Reduced magnetohydrodynamic theory of oblique plasmoid instabilities

S. D. Baalrud; A. Bhattacharjee; Y. -M. Huang

doi:10.1063/1.3678211

paper

Reduced magnetohydrodynamic theory of oblique plasmoid instabilities

arXiv:1201.0313 · doi:10.1063/1.3678211

Abstract

The three-dimensional nature of plasmoid instabilities is studied using the reduced magnetohydrodynamic equations. For a Harris equilibrium with guide field, represented by $\vc{B}_o = B_{po} \tanh (x/λ) \hat{y} + B_{zo} \hat{z}$, a spectrum of modes are unstable at multiple resonant surfaces in the current sheet, rather than just the null surface of the polodial field $B_{y o} (x) = B_{p o} tanh (x / λ)$ , which is the only resonant surface in 2D or in the absence of a guide field. Here $B_{p o}$ is the asymptotic value of the equilibrium poloidal field, $B_{z o}$ is the constant equilibrium guide field, and $λ$ is the current sheet width. Plasmoids on each resonant surface have a unique angle of obliquity $θ \equiv arctan (k_{z} / k_{y})$ . The resonant surface location for angle $θ$ is $x_s = - λ\arctanh (\tan θB_{zo}/B_{po})$, and the existence of a resonant surface requires $∣ θ ∣ < arctan (B_{p o} / B_{z o})$ . The most unstable angle is oblique, i.e. $θ \neq = 0$ and $x_{s} \neq = 0$ , 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 $N \sim S_{L}^{3/8} (1 - μ^{2})^{- 1/4} (1 + μ^{2})^{3/4}$ .

9 pages, 8 figures, to be published in Physics of Plasmas

Reduced magnetohydrodynamic theory of oblique plasmoid instabilities

Abstract

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