Recognising some of these scales as indicative of the reconnecting layer width requires instead further insight (and hypothesis) of physical nature.

This is the subject we discuss in this section, where we are going to provide an operational definition allowing the measurement of the reconnecting layer width, $\delta$, which is in agreement with previous theoretical assumptions (i.e. theoretical definitions based on some physical insight that have been used in the previous literature), and which can be useful for both numerical and experimental quantitative estimates. The appropriateness of this definition is then shown by direct comparison with boundary layer calculations of §§ 6 and 7 and by numerical verification of its asymptotic scaling.

By analysing some local properties of the derivatives of the eigenfunctions close to or on the neutral line, we also identify the asymptotic scalings of some microscopic scales, which are associated with the inverse of these spatial gradients. The relevance of these scale lengths will be shown again by comparison between the results of the boundary layer calculations and the numerical results.

To this purpose, we use the adaptive multi-precision solver discussed by Betar et al. (2020), which had been specifically developed to address the generalised eigenvalue problem in a slab periodic box of dimension $[-L_x/2, L_x/2]$, for $k$ assigned. This solver uses a compact finite difference scheme of tunable precision for the derivatives on a non-uniform grid along the $x$-direction. The use of a non-uniform grid and of tunable precision in the calculation allow us to resolve the inner layer even for microscopic, realistic values of the non-ideal parameters: the grid spacing in the inner layer can be so chosen to be much smaller than in the outer layer to save computing time without loosing accuracy. In that work, we verified the numerical scalings predicted by boundary layer calculations in different reconnection regimes. Although the scalings in the small-$\Delta'$ regime are insensitive of the magnetic equilibrium profile, this is not the case for the large-$\Delta'$ limit, as it had been already noted in a series of works by Cross & van Hoven (1971), Van Hoven & Cross (1971), Van Hoven & Cross (1973b) and Van Hoven & Cross (1973a): in both (Betar et al. 2020) and in the present article, the numerical results refer to the equilibrium profile (4.3) in a numerical box with $L_x = 4\pi$.

## 8.1. Notions of reconnecting layer and estimates of its width in previous literature

The reconnecting layer width $\delta$, meant as the extension of the interval around the neutral line, where the reconnection process takes place and is mostly localised, is not per se unequivocally defined via boundary layer calculations. Its identification requires further ansatz based on physical assumptions. Because of this, in most of the early reference papers about boundary layer calculations performed in the warm fluid-electron regime, the notion of ‘reconnecting layer width’ has not been explicitly used (cf. Pegoraro & Schep 1986; Pegoraro et al. 1989; Porcelli 1991). In this context, an early notion of ‘layer width’, identifiable as $\delta$, has just been used in some kinetic models for tearing modes (cf. Drake & Lee 1977; Mahajan et al. 1978, 1979; Cowley et al. 1986). In particular, in some of these works (Drake & Lee 1977; Cowley et al. 1986), it has been referenced as the ‘electron layer width’, because of its identification with the microscopic region dominated by electron dynamics, in contraposition to the broader ‘ion layer width’, where non-ideal effects are important, but at scales larger than those of electrons. In this sense, $\delta$ has been made to actually correspond to the innermost layer width, which we have named $\delta_1$ (the ‘ion layer’ being practically correspondent to $\delta_2$). This identification, which also agrees with the assumption made in cold-electron regimes (practically since the first work of Furth et al. (1963) – cf. also Ottaviani & Porcelli 1995) where a single non-ideal layer can
be identified, has been made more explicit in later works (Bhattacharjee et al. 2005; Zocco & Schekochihin 2011; Connor et al. 2012b).

The physical argument which is at the basis of the identification $\delta \rightarrow \delta_1$, and which has been more or less explicitly stated in different works on tearing mode analysis, grounds on the idea that it is the innermost subdomain which contains the essential non-ideal physics allowing the reconnection process. For example, for $\rho_s > \bar{d}_e$, the non-ideal region extending up to $x/\rho_s \sim 1$ is not of interest in this sense, coherently with the fact that $\rho_s$ does not allow, *per se*, the onset of tearing instabilities: it is only in the innermost layer of width $\delta_1$ that both electron inertia ($d_e$) or resistivity ($S^{-1}$) and ion-sound FLR effects ($\rho_s$) are important. Because of this, $\delta_1$ is the natural candidate to be identified as the characteristic microscopic region where magnetic lines dragged there by the flow can intersect, by thus violating the frozen-in condition. And since this cannot occur as long as magnetic lines are frozen in the electron flow (cf. comment on (2.7)), this layer is arguably dominated by electron physics, whence the appropriateness of naming it the ‘electron layer.’

Alternative, yet similar, expressions have been frequently used in the literature to characterise the layer where the electron frozen-in condition is violated. In this regard, it is worth mentioning the ‘dissipation region (or layer)’, named this way with reference to the dissipation of magnetic flux in resistive reconnection (see, e.g. among many others, Parker 1973; Mandt, Denton & Drake 1994; Shay et al. 1998), and the probably even more common expression, ‘(electron) diffusion region’. Introduced with initial reference to the resistive magnetic diffusion in Sweet-Parker-like steady reconnection processes – see, e.g. Sonnerup (1973) and Vasyliunas (1975) – the latter expression is currently used to identify the reconnecting layer in any reconnection processes, and thus also in all regimes of tearing-type modes (see, e.g. Drake & Kleva 1991; Hesse, Forbes & Birn 1999; Le et al. 2013) and several other works, especially connected to astrophysical plasma research). All these notions fit with the idea of identifying the extension of the reconnecting layer with that of the reconnecting current sheet. In practically all works on magnetic reconnection explicitly touching on the subject, the scale length $\delta$ is more or less explicitly assimilated also to the characteristic width of the ‘(reconnecting) current layer (or sheet)’.

However, in spite of the vast scientific literature, spanning almost seven decades, which addresses the subject of characterising the reconnecting layer in different reconnection scenarios and regimes, no precise and generally acquired operational definition seems to exist for it. Identifying the reconnecting region and measuring the spatial profile of physical quantities inside of it is a crucial element for the modelling of reconnection processes observed in Nature or in experiments (cf. e.g. Bratenahl & Yeates 1970; Yamada et al. 1997; Vaivads et al. 2004; Yamada et al. 2010). Even identifying its position and extension from measurements may be a non-trivial task, and indeed specific proxies are sought for this purpose in different reconnection regimes (e.g. the quadrupolar pattern of the magnetic field for Hall-reconnection, etc.). Also, when reconnection is known or is expected to occur because of tearing-type modes, for which quite accurate analytical estimates are available, quantitative information on $\delta$ obtained from measurements can give insight on important features of the reconnection process (we will dedicate §§9 and 10 of this work to this point and to the heuristic interpretation of the boundary layer calculation). Moreover, once compared with theoretical estimates, this quantitative information can give indication on the dominant non-ideal effects at play. We recall indeed that the scale $\delta_1$ obtained from boundary layer calculations is not ‘trivial’ (cf. §§9 and 10): for example, in the large-$\Delta'$ limit of the $\rho_s > \bar{d}_e$ regime, we have seen the scale $\delta_1$ to be *asymptotically smaller than both* $\bar{d}_e$ and $\rho_s$. At the same time, despite this information being available since the first boundary layer calculations performed in the Fourier space in the warm electron regime, its interpretation in physical terms has not
been univocal: while practically all early works agreed on recognising the non-ideal ion
region to have extension of the order of $\rho_s$, the width of the current layer has been instead
differently identified, on the basis of slightly different physical arguments, often depending
on further hypotheses or heuristic estimates. For example, the current width in the linear,
large-$\Delta'$, warm-collisionless regime has been also estimated to be larger than $d_e$ (see, e.g.
Drake & Kleva 1991; Ottaviani & Porcelli 1995), on the basis of the heuristic arguments
developed by Drake & Lee (1977), Cowley et al. (1986) and then re-discussed by Zocco
& Schekochihin (2011), or of the order of $\sim d_e$ (see, e.g. Grasso et al. 1999), based on
the simple argument that $d_e$ is the characteristic scale related to the reconnection process
in collisionless regimes (see, e.g. Vasyliunas 1975). The distinction between current layer
width and reconnecting layer width, which was done by Zocco & Schekochihin (2011), is
a subtle point to which we will return later, at the end of § 10.5.

From all these argument follows the interest in seeking a priori estimates of $\delta$, which
can be of general acceptance and do not ground on boundary layer calculations but can be
directly implemented by starting from experimental or numerical data.

In the context of nonlinear numerical simulations of tearing mode reconnection,
different examples of the estimate of $\delta$ have been proposed in the literature. Sometimes,
$\delta$ has been identified in terms of the global profile of the eigenmode superposed on the
equilibrium function. For example, Ali, Li & Kishimoto (2014) have measured the width
of the current sheet as the distance between the local minima of $J_{z,1}(x, y_0) + J_{z,0}(x)$ with
respect to the coordinate $x$, and for $y_0$ that corresponds to the ordinate of the X-point. The
current layer width has also been identified (Tenerani et al. 2015) by evaluating the distance
between the local minima of what here we would name $B_{x,1}(x, y_0) \sim \psi_1(x) \cos(k(y - y_0))$.
It can be noted that the estimate of Tenerani et al. (2015) relates to the numerical evaluation
we later give of the inverse scale length $D'$ (cf. (10.10) and figure 14) rather than of $\delta$. In
the context of the aforementioned work, this is however coherent since in the simulations
of Tenerani et al. (2015), the instability of a large aspect ratio current sheet is studied,
where a fastest growing mode exists (as evidenced already by Furth et al. 1963) for which
$\Delta' \sim 1/\delta$ (cf. Loureiro et al. 2007; Bhattacharjee et al. 2009; Del Sarto et al. 2016; Betar
et al. 2020). In the nonlinear simulations of Papini et al. (2019b), instead, the reference
value for $\delta$ has been taken by evaluating the full width (with respect to, let us say, the $x$
coordinate) at half-maximum of the total current $J_z$ minus its average background value,
that is, $\delta$ has been evaluated as the width at half-height of a local estimation of $J_{z,1}(x)$.

## 8.2. *Operational definition of the reconnecting layer width, $\delta$, and its scalings*

Here, we propose a quantitative, operational definition for the measurement of $\delta$, which
differs with respect to those previously suggested in the way we estimate the current layer
width, at least for tearing-type reconnection. We start indeed by noting that, in agreement
with the well-known identification ‘reconnecting layer $\leftrightarrow$ current sheet’, this layer can be
identified as the region around the neutral line in which the current density related to the
perturbation is concentrated. In our notation, this current density is $J_{z,1} = -\nabla^2\psi_1$. Inside
of this region, a velocity field is also concentrated, corresponding to an outflow parallel
to the neutral line outwardly directed from the X-point, which is a hyperbolic point of the
flow (see sketch in figure 11). In the notation used here, such a velocity field is $v_{y,1} = -\phi'_1$.

The profiles of $J_{z,1}(x)$ and of $v_{y,1}(x)$ close to the neutral line (cf. figure 12) are
qualitatively analogous, so that both their respective characteristic ‘thicknesses’ could
be taken as candidates for $\delta$. Therefore, by referring to the eigenfunctions only, we here
consider the distance from the neutral line of the local maxima (or minima) of the gradient
of the current density ($J'_z$) and to the distance from the neutral line of the local maxima
(or minima) of the vorticity ($\phi''$). Since these two distances can in principle differ, we
<figure><img src="image_22.png" /><figcaption>FIGURE 11. Sketch of the hyperbolic pattern of the velocity field associated with linear perturbations around the X-point. The yellow stripe represents the reconnecting layer in the vicinity of the neutral line, where a current sheet $J_{z,1}$ of width ~ $\delta$ is concentrated (the width of the magnetic island, instead, may well trespass the width of this region, during its nonlinear evolution – cf. caption of figure 2). Note that the estimate $v_{x,1} \sim \gamma \delta$, with $v_{x,1} = -ik\varphi_1$, holds inside of this region, whereas the $v_{y,1} = \varphi'_1$ component exhibits a change of sign on the neutral line and therefore displays a strong gradient across the $\Delta x \simeq \delta$ interval.</figcaption></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /><figcaption>FIGURE 12. Examples of evaluation of the half-width of the current layer as the distance of the local maxima (minima) of $\psi_1''' = -J_{z,1}'$ (a) and of $\varphi_1''$ (b) from the neutral line (the parameters of the numerical calculation are indicated at the top of the panels).</figcaption></figure>

respectively name them $\delta_{\psi}$ and $\delta_{\varphi}$:

$$ \delta_{\psi} : \quad J_{z,1}''|_{x=\delta_{\psi}} = \psi_1^{iv}|_{x=\delta_{\psi}} = 0, \quad \delta_{\varphi} : \quad v_{y,1}''|_{x=\delta_{\varphi}} = \varphi_1'''|_{x=\delta_{\varphi}} = 0. \qquad (8.1) $$

As shown in figure 12, $\delta_{\psi}$ and $\delta_{\varphi}$ can be easily calculated once the profile of the corresponding eigenfunction has been computed.

A numerical scan performed in the whole parameter range of the regimes considered by Betar et al. (2020) indicates that $\delta_{\psi}$ and $\delta_{\varphi}$ approximatively display the same asymptotic scaling, as shown in figure 13, as an example, for the warm-collisionless reconnection regime. We can so write $\delta_{\psi} \sim \delta_{\varphi}$, even if a proportionality factor not much different from unity is present (for example, we measured $\delta_{\varphi} \simeq 2\delta_{\psi}$ in warm-collisionless RMHD with $\rho_s/d_e \simeq 10$), which seems to display a weak dependence on the non-ideal parameters involved, at least in the parameter range we have numerically investigated.
<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /><figcaption>FIGURE 13. Scalings of $\delta_{\varphi}$ (a,c) and $\delta_{\psi}$ (b,d) calculated using the definitions given by (8.1). The scalings, shown on the different panels, prove that these two quantities follow the same scaling laws.</figcaption></figure>

Based on the coherence with the numerical results that show the correspondence of the scalings of $\delta_1$ with those of $\delta_{\psi}$ in all the reconnection regimes considered (also when electron-electron viscosity is included – see Betar et al. 2020),

we identify $\delta$ as twice the distance from the neutral line of the inflection point of $J_z$. (8.2)

Accordingly, we propose to operationally define the layer width as

$$ \delta \equiv \delta_{\psi} \sim \delta_{1}. \qquad (8.3) $$

The symbols $\delta_{SD}$ and $\delta_{LD}$ can be therefore used to indicate $\delta \equiv \delta_{\psi} \sim \delta_1$ in the small- and large-$\Delta'$ regimes.

### 8.3. Further micro-scales related to the gradients of the eigenmodes on the neutral line

A further ensemble of characteristic spatial scales of the system is provided by the normalised derivatives of the eigenfunctions evaluated on the neutral line. These can be shown to be related to combinations of powers of the non-ideal parameters and of $\delta_1$.
We define

$$ \delta_{\psi}^{(N)} \equiv \left| \frac{\psi_{1}^{(N)}}{\psi_{1}} \right|_{x=0}^{-\left(\frac{1}{N}\right)}, \quad N = 2, 4, 6, \dots, \qquad (8.4) $$

where N is the order of the derivative with respect to the shear variable x.

These length scales can be related to the local expansion of the eigenfunctions in a neighbourhood of x = 0. Using the fact that both $\psi_1$ and $v_{y,1}(x) = \phi'_1(x)$ are even and are continuous, at least up to the third derivative with respect to x, we can write

$$ \psi_1(x) \approx c_0 \left\{ 1 \pm \frac{1}{2} \left( \frac{x}{\delta_{\psi}^{(2)}} \right)^2 \pm \frac{1}{4!} \left( \frac{x}{\delta_{\psi}^{(4)}} \right)^4 \pm \dots \right\}, \qquad (8.5) $$

$$ v_y(x) = \phi'_1(x) \approx q_0 \left\{ 1 \pm \frac{1}{2} \left( \frac{x}{\delta_{v_y}^{(2)}} \right)^2 \pm \frac{1}{4!} \left( \frac{x}{\delta_{v_y}^{(4)}} \right)^4 \pm \dots \right\}. \qquad (8.6) $$

Note that the coefficients of (8.6) are related to those of (8.5). Using indeed $d\phi_1(\zeta)/d\zeta = \zeta d\psi_1(\zeta)/d\zeta - \psi_1(\zeta) - \chi_\infty$ (cf. (5.11) and (5.12)) for $\zeta = x/\delta_1$, whence $dx/d\zeta = \delta_1$, and taking into account the normalisation of $\phi_1(\zeta)$, i.e. $\phi_1(\zeta) = A\phi_1(x)$ with $A = -i\gamma/(k\delta_1 J_0)$ in the cold-electron regime and $A = -i\rho_s/d_e$ in the warm-electron regime, one obtains

$$ \frac{d\phi_1}{dx} = \frac{x}{A\delta_1} \frac{d\psi_1}{dx} - \frac{\psi_1}{A\delta_1} - \frac{\chi_\infty}{A\delta_1}. \qquad (8.7) $$

Direct substitution of (8.5) and (8.6)) into (8.7) and comparison of equal powers of x yields

$$ q_0 \sim -\frac{c_0}{A\delta_1} - \frac{\chi_\infty}{A\delta_1}, \quad \delta_{v_y}^{(N)} \sim \left(1 + \frac{\chi_\infty}{c_0}\right)^{1/N} (N-1)^{1/N} \delta_{\psi}^{(N)}. \qquad (8.8) $$

Using then (5.14) and (5.17), we can write

$$ q_0 \sim \frac{O(\Delta'\delta_1)}{A\delta_1} \quad \text{for } \Delta'\delta_1 \ll 1, \quad q_0 \sim -\frac{c_0}{A\delta_1} \quad \text{for } \Delta'\delta_1 \gg 1. \qquad (8.9) $$

Notice that the first of (8.9) is of scarce utility here, since not accurate enough: the estimate (5.14), which identifies $\chi_\infty \sim -c_0$ by neglecting possible corrections roughly estimable as being of the order of $\sim O(\Delta'\delta_1)$, cannot be indeed used for the present calculations, as it would always yield $v_y(0) = 0$. This does not generally agree with the analytical results of the boundary layer calculations in the small-$\Delta'$ limit (cf. e.g. figure 14c obtained via numerical integration of the eigenvalue problem). Therefore, when needed, depending on the reconnecting regime and wavelength limit considered next, we will rely on a more accurate estimate obtained from the boundary layer solution found in the case of interest.

The scalings of some of the length scales $\delta_{\psi}^{(N)}$ are easily found to be related to those of $\delta_1$ (and hence of $\delta \sim \delta_{\psi}$). This can be proven analytically. A possibility, quite generally applicable, consists in combining ((2.1) and (2.2)) by using $\phi_0(0) = 0$ with the local expansion

$$ \psi_0(x) \approx C_0 + \frac{C_2}{2} x^2 + \frac{C_4}{4!} x^4 + \mathcal{O}(x^4), \qquad (8.10) $$

and in differentiating the eigenfunctions the number of times which is needed. Note the correspondence $C_2 = J_0$ in the notation used in previous sections. The sought scalings
can be obtained by balancing the dominant terms of the equations while taking the limit $x \to 0$, and by using the scalings of $\gamma$ and $\delta_1$ obtained from boundary layer calculations. An alternative procedure consists in the direct evaluation of these derivatives from the approximated eigenmode solutions close to the neutral line, once they are obtained by integration of the boundary layer problem.

Here, below (§ 8.3.1), we use the second procedure to compute $\delta_{\psi}^{(2)}$ and the former to obtain $\delta_{\psi}^{(4)}$.

### 8.3.1. Asymptotic scalings of $\delta_{\psi}^{(2)}$ and of $\delta_{\psi}^{(4)}$

The scalings that can be obtained for $\delta_{\psi}^{(2)}$ and $\delta_{\psi}^{(4)}$ in the different regimes are summarised in table 2.

The estimate of $\delta_{\psi}^{(2)}$ is of interest, since it provides a direct estimate of the peak amplitude of the current density on the neutral line during the linear stage of the tearing mode evolution. According to the values in table 2, $J_z|_{x=0} \simeq c_0/\bar{d}_e$. Also, the scale of $\delta_{\psi}^{(2)}$ is of potential interest, since it gives the characteristic curvature with respect to $x$ of the velocity profile $v_{\psi}(x)$ on the neutral line. While, according to (8.8) and (8.9), $\delta_{\psi}^{(2)} \sim \bar{d}_e$ in the large-$\Delta'$ limit, in the small-$\Delta'$ limit, its value is typically asymptotically smaller, since it depends on the estimate of $(1 + \chi_{\infty}/c_0) \ll 1$ (cf. e.g. (8.21) for the cold-electron limit). These estimates are likely to be relevant for the study of the stability of the Bickley jet (Bickley 1937) related to the $v_{\psi}(x)$ velocity component, which, especially in the cold collisionless, large-$\Delta'$ regimes, nonlinearly develops along the neutral line and for $d_e^2 \gg \rho_s^2$, leads to a turbulent regime via the onset of secondary Kelvin-Helmholtz instabilities (Del Sarto et al. 2003, 2006). A similar feature had been observed also in early nonlinear simulations of the collisionless internal-kink mode in cylindrical geometry (Biskamp & Sato 1997) and had motivated dedicated studies of the stability of the Bickley jet in the presence of a background magnetic field aligned to it (Biskamp, Schwarz & Zeiler, 1998). The destabilisation via Kelvin-Helmholtz of a Bickley jet developing during the nonlinear stage of the collisionless reconnection process has been confirmed also in nonlinear simulations of tearing modes in three-dimensional geometry (Grasso, Borgogno & Pegoraro 2007; Grasso et al. 2009, 2020).

The estimate of $\delta_{\psi}^{(4)}$ is also of interest, since it intervenes in the modelling of the nonlinear current sheet evolution, which in the purely collisionless regime has been shown to shrink exponentially in time (Ottaviani & Porcelli 1993, 1995).

Below, we separately evaluate $\delta_{\psi}^{(2)}$ and $\delta_{\psi}^{(4)}$ so to prove the scalings reported in the table.

### 8.3.2. Asymptotic scalings of $\delta_{\psi}^{(2)}$

The scaling of $\delta_{\psi}^{(2)}$ follows from (2.1), which, in the purely collisionless limit, both ‘cold’ and ‘warm’, implies the conservation of the electron canonical momentum on the neutral line because of (8.10): $\partial_t(\psi - d_e^2\nabla^2\psi)|_{x=0} = 0$. The result can be formally extended to the resistive regime by relying again on the generalised electron skin depth $\bar{d}_e$. More precisely, regardless of the value of $\rho_s$, we obtain from (2.1)

$$ \delta_{\psi}^{(2)} = \frac{\bar{d}_e}{\sqrt{1 + k^2 \bar{d}_e^2}} \approx \bar{d}_e, \qquad (8.11) $$
<table><caption>TABLE 2. Scaling of the spatial scales associated with the second- and fourth-order derivative of $\psi_1(x)$, evaluated on the neutral line $x = 0$ in different reconnection regimes and wavelength limits. In each regime, $\delta_{SD}$ and $\delta_{LD}$ are those of **table 1** and correspond to the different values of $\delta_1$ evaluated in §§ 6 and 7.</caption><thead><tr><th rowspan="3">Length scale</th><th colspan="3">Cold regimes</th><th colspan="3">Warm regimes</th></tr><tr><th colspan="2">collisionless</th><th rowspan="2">resistive<br>(Sγ)<sup>-1/2</sup></th><th colspan="2">collisionless</th><th rowspan="2">resistive<br>(Sγ)<sup>-1/2</sup></th></tr><tr><th>Δ'<sub>δ1</sub> ≪ 1</th><th>Δ'<sub>δ1</sub> ≫ 1</th><th>Δ'<sub>δ1</sub> ≪ 1</th><th>Δ'<sub>δ1</sub> ≫ 1</th></tr></thead><tbody><tr><td>δ<sub>ψ</sub><sup>(2)</sup></td><td>d<sub>e</sub></td><td></td><td></td><td>d<sub>e</sub></td><td></td><td></td></tr><tr><td>δ<sub>ψ</sub><sup>(4)</sup></td><td>√d<sub>e</sub>δ<sub>SD</sub></td><td>√d<sub>e</sub>δ<sub>LD</sub></td><td>δ<sub>SD</sub><sup>-1/2</sup><br>(Sγ)<sup>1/4</sup></td><td>√d<sub>e</sub>δ<sub>SD</sub></td><td>√d<sub>e</sub>δ<sub>LD</sub></td><td>δ<sub>SD</sub><sup>-1/2</sup><br>(Sγ)<sup>1/4</sup></td></tr></tbody></table>
since in both the large- and small-Δ' regimes, purely resistive or purely inertial, $k^2 \bar{d}_e^2 \ll 1$. The above expression can also be approximatively written as

$$ \left. \frac{\psi_1''}{\psi_1} \right|_{x=0} \approx \frac{1}{\bar{d}_e^2} \implies \delta_{\psi}^{(2)} \approx \bar{d}_e. \qquad (8.12) $$

The corresponding scalings in table 2 descend from (8.12), after the relevant parameter and wavelength limits are considered.

### 8.3.3. Asymptotic scalings of $\delta_{\psi}^{(4)}$

To evaluate $\delta_{\psi}^{(4)}$, we should differentiate the solutions $\psi_1$ obtained in each reconnection regime and wavelength limit. Save for the warm-electron, large-Δ' limit, where the last equality of (7.38) immediately gives $|(d^4\psi_1/d\zeta^4)/\psi_1|_{\zeta=0} \simeq 2\delta_1^2/\bar{d}_e^2 + O(\delta_1^4/\bar{d}_e^4)$, whence $\delta_{\psi}^{(4)}$ can be rapidly evaluated by using $\zeta = x/\delta_1$ and then $\delta_1 = dx/d\zeta$, in all other cases, some more algebra is required. It is therefore interesting to look if it is possible to address all the regimes and cases in a unified way. To this purpose, we can combine (8.10) and (8.5) and (8.6) with the limit $x \to 0$ of the second-order derivative of (2.1) and of the first-order derivative of (2.2) with respect to $x$. Having introduced once more $\bar{d}_e$ so as to treat the inertial and resistive cases all together, we obtain

$$ \gamma \left( (1 + k^2 \bar{d}_e^2) \psi_1'' - \bar{d}_e^2 \psi_1^{iv} \right) \Big|_{x=0} \approx i2C_2k \left( \rho_s^2 \varphi_1''' - \varphi_1' \right) \Big|_{x=0}, \qquad (8.13) $$

$$ \gamma \varphi_1'''|_{x=0} \approx ik(C_2\psi_1'' - C_4\psi_1)|_{x=0}. \qquad (8.14) $$

Combining them and using (8.6),

$$ \gamma^2 \left( (1 + k^2 \bar{d}_e^2) \psi_1'' - \bar{d}_e^2 \psi_1^{iv} \right) \Big|_{x=0} \approx -2C_2k^2 \rho_s^2 (C_2\psi_1'' - C_4\psi_1) \Big|_{x=0} - i2C_2k\gamma q_0. \qquad (8.15) $$

Finally, using (8.12),

$$ -\frac{\bar{d}_e^4}{1 + k^2 \bar{d}_e^2} \left. \frac{\psi_1^{iv}}{\psi_1} \right|_{x=0} \approx -1 - 2(C_2)^2 \frac{k^2}{1 + k^2 \bar{d}_e^2} \frac{\rho_s^2}{\gamma^2} + 2C_2 \frac{k^2}{1 + k^2 \bar{d}_e^2} \frac{\bar{d}_e^2}{\gamma^2} \left( \rho_s^2 C_4 - i \frac{\gamma q_0}{k c_0} \right). \qquad (8.16) $$

At this point, we specialise the result to the cold and to the electron limits.

#### Cold-electron limit

In the cold limit $\rho_s = 0$, using $A = -i\gamma/(k\delta_1 J_0)$, $q_0 \sim -c_0/(A\delta_1)$ or $\sim -2c_0/(A\delta_1)$ (cf. (8.9)), we can so distinguish two cases.

(a) One case is for $k^2 \bar{d}_e^2 \delta_1 / \gamma^2 \sim d_e \ll 1$, which corresponds to the large-Δ' limit, in which we obtain the scaling

$$ \left. \frac{\psi_1^{iv}}{\psi_1} \right|_{x=0} \approx \frac{1}{\bar{d}_e^4} \implies \delta_{\psi}^{(4)} \approx \bar{d}_e \sim \delta_1 \sim \sqrt{\bar{d}_e \delta_{LD}}. \qquad (8.17) $$

(b) The second case is for $k\bar{d}_e^2 q_0 / (c_0 \gamma) \gg 1$, which is true in the small-Δ' limit, for which $\gamma \bar{d}_e / k \sim \delta_1^2$ (cf. (6.19)), but for which a better estimate of $q_0$, as given by the first part of (8.9), must be found. To this purpose, we can directly use the solution found for $\Phi(z) = (\bar{d}_e k^3 J_0^3 / \gamma^3)^{1/2} \varphi_1(x(kJ_0)^{1/2} / (\gamma \bar{d}_e)^{1/2})$ from boundary
layer calculations performed in this regime (§ 6.2). We first take the derivative of (6.15):

$$ \frac{d\Phi}{dz} = -\frac{1}{2} \int_{0}^{1} (1-t^2)^{-1/4} e^{-(1/2)tz^2} dt + z^2 \int_{0}^{1} t(1-t^2)^{-1/4} e^{-(1/2)tz^2} dt. \quad (8.18) $$

Since $z \propto x$, we can evaluate $Q_0 \equiv \Phi'(z)|_{z=0}$ and then relate it to $q_0 = \varphi'_1(x)|_{x=0}$. From (8.18), we obtain

$$ Q_0 = -\frac{1}{2} \int_0^1 \frac{dt}{(1-t^2)^{1/4}} = -\int_0^{\pi/4} \sqrt{1-2\sin^2(u)} du = -\mathcal{E}\left(\frac{\pi}{4}\middle|2\right), \quad (8.19) $$

where in the last passage, we have made the changes of variables of integration $t = \sin\theta$ and $u = \theta/2$, and where $\mathcal{E}$ is the incomplete elliptic integral of second kind (see Gradshteyn & Ryzhik 2015, (8.2) in § 8.111). Using (6.14), we thus find

$$ q_0 = i \frac{c_0 \gamma}{k \bar{d}_e J_0} Q_0 = -i \frac{c_0 \gamma}{k \bar{d}_e J_0} \mathcal{E} \left( \frac{\pi}{4} \middle| 2 \right). \quad (8.20) $$

Finally, using (6.21), one finds

$$ q_0 = -i \frac{c_0 (\Delta' \bar{d}_e)^2}{I^2} \mathcal{E} \left( \frac{\pi}{4} \middle| 2 \right). \quad (8.21) $$

This leads to

$$ \left. \frac{\psi_1^{iv}}{\psi_1} \right|_{x=0} \approx -i2 \frac{C_2}{I^2} \mathcal{E} \left( \frac{\pi}{4} \middle| 2 \right) \frac{k\Delta'}{\gamma}, \quad (8.22) $$

that is,

$$ \left. \frac{\psi_1^{iv}}{\psi_1} \right|_{x=0} \approx \frac{1}{\Delta' \bar{d}_e^3} \quad \Rightarrow \quad \delta_{\psi}^{(4)} \approx \sqrt{\bar{d}_e \delta_{SD}}. \quad (8.23) $$

### Warm-electron limit

In the warm limit $\rho_s > \bar{d}_e$, we can assume instead the first term of (8.22) to be always dominant at the right-hand side and greater than unity, so as to write

$$ \left. \frac{\psi_1^{iv}}{\psi_1} \right|_{x=0} \approx 2 C_2^2 \frac{k^2 \rho_s^2}{\gamma^2 \bar{d}_e^4}. \quad (8.24) $$

Using finally the definition (7.11), we obtain, in both the small- and large-$\Delta'$ limits,

$$ \left. \frac{\psi_1^{iv}}{\psi_1} \right|_{x=0} \approx \frac{1}{\bar{d}_e^2 \delta_1^2} \quad \Rightarrow \quad \delta_{\psi}^{(4)} \approx \sqrt{\bar{d}_e \delta_1}. \quad (8.25) $$

Specialising $\bar{d}_e$ in the purely inertial or purely resistive regime and taking the relevant scalings in the large-$\Delta'$ allows us to complete table 2.
## 9. Heuristic derivation of the scaling laws of tearing modes

In previous sections, we obtained the growth rate scaling law by analytically solving the eigenvalue problem in the warm- and cold-collisionless regimes. To solve the equations, we expanded the equilibrium profile around the neutral line using a Taylor series. Therefore, despite the cumbersome analysis developed to find them, the eigenfunctions so obtained are merely approximations and not ‘exact’ solutions of the problem. The complexity of the boundary layer approach is also evident, which leads to differential equations of hypergeometric nature.

A complementary approach is possible, which, in some cases as we are going to see, allows the estimate of the scalings of both the growth rate and of the reconnecting layer width by providing some more physical insight about the analytical assumption made in the boundary layer formalism. This method is based on some heuristic orderings of the terms in the inner layer equations, and on balancing these terms together to obtain the scaling laws via dimensional analysis – see Drake & Lee (1977) and Cowley et al. (1986) for an application to kinetic tearing, Betar et al. (2020) for an application to the cold resistive and viscous-resistive regimes, and Drake & Kleva (1991) for an application to secondary instabilities to a primary tearing-type mode. Although it has not been detailed, this approach has been also used to get the collisionless scalings of tearing modes in Ottaviani & Porcelli (1995). An analogous approach is at the basis of the available theoretical estimates of the scalings of the reconnecting rate in the whistler-mediated reconnection scenario (Mandt et al. 1994) and of the reconnecting rate in Hall-MHD reconnection (Biskamp, Schwarz & Drake 1995, 1997). Heuristic ansatz on the scaling of the gradients of the tearing eigenfunction have been revealed to be useful also for quantitative estimates – which have been numerically verified *a posteriori* – about the time and spatial behaviour of the reconnecting current sheet during its nonlinear, collisionless evolution (Ottaviani & Porcelli 1993, 1995). More in general, heuristic estimates are of fundamental importance to allow insight on the physical interpretation of less evident analytical results (see, e.g. Drake & Lee 1977; Cowley et al. 1986 for tearing modes and Grasso et al. (1999) for the interpretation of the physical meaning of $\rho_s$ in reduced MHD reconnection).

While this heuristic method, with some variations, is frequently presented in textbooks as a shortcut procedure to find the scalings of the cold-electron, resistive tearing mode (see, e.g. Biskamp 2000, § 4.1.1; Schnack 2009, Lecture 34; Boyd & Sanderson 2003, § 5.3.1, to give some examples), its application cannot be clear when more than one boundary layer exists, as it is in the case of warm-electron tearing modes. Discussing this point is therefore of general interest: this is what we are going to do in this section, where we compare the heuristic approach to the boundary layer analysis presented in previous sections. In particular, we are going to show that further information is required to get consistent results from the heuristic analysis to get the correct scalings when $\rho_s \gtrsim d_e$. From preliminary analysis, this information appears not to be immediately available from *a priori* arguments. This suggests that the heuristic approach should be carefully handled, when electron temperature effects (and, more in general, FLR effects) are included, since it could lead to incorrect estimates, as we are going to show below.

### 9.1. *General hypotheses in the heuristic approach to the scaling estimate*

Let us first outline the general hypotheses, which allow one to recover the correct scaling laws by dimensional analysis in the textbook-like examples of the purely resistive and of the purely inertial tearing mode analysis.
We first re-write the eigenvalue equations for $x \ll 1$ (i.e. $x/a \ll 1$ in dimensional units), in the non-ideal region:

$$ \psi_1 - i \frac{kx}{\gamma} J_0 \varphi_1 = \bar{d}_e^2 \psi_1'' + \frac{\rho_s^2}{\gamma^2} k^2 x^2 J_0^2 \psi_1'', \quad (9.1) $$

$$ \varphi_1'' = -i \frac{kx}{\gamma} J_0 \psi_1'', \quad (9.2) $$

where (9.2) was used to express the second term on the right-hand side of (9.1), and resistivity has been included in the parameter $\bar{d}_e^2$.

The usual heuristic estimations, as they have been successfully used for both the purely collisionless and purely resistive regimes, are based on the following ideas.

(i) There is a characteristic scale, say $l_c$, for the gradient of $\psi_1$, which we are going to quantitatively define in the following. It allows one to estimate $\psi_1' \sim \psi_1/l_c$ at some point $x$ in the neighbourhood of the neutral line.

(ii) A single characteristic microscopic scale exists for both the first derivative of $\psi_1'$ and $\varphi_1$: this corresponds to the inner layer width, $\delta$, which we operationally define according to (8.3), (this can be proven via numerical integration of the equations – see later).

(iii) We can generally assume $\delta \le l_c$.

We then add a further assumption that can be a *posteriori* verified (also numerically), and somewhat generalises the examples for which the heuristic approach has been successfully applied in the past.

(iv) $l_c$ is the largest characteristic scale length in the matching layer with the ideal-MHD solution. That is, in a neighbourhood of the neutral line, we write the estimates

$$ \psi_1' \sim \frac{\psi_1}{l_c}, \quad \psi_1'' \sim \frac{\psi_1}{\delta l_c}. \quad (9.3) $$

The two characteristic scales that naturally appear when a distinction has been made between the large-$\Delta'$ and small-$\Delta'$ limits are $(\Delta')^{-1}$ and $\delta$. Accordingly, in the small-$\Delta'$ limit, $l_c = (\Delta')^{-1}$ with $\Delta'\delta \ll 1$, while in the large-$\Delta'$ limit, $l_c = \delta$ and $\Delta'\delta \gg 1$. This argument suggests the following scalings (Ottaviani & Porcelli 1995):

$$ \frac{\psi_1''}{\psi_1} \sim \frac{1}{\delta^2} \quad \text{for } (\Delta'\delta \gg 1), \quad \frac{\psi_1''}{\psi_1} \sim \frac{\Delta'}{\delta} \quad \text{for } (\Delta'\delta \ll 1). \quad (9.4) $$

Note that, differently from (8.12), where the ratio $\psi_1''/\psi_1$ is evaluated exactly on the neutral line ($x \equiv 0$), in the estimates of (9.4), it is evaluated in the neighbourhood of the line (i.e. $x \simeq 0$).

After approximating $x \sim \delta$ in the inner layer, (9.2) gives

$$ \varphi_1'' \sim \frac{k\delta}{\gamma} J_0 \psi_1'', \quad (9.5) $$

which is true in all tearing regimes. All further estimates rely on assumptions about the relative ordering between the terms of the equations. In what follows, we will discuss the derivation of the scaling laws in the different regimes by using this method: it will prove to be successful in the cold-collisionless limit, but we will see that it fails in the warm-collisionless regime.
## 9.2. Heuristic derivation of the scaling laws in the cold-electron regime ($\rho_s^2 \ll \bar{d}_e^2$)

In these regimes, we can take $\rho_s = 0$. Therefore, the second term on the right-hand side of (9.1) vanishes.

For reasons of ‘economy of thought’ and of convenience about the generalisability of the heuristic approach, which will be discussed next (cf. comments on (10.4) in § 10.1), we now follow a procedure, which, although practically equivalent to those that can be found in classical textbook examples, slightly differs from most of them, as far as some ansatz are concerned: in particular, we are not going to make use of the estimate $\varphi_1'' \sim \varphi_1/\delta_1^2$, otherwise typically used, and which can be verified to be valid in the cold-electron regimes.

We then start by balancing the remaining terms of (9.1) and (9.2). This leads us to

$$ \psi_1 \sim \bar{d}_e^2 \psi_1'', \quad \varphi_1 \sim \frac{\gamma}{k\delta J_0} \psi_1, \quad \varphi_1 \sim \frac{\gamma \bar{d}_e^2}{k\delta J_0} \psi_1''. \qquad (9.6) $$

Differentiating twice the second part of (9.6) and using (9.5), one gets

$$ \gamma \sim k\delta J_0, \qquad (9.7) $$

which is valid for both wavelength limits and in both the collisionless and resistive regimes. Substituting (9.4) in the first part of (9.6), one obtains

$$ \delta \sim \bar{d}_e \quad \text{for } (\Delta'\delta \gg 1), \quad \delta \sim \bar{d}_e^2 \Delta' \quad \text{for } (\Delta'\delta \ll 1). \qquad (9.8) $$

It can be noticed that, looking at the physical aspects, the first of the conditions in (9.6) is the result of the balance between the two terms directly involved in the process of energy conversion that is related to magnetic reconnection: the magnetic potential $\psi_1$ on the one side, and, on the other side, the electron kinetic energy $d_e^2 \psi_1''$ in the collisionless limit or the energy dissipated by Ohm’s law in the resistive limit, $S^{-1}\psi_1''$.

At this point, it is convenient to treat the purely inertial and the purely collisionless case separately.

### (a) Collisionless case.

Substituting $\bar{d}_e \rightarrow d_e$ and (9.8) into (9.7) yields

$$ \gamma \sim J_0 k d_e \quad \text{for } \Delta'\delta \gg 1, \quad \gamma \sim J_0 k (\Delta')^2 d_e^3 \quad \text{for } \Delta'\delta \ll 1. \qquad (9.9) $$

The corresponding scalings of the width of the reconnecting layer read

$$ \delta \sim d_e \quad \text{for } \Delta'\delta \gg 1, \quad \delta \sim \Delta' d_e^2 \quad \text{for } \Delta'\delta \ll 1. \qquad (9.10) $$

These are the scaling laws of Porcelli (1991) that we have analytically obtained in § 6.

### (b) Resistive case.

Substituting instead $\bar{d}_e \rightarrow S^{-1/2}/\gamma^{1/2}$ and proceeding as above yields the scalings of Furth et al. (1963) and Coppi et al. (1976) (see also Ottaviani & Porcelli 1995), which we have also already obtained in § 6:

$$ \gamma \sim J_0^{2/3} k^{2/3} S^{-1/3} \quad \text{for } \Delta'\delta \gg 1, \quad \gamma \sim J_0^{2/5} k^{2/5} (\Delta')^{4/5} S^{-3/5} \quad \text{for } \Delta'\delta \ll 1, \qquad (9.11) $$

$$ \delta \sim J_0^{-1/3} k^{-1/3} S^{-1/3} \quad \text{for } \Delta'\delta \gg 1, \quad \delta \sim J_0^{-2/5} k^{-2/5} (\Delta')^{1/5} S^{-2/5} \quad \text{for } \Delta'\delta \ll 1. \qquad (9.12) $$
9.3. Heuristic derivation of the scaling laws in the warm-electron regime ($\rho_s \gtrsim \bar{d}_e$)

If we now follow an analogous approach in the warm-collisionless regime, problems arise suggesting some (or all) of the hypotheses made at points (i)–(iii) and in (9.3)–(9.5) to not be correct. Let us see why.

In this regime, the second term on the right-hand side of (9.1) does not vanish. Therefore, it is expected that this term will be of the same order of the first right-hand side term at the boundary of the inner layer (i.e. at $|x| \simeq \delta$). Balancing these two terms yields

$$ \frac{\gamma}{kJ_0} \sim \frac{\rho_s}{\bar{d}_e} \delta. \qquad (9.13) $$

By balancing the first left-hand side terms of (9.1), one estimates $\psi_1 \sim \bar{d}_e^2 \psi_1''$. For the small-$\Delta'$ limit, one has $\psi_1''/\psi_1 \sim \Delta'/\delta \sim 1/\bar{d}_e^2$, meaning $\delta \sim \Delta'\bar{d}_e^2$. Therefore, for the $\Delta'\delta \ll 1$ limit, one obtains

$$ \gamma \sim J_0 k \Delta' \rho_s \bar{d}_e, \quad \delta \sim \Delta' \bar{d}_e^2. \qquad (9.14) $$

These scaling laws for $\gamma$ and $\delta$ are identical to those given by (7.65) which we analytically derived in § 7.5. That is, they allow us to recover the small-$\Delta'$ limit of the scalings of (Pegoraro & Schep 1986).

Following the same line of thought to find the scaling laws of the large-$\Delta'$ limit, one would expect, as discussed in § 9, that the largest scale $l_c$ equals $\delta$ since this time, $\delta \gg (\Delta')^{-1}$. Therefore, $\psi_1 \sim \bar{d}_e^2 \psi_1''$ gives $\delta \sim \bar{d}_e$, which differs from the scaling in (7.53). Proceeding with this argument, after substituting $\delta \sim \bar{d}_e$ in (9.13), the scaling $\gamma \sim k\rho_s$ is obtained. This also differs from the scaling law obtained analytically in (7.54) and which, instead, has been numerically verified (Betar et al. 2020). No numerical evidence in the range $\rho_s \gtrsim \bar{d}_e$ has been found of the scalings $\delta \sim \bar{d}_e$ and $\gamma \sim k\rho_s$. Also note that $\gamma \sim k\rho_s$ does not display any explicit dependence on $\bar{d}_e$, which contains the non-ideal parameters that allow magnetic reconnection (and which should make $\gamma \to 0$ as $\bar{d}_e \to 0$). We therefore conclude the scalings $\delta \sim \bar{d}_e$ and $\gamma \sim k\rho_s$ to be wrong.

This implies that the generalisability of the heuristic approach to the warm regimes is not evident and further information about the estimates of the relevant quantities is needed.

Even if this problem is not solved yet, and a closed set of equations for the heuristic estimates is not available when $\rho_s \gtrsim \bar{d}_e$, in the next subsection, we investigate this possibility by introducing a new characteristic scale-length of the system associated with the gradient of the velocity component parallel to the neutral line, and which we postulate to be related to the gradient of the magnetic flux function at the boundaries of the ‘outer region’: although so far we must rely on its numerical estimate, we are going to show that this allows us to use a heuristic-type approach to get the correct scaling laws of the growth rate and of the inner layer width in all reconnection regimes here considered. In this sense, introducing this scale length at least allows us to generalise the heuristic procedure to warm-electron regimes, although this generalisation remains so far incomplete, due to the lack of a procedure apt to a priori estimate this scale length. Moreover, the asymptotic scaling of this quantity can be shown to correspond to that of a characteristic scale length, which in both Porcelli (1991) and Zocco & Schekochihin (2011) has been obtained as a normalisation length in boundary layer integration procedure. In these works, it had been related to the width $> \delta_1$ of the domain sub-interval in which the solution of the innermost equation is valid in the large-$\Delta'$ limit.
## 10. An ansatz about the ‘generalisation’ of the heuristic estimates: the role of the velocity gradient in the non-ideal region

A critical ingredient of the previous analysis is the estimate of the characteristic length, $l_c$, related to the first derivative of the magnetic stream function in the non-ideal region. Let us focus on the $\rho_s \gtrsim \bar{d}_e$ regime, where heuristic estimates display problems. Using $\psi_1 \sim \bar{d}_e^2 \psi_1''$ and (9.13), we see that the scale length $l_c$ enters in the estimates of the growth rate and of the reconnecting layer width as

$$ \gamma \sim J_0 \frac{k \rho_s \bar{d}_e}{l_c}, \quad \delta \sim \frac{\bar{d}_e^2}{l_c}. \qquad (10.1) $$

In both the collisionless and resistive regimes, the correct estimates are recovered in the small-$\Delta'$ limit when $l_c \sim (\Delta')^{-1}$, whereas they are incorrect in the large-$\Delta'$ limit if we assume $l_c \sim \delta$. This suggests looking for another reasonable estimate for $l_c$, in this case.

The strategy we pursue here is therefore to look for a third ‘effective’ scale length for $l_c$, different from $\Delta'$ and $\delta$, that would allow us to recover the correct scaling from (10.1) in the large-$\Delta'$ limit, and to verify its relevance and appropriateness by means of numerical calculations.

### 10.1. Velocity gradient in the innermost non-ideal region

The likely candidate we propose for an alternative definition of $l_c$ is the inverse of the logarithmic jump in the component of the derivative of the fluid velocity parallel to the neutral line and evaluated at $x = \pm \delta$, which we name $\Delta'_{v_y}$, in (loose) analogy with the usual $\Delta'$ defined for the magnetic stream function $\psi_1$:

$$ \Delta'_{v_y} = \frac{v'_y(\delta) - v'_y(-\delta)}{v_y(\delta)} = \frac{\varphi''_1(\delta) - \varphi''_1(-\delta)}{\varphi'_1(\delta)}. \qquad (10.2) $$

It must be noted that in (10.2), we have used the ‘whole’ eigenfunction $\varphi_1$, differently from what happens in the definition of $\Delta'$, in which only the ‘outer’ eigenfunction $\psi_{\text{out}}$ is involved. This fact is important for the numerical computation of both $\Delta'_{v_y}$ and $\Delta'$, as it will be discussed in § 10.3. Definition (10.2) and the identification $\delta = \delta_1$ means that $1/\Delta'_{v_y}$ represents the characteristic scale length of the velocity gradient at the boundary of the innermost layer, i.e. the ‘electron diffusion region’.

Evaluating (9.2) at $x = \delta$ and $x = -\delta$, and using the definition (10.2), we obtain

$$ \gamma \Delta'_{v_y} \varphi'_1|_{x=\delta} = ik\delta J_0 (\psi''_1|_{x=\delta} + \psi''_1|_{x=-\delta}). \qquad (10.3) $$

Using the fact that $\psi''_1(\delta) = \psi''_1(-\delta)$ and assuming the validity of condition (9.3) at $x = \delta$, we find

$$ \left. \frac{\gamma \varphi'_1}{k \psi_1} \right|_{x=\delta} = \frac{2i J_0}{l_c \Delta'_{v_y}}. \qquad (10.4) $$

Equation (10.4) expresses a constraint on the product $l_c \Delta'_{v_y}$ which depends on the scaling of $\gamma$, and on the profiles of $v_y = \varphi'_1$ and of $\psi_1$. It should be noted that in the cold-electron regimes, the correct scalings can be obtained via heuristic approach using the hypothesis $\varphi''_1|_{x=\delta} \sim \varphi_1/\delta_1^2$ (see, e.g. Biskamp 2000, § 4.1.1). Should this assumption be always correct, the scaling $l_c \sim (\Delta_{v_y})^{-1} \sim \delta_1$ would be always obtained in the large-$\Delta'$ limit regardless of the reconnection regime, but this estimate does not allow us to recover
the correct scalings when $\rho_s \gtrsim \bar{d}_e$. However, although so far the asymptotic scaling of $\Delta'_{v_y}$ is not known, considering the results that we have found to be valid in the cold-electron regimes, we can expect that, at least for $\rho_s = 0$ and in the large-$\Delta'$ limit, the quantity $\Delta'_{v_y}$ be related to (actually, 'be proportional to the inverse of') $\delta_1$.

## 10.2. A heuristic generalisation of the definition of the scale $l_c$

Based on the remarks above, we heuristically postulate the definition:

$$l_c \sim \max\{(\Delta')^{-1}, (\Delta'_{v_y})^{-1}\}, \qquad (10.5)$$

which we will later show (§ 10.4) to be indeed consistent with all other definitions and hypotheses, and thus, arguably correct. It can be a posteriori verified that the results would have been equally consistent even if we had developed the arguments which follow using the alternative definition $l_c \sim \max\{(D')^{-1}, (\Delta'_{v_y})^{-1}\}$, based on the further inverse scale length $D'$, which we introduce below, in (10.10), and whose scaling we numerically compute later, in different regimes. Definition (10.5) has been chosen, here, since $\Delta'$ is always a priori known, whereas the evaluation of $D'$, too, requires a numerical integration of the boundary layer problem. According to heuristic definition (10.5), in the small-$\Delta'$ limit, we expect $l_c \sim (\Delta')^{-1} \gg (\Delta'_{v_y})^{-1}$, whereas in the large-$\Delta'$ limit, we expect $l_c \sim (\Delta'_{v_y})^{-1} \gg \Delta'$. From (10.5), the transition between the two limits is therefore ruled by the asymptotic scaling of the ratio $\gamma\varphi'_1/(k\psi_1)$. In particular, regardless of the reconnection regime (resistive or collisionless, warm or cold), we must have

$$\left. \frac{\gamma \varphi'_1}{k \psi_1} \right|_{x=\delta} \sim \frac{\Delta'}{\Delta_{v_y}} \ll 1 \quad (\text{for } \Delta' \delta \ll 1), \qquad \left. \frac{\gamma \varphi'_1}{k \psi_1} \right|_{x=\delta} \sim O(1) \quad (\text{for } \Delta' \delta \gg 1). \qquad (10.6)$$

The first of the conditions in (10.6) can be however a posteriori refined and re-formulated as a condition on the ratio $(\gamma\varphi''_1)/(k\psi_1)$, which reads

$$\left. \frac{\gamma \varphi''_1}{k \psi_1} \right|_{x=\delta} \sim O(1) \quad (\text{for } \Delta' \delta \ll 1). \qquad (10.7)$$

This condition follows from (9.2) combined with the second part of (9.3) and from the knowledge we have about the scalings of $\delta$ and $\gamma$ in terms of $l_c$: using the estimates $l_c\delta \sim d_e^2$ and $l_c\delta \sim S^{-1}/\gamma$ that we obtain by specialising $\bar{d}_e$ to the collisionless and resistive regimes, respectively, we obtain

$$\left. \frac{\gamma \varphi''_1}{k \psi_1} \right|_{x=\delta} \sim \frac{\delta}{d_e^2} \quad (\text{collisionless}), \qquad \left. \frac{\gamma \varphi''_1}{k \psi_1} \right|_{x=\delta} \sim \frac{\gamma \delta}{S^{-1}} \quad (\text{resistive}). \qquad (10.8)$$

Equation (10.9) is then verified in all reconnection regimes once we substitute the relevant known scalings we have already evaluated from boundary layer theory into (10.8). Also, using the definition of $\Delta'_{v_y}$ of (10.2), the two equations in (10.8) result to be compatible with the second part of (10.6). In conclusion, we can therefore write the constraints:

$$\left\{ \begin{array}{l} \left. \frac{\gamma \varphi''_1}{k \psi_1} \right|_{x=\delta} \sim O(1) \\ \left. \frac{\gamma \varphi'_1}{k \psi_1} \right|_{x=\delta} \sim \frac{\Delta'}{\Delta_{v_y}} \ll 1 \quad (\text{for } \Delta' \delta \ll 1), \qquad \left. \frac{\gamma \varphi'_1}{k \psi_1} \right|_{x=\delta} \sim O(1) \quad (\text{for } \Delta' \delta \gg 1). \end{array} \right. \qquad (10.9)$$
These conditions can be taken to be generally discriminating for the transition from the small- to the large-$\Delta'$ scaling relations in any reconnection regimes.

It must be however emphasised that, while both (10.6) and (10.8) are self-consistently *deduced* from the specific hypotheses we have made so far, (10.7) must be *heuristically assumed*, since, although *a priori* reasonable and compatible with the first of conditions (10.6), it does not follow from the other hypotheses, but it is just verified once the scalings of $\gamma$ and $\delta$ are found.

The discussion and the analysis we are going to develop next (§ 10.4) seems to suggest that accomplishing this latter task in the context of the heuristic derivation may not be feasible. Nevertheless, we will numerically prove the correctness of the hypotheses (i)–(iii) of § 9 and we will show that definitions (10.4) and (10.5) are consistent with the estimates of the correct scalings, provided the scaling of $\Delta'_{v_y}$ is known. In doing so, we will elucidate (cf. Appendix E) the logical points of the heuristic approach in the different tearing regimes by pointing out when an *a priori* self-consistent estimate can be done or not, and, in the second case, which information is missing. Before doing so, we need however to first discuss how to numerically evaluate $\Delta'_{v_y}$, which must be compared with the numerical value of $\Delta'$ and of $\delta$, the latter of which has been already discussed in § 8. This is what we are going to do in § 10.3.

### 10.3. Numerical evaluation of $\Delta'$ and $\Delta'_{v_y}$

Let us first look at a numerical procedure that allows us to quantify the inverse scale lengths $\Delta'$ and $\Delta'_{v_y}$ defined by (3.8) and (10.2), respectively.

Of course, because of the definition in (3.8), the values of $\Delta'$ are always independent of the non-ideal parameters. The numerical evaluation of $\Delta'(k)$ becomes therefore trivial whenever an analytical formula that depends only on $k$ and on the equilibrium profile can be obtained (cf. (4.13)).

Then, for the evaluation of $\Delta'_{v_y}$, the definition in (10.2) may be operationally used, although it is procedurally quite demanding: it requires to compute first the eigenfunction $\varphi_1$, then its first and second derivative, and then to evaluate them at $x = \delta$, a value which can be numerically obtained by using the definition in (8.3) and by following the procedure sketched in figure 12.

It is however possible to 'speed up' the calculation of the scalings of both $\Delta'$ and $\Delta_{v_y}$, by relying on an alternative numerical procedure.

This procedure mimics the numerical evaluation of $\Delta'$ that is made possible *only in the small-$\Delta'$ limit*, thanks to the geometrical interpretation that can be given of the instability parameter in terms of a local expansion of the outer solution (Furth et al. 1963): we recall that by using $\psi_{\text{out}} \approx c_0 + c_1|x|$ as $|x| \to 0$, then $\Delta' = 2c_1/c_0$ (cf. §§ 5.3.1 and 5.3.2). These two coefficients can be evaluated by measuring the value of $\psi_{\text{out}}$ and of the slope of the tangent to $\psi_{\text{out}}$ close to $x = 0$.

This idea can be borrowed so as to evaluate analogous quantities defined with respect to the *total* solutions $\psi_1$ and $\varphi_1$: noting that, graphically speaking, both $\psi_1$ and $\varphi'_1$ still display a linear behaviour with respect to $x$ as $x \to \delta$, both in the small- and large-$\Delta'$ limits, we write $\psi_1|_{x \to \delta} \approx c_0 + c_1|x|$ and $v_y|_{x \to \delta} = \varphi'_1|_{x \to \delta} \approx q_0 + q_1|x|$. Note that these approximations are consistent with the local expansions (8.5) and (8.6), which are valid, instead, for $x \ll \delta$. Therefore, once the corresponding eigenfunctions have been numerically computed, at $|x| \sim \delta$, we can evaluate
<figure><img src="image_29.png" /><figcaption>FIGURE 14. (a) Example of numerical evaluation of the coefficients $c_0$ and $c_1$ of (10.10), once the profile of $\psi_1(x)$ and of its derivatives $\psi'_1(x)$ have been computed for $|x| \ge \delta$ in a neighbourhood of $x = \delta$. (b) Analogous example of evaluation of $q_0$ and $q_1$ of (10.10) from $\varphi'_1$ and $\varphi''_1$, calculated once the eigenfunction $\varphi_1$ has been computed; in this example, $\Delta'_{v_y} \approx 73.82$. (c) A zoomed version of panel (b) in which the inner region, here of width $\delta \approx 0.00184$ (computed according to the method outlined in § 8), is shaded in light-blue colour. All the values required to estimate $\Delta'_{v_y}$ using (10.2) are here shown; it is this way that $\Delta'_{v_y} \approx 74.48$ is obtained, in very good agreement with the result computed with the alternative method sketched in panel (b).</figcaption></figure>

(see figure 14)

$$D' \equiv 2\frac{c_1}{c_0}, \quad \Delta'_{v_y} = 2\frac{q_1}{q_0} \qquad (10.10)$$

In particular, due to the smallness of $\delta$, the coefficients $c_1, c_0$ and $q_0, q_1$ can be numerically computed as shown in figure 14, by measuring the values of $\psi_1$ and $\varphi_1$ and of the peak values of their derivatives close to $x = 0$.

We have verified that the scalings obtained for $\Delta'_{v_y}$ in this way agree with those directly computed by first evaluating $\delta$ and by then using the definition in (10.2), as shown in figure 14(b,c), and that $D' = \Delta'$ in the small-$\Delta'$ limit.

Figure 15 displays the scaling laws of $(D')^{-1}$, $(\Delta'_{v_y})^{-1}$ and $\delta$, numerically computed according to the operational definitions given by (10.10) and (8.3), respectively, in the cold-collisionless regime at $\rho_s = 0$: the different characteristic lengths are shown in
<figure><img src="image_30.png" /><figcaption>FIGURE 15. Examples of scaling laws of $D'$, $\Delta'_{v_y}$ (cf. (10.10)) and $\delta$ with respect to $d_e$, for $\rho_s = 0$. The large-$\Delta'$ limit is in panel (a), the small-$\Delta'$ in panel (b). The values of $D'$ and $\Delta'_{v_y}$ are computed numerically according to the procedure sketched in figure 14. In the small-$\Delta'$ limit, the values of $\Delta'$, obtained from (4.13), coincide with the orange line in panel (b).</figcaption></figure>

figure 15(a) for the large-$\Delta'$ limit, and in figure 15(b) for the small-$\Delta'$ limit. Figure 16 shows the corresponding scaling laws for the tearing modes in the warm-collisionless regimes at $\rho_s \neq 0$. The dependence of $(D')^{-1}$, $(\Delta'_{v_y})^{-1}$, and $\delta$ on the electron skin depth ($d_e$) is in figure 16(a,c), the dependence on the ion-sound Larmor radius ($\rho_s$) is in figure 16(b,d); both the large-$\Delta'$ limit (figure 16(a,b)) and the small-$\Delta'$ limit (figure 16(c,d)) are considered.

Numerical results prove that, as it could be expected, $D'$ of (10.10) coincides with the definition of $\Delta'$ only in the small-$\Delta'$ limit, where we can indeed state that $\psi_1(x)|_{x \to \delta^+} \simeq \psi_{\text{out}}(x)|_{x \to \delta^+}$ (cf. figure 15b, and figure 16c,d). In this limit, $D'$ can be therefore taken as an accurate estimate of $\Delta'$, in spite of the fact that the latter is formally defined by evaluating the derivatives of $\psi_{\text{out}}$ at a distance from the neutral line much larger than $\delta$. This is made possible by the fact that in this large wavelength limit, the outer solution must match the inner one in an overlapping region that gets sufficiently close to $|x| = \delta$. That is, the constant-$\psi$ hypothesis holds in the whole non-ideal region, down to the innermost layer.

Different is the result in the large-$\Delta'$ limit: in the cold-collisionless regime, $(D')^{-1}$ displays the same asymptotic scaling of $\delta$, except for a numerical factor (cf. figure 15a), and these scaling laws are the same as for $(\Delta'_{v_y})^{-1}$; in the warm-collisionless regime, instead, both $(D')^{-1}$ and $(\Delta'_{v_y})^{-1}$ display the same asymptotic scalings $(D')^{-1} \sim (\Delta'_{v_y})^{-1} \sim \rho_s^{1/3} d_e^{2/3}$ (figure 16a,b) which are non-trivial, since they differ from $\delta_{\text{LD}}^{-1} \sim \rho_s^{1/3} d_e^{4/3}$. Analogous results, not shown here, are found in the warm-resistive, large-$\Delta'$ limit, in which, for these quantities, we obtain also an explicit dependence on $k$: $(D')^{-1} \sim (\Delta'_{v_y})^{-1} \sim k^{-2/7} \rho_s^{-1/7} S^{-2/7}$ and $\delta \sim k^{-4/7} \rho_s^{-5/7} S^{-4/7}$.

For summary, we recall here the scalings of $\Delta'_{v_y}$ that have been numerically obtained in the different regimes:

$\Delta'_{v_y} \sim \delta_{SD}^{-1}$ (warm/cold and collisionless/resistive regimes at small-$\Delta'$);

$\Delta'_{v_y} \sim \delta_{LD}^{-1}$ (cold and collisionless regimes at large-$\Delta'$);
<figure><img src="image_31.png" /><figcaption>FIGURE 16. Examples of scaling laws of $D'$, $\Delta'_{vy}$ (cf. (10.10)) and $\delta$ for $\rho_s \neq 0$. Panels (a,b) correspond to the large-$\Delta'$ limit, whereas panels (c,d) correspond to the small-$\Delta'$ limit. The values of $D'$ and $\Delta'_{vy}$ are computed numerically according to the procedure sketched in figure 14. Notice that $D'$ and $\delta$ are independent of $\rho_s$ in the small-$\Delta'$ limit (cf. panel d). In the small-$\Delta'$ limit, the values of $\Delta'$ obtained from (4.13) coincide with the blue lines in panels(c,d).</figcaption></figure>

$\Delta'_{vy} \sim \rho_s^{-1/3} d_e^{-2/3}$ (warm-collisionless regime at large-$\Delta'$);

$\Delta'_{vy} \sim k^{-2/7} \rho_s^{1/7} S^{-2/7}$ (warm-resistive regime at large-$\Delta'$).

## 10.4. Role of $\Delta'_{vy}$ in heuristic-type estimates

It is easy to verify that combining the estimates in (10.1) with the definition in (10.5) and with the scalings numerically obtained for $\Delta'_{vy}$, the correct scaling laws obtained in §§6 and 7 can be recovered in both the warm-collisionless and warm-resistive regimes.

This is discussed in detail in Appendix F, in which the logical steps of the procedure are singled out and identified, thus providing insight on the physical interpretation of the analytical results of the boundary layer calculations. In particular, the main hypotheses on which this (partial) heuristic approach relies and the corresponding results in all collisionless regimes, both ‘warm’ and ‘cold’, are summarised in tables 3–5 of Appendix F, where the logical steps of the procedure are presented as statements and formulae.
There are two main results of this analysis.

(i) Introducing the scale length $\Delta'_{v_y}$ and postulating the definition in (10.5) make it possible to obtain the scaling laws, which are, in principle, correct in all regimes and wavelength limits; however, in the large-$\Delta'$ warm-electron regime, the procedure results to be not 'closed', in the sense that the estimation of the scaling law of $\Delta'_{v_y}$ seems to be not possible by simple dimensional analysis.

(ii) Quite interestingly, using the definition in (10.5), the scaling laws of all reconnection regimes can be cast in a form which is perfectly symmetric in the small- and large-$\Delta'$ limit, with respect to the substitution $\Delta' \leftrightarrow \Delta'_{v_y}$.

These results, on the one hand, suggest to us that introducing the inverse scale length $\Delta'_{v_y}$ is a promising ingredient in the attempt to extend the heuristic-type analysis. On the other hand, however, finding the further constraint that makes it possible to obtain a priori all the sought algebraic scalings without resorting to numerical analysis and by mere dimensional analysis seems an elusive task. The logical steps identified in the tables of Appendix F, as well as some further insight on the interpretation of the boundary layer results, which follows from the heuristic approach and which we are going to discuss below (§ 10.5), could imply the heuristic analysis to be intrinsically not applicable in some regimes. This problem becomes manifest, in particular, in the large-$\Delta'$ limit when $\rho_s \neq 0$, in which the scaling of $\Delta'_{v_y}$ is a non-trivial power law combination of the scales $\rho_s$ and $d_e$. Nevertheless, the coherence of the results summarised in tables 3–5 supports the consistency of the heuristic ‘ansätze’ we have made so far.

In particular, the logical steps identified in tables 3–5 show that the difference between the warm and cold regimes lies in the balance condition $\psi_1 \sim ikx\phi_1/\gamma\phi_1$, which is valid in the cold regimes only (hypothesis [T6.H2] of table 5), and which is replaced by the balance expressed by hypothesis [T5.H2] of table 4 when $\rho_s \neq 0$. Condition [T6.H2] expresses indeed the validity of (4.1), in turn related to the inverse scale $\Delta'$, down to the boundary layer at the frontier with the innermost region: this means that the newly introduced scale $(\Delta'_{v_y})^{-1}$ is redundant in this regime with respect to $\delta$ and $\Delta'^{-1}$ (as, however, it was evident already from the heuristic-type approach discussed in § 9.3). We have indeed seen from the numerical results summarised at the end of § 10.3 that the scaling of $(\Delta'_{v_y})^{-1}$ turns out to always coincide with that of $\delta$. Such a 'closure' condition for $\Delta_{v_y}$ is lost at $\rho_s \neq 0$, when hypothesis [T6.H2] is replaced by hypothesis [T5.H2], which leads to the general constraint $\gamma \sim k\rho_s(\delta/l_c)^{1/2}$: here, a value different from that of $\delta$ or $(\Delta')^{-1}$, with respect to which the transition from the small- to the large-$\Delta'$ limits is measured, is in principle admitted for $l_c$.

### 10.5. Significance of the inverse spatial scale $\Delta'_{v_y}$: coherence with boundary layer calculations and comparison with previous work

Some insight about the physical significance of the characteristic spatial scales associated with $\Delta'_{v_y}$ is obtained from comparison of their scalings with identical asymptotic scalings which can be obtained from some boundary layer results available in the previous literature (Pegoraro & Schep 1986; Cowley & Hastie 1988; Porcelli 1991; Zocco & Schekochihin 2011) and from related works discussing the implications of these results (Ottaviani & Porcelli 1995; Grasso et al. 1999).

Combination of (10.5) (i.e. hypothesis T4.H1) with hypotheses T5.H3 or T6.H3 allows the identification of the small- and large-$\Delta'$ limits, say $\Delta'\delta \ge 1$, expressed in each
reconnection regime in terms of $l_c$ and thus in terms of $\Delta'_{v_y}$:

$$ \Delta' \delta \ge 1 \Leftrightarrow \left\{ \begin{array}{l} \Delta' d_e \ge \max \left\{ \frac{1}{\Delta' d_e}, \frac{1}{\Delta'_{v_y} d_e} \right\} \quad \text{(collisionless)}, \\ \\ \Delta' S^{-1} \ge \max \left\{ \frac{\gamma}{\Delta'}, \frac{\gamma}{\Delta'_{v_y}} \right\} \quad \text{(resistive)}. \end{array} \right. \qquad (10.11) $$

In this regard, it is interesting to compare, e.g. the conditions for the warm-collisionless case with the similar conditions in Grasso et al. (1999) that have been written (cf. (17) and paragraphs below (18) therein) as

$$ \Delta' d_e > \min \left\{ 1, \left( \frac{d_e}{\rho_s} \right)^{1/3} \right\}, \quad \Delta' d_e < 1, \qquad (10.12) $$

based on the results of the boundary layer calculations in the Fourier representation by Porcelli (1991) (cf. conditions on $\lambda_H \equiv -\pi/\Delta'$ in between (7) and (9) therein). Substitution of the scalings numerically found for $\Delta'_{v_y}$ in § 10.3 for $\rho_s^2 \gg d_e^2$ into the collisionless condition of (10.11) gives

$$ \Delta' d_e > \max \left\{ \begin{array}{l} \frac{1}{\Delta' d_e} \left( \frac{\rho_s}{d_e} \right)^{1/3} \\ \text{(large-$\Delta'$)} \end{array} \right\}, \quad \Delta' d_e < \max \left\{ \frac{1}{\Delta' d_e}, \Delta' d_e \right\}, \qquad (10.13) $$

which are indeed compatible with the conditions in (10.12), once $(\Delta' d_e)^2 < 1$ is deduced from the second part of (10.13) in the small-$\Delta'$ limit and therefore $\Delta' d_e < 1$ is assumed, with $\Delta' d_e \sim 1$ fixing the threshold value also for the first inequality of (10.13).

In this regard, we notice that the non-trivial characteristic scale length associated with the asymptotic scaling, which we have numerically found for $\Delta'_{v_y}$ in the large-$\Delta'$, warm-electron limit, naturally emerges from the boundary layer analysis developed by Pegoraro & Schep (1986); Porcelli (1991): here, the general dispersion relation encompassing both the small- and large-$\Delta'$ limits can be written in terms of $\bar{d}_e$ as (to this purpose, we can just substitute $d_e \to \bar{d}_e$, e.g. in (25) of Ottaviani & Porcelli 1995)

$$ \frac{\pi}{2} \gamma^2 = -\frac{\pi \rho_s}{\Delta'} + \frac{\rho_s^2 \bar{d}_e}{\gamma}. \qquad (10.14) $$

Naming $\gamma_{LD} \sim (2/\pi)^{1/3} \bar{d}_e^{1/3} \rho_s^{2/3}$ the solution obtained in the $\Delta' \to \infty$ limit, which we already recovered in previous sections, one sees that the opposite, small-$\Delta'$ limit is obtained when the condition

$$ \frac{\pi \rho_s}{\Delta'} \gg \frac{\rho_s^2 \bar{d}_e}{\gamma_{LD}} \Leftrightarrow \Delta' \ll \frac{1}{2 \rho_s^{1/3} \bar{d}_e^{2/3}} \Rightarrow \Delta' \ll \Delta'_{v_y} \qquad (10.15) $$

is satisfied, where in the last passage, we have used the numerical result we previously found in this wavelength limit, $\Delta'_{v_y} \sim \rho_s^{-1/3} \bar{d}_e^{-2/3}$. The rightmost condition of (10.15) is consistent indeed with the constraints in (9.4), previously found via the heuristic estimates discussed in § 10.2. This suggests that $\Delta'_{v_y}$ may provide a physical interpretation of the
appearance of this characteristic scale length in boundary layer calculations, performed in the framework of the two-fluid model we consider here.

This interpretation is however somewhat different from that provided by Ottaviani & Porcelli (1995), and which was based on the previous boundary layer calculations (Cowley et al. 1986; Pegoraro & Schep 1986; Porcelli 1991) performed by starting from a charge density equation in which polarisation effects were taken into account through the gyrokinetic particle response. In that modelling framework, the scale-length $\rho_s^{1/3} \bar{d}_e^{2/3} \gg \delta_1$ was noted by Ottaviani & Porcelli (1995) to correspond to the distance from the neutral line at which $\gamma_{LD}$ becomes comparable to the phase-velocity $k_\| (x) v_{\text{th}}^e = (k \cdot B_0(x)/B_0) v_{\text{th}}^e$, and below which the isothermal electron closure would formally break down. As already noted in the same work, however, the appropriateness of the isothermal condition, assumed to be valid for the purpose of the boundary layer calculations, had been numerically verified to a good extent in some previous works (Berk, Mahajan & Zhang 1991; Coppi & Detragiache 1992) and, more recently, its validity has been supported by the numerical studies of Perona, Eriksson & Grasso (2010). Relating the scale $\rho_s^{1/3} \bar{d}_e^{2/3}$ to the failure of the isothermal closure is thus consistent with the interpretation provided in further – and in part preceding– works based on a kinetic approach (Drake & Lee 1977; Cowley et al. 1986; Zocco & Schekochihin 2011). In these works, such a critical distance was shown by heuristic arguments (see Drake & Lee 1977; Cowley et al. 1986, § V) to correspond to the characteristic width of the current layer, determined by the balancing of the total current generated by the parallel electron pressure gradients with the current generated by the (reconnecting) parallel electric field. Instead, in § 8 and in Betar et al. (2020), we have numerically proven the current sheet to be concentrated around the neutral line in a region of width $\delta_1$. The subtle point here may be in the meaning which can be given to the notion of ‘characteristic width of the current layer’, since it is true that, while the inflection points around the peak of $J_{z,1}$ are numerically found to be located at a distance $\delta_1$ from the neutral line (cf. definition in (8.3)), the profile of the inner solution (i.e. the solution found in the ‘electron region’) extends beyond this distance. In particular, the matching with the outermost non-ideal solution (i.e. the solution in the ‘ion region’) is not to be meant as a matching in a single point, but rather as an asymptotic matching valid over an intermediate layer, whose distance from the neutral line in the warm-electron regime can be of the order of $\rho_s^{1/3} \bar{d}_e^{2/3}$. This latter point of view is in agreement with the notion of ‘inner solution width’, to which Zocco & Schekochihin (2011) make reference: the characteristic width of the inner solution, which is named $\delta_{\text{in}}$ in the notation of their work, differs from the width of the inner layer, which we have here named $\delta_1 = \delta$, and satisfies $\delta_1 < \delta_{\text{in}} < \delta_2$.

If we look in detail at the boundary layer analysis carried out by these latter authors in the warm-electron regime, we see that the scalings they a posteriori obtained (cf. (B56) and (B101) in the ‘Collisionless two fluid limit’ and ‘Resistive two fluid limit’ of that work) for the normalisation scale $\delta_{\text{in}}$, coincide in the large-$\Delta'$ limit with the scalings of $(\Delta'_{v_y})^{-1}$ that we have detailed at the end of § 10.3. In particular, for the regime that we can write as $\rho_s \gtrsim \bar{d}_e$, these authors identified the small- and large-$\Delta'$ conditions, which we have here generally expressed as $\Delta'\delta_1 \ll 1$ and $\Delta'\delta_1 \gg 1$, respectively, via the conditions $\Delta'\delta_{\text{in}} \ll 1$ and $\Delta'\delta_{\text{in}} \gg 1$, instead. Using the correspondence $\delta_{\text{in}} \rightarrow (\Delta'_{v_y})^{-1}$, these conditions would map into $\Delta' \ll \Delta'_{v_y}$ and $\Delta' \gg \Delta'_{v_y}$, the former of which is consistent with (9.4). It should be however noted that the scale $\delta_{\text{in}}$ appears in the boundary layer analysis of Zocco & Schekochihin (2011) as a consequence of a normalisation choice of the non-ideal equations of their model ((B35)–(B36) therein), which differs with respect to the one we have detailed in § 7: while the scalings of their outermost and innermost non-ideal layers are a posteriori found to coincide with those of the scales $\delta_2$ and $\delta_1$ that we introduced in §§ 7.2.1
and 7.2.2, these authors chose instead $\delta_{\text{in}} \equiv (\sqrt{2}\rho_s\delta_1)^{1/2}$ as a normalisation scale of the innermost equations (our $\delta_1$ mapping into $\delta$ of (B28) of their work), and they subsequently ordered the terms of the tearing equations with respect to this spatial scale. The scalings of the relevant quantities have been thus obtained there, via some heuristic ansatz on the width of the integral at the left-hand side of the equivalent of (3.7), in the form in which they obtained it (cf. (B46), therein). To evaluate the integral, which depends on $\varphi_1''$ (cf. (3.10)), the terms in the corresponding auxiliary equation ((B42) therein), which are related to integrand via $\varphi_1'' = \chi'$, have been so ordered in relation to the characteristic width $\delta_{\text{in}}$. This approach is at the basis of the ordering of the wavelength regime in terms of the product $\Delta'\delta_{\text{in}}$, which differs from the ordering in terms of the product $\Delta'\delta_1$, which we have here adopted, instead. The coherence of the results obtained in the two approaches, also emphasises the margin of arbitrariness in the choice of the normalisation scale, with respect to which it is possible to define the width of the boundary layers and to perform the integration and matching, after some appropriate approximations of the terms in the equations are assumed on heuristic basis. This makes possible the interpretation of $\delta_{\text{in}}$ as the width of the solution in the innermost equation, information which is not evident if one follows instead the normalisation procedure we have adopted in this work. At the same time, it should be noticed that the scale $(\Delta'_{v_y})^{-1}$ seems to be not generally identifiable as the $\delta_{\text{in}}$ obtained by Zocco & Schekochihin (2011): although the correspondence between the scalings of $\delta_{\text{in}}$ obtained by these authors and the scaling of $(\Delta'_{v_y})^{-1}$ that we have obtained holds in the large-$\Delta'$ limit, it fails in the small-$\Delta'$ limits (cf. their (B54) and (B97) with the scalings at the end of § 10.3).

In summary, while on the one hand, we note the agreement of the results obtained by solving the boundary layer equations in the different models and with integration techniques that rely on slightly different heuristic hypotheses, on the other hand, we note that the appearance of the characteristic scale, which, in previous works, has been interpreted in terms of inherently kinetic features, in the MHD model of § 2, can be instead entirely related to ‘fluid-like’ features associated with the gradients of the velocity field in the non-ideal region. It should be also noticed, in this regard, the care with which conclusions drawn by heuristic estimates based on dimensional analysis must be dealt with: although it is comforting that different models of tearing mode analysis yield the same quantitative results, their physical interpretation is a more delicate issue, which requires further insight based on the consistency of the specific hypothesis of each model and therefore may not be univocal. In particular, the symmetry between the scaling laws in the small- and large-$\Delta'$ limits with respect to the substitution $\Delta' \leftrightarrow \Delta'_{v_y}$, which we have detailed in tables 3–5, suggests that the quantity $\Delta'_{v_y}$, in this modelling related to inherently ‘fluid’ features, play a general, important role in the tearing mode regimes, which may deserve further investigations.

## 11. Summary and conclusions

We have reviewed the solution of the boundary layer problem for collisionless and resistive tearing instabilities in slab geometry, in both the small- and large-$\Delta'$ limits. The calculations in the warm regime, in which two matching regions are required, have been solved in the coordinate space by using the integral representation of hypergeometric functions to integrate the differential equations of the boundary layer approach (§§ 6 and 7). To the best of our knowledge, this kind of analysis has not been presented before, elsewhere, and in the present work, emphasis is put on a pedagogical derivation of the results. In this way, we have recovered the results first obtained by Pegoraro & Schep (1986), Pegoraro et al. (1989) and Porcelli (1991) in a Fourier representation.
While developing this analysis, we have also been able to make a direct comparison (§ 6) with calculations in the coordinate space that had been earlier performed in the cold-collisionless regime and in the purely resistive regime of tearing modes, where a single matching region is required (Furth et al. 1963; Coppi 1964c; Ara et al. 1978), and with other calculations in the coordinate space that had been carried out in the warm-collisionless regime in the presence of warm ions (Zocco & Schekochihin 2011).

Then, by making reference to the results of the boundary layer calculations, we have been able to relate the inverse of the derivatives of the eigenfunctions evaluated on the neutral line to specific scalings with respect to the non-ideal parameters, which had not been noted before (§ 8.3). We have also shown the relation of the inverse of these derivatives with the reconnecting layer width, $\delta$, whose operational definition, which we had previously verified in different reconnection regimes (Betar et al. 2020), we have here discussed for the first time (§ 8). These characteristic length scales, which can be useful for numerical diagnostics, are summarised in table 5.

We have interpreted in § 9 the results of the boundary layer analysis in light of heuristic derivations for the scalings of the growth rate and for the characteristic width of the reconnection layer, by following a dimensional analysis procedure that had been already successfully used in previous works but only when $\rho_s = 0$. In this way, we have highlighted in § 9.3 how the heuristic approach alone fails to provide the correct scaling when the ion-sound Larmor radius is not negligible ($\rho_s \gtrsim \bar{d}_e$).

Then, thanks to the operational definition we have given of $\delta$, and by relying on both the heuristic estimates and the numerical solutions of the eigenvalue problem, we have shown a further non-trivial relation between the first derivative of $\psi_1$, evaluated close to the neutral line, and the gradients of the velocity component parallel to it (§ 10). We have, in this way, introduced an inverse characteristic scale length which we have named $\Delta'_{v_y}$, because of its analogy with the classical $\Delta'$ parameter. Using this (inverse) scale length, we have therefore shown, both analytically and numerically, that, for the purpose of heuristic estimates, we can generally assume $\psi'_1|_{x=\delta} \sim \psi_1|_{x=\delta}/l_c$ with $l_c = \max\{(\Delta')^{-1}, (\Delta'_{v_y})^{-1}\} \gtrsim \delta$. Knowing the asymptotic scaling of $\Delta'_{v_y}$, an estimate of both $\delta$ and $\gamma$ can be made in any RMHD reconnection regime (§ 10.4) by just using dimensional analysis.

It is interesting to note that, from an experimental point of view, density fluctuations $n_1$ are easier to be measured than magnetic perturbations $\psi_1$, and that the former can be related to the fluid stream function perturbation via $n_1 \sim \nabla^2\varphi$. In general, then, the estimate of the spatial gradients of $\varphi_1$ from experimentally measured profiles of the density may be more reliable than the evaluation of the spatial gradients of $\psi_1$. In this context, the characterisation of the large- and small-$\Delta'$ limits we have provided in § 10 with (10.6), or, more generally, the relation between the value of $\Delta'_{v_y} = (\varphi''_1/\varphi'_1)|_{x=\delta} \sim \Delta'_{v_y}/2$ (cf. definition in (10.2)) and the scaling of $\delta$ or of the other scales detailed at the end of § 10.3, may be of interest.

Finally, we note that the introduction of the inverse scale $\Delta'_{v_y}$ makes the scaling laws in the large-$\Delta'$ limit mirror those in the small-$\Delta'$ limit provided the substitution $\Delta' \leftrightarrow \Delta'_{v_y}$.

## Acknowledgements

This article was written in honour of F. Pegoraro's 70th birthday. It is our pleasure to recall his fundamental contributions to the linear and nonlinear theory of reconnecting instabilities in plasmas. The authors thank Miho Janvier (IAS CNRS – Université Paris-Saclay, France) and Chiara Marchetto (ISC-CNR Milano, Italy) for useful discussions about some bibliographical references. The authors are grateful to the 'Maison
de la simulation Lorraine' for partial time allocation on the cluster Explor (Project No. 2019M4XXX0978).

Editor Alex Schekochihin thanks the referees for their advice in evaluating this article.

## Funding

This work was carried out within the framework of the French Federation for Magnetic Fusion Studies (FR-FCM) and of the Eurofusion consortium and received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under grant agreement No. 633053 (WPEDU fundings obtained through FR-FCM AAP 2017–2021 ‘Evolution of current sheets in low-collision plasmas’, in particular, are gratefully acknowledged). The views and opinions expressed herein do not necessarily reflect those of the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

## Declaration of interests

The authors report no conflict of interest.

## Appendix A. Derivation of the model equations for tearing modes in slab RMHD

Different derivations exist in the literature of the set of tearing equations we have considered in this paper. For the inclusion of finite ion-sound Larmor radius effects in a two-field model under the strong (infinite) guide field hypothesis, we direct the reader for example to Schep et al. (1994), Kuvshinov, Pegoraro & Schep (1994), Bergmans (2001) and Del Sarto et al. (2006). Here below, we repropose however their complete derivation, with more details than in previous articles.

Despite the different procedures proposed over the years to obtain (2.1) and (2.2), it is agreed that their nonlinear form reads

$$ \frac{\partial}{\partial t}(\psi - d_e^2 \nabla^2 \psi) + [\varphi, \psi - d_e^2 \nabla^2 \psi] = \rho_s^2[\nabla^2 \varphi, \psi] + S^{-1}\nabla^2 \psi, \quad (A1) $$

$$ \frac{\partial}{\partial t}(\nabla^2 \varphi) + [\varphi, \nabla^2 \varphi] = [\psi, \nabla^2 \psi]. \quad (A2) $$

The above equations have been written using the standard ‘Poisson’s bracket’ notation, $[f(x, y, t), g(x, y, t)] = \partial_x f \partial_y g - \partial_y f \partial_x g$. Each bracket term can be thus related to a convection term associated with one of the two scalar function involved, e.g.

$$ [f, g] = (\nabla f \times \nabla g) \cdot e_z = \underbrace{(e_z \times \nabla f)}_{u_f} \cdot \nabla g. \quad (A3) $$

Also note that in (2.1), the equilibrium contribution has been removed from the $S^{-1}\nabla^2 \psi$ term for the sake of linear analysis, since, in the asymptotic limit $S^{-1} \ll 1$, the time scale of resistive dissipation of the magnetic equilibrium is a posteriori found to be much longer than that of the tearing-type instability.

The appropriateness of (A1) and (A2), and notably of the $\rho_s^2$-related contribution, is supported by comparison of the linear dispersion relation obtained in the collisionless regime ($\eta = \nu = 0$) with that obtained from the full Vlasov–Maxwell system in the same geometry configuration, that is, $B^0 = B_{\perp}^0 + B_z^0 e_z$ and $k = k_{\perp} + k_z e_z$: the dispersion relation of shear kinetic-Alfvén waves (see e.g. Hasegawa & Uberoi 1982, pp. 19–21) is
indeed recovered,

$$ \omega^2 = c_{A,\perp} k_{\perp} \sqrt{\frac{1+k^2 \rho_s^2}{1+k^2 d_e^2}} \quad (A4) $$

where $c_{A,\perp}$ is the Alfvén velocity evaluated with the perpendicular magnetic component, only. Also note that (A1) and (A2) can be combined to give an equation for the energy conservation in the form

$$ \frac{\partial}{\partial t} \iint \left( \underbrace{|\nabla \psi|^2}_{\mathcal{E}_B} + \underbrace{d_e^2 |\nabla^2 \psi|^2}_{\mathcal{E}_J} + \underbrace{|\nabla \varphi|^2}_{\mathcal{E}_{\text{kin}}} + \underbrace{\rho_s^2 |\nabla^2 \varphi|^2}_{\mathcal{E}_{\text{int}}} \right) \, dx \, dy = 2S^{-1} \iint |\nabla^2 \psi|^2 \, dx \, dy, \quad (A5) $$

where $\mathcal{E}_B, \mathcal{E}_J, \mathcal{E}_{\text{kin}}, \mathcal{E}_{\text{int}}$ stand for the energy contributions respectively related to the in-plane magnetic field, to the current density (or electron kinetic energy), to the ion kinetic energy and to the internal energy (in turn related to the electron thermal energy and to their parallel compressibility along magnetic lines – see Grasso et al. 1999).

The difference among the ‘different’ RMHD models which lead to the same set of (A1) and (A2) is in the weight that must be attributed to the different terms of the nonlinear equations with respect to the expansion parameters that have been adopted.

Here below, we focus on the first derivation proposed by Pegoraro & Schep (1986) and Schep et al. (1994); Kuvshinov et al. (1994) and that has been later re-discussed by Bergmans (2001) and Del Sarto et al. (2006). In this framework, (A1) and (A2) can be shown to follow from the z-component of the electron momentum equation, which gives (A1), and from the charge continuity equation, which gives (A2), under the assumption that $\nabla = (\partial_x, \partial_y, 0)$ and that the gradient of the equilibrium density, and of its fluctuations as well, are smaller than the equilibrium quantity.

We start from the fluid equations for the species $\alpha$, and we assume the ion pressure to be negligible with respect to the electron temperature (i.e. we take the cold-ion limit). The continuity and momentum equations can be written in dimensionless form, normalised to the MHD scales, as

$$ \frac{\partial n^\alpha}{\partial t} + \nabla \cdot (n^\alpha \mathbf{u}^\alpha) = 0, \quad (A6) $$

$$ d_e^2 \left( \frac{\partial \mathbf{u}^e}{\partial t} + \mathbf{u}^e \cdot \nabla \mathbf{u}^e \right) = -d_i \left( E + \mathbf{u}^e \times \mathbf{B} - \frac{\mathbf{J}}{S} \right) - \rho_s^2 \frac{\nabla \cdot \mathbf{\Pi}^e}{n^e}, \quad (A7) $$

$$ d_i^2 \left( \frac{\partial \mathbf{u}^i}{\partial t} + \mathbf{u}^i \cdot \nabla \mathbf{u}^i \right) = d_i \left( E + \mathbf{u}^i \times \mathbf{B} - \frac{\mathbf{J}}{S} \right). \quad (A8) $$

Above, the ion skin depth $d_i = d_e \sqrt{m_i/m_e}$ has been further introduced and the pressure tensor $\mathbf{\Pi}^e$ has been normalised to the electron density times an electron reference temperature. This, after normalisation to the MHD scales, makes the squared ion-sound Larmor radius appear (normalised to the equilibrium shear length).

From now on, we will refer the symbol $\perp$ to the components that are orthogonal to the guide field direction (i.e. $e_z$), that is, that lie on the $(x, y)$-plane. Note that this differs from another notation, frequently used for example in tokamak geometry, in which the direction ‘parallel’ and ‘perpendicular’ are referred to the total magnetic field. The latter choice is generally more appropriate for the most general RMHD modelling, which we recall has
been first introduced by Strauss (1976, 1977) in toroidal geometry and then in the so-called ‘cylindrical tokamak’ approximation, by ordering the ratio between the toroidal and the poloidal gradients to be comparable to the ratio between the poloidal and toroidal magnetic components. Our choice of notation is here justified by the assumption of translation invariance for the equivalent to the ‘toroidal’ component (i.e. $\partial_z = 0$), corresponding to the large guide field component too (cf. figure 5).

One of the two equations we will focus on is the z-component of Ohm’s law, which, for reasons we will show next, is more convenient to rewrite in the form:

$$ -E_z - \frac{d_e^2}{d_i} \frac{\partial u_z^e}{\partial t} - \mathbf{u}_\perp^e \cdot \left[ \mathbf{B} \times \mathbf{e}_z + \frac{d_e^2}{d_i} \nabla u_z^e \right] = -S^{-1} J_z + \frac{\rho_s^2}{d_i} \frac{\nabla \cdot \mathbf{\Pi}^e}{n^e} \cdot \mathbf{e}_z. \quad (A9) $$

The other one is the charge continuity equation (i.e. the sum over species of (A6) multiplied by $q^\alpha/n^\alpha$), which, using the quasi-neutrality condition $q^e n^e + q^i n^i = 0$, we can rewrite as

$$ q^e \mathbf{u}_\perp^e \cdot \nabla (\ln n^e) + q^i \mathbf{u}_\perp^i \cdot \nabla (\ln n^i) + \nabla \cdot (q^e \mathbf{u}_\perp^e + q^i \mathbf{u}_\perp^i) = 0. \quad (A10) $$

The two equations above will be then specialised by approximating the fluid equations in terms the expansion parameters:

$$ \epsilon_B \equiv \frac{|\mathbf{B}_z^0|}{B_z^0}, \quad \epsilon_m \equiv \frac{m_e}{m_i}. \quad (A11) $$

The electromagnetic field components are written in terms of the scalar quantities of interest as

$$ \mathbf{B} = \underbrace{B_z^0 \nabla \psi \times \mathbf{e}_z}_{B_\perp} + \underbrace{B_z^0 (1 + b_1)}_{B_z^0 + B_z^1} \mathbf{e}_z, \quad E = - \underbrace{\nabla \phi}_{E_\perp} - \frac{1}{c} \underbrace{\frac{\partial \psi}{\partial t}}_{E_z} \mathbf{e}_z. \quad (A12) $$

Since we assume $J = (c/4\pi)\nabla \times \mathbf{B}$, it follows $\nabla \cdot \mathbf{B} = 0$ and

$$ J_\perp = \frac{c}{4\pi} \nabla \times b \mathbf{e}_z, \quad J_z = -\frac{c B_z^0}{4\pi} \nabla^2 \psi. \quad (A13) $$

After normalisation to the MHD scales, the fluid stream function $\varphi = \phi/(cB_z^0)$ appears as related to the electrostatic potential. Because of the uniform guide field hypothesis, the scalar function $b$ also coincides with the first-order perturbation of the $B_z$ component.

We furthermore assume $|\mathbf{B}_z^1| \sim |\mathbf{B}_z^0| \sim \epsilon_B B_z^0$, whence it follows

$$ \frac{\partial}{\partial t} \sim \frac{c_{A,\perp}}{L_0} \sim \epsilon_B \frac{B_z^0}{L_0}, \quad (A14) $$

and we assume the charged particle dynamics along $z$ to be mostly due to the electrostatic acceleration. This means that the ion velocity component along $z$, $u_z^i$, is $\epsilon_m$ times smaller than the electron velocity component, $u_z^e$, which is instead comparable in amplitude to both $\mathbf{u}_\perp^e$ and $\mathbf{u}_\perp^i$. This also means that $J_z \equiv q_i n_i u_z^i + q_e n_e u_z^e = q_e n_e u_z^e + \mathcal{O}(\epsilon_m)$. In normalised units, $J = n^e(\mathbf{u}^i - \mathbf{u}^e)$, and therefore

$$ u_z^e = d_i \frac{\nabla^2 \psi}{n_e} + \mathcal{O}(\epsilon_m). \quad (A15) $$

The idea is then to use these hypotheses combined with a strong guide field limit so to consider the drift-ordering expansion for the $\mathbf{u}_\perp^\alpha$ fluid velocities. Using the standard
procedure, we obtain from (A7) and (A8)) and (A12) the normalised velocity components:

$$ \boldsymbol{u}_{\perp}^{e} = \underbrace{\boldsymbol{e}_{z} \times \nabla \varphi + (\boldsymbol{e}_{z} \times \nabla \psi) \boldsymbol{u}_{z}^{e} - \frac{\rho_{s}^{2}}{d_{i}} \boldsymbol{e}_{z} \times \frac{\nabla \cdot \boldsymbol{\Pi}^{e}}{n_{e}} - S^{-1} \boldsymbol{e}_{z} \times \boldsymbol{J}_{\perp}}_{\text{1st order}} + \underbrace{\mathcal{O}(\varepsilon_{m}^{1/2} \varepsilon_{B}^{2})}_{\text{2nd order}}, \quad (\text{A16}) $$

$$ \boldsymbol{u}_{\perp}^{i} = \underbrace{\boldsymbol{e}_{z} \times \nabla \varphi - S^{-1} \boldsymbol{e}_{z} \times \boldsymbol{J}_{\perp} + \mathcal{O}(\varepsilon_{m} \varepsilon_{B})}_{\text{1st order}} - \underbrace{d_{i} \left( \frac{\partial}{\partial t} \nabla \varphi + [(\boldsymbol{e}_{z} \times \nabla \varphi) \cdot \nabla] \nabla \varphi \right)}_{\text{2nd order}} + \mathcal{O}(\varepsilon_{B}^{3}). \quad (\text{A17}) $$

The $\mathcal{O}(\varepsilon_m \varepsilon_B)$ neglected terms in (A17) are due to the $(\boldsymbol{e}_z \times \nabla \psi) \boldsymbol{u}_z^i$ contribution of the first-order drift-expansion. We note that the terms neglected in both equations are comparable if we order $\varepsilon_m \sim \varepsilon_B^2$.

Equations (A1) and (A2)) follow then from substitution of (A16) and (A17)) into (A9) and (A10) after a few further specific hypotheses. They are:

(i) the ordering $B_z^1/B_z^0 = b \sim \varepsilon_B^2$, which also implies $|\boldsymbol{J}_{\perp}| \sim \varepsilon_B^2$; and

(ii) the assumption that the anisotropic electron pressure components are given by the first-order FLR corrections to a double adiabatic-type pressure tensor.

The original derivation had been obtained by Schep et al. (1994) in the strictly collisionless limit, in which, in the geometry of interest to us, the pressure tensor components including FLR 'gyrofluid' corrections can be written in dimensional units as

$$ \boldsymbol{\Pi}^{e} = \begin{pmatrix} P_{\perp}^{e} + \frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{x}^{e}}{\partial y} + \frac{\partial u_{y}^{e}}{\partial x} \right) & \frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{y}^{e}}{\partial y} - \frac{\partial u_{x}^{e}}{\partial x} \right) & \frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{z}^{e}}{\partial y} \right) \\ \frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{y}^{e}}{\partial y} - \frac{\partial u_{x}^{e}}{\partial x} \right) & P_{\perp}^{e} - \frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{x}^{e}}{\partial y} + \frac{\partial u_{y}^{e}}{\partial x} \right) & -\frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{z}^{e}}{\partial x} \right) \\ \frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{z}^{e}}{\partial y} \right) & -\frac{P_{\perp}^{e}}{\Omega_{e}} \left( \frac{\partial u_{z}^{e}}{\partial x} \right) & P_{\parallel}^{e} \end{pmatrix}. \quad (\text{A18}) $$

The components of matrix (A18) can be obtained by following a standard procedure (see e.g. Thompson 1961; Roberts & Taylor 1962; MacMahon 1965 for the ion case and Cerri et al. (2013) for a recent re-derivation for both species in the strictly collisionless limit). We make the further hypothesis that

$$ P_{\parallel}^{e} \approx P_{\perp}^{e} = n^{e} T^{0} \quad (\text{A19}) $$

with $T^0$ reference electron temperature (uniform in space and constant in time). Since normalisation to the MHD reference quantities means that $\boldsymbol{u}^e$ is normalised to $c_{A,\perp} = d_i \Omega_i$ and spatial derivatives are normalised to $1/a$, using $\Omega_i/\Omega_e \simeq \varepsilon_m$, the combination of (A3),
(A15), (A18) and (A19) gives:

$$ \frac{\rho_s^2 (\nabla \cdot \mathbf{\Pi}^e)}{d_i n^e} \cdot \mathbf{e}_z = \varepsilon_m d_i \frac{\rho_s^2}{n^e} \left( \frac{\partial n^e}{\partial x} \frac{\partial}{\partial y} \left( \frac{\nabla^2 \psi}{n^e} \right) - \frac{\partial n^e}{\partial y} \frac{\partial}{\partial x} \left( \frac{\nabla^2 \psi}{n^e} \right) \right) \\ = \varepsilon_m d_i \rho_s^2 \left[ \ln n^e, \frac{\nabla^2 \psi}{n^e} \right], \qquad (A20) $$

$$ \frac{\rho_s^2}{d_i} \mathbf{e}_z \times \frac{\nabla \cdot \mathbf{\Pi}^e}{n^e} = \frac{\rho_s^2}{d_i} \mathbf{e}_z \times \nabla (\ln n^e) + \mathcal{O} \left( \varepsilon_m \varepsilon_B \left| \frac{\nabla n^e}{n^e} \right| \right). \qquad (A21) $$

Note that, in (A21), FLR corrections contribute with terms ~ $\varepsilon_m \varepsilon_B$ smaller than the isotropic diamagnetic drift term: comparison with the neglected terms of (A16) and (A17) shows that they can be disregarded.

We can now rewrite (A9) without collisions as

$$ \frac{\partial}{\partial t} \left( \psi - d_e^2 \frac{\nabla^2 \psi}{n^e} \right) + \mathbf{u}_{\perp}^e \cdot \nabla \left( \psi - d_e^2 \frac{\nabla^2 \psi}{n^e} \right) = \varepsilon_m d_i \rho_s^2 \left[ \ln n^e, \frac{\nabla^2 \psi}{n^e} \right]. \qquad (A22) $$

From substitution of (A21) into (A16), we get, using $d_e^2 = \varepsilon_m d_i^2$,

$$ \mathbf{u}_{\perp}^{e} \cdot \nabla \left( \psi - d_{e}^{2} \frac{\nabla^{2} \psi}{n^{e}} \right) = \left[ \varphi, \psi - d_{e}^{2} \frac{\nabla^{2} \psi}{n^{e}} \right] - \frac{\rho_{s}^{2}}{d_{i}} [\ln n^{e}, \psi] \\ + \varepsilon_{m} d_{i} \rho_{s}^{2} \left[ \ln n^{e}, \frac{\nabla^{2} \psi}{n^{e}} \right] + \mathcal{O}(\varepsilon_{B}^{3}) + \mathcal{O}(\varepsilon_{m}^{1/2} \varepsilon_{B}^{2}). \qquad (A23) $$

Note the presence of a $[\ln n^e, \nabla^2 \psi/n^e]$ term, identical to that of (A21), which is here due the diamagnetic drift contribution in the convection term $\mathbf{u}_{\perp}^e \cdot \nabla \mathbf{u}_z^e$ of the electron momentum equation. We can then write

$$ \frac{\partial}{\partial t} \left( \psi - d_e^2 \frac{\nabla^2 \psi}{n^e} \right) + \left[ \varphi, \psi - d_e^2 \frac{\nabla^2 \psi}{n^e} \right] = \frac{\rho_s^2}{d_i} [\ln n^e, \psi] + \mathcal{O}(\varepsilon_B^3) + \mathcal{O}(\varepsilon_m^{1/2} \varepsilon_B^2). \qquad (A24) $$

The exact cancellation of the bracket terms proportional to $\varepsilon_m d_i \rho_s^2$ is known as ‘gyroviscous’ (or, better ‘gyrofluid’) cancellation’ (see Roberts & Taylor 1962). The terms involved are however at least $\varepsilon_m^{1/2}$ smaller than the other ones, even when $d_e$ and $\rho_s$ are left unordered with respect to $\varepsilon_B$ or $\varepsilon_m$. Equation (A1) in the collisionless limit is finally recovered once we use continuity equations for ions and quasi-neutrality. From (A6), the continuity equation for ions can be rewritten as

$$ \frac{\partial}{\partial t} \ln n^i + [\varphi, \ln n^i] = d_i \frac{\partial}{\partial t} \nabla^2 \varphi + d_i [\varphi, \nabla^2 \varphi] + \mathcal{O}(\varepsilon_B^3). \qquad (A25) $$

Only the first-order $E \times B$ term of (A17) survives as a contribution to the convection term $\mathbf{u}_{\perp}^i \cdot \nabla \ln n^i$, since we can heuristically order $|\nabla \ln n^i| \sim \varepsilon_B$. Equation (A25) leads us to solve $d(\ln n^i)/dt = d_i d(\nabla^2 \varphi)/dt$, of which we can take the solution $\ln n^i = d_i \nabla^2 \varphi$ that verifies the heuristic hypothesis about the ordering of the spatial gradient of $\ln n^i$. From quasi neutrality, we finally write

$$ \ln n^e = d_i \nabla^2 \varphi, \qquad (A26) $$

which brings (A24) to the form (A1) once we approximate $n^e \simeq n^0$ (i.e. $n^e \simeq 1$ in normalised units) in the left-hand side gradients of $\nabla^2 \psi/n^e$.
The equation for $\nabla^2\varphi$ is finally obtained from (A10) using the same approximation of the density $n^e$. Such equation follows from combining (A25) with the electron continuity equation, which, using (A16), reads

$$ \frac{\partial}{\partial t} \ln n^e + [\varphi, \ln n^e] + d_i \left[ \psi, \frac{\nabla^2 \psi}{n^e} \right] + \mathcal{O}(\varepsilon_B^3) = \mathcal{O}(\varepsilon_m^{1/2} \varepsilon_B^2). \quad (A27) $$

Note that the $[\psi, \nabla^2\psi/n^e]$ contribution comes from the only non-null divergence term of (A16), that is, from $\nabla \cdot ((e_z \times \nabla\psi)u_z^e)$.

When collisions are included, the $+S^{-1}\nabla^2\psi$ contribution is recovered at the right-hand side of (A22) and (A23). In principle, the pressure tensor (A18) should be modified so as to include ‘gyroviscous’ corrections. These are provided, for example, by the model of Braginskii (1958). However, as long as the cold ion limit is formally assumed, so as to get rid of the ion temperature contribution, the corrections to the components of $\Pi^e$ due to electron-ion viscosity are $\varepsilon_m$ smaller than $S^{-1}$, and thus are also negligible in (A18). The last, further contribution we should care about is an additional $+S^{-1}\nabla^2 b \sim \mathcal{O}(S^{-1}\varepsilon_B^2)$ term at the right-hand side of (A25), which comes from the divergence of the $-S^{-1}e_z \times J_\perp$ term of ion drift velocity, (A17). This term is negligible under the further assumption $S^{-1} \ll d_i$. It must be noted that the latter choice is consistent with the fact that this two-field set of equations correspond to the extended MHD model in which the Hall term is neglected.

The quantitative comparison between the derivation here presented and that suggested by Kleva et al. (1995), Wang & Bhattarjee (1995) and Bian & Tsiklauri (2009), in which the $\rho_s^2$-related Poisson bracket is interpreted as associated with the Hall-term contribution in Ohms’ law – see also Del Sarto et al. (2006) and appendices of Del Sarto et al. (2016) – will be discussed elsewhere.

We finally recall that although in this modelling we assumed, since the beginning, strict translational invariance along $z$, i.e. $\partial/\partial z = 0$, the RMHD modelling in a strong guide field, large aspect ratio tokamak with toroidal coordinates $(r, \theta, \varphi)$ allows for the inclusion of the derivatives along $\varphi$ that are ordered $\partial/\partial\varphi \sim \varepsilon_B\partial/\partial\theta$, i.e. $k_\varphi \sim \varepsilon_B k_\theta$ (Strauss 1976, 1977). This maps into a Cartesian ‘extended-slab’ RMHD modelling that includes the derivatives along $z$ as corrections of order $\varepsilon_B$ with respect to the perpendicular derivatives, i.e. $\partial/\partial z \sim \varepsilon_B\partial/\partial y$ or $k_z \sim \varepsilon_B k_y$ (see, e.g. Matthaeus & Lamkin 1986 or Biskamp 2000, p. 17). In the simplest, cold-electron limit, these corrections lead to further terms $+\partial\varphi_1/\partial z$ and $+\partial\nabla^2\psi_1/\partial z$ on the right-hand side of (A1) and (A2), respectively. The role of these terms, as well as of further $\rho_s^2$-related contributions linked to the parallel electron compressibility in the nonlinear evolution of tearing modes with different helicities, has been first investigated in an extended version of model equations (A1) and (A2) by Grasso et al. (2004) and Borgogno et al. (2005).

## Appendix B. About alternative definitions of the reconnection rate

In the literature, alternative quantitative definitions exist, which, for historical reasons, are typically associated with the term ‘reconnection rate’, although they have been formulated under the hypothesis of steady reconnection. In the notation and geometry choice we make here, this alternative estimate of the reconnection rate can be written as

$$ \mathcal{R}_{\text{steady}} = \frac{1}{L_0} U_x^{|\text{at some } x \neq 0|} \quad (\text{B1}) $$

This definition originates from the Sweet-Parker model (Parker 1957), in which the rate at which the magnetic flux is reconnected, normalised with respect to the reference length
$L_0$, is estimated as the component of the velocity $U$ perpendicular to the neutral line such that

$$ (U_x B_y)|_{\text{near } X\text{-point}}^{\text{at some } x \neq 0} = \mathcal{F}_z|_{\text{near } X\text{-point}}^{\text{at some } x \neq 0}. \quad (B2) $$

This relation holds in the neighbourhood of the neutral line under steadiness assumptions, for which the out-of-plane component of the electric field ($E_z$), which in planar reconnection is completely inductive (i.e. $E_z = -\partial\psi/\partial t$), is zero – cf. (2.7). In this case, the velocity $U_x$ gives the rate per unit length $L_0$ at which the magnetic field lines are pushed and merged at the $X$-point.

Quantitative estimates of $R_{\text{steady}}$ can be obtained by arguments relying on the continuity of the flow, which relate the velocity $U_x$, at which the magnetic field is dragged to the $X$ point, to the velocity $U_x$ at which, after reconnection has occurred, the field is transported out of it, along the neutral line ($U_x L \simeq U_y a$ – cf. figure 2a). These arguments also require a force balance condition implying that the upstream velocity $U_y$, evaluated close to the neutral line and sufficiently far from the $X$-point, is of the order of the Alfvén velocity evaluated with respect to $B_y$; these arguments allow one to relate $U_x|_{\text{near } X\text{-point}}$ to the dissipation mechanism given by $\mathcal{F}_z$ (Parker 1957; Petschek 1964; Parker 1973; Park et al. 1984; Wesson 1990).

Note that in the tearing mode scenario the reference length $L_0$ is a macroscopic quantity associated with the shear length $a$ of the magnetic equilibrium, i.e. $L_0 = a$. In the first astrophysical applications of the steady reconnection scenario, instead, in which a distinction between equilibrium and perturbed quantities is not required, the macroscopic reference length was usually assumed to be of the order of the current sheet length $L$, i.e. $L_0 = L \gg a$, since the shear length $a$ of the magnetic field $B_y(x)$ associated with the reconnecting current sheet was, in those contexts, a microscopic quantity. Thus, an alternative definition of ‘reconnection rate per unit length $L$’ is often met in the literature. This was defined by Petschek (1964) analogously to the Alfvénic Mach-number, from which it takes the symbol, as

$$ M_{\text{rec}} = \frac{U_y|_{\text{near } X\text{-point}}}{U_x|_{\text{far from } X\text{-point}}} \approx \frac{U_y|_{\text{near } X\text{-point}}}{c_A(L)}, \quad (B3) $$

where $c_A(L)$ is the Alfvén velocity evaluated in terms of $B_y$, measured sufficiently far from the neutral line along the neutral line $x = 0$, in a region where ideal MHD is valid. In the steady reconnection scenario, this definition is practically equivalent to (B1).

Finally, a further alternative definition of the reconnection rate is also often adopted,

$$ \tilde{\mathcal{R}}_{\text{rec}} = \frac{1}{E_{\parallel}} \frac{dE_{\parallel}}{dt} \Big|_{X\text{-point}}, \quad E_{\parallel} \equiv \frac{E \cdot B}{B}. \quad (B4) $$

We conclude by noting that while no ambiguity exists for the evaluation of the reconnection rate in the tearing-type instabilities with fixed wavenumber on which we are interested in this work, the situation becomes of course more complex in experimental contexts or in nonlinear numerical simulations of reconnection processes, especially if several reconnection sites ($X$-points) are simultaneously present and dynamically evolve in time. In those cases, estimates based on local measures like those provided by (2.13), (B3) and (B4) are of course more appealing for practical reasons. Their use for getting numerical values to be compared with scalings predicted by specific reconnection models should be however handled with care, since the way precision issues are determined in the measurement of numerical or experimental values is often unlikely to be sufficient to discriminate between a reconnection scenario or the other, e.g. in assessing the regime
of reconnection events secondary to primary ones. Conversely, theoretical arguments supporting the existence or not of some specific regime are likely to provide a more accurate guide for the operational definition of reconnection rate that should be adopted in each case (cf. reference quoted just before (2.10)).

# Appendix C. Integration of the boundary layer equation via Fourier transformation

Tackling the integration of tearing modes in the Fourier space with respect to the variable $x$ is particularly useful since the overall order of the differential equations results is lowered (note that $\psi_0 \sim x \rightarrow d/dk_x$, whereas $\psi_1^{(N)} \rightarrow \sim k_x^N \hat{\psi}_1$). This approach is possible since the eigenfunctions we seek are bounded at each time with respect to the $x$ variable. Also note that Fourier representation is quite natural in the mathematical treatment of the gyrokinetic operator in the Vlasov-Poisson equation which, at some level of the analysis, must be treated when FLR effects of electrons and/or of ions are included. However, while the integration in the Fourier space may be more efficient, from an analytical point of view, for obtaining the dispersion relation and the asymptotic scalings of $\gamma$ and $\delta \sim \delta_1$, it has the drawback of requiring further analysis – not always trivial – for the computation of the eigenfunction profile in closed form in the coordinate space (i.e. the inverse Fourier transforms of the eigenfunctions integrated in the $k_x$-space must be evaluated).

## C.1. *A brief historical review on the boundary layer approach to tearing-type equations in the Fourier space*

Fourier analysis was probably first applied to integrate interchange-type eigenmodes of MHD-type equations by Coppi (1964b), and was used in the tearing mode problem by Ara et al. (1978) to find approximate solutions of the inner equations from which to estimate the weight of ion-ion viscosity in the dispersion relation via a variational approach. A more detailed Fourier approach to tearing-type equations has been developed by Pegoraro & Schep (1981) to model low-frequency modes in toroidal tokamak geometry. This approach, which has been later used in several other specific reconnection models, consists in performing the Fourier transformation with respect to the stretched variable $\zeta$ of (5.9) and (5.10), and then combining the inner-layer equations for $\psi_1$ and $\phi_1$ into a single equation for $\psi_1$. The integration of the tearing-type equations including FLR effect of both ions (for which a continuity equation was derived from a gyrokinetic one) and electrons (of which the isothermal limit was taken) has been first detailed by Pegoraro & Schep (1986), using the ballooning representation, and then, starting from a different kinetic model and using a somewhat different approach, by Cowley et al. (1986). The technique detailed by Pegoraro & Schep (1986) has been further used for the internal $m=1$ kink mode in high-$\beta$ regimes (Pegoraro et al. 1989). Relying on the same type of analysis, Porcelli (1991) derived the scaling laws for the collisionless regime in which magnetic reconnection occurs due to finite electron inertia and in combination with FLR effects. The Fourier analysis was also applied to investigate the scaling laws of the collisionless reconnection instabilities occurring in high frequency fluid regimes where ions form a neutralising background (Bulanov et al. 1992; Attico, Califano & Pegoraro 2000, 2001). The integration of tearing-type modes in Fourier space was then extended to other regimes in which electron temperature gradients at equilibrium are admitted and finite electron $\beta$ effects are included, while a full gyro-kinetic model is taken for describing the ion response: motivated by the operational regime of large-size tokamaks, which is expected to fall in the semi-collisional regimes where the width of the semi-collisional region is much smaller than the ion Larmor radius and finite $\beta$-effects are expected to play an important role, Connor et al. (2012b) investigated tearing-type instabilities in low-$\beta$ and high-$\beta$
regimes and the transition between them by relying on integration in the Fourier-space. In doing so, these authors also made a comparison of the solutions they obtained with those previously found in the coordinate space by Cowley et al. (1986). Here, solutions of tearing equations in the semi-collisional limit were obtained by starting from a full kinetic model and by performing an expansion of the eigenfunctions in powers of the ratio between the integration layer width over the ion Larmor radius. The technique used by Connor et al. (2012b) is analogous to the boundary layer approach discussed by Pegoraro & Schep (1981, 1986): using the fact that the dynamics of ions decouples from that of electrons at a distance of the order of the ion gyro-radius, the system of equations in each region was first transformed into a single equation for the current density in the Fourier space. Then, for the low-$\beta$ regime, the current density equation for ions was solved by Connor et al. (2012b) in Fourier space, while that for electrons was solved in real space. These solutions were made to match each other at some intermediate layer, while the solution in the ion region was also required to match that in the ideal MHD region, located far from the resonance surface. The technique to find the solution in the ion and electron regions consisted in expanding and matching the current density in powers of $\beta$. This method leads to a general dispersion relation that extends that of previous models by including four branches: the ion drift mode, the drift tearing mode, and other two branches corresponding to the kinetic Alfvén waves which couple to drift tearing modes for finite values of $\beta$.

### C.2. Comparison of the analysis of §§ 4–6 with the Fourier approach

In the analysis of §§ 4–6, we have considered the low-$\beta$ regime with isothermal electrons, no equilibrium density fluctuations and cold ions, which rules out the coupling with drift modes while the coupling with kinetic Alfvén waves is restricted to the branch described by (A4). It is easy to show the essential points of the Fourier approach of Pegoraro & Schep (1986) discussed above, by making a direct comparison with the equations discussed in §§ 4–6 by Fourier transforming (5.9) and (5.10), and to combine them together so as to write a single equation for the Fourier components of the current density. To this purpose, let us first define Fourier transform for a function $f(\zeta)$ as

$$ \hat{f} = \int_{-\infty}^{+\infty} f(\zeta) e^{iq\zeta} d\zeta, \quad \zeta = \frac{x}{\sigma}, \quad q = k_x \sigma, \qquad (C1) $$

where we have re-called the definition of the stretched variable, and therefore the relation of the ‘stretched’ Fourier coordinate $q$ with $k_x$, the standard Fourier-conjugate wavenumber of $x$. For the scale $\sigma$, we have used the same notation of Porcelli (1991). In the notation adopted throughout the present manuscript, it would read indeed $\delta_1$ or $\delta_2$. Multiplying (5.9) and (5.10) by $e^{iq\zeta}$ and integrating from $-\infty$ to $+\infty$ while using the definition (C1), one obtains respectively

$$ \frac{q^2}{\hat{G}^2} \frac{d^2 \hat{\psi}_1}{dq^2} - \left(1 + \frac{q^2}{\hat{G}^2}\right) \hat{\psi}_1 = -i d_e^2 \frac{d\hat{\phi}}{dq}, \quad \hat{\phi} = -i \frac{d\hat{\psi}_1}{dq}. \qquad (C2) $$

Taking the derivative of the second equation in (C2) with respect to $q$ and substituting the results in the former, one finds

$$ \left(d_e^2 + \frac{q^2}{\hat{G}^2}\right) \frac{d^2 \hat{\psi}_1}{dq^2} - \left(1 + \frac{q^2}{\hat{G}^2}\right) \hat{\psi}_1 = 0. \qquad (C3) $$

This equation represents an eigenvalue problem for the eigenfunction $\hat{\psi}_1$. For analytical convenience, it is however useful to transform it in an eigenvalue problem for the current
density $\hat{J}_1 = q^2 \hat{\psi}_1$. After this substitution, one obtains

$$ \left(d_e^2 \hat{G}^2 + q^2\right) \frac{d^2}{dq^2} \left(\frac{\hat{J}_1}{q^2}\right) - (q^2 \hat{G}^2 + 1) \hat{J}_1 = 0. \qquad (C4) $$

This is the cold-ion limit of (3) of Porcelli (1991). The equivalence is evident once the following correspondence between variables and parameters of the aforementioned reference and those of §§ 4–6 are established: $\Gamma = \gamma$, $\Delta^2 = d_e^2(1 + S^{-1}/\gamma) \to d_e^2$, and $\rho_\tau = (\rho_s^2 + \rho_i^2)^{1/2} \to \rho_s$.

Here, we do not discuss further the solution of (C4), since it represents a specific case of the equation already solved by Pegoraro & Schep (1981, 1986) and further used by Pegoraro et al. (1989), Porcelli (1991), Attico et al. (2000, 2001) and Connor et al. (2012b).

## Appendix D. An example of renormalisation of a differential equation

Consider (6.12) that we rewrite here for convenience as

$$ \frac{\tilde{\varphi}_1''}{c_0} - \frac{A}{BC} \zeta^2 \frac{\tilde{\varphi}_1}{c_0} = \frac{A}{B} \zeta, \qquad (D1) $$

where

$$ A \equiv \ell^2, \quad B \equiv \bar{d}_e^2, \quad C \equiv \mathcal{G}^2. \qquad (D2) $$

As an example, let us consider the case in which also the scale $\ell$ is non-trivial (i.e. $\ell \neq 1$). From the point of view of physical dimensions, (6.12) and (D1) are already written in dimensionless form but we want to check if it is possible to bring them to the ‘cleaner’ form of (6.13), after further renormalisation of quantities. Suppose then to multiply (D1) by $A^a B^b C^c$, where $a, b, c$ are here numerical coefficients to be determined a posteriori. Estimating, just for the purpose of dimensional analysis, $\tilde{\varphi}_1'' \sim \tilde{\varphi}_1/\zeta^2$, we obtain the following system:

$$ A^a B^b C^c \left( \frac{\tilde{\varphi}_1}{c_0} \right) \zeta^{-2} \sim 1, \quad A^{a+1} B^{b-1} C^{c-1} \left( \frac{\tilde{\varphi}_1}{c_0} \right) \zeta^2 \sim 1, \quad A^{a+1} B^{b-1} C^c \zeta \sim 1. \qquad (D3) $$

The choice of fixing all right-hand side coefficients equal to a number is made possible by the fact that the number of independent conditions is equal to or larger than the number of unknown coefficients $a, b, c$: the parameters $A, B$ and $C$ can be thus completely absorbed in the redefinition of $\zeta$ and $\tilde{\varphi}_1/c_0$. This is the case also when the scale $\ell$ is absent, i.e. when $\ell = A = 1$. When the number of parameters is larger than the number of constraints, some further parameter(s) corresponding to a dimensional combination(s) of powers of $A, B, \dots$ will appear in the renormalised equations. The system above can be therefore solved to find that the values of the sought coefficients are $a = -3/4, b = 3/4, c = -1/4$. This means that we can define

$$ \left. \begin{array}{l} z \equiv A^{a+1} B^{b-1} C^c \zeta = A^{1/4} B^{-1/4} C^{-1/4} \zeta, \\ \Phi \equiv A^{-a-1} B^{-b+1} C^{-c-1} \left( \frac{\tilde{\varphi}_1}{c_0} \right) = A^{-1/4} B^{1/4} C^{-3/4} \left( \frac{\tilde{\varphi}_1}{c_0} \right). \end{array} \right\} \qquad (D4) $$

For $\ell = 1$, this corresponds to (6.13).
<figure><img src="image_32.png" /><figcaption>FIGURE 17. The region where the solutions $\psi_\alpha$ and $\psi_\beta$ diverge is here displayed in a grey colour inside two lobes that join each other at $\zeta = 0$. Outside these two lobes, both series converge.</figcaption></figure>

## Appendix E. Convergence and independence of the solutions of the hypergeometric form of the inner equation

We here discuss some technical details about the solutions (7.25) and (7.27)) of the innermost equation, which we have obtained by evaluation of the residues of its hypergeometric form.

### E.1. Convergence of $\psi_\alpha$ and $\psi_\beta$

Following the ratio test of sequences for the coefficients of $(1 + \zeta^2)^s$, one finds $\lim_{m \to \infty} g_{m+1}/g_m = 1$, meaning that the power series (7.25) and (7.27) converge when $\|1 + \zeta^2\| > 1$. This region covers all the complex plane, except when $\zeta = 0$. In the complex plane, the boundary corresponding to the convergence radius $\|1 + \zeta^2\| = 1$ is given by

$$ (1 + \zeta_R^2 - \zeta_I^2)^2 + 4\zeta_R^2\zeta_I^2 = 1, \qquad (E1) $$

which is a curve forming two lobes joining at $\zeta = 0$ and intersecting the $\zeta_R = 0$ axis at $\zeta_I = \sqrt{2}$, while the maximum width of the lobes occur at $\zeta_R = \pm 1/2$ and $\zeta_I = \pm \sqrt{3}/2$.

The region inside the lobes of figure 17 represents the divergence region, which includes the singular points at $\zeta_I = \pm i$.

### E.2. Linear independence of $\psi_\alpha$ and $\psi_\beta$

It is not difficult to prove that the two solutions given by (7.25) and (7.27) are linearly independent in the domain of convergence, say $\mathcal{D}_{\alpha,\beta}$, where both $\psi_\alpha$ and $\psi_\beta$ are defined. One way to prove this is to show that their Wronskian is non-zero everywhere in $\mathcal{D}_{\alpha,\beta}$. An easier way to proceed is to investigate the independence of the leading terms in both
series. Taking $m = 0$, (7.25) and (7.27) become

$$ \psi_{\alpha,0} = \frac{\Gamma(-\nu)}{\Gamma\left(\frac{5}{4} - \frac{\nu}{2}\right) \Gamma\left(\frac{1}{4} - \frac{\nu}{2}\right)} (1 + \zeta^2)^{\alpha/2}, \quad \psi_{\beta,0} = \frac{\Gamma(\nu)}{\Gamma\left(\frac{5}{4} + \frac{\nu}{2}\right) \Gamma\left(\frac{1}{4} + \frac{\nu}{2}\right)} (1 + \zeta^2)^{\beta/2}. \qquad (E2) $$

The coefficients in the two previous relations are non-zero. Therefore, if $\alpha \neq \beta$, then $\psi_{\alpha,0}$ and $\psi_{\beta,0}$ are linearly independent, and so are $\psi_{\alpha}$ and $\psi_{\beta}$. However, if $\alpha = \beta$, then $\nu = 0$. Using the definition of the $\Gamma$ function for positive and negative arguments, this means that the growth rate $\gamma$ is purely imaginary. Therefore, in this case, the configuration is stable to tearing-type modes, which is not a case of interest here.

Another interesting remark related to (7.25) and (7.27) is that they are even functions with respect to $\zeta$. Due to this symmetry, we expect that there is also an odd solution with respect to $\zeta$ which will be obviously linearly independent of the even solutions. The existence of this additional solution would apparently raise a paradox, because we would now have three independent solutions: the two independent even solutions $\psi_{\alpha}$ and $\psi_{\beta}$, and one odd solution, whereas a second-order ODE cannot have more than two linearly independent solutions. This apparent paradox can be solved by observing that going from $\zeta = 0^+$ to $\zeta = 0^-$ in the complex plane by passing through $\zeta = 0$ is not possible for any of the two series, because both $\psi_{\alpha}$ and $\psi_{\beta}$ diverge there. Therefore, one should follow one of the lobes drawing the boundary of the divergence region in the complex plane (see figure 17), that is, one of the lobes encircling the singularities at $\zeta = \pm i$. Thus, to construct this solution, one can use a linear combination of the two series. This combination can be obtained using the integral representation and by choosing a path $C_{\alpha}$ (or $C_{\beta}$, alternatively) that goes around the poles associated with the exponent $\alpha$ (or $\beta$) by thus avoiding those of $\beta$ (or $\alpha$).

## Appendix F. Heuristic estimation of the scaling laws by making use of the inverse gradient scale $l_c$ and of $\Delta'_{v_y}$

Here we discuss the logical steps of a unified heuristic procedure which would make it possible to deduce the correct scalings in any regime and wavelength limit among those considered in this work, were the scaling of $\Delta'_{v_y}$ be always deductible: in any regime, we obtain a scaling law, which is symmetric with respect to that of the small-$\Delta'$ limit prior to the substitution of $\Delta' \leftrightarrow \Delta'_{v_y}$. The proposed procedure, however, is ‘self-consistent’ in all regimes with the exception of the large-$\Delta'$, warm-electron limit, where the scaling of $\Delta'_{v_y}$ appears to be a priori non-deductible via dimensional analysis. To facilitate the identification and presentation of the logical steps of the proposed heuristic procedure, they are summarised as statements and formulae in tables 3–5.

In particular, in table 3, we recall the approximated linear equations, definitions and general constraints we rely upon, as well as the conclusions that can be drawn for them regardless of the reconnection regime considered.

Finally, in table 5, we consider the cold-tearing regimes ($\rho_s = 0$), both collisionless and resistive, by showing how the heuristic approach that takes $\Delta'_{v_y}$ into account works and still provides the good scalings. Although having introduced the new scale $\Delta'_{v_y}$ makes the heuristic procedure slightly longer than the one discussed in § 9.3, it allows one to appreciate the ‘symmetry’ between $\Delta'$ in the small-$\Delta'$ limit and $\Delta'_{v_y}$ in the large-$\Delta'$ limit also in these cold-electron regimes. More importantly, and differently from the warm-reconnection regimes, it becomes manifest that for $\rho_s = 0$, it is possible to close
General hypotheses and conditions for a heuristic-type approach

<table><caption>TABLE 3. Starting equations, definitions, and assumptions based on general constraints that we take for the heuristic-type approach in all reconnection regimes. Equations [T4.E1–T4.E2] are the eigenmode equations ((2.1) and (2.2)) in which we have assumed $k$ to be negligible with respect to the inverse scale of the spatial gradients of the eigenfunctions. Equations [T4.D1–T4.D3] are the operational definitions we have given in §§ 8, 9 and 10 of $\delta$, $l_c$ and $\Delta'_{v_y}$, respectively. Equations [T4.C1, T4.C2] and [T4.H1] represent further conditions on the parameters that must be fulfilled regardless of the reconnection regime: [T4.C1] is a constraint that follows from [T4.E2] and definitions [T4.D2–T4.D3]; conditions [T4.C2a] and [T4.C2] correspond to (10.9) that we have discussed in the text; condition [T4.H1] corresponds to the ansatz (10.5) discussed in the text. Equations [T4.R1a] and [T4.R1b] are two immediate conclusions that can be drawn and allow us to specify hypothesis [T4.H1] in the small- and large-$\Delta'$ limits, respectively.</caption><tr><td>Approximation</td><td>[T4.E1]</td></tr><tr><td>Equations</td><td>[T4.E2]</td></tr><tr><td>Definitions</td><td>[T4.D1]<br>[T4.D2]<br><br>δ : ψ<sub>1</sub><sup>(iv)</sup>(δ) = 0<br>l<sub>c</sub> : ψ'<sub>1</sub>(δ) ~ ψ<sub>1</sub>(δ)<br><br>Δ'<sub>v<sub>y</sub></sub> ≡ <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">φ''<sub>1</sub>(δ) - φ''<sub>1</sub>(-δ)</div><div style="padding: 0 5px;">φ'<sub>1</sub>(δ)</div></div> = <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">2φ''<sub>1</sub>(δ)</div><div style="padding: 0 5px;">φ'<sub>1</sub>(δ)</div></div> (from T4.E2, T4.D2–T4.D3)</td></tr><tr><td>Constraints</td><td>[T4.C1]<br><br><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">γφ'<sub>1</sub></div><div style="padding: 0 5px;">kψ<sub>1</sub></div></div>|<sub>x=δ</sub> ~ <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">2</div><div style="padding: 0 5px;">l<sub>c</sub>Δ'<sub>v<sub>y</sub></sub></div></div></td></tr><tr><td>Hypotheses</td><td>[T4.H1]<br><br><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">γφ'<sub>1</sub></div><div style="padding: 0 5px;">kψ<sub>1</sub></div></div>|<sub>x=δ</sub> ~ O(1)<br>(Δ'δ ≪ 1)<br><br>l<sub>c</sub> ~ max{⟨(Δ')⁻¹, (Δ'<sub>v<sub>y</sub></sub>)⁻¹⟩ ≥ δ<br>ψ''(δ) ~ <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">ψ<sub>1</sub>(δ)</div><div style="padding: 0 5px;">l<sub>c</sub>δ</div></div></td></tr><tr><td>Results 1</td><td>[T4.R1a] l<sub>c</sub> ~ O(1) ⇒ l<sub>c</sub> ~ (Δ')⁻¹ ≥ (Δ'<sub>v<sub>y</sub></sub>)⁻¹<br>(from T4.C2a, T4.D3, T4.C1 and T4.H1)<br><br>[T4.H2] l<sub>c</sub> ~ O(1) ⇒ l<sub>c</sub> ~ (Δ')⁻¹ ≥ (Δ'<sub>v<sub>y</sub></sub>)⁻¹<br>(from T4.C21 and T4.Cb)</td></tr></table>
<table><caption>TABLE 4. Combining the hypotheses and results reported in table 2 with hypotheses [T5.H2–T5.H3] that are specific of the warm regimes, it is possible to deduce in a self-consistent way the set of equations [T5.R2–T5.R5]. It must be emphasised, however, that the system of condition is not closed in the large-Δ' regime and additional information about Δ'_{vy} is required to obtain the corresponding scaling laws.</caption><thead><tr><th colspan="3"></th><th colspan="2">Warm-reconnection regimes</th></tr><tr><th colspan="3"></th><th>warm-collisionless regime</th><th>warm-resistive regime</th></tr></thead><tbody><tr><td>Hypothesis 2</td><td>[T5.H2]</td><td>kδ φ<sub>1</sub> / γ ψ<sub>1</sub> Δ<sub>vy</sub> ∼ 2 / ρ<sub>s</sub></td><td>(ψ<sub>1</sub> ∼ k<sup>2</sup>ρ<sub>s</sub><sup>2</sup> / γ<sup>2</sup> x<sup>2</sup>ψ<sub>1</sub><sup>′′</sup> in T4.E1, using T4.E2)</td><td></td></tr><tr><td>Hypotheses 3</td><td>[T5.H3a]</td><td>l<sub>c</sub>δ ∼ d<sub>e</sub><sup>2</sup></td><td>[T5.H3b]</td><td>γl<sub>c</sub>δ ∼ S<sup>-1</sup></td></tr><tr><td></td><td>(ψ<sub>1</sub> ∼ d<sub>e</sub><sup>2</sup>ψ<sub>1</sub><sup>′′</sup> in T4.E1)</td><td></td><td></td><td>(ψ<sub>1</sub> ∼ S<sup>-1</sup> / γ ψ<sub>1</sub><sup>′′</sup> in T4.E1)</td></tr><tr><td>Result 2</td><td>[T5.R2]</td><td>γ ∼ kρ<sub>s</sub> (δ / l<sub>c</sub>)<sup>1/2</sup></td><td>[T5.R3b]</td><td>γ ∼ k<sup>2/3</sup> / ρ<sub>s</sub><sup>2/3</sup> S<sup>-1/3</sup> / l<sub>c</sub><sup>2/3</sup></td></tr><tr><td>Result 3</td><td>[T5.R3a]<br>(from T5.H3a, T5.R2)</td><td>γ ∼ kρ<sub>s</sub> d<sub>e</sub> / l<sub>c</sub></td><td></td><td>(from T5.H3b, T5.R2)</td></tr><tr><td>Result 4</td><td>[T5.R4a]<br>γ ∼ kΔ<sub>′</sub>ρ<sub>s</sub>d<sub>e</sub> / (Δ'<sub>s</sub> ≪ 1)</td><td>γ ∼ kΔ<sub>′</sub>ρ<sub>s</sub>d<sub>e</sub> / (Δ'<sub>s</sub> ≫ 1)</td><td>[T5.R4c]<br>γ ∼ (kΔ<sub>′</sub>)<sup>2/3</sup> / ρ<sub>s</sub><sup>2/3</sup> S<sup>-1/3</sup> / (Δ'<sub>s</sub> ≪ 1)</td><td>[T5.R4d]<br>γ ∼ (kΔ<sub>′</sub>)<sup>2/3</sup> / ρ<sub>s</sub><sup>2/3</sup> S<sup>-1/3</sup> / (Δ'<sub>s</sub> ≫ 1)</td></tr><tr><td>Result 5</td><td>[T5.R5a]<br>δ ∼ Δ'<sub>vy</sub>d<sub>e</sub><sup>2</sup> / (Δ'<sub>s</sub> ≪ 1)</td><td>δ ∼ Δ'<sub>vy</sub>d<sub>e</sub><sup>2</sup> / (Δ'<sub>s</sub> ≫ 1)</td><td>[T5.R5c]<br>δ ∼ (kρ<sub>s</sub>S)<sup>-2/3</sup> Δ<sub>vy</sub><sup>1/3</sup> / (Δ'<sub>s</sub> ≪ 1)</td><td>(from T4.R1a)<br>[T5.R5d]<br>δ ∼ (kρ<sub>s</sub>S)<sup>-2/3</sup> Δ<sub>vy</sub><sup>1/3</sup> / (Δ'<sub>s</sub> ≫ 1)</td></tr><tr><td colspan="5">(δ is found by reversing T5.R2 for the relevant value of l<sub>c</sub> and γ : δ ∼ (γ/(kρ<sub>s</sub>))<sup>2</sup>l<sub>c</sub>)</td></tr></tbody></table>
<table><caption>TABLE 5. Combining the hypotheses and results reported in table 2 with hypotheses [T6.H2] and [T6.H3] that are specific of the cold regimes, it is possible to deduce in a self-consistent way the set of equations [T6.C3]–[T6.R4]. Note that, in this case, the set of conditions is closed also in the large-Δ' regime thanks to [T6.H2], which leads to the constraint [T6.C4] that sets the scaling of Δ'<sub>v<sub>ij</sub></sub> to be the same of δ<sup>-1</sup>. All scalings can be this way a priori determined without resorting to numerical calculations.</caption><thead><tr><th colspan="3">Cold-reconnection regimes</th></tr><tr><th></th><th>cold-collisionless regime</th><th>cold-resistive regime</th></tr></thead><tbody><tr><td>Hypothesis 2</td><td>[T6.H2]</td><td>kδφ<sub>1</sub>/γψ<sub>1</sub> ~ O(1)</td><td>(ψ<sub>1</sub> ~ i<sup>k</sup>/γxφ<sub>1</sub> in T4.E1)</td></tr><tr><td>Hypotheses 3</td><td>[T6.H3a]</td><td>[T6.H3b]</td><td>γl<sub>c</sub>δ ~ S<sup>-1</sup></td></tr><tr><td></td><td>(ψ<sub>1</sub> ~ d<sub>e</sub><sup>2</sup>ψ<sub>1</sub><sup>′′</sup> in T4.E1)</td><td>(ψ<sub>1</sub> ~ i<sup>k</sup>/γ(xφ<sub>1</sub><sup>′</sup>) and T4.D2, T6.H2)</td><td>(ψ<sub>1</sub> ~ S<sup>-1</sup>/γψ<sub>1</sub><sup>′′</sup> in T4.E1)</td></tr><tr><td>Constraint 3</td><td>[T6.C3]</td><td>(ψ<sub>1</sub> ~ i<sup>k</sup>/γ(xφ<sub>1</sub><sup>′</sup>) and T4.D2, T6.H2)</td><td>(ψ<sub>1</sub> ~ S<sup>-1</sup>/γxφ<sub>1</sub><sup>′</sup> in T4.E1)</td></tr><tr><td>Constraint 4</td><td>[T6.C4]</td><td>((ψ<sub>1</sub><sup>′</sup>)<sup>′</sup> ~ i<sup>k</sup>/γφ<sub>1</sub><sup>′′</sup> and T4.D2, T4.H2)</td><td></td></tr><tr><td>Constraint 5</td><td>[T6.C5]</td><td>γ ~ kδ<sup>3/2</sup>Δ<sup>1/2</sup> for δΔ' &lt;&lt; 1</td><td>(from T6.C3, T4.C1 using T6.C4)</td></tr><tr><td>Results 2</td><td>[T6.R2a]</td><td>γ ~ kδ for δΔ' &gt;&gt; 1</td><td>(T6.C5 and T4.R1a)</td></tr><tr><td>Results 3</td><td>[T6.R2b]</td><td>[T6.R3b]</td><td>(T6.C5, T4.R1b, T6.C5)</td></tr><tr><td></td><td>γ ~ kΔ<sup>2</sup>d<sub>e</sub><sup>3</sup></td><td>γ ~ kΔ<sup>2</sup>d<sub>e</sub><sup>3</sup></td><td>[T6.R3d]</td></tr><tr><td></td><td>(Δ'δ &lt;&lt; 1)</td><td>~ kd<sub>e</sub></td><td>γ ~ (kΔ<sup>2</sup>)<sup>2/5</sup>S<sup>-3/5</sup></td></tr><tr><td></td><td>(from T4.R1a)</td><td>(Δ'δ &gt;&gt; 1)</td><td>~ k<sup>2/3</sup>S<sup>-1/3</sup></td></tr><tr><td></td><td>(from T4.R1b)</td><td>(Δ'δ &lt;&lt; 1)</td><td>(Δ'δ &gt;&gt; 1)</td></tr><tr><td>Result 4</td><td>[T6.R4a]</td><td>~ k<sup>2/5</sup>Δ<sup>1/5</sup>S<sup>-2/5</sup></td><td>(from T4.R1a)</td></tr><tr><td></td><td>δ ~ Δ'<sub>v<sub>ij</sub></sub>d<sub>e</sub><sup>2</sup></td><td>~ d<sub>e</sub></td><td>δ ~ k<sup>-2/5</sup>Δ<sup>1/5</sup>S<sup>-2/5</sup></td></tr><tr><td></td><td>(Δ'δ &lt;&lt; 1)</td><td>(Δ'δ &gt;&gt; 1)</td><td>~ k<sup>-1/3</sup>S<sup>-1/3</sup></td></tr><tr><td></td><td>(δ is found by reversing T6.C5 using T6.H3a,b for the relevant value of γ)</td><td>(Δ'δ &lt;&lt; 1)</td><td>(Δ'δ &gt;&gt; 1)</td></tr></tbody></table>
the system of heuristic conditions also in the large-$\Delta'$ limit, so as to determine a *priori* the corresponding scaling of $\Delta'_{v_y}$, which here always results to be the same of $\delta$.

In table 4, we specialise the results to the warm-tearing regimes ($\rho_s \neq 0$), both collisionless and resistive: in each regime, we report the further ansatz required to get the sought scalings in both the small- and large-$\Delta'$ limits. The 'symmetry' of the scalings written in the small-$\Delta'$ limit in terms of $\Delta'$ with respect to the scaling written in the large-$\Delta'$ limit in terms of $\Delta'_{v_y}$ can be in this way appreciated, although the system of equations is not closed in the large-$\Delta'$ limit and we must rely on additional information (e.g. numerical calculations performed by following the procedure detailed in previous § 10.3) to explicitly evaluate $\Delta'_{v_y}$.

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