# Normal modes of transverse coronal loop oscillations from numerical simulations: I. Method and test case

## TL;DR

This paper introduces an iterative numerical method combining time-dependent simulations and CEOF analysis to accurately determine the normal modes of coronal loops, validated against an analytical test case.

## Contribution

The paper presents a novel iterative approach that integrates numerical simulations and CEOF analysis to extract coronal loop normal modes, improving accuracy over previous methods.

## Key findings

- Method achieves less than 0.7% error in frequency and eigenfunctions after 6 iterations.
- Eigenfunctions with discontinuities show larger errors at discontinuity positions.
- Validated approach with an analytical test case before applying to real data.

## Abstract

The purpose of this work is to develop a procedure to obtain the normal modes of a coronal loop from time-dependent numerical simulations with the aim of better understanding observed transverse loop oscillations. To achieve this goal, in this paper we present a new method and test its performance with a problem for which the normal modes can be computed analytically. In a follow-up paper, the application to the simulations of \citet{rial2013} is tackled. The method proceeds iteratively and at each step consists of (i) a time-dependent numerical simulation followed by (ii) the Complex Empirical Orthogonal Function (CEOF) analysis of the simulation results. The CEOF analysis provides an approximation to the normal mode eigenfunctions that can be used to set up the initial conditions for the numerical simulation of the following iteration, in which an improved normal mode approximation is obtained. The iterative process is stopped once the global difference between successive approximate eigenfunctions is below a prescribed threshold. The equilibrium used in this paper contains material discontinuities that result in one eigenfunction with a jump across these discontinuities and two eigenfunctions whose normal derivatives are discontinuous there. After 6 iterations, the approximation to the frequency and eigenfunctions are accurate to $\lesssim 0.7$\% except for the eigenfunction with discontinuities, which displays a much larger error at these positions.