3.1. The coupling to small scales is always formally non-negligible

Let us first ask whether it is possible to find a regime in which the coupling to small scales caused by the $\partial _j \bar {\mathcal {D}}_{ij} ( \tilde {M}_0 , \tilde {\xi } )$ term in (3.2) can be neglected in comparison with the isotropic restoring force $v_E^2 \partial _j \partial _j \bar {\xi }_i$. This would require $k_{\rm t}\,\tilde {\xi } \ll k_{\rm w}\,\bar {\xi }$, i.e. $\tilde {\xi } \ll \epsilon \, \bar {\xi }$. Under this ordering, there are no terms in (3.3) that can balance $\partial _j \mathcal {D}_{ij} ( \tilde {M}_0 , \bar {\xi } )$ apart from the viscous term. Such a balance implies

(3.4)\begin{equation} k_{\rm w} k_{\rm t} v_E^2 \bar{\xi} \sim \nu k_{\rm t}^2 \,\omega \tilde{\xi} \ll \nu k_{\rm t}^2 \,\omega \epsilon\, \bar{\xi}, \end{equation}

i.e. $v_E^2 \ll \nu \omega$. Assuming that the large-scale response is indeed elastic gives $\omega \sim k_{\rm w} v_E$ as before, which leaves us with $v_E \ll \nu k_{\rm w}$. However, this is precisely the condition for the magnetoelastic wave's viscous damping rate to be large compared with the wave frequency, a contradiction to the assumed scaling $\omega \sim k_{\rm w} v_E$. Hence we find that the coupling to small scales is always non-negligible if the field is to respond elastically, i.e. if $\omega \sim k_{\rm w} v_E$.

This conclusion is a result of the arithmetics of powers of $k$ in each of the terms in (3.2) and (3.3): essentially, viscosity with its $k^2$ scaling does not ‘switch on’ fast enough at larger $k$ to prevent the driving of dynamically important small-scale motions. Quenching them requires a viscous damping that scales with $k$ faster than $k^2$, i.e. a hyperviscosity.Footnote ¹ In § 4.3.1, we will present numerical experiments with tangle scales hyperviscously damped, which do indeed show precise agreement with (2.8).

That the coupling to small scales should be formally non-negligible is, in fact, clear on intuitive physical grounds; a set of disconnected ‘blobs’ of magnetic field may well satisfy $\bar {M}_{0ij} \propto \delta _{ij}$, but will not be able to support a net tension on scales much larger than the typical blob size. In this case, the term describing the coupling to small scales $\partial _j \bar {\mathcal {D}}_{ij} ( \tilde {M}_0 , \tilde {\xi } )$ in (3.2) will be non-negligible and its effect will be to cancel the large-scale elasticity term, $v_E^2 \partial _j \partial _j \bar {\xi }_i$. This intuition also suggests that the effect of the coupling to small scales should reduce the effective elasticity, an expectation that is confirmed by the analysis in the next section. Hyperviscosity modifies this picture by preventing any differential motion on small scales, so the fluid behaves as though the magnetic blobs were connected by rigid rods that allow the large-scale tension to be maintained.

3.2. Normal-mode analysis

Some general statements can be made regarding the elastic response of an inviscid tangle by conducting a normal-mode analysis. The essential result of this section is that for a stable field configuration, the frequency of magnetoelastic waves is always decreased from $k_{\rm w} v_E$ as a result of intermittency of the Maxwell stress.

We expand the displacement field and Maxwell stress in Fourier modes, viz., $\xi _i=\sum _{\boldsymbol {k}} \xi _i(\boldsymbol {k})\exp ({\textrm {i}(\boldsymbol {k}\boldsymbol {\cdot }\boldsymbol {r}-\omega t)})$ and $M_{0ij}=\sum _{\boldsymbol {k}} M_{0ij}(\boldsymbol {k})\exp ({\textrm {i}\boldsymbol {k}\boldsymbol {\cdot }\boldsymbol {r}})$, define the projection operator $\mathcal {P}_{ij}(\boldsymbol {k})=\delta _{ij}-k_i k_j/k^2$, and define the matrix elements $A_{ij}(\boldsymbol {k}, \boldsymbol {k}')$ of the operator $\mathcal {P}_{il}\partial _j \mathcal {D}_{lj} ( M_0 , \bullet )$ in the Fourier basis so that

(3.5)\begin{equation} -\omega^2 \xi_i(\boldsymbol{k}) = \sum_{\boldsymbol{k}'} A_{ij}(\boldsymbol{k}, \boldsymbol{k}') \xi_j(\boldsymbol{k}'). \end{equation}

The explicit form of $A_{ij}(\boldsymbol {k}, \boldsymbol {k}')$ can be obtained straightforwardly from the definition (2.6) of $\mathcal {D}$, viz.,

(3.6)\begin{align} &A_{in}(\boldsymbol{k}, \boldsymbol{k}') = - \mathcal{P}_{il}\left(\boldsymbol{k}\right)k_j \left[-\left(k_m-k'_m\right) M_{0lj}\left(\boldsymbol{k}-\boldsymbol{k}'\right)\mathcal{P}_{mn}\left(\boldsymbol{k}'\right) \right.\nonumber\\ &\quad + \left. M_{0lm}\left(\boldsymbol{k}-\boldsymbol{k}'\right)k_m' \mathcal{P}_{jn}\left(\boldsymbol{k}'\right)+M_{0jm}\left(\boldsymbol{k}-\boldsymbol{k}'\right)k_m' \mathcal{P}_{ln}\left(\boldsymbol{k}'\right)\right]. \end{align}

Importantly, $A_{ij}(\boldsymbol {k}, \boldsymbol {k}')$ is Hermitian, i.e. $A_{ij}(\boldsymbol {k}, \boldsymbol {k}')=[ A_{ji}(\boldsymbol {k}', \boldsymbol {k}) ]^{*}$.Footnote ² Let us prove this explicitly. The Hermitian conjugate of (3.6) is

(3.7)\begin{align} \left[ A_{ni}(\boldsymbol{k}', \boldsymbol{k}) \right]^{*} & = - \mathcal{P}_{nl}\left(\boldsymbol{k}'\right)k_j' \left[-\left(k'_m-k_m\right) M_{0lj}\left(\boldsymbol{k}-\boldsymbol{k}'\right)\mathcal{P}_{mi}\left(\boldsymbol{k}\right) \right.\nonumber\\ &\quad + \left. k_m M_{0lm}\left(\boldsymbol{k}-\boldsymbol{k}'\right) \mathcal{P}_{ji}\left(\boldsymbol{k}\right)+k_m M_{0jm}\left(\boldsymbol{k}-\boldsymbol{k}'\right) \mathcal{P}_{li}\left(\boldsymbol{k}\right)\right]. \end{align}

Taking $m\leftrightarrow l$ in the first term of (3.6), and $j\leftrightarrow l$ in the second term, then subtracting (3.7) from (3.6) gives

(3.8)\begin{align} & A_{in}(\boldsymbol{k}, \boldsymbol{k}') -\left[ A_{ni}(\boldsymbol{k}', \boldsymbol{k}) \right]^{*}\nonumber\\ &\quad = \mathcal{P}_{il}\left(\boldsymbol{k}\right) \mathcal{P}_{nj}\left(\boldsymbol{k}'\right)\left(\delta_{lp}\delta_{jq}-\delta_{lq}\delta_{jp}\right) \left(k_q-k'_q\right) \left(k_m-k'_m\right) M_{0pm}\left(\boldsymbol{k}-\boldsymbol{k}'\right), \end{align}

which is zero, as $\delta _{lp}\delta _{jq}-\delta _{lq}\delta _{jp}$ = $\epsilon _{rlj}\epsilon _{rpq}$, and $\epsilon _{rpq} (k_q-k'_q)(k_m-k'_m) M_{0pm}(\boldsymbol {k}- \boldsymbol {k}') = 0$ because the equilibrium field must satisfy $\boldsymbol {\nabla }\times (\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {M}_0)=0$ (see equation 2.5). Therefore, $A_{ij}(\boldsymbol {k}, \boldsymbol {k}')=[ A_{ji}(\boldsymbol {k}', \boldsymbol {k}) ]^{*}$, q.e.d.

When $\boldsymbol {k}'=\boldsymbol {k}$, (3.6) reduces to $A_{ij}(\boldsymbol {k}, \boldsymbol {k})=-M_{0ll}(\boldsymbol {0})k^2 \delta _{ij}=-k^2v_E^2 \delta _{ij}$, so each Fourier mode is subject to the large-scale isotropic restoring force, as in (3.3). Terms with $\boldsymbol {k}'\neq \boldsymbol {k}$ describe the coupling of different Fourier modes; indeed, from (3.6), we see that Fourier modes of the displacement field $\boldsymbol {\xi }$ with wavevectors $\boldsymbol {k}$ and $\boldsymbol {k}'$ are coupled to each other only when $M_{0jm}(\boldsymbol {k}-\boldsymbol {k}')$ is non-zero. This observation shows that two large-scale Fourier modes with $\boldsymbol {k}, \boldsymbol {k}'\sim k _{\rm w}$ are not coupled by (3.6), because $M_{0ij}(\boldsymbol {k})=0$ for $\boldsymbol {k}\sim k _{\rm w}$, by (3.1). In principle, these large-scale Fourier modes can still be coupled as a result of each of them coupling individually to small scales. The condition for two large-scale Fourier modes to be coupled in this way is the existence of a path in Fourier space between them, along the wavevectors of $M_0$. This is precisely the condition for the Fourier modes of $M_0$ to ‘beat’ at the magnetoelastic-wave scale, which is in contradiction to the assumption of statistical homogeneity at large scales. Henceforth, we will assume that such beating is absent from $M_0$, and, therefore, that different large-scale Fourier modes are completely decoupled.Footnote ³ We will also assume that large-scale modes with the same $\boldsymbol {k}$ but different spatial directions are decoupled, which is a natural consequence of statistical isotropy. This discussion implies that it is sensible to decompose the large-scale perturbation into its constituent Fourier modes, each of which will independently drive small-scale motions that feed back on the particular large-scale Fourier mode that caused them, but not on any other large-scale modes.

If we take $M_0$ to have a finite number of non-zero Fourier modes,Footnote ⁴ then $A_{ij}(\boldsymbol {k}, \boldsymbol {k}')$ can be considered as a Hermitian matrix whose elements $A_{(\boldsymbol {k}, i)(\boldsymbol {k}',\, j)}$ describe the coupling of the mode $\xi _i (\boldsymbol {k})$ with $\xi _j (\boldsymbol {k}')$. Since a large-scale mode may couple to small-scale modes but not to other large-scale modes, the matrix representation of $A$ is block diagonal, with each block corresponding to one particular large-scale mode and the small-scale modes to which it couples. Let $\mathcal {A}$ be the block of size $(N+1)\times (N+1)$ corresponding to a particular large-scale mode $\xi _z (\boldsymbol {k}_{\rm w})$, taken to be in the $z$-direction without loss of generality, that is coupled to $N$ small-scale modes. The general structure of $\mathcal {A}$ is

(3.9)\begin{equation} \mathcal{A} = \begin{pmatrix} -k_{\rm w}^2 v_E^2 & \boldsymbol{f}^{\dagger}\\ \boldsymbol{f} & \mathcal{B} \end{pmatrix}, \end{equation}

where the element $\mathcal {A}_{(\boldsymbol {k}_{\rm w}, z)(\boldsymbol {k}_{\rm w}, z)}\equiv A_{zz}(\boldsymbol {k}_{\rm w}, \boldsymbol {k}_{\rm w})=- k_{\rm w}^2 v_E^2$ describes the isotropic elastic restoring force on the large-scale mode; $\mathcal {A}_{(\boldsymbol {k}, i)(\boldsymbol {k}_{\rm w}, z)}\equiv A_{iz}(\boldsymbol {k}, \boldsymbol {k}_{\rm w})\equiv f_{(\boldsymbol {k}, i)}$ for $|\boldsymbol {k}|\sim k_{\rm t}$ is an $N$-dimensional vector that gives the coupling of the small-scale modes to the large-scale mode; and $\mathcal {A}_{(\boldsymbol {k}, i)(\boldsymbol {k}', j)}\equiv A_{ij}(\boldsymbol {k},\boldsymbol {k}')\equiv \mathcal {B}_{(\boldsymbol {k}, i)(\boldsymbol {k}', j)}$ for $|\boldsymbol {k}|,\,|\boldsymbol {k}'|\sim k _{\rm t}$ is the $N\times N$ Hermitian matrix representing the coupling of the relevant small-scale modes to each other. The sizes of these three components are, respectively, $k_{\rm w}^2 v_E^2$, $k_{\rm w} k_{\rm t} v_E^2$ and $k_{\rm t}^2 v_E^2$.

The $N+1$ normal modes of the system have frequencies $\omega _{\mu }$ that satisfy $\det (\mathcal {A}+\omega _{\mu }^2 I_{N+1})=0$, where $I_{N+1}$ is the identity matrix of size $N+1$. To leading order in $\epsilon$, the fast, small-scale motions with frequency $\omega _{\mu }\sim k_{\rm t} v_E$ satisfy

(3.10)\begin{equation} \det \begin{pmatrix} \omega_{\mu}^2 & \boldsymbol{f}^{\dagger}\\ \boldsymbol{f} & \mathcal{B} + \omega_{\mu}^2 I_N \end{pmatrix}= \omega_{\mu}^2 \det \left(\mathcal{B}+\omega_{\mu}^2 I_N\right)=0\implies \det \left(\mathcal{B}+\omega_{\mu}^2 I_N\right)=0, \end{equation}

so that the fast frequencies are unaffected by the coupling of large and small scales. Equation (3.10) has $N$ solutions for $\omega _{\mu }^2$, which we denote by $\mu =1,\ldots ,N$. The remaining slow, large-scale solution with frequency $\omega _0\sim k _{\rm w} v_E$ represents a magnetoelastic wave, and satisfies

(3.11)\begin{equation} \det \begin{pmatrix} \omega_0^2- k_{\rm w}^2 v_E^2 & \boldsymbol{f}^{\dagger}\\ \boldsymbol{f} & \mathcal{B} \end{pmatrix}=0. \end{equation}

Defining the block matrix

(3.12)\begin{equation} T=\begin{pmatrix} 1 & 0\\ 0 & U \end{pmatrix}, \end{equation}

where $U$ is the $N\times N$ unitary matrix that diagonalises the matrix $\mathcal {B}$, and using the invariance of the determinant under the basis transformation defined by $T$, we obtain

(3.13)\begin{equation} \det \begin{pmatrix} \omega_0^2- k_{\rm w}^2 v_E^2 & \left(U\boldsymbol{f}\right)^{\dagger}\\ U\boldsymbol{f} & \varLambda \end{pmatrix}=0,\end{equation}

where $\varLambda =U\mathcal {B}\,U^{\dagger}=\text {diag}(\lambda _1, \ldots ,\lambda _N )=\text {diag}(-\omega _1^2, \,\dots ,-\omega _N^2 )$ is the diagonal matrix of eigenvalues of $\mathcal {B}$, which are the $N$ (negative squared) frequencies of the small-scale system. Since $\varLambda$ is diagonal, the determinant is simple to evaluate, and (3.13) leads to

(3.14)\begin{equation} \omega_0^2 = k_{\rm w}^2 v_E^2 - \sum_{\mu=1}^N \frac{\large|(U \boldsymbol{f})_{\mu}\large|^2}{\omega_{\mu}^2}. \end{equation}

Equation (3.14) shows that the effect of the coupling to small-scale motions is to reduce the frequency of large-scale waves, as long as the equilibrium is stable, i.e. as long as all $\omega _{\mu }^2>0$. Furthermore, since the components of $U$ and $\boldsymbol {f}$ are ${\sim }1$ and ${\sim }k_{\rm w} k_{\rm t} v_E^2$, respectively, while $\omega _{\mu }\sim k_{\rm t} v_E$ for $\mu \geq 1$, the frequency is reduced by a factor of order unity. Physically, this is akin to the elastic response of a tangled ball of elastic string compared with a solid elastic block: a deformation will generally produce a smaller restoring force in the elastic ball because the strings can move relative to each other to reduce the elastic energy (this is not possible in a hyperviscous fluid as small-scale motions are suppressed).

Of course, there is no reason that the right-hand side of (3.14) should be positive; if it is not, (3.14) describes a growing perturbation at large scales. This result shows that it is in principle possible for an unstable magnetic tangle to relax via motions on large scales rather than small scales. However, we have found no example of this in our numerical studies, where the effect of the tangled field on large-scale modes was always restoring, with instability proceeding only via motions at small scales.

The eigenvectors of $\mathcal {A}$ can be obtained to leading order by noting that $\mathcal {A} (0, \boldsymbol {e}_{\mu })^{\text {T}} = \lambda _{\mu }(0, \boldsymbol {e}_{\mu })^{\text {T}} + O(\epsilon )$, where $\boldsymbol {e}_{\mu }$ is the $\mu$th eigenvector of the matrix $\mathcal {B}$. The final eigenvector with associated frequency $\omega _0$ is uniquely constrained by orthogonality to be $(1, \boldsymbol {0})^{\text {T}}+O(\epsilon )$. Therefore, a magnetoelastic wave consists primarily of a large scale oscillation, together with small-amplitude (vanishing as $\epsilon \rightarrow 0$), slow (frequency $\omega _0$), small-scale oscillations. Physically, these small-scale oscillations represent the rearrangement of small-scale structures to reduce the elastic energy and, therefore, the large-scale tension. Despite having small amplitude, these small-scale motions are by no means negligible – as we have found, the magnetoelastic wave frequency is changed by a factor of order unity in their absence.

3.3. The FOSA

In this section, we describe an approximate method to obtain the wave frequency $\omega _0$ in terms of statistical properties of the magnetic tangle, by assuming that the coupling between different small-scale Fourier modes is small. This approximation is equivalent to neglecting the term $\partial _j \tilde {\mathcal {D}}_{ij} ( \tilde {M}_0 , \tilde {\xi })$ in (3.3); in the context of large-scale kinematic dynamo theory, it is often called the first-order smoothing approximation (FOSA) or the second-order correlation approximation (SOCA). In the dynamo-theory context, it is employed in the small-scale part of the induction equation to neglect a similar ‘fluctuating part of the product of two fluctuations’ term, allowing the small-scale induction equation to be solved for the small-scale magnetic field, which can then be used to compute the growth of the large-scale field (for a review, see Rincon (Reference Rincon2019)). Our use of this approximation is directly complementary: instead of solving an equation for the small-scale magnetic field given a prescribed flow, we use the FOSA to solve an equation for a small-scale flow given a known small-scale magnetic-field configuration, $M_0$.

Much like in the dynamo-theory context, where the FOSA is rigorously justified only when either the velocity correlation time is small, or $\mathrm {Rm}\ll 1$, the assumption of weak coupling between modes is unlikely to be well satisfied in any real magnetic tangle. However, the FOSA remains a useful tool, and we show in § 4 that it provides a remarkably good description of numerical simulations of large-scale waves in tangled-magnetic-field equilibria.

Under the FOSA, we neglect all off-diagonal terms of the matrix $\mathcal {B}$ in (3.11), so that the coupling of large and small scales is retained, but the small-scale modes do not couple to each other. This simplifies (3.14) considerably because now $U=I_N$ and $\omega _{\mu }= k_{\mu } v_E$ for $\mu =1,\ldots ,N$, where $k_{\mu }$ are the wavenumbers of the $N$ small-scale modes, so that (3.14) becomes

(3.15)\begin{equation} \omega_0^2 = k^2 v_E^2 - \sum_{\boldsymbol{k}'\neq\boldsymbol{k}_{\rm w}} \frac{A_{zp}(\boldsymbol{k}_{\rm w}, \boldsymbol{k}')A_{pz}(\boldsymbol{k}', \boldsymbol{k}_{\rm w}) }{k'^2 v_E^2}. \end{equation}

With $i \rightarrow z$ and $\boldsymbol {k}\rightarrow \boldsymbol {k}_{\rm w}$, (3.6) gives

(3.16)\begin{align} A_{zn}(\boldsymbol{k}_{\rm w}, \boldsymbol{k}') &= - \mathcal{P}_{zl}\left(\boldsymbol{k}_{\rm w}\right)k_{{\rm w}j} \left[-\left(k_{{\rm w}m}-k'_{m}\right) M_{0lj}\left(\boldsymbol{k}_{\rm w}-\boldsymbol{k}'\right)\mathcal{P}_{mn}\left(\boldsymbol{k}'\right) \right.\nonumber\\ &\quad + \left. M_{0lm}\left(\boldsymbol{k}_{\rm w}-\boldsymbol{k}'\right)k_m' \mathcal{P}_{jn}\left(\boldsymbol{k}'\right)+M_{0jm}\left(\boldsymbol{k}_{\rm w}-\boldsymbol{k}'\right)k_m' \mathcal{P}_{ln}\left(\boldsymbol{k}'\right)\right]. \end{align}

With $|\boldsymbol {k}'| \sim k_{\rm t}$, the first term in the square brackets vanishes to leading order in $\epsilon$ because $k'_m \mathcal {P}_{mn}(\boldsymbol {k}')=0$. Relabelling $n\rightarrow p$, then swapping $m\leftrightarrow j$ and $m\leftrightarrow l$ in the second term, we obtain

(3.17)\begin{align} A_{zp}(\boldsymbol{k}_{\rm w}, \boldsymbol{k}') &= - \mathcal{P}_{zl}\left(\boldsymbol{k}_{\rm w}\right)k_{{\rm w}m} M_{0lj}\left(\boldsymbol{k}_{\rm w}-\boldsymbol{k}'\right)k_j' \mathcal{P}_{mp}\left(\boldsymbol{k}'\right)\nonumber\\ &\quad-\mathcal{P}_{zm}\left(\boldsymbol{k}_{\rm w}\right)k_{{\rm w}j} M_{0jl}\left(\boldsymbol{k}_{\rm w}-\boldsymbol{k}'\right)k_l' \mathcal{P}_{mp}\left(\boldsymbol{k}'\right)\nonumber\\ &= - k_{{\rm w}n} k_l' \mathcal{P}_{zq}\left(\boldsymbol{k}_{\rm w}\right)\mathcal{P}_{mp}\left(\boldsymbol{k}'\right)\left(\delta_{nm}\delta_{jq}+\delta_{nj}\delta_{mq}\right)M_{0lj}\left(\boldsymbol{k}_{\rm w}-\boldsymbol{k}'\right). \end{align}

An analogous expression for $A_{pz}(\boldsymbol {k}', \boldsymbol {k}_{\rm w})$ can be obtained directly from (3.7), but the derivation is much more involved. Instead, we can use the fact that $A$ is Hermitian:

(3.18)\begin{align} A_{pz}(\boldsymbol{k}', \boldsymbol{k}_{\rm w})&=\left[A_{zp}(\boldsymbol{k}_{\rm w}, \boldsymbol{k}')\right]^{*}\nonumber\\ &=- k_{{\rm w}n} k_l' \mathcal{P}_{zq}\left(\boldsymbol{k}_{\rm w}\right)\mathcal{P}_{mp}\left(\boldsymbol{k}'\right)\left(\delta_{nm}\delta_{jq}+\delta_{nj}\delta_{mq}\right)M_{0lj}\left(\boldsymbol{k}'-\boldsymbol{k}_{\rm w}\right)\nonumber\\ & =- k_{{\rm w}n} k_j' \mathcal{P}_{zm}\left(\boldsymbol{k}_{\rm w}\right)\mathcal{P}_{qp}\left(\boldsymbol{k}'\right)\left(\delta_{nq}\delta_{lm}+\delta_{nl}\delta_{mq}\right)M_{0lj}\left(\boldsymbol{k}'-\boldsymbol{k}_{\rm w}\right), \end{align}

where to obtain the final expression we have swapped $l \leftrightarrow j$ and $q \leftrightarrow m$.

With (3.17) and (3.18), (3.15) becomes

(3.19)\begin{equation} \omega_0^2 = k_{\rm w}^2 v_E^2 - \left(\delta_{nm}\delta_{jz}+\delta_{nj}\delta_{mz}\right)\left(\delta_{zl'}\delta_{n'q'}+\delta_{l'n'}\delta_{zq'}\right)k_{{\rm w}n} k_{{\rm w}n'}R_{mq'jl'}, \end{equation}

where

(3.20)\begin{equation} R_{mq'jl'}=\sum_{\boldsymbol{k}'} \frac{k_l' k_{j'}'}{k'^2 v_E^2} \mathcal{P}_{q'm}\left(\boldsymbol{k}'\right)\tilde{M}_{lj}\left(-\boldsymbol{k}'\right) \tilde{M}_{l'j'}\left(\boldsymbol{k}'\right), \end{equation}

to leading order in $\epsilon$. The rank-four tensor $R_{pq'jl'}$ is a statistical property (in the sense of volume averaging) of the magnetic tangle. Since we have assumed statistical isotropy, the most general form it can take is $R_{pq'jl'}=a\delta _{pq'}\delta _{jl'}+b\delta _{pj}\delta _{q' l'}+c\delta _{pl'}\delta _{q'j}$. Noting from (3.20) that $R_{pq'jl'}$ is symmetric in $p$ and $q'$ and vanishes on contraction of $p$ with $l'$, we find

(3.21)\begin{equation} R_{pq'jl'}=\frac{1}{30v_E^4}\left(4\delta_{l'j}\delta_{pq'}-\delta_{l'p}\delta_{jq'}-\delta_{l'q'}\delta_{jp}\right)\left(\left\langle P_0^2\right\rangle - \left\langle P_0\right\rangle^2\right). \end{equation}

where we have used the equilibrium condition $k_j M_{0ij}(\boldsymbol {k}) + k_i P_0(\boldsymbol {k})=0$, where $P_0$ is the equilibrium total pressure distribution, and identified

(3.22)\begin{equation} \sum_{\boldsymbol{k}'}\left|\tilde{P}_0 \left(\boldsymbol{k}'\right)\right|^{2} = \left \langle P_0^2 \right \rangle - \left \langle P_0 \right \rangle^2, \end{equation}

where angled brackets indicate spatial averages. On substitution of this result into (3.19) and contraction of the many Kronecker deltas, we finally arrive at

(3.23)\begin{equation} \omega_0^2 = k^2 v_E^2\left(1-\chi\right), \end{equation}

where

(3.24)\begin{equation} \chi = \frac{1}{5v_E^4}\left(\left\langle P_0^2\right\rangle - \left\langle P_0\right\rangle^2\right). \end{equation}

We therefore find that under the FOSA, the effect of the small-scale structure of the magnetic tangle is to reduce the frequency of magnetoelastic waves by an amount proportional to the variance of the total pressure. This is really a statement about the magnetic field, since the total pressure must balance the magnetic tension force in equilibrium, so (3.24) shows that more intermittent magnetic fields are less elastic. This is precisely in agreement with the intuitive reasoning of § 3.1, where we argued that a field configuration consisting of disconnected magnetic ‘blobs’ would not support a large-scale tension – such a field would have a large value of $\chi$.

In the special case of a force-free magnetic tangle, i.e. when only magnetic pressure balances the magnetic tension, $P_0=B_0^2/2$, so

(3.25)\begin{equation} \chi = \frac{9}{20}\left(\frac{\left\langle B_0^4\right\rangle}{\left\langle B_0^2\right\rangle^2} - 1\right). \end{equation}

Like (3.14), (3.23) permits an instability on large scales, occurring when $\chi >1$. Note that the FOSA treats the small-scale system as stable; this is a consequence of neglecting the off-diagonal terms in the matrix $\mathcal {B}$. Formally, therefore, the FOSA predicts that when the variance of the total pressure becomes large enough (so that $\chi >1$), a statistically homogeneous and isotropic magnetic tangle can be unstable to large-scale perturbations, even while remaining stable to those with scale similar to that of the tangle itself. This prediction is contrary to the intuitive picture that we have developed so far. Consider again a set of disconnected magnetic blobs. While it is intuitive that this configuration should not produce an elastic response to a large-scale perturbation, it is not clear that increasing the variance of the field strength (e.g. by reducing the size of the blobs while increasing their field strength to maintain a constant spatial average) should introduce an instability, causing the large-scale perturbation to grow. The source of the discrepancy between formalism and physical intuition is that strongly intermittent configurations with large $\chi$ are poorly described by the FOSA. From (3.15), we see that $\chi$ can be thought of as a measure of the strength of the coupling between large and small scales. When this coupling is strong (large $\chi$), describing the small-scale dynamics accurately is essential to obtaining correct large-scale dynamics. Indeed, in § 4, we show with numerical simulations of linear force-free fields that the FOSA becomes inaccurate for large values of $\chi$.

3.4. Magnetoelastic waves in a viscous fluid

In this section, we develop a theory for magnetoelastic waves propagating through a viscous fluid. As we found in § 3.2, a magnetoelastic wave in an inviscid fluid is a large-scale oscillation, accompanied by slow ($\omega \sim k_{\rm w} v_E$), small-amplitude oscillations associated with the relaxation of small-scale structures in response to the large-scale tension. In the viscous case, these small-scale structures will be damped by viscosity; their presence therefore results in an anomalous viscous damping of magnetoelastic waves.

For definiteness, we take the initial condition to be a large-scale velocity perturbation $u_z$ along $z$ with wavevector $\boldsymbol {k}_{\rm w}$. We solve the initial-value problem by taking a Laplace transform in time,

(3.26)\begin{equation} p^2 \xi_z(\boldsymbol{k}_{\rm w}) - u_z\left(\boldsymbol{k}_{\rm w},t=0\right) = -k_{\rm w}^2v_E^2 \xi_z(\boldsymbol{k}_{\rm w})+ \sum_{\boldsymbol{k}'} A_{zj}(\boldsymbol{k}_{\rm w}, \boldsymbol{k}') \xi_j(\boldsymbol{k}') +\nu k_{\rm w}^2 p \xi_z(\boldsymbol{k}_{\rm w}), \end{equation}

where $p$ is the Laplace conjugate variable to time (analogous to $\textrm {i}\omega$).

In the FOSA, an initially unperturbed small-scale mode with wavenumber $k' \gg k_{\rm w}$ satisfies

(3.27)\begin{equation} p^2 \xi_j(\boldsymbol{k}') = -v_E^2 k'^2 \xi_j(\boldsymbol{k}') + A_{jz}(\boldsymbol{k}', \boldsymbol{k}_{\rm w}) \xi_z(\boldsymbol{k}_{\rm w}) +\nu k'^2 p \xi_j(\boldsymbol{k}'). \end{equation}

Solving this algebraic equation for $\xi _j(\boldsymbol {k}')$, and substituting into (3.26), we obtain

(3.28)\begin{align} & p^2 \xi_z(\boldsymbol{k}_{\rm w}) - u_z\left(\boldsymbol{k}_{\rm w},t=0\right) =-k_{\rm w}^2v_E^2 \xi_z(\boldsymbol{k}_{\rm w})\nonumber\\ &\quad+ \sum_{\boldsymbol{k}'} \frac{A_{zj}(\boldsymbol{k}_{\rm w}, \boldsymbol{k}') A_{jz}(\boldsymbol{k}', \boldsymbol{k}_{\rm w})}{p^2 + v_E^2 k'^2 + \nu k'^2 p}\xi_z (\boldsymbol{k}_{\rm w}) +\nu k_{\rm w}^2 p \xi_z(\boldsymbol{k}_{\rm w}), \end{align}

where the sum is over all small-scale modes.

With $p\sim \omega _0\sim k_{\rm w} v_E$, the $p^2$ in the denominator of the coupling term in (3.28) is small compared with $v_E^2 k'^2$.Footnote ⁵ Neglecting it, we can use (3.17) and (3.18) analogously to the inviscid case to write

(3.29)\begin{equation} p^2 \xi_z(\boldsymbol{k}_{\rm w}) - u_z\left(\boldsymbol{k}_{\rm w},t=0\right) =-k_{\rm w}^2v_E^2 \xi_z(\boldsymbol{k}_{\rm w})+\frac{\chi}{1 + \nu p/v_E^2}k_{\rm w}^2v_E^2\xi_z(\boldsymbol{k}_{\rm w}) +\nu k_{\rm w}^2 p \xi_z(\boldsymbol{k}_{\rm w}). \end{equation}

Note that, as predicted in § 3.1, the term representing coupling to small scales remains finite as $\epsilon \rightarrow 0$, and can only be neglected when $\nu p / v_E^2 \sim \nu k_{\rm w}^2 / k_{\rm w} v_E \gg 1$, i.e. when wave motions are strongly damped. In contrast, in the hyperviscous case, the number of powers of $k'$ is increased in the viscosity term in the denominator of the coupling term in (3.28), so that it is sufficient for small scales to be strongly damped for the coupling term to be negligible as $\epsilon \rightarrow 0$. We therefore recover the simple wave equation (2.7) in the hyperviscous case, as predicted in § 3.1.

Finally, we invert the Laplace transform using Cauchy's residue theorem. The solution is a sum of Laplace modes $\bar {u}_i (t) = \left (\sum _n A_n e^{p_n t}\right ) \bar {u}_i (0)$ with amplitudes

(3.30)\begin{equation} A_{n} = \text{Res} \left[ \frac{p}{D(p )} , p\rightarrow p_n \right], \end{equation}

where ‘$\mathrm {Res}$’ denotes the residue and $\{p_n\}$ are the roots of the dispersion relation

(3.31)\begin{equation} D(p_n) \equiv p_n^2+\left(1-\frac{\chi}{1 + \nu p_n/v_E^2}\right) v_E^2k^2+\nu k^2 p_n = 0. \end{equation}

It is instructive to consider the case where the viscous damping rate is finite but small compared with the wave frequency, i.e. $\hat {\nu }\equiv \nu k_{\rm w}/v_E\ll 1$. The solutions of (3.31) in this limit are

(3.32)\begin{align} p_1 &= +\textrm{i} v_E k_{\rm w} \sqrt{1-\chi} - \tfrac{1}{2}\nu k_{\rm w}^2 \left(1+\chi\right) + O\left(\hat{\nu}^2\right), \end{align}

(3.33)\begin{align}p_2 &= -\textrm{i} v_E k_{\rm w} \sqrt{1-\chi} - \tfrac{1}{2}\nu k_{\rm w}^2 \left(1+\chi\right) + O\left(\hat{\nu}^2\right), \end{align}

(3.34)\begin{align}p_3 &= -v_E^2 /\nu + O\left(\hat{\nu}\right), \end{align}

with corresponding amplitudes

(3.35a–c)\begin{equation} A_{1} = \tfrac{1}{2} + O\left(\hat{\nu}\right), \quad A_{2} = \tfrac{1}{2} + O\left(\hat{\nu}\right), \quad A_{3} = \chi \hat{\nu}^2 + O\left(\hat{\nu}^3 \right). \end{equation}

In the first two solutions, $p_1$ and $p_2$, given by (3.32) and (3.33), we find the wave modes of the previous section, but now with a damping rate that is larger than in the perfectly homogeneous case by a factor of $1+\chi$. This increased damping rate is expected: it is a consequence of the viscous damping of the small-scale motions associated with magnetoelastic waves. It may appear odd that a damping associated with small-scale motions has a rate $\propto \nu k_{\rm w}^2$, especially as the condition $\hat {\nu }\ll 1$ does not exclude the possibility of fast, small scale motions being overdamped by viscosity. The resolution is that the small-scale motions associated with a magnetoelastic wave are slow (frequency $\sim k_{\rm w} v_E$), not fast, and therefore they are not viscously dominated when $\hat {\nu }\ll 1$, as $\nu \partial _j \partial _j \partial _{\rm t} \tilde {\xi }_i\sim \nu k_{\rm t}^2 \omega _0 \tilde {\xi } \sim \hat {\nu }k_{\rm t}^2v_E^2 \tilde {\xi } \ll v_E^2 \partial _j \partial _j \tilde {\xi }_i\sim k_{\rm t}^2v_E^2 \tilde {\xi }$. These motions satisfy the balance $v_E^2 \partial _j \partial _j \tilde {\xi }_i \sim \partial _j \mathcal {D}_{ij} ( \tilde {M}_0 , \bar {\xi } )$, which gives $\tilde {\xi }\sim \epsilon \bar {\xi }$. The rate of energy dissipation associated with them is therefore $\sim \nu k_{\rm t}^2 \omega _0^2 \tilde {\xi }^2\sim \nu k_{\rm w}^2 \omega _0^2 \bar {\xi }$, which is consistent with the anomalous damping rate in (3.32) and (3.33).

The third mode, $p_3$, given by (3.34), describes the case where the slow, small-scale motions associated with the waves do become viscously dominated, so obey the balance $v_E^2 \partial _j \partial _j \tilde {\xi }_i \sim k_{\rm t}^2 \partial _{\rm t} \tilde {\xi }$, giving a damping rate of $v_E^2/\nu$. As above, these motions become viscously dominated only when $\hat {\nu }\sim 1$, explaining the vanishing amplitude of this mode when $\hat {\nu }\ll 1$.Footnote ⁶

4.1. Ideal instability of tangled-magnetic-field equilibria

The analysis in the preceding sections has been idealised because real tangled magnetic equilibria are generically unstable, even to ideal perturbations (Er-Riani, Naji & El Jarroudi Reference Er-Riani, Naji and El Jarroudi2014; East et al. Reference East, Zrake, Yuan and Blandford2015). The instability typically proceeds via motions at small scales that are fast compared with the magnetoelastic wave motions, but are inhibited by the presence of viscosity. However, the growth rate associated with viscously dominated, small-scale unstable modes is ${\sim }v_E^2/\nu$, which becomes comparable with the wave frequency when $\hat {\nu }=\nu k_{\rm w}^2 / k_{\rm w} v_E \sim 1$, i.e. when magnetoelastic waves are strongly damped by viscosity. The question then arises as to whether any static, tangled-magnetic-field equilibrium will persist long enough for waves to propagate through it, even with strong viscosity. In fact, we have found in our numerical study that when the equilibrium state is an linear force-free magnetic field (see next section) the growth rate of the instability is sufficiently slow for the equilibrium configuration to persist for many wave periods.

The existence of an instability of linear force-free magnetic fields in ideal MHD has only recently been appreciated, and indeed many inaccurate statements have historically been presented in the literature. For the interested reader, we present a short review of this instability in appendix A.

4.2. Simulation set-up

In order to test the analytic predictions derived in the previous sections, we conduct simulations of standing magnetoelastic waves on periodic, linear, force-free magnetic-field equilibria with different values of the parameter $\chi$.

The linear force-free magnetic-field condition is $\boldsymbol {\nabla }\times \boldsymbol {B}=\alpha \boldsymbol {B}$, where $\alpha$ is a constant. Taking the curl and using $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B}=0$, we get a Helmholtz equation for the magnetic field, $\nabla ^2\boldsymbol {B}=-\alpha ^2 \boldsymbol {B}$, which shows that the modes in the Fourier expansion of $\boldsymbol {B}$ must all have $|\boldsymbol {k}|=\alpha$. The linear force-free equilibrium must then have $\textrm {i} \boldsymbol {k}\times \boldsymbol {B_k} = \alpha \boldsymbol {B_k}$, or

(4.1)\begin{equation} \hat{\boldsymbol{k}}\times \text{Re}\left(\boldsymbol{B_k}\right)= \text{Im}\left(\boldsymbol{B_k}\right). \end{equation}

To generate tangled magnetic equilibria, we take $\boldsymbol {k}L=2{\rm \pi} (12, 6, 0)^{\mathrm {T}}$, together with all its permutations and negations, where $L$ is the box size. This means that the magnetic-field structure is periodic on a scale of $L/6$, which ensures the scale separation between the box scale and the scale of the tangled field. We then generate real vectors $\text {Re}(\boldsymbol {B_k})$ subject to the condition $\text {Re}(\boldsymbol {B_k})= \text {Re}(\boldsymbol {B}_{-\boldsymbol {k}})$, so that $\boldsymbol {B}$ is real, and to (4.1), so that it is force free, using a numerical optimisation procedure to ensure $\bar {M}_{0ij}\propto \delta _{ij}$ and to produce fields with different values of the parameter $\chi$.Footnote ⁷ The magnetic-field configurations thus obtained with $\chi =0.17$ and $\chi =0.98$ are shown in figure 1. Curiously, we have been unable to generate a field with $\chi >1$ using this optimisation procedure, which is the condition for the large-scale mode to become unstable in the FOSA (however, see the discussion in § 3.3).

Figure 1.

Magnetic-field structure of two force-free equilibria resulting from the numerical optimisation procedure described in § 4.2. The upper colour bar describes the plots of $B_z$, while the lower colour bar describes the plots of $B^2$, in units of $\langle v_A^2 \rangle ^{1/2}$ and $\langle v_A^2 \rangle$, respectively. Note that the configuration with $\chi = 0.17$ has smaller variation in the magnetic-field strength than the configuration with $\chi = 0.98$. Indeed, the $\chi =0.98$ configuration is hardly ‘tangled’ at all; it more closely resembles a collection of magnetic ‘blobs’ (cf. the discussion in § 3.1).

For each equilibrium configuration, we introduce a sinusoidal velocity shear $\boldsymbol {u} = 0.05\, \langle v_A^2\rangle ^{1/2}\sin {(2{\rm \pi} x/\lambda )}\hat {\boldsymbol {z}}$, where $\lambda =2{\rm \pi} /k_{\rm w}=L$ and $\langle v_A^2 \rangle ^{1/2}=\langle B_0^2 \rangle ^{1/2}=\sqrt {3}\,v_E$ is the root-mean-square Alfvén speed, and we measure the evolution of the amplitude of this mode in the Fourier representation of $\boldsymbol {u}$.

The three-dimensional, incompressible MHD equations are solved using the DedalusFootnote ⁸ code (Burns et al. Reference Burns, Vasil, Oishi, Lecoanet and Brown2020) at $128^3$ resolution (padded for dealiasing according to the 2/3 rule). We take the resistivity $\eta = 0$, and thus our simulations are only valid in describing the evolution at early times when the small-scale instability of the equilibrium field configuration has not yet developed and hence the magnetic field structure is still well-resolved.