The Behavior of Transverse Waves in Nonuniform Solar Flux Tubes. I. Comparison of Ideal and Resistive Results

Abstract

Magnetohydrodynamic (MHD) waves are ubiquitously observed in the solar atmosphere. Kink waves are a type of transverse MHD waves in magnetic flux tubes that are damped due to resonant absorption. The theoretical study of kink MHD waves in solar flux tubes is usually based on the simplification that the transverse variation of density is confined to a nonuniform layer much thinner than the radius of the tube, i.e., the so-called thin boundary approximation. Here, we develop a general analytic method to compute the dispersion relation and the eigenfunctions of ideal MHD waves in pressureless flux tubes with transversely nonuniform layers of arbitrary thickness. Results for kink waves are produced and are compared with fully numerical resistive MHD eigenvalue computations in the limit of small resistivity. We find that the frequency and resonant damping rate are the same in both ideal and resistive cases. The actual results for thick nonuniform layers deviate from the behavior predicted in the thin boundary approximation and strongly depend on the shape of the nonuniform layer. The eigenfunctions in ideal MHD are very different from those in resistive MHD. The ideal eigenfunctions display a global character regardless of the thickness of the nonuniform layer, while the resistive eigenfunctions are localized around the resonance and are indistinguishable from those of ordinary resistive Alfvén modes. Consequently, the spatial distribution of wave energy in the ideal and resistive cases is dramatically different. This poses a fundamental theoretical problem with clear observational consequences. Subject headings: Sun: oscillations -Sun: atmosphere -Sun: magnetic fields -waves -Magnetohydrodynamics (MHD) omit the intermediate steps and give the final expression of the dispersion relation, namely k ⊥,e ρ e ω 2 −k 2 z v 2 A,e K ′ m [k⊥,e(R+l/2)] K m[ k ⊥,e (R+l/2)] G e − Ξ e k ⊥,e ρ e ω 2 −k 2 z v 2 A,e K ′ m[ k ⊥,e (R+l/2)] K m[ k ⊥,e (R+l/2)] F e − Γ e

Magnetohydrodynamic Oscillations in the Solar Corona and Earth’s Magnetosphere: Towards Consolidated Understanding

“…The analytical theory for resonant absorption in notso-thin layers was constructed by Hollweg (1990), and further developed in a number of follow-up studies, e.g. Soler et al (2013). The mathematical approach is base upon the expansion in the Frobenius series near the resonant point.…”

Model Comparison for the Density Structure Across Solar Coronal Waveguides

“…This is the socalled TTTB approximation, which assumes that the wavelength is long and that the thickness of the nonuniform transitional layer is short in comparison to the radius of the waveguide (R 1, l  l R 1  ) (see, e.g., Hollweg & Yang 1988;Goossens et al 1995Goossens et al , 2002Ruderman & Roberts 2002). The analysis was then extended to include fully nonuniform layers in both one-dimensional (Van Doorsselaere et al 2004;Soler et al 2013) and two-dimensional ) density models.…”

“…Soler et al (2013) describethe basic theory behind the application of the Frobenius method implementation to compute resonantly damped oscillations in nonuniform waveguides. The seismology analysis by Soler et al (2014) for parameters such as the Alfvén speed, the density contrast, and the transverse inhomogeneity length scale shows that significantly different inversion results are obtained depending on the adopted density model.…”

Resonantly damped oscillations of a system of two coronal loops

“…The introduction of a thin inhomogeneous layer leads to a continuous Alfvén spectrum. At the Alfvén resonant magnetic surface, defined by Re ω = ω A , the ideal MHD equations become "quasi-singular" (Soler et al 2013). For the two-loop system, we calculated that just as in cylindrical geometry, the eigenfrequencies are complex and hence the resonant position does not lie on the real axis in the complex plane.…”

“…One needs determine a posteriori whether this dependence still holds in bicylindrical coordinates. It is important to note that the density profile does matter when we consider thick inhomogeneous layers (Soler et al 2013).…”