Reduced magnetohydrodynamic theory of oblique plasmoid instabilities

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 ) y ̂ + B zo z ̂ , 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 = tan θ B 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 .