2. Forced magnetic reconnection in Taylor’s model

Magnetic reconnection in a given system is conventionally categorized as spontaneous or forced. Spontaneous magnetic reconnection refers to the case in which the reconnection arises by some internal instability of the system or loss of equilibrium, with the most typical example being the tearing mode. Forced magnetic reconnection instead refers to the cases in which the reconnection is driven by some externally imposed flow or magnetic perturbation. In this case, one of the most important paradigms is the so-called ‘Taylor problem’, which consists in the study of the evolution of the magnetic reconnection process in a tearing-stable slab plasma equilibrium which is subject to a small amplitude boundary perturbation. This situation is depicted in figure 1, where the shared equilibrium magnetic field has the form

(2.1) $$\begin{eqnarray}\boldsymbol{B}=B_{z}\boldsymbol{e}_{z}+(x/L)B_{0}\boldsymbol{e}_{y},\end{eqnarray}$$

with $B_{z}$ , $B_{0}$ and $L$ as constants, and the perfectly conducting walls which bound the plasma are located at $x=\pm L$ . Magnetic reconnection is driven at the resonant surface $x=0$ by a deformation of the conducting walls such that

(2.2) $$\begin{eqnarray}x_{w}\rightarrow \pm L\mp {\it\Xi}_{0}\cos (ky),\end{eqnarray}$$

where $k=2{\rm\pi}/L_{y}$ is the perturbation wave number and ${\it\Xi}_{0}$ is a small ( $\ll L$ ) displacement amplitude. The boundary perturbation is assumed to be set up in a time scale that is long compared to the Alfvén time ${\it\tau}_{A}=L/v_{A}$ , with $v_{A}=B_{0}/\sqrt{4{\rm\pi}{\it\rho}}$ , but short compared to any characteristic reconnection time scale. Hence, the plasma can be considered in magnetostatic equilibrium everywhere except near the resonant surface at $x=0$ .

Figure 1. Geometry of the Taylor model. The equilibrium magnetic field component $B_{y}$ is sheared in the $x$ direction, being null at $x=0$ . The plasma is bounded by perfectly conducting walls at $x=\pm L$ , while it is periodic in the $y$ direction. Magnetic reconnection is driven at $x=0$ by the perturbation ${\it\Xi}_{0}\cos (ky)$ at the perfectly conducting walls.

The first and probably most important contribution to unveiling the behaviour of forced magnetic reconnection in Taylor’s model is due to Hahm & Kulsrud (Reference Hahm and Kulsrud1985), who showed that very small amplitude boundary perturbations cause an initial linear phase in which a current sheet builds up at the resonant surface, and successive phases in which the reconnection process evolves according to a linear resistive regime and a nonlinear Rutherford regime (Rutherford Reference Rutherford1973). The scenario discussed by Hahm & Kulsrud (Reference Hahm and Kulsrud1985), which is characterized by a very slow evolution of the reconnection process, was complemented some years later by Wang & Bhattacharjee (Reference Wang and Bhattacharjee1992a ), who showed that larger perturbations may foster reconnection to proceed through the nonlinear regime according to a Sweet–Parker-like evolution (Waelbroeck Reference Waelbroeck1989), which gives way to a Rutherford evolution only on the long time scale of resistive diffusion. The scenario outlined by Wang & Bhattacharjee (Reference Wang and Bhattacharjee1992a ) is characterized by a reconnection evolution faster than that presented by Hahm & Kulsrud (Reference Hahm and Kulsrud1985), but it could still be slow for very small values of plasma resistivity, since in both the Sweet–Parker-like (Waelbroeck Reference Waelbroeck1989) and Rutherford (Rutherford Reference Rutherford1973) regimes, the reconnection rate is strongly dependent on the resistivity, which is known to be extremely small in many laboratory fusion plasmas and space/astrophysical plasmas. However, recent works (Comisso et al. Reference Comisso, Grasso and Waelbroeck2014, Reference Comisso, Grasso and Waelbroeck2015) have shown that relatively large boundary perturbations lead to a different reconnection evolution in plasmas with small resistivity and viscosity. In these cases, after a linear inertial phase and an initial nonlinear regime characterized by a gradually evolving current sheet, the reconnection suddenly enters into a fast reconnection regime distinguished by the disruption of the current sheet due to the development of secondary magnetic islands (usually called plasmoids: see Biskamp Reference Biskamp2000 or Loureiro et al. Reference Loureiro, Schekochihin and Cowley2007).

In addition to the works discussed above, which adopt a magnetohydrodynamic (MHD) description of the plasma, we emphasize that many other efforts have been devoted to investigating the Taylor problem assuming MHD, two-fluid and kinetic descriptions (see Wang & Bhattacharjee Reference Wang and Bhattacharjee1992b ; Ma et al. Reference Ma, Wang and Bhattacharjee1996; Avinash et al. Reference Avinash, Bulanov, Esirkepov, Kaw, Pegoraro, Sasorov and Sen1998; Rem & Schep Reference Rem and Schep1998; Valori et al. Reference Valori, Grasso and de Blank2000; Fitzpatrick Reference Fitzpatrick2003; Fitzpatrick Reference Fitzpatrick2004a ,Reference Fitzpatrick b , Reference Fitzpatrick2008; Fitzpatrick et al. Reference Fitzpatrick, Bhattacharjee, Ma and Linde2003; Cole & Fitzpatrick Reference Cole and Fitzpatrick2004; Bian & Vekstein Reference Bian and Vekstein2005; Birn et al. Reference Birn, Galsgaard, Hesse, Hoshino, Huba, Lapenta, Pritchett, Schindler, Yin, Büchner, Neukirch and Priest2005; Vekstein & Bian Reference Vekstein and Bian2006; Birn & Hesse Reference Birn and Hesse2007; Hosseinpour & Vekstein Reference Hosseinpour and Vekstein2008; Gordovskyy et al. Reference Gordovskyy, Browning and Vekstein2010a ,Reference Gordovskyy, Browning and Vekstein b ; Lazzaro & Comisso Reference Lazzaro and Comisso2011; Dewar et al. Reference Dewar, Bhattacharjee, Kulsrud and Wright2013; Hosseinpour Reference Hosseinpour2013). Indeed, the Taylor problem has important applications besides being interesting from the point of view of basic physics. For instance, in laboratory fusion plasmas the Taylor model represents a convenient way to study magnetic reconnection processes driven by resonant magnetic perturbations, while in astrophysical plasmas this model can be adopted to the study of magnetic reconnection forced by the motions of photospheric flux tubes.

3. Evolution of the reconnection process in Taylor’s model

In this section we review the present understanding of the forced magnetic reconnection dynamics in Taylor’s model focusing on a visco-resistive plasma with $P_{m}$ greater than 1. As shown by Hahm & Kulsrud (Reference Hahm and Kulsrud1985), this dynamics always starts with a linear inertial phase in which a current sheet builds up at the resonant surface and shrinks inversely in time. Concurrently, the current density at the $X$ -point increases linearly in time. The reconnection rate during this phase can be evaluated by recalling that the current density is proportional to the out-of-plane electric field at the $X$ -point, which is equal to

(3.1) $$\begin{eqnarray}\left.\partial _{t}{\it\psi}\right|_{X}=\frac{2}{{\rm\pi}}{\it\Delta}_{s}^{\prime }kL^{2}B_{0}{\it\Xi}_{0}\frac{t}{{\it\tau}_{A}{\it\tau}_{{\it\eta}}}\end{eqnarray}$$

for $t\ll {\it\tau}_{{\it\nu}}^{1/3}({\it\tau}_{A}/kL)^{2/3}$ (Fitzpatrick Reference Fitzpatrick2003; Comisso et al. Reference Comisso, Grasso and Waelbroeck2015). Here, ${\it\psi}$ stands for the magnetic flux function of the perturbed magnetic field in the reconnection plane ( ${\it\delta}\boldsymbol{B}_{\bot }=\boldsymbol{{\rm\nabla}}{\it\psi}\times \boldsymbol{e}_{z}$ ), ${\it\tau}_{{\it\nu}}=L^{2}/{\it\nu}$ and ${\it\tau}_{{\it\eta}}=L^{2}/D_{{\it\eta}}$ indicate the characteristic time for viscous and resistive diffusion, respectively, while ${\it\Delta}_{s}^{\prime }=2k/\sinh \,(kL)$ parametrizes the contribution of the external source perturbation to the gradient discontinuity of the magnetic flux function at the resonant surface. It is important to point out that this phase is characterized by a non-constant- ${\it\psi}$ behaviour of the magnetic flux function across the island. Depending on whether or not this property persists until the beginning of the nonlinear regime, different scenarios may occur.

3.1. Hahm–Kulsrud scenario

If the boundary perturbation is such that (Fitzpatrick Reference Fitzpatrick2003; Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.2) $$\begin{eqnarray}{\it\Psi}_{0}=B_{0}{\it\Xi}_{0}\ll \left({\it\tau}_{{\it\nu}}{\it\tau}_{{\it\eta}}\right)^{-1/6}\left(\frac{{\it\tau}_{A}}{kL}\right)^{1/3}\frac{B_{0}}{{\it\Delta}_{s}^{\prime }}\equiv {\it\Psi}_{W},\end{eqnarray}$$

after the inertial phase the reconnection process evolves through a visco-resistive phase, which is a linear regime characterized by a constant- ${\it\psi}$ behaviour, i.e. the perturbed magnetic flux function can be treated as a constant in $x$ over the width of the reconnection layer. During this phase the reconnection rate is given by

(3.3) $$\begin{eqnarray}\left.\partial _{t}{\it\psi}\right|_{X}=B_{0}{\it\Xi}_{0}\frac{{\it\Delta}_{s}^{\prime }L}{{\it\tau}_{\ast }}\text{e}^{{\it\Delta}_{0}^{\prime }Lt/{\it\tau}_{\ast }},\end{eqnarray}$$

where ${\it\Delta}_{0}^{\prime }=2k/\tanh \,(kL)$ is the standard tearing stability parameter and ${\it\tau}_{\ast }$ is a characteristic time defined as (Fitzpatrick Reference Fitzpatrick2003; Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.4) $$\begin{eqnarray}{\it\tau}_{\ast }\equiv {\rm\pi}6^{2/3}\frac{{\it\Gamma}\left(\frac{5}{6}\right)}{{\it\Gamma}\left(\frac{1}{6}\right)}\frac{{\it\tau}_{{\it\eta}}^{5/6}}{{\it\tau}_{{\it\nu}}^{1/6}}\left(\frac{{\it\tau}_{A}}{kL}\right)^{1/3},\end{eqnarray}$$

with ${\it\Gamma}$ indicating the Gamma function. Equation (3.3) is valid for $t\gg {\it\tau}_{{\it\nu}}^{-1/3}{\it\tau}_{{\it\eta}}^{2/3}$ $({\it\tau}_{A}/kL)^{2/3}$ (Fitzpatrick Reference Fitzpatrick2003; Comisso et al. Reference Comisso, Grasso and Waelbroeck2015) and a magnetic island width much smaller than the linear layer width, i.e. $w\ll {\it\delta}_{{\it\nu}{\it\eta}}\sim ({\it\tau}_{{\it\nu}}{\it\tau}_{{\it\eta}})^{-1/6}({\it\tau}_{A}/kL)^{1/3}L$ (Porcelli Reference Porcelli1987; Fitzpatrick Reference Fitzpatrick1993). If the perturbation is sufficient to drive the magnetic island into the nonlinear regime ( $w\gtrsim {\it\delta}_{{\it\nu}{\it\eta}}$ ), the visco-resistive phase ends up in a Rutherford evolution, whose island width growth is governed by the Rutherford equation

(3.5) $$\begin{eqnarray}{\mathcal{I}}\frac{{\it\tau}_{{\it\eta}}}{L^{2}}\frac{\text{d}w}{\text{d}t}={\it\Delta}_{0}^{\prime }+{\it\Delta}_{s}^{\prime }\frac{{\it\Psi}_{0}}{{\it\psi}|_{X}},\end{eqnarray}$$

where ${\mathcal{I}}=0.823$ (Rutherford Reference Rutherford1973; Fitzpatrick Reference Fitzpatrick1993). This is a very slow reconnection evolution in which the reconnection rate can be evaluated analytically in the two limits (Hahm & Kulsrud Reference Hahm and Kulsrud1985; Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.6) $$\begin{eqnarray}\displaystyle \left.\partial _{t}{\it\psi}\right|_{X}=\frac{2{\it\Delta}_{s}^{\prime }{\it\Psi}_{0}}{(-{\it\Delta}_{0}^{\prime }){\it\tau}_{NL}}\left(\frac{3t}{{\it\tau}_{NL}}\right)^{-1/3}\quad \text{for }t\ll {\it\tau}_{NL}, & & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle \left.\partial _{t}{\it\psi}\right|_{X}=\frac{2{\it\Delta}_{s}^{\prime }{\it\Psi}_{0}}{(-{\it\Delta}_{0}^{\prime }){\it\tau}_{NL}}\tanh \left(\frac{t}{{\it\tau}_{NL}}\right)\cosh ^{-2}\left(\frac{t}{{\it\tau}_{NL}}\right)\quad \text{for }t\gg {\it\tau}_{NL}, & & \displaystyle\end{eqnarray}$$

where

(3.8) $$\begin{eqnarray}{\it\tau}_{NL}=\frac{4{\mathcal{I}}}{(-{\it\Delta}_{0}^{\prime })L}\left(\frac{{\it\Delta}_{s}^{\prime }}{(-{\it\Delta}_{0}^{\prime })}\frac{{\it\Xi}_{0}}{L}\right)^{1/2}{\it\tau}_{{\it\eta}}.\end{eqnarray}$$

3.2. Wang–Bhattacharjee scenario

If the boundary perturbation is such that (Fitzpatrick Reference Fitzpatrick2003; Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.9) $$\begin{eqnarray}{\it\Psi}_{0}\gtrsim {\it\Psi}_{W},\end{eqnarray}$$

the non-constant- ${\it\psi}$ behaviour characteristic of the inertial phase lingers until the nonlinear regime is entered. Therefore, since in this case the magnetic island grows faster than the current can diffuse out of the reconnecting layer, the evolution of the reconnection process is distinguished by a strong current sheet at the resonant surface (Waelbroeck Reference Waelbroeck1989). The reconnecting current sheet turns out to be stable if the boundary perturbation is such that (Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.10) $$\begin{eqnarray}{\it\Psi}_{0}=B_{0}{\it\Xi}_{0}<CB_{0}L\frac{k}{{\it\Delta}_{s}^{\prime }}\frac{{\it\tau}_{A}}{{\it\tau}_{{\it\eta}}}\left(1+\frac{{\it\tau}_{{\it\eta}}}{{\it\tau}_{{\it\nu}}}\right)^{1/2}\equiv {\it\Psi}_{c},\end{eqnarray}$$

where the multiplicative constant $C\sim 2{\it\epsilon}_{c}^{-2}$ depends on the critical inverse aspect ratio of the reconnecting current sheet (specified later). In this case, the reconnection process follows a Sweet–Parker evolution (modified by plasma viscosity: Park et al. Reference Park, Monticello and White1984), whose reconnection rate in Taylor’s model is (Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.11) $$\begin{eqnarray}\left.\partial _{t}{\it\psi}\right|_{X}\approx \frac{1}{3}B_{0}L\left({\it\Delta}_{s}^{\prime }{\it\Xi}_{0}\right)^{3/2}\left(\frac{kL}{{\it\tau}_{A}{\it\tau}_{{\it\eta}}}\right)^{1/2}\left(1+\frac{{\it\tau}_{{\it\eta}}}{{\it\tau}_{{\it\nu}}}\right)^{-1/4}.\end{eqnarray}$$

Finally, the Sweet–Parker type of evolution gives way to a Rutherford evolution on the time scale of resistive diffusion.

3.3. Our scenario

If the boundary perturbation satisfies (Fitzpatrick Reference Fitzpatrick2003; Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.12) $$\begin{eqnarray}{\it\Psi}_{0}\gtrsim \left({\it\tau}_{{\it\nu}}{\it\tau}_{{\it\eta}}\right)^{-1/6}\left(\frac{{\it\tau}_{A}}{kL}\right)^{1/3}\frac{B_{0}}{{\it\Delta}_{s}^{\prime }}\equiv {\it\Psi}_{W}\end{eqnarray}$$

and also the condition (Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.13) $$\begin{eqnarray}{\it\Psi}_{0}>CB_{0}L\frac{k}{{\it\Delta}_{s}^{\prime }}\frac{{\it\tau}_{A}}{{\it\tau}_{{\it\eta}}}\left(1+\frac{{\it\tau}_{{\it\eta}}}{{\it\tau}_{{\it\nu}}}\right)^{1/2}\equiv {\it\Psi}_{c},\end{eqnarray}$$

the reconnection process does not reach a stable Sweet–Parker regime, but a different situation occurs. A gradually thinning current sheet evolves until its aspect ratio reaches the limit that allows the plasmoid instability to develop. The growth of the plasmoids leads to the disruption of the current sheet, and therefore to a dramatic increase in the reconnection rate. The reconnection rate during this plasmoid-dominated phase has been evaluated in a statistical steady state as (Comisso et al. Reference Comisso, Grasso and Waelbroeck2015)

(3.14) $$\begin{eqnarray}\partial _{t}{\it\psi}_{p}\approx {\it\epsilon}_{c}B_{0}L\left({\it\Delta}_{s}^{\prime }{\it\Xi}_{0}\right)^{2}{\it\tau}_{A}^{-1}\left(1+\frac{{\it\tau}_{{\it\eta}}}{{\it\tau}_{{\it\nu}}}\right)^{-1/2},\end{eqnarray}$$

where ${\it\epsilon}_{c}={\it\delta}_{c}/L_{c}$ is the critical inverse aspect ratio of the reconnecting current sheet. This quantity, whose value has been found to lie in the range $1/100$ – $1/200$ by means of numerical simulations (Bhattacharjee et al. Reference Bhattacharjee, Huang, Yang and Rogers2009; Cassak et al. Reference Cassak, Shay and Drake2009; Samtaney et al. Reference Samtaney, Loureiro, Uzdensky, Schekochihin and Cowley2009; Huang & Bhattacharjee Reference Huang and Bhattacharjee2010; Skender & Lapenta Reference Skender and Lapenta2010), represents the threshold below which the reconnecting current sheet becomes unstable to the plasmoid instability (Loureiro et al. Reference Loureiro, Schekochihin and Cowley2007). This threshold condition as a function of the Lundquist and magnetic Prandtl numbers has been recently discussed by Loureiro et al. (Reference Loureiro, Schekochihin and Uzdensky2013), Comisso et al. (Reference Comisso, Grasso and Waelbroeck2015) and Tenerani et al. (Reference Tenerani, Rappazzo, Velli and Pucci2015). In particular, it has been shown (Comisso et al. Reference Comisso, Grasso and Waelbroeck2015) that a reconnecting current sheet becomes unstable when the Lundquist number based on its length exceeds the critical value $S_{c}\approx {\it\epsilon}_{c}^{-2}\left(1+{\it\tau}_{{\it\nu}}/{\it\tau}_{{\it\eta}}\right)^{1/2}$ .

4. Phase diagrams

In this section we illustrate the domain of existence of the three different scenarios described above, with the help of appropriate parameter space maps. For the sake of clarity we restate the three types of reconnection evolutions we are referring to:

1. (1) the Hahm–Kulsrud scenario (Hahm & Kulsrud Reference Hahm and Kulsrud1985; Fitzpatrick Reference Fitzpatrick2003);

2. (2) the Wang–Bhattacharjee scenario (Wang & Bhattacharjee Reference Wang and Bhattacharjee1992a ; Fitzpatrick Reference Fitzpatrick2003);

3. (3) our scenario (Comisso et al. Reference Comisso, Grasso and Waelbroeck2015).

Since each of these scenarios includes different phases/regimes of reconnection, the concept of ‘phase diagrams’ is intended here in a broader sense. Due to this fact, they could also be defined in a more general way as ‘scenario diagrams’. This type of diagrams can be constructed from the conditions summarized in the previous section. Therefore, the possible evolutions of the reconnection process may be organized in a four-dimensional parameter space map with $\hat{{\it\Psi}}_{0}={\it\Psi}_{0}/B_{0}L$ , $\hat{k}=kL$ , $S=Lv_{A}/D_{{\it\eta}}$ , and $P_{m}={\it\nu}/D_{{\it\eta}}$ on the four axes. However, due to the difficulty in visualizing such a four-dimensional diagram, it is convenient to consider two-dimensional slices for fixed values of two of the four parameters.

Figure 2. Two-dimensional slices of a phase/scenario diagram for forced magnetic reconnection in the magnetohydrodynamical Taylor model. Fixed parameters are (a) $\hat{k}=1/8$ , $P_{m}=5$ , (b) $\hat{k}=1/8$ , $P_{m}=500$ , (c) $\hat{k}=2$ , $P_{m}=5$ , and (d) $\hat{k}=2$ , $P_{m}=500$ . The numerical labels indicate (1) the Hahm–Kulsrud scenario, (2) the Wang–Bhattacharjee scenario, and (3) our scenario. The boundaries between the different scenarios are identified by the functions $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{W}/3$ and $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{c}$ for $\hat{{\it\Psi}}_{c}>\hat{{\it\Psi}}_{W}/3$ .

Figure 3. (a) Thresholds $\hat{{\it\Psi}}_{W}/3$ (red line) and $\hat{{\it\Psi}}_{c}$ (blue line) as a function of the magnetic Prandtl number $P_{m}$ for $S=10^{8}$ , $\hat{k}=0.5$ and $C=2(150)^{2}$ . (b) Corresponding two-dimensional slice of the phase/scenario diagram identifying (1) the Hahm–Kulsrud scenario, (2) the Wang–Bhattacharjee scenario, and (3) our scenario.

Figure 4. Boundaries (identified by the functions $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{W}/3$ and $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{c}$ for $\hat{{\it\Psi}}_{c}>\hat{{\it\Psi}}_{W}/3$ ) of the different possible evolutions of the reconnection process for $\hat{k}=0.5$ and various values of the magnetic Prandtl number.

Let us first consider four two-dimensional slices with fixed values of the magnetic Prandtl number and perturbation wave number. Assuming that the Hahm–Kulsrud scenario (which occurs if ${\it\Psi}_{0}\ll {\it\Psi}_{W}$ ) holds until ${\it\Psi}_{0}={\it\Psi}_{W}/3$ , the corresponding diagrams for (a)  $\hat{k}=1/8$ , $P_{m}=5$ , (b)  $\hat{k}=1/8$ , $P_{m}=500$ , (c)  $\hat{k}=2$ , $P_{m}=5$ , and (d)  $\hat{k}=2$ , $P_{m}=500$ are shown in figure 2(a–d). From these plots it is clear that the Wang–Bhattacharjee scenario is limited to a small range of values of the Lundquist number and the source perturbation amplitude. Increasing values of the magnetic Prandtl number and perturbation wave number extend the domain of existence of this possible type of evolution of the system. However, after a threshold value of the Lundquist number (identified by the intersection of the two black lines representing $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{W}/3$ and $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{c}$ ), the Wang–Bhattacharjee scenario cannot occur because it is not possible to obtain a stable Sweet–Parker-type evolution. In these cases the Hahm–Kulsrud scenario is facilitated by very small perturbation amplitudes, whereas larger perturbations lead the system to a fast reconnection regime as described in § 3.3. Note that while previously proposed phase diagrams always predict fast reconnection (Huang et al. Reference Huang, Bhattacharjee and Sullivan2011; Ji & Daughton Reference Ji and Daughton2011; Daughton & Roytershteyn Reference Daughton and Roytershteyn2012; Cassak & Drake Reference Cassak and Drake2013; Huang & Bhattacharjee Reference Huang and Bhattacharjee2013; Karimabadi & Lazarian Reference Karimabadi and Lazarian2013), in clear contrast to what happens in nature, our diagrams show that reconnection proceeds very slowly (region (1)) if the source perturbation is not sufficiently large.

Let us now examine the effect of the plasma viscosity by considering the domain of existence of the different scenarios as a function of the parameters $\hat{{\it\Psi}}_{0}$ and $P_{m}$ . Figure 3(a) shows the functions $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{W}/3$ and $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{c}$ for $S=10^{8}$ and $\hat{k}=0.5$ . For $\hat{{\it\Psi}}_{c}<\hat{{\it\Psi}}_{W}/3$ the threshold for plasmoid formation coincides with that for the nonlinear evolution characterized by a strong reconnecting current sheet. Therefore, for $\hat{{\it\Psi}}_{c}<\hat{{\it\Psi}}_{W}/3$ an increase in the amplitude perturbation $\hat{{\it\Psi}}_{0}$ drives the system directly from scenario (1) to scenario (3). This situation is depicted in figure 3(b), where it is clearly shown that the increase of the magnetic Prandtl number has the effect of making possible or extending the domain of existence of scenario (2), as recently pointed out in Tenerani et al. (Reference Tenerani, Rappazzo, Velli and Pucci2015) and Comisso et al. (Reference Comisso, Grasso and Waelbroeck2015).

To clarify the effect of the plasma viscosity we also delineate the boundaries of the diverse evolutions (1)–(3) in a parameter space map $(\hat{{\it\Psi}}_{0},S)$ (as in figure 2) for fixed $\hat{k}=0.5$ but different values of $P_{m}$ . This is shown in figure 4 for $P_{m}=5-5\times 10^{5}$ . The increase of the magnetic Prandtl number extends the domain of existence of the slow reconnection scenario (1) at the expense of the fast reconnection scenario (3). The area of existence of scenario (2) remains almost unchanged, but shifted towards higher values of the Lundquist number.

Figure 5. (a) Thresholds $\hat{{\it\Psi}}_{W}/3$ (red line) and $\hat{{\it\Psi}}_{c}$ (blue line) as a function of the perturbation wave number $\hat{k}$ for $S=10^{8}$ , $P_{m}=5$ and $C=2(150)^{2}$ . (b) Corresponding two-dimensional slice of the phase/scenario diagram identifying (1) the Hahm–Kulsrud scenario, (2) the Wang–Bhattacharjee scenario, and (3) our scenario.

Figure 6. Boundaries (identified by the functions $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{W}/3$ and $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{c}$ for $\hat{{\it\Psi}}_{c}>\hat{{\it\Psi}}_{W}/3$ ) of the different possible evolutions of the reconnection process for $P_{m}=5$ and various values of the perturbation wave number.

We now examine in more detail how the possible evolutions of the forced magnetic reconnection process depend on the wave number of the boundary perturbation. Figure 5(a) shows the thresholds $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{W}/3$ and $\hat{{\it\Psi}}_{0}=\hat{{\it\Psi}}_{c}$ as a function of $\hat{k}$ for fixed values of $S=10^{8}$ and $P_{m}=5$ . Below a critical perturbation wave number $\hat{k}^{\ast }$ (corresponding to $\hat{k}^{\ast }\approx 1$ for the fixed parameters used in figure 5 a), every time the non-constant- ${\it\psi}$ magnetic island passes into the nonlinear regime, the evolution of the system leads to the plasmoid-dominated phase predicted in scenario (3). The domains of existence of the possible evolutions (1)–(3) are illustrated in figure 5(b). The scenario discussed by Wang and Bhattacharjee happens only for a small range of ( $\hat{{\it\Psi}}_{0},\hat{k}$ ) parameters. Note also that scenario (3) is facilitated for $\hat{k}\lesssim \hat{k}^{\ast }$ , while scenario (1) may occur for large amplitude boundary perturbations if $\hat{k}\gg \hat{k}^{\ast }$ .

To better evaluate the effects of $\hat{k}$ on the possible evolutions of the reconnection process, we plot in figure 6 the boundaries between scenarios (1)–(3) in a parameter space map $(\hat{{\it\Psi}}_{0},S)$ (as in figures 2 and 4) for fixed $P_{m}=5$ but different values of $\hat{k}$ . The maximum area of existence of scenario (2) occurs for $\hat{k}\sim 1$ , while for $\hat{k}\ll 1$ and $\hat{k}\gg 1$ , scenario (2) appears for a very limited range of ( $\hat{{\it\Psi}}_{0},\hat{k}$ ) parameters. Note also that scenario (1) is greatly facilitated in the case of very large perturbation wave numbers ( $\hat{k}\gg 1$ ), while scenario (3) is facilitated by relatively large amplitude perturbations with $\hat{k}\lesssim 1$ .