Sensitivity of Coronal Loop Sausage Mode Frequencies and Decay Rates to Radial and Longitudinal Density Inhomogeneities: A Spectral Approach

TL;DR

This paper develops a spectral method to analyze how radial and longitudinal density inhomogeneities affect the frequencies and decay rates of coronal loop sausage modes, providing a way to infer internal structure from observations.

Contribution

It introduces a spectral approach and perturbation method to explicitly relate observed eigenfrequencies to density inhomogeneities in coronal loops.

Findings

Eigenvalues are mainly influenced by diagonal matrix terms in the spectral description.

Knowledge of eigenfrequency real and imaginary parts allows inference of boundary layer properties.

Fundamental modes are sensitive only to the loop's outer regions, especially for small density enhancements.

Abstract

Fast sausage modes in solar magnetic coronal loops are only fully contained in unrealistically short dense loops. Otherwise they are leaky, losing energy to their surrounds as outgoing waves. This causes any oscillation to decay exponentially in time. Simultaneous observations of both period and decay rate therefore reveal the eigenfrequency of the observed mode, and potentially insight into the tubes' nonuniform internal structure. In this article, a global spectral description of the oscillations is presented that results in an implicit matrix eigenvalue equation where the eigenvalues are associated predominantly with the diagonal terms of the matrix. The off-diagonal terms vanish identically if the tube is uniform. A linearized perturbation approach, applied with respect to a uniform reference model, is developed that makes the eigenvalues explicit. The implicit eigenvalue problem is easily solved numerically though, and it is shown that knowledge of the real and imaginary parts of the eigenfrequency is sufficient to determine the width and density contrast of a boundary layer over which the tubes' enhanced internal densities drop to ambient values. Linearized density kernels are developed that show sensitivity only to the extreme outside of the loops for radial fundamental modes, especially for small density enhancements, with no sensitivity to the core. Higher radial harmonics do show some internal sensitivity, but these will be more difficult to observe. Only kink modes are sensitive to the tube centres. {Variation in internal and external Alfv\'en speed along the loop is shown to have little effect on the fundamental dimensionless eigenfrequency, though the associated eigenfunction becomes more compact at the loop apex as stratification increases, or may even displace from the apex.