Temporal evolution of axially standing kink motions in solar coronal slabs: An eigenfunction expansion approach

TL;DR

This study investigates the evolution of kink motions in solar coronal slabs using an eigenfunction expansion approach, revealing how proper and improper eigenmodes influence oscillation periodicities and their potential for solar seismology.

Contribution

It introduces a novel eigenfunction expansion method to analyze 2D axial kink motions, distinguishing contributions from proper and improper eigenmodes in open systems.

Findings

Proper eigenmodes dominate long periodicities, especially with higher density contrast.

Improper continuum modes can cause short periodicities depending on initial conditions.

Oscillation frequencies align with discrete leaky modes despite different boundary conditions.

Abstract

We aim to provide more insights into the applicability to solar coronal seismology of the much-studied discrete leaky modes (DLMs) in classic analyses. Under linear ideal pressureless MHD, we examine two-dimensional (2D) axial fundamental kink motions that arise when localized velocity exciters impact some symmetric slab equilibria. Continuous structuring is allowed for. A 1D initial value problem (IVP) is formulated in conjunction with an eigenvalue problem (EVP) for laterally open systems, with no strict boundary conditions (BCs) at infinity. The IVP is solved by eigenfunction expansion, allowing a clear distinction between the contributions from proper eigenmodes and improper continuum eigenmodes. Example solutions are offered for parameters typical of active region loops. Our solutions show that the system evolves towards long periodicities due to proper eigenmodes (of order the axial Alfven time), whereas the interference of the improper continuum may lead to short periodicities initially (of order the lateral Alfven time). Specializing to the slab axis, we demonstrate that the proper contribution strengthens with the density contrast, but may occasionally be stronger for less steep density profiles. Short periodicities are not guaranteed in the improper contribution, the details of the initial exciter being key. When identifiable, these periodicities tend to agree with the oscillation frequencies expected for DLMs, despite the differences in the BCs between our EVP and classic analyses. The eigenfunction expansion approach enables all qualitative features to be interpreted as the interplay between the initial exciter and some response function, the latter solely determined by the equilibria. Classic theories for DLMs can find seismological applications, with time-dependent studies offering additional ways for constraining initial exciters.