Standing oscillations in a resonant sunspot atmosphere captured by integral field spectroscopy

The solar atmosphere is replete with magnetohydrodynamic wave activity, with magnetic structures such as sunspots channelling wave energy flux efficiently into the outer atmosphere. Steep density and temperature gradients between the photosphere and chromosphere provide ideal conditions for magnetoacoustic resonance cavities, amplifying $\sim 5$ mHz oscillatory power in sunspot atmospheres. However, diagnosis of such cavities has largely been limited to lines such as Ca II H/K and He I 10830 Å, with no evidence yet from layers probed by the Na I D$_1$/D$_2$ doublet. Here we use the newly commissioned integral field unit FRANCIS to examine oscillations spanning the formation heights of the Na I D$_1$/D$_2$ lines and determine whether propagating and/or standing modes are present within a sunspot umbra. The RH1.5D code estimated formation heights for three windows: the Na I D$_1$ wing (core $-300$ mÅ; $\approx 355$ km), the Na I D$_1$ core ($\approx 750$ km), and the Na I D$_2$ core ($\approx 850$ km). Wavelet cross-correlation of line-core and bisector Doppler velocities yielded phase spectra versus height, classifying the dominant $\sim 5.5$ mHz oscillations as propagating or standing-like. At the umbra-penumbra boundary we find propagating modes with energy fluxes of $\sim 1.3 \times 10^{4}$ W m$^{-2}$ in the upper photosphere, falling to $\sim 3.1 \times 10^{3}$ W m$^{-2}$ in the lower chromosphere, implying a damping length $L_d \approx 363$ km, comparable to the local density scale height. In contrast, near-zero phase differences dominate regions of enhanced chromospheric power at the umbral centre, evidencing standing-wave behaviour and resonance-cavity dynamics. These results demonstrate the suitability of solar integral field units for mapping sunspot wave properties, with the Na I D$_1$/D$_2$ lines offering a novel diagnostic of resonance cavities and energy flux.