Viscous effects on plasmoid formation from nonlinear resistive tearing growth in a Harris sheet

Our nonlinear simulations find that the onset of secondary tearing instability and plasmoid formation occur only when the ${\Delta^{\prime}}$ is above a certain threshold value. In our nonlinear simulation, we mostly employ ${64\times 48}$ 2D finite elements with a polynomial degree of 5, which ensures the numerical convergence (Figure 3).

The evolution of kinetic energy reaches its maximum sooner as we increase the value of ${\Delta^{\prime}}$ (Figure 4). The minimum value of ${\Delta^{\prime}}$ at which the X-point evolves into the Y-type is ${14.04}$. For ${\Delta^{\prime}=13.42}$, the current sheet remains the shape of X-point over the entire time (Figure 5). As the value of ${\Delta^{\prime}}$ increases to above 14.05, the X-point evolves into a Y-type current sheet as shown in Figures 6 and 7. Waelbroeck [34] first predicted the criterion for the collapse of X-point into Y-type current sheet to be ${W>W_{c}\approx 25/\Delta^{\prime}}$, where ${W}$ represents the width of primary island and ${W_{c}}$ represents the critical width at which X-point collapses. This conversion of X-point into Y-type current sheet is termed as the secondary instability (SI)[34]. The subsequent collapse of the Y-type current sheet into plasmoids is known as the plasmoid instability (PI) [35].

For ${\eta=2.8\times 10^{-4}}$ and ${P_{r}=1}$ the critical ${\Delta^{\prime}}$ value for the onset of plasmoid instability (PI) is ${\Delta^{\prime}=17.31}$, which is in agreement with the previous reduced MHD simulation [18]. In this case, at time 200 Sec, Figure 8(a), an X-point is formed as shown. At time 340 Sec, Figure 8(b), the Y-type current sheet develops during the nonlinear stage [3, 6, 36]. At time 360 Sec, Figure 8(c) the current density reaches maximum and the current sheet becomes more stretched and thinner. Finally, at time 365 Sec, Figure 8(d), formation of plasmoid takes place.

In our cases, we only vary the viscosity in terms of ${P_{r}}$ by keeping the resistivity and instability parameter constant, which are ${\eta=2.8\times 10^{-4}}$ and ${\Delta^{\prime}=49.66}$ respectively. To study the effect of viscosity, we choose four different cases with ${P_{r}=0.33,0.5,1}$ and ${10}$ (Figure 9). In Figure 10, the dynamics of the visco-resistive tearing mode growth are divided in 5-stages. The first stage is the initial transient stage when the linear instability starts to grow. The second stage is the ${FKR}$ stage, during which both the reconnection rate and the magnetic island width grow exponentially. In the third stage, the so-called Rutherford stage, the island evolves toward saturation and subsequent decay. The X-point collapse and the Y-type, SP-like current sheet forms during the fourth stage (Figure 11 (a)-(c)). The transition from the X-type geometry to the Y-type current sheet is known as the secondary instability. The first peak that appears in the kinetic energy evolution is due to the X-point collapse with the onset of secondary instability at t = 160 sec as shown in Figure 9. After the X-point collapse, the SP-like current sheet starts to become elongated in the poloidal direction. During the fifth stage, the Y-type current sheet starts to become more elongated and SP-like. After the collapse of SP-like current sheet, the secondary island appears along with two X-points on both ends (Figure 11(e)). After t = 160 sec, the growth rate starts to decrease up to t = 170 sec, and a significant change in the growth rate occurs due to the collapse of the Y-type current sheet and the formation of small plasmoid chain. The size of the secondary island increases up to some extent and both X-points collapse to form two tertiary current sheets with the passage of time. The second large peaks in the growth rate and the kinetic energy plots represent the collapse of these two tertiary current sheets. As the width of the secondary island approaches some critical value, the ejection of the secondary island takes place. At this point kinetic energy increases abruptly and the secondary island coalesces with the primary one (Figure 12(d)). As the Y-type current sheet collapse and the plasmoid instability (PI) appears, a drastic increase in the growth rate takes place (Figure 10) which is much larger as compared to the growth rate of the secondary instability.

The nonlinear stages of our simulation results described above are quite similar to the nonlinear secondary island evolution reported by N. F. Loureiro et al. [2]. But the collapse of the secondary island and direction of ejection are different in our cases. For example in our simulations for the ${P_{r}=0.5}$ and the ${P_{r}=5}$ cases, the direction of plasmoid ejection is upward, whereas for the ${P_{r}=1}$ and the ${P_{r}=10}$ cases the direction is downward. The directions of plasmoid ejection are different for different ${P_{r}}$ cases.

Our simulations for various ${P_{r}}$ numbers show viscous effects on both the timing and the spatial structure of plasmoid instability. In the ${P_{r}}$ = 0.33 (i.e. very low viscosity), we find secondary islands that saturate early at smaller size (Figure 11(f)). As we increase the viscosity further (${P_{r}}$ = 0.5), the appearance and ejection of the secondary island becomes more prominent, along with the larger saturated island and plasmoid (Figure 12). A clear transition occurs at ${P_{r}=1}$, when the size of the primary plasmoid becomes the maximum, and the onsets of the secondary instability, the PI and the island saturation are significantly postponed. The direction of plasmoid ejection also switches to the opposite. The second peak of kinetic energy increases with viscosity and reaches maximum at ${P_{r}=1}$ too. For higher Prandtl number (${P_{r}}$ = 10, with ${\eta=2.8\times 10^{-4}}$ and viscosity = 0.0028) the size of the plasmoid becomes smaller (Figure 14).

Figure 15 summarizes the relationships among plasmoid width, saturated island width, and ${P_{r}}$ number. At lower ${P_{r}}$, the width of saturated island is small, but as we increase viscosity, the width of saturated island increases sharply up to ${P_{r}=1}$, beyond which the saturated island width becomes almost independent of the ${P_{r}}$ number. The ${P_{r}=1}$ number also separates two regimes for the plasmoid width. In the ${P_{r}<1}$ regime, the width of plasmoid increases drastically with viscosity, whereas in the ${P_{r}>1}$ regime, the width of plasmoid slowly decreases with the viscosity.