A THEORETICAL MODEL OF A THINNING CURRENT SHEET IN THE LOW-β PLASMAS

It has been known that magnetic reconnection plays important roles in many energetic explosions such as solar flares (Zweibel & Yamada 2009; Shibata & Magara 2011). The knowledge about solar flares has been applied to other explosive phenomena, such as superflares on solar-type stars (Maehara et al. 2012) and soft gamma repeaters (Lyutikov 2006; Masada et al. 2010; Meng et al. 2014). In a classical standard model of solar flares, the magnetic energy is rapidly converted to the thermal and kinetic energies of plasmas through magnetic reconnection (Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976; Shibata et al. 1995; Shibata & Magara 2011). Such rapid reconnection requires a much shorter current sheet width than its length (Yamada et al. 2010). Therefore, how thin current sheets are established is a central issue for understanding the origin of explosive phenomena.

Magnetohydrodynamics (MHD) processes to form a thin current sheet have been theoretically studied by many authors. In Forbes & Priest (1995) and Lin & Forbes (2000), they numerically and analytically investigated the formation of a current sheet below an erupting flux rope during a solar flare. There are many observational supports of the formation and the evolution of a current sheet between a cusp-shaped flaring loop and an erupting flux rope (Liu et al. 2010). The formation of a current sheet between an emerging flux and a preexisting magnetic field in the solar corona has been numerically studied (Forbes & Priest 1984; Shibata et al. 1992; Yokoyama & Shibata 1996). In McLaughlin et al. (2009), the collapse of a null point to a current sheet triggered by MHD fast-mode waves was investigated.

A long and thin current sheet can form many fine-scale current sheets inside it through some MHD instabilities. Such a current sheet can be fragmented to form magnetic islands or plasmoids through the tearing or plasmoid instabilities (Furth et al. 1963; Loureiro et al. 2007). The plasmoids are pulled against each other by a Lorentz force. When plasmoids collide with each other, new current sheets will be formed between them. This process can operate at multi-spatial scale, depending on the Lundquist number (Tajima et al. 1987; Shibata & Tanuma 2001; Bárta et al. 2011). Observational support from the direct imaging observation of the plasmoid formation in a solar flare was given by Takasao et al. (2012) for the first time. They also found that the plasma blobs (possibly plasmoids) in the current sheet collided or merged with each other before they were ejected from the current sheet. The pinching process of a current sheet by a Lorentz force like the last two processes above is the main subject of this paper.

In this study, we investigated the implosion process in which current sheets are pinched by the magnetic pressure gradient. In a previous study, the self-similar solution in the implosion process was analytically derived under the assumption that the gas pressure can be ignored (Imshennik & Syrovatski 1967; Bulanov & Olshanetskii 1984; Tajima et al. 1987). Forbes (1982) investigated the implosion process, including the gas pressure, and found the generation of two MHD fast-mode shocks as a result of a significant increase in the gas pressure at the center of the current sheet. However, in that simulation, the current sheet was incompletely pinched because of the low spatial resolution. To study the pinching process of the thin current sheet formed behind the shocks in detail, we performed a series of numerical simulations with a sufficient spatial resolution.

In Section 2, the self-similar solution proposed by Tajima et al. (1987) is introduced. In Section 3, we show the basic equations and initial conditions of the numerical simulations. In Section 4, we show the results of the numerical simulations and the power-law scalings of the physical quantities in the pinched current sheets. In Section 5, we analytically derive the power-law relation shown in Section 4. Finally, we summarize our conclusions in Section 5.

The situation that we consider is similar to that of Tajima et al. (1987), where a current sheet is pinched by a Lorentz force (see Figure 1). They derived a self-similar solution using the two-fluid equations and Maxwell's equations. Here we derive the same self-similar solution from 1D MHD equations to review the basic physics and assumptions of the solution. We start from the basic 1D MHD equations:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\rho =-\displaystyle \frac{\partial }{\partial x}\left(\rho {v}_{x}\right),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\rho {v}_{x}\right)=-\displaystyle \frac{\partial }{\partial x}\left(p+\rho {v}^{2}\right)-\displaystyle \frac{{j}_{z}{B}_{y}}{c},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}{B}_{y}=-\displaystyle \frac{\partial }{\partial x}\left({v}_{x}{B}_{y}\right),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{j}_{z}=\displaystyle \frac{c}{4\pi }\displaystyle \frac{\partial }{\partial x}{B}_{y}.\end{eqnarray} \tag{ 4 }$$

Now we introduce dimensionless physical quantities as below:

$$\begin{eqnarray}{x}^{*} & \equiv & x/L,\\ {t}^{*} & \equiv & t/{\tau }_{{\rm{A}}},\\ {\rho }^{*}({t}^{*}) & \equiv & \rho (t)/{\rho }_{0},\\ {p}^{*}({x}^{*},{t}^{*}) & \equiv & p(x,t)/{p}_{0},\\ {B}_{y}^{*}({x}^{*},{t}^{*}) & \equiv & {B}_{y}(x,t)/{B}_{0},\\ {v}^{*}({x}^{*},{t}^{*}) & \equiv & {v}_{x}(x,t)/{V}_{{\rm{A}}},\end{eqnarray}$$

where L is the initial current sheet thickness, V_A is the initial Alfvén speed, ${\tau }_{{\rm{A}}}\equiv L/{V}_{{\rm{A}}}$ is an Alfvén timescale, and the initial values of the physical quantities are denoted by 0. The dimensionless equations can be written as

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial {t}^{*}}{\rho }^{*}=-\displaystyle \frac{\partial }{\partial {x}^{*}}({\rho }^{*}{v}^{*}),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial {t}^{*}}({\rho }^{*}{v}^{*})=-{\beta }_{0}\displaystyle \frac{\partial }{\partial {x}^{*}}{p}^{*}-\displaystyle \frac{\partial }{\partial {x}^{*}}({\rho }^{*}{v}^{*2})-\displaystyle \frac{\partial }{\partial {x}^{*}}\left(\displaystyle \frac{{B}_{y}^{*2}}{2}\right),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial {t}^{*}}{B}_{y}^{*}=-\displaystyle \frac{\partial }{\partial {x}^{*}}({v}^{*}{B}_{y}^{*}),\end{eqnarray} \tag{ 7 }$$

where ${\beta }_{0}={p}_{0}/({B}_{0}^{2}/2)$ is the initial plasma beta.

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We introduce a scale factor a(t) as follows

$$\begin{eqnarray}&&{v}^{*}=\displaystyle \frac{\dot{a}({t}^{*})}{a({t}^{*})}{x}^{*},\end{eqnarray} \tag{ 8 }$$

where a dot represents the time derivative. An ansatz is that the velocity is linear in $x*$. For simplicity, we assume that ${\rho }^{*}$, the density in the current sheet, is spatially uniform. From Equations (5) and (8), we obtain

$$\begin{eqnarray}&&{\rho }^{*}=\displaystyle \frac{1}{a({t}^{*})}.\end{eqnarray} \tag{ 9 }$$

We also assume that ${B}_{y}^{*}$ is linear in x^*: ${B}_{y}^{*}({x}^{*},{t}^{*})={B}^{*}({t}^{*}){x}^{*}$. Then from Equations (7) and (8), we obtain

$$\begin{eqnarray}&&{B}^{*}=\displaystyle \frac{1}{a{({t}^{*})}^{2}}.\end{eqnarray} \tag{ 10 }$$

We take the following form of the scaling factor

$$\begin{eqnarray}&&a({t}^{*})\propto {\left(\displaystyle \frac{{t}_{0}-t}{{t}_{0}}\right)}^{k}={\left(1-\displaystyle \frac{{t}^{*}}{{t}_{0}^{*}}\right)}^{k},\end{eqnarray} \tag{ 11 }$$

where t₀ is the explosion time when the thickness of the current sheet goes to 0. Neglecting the gas pressure term in Equation (6), we obtain the index k and explosion time t₀ from Equations (6) and (11)

$$\begin{eqnarray}&&a({t}^{*})\propto {\left(1-\displaystyle \frac{{t}^{*}}{{t}_{0}^{*}}\right)}^{2/3},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{t}_{0}^{*}=\displaystyle \frac{\sqrt{2}}{3}.\end{eqnarray} \tag{ 13 }$$

From Equation (12), we obtain the following expressions:

$$\begin{eqnarray}&&{v}^{*}({x}^{*},{t}^{*})\propto {\left(1-\displaystyle \frac{{t}^{*}}{{t}_{0}^{*}}\right)}^{-1}{x}^{*},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\rho }^{*}({t}^{*})\propto {\left(1-\displaystyle \frac{{t}^{*}}{{t}_{0}^{*}}\right)}^{-2/3},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{B}_{y}^{*}({x}^{*},{t}^{*})\propto {\left(1-\displaystyle \frac{{t}^{*}}{{t}_{0}^{*}}\right)}^{-4/3}{x}^{*}.\end{eqnarray} \tag{ 16 }$$

Note that these scaling laws are the same as those originally obtained by Imshennik & Syrovatski (1967).

If the plasma adiabatically evolves, we obtain the gas pressure in the following form:

$$\begin{eqnarray}&&{p}^{*}({t}^{*})\propto {\left(1-\displaystyle \frac{{t}^{*}}{{t}_{0}^{*}}\right)}^{-10/9}.\end{eqnarray} \tag{ 17 }$$

This indicates that the gas pressure at the center should explode as $t\to {t}_{0}$. Therefore, the pinching by a Lorentz force must stop, which cannot be described by the linear theory. To study the evolution near and after the explosion time, we numerically investigated the implosion of a current sheet using 1D MHD simulations.

All the physical quantities are functions of x and t, and a magnetic field is considered only in the y-direction (B_y). The basic equations are as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+\displaystyle \frac{\partial }{\partial x}\left(\rho {v}_{x}\right)=0,\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\rho {v}_{x}\right)+\displaystyle \frac{\partial }{\partial x}\left(\rho {v}_{x}^{2}+p-\displaystyle \frac{{B}_{y}^{2}}{8\pi }\right)=0,\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {B}_{y}}{\partial t}+\displaystyle \frac{\partial }{\partial x}\left({v}_{y}{B}_{x}-{B}_{y}{v}_{x}\right)=0,\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial e}{\partial t}+\displaystyle \frac{\partial }{\partial x}\left[\left(e+p+\displaystyle \frac{{B}_{y}^{2}}{8\pi }\right){v}_{x}\right]=0,\end{eqnarray} \tag{ 21 }$$

where the total energy density e is defined as

$$\begin{eqnarray}&&e\equiv \displaystyle \frac{p}{\gamma -1}+\displaystyle \frac{\rho {v}_{x}^{2}}{2}+\displaystyle \frac{{B}_{y}^{2}}{8\pi }.\end{eqnarray} \tag{ 22 }$$

We adopted the self-similar solution of Tajima et al. (1987) as the initial condition of the numerical simulations. The initial condition is not an MHD equilibrium, and after the simulation's start, the current sheet is pinched by a Lorentz force as shown in Figure 1. The calculation domain is in the range of $0\leqslant x/L\leqslant 2$, where L is the initial width of the current sheet. An antiparallel magnetic field is given by

$$\begin{eqnarray}&&{B}_{y}(x)={B}_{0}\mathrm{tanh}\left(\displaystyle \frac{x}{L}\right),\end{eqnarray} \tag{ 23 }$$

where B₀ is defined as $| {B}_{y}(\pm L)|$. For simplicity, we set the initial gas pressure p and the initial electron density ρ to be spatially uniform. The initial gas pressure is normalized by the initial magnetic pressure ${B}_{0}^{2}/8\pi$. The initial velocity field is given by

$$\begin{eqnarray}&&{v}_{x}(x)={v}_{0}\mathrm{tanh}\left(\displaystyle \frac{x}{L}\right).\end{eqnarray} \tag{ 24 }$$

In our numerical simulations, the total grid number is fixed to 40,000, which is adequately larger than 512 in Forbes (1982). The numerical scheme we adopted is based on the HLLD scheme, which is a fully shock-capturing scheme (Miyoshi & Kusano 2005).

4.1. Time Evolution of a Current Sheet in Numerical Simulations

As a typical case, here we show the time evolution of the case with ${\beta }_{0}={10}^{-2}$ (see Figure 2). During $t\lt {t}_{0}$, the self-similar evolution predicted in Tajima et al. (1987) is confirmed with plasma gas pressure (see Figure 3). At $t\sim {t}_{0}$, the implosion leads to the formation of a shock (MHD fast-mode shock) as a result of a significant increase in the gas pressure at the center of the current sheet. As shown in Figure 2, the compressed plasma is continuously left behind the shock as the shock propagates outward. Note that $p/{\rho }^{\gamma }$ is almost kept constant at the center of the current sheet, which indicates that the physical quantities there adiabatically evolve. To investigate the implosion process with the gas pressure, we performed the numerical simulations with various values of initial gas pressure and the spatial constant value of initial velocity. As a result of numerical simulations, we found the power-law behavior of physical quantities in the pinched current sheet behind the shock. As shown in the next section, we were able to analytically derive the scaling law with respect to β₀.

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4.2.  β₀ Dependence of Physical Quantities the in the Current Sheet

We investigate the β₀ dependence of the physical quantities in the current sheet. The range of β₀ is $-3\leqslant \mathrm{log}{\beta }_{0}\leqslant -1$. The β₀ dependence of the density, pressure, and magnetic field strength just behind the shock is shown in Figure 4. Figure 4 implies the following power-law relations:

$$\begin{eqnarray}&&\rho /{\rho }_{0}\propto {\beta }_{0}^{-0.58},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&p/{p}_{0}\propto {\beta }_{0}^{-0.97},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{B}_{y}/{B}_{y0}\sim \mathrm{const}.\end{eqnarray} \tag{ 27 }$$

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We also investigated the β₀ dependence of the thickness of a current sheet, ${L}_{\mathrm{min}}$. We define ${L}_{\mathrm{min}}$ as follows

$$\begin{eqnarray}&&L(t)=\displaystyle \frac{{B}_{\mathrm{max}}(t)}{{J}_{\mathrm{max}}(t)},\end{eqnarray} \tag{ 28 }$$

where B_max(t) and J_max(t) are, respectively,

$$\begin{eqnarray}{B}_{\mathrm{max}} & \equiv & \mathrm{max}\left({B}_{y}(x,t)\right),\\ {J}_{\mathrm{max}} & \equiv & \mathrm{max}\left({J}_{z}(x,t)\right).\end{eqnarray}$$

As a result of the numerical simulations, we found a scaling law of ${L}_{\mathrm{min}}$ (see Figure 5) as follows:

$$\begin{eqnarray}&&{L}_{\mathrm{min}}\propto {\beta }_{0}^{1.177}.\end{eqnarray} \tag{ 29 }$$

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To study the formation of the pinched current sheet in detail, we tracked selected Lagrangian particles that are initially located in the initial current sheet. Figure 6 shows the trajectories of the Lagrangian particles on the time–distance diagrams of ${v}_{x},{\mathrm{log}}_{10}p$, and B_y. From Figure 6, we found that the pinched current sheet is formed from a part of the initial current sheet, not from the whole of it.

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In this section, we aim to derive the β₀ dependence of the physical quantities in the pinched current sheet. As shown in Section 4, we confirmed the pinching process by the Lorentz force and the propagation of the MHD fast-mode shocks, as shown in Forbes (1982). Behind the fast-mode shock, the physical quantities are determined by the Rankine–Hugoniot equations

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{x2}}{{v}_{x1}}=\displaystyle \frac{1}{X},\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{y2}}{{B}_{y1}}=X,\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{p}_{2}}{{p}_{1}}=\gamma {\bar{{M}_{1}}}^{2}\left(1-\displaystyle \frac{1}{X}\right)+\displaystyle \frac{1-{X}^{2}}{{\beta }_{1}}+1,\end{eqnarray} \tag{ 32 }$$

where the physical quantities in front of and behind the shock are denoted by 1 and 2, respectively, $\bar{{M}_{1}}={v}_{x1}/{c}_{s1}$ is the Mach number, and $X={\rho }_{2}/{\rho }_{1}$ is the density ratio, which is the positive solution of

$$\begin{eqnarray}&&2(2-\gamma ){X}^{2}+\left[2{\beta }_{1}+(\gamma -1){\beta }_{1}{\bar{{M}_{1}}}^{2}+2\right]\gamma X\\ &&\quad -\gamma (\gamma +1){\beta }_{1}{\bar{{M}_{1}}}^{2}=0.\end{eqnarray} \tag{ 33 }$$

From Equations (31) and (33), the compression ratio is limited in the range of

$$\begin{eqnarray}&&1\lt \displaystyle \frac{{B}_{2}}{{B}_{1}}\lt \displaystyle \frac{\gamma +1}{\gamma -1},\end{eqnarray} \tag{ 34 }$$

where the upper limit is 4 for $\gamma =5/3$. Therefore, the maximum value of the strength of the magnetic field should weakly depend on β₀ (${B}_{\mathrm{max}}/{B}_{0}\sim \mathrm{const}$ with respect to β₀, Equation (27)). The current sheet, which is pinched by the shocks, is in the MHD equilibrium (i.e., v_x = 0). This means that the gas pressure at the center of the current sheet, ${p}_{\mathrm{max}}$, should be comparable to the magnetic pressure outside of the current sheet:

$$\begin{eqnarray}&&{p}_{\mathrm{max}}\sim \displaystyle \frac{{B}_{\mathrm{max}}^{2}}{8\pi }\sim \displaystyle \frac{{B}_{0}^{2}}{8\pi },\end{eqnarray} \tag{ 35 }$$

which gives the relation (26)

$$\begin{eqnarray}&&\displaystyle \frac{{p}_{\mathrm{max}}}{{p}_{0}}\sim {\beta }_{0}^{-1},\end{eqnarray} \tag{ 36 }$$

where p₀ is the initial gas pressure. From Equation (36) and the adiabatic condition, the density at the center (which takes the maximum value, Equation (25)) can be written by

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{\mathrm{max}}}{{\rho }_{0}}\sim {\left(\displaystyle \frac{{p}_{\mathrm{max}}}{{p}_{0}}\right)}^{1/\gamma }\sim {\beta }_{0}^{-3/5},\end{eqnarray} \tag{ 37 }$$

where ρ₀ is the initial plasma density and $\gamma =3/5$ is the adiabatic index.

To derive the β₀ dependence of ${L}_{\mathrm{min}}$, relation (29), we defined B₁ as the magnetic field strength of the maximum value in the pinched current sheet. As shown in Lagrangian particle trajectories, note that the pinched current sheet is formed from not the whole of the initial current but a part of it. We also defined the length of the part in the initial current sheet as ${L}_{0}^{*}$ and the maximum value of the magnetic field strength in the part as ${B}_{0}^{*}$. In our numerical simulations, we confirmed ${L}_{0}^{*}\ll {L}_{0}$ by the Lagrangian particle motions. Near the center of the initial current sheet, the magnetic field strength linearly depends on x,

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{0}^{*}}{{L}_{0}}\sim \displaystyle \frac{{B}_{0}^{*}}{{B}_{0}}.\end{eqnarray} \tag{ 38 }$$

We considered the conservation law of a magnetic flux,

$$\begin{eqnarray}{B}_{0}^{*}{L}_{0}^{*} & = & {B}_{1}{L}_{\mathrm{min}},\\ & = & {B}_{0}{L}_{\mathrm{min}}.\end{eqnarray} \tag{ 39 }$$

From Equations (38) and (39), we obtained

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{min}}}{{L}_{0}}\sim {\left(\displaystyle \frac{{L}_{0}^{*}}{{L}_{0}}\right)}^{2}.\end{eqnarray} \tag{ 40 }$$

Finally, from the mass conservation law,

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{0}}{{L}_{\mathrm{min}}}=\displaystyle \frac{{\rho }_{\mathrm{max}}}{{\rho }_{0}}\sim {\beta }_{0}^{-3/5}.\end{eqnarray} \tag{ 41 }$$

From Equations (40) and (41), we derived

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{min}}}{{L}_{0}}\sim {\beta }_{0}^{6/5},\end{eqnarray} \tag{ 42 }$$

which is consistent with Equation (29).

In this paper, we studied the implosion process of current sheets in a low-β plasma using one-dimensional ideal MHD simulations. We confirmed that the self-similar solution by Tajima et al. (1987) holds before the explosion time. The plasma adiabatically evolves until the explosion time. At $t\sim {t}_{0}$, the behavior of the current sheet deviates from the analytical self-similar solution, and the MHD fast-mode shocks are formed because of the increase in the gas pressure at the center of the current sheet. After a pair of MHD fast-mode shocks propagate away from the current sheet, the pinched current sheet is formed between a pair of the shocks. We studied the β₀ dependence of the maximum values of the physical quantities inside the current sheet and the minimum thickness of the current sheet (Equations (36), (37), and (42)).

In contrast to our study, the η dependence of the physical quantities and of the reconnection rate was investigated by McClymont & Craig (1996) in MHD simulations similar to ours, where η is the magnetic diffusivity. They found that the effect of the gas pressure does not significantly change the dependence of the thickness of the current sheet on η, which means that our scaling laws can be easily extended to the cases with the resistivity.

In McLaughlin et al. (2009), the collapse of a null point to a current sheet triggered by MHD fast-mode waves was investigated. In their simulations, successive current sheet pinching was observed. The pinching is done by the gas pressure of the heated plasma in the outflow region, while the pinching in our study is done by a Lorentz force. To extend this study, we are currently performing two-dimensional MHD simulations of the coalescence process of plasmoids and studying the pinching process of the current sheet formed between plasmoids in detail.

We analyzed the β₀ dependence of the fast-mode Mach number and the compression ratio, ${p}_{2}/{p}_{1}$ (see Figure 7). Figure 7 shows that the fast-mode Mach number, $\bar{{M}_{1}}$, weakly depends on β₀. We also found that the compression ratio of the density and magnetic field, X, weakly depends on β₀. Therefore, from Equation (32) we obtain

$$\begin{eqnarray}&&{\mathrm{log}}_{10}\left(\displaystyle \frac{{p}_{2}}{{p}_{1}}\right)\propto {\beta }_{1}\sim {\beta }_{0}.\end{eqnarray} \tag{ 43 }$$

This β₀ dependence of the compression ratio, ${p}_{2}/{p}_{1}$, is similar to Equation (43) with ${\beta }_{0}\lt {10}^{-2}$. In recent studies, it is indicated that particles can be accelerated in contracting plasmoids (Drake et al. 2006) and in plasmoids crossing fast-mode shocks (Nishizuka & Shibata 2013). Figure 8(a) shows an overview of the implosion process of the current sheet between plasmoids and the formation of the MHD fast-mode shocks. In our study, we found that fast-mode shocks can be formed during the implosion process in low-β plasmas. Applying the shock formation in the implosion process to a coalescence process, we conjecture that particles in the plasmoids are efficiently accelerated at shocks generated by the coalescence of plasmoids through a Fermi acceleration process (see Figure 8(b)).

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We investigated the parameter region where our scaling laws are valid. Since we assume the framework of non-relativistic ideal MHD, (1) the drift velocity of the electrons needs to be sufficiently smaller than the light speed and (2) the thickness of the current sheet needs to be larger than the ion skin depth. The electron drift velocity can be written as follows

$$\begin{eqnarray}&&J={{nev}}_{\mathrm{drift},e},=\displaystyle \frac{c}{4\pi }{\rm{\nabla }}\times {\boldsymbol{B}}\sim \displaystyle \frac{c}{4\pi }\displaystyle \frac{B}{L},\end{eqnarray} \tag{ 44 }$$

where L is the thickness of the current sheet. From Equation (44), ${v}_{\mathrm{drift},e}$, the drift velocity of electrons, is given as

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{\mathrm{drift},e}}{c}\sim \displaystyle \frac{1}{4\pi e}\displaystyle \frac{B}{{nL}}.\end{eqnarray} \tag{ 45 }$$

Applying the scaling laws of relations (36), (37), and (42) to Equation (45), we obtain

$$\begin{eqnarray}\displaystyle \frac{{v}_{\mathrm{drift},e}}{c} & \sim & {10}^{-4}{\left(\displaystyle \frac{{\beta }_{0}}{{\beta }_{-2}}\right)}^{-3/5}\\ & & \times \left(\displaystyle \frac{{B}_{0}}{{B}_{1}}\right){\left(\displaystyle \frac{{n}_{0}}{{n}_{9}}\right)}^{-1}{\left(\displaystyle \frac{{L}_{0}}{{L}_{6}}\right)}^{-1},\end{eqnarray} \tag{ 46 }$$

where ${\beta }_{-2}={10}^{-2},{B}_{1}=10\;{\rm{G}},{n}_{9}={10}^{9}\;{\mathrm{cm}}^{-3}$, and ${L}_{6}={10}^{6}\mathrm{cm}$. Since we study non-relativistic MHD processes, ${v}_{\mathrm{drift},e}/c\lt 1$ is required. This leads to

$$\begin{eqnarray}{\mathrm{log}}_{10}\left(\displaystyle \frac{{L}_{0}}{{L}_{6}}\right) & \gt & -{\mathrm{log}}_{10}\left(\displaystyle \frac{{n}_{0}}{{n}_{9}}\right)-\displaystyle \frac{3}{5}{\mathrm{log}}_{10}\left(\displaystyle \frac{{\beta }_{0}}{{\beta }_{-2}}\right)\\ & & +{\mathrm{log}}_{10}\left(\displaystyle \frac{{B}_{0}}{{B}_{1}}\right)-4.\end{eqnarray} \tag{ 47 }$$

Since the thickness of the current sheet must be larger than the ion skin depth, we similarly obtain

$$\begin{eqnarray}\displaystyle \frac{{\lambda }_{i}}{d} & = & \displaystyle \frac{1}{c}\sqrt{\displaystyle \frac{4\pi {e}^{2}}{{m}_{i}}}\displaystyle \frac{\sqrt{n}}{d},\\ & \sim & {10}^{-4}{\left(\displaystyle \frac{{\beta }_{0}}{{\beta }_{-2}}\right)}^{-3/2}{\left(\displaystyle \frac{{n}_{0}}{{n}_{9}}\right)}^{1/2}{\left(\displaystyle \frac{{L}_{0}}{{L}_{6}}\right)}^{-1}\lt 1,\end{eqnarray} \tag{ 48 }$$

where λ_i is the ion skin depth. We can transform Equation (48) as

$$\begin{eqnarray}{\mathrm{log}}_{10}\left(\displaystyle \frac{{L}_{0}}{{L}_{6}}\right) & \gt & \displaystyle \frac{1}{2}{\mathrm{log}}_{10}\left(\displaystyle \frac{{n}_{0}}{{n}_{9}}\right)\\ & & -\displaystyle \frac{3}{2}{\mathrm{log}}_{10}\left(\displaystyle \frac{{\beta }_{0}}{{\beta }_{-2}}\right)-4.\end{eqnarray} \tag{ 49 }$$

In Figure 9, we show the parameter region using inequalities (47) and (49) with the values in the solar corona, ${\beta }_{0}={10}^{-2}$ and B₀ = 10 G. It is found that our scaling laws are applicable in the solar corona. These scaling laws will help us estimate physical quantities in a current sheet formed in a low-β plasma. Describing the evolution of the pinching process in an extremely low-β plasma like the atmosphere of the neutron stars will be our future work.

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We thank Dr. Jin Matsumoto for fruitful discussions and comments. Shinsuke Takasao acknowledges support from the Research Fellowship of the Japan Society for the Promotion of Science (JSPS). This work was supported by the Grant-in-Aids from the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 25287039).