Tearing stability of the two-dimensional magnetotail

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Abstract
The linearized incompressible magnetohydrodynamic equations that include a generalized Ohm’s law are solved for tearing eigenmodes of a plasma sheet with a normal magnetic field (B n ). In contrast to the Harris sheet with the equilibrium magnetic field [B=B 0 tanh(z/a)x̂], the two‐dimensional plasma sheet with the field [B=B 0 tanh(z/a)x̂+B n z ̂], in which the B n field lies in the plane of the B x field, has no neutral line if B n ≠0. Such a geometry is intrinsically resilient to tearing because it cannot change topology by means of linear perturbations. This qualitative geometrical idea is supported by calculations of growth rates using a generalized Ohm’s law that includes collisional resistivity and finite electron inertia as the mechanisms for breaking field lines. The presence of B n reduces the resistive tearing mode growth rate by several orders of magnitude (assuming B n /B 0∼0.1) compared with that in the Harris sheet model (B n =0). The growth rate scaling with Lundquist number (S) has the typical S −3/5 (S −1/3) dependence for large (small) wave numbers and very small values of B n . For larger values of B n , all modes behave diffusively, scaling as S −1. The collisionless electron tearing mode growth rate is found to be proportional to δ2 e in the presence of significant B n (≳10−2 B 0) and large k x (∼0.1a −1–0.5a −1), and becomes completely stable (γ<0) for B n /B 0≥0.2. Implications for magnetospheric substorms are discussed.