General theory of the plasmoid instability

Abstract

A general theory of the onset and development of the plasmoid instability is formulated by means of a principle of least time. The scaling relations for the final aspect ratio, transition time to rapid onset, growth rate, and number of plasmoids are derived and shown to depend on the initial perturbation amplitude (ŵ0), the characteristic rate of current sheet evolution (1/τ ), and the Lundquist number (S). They are not simple power laws, and are proportional to S α τ β [ln f (S, τ,ŵ0)] σ . The detailed dynamics of the instability is also elucidated, and shown to comprise of a period of quiescence followed by sudden growth over a short time scale.The rapid conversion of magnetic energy into plasma particle energy through the process of magnetic reconnection is of great importance in the realm of plasma physics and astrophysics [1][2][3][4]. Sawtooth crashes, magnetospheric substorms, stellar and gamma-ray flares are just a few examples of pheneomena in which magnetic reconnection plays an essential role.In large systems, such as those found in space and astrophysical environments, the potential formation of highly elongated current sheets would result in extremely low reconnection rates, which fail to account for the observed fast energy release rates [5][6][7]. However, such current sheets are subject to a violent linear instability that leads to their breakup, giving rise to a tremendous increase in the reconnection rate that appears to be very weakly dependent on the Lundquist number of the system in the nonlinear regime [8][9][10][11][12][13][14][15][16][17]. This crucial instability, which serves as a trigger of fast reconnection, is the plasmoid instability [2], thus dubbed as it leads to the formation of plasmoids.In the widely studied Sweet-Parker current sheets, which are characterized by an inverse aspect ratio a/L ∼ S −1/2 , Tajima and Shibata [1], as well as Loureiro et al.[18], have found that the growth rate γ and the wavenumber k of the plasmoid instability obey γτ A ∼ S 1/4 and kL ∼ S 3/8 , where τ A is the Alfvénic timescale based on the length of the current sheet. Since the Lundquist number S is extremely large in most space and astrophysical plasmas [19], the linear growth of the instability turns out to be surprisingly fast, and the number of plasmoids produced is also very high. Other notable works have since followed, which have verified and extended the work on the plasmoid instability in different contexts [20][21][22][23][24].Despite the success of the theory, its limitations soon became evident. For sufficiently high growth rates, Sweet-Parker current sheets cannot be attained as current layers are linearly unstable and disrupt before this state is achieved. In order to bypass this limitation, Pucci and Velli [25] conjectured that current sheets break up when γτ A ∼ 1. Later, Uzdensky and Loureiro [26] considered a similar criterion (γτ = 1) as the end-point of the linear stage of the instability, presenting an appealing but heuristic discussion for the case of a current sheet evolving...

Magnetic Reconnection in Strongly Magnetized Regions of the Low Solar Chromosphere

“…Since the length scale is very small, the Lundquist number during the magnetic reconnection process is only around 2000 in this work. In agreement with previous simulations of reconnecting current sheets (e.g., Huang et al 2017;Comisso et al 2016;Ni et al 2013Ni et al , 2012Leake et al 2012;Huang & Bhattacharjee 2010), no plasmoid instability appears in our simulations.…”

Role of Magnetic Reconnection in Magnetohydrodynamic Turbulence

“…[36] for a recent review). One straightforward implication of this instability [37][38][39][40] is that the Sweet-Parker current sheets cannot be formed in the first place [37,[41][42][43]. We now demonstrate that the MHD turbulent cascade will be affected by this instability before it has a chance to form Sweet-Parker current sheets at small scales, thus qualitatively changing the route to energy dissipation in MHD turbulence.…”

Collisionless magnetic reconnection in curved spacetime and the effect of black hole rotation

“…The latter condition is satisfied not only in steady-state, but also at the time of maximum reconnection rate. The reconnection layer is implicitly assumed to be stable to the plasmoid instability [45]. If this is not the case, the global reconnection layer would be replaced by a chain of plasmoids of different sizes separated by smaller current sheets [46], but our analysis would still be valuable for understanding the properties of the basic current sheets composing the global reconnection layer [47][48][49][50][51].…”