The three-dimensional nature of plasmoid instabilities is studied using the reduced magnetohydrodynamic equations. For a Harris equilibrium with guide field, represented by B o = B po tanh (x/λ) ŷ + B zoẑ, a spectrum of modes are unstable at multiple resonant surfaces in the current sheet, rather than just the null surface of the poloidal field B yo (x) = B po tanh (x/λ), which is the only resonant surface in 2D or in the absence of a guide field. Here, B po is the asymptotic value of the equilibrium poloidal field, B zo is the constant equilibrium guide field, and λ is the current sheet width. Plasmoids on each resonant surface have a unique angle of obliquity θ arctan (k z/k y). The resonant surface location for angle θ is x s = λ arctanh (μ), where μ = tanθB zo/B po and the existence of a resonant surface requires |θ| ≤ arctan (B po/B zo). The most unstable angle is oblique, i.e., θ ≠ 0 and x s ≠ 0, in the constant-ψ regime, but parallel, i.e., θ = 0 and x s = 0, in the nonconstant-ψ regime. For a fixed angle of obliquity, the most unstable wavenumber lies at the intersection of the constant-ψ and nonconstant-ψ regimes. The growth rate of this mode is γ max/Γ o≃ S L 1/4 (1 - μ 4) 1/2, in which Γ o = V A/L, V A is the Alfvén speed, L is the current sheet length, and S L is the Lundquist number. The number of plasmoids scales as N ∼ S L 3/8 (1 - μ 2) -1/4 (1 + μ 2) 3/4.
All Science Journal Classification (ASJC) codes
- Condensed Matter Physics