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/lambda)(y) over cap + B-zo(z) over cap, 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/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 equivalent to arctan(k(z)/k(y)). The resonant surface location for angle 0 is x(s) = lambda arctanh(mu), where mu = tan theta B-zo/B-po and the existence of a resonant surface requires vertical bar theta vertical bar < arctan(B-po/B-zo). The most unstable angle is oblique, i.e., theta not equal 0 and x(s) not equal 0, in the constant-psi regime, but parallel, i.e., theta = 0 and x(s) = 0, in the nonconstant-psi regime. For a fixed angle of obliquity, the most unstable wavenumber lies at the intersection of the contstant-psi and nonconstant-psi regimes. The growth rate of this mode is gamma(max)/Gamma(o) similar or equal to S-L(1/4) (1-mu(4))(1/2), in which Gamma(o)=V-A/L, V-A is the Alfven speed, L is the current sheet length, and S-L is the Lundquist number. The number of plasmoids scales as N similar to S-L(3/8)(1-mu(2))(-1/4)(1+mu(2))(3/4). (C) 2012 American Institute of Physics. [doi:10.1063/1.3678211]