuss pulsar rotation measures as measured from inside our galaxy. For simplicity, we omit the $z$ -dependence and assume that both the observer and pulsars are in the galactic mid-plane. A schematic plot is shown in figure 1. The distance of the observer to the galactic centre is denoted by $r_{1}$ , and for this simple example, we assume pulsars to have a fixed distance $L=r_{2}<r_{1}$ from the observer and lie in the galactic mid-plane. We use $r_{1}=8~\text{kpc}$ and $r_{2}=3~\text{kpc}$ for typical values in calculations. The line-of-sight average of magnetic fields is also a function of $\unicode[STIX]{x1D703}$ , the azimuthal angle for a polar coordinate system centred at the Earth which denotes the positions of pulsars, and $\unicode[STIX]{x1D703}=0$ points to the galactic centre. The line-of-sight average of the mean field is then

(6.21) $$\begin{eqnarray}(\overline{B})_{L}(\unicode[STIX]{x1D703})=-\frac{r_{1}\sin \unicode[STIX]{x1D703}}{r_{2}}\int _{0}^{r_{2}}\,\text{d}r\frac{\overline{B}_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D70C})}{\unicode[STIX]{x1D70C}}+\frac{1}{r_{2}}\int _{0}^{r_{2}}\,\text{d}r\frac{r-r_{1}\cos \unicode[STIX]{x1D703}}{\unicode[STIX]{x1D70C}}\overline{B}_{r}(\unicode[STIX]{x1D70C}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}^{2}=r_{1}^{2}-2r_{1}r\cos \unicode[STIX]{x1D703}+r^{2}$ is the radial coordinate in the galactocentric coordinate system (see figure 1). The line-of-sight-averaged $\overline{B}^{2}$ is given by

(6.22) $$\begin{eqnarray}(\overline{B}^{2})_{L}(\unicode[STIX]{x1D703})=\frac{1}{r_{2}}\int _{0}^{r_{2}}\,\text{d}r\,\overline{B}^{2}(\unicode[STIX]{x1D70C}).\end{eqnarray}$$

The intrinsic error is given by

(6.23) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\text{IE}}^{2}(\unicode[STIX]{x1D703})=\frac{r_{1}^{2}\sin ^{2}\unicode[STIX]{x1D703}}{r_{2}}\int _{0}^{r_{2}}\,\text{d}r\frac{\unicode[STIX]{x1D70E}_{\text{IE},\overline{B}_{\unicode[STIX]{x1D719}}}^{2}}{\unicode[STIX]{x1D70C}^{2}}+\frac{1}{r_{2}}\int _{0}^{r_{2}}\,\text{d}r\left(\frac{r-r_{1}\cos \unicode[STIX]{x1D703}}{\unicode[STIX]{x1D70C}}\right)^{2}\unicode[STIX]{x1D70E}_{\text{IE},\overline{B}_{r}}^{2}.\end{eqnarray}$$

The resultant curve is shown in figure 6 in the same plot style as those in the previous subsections. In this case, stochastic fluctuations introduce only small $\unicode[STIX]{x1D70E}_{\text{IE}}^{2}$ and moderate $\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ , the latter being dominant because the line-of-sight average yields a large $(\overline{B})_{L}$ and the number of eddy cells along the line of sight is small as a consequence of small  $L$ . Thus $\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ dominates the total uncertainty $\unicode[STIX]{x1D70E}^{2}=\unicode[STIX]{x1D70E}_{\text{IE}}^{2}+\unicode[STIX]{x1D70E}_{\text{FE}}^{2}$ , and therefore in figure 7 which again shows relative errors at different directions of observation as a function of  $l$ , most curves are monotonic and reach their minima when $l\rightarrow l_{s}$ . Since $l$ is physically constrained in the region $[l_{s},l_{L}]$ (otherwise the statistical prescriptions of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ break down), this implies that $l\simeq l_{s}$ is the optimal choice of average scale in this case.

Figure 6. Line-of-sight predictions and error bars of pulsar rotation measures for our view from within our Galaxy based on the analytically solvable dynamo model, equations (6.3) to (6.5), taken from § 4.5 in Zhou & Blackman (Reference Zhou and Blackman2017). (a,b) Show error bars corresponding to the intrinsic error and filtering error, respectively.

Figure 7. The total relative error for pulsar RMs at different azimuthal angle (centred at the Earth) as a function of averaging scale  $l$ . Filtering error dominates as a result of short length of the line of sight.

It cannot be excluded that for different parameters, e.g. if $q\equiv \langle b^{2}\rangle /\langle B\rangle ^{2}$ were to exceed some critical value, the errors might dominate mean field variations making it difficult to statistically identify mean field reversals.