TY - JOUR

T1 - Reduced magnetohydrodynamic theory of oblique plasmoid instabilities

AU - Baalrud, S. D.

AU - Bhattacharjee, A.

AU - Huang, Y. M.

N1 - Funding Information:
The authors thank Dr. Will Fox, Dr. Robert Dewar, Dr. Bill Daughton, and Dr. Carl Sovinec for helpful discussions. This research was supported in part by an appointment to the U.S. Department of Energy Fusion Energy Postdoctoral Research Program administered by the Oak Ridge Institute for Science and Education (S.D.B.), and DOE Grant No. DE-FG02-07ER46372, NSF Grant Nos. ATM-0802727, ATM-0903915, and AGS-0962698, and NASA Grant Nos. NNX09AJ86G and NNX10AC04G.

PY - 2012/2

Y1 - 2012/2

N2 - 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 zoẑ, 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.

AB - 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 zoẑ, 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.

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U2 - 10.1063/1.3678211

DO - 10.1063/1.3678211

M3 - Article

AN - SCOPUS:84863272876

SN - 1070-664X

VL - 19

JO - Physics of Plasmas

JF - Physics of Plasmas

IS - 2

M1 - 022101

ER -