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 θarctan(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., θ0 and xs0, 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/ΓoSL1/4(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.

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