3.1. Flux conservation

Alfvén’s celebrated result for ideal MHD (Alfvén Reference Alfvén1942) shows that magnetic flux is frozen into the flow and, as a consequence of this, field line topology is preserved (field lines behave as material lines). For the TRM, there are similarities to this situation, but also fundamental differences.

Let us assume that the mean velocity and magnetic fields are suitably smooth (possessing as many derivatives as needed – a reasonable assumption for the mean fields). Let $\mathcal {C}$ be a closed loop, with unit tangent vector ${\boldsymbol{t}}$ , that is (Lie-)transported by the mean velocity ${\boldsymbol{U}}$ . Let the corresponding surface, bounded by $\mathcal {C}$ , be denoted $\mathcal S$ and let the unit normal vector to this surface be denoted ${\boldsymbol{n}}$ . The mean magnetic flux $\Phi$ through $\mathcal S$ is

\begin{equation*} \Phi = \int _{\mathcal S}{\boldsymbol{B}}\cdot {\boldsymbol{n}}\,\textrm {d}^2x. \end{equation*}

It is not difficult to show that, with magnetic diffusion ignored,

\begin{align*} \frac {\textrm {d}\Phi }{\textrm {d} t} &= \int _{\mathcal S}\left [\frac {\partial {\boldsymbol{B}}}{\partial t} - \nabla \times ({\boldsymbol{U}}\times {\boldsymbol{B}})\right ]\cdot {\boldsymbol{n}}\,\textrm {d}^2x\\ &=\oint _{\mathcal {C}}{\boldsymbol{E}}_M\cdot {\boldsymbol{t}}\,\textrm {d} x, \end{align*}

where the last line follows from an application of Stokes’ theorem. It is, therefore, clear that the mean magnetic flux is conserved if the electromotive force has the form ${\boldsymbol{E}}_M=\nabla \phi$ , for some scalar function $\phi$ . In this situation, we have a modified form of Alfvén’s theorem, although there are important physical differences. For simplicity, let us consider the case of $\phi =\textrm {const.}$ , i.e. ${\boldsymbol{E}}_M=\boldsymbol {0}$ . Superficially, this choice seems to satisfy flux conservation trivially, akin to removing magnetic diffusion from the induction equation. However, on consideration of (2.4) for this case, namely

(3.1) \begin{equation} \boldsymbol {0}=\alpha {\boldsymbol{B}}-\beta {\boldsymbol{J}}+\gamma \boldsymbol {\Omega }, \end{equation}

this situation is more complex. First, the trivial solution $\alpha =\beta =\gamma =0$ means that there is no turbulence and we are back to laminar MHD. If the turbulent transport coefficients are non-zero, however, equation (3.1) represents a dynamic balance (Yokoi Reference Yokoi2020). This means that, although the mean flux is conserved, it is not simply in a passive manner as for the magnetic field in laminar ideal MHD, in which the full, rather than just the mean, magnetic flux is frozen into the flow. Instead, turbulence acts continuously in a particular way (satisfying equation (3.1)) to conserve the mean flux. Thus, although the derivations of flux conservation in laminar ideal MHD and mean flux conservation in the TRM are very similar, the underlying physics of both is markedly different.

In general, the electromotive force is not likely to simply be zero or a potential field (we will show such an example later), so the effects of turbulence will violate Alfvén’s theorem for mean fields. Thus, the mean magnetic flux is not, in general, conserved in the TRM.

3.2. Field line topology

For laminar ideal MHD, a corollary of Alfvén’s theorem is that the topology of field lines is also conserved. Translating this result to the TRM, the condition for the conservation of (mean) field line topology can be written as

\begin{equation*} \frac {\partial {\boldsymbol{B}}}{\partial t} - \nabla \times ({\boldsymbol{U}}\times {\boldsymbol{B}}) = \lambda {\boldsymbol{B}}, \end{equation*}

for some scalar function $\lambda$ (Stern Reference Stern1966; Hornig & Schindler Reference Hornig and Schindler1996). We thus have more options for mean field line conservation than just setting ${\boldsymbol{E}}_M=\nabla \phi$ , which corresponds to setting $\lambda =0$ . In other words, the mean field line topology can be conserved even if the mean flux is not. We now follow a standard approach (see, for example, § 2.2 in Birn & Priest (Reference Birn and Priest2007) by Hornig) but applied to the mean fields. Let us assume that there exists a velocity field ${\boldsymbol{V}}$ such that

\begin{equation*} \frac {\partial {\boldsymbol{B}}}{\partial t} = \nabla \times ({\boldsymbol{V}}\times {\boldsymbol{B}}). \end{equation*}

That is, the mean magnetic flux is frozen into the flow determined by ${\boldsymbol{V}}$ . The ideal and mean induction equation for a general electromotive force is

\begin{equation*} \frac {\partial {\boldsymbol{B}}}{\partial t} = \nabla \times ({\boldsymbol{U}}\times {\boldsymbol{B}})-\nabla \times {\boldsymbol{E}}_M, \end{equation*}

where we assume no dynamic balance. Equating these two induction equations, it is clear that

\begin{equation*} \nabla \times [({\boldsymbol{U}}-{\boldsymbol{V}})\times {\boldsymbol{B}}] = \nabla \times {\boldsymbol{E}}_M. \end{equation*}

If we write ${\boldsymbol{W}}={\boldsymbol{U}}-{\boldsymbol{V}}$ , then for the electromotive force to preserve the mean field line topology, it must have the form

(3.2) \begin{equation} {\boldsymbol{E}}_M = {\boldsymbol{W}}\times {\boldsymbol{B}}+\nabla \phi . \end{equation}

The individual parts of ${\boldsymbol{E}}_M$ in (2.4) do not have the form of the first or second terms on the right-hand side of (3.2). Therefore, in general, the effect of turbulence is to not preserve mean field line topology but to cause reconnection. In particular, the mathematical form of the term related to the turbulent energy in (2.4) is very similar to that of the non-ideal term of resistive MHD (magnetic diffusion), even though the physics represented by the two terms is different. Thus, in general, turbulence leads to the reconnection of mean field lines.

These general theoretical considerations imply that reconnection in the TRM depends on the dominance of one or more of the terms that compose the electromotive force. The efficiency of reconnection depends on how well $|{\boldsymbol{E}}_M|$ of sufficient size can be localised, similar to the localisation of magnetic diffusion at magnetic topological boundaries (e.g. separators and null points) for reconnection in laminar resistive MHD. We thus have a mathematical correspondence between laminar and turbulent reconnection, even if the equations represent different physics. This suggests that for the phenomena of laminar reconnection can find turbulent analogues within the context of the TRM. We now investigate this claim for 2-D steady-state reconnection in a current sheet.

4.1. Set-up

We solve the TRM equations numerically using the method of lines solver Bout++ (Dudson et al. Reference Dudson, Umansky, Xu, Snyder and Wilson2009), a semi-discrete simulation framework that solves systems of partial differential equations in general curvilinear coordinates. In our simulations, we discretise the spatial coordinates $(z,x)$ into a $N_z\times N_x$ uniform mesh in a simulation domain of (non-dimensional) size $L_z \times L_x$ (the $z$ -direction is horizontal and the $x$ -direction is vertical). We solve the momentum and induction equations in stream-function form, so the full system of equations is

(4.1) \begin{align} \frac {\partial }{\partial t} \nabla ^2 \phi &= \left \{\nabla ^2 \phi, \phi \right \} + \left \{ \psi, \nabla ^2 \psi \right \} + \frac {1}{\textrm {Re}} \nabla ^4 \phi, \end{align}

(4.2) \begin{align} \frac {\partial }{\partial t} \psi &= \left \{\psi, \phi \right \} + \left ( C_\beta \tau K + \frac {1}{\textrm {Rm}} \right ) \nabla ^2 \psi - C_\gamma \tau W \nabla ^2 \phi, \end{align}

(4.3) \begin{align} \frac {\partial }{\partial t} K &= \left \{K, \phi \right \} + \left \{\psi, W \right \} +C_\beta \tau K \left [ \nabla ^2 \psi \right ]^2 - C_\gamma \tau W \nabla ^2 \psi \nabla ^2 \phi \nonumber \\ & - \frac {K}{\tau } - \chi \nabla ^4 K, \end{align}

(4.4) \begin{align} \frac {\partial }{\partial t} W &= \left \{W, \phi \right \} + \left \{\psi, K \right \} +C_\beta \tau K \nabla ^2 \psi \nabla ^2 \phi - C_\gamma \tau W \left [ \nabla ^2 \phi \right ]^2 \nonumber \\ & - C_W\frac {W}{\tau } - \chi \nabla ^4 W, \end{align}

where ${\boldsymbol{U}} = \nabla \times ( \phi {\boldsymbol{e}}_{\boldsymbol{y}} )$ and ${\boldsymbol{B}}=\nabla \times ( \psi {\boldsymbol{e}}_{\boldsymbol{y}} )$ . We have included hyper-diffusivity on the turbulent quantities for stability with a hyper-diffusion coefficient of $\chi =10^{-7}$ . The curly braces are Poisson brackets defined as

\begin{equation*}\left \{ A,B \right \}= \frac {\partial A}{\partial x}\frac {\partial B}{\partial z}-\frac {\partial B}{\partial x}\frac {\partial A}{\partial z}.\end{equation*}

The brackets are discretised via Arawaka’s method (Arakawa Reference Arakawa1966) while the laplacian terms are solved by second-order central differences. This reduces the entire problem to a system of ordinary differential equations that are then integrated in time via the implicit pvode scheme, a backwards differencing method (Byrne & Hindmarsh Reference Byrne and Hindmarsh1999). The initial form of the magnetic field in all simulations takes the form of a Harris current sheet

(4.5) \begin{align} {\boldsymbol{B}}_0 = \tanh \left (\pi \frac {x}{\delta }\right ) {\boldsymbol{e}}_z, \end{align}

where $\delta$ is the half-thickness of the current sheet. There is no initial flow and the initial turbulent energy is set to $K_0 = K_{min }$ , the smallest value the turbulent energy can take in the simulations. We set this minimum value to $K_{min }=10^{-8}$ , which is as small as floating point precision allows. The initial condition for $K_0=K_{min }$ then corresponds to a laminar current sheet.

The turbulent time scale is determined as in Higashimori et al. (Reference Higashimori, Yokoi and Hoshino2013), in which we first consider the dominant balance of terms in the steady-state form of (2.8). If ${\boldsymbol{J}}_0$ is the initial current density and $J_0$ represents its magnitude at the centre of the current sheet, for a steady state we have

(4.6) \begin{equation} \tau _0 \approx \frac {1}{\sqrt {C_\beta }J_0}. \end{equation}

The turbulent time scale is varied from its steady-state value by rescaling as $\tau =C_\tau \tau _0$ . We will make use of the scale factor $C_\tau$ later when characterising reconnection solutions.

In the following simulations, the current sheet (half-)thickness is set as $\delta =1$ . Unless otherwise specified, the lengths of the domain are, $L_x=10$ and $L_z=80$ . Our simulations are run at a resolution of $1024\times 2048$ with periodic boundary conditions in the $z$ -direction and perfectly conducting, no-slip conditions in the $x$ -direction. The background Reynolds numbers (fluid and magnetic) are both set to $5 \times 10^{4}$ .

To perturb the current sheet, we introduce a mix of modes that leads to reconnection at the centre of the domain. In terms of the magnetic flux function, the perturbation has the form

\begin{equation*} \phi _0= \frac {1}{1000}\: \exp [-100x^2] \sum _{n=1}^{13} \frac {1}{n}\cos (nz). \end{equation*}

This expression is designed to produce a small-amplitude perturbation that is limited to near the centre of the current sheet.

4.2. Reconnection rates

In order to determine whether or not a particular reconnection solution is fast, we need to determine the rate of reconnection. There are various ways to measure the reconnection rate but the standard method, particularly for 2-D reconnection, is to measure the Alfvén Mach number at the edge of the current sheet, $M_{{in}}\;:\!= {U_{x,{ in}}}/{U_{A,{in}}}$ . Theories of steady-state reconnection provide expressions for the reconnection rate based on particular scalings of the diffusion region. For example, in the Sweet–Parker scaling for resistive MHD, the reconnection rate is given by

(4.7) \begin{equation} M_{{in}} \sim \frac {1}{\sqrt { \textrm {Rm}}} = \sqrt {\eta }, \end{equation}

where we introduce the $\eta$ -notation for the constant background magnetic diffusion coefficient, in order to simplify notation later.

Figure 1. A representation of the identification of the diffusion region. The central figure shows a picture of the current density magnitude $J$ for a simulation with $C_\tau =1$ . The left panel shows in inflow velocity across the centre of the sheet and the bottom panel displays the magnitude of $\eta _{{eff}}$ . The dotted lines correspond to where we identify the boundaries of the diffusion region to be, as described in the main text.

In our simulations, the size and shape of the diffusion region depends on the dominant physics controlling it, and we will describe this in more detail later. However, for the practical identification of the diffusion region’s boundaries, we identify characteristics that apply to all the turbulent solutions under study. The height of the diffusion region is determined by the rapid change in the mean velocity, as shown in figure 1. For the width, we select the range in which the effective magnetic diffusivity $\eta _{{eff}}$ is more than twice the average over the entire length of the current sheet. In order to define the effective magnetic diffusivity, we combine the effects of magnetic diffusion and turbulent energy from the electromotive force to identify

(4.8) \begin{equation} \eta _{{eff}} = \eta + C_\beta \tau K. \end{equation}

This quantity can be derived by expanding the induction equation (2.2) using (2.4) and (2.6) and ignoring the contribution of cross-helicity, which is small in the diffusion region. It is clear from figure 1 that this approach selects a reasonable choice for the width of the diffusion region.

Since we solve an initial value problem, we never produce a perfect steady state (and so, in all later discussion, the term steady state refers to quasi-steady state, for which there is weak decay with time). In practice, this means that the diffusion region changes in size by a small amount as the total magnetic flux is depleted over time. Thus, all of our reconnection measurements are based on an average of the five closest grid cells to the diffusion region boundary.

Figure 2. This figure shows the reconnection rate $M_{{in}}$ for various values of $C_\tau$ .

Figure 2 shows the reconnection rate of the TRM over time for various values of the turbulent time scale factor $C_\tau$ (equivalent to considering different $\tau$ ). What is clear is that, for values of $C_\tau$ large enough, the reconnection rate evolves to a unique steady rate just below the fast rate of $M_{{in}}\approx 0.1$ (Cassak, Liu & Shay Reference Cassak, Liu and Shay2017). For lower values of $C_\tau$ the steady-state reconnection rate remains low, typically less than 0.01. Remarkably, for this complex system, there are only two general types of steady reconnection solution. To justify the existence of these two steady solutions, we will identify the critical balances that lead to their formation. Before this, however, we now present an overview of the nature of all reconnection solutions.

4.3. Overview of all solutions

As mentioned earlier, it has been reported that three distinct reconnection solutions exist for the TRM model: laminar reconnection, turbulent reconnection and turbulent diffusion. Here we show that only the first two of these solutions are distinct, with the third solution (turbulent diffusion) revealing itself as a version of the second (turbulent reconnection) on a long time scale. Representations of these solutions are displayed in figure 3.

Figure 3. Maps of the current density magnitude $J$ displaying different phases of the reconnection solutions for different values of $C_\tau$ . The images on the left (a, c, e) depict an early stage of reconnection, before significant deformation of the current sheet. The images on the right (b, d, f) depict when steady-state reconnection has been established. Note that we have zoomed in on the diffusion region so $x \in [-2.5,2.5]$ . The domain for the $C_\tau =2.5$ case has been extended to $L_z=160$ while retaining the same resolution so that a steady state can be achieved before the outflow impinges upon the boundary.

Focussing on each solution in turn, figures 3(a) and 3(b) display initial and late times of laminar reconnection. This solution is laminar in the context of the TRM as there is no growth of turbulent energy $K$ (to be discussed in more detail shortly) and the current sheet evolves to the classic Sweet–Parker scenario of resistive MHD.

For the other solutions, shown in figures 3(c–f), respectively, the form of reconnection is governed by turbulence. Comparing the early phases in (c) and (e), the solution with large $C_\tau$ has led to a much thicker and diffuse current sheet. The initial reconnection rate for this $C_\tau =2.5$ case is much smaller than that of the $C_\tau =1$ case on the time scale which the latter develops fast reconnection (see figure 2). This slower reconnection has been previously described as turbulent diffusion (Higashimori et al. Reference Higashimori, Yokoi and Hoshino2013). However, if the solution is allowed to evolve to much later times, it develops the same fast reconnection as for smaller $C_\tau$ (turbulent reconnection) cases.

The magnetic field of the fast turbulent reconnection solutions follows the structure of Petschek reconnection, as opposed to laminar Sweet–Parker reconnection. As is clear from figures 3(d) and 3(f), the diffusion region is localised in the centre of the domain with four thin current layersFootnote ¹ emanating diagonally outwards. Beyond this visual inspection, we also present evidence that the localised diffusion region behaves approximately like a Sweet–Parker region with a local reconnection rate

\begin{equation*} M_{{in}}\sim \textrm {Rm}_{{eff}}^{-1/2}, \end{equation*}

where the localised magnetic Reynolds number at the diffusion region is $\textrm {Rm}_{\textrm {diff}} = L U_{A,\textrm { in}}/\eta _{{eff}}$ (e.g. Baty, Priest & Forbes Reference Baty, Priest and Forbes2006). Table 1 shows a close match for the Sweet–Parker scaling across a range of $C_\tau$ .

Table 1. A comparison of the Mach number at the edge of the diffusion region, $M_{{in}}$ , and the Sweet–Parker reconnection rate in the diffusion region, $\textrm {Rm}_{{eff}}^{-1/2}$ , for a range of values of $C_\tau$ . The values are taken from a time average starting at $t_{{sim}}=t_{{end}}-40$ and proceeding to the end of the simulation, $t_{{sim}}=t_{{end}}$ . The error bars come from the error in measuring the inflow region across five grid cells of the boundary, as mentioned in the main text.

In summary, the TRM supports two steady-state reconnection solutions, a slow and laminar Sweet–Parker solution and a fast and turbulent Petschek solution. We now consider how these solutions are selected.

4.4. Critical current balances

The two steady-state reconnection solutions are due to intrinsic balances within the system. In searching for these balances, there are two key physical elements to consider. First, the inclusion of a constant background magnetic diffusivity $\eta$ leads to a natural length scale. In a steady-state current sheet, the classical Sweet–Parker scaling provides an estimate of the current sheet thickness as $\delta \sim \eta ^{1/2}$ . This estimate provides a lower length scale that can be reached by the system in a steady state when the presence of turbulence is absent. For our purposes, it will be useful to provide an estimate of the current density magnitude at the reconnecting X-point. Under the Sweet–Parker scaling this quantity can be approximated as

(4.9) \begin{equation} J_\eta \sim \frac {B_{in}}{\eta ^{1/2}}, \end{equation}

where $B_{{in}}$ is the magnitude of the magnetic field at either of the upper or lower edges of the diffusion region.

For a mean steady state with turbulence, we look again to (2.8). Similar to the derivation of (4.6), and now combining this with the expression $\tau =C_\tau \tau _0$ , we arrive at the following estimate for the magnitude of the current density at the centre of the diffusion region

(4.10) \begin{equation} J_c=J_\tau \sim \frac {J_0}{C_\tau }, \end{equation}

where $J_\tau$ is the critical current density (this is made clear below). In order to examine the relevance of the estimates given in (4.9) and (4.10), we first consider how the magnitude of the current density at the centre of the diffusion region, denoted $J_c$ , relates to these quantities. Figure 4 displays the temporal development of $J_c$ and $K_c$ (the value of $K$ at the centre of the diffusion region) for different values of $C_\tau$ . The value of $J_0/C_\tau$ is displayed for each case as a horizontal dashed line.

Figure 4. Time evolutions of $J_c$ amd $K_c$ for different values of $C_\tau$ . For each case, the value of $J_\tau$ is represented by a dashed line.

Figure 4(a) displays a case of Sweet–Parker reconnection ( $C_\tau =0.1$ ). The current density $J_c$ increases until shortly after $t=50$ and then decreases at a slow rate (this is the quasi-steady Sweet–Parker solution). There is no growth in the turbulent energy $K_c$ . One feature that is clear from this figure is that $J_c\lt J_\tau$ for the entire evolution. In figure 4(b), however, which displays a case of Petschek reconnection ( $C_\tau =0.5$ ), $J_c$ breaks through the $J_\tau$ barrier. As the current density grows, it eventually (shortly after $t=20$ for this case) becomes strong enough to generate turbulent energy $K_c$ (see 2.8). The current density reaches a maximum value and then drops rapidly to a value just above $J_\tau$ . Co-temporal with this rapid decrease, the turbulent energy $K_c$ rises rapidly until holding an approximately steady value. Beyond $t\approx 30$ , both $J_c$ and $K_c$ are approximately steady (this is the quasi-steady Petscheck solution). In figure 4(c), we again have a case of Petschek reconnection ( $C_\tau =1.5$ ), but the difference to the previous case is that $J_c\gtrapprox J_\tau$ for the entire evolution. Here, the current density is strong enough to cause $K_c$ to increase from the initial condition. Later, the behaviour of $J_c$ and $K_c$ is qualitativly similar to the $C_\tau =0.5$ case, i.e. both achieve quasi-steady values with $J_c$ resting just above $J_\tau$ . What the cases in figure 4 show is that $J_\tau$ provides a critical barrier for the selection of a Petschek solution. If $J_c$ can become larger than $J_\tau$ , then turbulent energy can be produced and a steady state can form with $J_\tau$ acting as a lower boundary for the current density at the centre of the diffusion region.

Figure 5 displays the relationship of the average the critical current balances $J_\eta$ and $J_\tau$ as a function of $C_\tau$ , once a steady state has been reached. There is clearly an excellent match between the simulation values of $J_c$ , $J_\eta$ and $J_\tau$ . For $C_\tau \gtrapprox 0.2$ , $J_c\approx J_\tau$ and the turbulent energy reaches an approximately constant value of 0.1. These are all Petschek solutions. For $C_\tau \lessapprox 0.2$ , $J_\tau$ is not reached and $J_c\approx J_\eta$ . These are all Sweet–Parker solutions.

Figure 5. A representation of the steady-state values of $J_c$ and $K_c$ for a range of $C_\tau$ . The solid line shows the critical current selection $\min (J_\tau, J_\eta )$ , which identifies the selection of either Sweet–Parker or Petschek solutions.

4.5. Self-similarity of turbulent petschek solutions

We have shown that for turbulent Petschek solutions, the steady reconnection rate is $M_{{in}}\approx 0.1$ with $J_c\approx J_\eta$ . Since $J_\tau \propto 1/\tau$ , this suggests a self-similarity for the turbulent Petschek solutions, all depending on the turbulent time scale. For example, compare the solutions displayed in figures 3(d) and 3(e). The latter has a value of $J_c$ 2.5 times less than that of the former. However, it is clear that the size of the diffusion region of the latter is greater than the former. Based on $J_\tau$ , we can estimate the thickness of the diffusion region to be

(4.11) \begin{equation} \delta \sim \frac {C_\tau B_{in}}{J_0}. \end{equation}

Figure 6. Diffusion region thicknesses $\delta$ , as a function of $B_{{in}}$ , for three values of $C_\tau$ . Crosses are determined from simulations and lines of best fit are overplotted on the points of each case. The gradients of these lines follow the estimate in (4.11). The discrete steps in the data are due to the resolution. In the $x$ -direction the resolution is $\Delta x \approx 0.01$ so, as the sheet decreases in width, $\delta$ decreases in integer multiples of $\Delta x$ .

Figure 6 displays how the diffusion region thickness $\delta$ varies as a function of $B_{in}$ for three different values of $C_\tau$ . For each case, a line of best fit is plotted over the points determined from the simulations. The gradient of each line is $C_\tau /J_0$ , confirming the estimate in (4.11). Thus, the self-similarity of the turbulent Petschek solutions have diffusion region thicknesses and current densities scaling with $\tau$ and $1/\tau$ , respectively. The solutions are self-similar, with the turbulent time scale acting as the fundamental scaling parameter.

4.6. Theoretical justification of the steady Petschek reconnection rate

Through the simulations described above, we have shown that a universal steady Petschek reconnection rate is possible and that different Petschek solutions are self-similar. We now provide some basic theoretical justification that a universal Petschek rate is a natural consequence of some standard balances combined with the behaviour of the turbulent energy at the null point.

First, let the localised diffusion region have a thickness and width denoted by $\delta$ and $L$ respectively. Let $U_{{in}}$ denote the inflow speed, $U_{{out}}$ denote the outflow speed and $K_c$ denote the turbulent energy at the null point, as before. From mass conservation, it follows that

(4.12) \begin{equation} U_{{in}}L \sim U_{{out}}\delta . \end{equation}

Looking now to the momentum equation, balancing inertia with tension produces

(4.13) \begin{equation} U_{{out}} \sim U_A. \end{equation}

In the induction equation, it is the turbulent diffusivity $\beta _c\sim C_\beta \tau K_c$ , which is important for reconnection. In the steady state, we have the balance

(4.14) \begin{equation} U_{{in}} \sim \frac {\beta _c}{\delta } \sim \frac {C_\beta \tau K_c}{\delta }. \end{equation}

Two more relations are required to determine the five independent variables, and these can be found by considering the behaviour of turbulence in the diffusion region. Close to the null point, the cross-helicity effects can be ignored. Further, from (4.6) and (4.10), the critical current density can be written as $J_\tau = 1/(\sqrt {C_\beta }\tau )$ . Combining these properties, it is a straightforward consequence that

(4.15) \begin{equation} \frac {\partial K}{\partial t} +({\boldsymbol{U}}\cdot \nabla )K \sim C_\beta \tau K(J^2-J_\tau ^2). \end{equation}

At the null (and stagnation) point, we have that $J=J_\tau$ , which we have confirmed numerically. From this fact, scaling (4.11) was derived and can also be written as

(4.16) \begin{equation} \delta \sim \frac {B_{{in}}}{J_\tau } \sim \sqrt {C_\beta }\tau B_{{in}}. \end{equation}

Close to the null point, the right-hand side of (4.15) may be ignored, and the turbulent energy is simply advected by the flow. At larger distances, $J\ll J_\tau$ , representing the decay of turbulent energy. Given that the turbulent energy plays a controlling role in the definition of the diffusion region ( $\beta _c\propto K_c$ ), the distance that an Alfvénic flow (for fast reconnection) travels in a decay time can be described as

(4.17) \begin{equation} L \sim U_A\tau . \end{equation}

Putting all of this information together, the dominant scalings for steady Petschek reconnection can be written as the following five relations:

(4.18) \begin{equation} L\sim U_A\tau, \quad \delta \sim \sqrt {C_\beta }\tau B_{{in}}, \quad U_{{in}} \sim \sqrt {C_\beta } U_A, \quad U_{{out}} \sim U_A, \quad K_c\sim U_A^2. \end{equation}

Thus, it follows that the steady Petschek reconnection rate is

(4.19) \begin{equation} M_{{in}} = \frac {U_{{in}}}{U_A} \sim \sqrt {C_\beta }. \end{equation}

Since $C_\beta$ is a universal constant, it follows that the steady rate in (4.19) is also universal, matching qualitatively the behaviour found numerically.

Although $C_\beta$ is not formally a variable but rather a fixed universal property of turbulence within the framework of Higashimori et al. (Reference Higashimori, Yokoi and Hoshino2013), we may vary $C_\beta$ to test scaling (4.19) . We can confirm that, as suggested by (4.19), $M_{{in}}$ follows approximately the above scaling. This feature is displayed in figure 7 by considering ‘high’ values of $C_\beta$ , i.e. $O(0.3)$ , and ‘low’ values, i.e. $O(0.01-0.1)$ .

Figure 7. The behaviour of $M_{{in}}$ as a function of the (given) parameter $C_\beta$ . Note that $C_\gamma =C_\beta$ for each case. Here, $C_\tau =1$ .

Given the simplicity of the above scaling argument (i.e. purely local, ignoring the internal diffusion region structure, ignoring $W$ , etc.), the agreement shown in figure 7 is reasonable. Although the cross-helicity $W$ is missing from the scaling argument, it does play a role in figure 7 as, for each case, $C_\gamma =C_\beta$ . Since both of these parameters represent properties of MHD turbulence, they need to be varied together.