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) 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.