Standing Sausage Perturbations in solar coronal loops with diffuse boundaries: An initial-value-problem perspective

TL;DR

This paper investigates the dispersive properties of fast sausage modes in solar coronal loops with continuous density profiles, revealing that trapped modes may not significantly influence observable oscillations due to their spatial extent and energy reception limitations.

Contribution

It demonstrates that the absence of cutoff wavenumbers does not ensure observable trapped modes in coronal loops, challenging previous assumptions based on classical theory.

Findings

Trapped modes can exist for any axial wavenumber in certain density profiles.

Eigenfunctions of trapped modes have large spatial extents exceeding observational ranges.

The energy transfer to trapped modes is negligible due to their spatial characteristics.

Abstract

Working in pressureless magnetohydrodynamics, we examine the consequences of some peculiar dispersive properties of linear fast sausage modes (FSMs) in one-dimensional cylindrical equilibria with a continuous radial density profile ($\rho_0(r)$). As recognized recently on solid mathematical grounds, cutoff axial wavenumbers may be absent for FSMs when $\rho_0(r)$ varies sufficiently slowly outside the nominal cylinder. Trapped modes may therefore exist for arbitrary axial wavenumbers and density contrasts, their axial phase speeds in the long-wavelength regime differing little from the external Alfv$\acute{\rm e}$n speed. If these trapped modes indeed show up in the solutions to the associated initial value problem (IVP), then FSMs have a much better chance to be observed than expected with classical theory, and can be invoked to account for a considerably broader range of periodicities than practiced. However, with axial fundamentals in active region loops as an example, we show that this long-wavelength expectation is not seen in our finite-difference solutions to the IVP, the reason for which is then explored by superposing the necessary eigenmodes to re-solve the IVP. At least for the parameters we examine, the eigenfunctions of trapped modes are characterized by a spatial extent well exceeding the observationally reasonable range of the spatial extent of initial perturbations, meaning a negligible fraction of energy that a trapped mode can receive. We conclude that the absence of cutoff wavenumbers for FSMs in the examined equilibrium does not guarantee a distinct temporal behavior.