### Abstract

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/λ) ŷ + B _{zo}ẑ, 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/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 arctan (k _{z}/k _{y}). The resonant surface location for angle θ is x _{s} = λ arctanh (μ), where μ = tanθB _{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 ≠ 0 and x _{s} ≠ 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 _{max}/Γ _{o}≃ 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 ∼ S _{L} ^{3/8} (1 - μ ^{2}) ^{-1/4} (1 + μ ^{2}) ^{3/4}.

Original language | English (US) |
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Article number | 022101 |

Journal | Physics of Plasmas |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2012 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Condensed Matter Physics

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## Cite this

*Physics of Plasmas*,

*19*(2), [022101]. https://doi.org/10.1063/1.3678211