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 $\vc{B}o = B{po} \tanh (x/\lambda) \hat{y} + B_{zo} \hat{z}$, a spectrum of modes are unstable at multiple resonant surfaces in the current sheet, rather than just the null surface of the polodial field $B_{yo} (x) = B_{po} \tanh (x/\lambda)$, 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 $\lambda$ is the current sheet width. Plasmoids on each resonant surface have a unique angle of obliquity $\theta \equiv \arctan (k_z\mathrm{/}{k}_y)$. The resonant surface location for angle $\theta$ is $x_s = - \lambda \arctanh (\tan \theta B_{zo}/B_{po})$,andtheexistenceofaresonantsurfacerequires$|\theta| < \arctan (B_{po} / B_{zo})$\mathrm{.}Themostunstableangleisoblique,i\mathrm{.}e\mathrm{.}$\theta \neq 0$and$x_s \neq 0$,intheconstant-$\psi$regime,butparallel,i\mathrm{.}e\mathrm{.}$\theta = 0$and$x_s = 0$,inthenonconstant-$\psi$ regime. For a fixed angle of obliquity, the most unstable wavenumber lies at the intersection of the constant-$\psi$ and nonconstant-$\psi$ regimes. The growth rate of this mode is ${\gamma }_{\text{max}}\mathrm{/}{\mathrm{\Gamma }}_o\simeq S_L^{1\mathrm{/}4}(1-{\mu }^4{)}^{1\mathrm{/}2}$, in which ${\mathrm{\Gamma }}_o=V_A\mathrm{/}L$, $V_A$ is the Alfv'{e}n speed, $L$ is the current sheet length, and $S_L$ is the Lundquist number. The number of plasmoids scales as $N \sim S_L^{3/8} (1-\mu^2)^{-1/4} (1 + \mu^2)^{3/4}$.