The effect of line-tying on tearing modes

Abstract

Cylindrical magnetohydrodynamic (MHD) constant-ψ or nonconstant-ψ tearing modes that are linearly unstable with periodic axial boundary conditions are studied in a line-tied cylinder. Examples of these two respective classes of modes, with m=1 and m=2 (m being the azimuthal mode number), are studied. With a suitable MHD equilibrium, the former modes are marginally stable in ideal MHD for periodic axial boundary conditions, and occur as fast tearing modes (resistive kinks) in the presence of resistivity η. The latter modes are stable in ideal MHD for periodic axial boundary conditions, and with resistivity occur as constant-ψ tearing modes, unstable in a range of parameters. In both cases, the results for the line-tied modes show the expected tearing scaling with η for very long plasmas, but the scaling becomes γ∝η for smaller cylinder lengths L. These results are consistent with the following interpretation: For L→∞, the modes have a tearing width characteristic of tearing, leading to characteristic tearing mode growth. As L decreases, the modes develop a geometric width, which increases as L decreases; the γ∝η scaling occurs when L is small enough that the geometric width exceeds the tearing width.

The effect of plasma flow on line-tied magnetohydrodynamic modes

“…Indeed, the modes for L → ϱ are tearing modes, and in Ref. 8 it was concluded that for long cylinder lengths the line-tied kink modes still behave as tearing modes, with growth rates proportional to the appropriate fractional power of resistivity. Close to marginal stability, on the other hand, the line-tied tearing modes evolve to global resistive diffusion modes, with ␥ ϰ .…”

“…Further, the threshold for this change occurs when the "geometric width" equals the tearing layer width. 8,9 The investigation of the stability of line-tied kink modes for equilibria 1-3 with axial flow allows one to explore the effects of the flow for the following relevant situations: ͑1͒ When the line-tied kink modes behave as either ideal modes or as tearing modes for long cylinder lengths and ͑2͒ when the modes behave as either ideal MHD modes or as resistive diffusion modes for short cylinder lengths close to marginal stability. In the next subsection, we briefly discuss the methods used for the stability analysis.…”

“…Equilibrium 3 was used in Ref. 8 to investigate the existence of line-tied kink-tearing modes. Indeed, the modes for L → ϱ are tearing modes, and in Ref.…”

“…The theoretical work has focused on ideal and resistive MHD 1-7 with a strong focus on the effect of plasma resistivity, and whether magnetic reconnection and modes with a tearing-like character can exist in the presence of line-tying. 8,9 Another line of work has dealt with the effect of boundary conditions that are more general than simple line-tying boundary conditions. 10,11 This theoretical work has been motivated by results from two linear experiments in cylindrical devices.…”

Three-dimensional magnetic reconnection and its application to solar flares

“…Pritchett, Lee & Drake 1980;Ishii, Azumi & Kishimoto 2002;Bierwage et al 2005;Janvier, Kishimoto & Li 2011;Wang, Yokoyama & Isobe 2015). Finally the effect of line-tying, relevant in the context of coronal magnetic field, was investigated in Delzanno & Finn (2008), which found that the growth rate scaled with η for small structure lengths (compared with the system size), while it was tearing-like for long structures. This may be of interest considering different reconnecting coronal loops in the context of say, nanoflares involving small loops versus the larger structures generally seen in flares.…”