We ignore the density stratification effect in Case I, II, and IIa, because the width of the horizontal current sheet in our simulations is much shorter than the length. The simulation domain extends from x = 0 to x = L0 in the x-direction, and from 




y
=
−
0.5


L


0




 to 




y
=
0.5


L


0




 in the y-direction, in the three cases, with 






L


0


=


10


6




 m. Outflow boundary conditions are used in the x-direction and inflow boundary conditions in the y-direction. For the inflow boundary conditions, the fluid is allowed to flow into the domain but not to flow out; the gradient of the plasma density vanishes; the total energy is set such that the gradient in the thermal energy density vanishes; a vanishing gradient of parallel components plus divergence-free extrapolation of the magnetic field. For the outflow boundary conditions, the fluid is allowed to flow out of the domain but not to flow in, and the other variables are set by using the same method as the inflow boundary conditions. The horizontal force-free Harris current sheet is used as the initial equilibrium configuration of magnetic fields in Case I,
13






B


x
0


=
−


b


0


tanh
[
y

/

(
0.05


L


0


)
]




14






B


y
0


=
0




15






B


z
0


=


b


0



/

cosh
[
y

/

(
0.05


L


0


)
]
.


The magnetic fields in the low solar atmosphere could be very strong (Jin et al. 2009, 2012; Khomenko et al. 2014; Peter et al. 2014; Vissers et al. 2015) and the magnetic field can exceed 0.15 T in both the intranetwork and the network quiet region (e.g., Orozco Suárez et al. 2007; Martínez González et al. 2008; Jin et al. 2009, 2012). In the work by Jin et al. 2012, the maximum of the field strength was found to be 0.15 T. The magnetic field could be even stronger in the active region near the sunspot. Therefore, we set b0 = 0.05 T in Case I and Case II, and b0 = 0.15 T in Case IIa. Due to the force-freeness and neglect of gravity, the initial equilibrium thermal pressure is uniform. The initial temperature and plasma density are set as T0 = 4200 K and ρ0 = 1.66057 × 10−6 kg m−3 in Case I, and T0 = 4800 K and ρ0 = 3.32114 × 10−5 kg m−3 in Case II and Case IIa. Therefore, the initial plasma β is calculated as β ≃ 0.0583 in Case I, β ≃ 1.332 in Case II, and β ≃ 0.148 in Case IIa. The initial ionization degree is assumed as Yi = 10−3 in Case I, and Yi = 1. 2 × 10−4 in Case II and IIa. The magnetic diffusion in this work matches the form computed from the solar atmosphere model in Khomenko & Collados (2012), and we set 




η
=

[

5
×


10


4




(
4200

/

T
)


1.5


+
1.76
×


10


−
3




T


0.5




Y


i


−
1



]



 m2 s−1 in Case I, and 




η
=
[
5
×


10


4




(
4800

/

T
)


1.5



+
1.76

×



10


−
3




T


0.5




Y


i


−
1


]


 m2 s−1 in Case II and IIa. The first part ∼ T−1.5 is contributed by collisions between ions and electrons, the second part 




∼


T


0.5




Y


i


−
1




 is contributed by collisions between electrons and neutral particles. Small perturbations for both magnetic fields and velocities at t = 0 make the current sheet to evolve and secondary instabilities start to appear later in the three cases. The forms of perturbations are listed below:
16






b


x
1


=
−
pert
·


b


0


·
sin


2
π



y
+
0.5



L


0






L


0







·
cos


2
π



x
+
0.5


L


0






L


0











17






b


y
1


=
pert
·


b


0


·
cos


2
π



y
+
0.5


L


0






L


0







·
sin


2
π



x
+
0.5


L


0






L


0











18






v


y
1


=
−
pert
·


v


A
0


·
sin


π



y




L


0







·





random


n




Max

(

∣


random


n


∣

)




,


where pert = 0.08, vA0 is the initial Alfvén velocity, randomn is the random noise function in our code, and 




Max
(
∣


random


n


∣
)


 is the maximum of the absolute value of the random noise function. This random noise function makes the initial perturbations for the velocity in the y-direction to be asymmetric, and such an asymmetry makes the current sheet gradually become more tilted, especially after secondary islands appear. The reconnection process is not really symmetrical in nature (Murphy et al. 2012), this is one of the reasons that we use such a noise function. Another reason is that the asymmetric noise function makes the secondary instabilities develop faster. Figure 1(a) shows the distributions of the current density and magnetic fields at t = 0 in case I.
Citation: Jin et al.; 2012
Citation function: Uses
Citation: Jin et al.; 2012
Citation function: Uses
Citation: Jin et al. 2012
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