# An Inside Look at Sunspot Oscillations with Higher Azimuthal Wavenumbers

## TL;DR

This study provides the first direct observational evidence of an m=1 slow magneto-acoustic wave mode in sunspot chromospheric umbrae, using Fourier filtering and theoretical modeling to analyze wave signatures.

## Contribution

The paper identifies and characterizes the m=1 slow magneto-acoustic mode in sunspot umbrae, a novel observation not previously described.

## Key findings

- Detection of m=1 mode with ~170 s period
- Fourier filtering isolates specific wave components
- Theoretical models confirm observational interpretation

## Abstract

Solar chromospheric observations of sunspot umbrae offer an exceptional view of magneto-hydrodynamic wave phenomena. In recent years, a wealth of wave signatures related to propagating magneto-acoustic modes have been presented, which demonstrate complex spatial and temporal structuring of the wave components. Theoretical modelling has demonstrated how these ubiquitous waves are consistent with an m=0 slow magneto-acoustic mode, which are excited by trapped sub-photospheric acoustic (p-mode) waves. However, the spectrum of umbral waves is broad, suggesting that the observed signatures represent the superposition of numerous frequencies and/or modes. We apply Fourier filtering, in both spatial and temporal domains, to extract chromospheric umbral wave characteristics consistent with an m=1 slow magneto-acoustic mode. This identification has not been described before. Angular frequencies of 0.037 +/- 0.007 rad/s (2.1 +/- 0.4 deg/s), corresponding to a period approximately 170 s for the m=1 mode are uncovered for spatial wavenumbers in the range of 0.45<k<0.90 arcsec^-1 (5000-9000 km). Theoretical dispersion relations are solved, with corresponding eigenfunctions computed, which allows the density perturbations to be investigated and compared with our observations. Such magnetohydrodynamic modelling confirms our interpretation that the identified wave signatures are the first direct observations of an m=1 slow magneto-acoustic mode in the chromospheric umbra of a sunspot.