The onset of electron-only reconnection

Abstract

Motivated by recent observations of ‘electron-only’ magnetic reconnection, without an ion-scale sheet or ion outflows, in both the Earth’s magnetosheath and in numerical simulations, we study the formation and reconnection of electron-scale current sheets at low plasma $\unicode[STIX]{x1D6FD}$. We first show that ideal sheets collapse to thicknesses much smaller than the ion scales, by deriving an appropriate analogue of the Chapman–Kendall collapse solution. Second, we show that, in practice, reconnection onset happens in these collapsing sheets once they reach a critical aspect ratio, because the tearing instability then becomes faster than their collapse time scale. We show that this can happen for sheet thicknesses larger than the ion scale or at only a few times the electron scale, depending on plasma parameters and the aspect ratio of the collapsing structure, thereby unifying the usual picture of ion-coupled reconnection and the new regime of electron-only reconnection. We derive relationships between plasma $\unicode[STIX]{x1D6FD}$, ion-to-electron temperature ratio, the aspect ratio, electron outflow velocity and the final thickness of the sheets, and thus determine under what circumstances electron-scale sheets form and reconnect.

1 Introduction

Magnetic reconnection in current sheets is present in many astrophysical and laboratory settings and is an important process for converting magnetic energy into thermal and non-thermal kinetic energy (Zweibel & Yamada 2009; Ji et al. 2019; Yamada, Kulsrud & Ji 2010). In low-$\unicode[STIX]{x1D6FD}$ collisionless guide-field reconnection, field lines are broken within a microscopic electron diffusion region, of size comparable to the electron inertial length $d_{e}=c/\unicode[STIX]{x1D714}_{pe}$, where $\unicode[STIX]{x1D714}_{pe}=\sqrt{4\unicode[STIX]{x03C0}n_{e}e^{2}/m_{e}}$ is the electron plasma frequency. In the usual picture of (guide-field) reconnection, this electron region is embedded in a larger-scale ‘ion region’ of thickness of order the ion sound scale $\unicode[STIX]{x1D70C}_{s}=\unicode[STIX]{x1D70C}_{i}\sqrt{ZT_{e}/2T_{i}}$, where the ion gyroradius $\unicode[STIX]{x1D70C}_{i}=v_{\text{th}i}/\unicode[STIX]{x1D6FA}_{i}$, the ion thermal speed $v_{\text{th}i}=\sqrt{2T_{i}/m_{i}}$ and the ion gyrofrequency $\unicode[STIX]{x1D6FA}_{i}=ZeB_{0}/m_{i}c$. Associated with this region one expects bidirectional jets of plasma flowing away from the reconnection site at around the in-plane Alfvén speed $v_{\text{A}y}=B_{y}/\sqrt{4\unicode[STIX]{x03C0}n_{i}m_{i}}$, where $B_{y}$ is the reconnecting (in-plane) magnetic field. We will term this standard picture ‘ion-coupled’ reconnection.

However, recent analysis of data from the MMS spacecraft (Phan et al. 2018) has shown that, at least in Earth’s turbulent magnetosheath, one overwhelmingly finds current sheets with thicknesses of only a few $d_{e}$ (i.e. much thinner than $\unicode[STIX]{x1D70C}_{s}$) without associated ion regions and ion jets, but with clear electron jets moving at appreciable fractions of the electron Alfvén velocity $v_{\text{A}ey}=B_{y}/\sqrt{4\unicode[STIX]{x03C0}n_{e}m_{e}}$. This new regime has also recently been reproduced in the numerical simulations of Sharma Pyakurel et al. (2019). A similar situation has been observed in the hybrid Vlasov–Maxwell turbulence simulations of Califano et al. (2018), who observed that when energy is injected at scales much larger than the ion scales, ion-coupled reconnection is observed, while if energy is injected closer to the ion scales, only electron jets and electron-scale sheets are observed. Reconnection of current sheets with thicknesses of a few $d_{e}$ is also observed in the VINETA-II experiment (Jain et al. 2017). We will follow the terminology of Califano et al. (2018) and term this new regime ‘electron-only’ reconnection.

The goal of this paper is to develop a theoretical picture that can explain the onset of this new regime of electron-only reconnection, and under what circumstances it occurs in place of the standard ion-coupled reconnection. To achieve this, we will make use of a simplified model, rigorously derived from gyrokinetics in the limit $\unicode[STIX]{x1D6FD}_{e}\sim m_{e}/m_{i}\ll 1$, the so-called ‘kinetic reduced electron heating model’, henceforth KREHM (Zocco & Schekochihin 2011). The details of this model, as well as its utility and limitations, are described in § 2.

To make progress in understanding electron-only reconnection onset, there are two main questions which should be answered. First, can current sheets collapse to thicknesses of order a few $d_{e}$, significantly below the ion scales? We answer this in § 3 by deriving a collapsing X-point solution analogous to Chapman–Kendall collapse (Chapman & Kendall 1963; Biskamp 2000), and showing that the solution is valid both above and below the ion scales. This means that ideal (i.e. with field lines frozen into an effective electron flow) sheets can indeed collapse well past the ion scale.

Second, at what sheet thickness do these collapsing sheets begin to reconnect, and under what conditions is this thickness above (obtaining ion-coupled reconnection) or well below (obtaining electron-only reconnection) the ion scales? We answer this in § 4, showing that the sheets are disrupted when they collapse to a critical thickness¹ at which the linear tearing growth rate² is faster than the collapse or formation rate of the sheet derived in § 3. Our analysis allows us to derive relationships between $\unicode[STIX]{x1D6FD}_{e}$ and the aspect ratio and thickness of the sheet at disruption (reconnection onset), and we thus determine the physical conditions under which one will observe ion-coupled or electron-only reconnection. The essential physical result is that if the initial collapsing solution is not sufficiently anisotropic when its thickness is above the ion scales, the collapse continues down to a thickness well below the ion scale, and electron-only reconnection will occur. Remarkably (considering the formal limitation of KREHM to low $\unicode[STIX]{x1D6FD}$), we will see that our model seems to agree rather well with the magnetosheath observations of electron-only reconnection (Phan et al. 2018; Stawarz et al. 2019).

2 Equations

Our starting point is the KREHM derived by Zocco & Schekochihin (2011) (henceforth ZS11) from gyrokinetics for electron $\unicode[STIX]{x1D6FD}$ of order the mass ratio $\unicode[STIX]{x1D6FD}_{e}\sim Zm_{e}/m_{i}$, where $Z=q_{i}/e$ (and the temperature ratio $\unicode[STIX]{x1D70F}=T_{0i}/T_{0e}\sim 1$). The spatial scales of the fluctuations are ordered to be comparable to the relevant ion kinetic scales, $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim k_{\bot }\unicode[STIX]{x1D70C}_{s}\sim 1$, and so KREHM is able to capture both the large-scale ($k_{\bot }\unicode[STIX]{x1D70C}_{s}\ll 1$) and small-scale ($k_{\bot }\unicode[STIX]{x1D70C}_{s}\gg 1$) behaviour of the plasma, as well as the transition between them. This is essential for our present purposes, since we are attempting to diagnose the transition between ion-coupled and electron-only reconnection onset.

KREHM is in some sense a minimal model for describing collisionless guide-field reconnection, since the equations incorporate the dispersive effects entering at the ion sound scale $\unicode[STIX]{x1D70C}_{s}$ that are thought to be essential to enable fast reconnection (Rogers et al. 2001), a flux-unfreezing mechanism (electron inertia) entering at the electron inertial scale $d_{e}$, and a rigorous treatment of the (electron) heating channel. The equations, written in terms of the electrostatic potential $\unicode[STIX]{x1D719}$, the magnetic potential $A_{\Vert }$ and the reduced electron distribution function $g_{e}$, are

(2.1)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\frac{Z}{\unicode[STIX]{x1D70F}}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\frac{e\unicode[STIX]{x1D719}}{T_{0e}}=\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\frac{e}{cm_{e}}d_{e}^{2}\unicode[STIX]{x1D735}_{\bot }^{2}A_{\Vert },\quad & \displaystyle\end{eqnarray}$$

(2.2)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}(A_{\Vert }-d_{e}^{2}\unicode[STIX]{x1D735}_{\bot }^{2}A_{\Vert })=-c\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}z}-\frac{cT_{0e}}{e}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left[\frac{Z}{\unicode[STIX]{x1D70F}}\left(1-\hat{\unicode[STIX]{x1D6E4}}_{0}\right)\frac{e\unicode[STIX]{x1D719}}{T_{0e}}-\frac{\unicode[STIX]{x1D6FF}T_{\Vert e}}{T_{0e}}\right],\quad & \displaystyle\end{eqnarray}$$

(2.3)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}g_{e}}{\text{d}t}+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(g_{e}-\frac{\unicode[STIX]{x1D6FF}T_{\Vert e}}{T_{0e}}F_{0\text{e}}\right)=C[g_{e}]+\left(1-\frac{2v_{\Vert }^{2}}{v_{\text{th}e}^{2}}\right)F_{0\text{e}}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\frac{e}{cm_{e}}d_{e}^{2}\unicode[STIX]{x1D735}_{\bot }^{2}A_{\Vert },\quad & \displaystyle\end{eqnarray}$$

where

(2.4)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\frac{c}{B_{0}}\{\unicode[STIX]{x1D719},...\,\}, & \displaystyle\end{eqnarray}$$

(2.5)$$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}-\frac{1}{B_{0}}\{A_{\Vert },...\,\}, & \displaystyle\end{eqnarray}$$

(2.6)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FF}T_{\Vert e}}{T_{0e}}=\frac{1}{n_{0\text{e}}}\int d^{3}\boldsymbol{v}\frac{2v_{\Vert }^{2}}{v_{\text{th}e}^{2}}g_{e}, & \displaystyle\end{eqnarray}$$

and $\hat{\unicode[STIX]{x1D6E4}}_{0}$, encoding the ion finite Larmor radius effects, is the inverse Fourier transform of

(2.7)$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{0}\left({\textstyle \frac{1}{2}}k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}\right)=\text{I}_{0}\left({\textstyle \frac{1}{2}}k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}\right)\text{e}^{-1/2k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}},\end{eqnarray}$$

where $\text{I}_{0}$ is the modified Bessel function. The electron density perturbation $\unicode[STIX]{x1D6FF}n_{e}$ is related to $\unicode[STIX]{x1D719}$ via the relation

(2.8)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}=-\frac{Z}{\unicode[STIX]{x1D70F}}\left(1-\hat{\unicode[STIX]{x1D6E4}}_{0}\right)\frac{e\unicode[STIX]{x1D719}}{T_{0e}}.\end{eqnarray}$$

The Poisson bracket appearing in the equations is $\{f,g\}=\hat{\boldsymbol{z}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }f\times \unicode[STIX]{x1D735}_{\bot }g$. We have taken the limit $\unicode[STIX]{x1D708}_{ei}\ll \unicode[STIX]{x1D714}$, which allows us to neglect a resistive term in (2.2) (but not the collisions $C[g_{e}]$ in (2.3)).

The function $g_{e}$ is a mathematically convenient reduced electron-gyrocentre parallel velocity distribution function, which encodes the electron heating channel. This is perhaps all the information that is needed to understand the present paper: in more detail, it is related to the perturbed distribution function $\unicode[STIX]{x1D6FF}f_{e}$ by

(2.9)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}f_{e}(\boldsymbol{r},\boldsymbol{v},t)=\left(1+\frac{\unicode[STIX]{x1D6FF}n_{e}}{n_{0e}}+\frac{2v_{\Vert }u_{\Vert e}}{v_{\text{th}e}^{2}}\right)F_{0e}+g_{e}(\boldsymbol{R}_{e},\boldsymbol{v},t),\end{eqnarray}$$

where $\boldsymbol{R}_{e}=\boldsymbol{r}+\boldsymbol{v}_{\bot }\times \hat{\boldsymbol{z}}/\unicode[STIX]{x1D6FA}_{e}$ is the electron gyrocentre. Integrating the zeroth and first $v_{\Vert }$-moments of $\unicode[STIX]{x1D6FF}f_{e}$ over all velocity (perturbed electron density and parallel flow), we can see that contribution from $g_{e}$ to these vanishes, i.e. $g_{e}$ contains the information of all higher moments, but not density or parallel flow. It turns out (see ZS11 for details) that the non-trivial $v_{\bot }$ dependence of $g_{e}$ can be safely ignored, and so $g_{e}$ can be considered a function of $v_{\Vert }$, space and time only.

We may write (2.2) as

(2.10)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}A_{\Vert }}{\unicode[STIX]{x2202}t}=-c\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left[\unicode[STIX]{x1D719}+\frac{Z}{\unicode[STIX]{x1D70F}}\left(1-\hat{\unicode[STIX]{x1D6E4}}_{0}\right)\unicode[STIX]{x1D719}-\frac{\unicode[STIX]{x1D6FF}T_{\Vert e}}{e}\right]+\frac{\text{d}}{\text{d}t}d_{e}^{2}\unicode[STIX]{x1D735}_{\bot }^{2}A_{\Vert },\end{eqnarray}$$

which defines an effective velocity

(2.11)$$\begin{eqnarray}\boldsymbol{u}_{\text{eff}}=\hat{z}\times \unicode[STIX]{x1D735}_{\bot }\frac{c}{B_{0}}\left[\unicode[STIX]{x1D719}+\frac{Z}{\unicode[STIX]{x1D70F}}\left(1-\hat{\unicode[STIX]{x1D6E4}}_{0}\right)\unicode[STIX]{x1D719}-\frac{\unicode[STIX]{x1D6FF}T_{\Vert e}}{e}\right],\end{eqnarray}$$

into which the field lines are frozen, apart from the effects of electron inertia (Zocco & Schekochihin 2011). Thus, field lines are only broken when gradients are of order $d_{e}^{-1}$.

These equations have various interesting limits. First, for fluctuations on long length scales (compared to the ion gyroradius) $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$, and $\unicode[STIX]{x1D6E4}_{0}\approx 1-k_{\bot }^{2}\unicode[STIX]{x1D70C}_{i}^{2}/2$, and one can show that the equations reduce to those of RMHD, reduced magnetohydrodynamics (Strauss 1976), describing nonlinearly interacting Alfvénic fluctuations propagating up and down the magnetic field. Second, if one takes $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$, $\unicode[STIX]{x1D6E4}_{0}\approx 0$, and subsequently expand in the square root of the mass ratio $\sqrt{m_{e}/m_{i}}$, i.e. $k_{\bot }d_{e}\ll 1$, one obtains the equations of ERMHD, electron reduced magnetohydrodynamics (Schekochihin et al. 2009), which involve nonlinearly interacting kinetic-Alfvén fluctuations propagating up and down the field.

It is important to note that, as mentioned earlier, formally these equations are only valid for $\unicode[STIX]{x1D6FD}_{i}\sim \unicode[STIX]{x1D6FD}_{e}\ll 1$. This rather limits their formal applicability to some physical regimes, including the magnetosheath, where electron-only reconnection has recently been observed (Phan et al. 2018; Stawarz et al. 2019), where $\unicode[STIX]{x1D6FD}_{i}\gtrsim 1$ and $\unicode[STIX]{x1D6FD}_{e}\lesssim 1$. Nonetheless, we believe that their simplicity allows us to model the essential physics of current sheet collapse and reconnection onset in a useful way, for three main reasons. First, simulations of turbulence using KREHM have revealed that it seems to compare rather well with more complete models, even at higher $\unicode[STIX]{x1D6FD}$ than its formal regime of applicability (Grošelj et al. 2017). Second, we will show in § 3 that the Chapman–Kendall-like current sheet collapse solution also exists in ERMHD at arbitrary $\unicode[STIX]{x1D6FD}_{i}$ and $\unicode[STIX]{x1D70F}$, meaning that this part of the analysis remains true even without the formal $\unicode[STIX]{x1D6FD}_{i}\ll 1$ requirement. Finally, the collisionless tearing mode (which we will show enables the onset of reconnection in § 4) has a growth rate that decreases with $\unicode[STIX]{x1D6FD}_{e}$ (Numata & Loureiro 2015) for constant ratio of sheet thickness to $\unicode[STIX]{x1D70C}_{i}$, meaning that at higher $\unicode[STIX]{x1D6FD}_{e}$, the basic physics of our picture will remain, but with the sheets potentially attaining even smaller (relative to $\unicode[STIX]{x1D70C}_{i}$) thicknesses.

We should also mention here that Boldyrev & Loureiro (2019) have used a set of fluid equations derived for $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$, $\unicode[STIX]{x1D6FD}_{i}\sim 1$, $\unicode[STIX]{x1D6FD}_{e}\ll 1$ to study the small-scale tearing instability and its effect on turbulence, obtaining some results similar to ours (which is an additional source of reassurance that our low-$\unicode[STIX]{x1D6FD}_{i}$ ordering does not invalidate the basic picture of tearing-induced reconnection onset): we have used KREHM instead because, first, we want a set of equations covering the ion-scale transition at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ to describe the transition between the ion-coupled and electron-only reconnection regimes, and second, the electron heating channel as encoded by (2.3) is important for a realistic picture of reconnection.

3 X-point collapse

For electron-only reconnection to occur, it must first be possible to form current sheets with thicknesses well below the ion scale. In ideal MHD (magnetohydrodynamics), it is well known that X-points tend to collapse into sheets (Syrovatskii 1981). It turns out that this is still true in our system of equations, even at scales well below the ion gyroradius. If we require

(3.1)$$\begin{eqnarray}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D735}_{\bot }^{2}A_{\Vert }=0,\end{eqnarray}$$

then $g_{e}=0$ is a solution. Let us also impose a ‘vorticity-free’ condition on the system,

(3.2)$$\begin{eqnarray}\unicode[STIX]{x1D735}_{\bot }^{2}\unicode[STIX]{x1D719}=0.\end{eqnarray}$$

Under this condition, $\hat{\unicode[STIX]{x1D6E4}}_{0}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}$³, and along with (3.1), the whole of (2.1) vanishes. Our flow must, of course, be consistent with maintaining the condition (3.1).

3.1 Chapman–Kendall solution

One (two-dimensional) solution which satisfies the conditions (3.1), (3.2) is a slightly modified version of ‘Chapman–Kendall’ collapse (Chapman & Kendall 1963; Biskamp 2000), originally derived for MHD (but which turns out to still be applicable here),

(3.3)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}=\frac{B_{0}}{c}\frac{\unicode[STIX]{x1D6EC}(t)}{2}xy, & \displaystyle\end{eqnarray}$$

(3.4)$$\begin{eqnarray}\displaystyle & \displaystyle A_{\Vert }=\frac{B_{\bot }}{2}\left(\frac{x^{2}+2d_{e}^{2}}{a(t)}-\frac{y^{2}+2d_{e}^{2}}{L(t)}\right), & \displaystyle\end{eqnarray}$$

where $a(t)$ and $L(t)$ are the thickness and length of the current sheet respectively. This solution is an approximation valid in the region close to the X-point in the interior of a realistic current sheet. Inserting this into (2.10), we obtain

(3.5)$$\begin{eqnarray}\displaystyle & \displaystyle {\dot{a}}=-\unicode[STIX]{x1D6EC}(t)a, & \displaystyle\end{eqnarray}$$

(3.6)$$\begin{eqnarray}\displaystyle & \displaystyle \dot{L}=\unicode[STIX]{x1D6EC}(t)L. & \displaystyle\end{eqnarray}$$

Thus, in the absence of reconnection, current sheets can thin to scales much smaller than the ion scales. However, this does not show that such thin sheets are always physically realisable: we will show in § 4 that, in fact, localised current configurations with $L>a$ are disrupted by the onset of reconnection at scales larger than $d_{e}$ (but, importantly, sometimes smaller than $\unicode[STIX]{x1D70C}_{s}$)⁴. White, Hazeltine & Loureiro (2018) have found that KREHM also allows a more general class of Chapman–Kendall-like solutions, for example a helically twisted version of the simple solution presented here.

3.2 Choosing $\unicode[STIX]{x1D6EC}(t)$

It remains for us to decide a reasonable form for $\unicode[STIX]{x1D6EC}(t)$. We do this in two limiting cases, $a\gg \unicode[STIX]{x1D70C}_{i}$ and $a\ll \unicode[STIX]{x1D70C}_{i}$, and ask that the outflows match the characteristic wave speed involved in maintaining the structure of the sheet in the $y$-direction in both cases. In the case where $a\gg \unicode[STIX]{x1D70C}_{i}$, the outflows should attain the constant in-plane Alfvén speed $v_{\text{A}y}=B_{y}(a)/\sqrt{4\unicode[STIX]{x03C0}n_{0i}m_{i}}$ at $y\sim L(t)$. In the small-scale case where $a\ll \unicode[STIX]{x1D70C}_{i}$, the outflow speed at $y\sim L(t)$ should be the in-plane kinetic-Alfvén wave speed (see ZS11 or Schekochihin et al. 2009),

(3.7)$$\begin{eqnarray}u_{\text{KAW}y}=\sqrt{\frac{1}{2}\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)}v_{\text{A}y}\frac{\unicode[STIX]{x1D70C}_{i}}{a(t)}.\end{eqnarray}$$

Importantly, this is inversely proportional to $a$. Thus, it is physically reasonable to choose

(3.8)$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(t)=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{2v_{\text{A}y}}{L(t)}}, & a\gg \unicode[STIX]{x1D70C}_{i},\\[12.0pt] {\displaystyle \frac{\sqrt{2(1+Z/\unicode[STIX]{x1D70F})}v_{\text{A}y}\unicode[STIX]{x1D70C}_{i}}{a(t)L(t)}}, & a\ll \unicode[STIX]{x1D70C}_{i},\\ \end{array}\right.\end{eqnarray}$$

which defines (inverse) time scales associated with forming sheets of thickness $a$ and length $L$. With these choices, equations (3.5) and (3.6) have the solutions

(3.9)$$\begin{eqnarray}\displaystyle & \displaystyle a\gg \unicode[STIX]{x1D70C}_{i}:\quad L=L_{0}+2v_{\text{A}y}t,\quad a=a_{0}L_{0}/L, & \displaystyle\end{eqnarray}$$

(3.10)$$\begin{eqnarray}\displaystyle & \displaystyle a\ll \unicode[STIX]{x1D70C}_{i}:\quad L=L_{0}\exp \left(\frac{\sqrt{2(1+Z/\unicode[STIX]{x1D70F})}v_{\text{A}y}\unicode[STIX]{x1D70C}_{i}}{a_{0}L_{0}}t\right),\quad a=a_{0}L_{0}/L. & \displaystyle\end{eqnarray}$$

The aspect ratio as a function of $a$ is given by

(3.11)$$\begin{eqnarray}\frac{L}{a}=\frac{a_{0}L_{0}}{a^{2}},\end{eqnarray}$$

simply from continuity. It is important to note that, if the sheets are formed as part of turbulence, this scaling is not accurate, because structures will interact with each other as well as undergoing their own nonlinear evolution. Instead, one might expect, for $a\gg \unicode[STIX]{x1D70C}_{i}$, $L/a\propto a^{-1/4}$ (cf. Boldyrev 2006; Chandran, Schekochihin & Mallet 2015; Mallet & Schekochihin 2017). The aspect ratio scaling for $a\ll \unicode[STIX]{x1D70C}_{i}$ in turbulence is as yet unknown.

3.3 Collapse time

Let us start with a structure of thickness $a_{0}>\unicode[STIX]{x1D70C}_{i}$, and ask how long it takes for it to collapse to a thickness $a_{\ast }<\unicode[STIX]{x1D70C}_{i}$. First, using (3.9), the time taken to collapse to $\unicode[STIX]{x1D70C}_{i}$ is

(3.12)$$\begin{eqnarray}t_{\unicode[STIX]{x1D70C}}\sim \left(\frac{a_{0}}{\unicode[STIX]{x1D70C}_{i}}-1\right)\frac{L_{0}}{2v_{\text{A}y}}\approx \frac{a_{0}L_{0}}{2v_{\text{A}y}\unicode[STIX]{x1D70C}_{i}}.\end{eqnarray}$$

Subsequent collapse from $\unicode[STIX]{x1D70C}_{i}$ down to $a_{\ast }$ occurs according to (3.10) in a time

(3.13)$$\begin{eqnarray}t_{\ast }\sim \frac{a_{0}L_{0}\ln (\unicode[STIX]{x1D70C}_{i}/a_{\ast })}{\sqrt{2\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)}v_{\text{A}y}\unicode[STIX]{x1D70C}_{i}},\end{eqnarray}$$

which is, in practice, usually of order $t_{\unicode[STIX]{x1D70C}}$. Thus, the total collapse time is of order $t_{\unicode[STIX]{x1D70C}}$, and mainly depends on the initial size and reconnecting field within the structure.

3.4 Current sheet collapse at higher $\unicode[STIX]{x1D6FD}$

Finally, the question of whether or not this collapse process depends crucially on the low-$\unicode[STIX]{x1D6FD}$ ordering used to derive KREHM may be of interest to the reader. First, for $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$, the Chapman–Kendall collapse obviously works, since the RMHD equations are valid at arbitrary $\unicode[STIX]{x1D6FD}$ and can be derived from gyrokinetics under the assumption $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ (Schekochihin et al. 2009). Second, for $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gg 1$, we can examine the (two-dimensional) electron reduced MHD equations (Schekochihin et al. 2009), which may be manipulated into the following form, chosen for their similarity to (2.1) and (2.2),

(3.14)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\frac{Z}{\unicode[STIX]{x1D70F}}\frac{e\unicode[STIX]{x1D719}}{T_{0e}}=\frac{2}{2+\unicode[STIX]{x1D6FD}_{i}\left(1+\displaystyle \frac{Z}{\unicode[STIX]{x1D70F}}\right)}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\frac{e}{cm_{e}}d_{e}^{2}\unicode[STIX]{x1D735}_{\bot }^{2}A_{\Vert }, & \displaystyle\end{eqnarray}$$

(3.15)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}A_{\Vert }=-\frac{cT_{0e}}{e}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left[\frac{Z}{\unicode[STIX]{x1D70F}}\frac{e\unicode[STIX]{x1D719}}{T_{0e}}\right]. & \displaystyle\end{eqnarray}$$

The effective velocity into which the field lines are frozen is

(3.16)$$\begin{eqnarray}\boldsymbol{u}_{\text{eff}}=\hat{z}\times \unicode[STIX]{x1D735}_{\bot }\frac{c}{B_{0}}\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)\unicode[STIX]{x1D719},\end{eqnarray}$$

which can be compared with (2.11). Under the condition (3.1), equation (3.14) vanishes⁵. Inserting (3.3) and (3.4) with $d_{e}=0$ into (3.15), and requiring the outflow velocity to be the in-plane kinetic-Alfvén velocity (at larger $\unicode[STIX]{x1D6FD}_{i}$, see Schekochihin et al. 2009),

(3.17)$$\begin{eqnarray}u_{\text{KAW}y}=\sqrt{\frac{1+Z/\unicode[STIX]{x1D70F}}{2+\unicode[STIX]{x1D6FD}_{i}(1+Z/\unicode[STIX]{x1D70F})}}v_{\text{A}y}\frac{\unicode[STIX]{x1D70C}_{i}}{a(t)},\end{eqnarray}$$

we find that the solutions for $a(t)$ and $L(t)$ are

(3.18a,b)$$\begin{eqnarray}\displaystyle L=L_{0}\exp \left(\frac{\sqrt{\displaystyle \frac{2(1+Z/\unicode[STIX]{x1D70F})}{1+\unicode[STIX]{x1D6FD}_{i}(1+Z/\unicode[STIX]{x1D70F})/2}}v_{\text{A}y}\unicode[STIX]{x1D70C}_{i}}{a_{0}L_{0}}t\right),\quad a=a_{0}L_{0}/L,\end{eqnarray}$$

only mildly different to the KREHM case. Thus, the fact that current sheets collapse even at thicknesses $a\ll \unicode[STIX]{x1D70C}_{i}$ does not depend on the $\unicode[STIX]{x1D6FD}\ll 1$ ordering of KREHM.

Table 1. Scalings for the collisionless tearing mode in various limits, as discussed in appendix B. The final row, $a/\unicode[STIX]{x1D70C}_{s}\ll 1$ and $\unicode[STIX]{x1D6E5}^{\prime }\unicode[STIX]{x1D6FF}_{\text{in}}\gg 1$, is probably inaccurate since we do not have an analytic solution. The quantity $H=\sqrt{1+\unicode[STIX]{x1D70F}/Z}$; please see the appendix for details.

4 Disruption of forming sheets by the tearing instability

A useful model of the disruption of forming current sheets by reconnection has been developed recently by Uzdensky & Loureiro (2016). In this model, collapsing sheets are disrupted if their linear tearing mode is faster than their formation time scale, i.e.

(4.1)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6EC}(t)>1,\end{eqnarray}$$

assuming that the nonlinear stage of the tearing mode is at least as fast as the linear stage.

In general, the tearing mode that disrupts the sheet has the maximum tearing growth rate $\unicode[STIX]{x1D6FE}_{\text{max}}$ if it fits into the sheet, i.e. if $k_{\text{max}}L>1$, and by the growth rate of the longest-wavelength mode that fits in the sheet otherwise, i.e. the mode with $k_{y}L=1$. The relevant tearing growth rates for our system are summarised in table 1: the growth rates for the case $a\gg \unicode[STIX]{x1D70C}_{s}$ are derived in ZS11, while we derive the growth rates for the opposite case with $a\ll \unicode[STIX]{x1D70C}_{s}$ in appendix B.

4.1 Critical aspect ratio

One can show that for our choice of $\unicode[STIX]{x1D6EC}(t)$, in the case $a\gg \unicode[STIX]{x1D70C}_{s}$, $k_{\text{max}}$ becomes accessible before the mode with $k_{y}L=1$ goes unstable, while in the case where $a\ll \unicode[STIX]{x1D70C}_{s}$, the sheet is disrupted when $k_{\text{max}}L\sim 1$, and thus in practice we may always use $\unicode[STIX]{x1D6FE}_{\text{max}}$. Using (4.1), we find that sheets are disrupted for aspect ratios

(4.2)$$\begin{eqnarray}\displaystyle \frac{L}{a}>\left(\frac{L}{a}\right)_{\text{crit}}\sim \left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{1}{\sqrt{1+\unicode[STIX]{x1D70F}/Z}}}\left({\displaystyle \frac{a}{d_{e}}}\right)^{1/n}\left({\displaystyle \frac{a}{\unicode[STIX]{x1D70C}_{s}}}\right)\left({\displaystyle \frac{\unicode[STIX]{x1D70C}_{s}}{d_{e}}}\right)^{(1-1/n)/3}, & a\gg \unicode[STIX]{x1D70C}_{s}\\[6.0pt] \left({\displaystyle \frac{a}{d_{e}}}\right)^{1/n}, & a\ll \unicode[STIX]{x1D70C}_{s}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

The $a\gg \unicode[STIX]{x1D70C}_{s}$ expression was previously found by Del Sarto et al. (2016)⁶.

We have shown that there is a maximum achievable aspect ratio for sheets with both $a\gg \unicode[STIX]{x1D70C}_{s}$ and $a\ll \unicode[STIX]{x1D70C}_{s}$. Inserting $a\sim \unicode[STIX]{x1D70C}_{s}$ into (4.2), sheets will become unstable before they reach the ion scales ($a\sim \unicode[STIX]{x1D70C}_{s}$) only if their aspect ratio becomes larger than $(\unicode[STIX]{x1D70C}_{s}/d_{e})^{\unicode[STIX]{x1D708}}$, where $1/n\leqslant \unicode[STIX]{x1D708}\leqslant 1/3+2/3n$. Once the sheet reaches the thickness at which it disrupts, it reconnects at that thickness⁷ until all the available flux has been used up. Thus, if one observes a reconnecting (i.e. disrupting) sheet with length

(4.3)$$\begin{eqnarray}L>\frac{\unicode[STIX]{x1D70C}_{s}}{\sqrt{1+\unicode[STIX]{x1D70F}/Z}}\left(\frac{m_{e}}{m_{i}}\frac{Z}{\unicode[STIX]{x1D6FD}_{e}}\right)^{\unicode[STIX]{x1D708}/2},\end{eqnarray}$$

one expects to see that the sheet thickness is larger than the ion scale and that there should be bidirectional ion jets moving at $v_{\text{A}y}$. This roughly agrees with the numerical simulations of Sharma Pyakurel et al. (2019), who found that to obtain fully ion-coupled guide-field reconnection, $L$ needed to be of order $10d_{i}$, and is reminiscent of the results of Mandt, Denton & Drake (1994) for reconnection without a guide field, who observed a change in the scaling of the reconnection rate once the length of sheets became comparable to a few times the ion inertial length. In the magnetosheath interval studied by Phan et al. (2018) and subsequently Stawarz et al. (2019), $\unicode[STIX]{x1D6FD}_{e}\approx 0.5$ and $\unicode[STIX]{x1D70F}/Z\approx 10$; choosing $\unicode[STIX]{x1D708}=1$ and using the $a\gg \unicode[STIX]{x1D70C}_{s}$ expression gives the critical length for ion-coupled reconnection $L/\unicode[STIX]{x1D70C}_{s}\approx 10$ (this is also effectively the critical aspect ratio of such ion-coupled reconnecting sheets, since it is the point at which $a\approx \unicode[STIX]{x1D70C}_{s}$). In other words, the fact that ion-scale sheets and ion jets are not often observed in the magnetosheath means that the aspect ratios of the sheets formed (presumably by the turbulent dynamics) are not extremely high.

As $a\rightarrow d_{e}$, current sheets of any appreciable aspect ratio greater than 1 are disrupted – so in practice, as one might have expected, the steady-state Sweet–Parker-like configuration, with width $d_{e}$ (which we derive in appendix A) is not accessible. Reconnecting sheets for $a\ll \unicode[STIX]{x1D70C}_{s}$ should have the aspect ratio given by the second expression in (4.2), and so we can estimate the aspect ratio of the sheets observed by Phan et al. (2018), with thicknesses $a\approx 4d_{e}$, to be no more than ${\approx}4$, using the $n=1$ expression.

A simple estimate of the disruption time scale of the current sheets observed by Phan et al. (2018) is possible using (3.13). Given that $aL$ is constant during collapse, we can estimate $a_{0}L_{0}=aL\approx 64d_{e}^{2}$. Using $d_{e}\approx 1~\text{km}$, $v_{\text{A}y}\approx 25~\text{kms}^{-1}$ and $\unicode[STIX]{x1D70C}_{i}\approx 100~\text{km}$, $t_{\ast }\approx 0.06~\text{s}$, which suggests that these electron-only reconnection events are individually rather short lived.

4.2 Outflow velocity

One can obtain an estimate of the electron outflow velocity in the reconnecting sheets by a simple scaling argument. Our configuration is a current sheet of thickness $a$ and length $L\gg a$, and a known upstream field $B_{\text{in}}$. The electrons flow into the sheet at $u_{\text{in}}$, and out at $u_{\text{out}}$ (which are related to $\unicode[STIX]{x1D719}$ via (2.11)). We will assume that $g_{e}=0$ for simplicity; although this is not realistic, on the level of the scaling arguments presented here it is irrelevant.

First, because upon inspecting (2.11), $\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u}_{\text{eff}}=0$,

(4.4)$$\begin{eqnarray}Lu_{\text{in}}\sim au_{\text{out}}.\end{eqnarray}$$

Second, because $L\gg a$, the upstream energy is dominated by the magnetic field, while the energy downstream is dominated by the ion flow and the density perturbations (Zocco & Schekochihin 2011). Balancing these,

(4.5)$$\begin{eqnarray}\frac{B_{\text{in}}^{2}}{8\unicode[STIX]{x03C0}}\sim \left[1+\frac{Z}{\unicode[STIX]{x1D70F}}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\right]\frac{Z}{\unicode[STIX]{x1D70F}}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\frac{e^{2}n_{0e}}{2T_{0e}}\unicode[STIX]{x1D719}_{\text{out}}^{2}.\end{eqnarray}$$

In the limit $\unicode[STIX]{x1D70C}_{i}\ll a$, $Z/\unicode[STIX]{x1D70F}(1-\hat{\unicode[STIX]{x1D6E4}}_{0})\approx (\unicode[STIX]{x1D70C}_{s}/a)^{2}$, and using (2.11), this just results (reassuringly) in Alfvénic outflows, $u_{\text{out}}\sim v_{\text{A}y}$, where $v_{\text{A}y}=B_{\text{in}}/\sqrt{4\unicode[STIX]{x03C0}n_{0i}m_{i}}$. In the case where $\unicode[STIX]{x1D70C}_{i}\gg a$, using $\hat{\unicode[STIX]{x1D6E4}}_{0}\approx 0$, we find

(4.6)$$\begin{eqnarray}u_{\text{out}}\sim \sqrt{\frac{1}{2}\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)}v_{\text{A}y}\frac{\unicode[STIX]{x1D70C}_{i}}{a},\end{eqnarray}$$

which is just the in-plane kinetic-Alfvén wave speed. So, rather different from the MHD case, the outflow velocity depends on the thickness of the sheet. Written in terms of the electron Alfvén speed, this is

(4.7)$$\begin{eqnarray}u_{\text{out}}\sim \sqrt{\frac{\unicode[STIX]{x1D6FD}_{i}}{2}\left(1+\frac{Z}{\unicode[STIX]{x1D70F}}\right)}v_{\text{A}ey}\frac{d_{e}}{a}.\end{eqnarray}$$

For the Phan et al. (2018) sheets, our model predicts (with $\unicode[STIX]{x1D6FD}_{i}\approx 5$, cf. Stawarz et al. 2019) $u_{\text{out}}/v_{\text{A}ey}\approx 0.4$, while the observed outflow speed in Phan et al. (2018) is $u_{\text{out}}/v_{\text{A}ey}\approx 0.25-0.45$; given the idealisations involved (not to mention the fact that we are pushing our equations rather beyond their low-$\unicode[STIX]{x1D6FD}$ regime of validity), this is reasonable agreement. More generally, we plot the dependence of the thickness at disruption, $a_{\text{crit}}$, and the electron outflow velocity $u_{\text{out},\text{crit}}$, as functions of $(L/a)_{\text{crit}}$ in figure 1.

Figure 1. In black, the disrupted thicknesses (a) and outflow velocities (b) of reconnecting sheets as a function of aspect ratio (cf. (4.2) and (4.6)). Because we have only derived scalings in the asymptotic limits $a\gg \unicode[STIX]{x1D70C}_{s}$ and $a\ll \unicode[STIX]{x1D70C}_{s}$, for $0.5<a_{\text{crit}}/\unicode[STIX]{x1D70C}_{s}<2$ we have plotted these two scalings as dotted lines; the true scaling must lie between them. Red dashed lines show the positions of $\unicode[STIX]{x1D70C}_{s}$ and $d_{e}$ (a) and $v_{\text{A}ey}$ (b). We have used parameters taken from Stawarz et al. (2019), $\unicode[STIX]{x1D6FD}_{e}=0.5$, $\unicode[STIX]{x1D70F}=10$.

Finally, it is worth asking what the effect of relaxing the low-$\unicode[STIX]{x1D6FD}$ ordering of the KREHM equations would be on the tearing mode and on the critical aspect ratio we obtain here. We have shown in § 3.4 that $\unicode[STIX]{x1D6FD}$ does not make a difference to the collapse process. Numata & Loureiro (2015) have shown that, if $\unicode[STIX]{x1D70C}_{i}/a$ and $\unicode[STIX]{x1D70C}_{s}/a$ are kept constant, the gyrokinetic tearing mode growth rate basically agrees with KREHM, i.e. proportional to $\unicode[STIX]{x1D6FD}_{e}^{-1/2}$. Thus, we believe that using the KREHM tearing mode scalings is physically (if not quite mathematically) reasonable up to $\unicode[STIX]{x1D6FD}_{e}\sim 1$.

5 Conclusions

We have developed a theoretical model for low-$\unicode[STIX]{x1D6FD}$ current sheet formation and disruption by reconnection, and predict that sheets that are not sufficiently anisotropic will collapse to scales much smaller than the ion kinetic scales ($\unicode[STIX]{x1D70C}_{s}$ and $\unicode[STIX]{x1D70C}_{i}$), only reconnecting when their thickness is a few times the electron inertial scale $d_{e}$. For such sheets, we predict electron jets with velocities an appreciable fraction of the electron Alfvén velocity (i.e. much larger than the ion Alfvén velocity), and that there will be no ion-scale region and very slow or no ion outflows. As the aspect ratio of the sheets is varied, there is a gradual transition between the usual ‘ion-coupled’ regime and this new regime of ‘electron-only’ reconnection, with sheets reconnecting at thickness $a\approx \unicode[STIX]{x1D70C}_{s}$ when their aspect ratio $L/a\approx 10$. We find that in the magnetosheath, this means that the aspect ratio of reconnecting sheets, observed to be of thickness $a\approx 4d_{e}$, is no more than a factor of ${\approx}4$. Our results may thus help to explain recent observations of electron-only reconnection in the magnetosheath (Phan et al. 2018).

Our picture of reconnection onset is based on two phenomena. First, ideal current sheets tend to collapse (Syrovatskii 1981; Chapman & Kendall 1963; Biskamp 2000), and we show in § 3 that this collapse does not have to stop at the ion scale. Second, this ideal collapse process breaks down when the tearing mode growth rate becomes larger than the collapse (or sheet formation) rate, an application of the ideas of Uzdensky & Loureiro (2016) to a collisionless plasma. This implies a critical aspect ratio at which reconnection onset occurs, which we derive in § 4. Importantly, the sheet thickness at disruption can be above or below the ion scales, and so we can predict the physical conditions in which we expect ion-coupled reconnection to give way to electron-only reconnection.

Our analysis is based on the KREHM equations, which are derived from gyrokinetics for $\unicode[STIX]{x1D6FD}_{e}\sim \unicode[STIX]{x1D6FD}_{i}\sim Zm_{e}/m_{i}\ll 1$. This low-$\unicode[STIX]{x1D6FD}$ ordering is not usually formally valid in the magnetosheath. However, we can show that, in fact, the collapse process does not depend on the low $\unicode[STIX]{x1D6FD}$, and KREHM does a surprisingly good job of predicting the full gyrokinetic tearing rates even at moderate $\unicode[STIX]{x1D6FD}_{e}$. This perhaps explains why our estimates of electron outflow velocities agree reasonably well with the MMS observations. More generally, we believe that KREHM, while not always formally valid, is a useful minimal physical model of collisionless reconnection, which allows us to make theoretical progress in understanding electron-only reconnection and the transition from the ion-coupled to electron-only regimes.

Boldyrev & Loureiro (2019) use a different approach to the one used in this work: instead of studying a nonlinear collapsing solution, they show that if helicity in a structure is held constant, the nonlinear interaction is minimised by making the structure as sheet like as possible: thus, sheet-like structures last longer. They then argue that this means that the turbulence favours the creation of highly sheet-like structures, but that this is limited by the tearing instability, which then sets a particular critical aspect ratio (as does our analysis). Despite the overall similarity of the results, there are some important differences in the two pictures: in ours, the nonlinear interaction actively produces increasingly sheet-like structures, while in Boldyrev & Loureiro (2019), its role is to rapidly remove non-sheet-like structures. Our method has the advantage that it also works for sheets thicker than the ion scales, and we can therefore predict when one expects to see electron-only versus ion-coupled reconnection, depending on the characteristic aspect ratios of sheets as a function of their thickness.

In turbulence models that include the phenomenon of dynamic alignment (Boldyrev 2006; Chandran et al. 2015; Mallet & Schekochihin 2017), the turbulent structures at scale $\unicode[STIX]{x1D706}$ are sheet like in the perpendicular plane, with aspect ratios $\propto (\unicode[STIX]{x1D706}/L_{\bot })^{-1/4}$, where $L_{\bot }$ is the outer scale at which energy is injected into the system. Thus, for electron-only reconnection to be observed in turbulence, energy must be injected relatively close to the ion scales: in the magnetosheath, this injection could potentially come in the form of Alfvén vortices (Alexandrova 2008), perhaps driven by density gradients (Sundkvist & Bale 2008). If this picture is correct, electron-only reconnection may be relevant in other astrophysical environments with large density inhomogeneities, for example close to shocks. The appearance of electron-only reconnection in turbulent systems where the driving scale is relatively close to the ion scales has been observed rather clearly in the (two-dimensional) hybrid turbulence simulations of Califano et al. (2018), who observed electron-only reconnection when $d_{i}/L_{\bot }\approx 0.6$, but ion-coupled reconnection when $d_{i}/L_{\bot }\approx 0.3$. Our results therefore provide a theoretical context for these numerical results.

Another low-$\unicode[STIX]{x1D6FD}_{e}$ environment of interest is the inner heliosphere and solar corona, currently being explored by the Parker Solar Probe (Fox et al. 2016). Our results show that whether ion-coupled or electron-only reconnection, or a mixture of the two, occur in this turbulent setting depends crucially on the length of the inertial range between the injection scale and the ion scales.

However, a detailed picture of electron-only reconnection onset in turbulence in the range of scales between $\unicode[STIX]{x1D70C}_{s}$ and $d_{e}$ requires a model of how the aspect ratios of typical turbulent structures evolve in this range in the absence of reconnection (or, indeed, after ion-coupled reconnection decreases the typical aspect ratio, see e.g. Mallet, Schekochihin & Chandran 2017): this will be the focus of future work.