The velocity distribution of the plasma motion is shown in Fig. 12, where the magnitude is scaled with respect to the Alfvén velocity, vA, which is measured based on the magnetic field strength, B0 = 2 G, and the mass density of the equilibrium current sheet, ρc = 2.81 × 10−15 g cm−3. The Alfvén timescale is measured by 




t
A

=


L


¯


/

v
A



$ t_A = \bar{L}/\mathit{v}_A $


, where, 




L


¯



$ \bar{L} $


 is the unit length of 109 cm. It is evident from Figs. 12a, b and c that the velocity, vx remains localized in the vicinity of the current sheet (y = 0). We note that in line with our initial single-island magnetic field perturbation, we see a pronounced rightward motion in the right half of the domain, and a leftward motion in the left half. We later see typical Petschek-like signatures in the flow fields between islands, especially about the middle x = 0, with super-Alfvénic outflow speeds bounded by slow shocks. Figure 13a represents the evolutionary nature of a current sheet in adiabatic and non-adiabatic conditions. It is clear from the figure that the instantaneous maximum velocity growth for the non-adiabatic case is more rapid than for the adiabatic conditions. The evolutionary behavior of the current sheet configuration due to thermal and tearing instabilities is shown by the black curve in Fig. 13a for plasma-β = 0.2, and a given resistivity value, η = 0.001, while the evolution for different η values are shown in Fig. 13b. As a diagnostic measurement of the instability, we determine the evolution of the instantaneous maximum absolute velocity, |vx|max. Figure 13a shows that this evolution exhibits three distinct phases: (i) the early phase (between t = 0 and 250 s), where the velocity growth occurs exponentially (linearly on the logarithmic-linear scale), which is called the linear growth regime; (ii) the middle phase between t = 250 and 665 s, where the growth rate is slower compared to the linear phase, which is called the Rutherford regime (Rutherford 1903), and (iii) the final phase, which starts at t = 665 s, where the instability suddenly develops in an explosive way, and finally saturates at a later time, which we call the post-Rutherford regime. To infer the evolution rates quantitatively for all the different phases, we calculate the growth rates by scaling them with respect to the Alfvén timescale, tA. We define the growth rate as γ = d(ln(|vx|max))/dt. To estimate the linear growth rate, γlin, we calculate the growth rate in the linear regime by taking the mean value of the slope, which gives 



γ

l
i
n


=
3.76
×

10

−
1



t
A

−
1



$ \gamma_{\rm lin}=3.76 \times 10^{-1} t_A^{-1} $


. This value is larger by an order of magnitude compared to the studies of the double current sheet problem (Otto & Birk 1992; Zhang & Ma 2011; Akramov & Baty 2017; Paul & Vaidya 2021), where the radiative cooling effect (or other non-adiabatic effects, e.g., thermal conduction) is not incorporated. This implies that the higher linear growth rate can be ascribed to the non-adiabatic effects of the radiative cooling and background heating. This is also in agreement with our own study for a single current layer model reflected in Fig. 13a, where the average growth rate for the adiabatic medium is lower than the non-adiabatic case. Similarly, we estimate the average growth rates for the Rutherford regime (γRuth) and the post-Rutherford regime (γPR) for different resistivity values within the range of η = 0.0001 to 0.005. The velocity evolution for some selected resistivity values are shown in Fig. 13b. This shows that the explosive phase of the evolution starts at later times for higher resistivity values, and converges at the final stage. For a Sweet–Parker type current sheet (where the inverse aspect ratio of the current sheet follows the scaling relation 




l
s

/
L
∼

S

L


−
1
/
2




$ l_s/L \sim S_{\mathrm{L}}^{-1/2} $


), the thickness of the current sheet increases with the resistivity (Loureiro et al. 2007), which reduces the growth rate of the tearing mode when it is normalized with respect to the Alfvén crossing time along the length of the current sheet (in the x-direction in our case). Hence, the explosive phase of the evolution in our simulation starts at later times for higher resistivity values. We estimated the absolute current density, |Jz| (normalized to unity) before the fragmentation stage of the current sheet (t = 214 s) by taking a vertical cut along the y-direction at x = 0 for two different resistivities, η = 0.0001 and 0.001, to confirm that the thickness of the current sheet increases with resistivity (see Fig. 14). The resistivity dependence for the different evolution phases is shown in Fig. 15. Figure 15a shows that γRuth follows a power-law dependence with the resistivity, γRuth ≈ η−0.1, with a correlation coefficient (CC) of −64.1%. The resistivity scaling relation for the post-Rutherford and the entire nonlinear regimes are shown in Figs. 15b and c respectively. We estimate the growth rate scaling relations for the post-Rutherford regime, γPR ≈ η0.03 (with CC = 59.9%), and the entire non-linear regime, γavg ≈ η0.017 (with CC = 66.7%). Previous studies by Zhang & Ma (2011), Akramov & Baty (2017), and Guo et al. (2017 and references therein) have reported the resistivity scaling relation of the non-linear growth rates for the DTM setup in the adiabatic environment, which have larger power-law indices compared to our estimation. Hence, our study infers that the resistivity dependence on the nonlinear growth rates is weaker when the thermal instability reinforces the tearing mode.
Citation: Rutherford 1903
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