Nonlinear Acceleration Mechanism of Collisionless Magnetic Reconnection

University of Texas at Austin, Austin, Texas 78712 USAe-mail: hirota.makoto@jaea.go.jpA mechanism for fast magnetic reconnection in collisionless plasma is studied for understand-ing sawtooth collapse in tokamak discharges. Nonlinear growth of the tearing mode drivenby electron inertia is analytically estimated by invoking the energy principle for the first time.Decrease of potential energy in the nonlinear regime (where the island width exceeds the elec-tron skin depth) is found to be steeper than in the linear regime, resulting in acceleration of thereconnection. Release of free energy by such ideal fluid moti on leads to unsteady and strongconvective flow, which theoretically corroborates the iner tia-driven collapse model of the saw-tooth crash [D. Biskamp and J. F. Drake, Phys. Rev. Lett. 73, 971 (1994)].1 IntroductionSawtooth collapse in tokamak plasmas has been a puzzling phenomena for decades. Althoughthe m=1kink-tearing mode is essential for onset of this dynamics, Kadomtsev’sfull reconnec-tion model [1] and nonlinear growth of the resistive m=1mode [2] (both based on resistivemagnetohydrodynamic theory) fails to explain the short collapse times (∼ 100µs) as well aspartial reconnections observed in experiments. Since resistivity is small in high-temperaturetokamaks, two-fluid effects are expected to play an important role for triggering fast (or explo-sive) magnetic reconnection as in solar flares and magnetospheri c substorms.In earlier works [3, 4], the linear growth rate of the kink-tearing mode in the collisionlessregime has been analyzed extensively by using asymptotic matching, which shows an enhance-ment of the growth rate due to two-fluid effects, even in the absence of resistivity. Furthermore,direct numerical simulations [5, 6, 7] of two-fluid models show acceleration of reconnectionin the nonlinear phase, which indicates explosive tendencies until numerical error or artificialdissipation terminates them.However, theoretical understanding of such explosive phenomena is not yet established dueto the lack of analytical development. In contrast to the quasi-equilibrium analysis developedfor resistive reconnections [2, 8], the explosive process of collisionless reconnection should bea nonequilibrium problem, in which inertia is not negligible in the force balance and henceleads to acceleration of flow. The convenient assumption of steady reconnection is no longerappropriate.Recent theories [9, 10, 11] emphasize the Hamiltonian nature of two-fluid models and try togain deeper understanding of collisionless reconnection in the ideal limit.The purpose of the present work is to predict explosive growth of the kink-tearing modeanalytically by developing a new approach that is based on the energy principle [12]. For sim-plicity, we will consider only the effect of electron inertia, which is an attractive mechanism fortriggering fast reconnection in tokamaks; estimates of the reconnection rate are favorable [13],nonlinear acceleration is possible [6], and even the mysterious partial reconnection may be ex-plained by an inertia-driven collapse model [14, 15]. Whilewe address the same problem as