Visco-resistive plasmoid instability

Abstract

The plasmoid instability in visco-resistive current sheets is analyzed in both the linear and nonlinear regimes. The linear growth rate and the wavenumber are found to scale as S 1/4 (1 + Pm) −5/8 and S 3/8 (1 + Pm) −3/16 with respect to the Lundquist number S and the magnetic Prandtl number Pm. Furthermore, the linear layer width is shown to scale as S −1/8 (1 + Pm) 1/16 . The growth of the plasmoids slows down from an exponential growth to an algebraic growth when they enter into the nonlinear regime. In particular, the time-scale of the nonlinear growth of the plasmoids is found to be τNL ∼ S −3/16 (1 + Pm) 19/32 τA,L. The nonlinear growth of the plasmoids is radically different from the linear one and it is shown to be essential to understand the global current sheet disruption. It is also discussed how the plasmoid instability enables fast magnetic reconnection in visco-resistive plasmas. In particular, it is shown that the recursive plasmoid formation can trigger a collisionless reconnection regime if S Lcs(ǫcl k ) −1 (1 + Pm) 1/2 , where Lcs is the half-length of the global current sheet and l k is the relevant kinetic length scale. On the other hand, if the current sheet remains in the collisional regime, the global (time-averaged) reconnection rate is shown to be dψ/dt| X ≈ ǫcvA,uBu(1 + Pm) −1/2 , where ǫc is the critical inverse aspect ratio of the current sheet, while vA,u and Bu are the Alfvén speed and the magnetic field upstream of the global reconnection layer.

Plasmoid Instability in Forming Current Sheets

“…An important consequence is that the end of the linear phase practically corresponds to the disruption of the reconnecting current sheet when it is defined by the conditionŵ ∼â (Comisso & Grasso 2016;Uzdensky & Loureiro 2016). One could also define the disruption of the current sheet to correspond exactly to the end of the linear phase.…”

“…Therefore, after the linear regime the plasmoid instability will evolve only through a fast Waelbroeck phase (Waelbroeck 1989), as detailed in Comisso & Grasso (2016) for the case of a time-independent current sheet.…”

“…as in Loureiro et al (2013) and Comisso & Grasso (2016). Note that plasma viscosity leads to a decrease of the asymptotic aspect ratio (1/â * ) and of the growth rate associated to it.…”

“…However, reconnecting current sheets exceeding a certain aspect ratio cannot form because they become unstable to the formation of plasmoids, which break the reconnection layer into shorter elements, consequently leading to a significant increase in the reconnection rate (Daughton et al 2006(Daughton et al , 2009. Hence, the predictions of the classical Sweet-Parker reconnection model (Sweet 1958;Parker 1957) break down for sufficiently large systems such as those typically encountered in astrophysical environments -in these cases, it was shown that the reconnection rate becomes nearly independent of the magnetic diffusivity (Bhattacharjee et al 2009;Huang & Bhattacharjee 2010;Uzdensky et al 2010;Loureiro et al 2012;Takamoto 2013;Comisso et al 2015a;Ebrahimi & Raman 2015;Comisso & Grasso 2016).…”

“…Surprisingly, these scaling relations remained overlooked until they were independently rederived by Loureiro et al (2007) a decade later. They were also generalized further through the inclusion of threedimensional (Baalrud et al 2012), Hall (Baalrud et al 2011) and plasma viscosity (Loureiro et al 2013;Comisso & Grasso 2016) effects. However, it soon became apparent that the results obtained by assuming a Sweet-Parker current sheet were problematic because of the growth rate being proportional to S raised to a positive exponent, which implies that the growth rate approaches infinity in the limit S → ∞.…”