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 $Bo=Bpotanh(x/λ)ŷ+Bzoẑ$⁠, a spectrum of modes are unstable at multiple resonant surfaces in the current sheet, rather than just the null surface of the poloidal field $Byo(x)=Bpotanh(x/λ)$⁠, which is the only resonant surface in 2D or in the absence of a guide field. Here, $Bpo$ is the asymptotic value of the equilibrium poloidal field, $Bzo$ is the constant equilibrium guide field, and $λ$ is the current sheet width. Plasmoids on each resonant surface have a unique angle of obliquity $θ≡arctan(kz/ky)$⁠. The resonant surface location for angle $θ$ is $xs=λarctanh(μ)$⁠, where $μ=tanθBzo/Bpo$ and the existence of a resonant surface requires $|θ|<arctan(Bpo/Bzo)$⁠. The most unstable angle is oblique, i.e., $θ≠0$ and $xs≠0$⁠, in the constant-$ψ$ regime, but parallel, i.e., $θ=0$ and $xs=0$⁠, 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 $γmax/Γo≃SL1/4(1-μ4)1/2$⁠, in which $Γo=VA/L$⁠, $VA$ is the Alfvén speed, L is the current sheet length, and $SL$ is the Lundquist number. The number of plasmoids scales as $N~SL3/8(1-μ2)-1/4(1+μ2)3/4$⁠.