Quasimodes in the cusp continuum in nonuniform magnetic flux tubes

Abstract

Context. The study of magnetohydrodynamic (MHD) waves is important both for understanding heating in the solar atmosphere (and in particular the corona) and for solar atmospheric seismology. The analytical investigation of wave mode properties in a cylinder is of particular interest in this domain because many atmospheric structures can be modeled as such in a first approximation. Aims. The aim of this study is to use linearized ideal MHD to investigate quasimodes (global modes that are damped through resonant absorption) with a frequency in the cusp continuum, in a straight cylinder with a circular base and an inhomogeneous layer at its boundary that separates two homogeneous plasma regions inside and outside. We are particularly interested in the damping of these modes, and therefore try to determine their frequency as a function of background parameters. Methods. After linearizing the ideal MHD equations, we found solutions to the second-order differential equation for the perturbed total pressure in the inhomogeneous layer in the form of (1) Frobenius series around the regular singular points that are the Alfvén and cusp resonant positions, and (2) power series around regular points. By connecting these solutions appropriately through the inhomogeneous layer and with the solutions of the homogeneous regions inside and outside the cylinder, we derive a dispersion relation for the frequency of the eigenmodes of the system. Results. From the dispersion relation, it is also possible to find the frequency of quasimodes, even though they are not eigenmodes. As an example, we find the frequency of the slow surface sausage quasimode as a function of the width of the inhomogeneous layer for values of the longitudinal wavenumber relevant for photospheric conditions. The results closely match findings by other authors who studied the resistive slow surface sausage eigenmode. We also discuss the perturbation profiles of the quasimode and the eigenfunctions of continuum modes.