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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 $\vc{B}_o = B_{po} \tanh (x/λ) \hat{y} + B_{zo} \hat{z}$, a spectrum of modes are unstable at multiple resonant surfaces in the current sheet, rather than just the null surface of the polodial 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 $θ\equiv \arctan(k_z/k_y)$. The resonant surface location for angle $θ$ is $x_s = - λ\arctanh (\tan θB_{zo}/B_{po})$, and the existence of a resonant surface requires $|θ| < \arctan (B_{po} / B_{zo})$. The most unstable angle is oblique, i.e. $θ\neq 0$ and $x_s \neq 0$, 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 $γ_{\textrm{max}}/Γ_o \simeq S_L^{1/4} (1-μ^4)^{1/2}$, 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 $N \sim S_L^{3/8} (1-μ^2)^{-1/4} (1 + μ^2)^{3/4}$.