Cycle dependence of helioseismic oscillations above the acoustic cut-off frequency

TL;DR

This work addresses why helioseismic pseudomodes above the acoustic cut-off vary with the solar cycle. It develops an analytical Klein-Gordon cavity model in which the subsurface cavity acts as a Fabry-Pérot interferometer for high-frequency waves, deriving a dispersion relation that isolates the influence of the source depth $r_0$ and the photospheric cut-off $α$ on the pseudomode spectrum. The model, validated by Bayesian MCMC fits to GONG observations, shows that pseudomode peak frequencies shift in ways that depend on both parameters, with $r_0$-driven changes producing robust anti-phase modulations relative to the 11-year cycle and $α$-driven changes yielding harmonic-dependent phase behavior. The findings imply pseudomodes are a powerful diagnostic for subsurface solar and stellar structure and dynamics, while highlighting the need for more realistic profiles and multidimensional modelling for precise inference.

Abstract

Helioseismic and recent asteroseismic observations reveal fine structure in the power spectrum with alternating peaks and troughs above the acoustic cut-off frequency. This structure is interpreted as the interference patterns of high-frequency acoustic waves excited in the solar interior and propagating into the atmosphere, known as pseudomodes. Pseudomodes exhibit clear solar-cycle variability, with frequency shifts that occur predominantly in anti-phase with the activity cycle, although the underlying mechanism remains uncertain. This work investigates how the subsurface excitation source location and the photospheric acoustic cut-off frequency influence the formation, frequency distribution, and solar-cycle variability of pseudomodes. We employ an analytical Klein-Gordon subsurface cavity model, which is shown to act as an effective Fabry-Pérot interferometer for high-frequency waves that experience constructive and destructive interference between the source location and the lower turning point. We derive an effective dispersion relation isolating the effects of the source location and photospheric cut-off on the pseudomode frequency. The model reproduces the observed peak-trough pseudomode spectrum for reasonable parameter values constrained by Bayesian MCMC best-fitting to GONG observations. We also find that solar-cycle-associated 11-year modulations of the source location result in anti-phase pseudomode frequency shifts, whereas similar cyclic variations in the cut-off frequency produce harmonic-dependent behaviour, yielding both in-phase and anti-phase shifts. As the acoustic cut-off and mode excitation relate to stratification and flows in the solar interior, the results highlight pseudomodes as a powerful diagnostic tool for changes in subsurface solar and stellar structure through the solar cycle.

Figures (8)

• Figure 1: Acoustic cut-off frequency radial profile (the thick black line) given by Eq. (\ref{['eq:cutoff_profile']}), normalised to $\nu_0=833$$\mu$Hz (corresponding to the 20-min acoustic travel time through the subsurface acoustic cavity of $0.1R_\odot$ depth). The radial distance $r$ is normalised to $a=0.1R_\odot$ corresponding to the acoustic cavity depth of $\ell=100$ modes with frequencies near the 3-mHz spectral peak. The height $r/a=1$ is the photospheric level, and $r=0$ mimicks the wave's lower turning point. The oscillatory signals (the thin grey lines) illustrate the spatial structure of low-frequency (below the acoustic cut-off, $\alpha=40$ which is about 5300 $\mu$Hz if one accounts for normalisation to $\nu_0=833\,\mu$Hz, $\alpha\nu_0/2\pi$) resonant acoustic waves of radial harmonics $n=1$, 5, and 10 (with $\omega/\nu_0 = 3.06$, 15.3, and 30.48, respectively) trapped within the cavity; and the high-frequency (above the acoustic cut-off) propagating pseudomode with $\omega/\nu_0 = 53.65$. The wave forms are obtained as numerical solutions of the governing Eq. (\ref{['eq:KG']}).
• Figure 2: The Fourier power spectrum of low-frequency (below the acoustic cut-off frequency $\omega<\alpha$) and high-frequency ($\omega>\alpha$) acoustic waves in a subphotospheric cavity shown in Fig. \ref{['fig:cutoff_profile']}. The spectrum is given by Eq. (\ref{['eq:spectrum']}) with the acoustic cut-off frequency $\alpha/\nu_0 = 40$ and the wave source height $r_0/a=0.91$. The normalisation parameters $\nu_0$ and $a$ are the same as in Fig. \ref{['fig:cutoff_profile']}.
• Figure 3: The snapshots of the high-frequency (above acoustic cut-off, $\omega>\alpha$) acoustic perturbation function $\psi(r)$ taken at $t=4.62$ (in solid red), obtained numerically from the governing Eq. (\ref{['eq:KG']}) with the acoustic cut-off radial profile given by Eq. (\ref{['eq:cutoff_profile']}), driver given by Eq. (\ref{['eq:driver']}), and $\alpha/\nu_0=40$ and $r_0/a=0.91$. The top and bottom panels show the corresponding wave solutions in the regime of constructive (top) and destructive (bottom) interference with $\omega/\nu_0 = 53.65 + \pi/4r_0$ and $\omega/\nu_0 = 55.25 + \pi/4r_0$ (cf. Fig. \ref{['fig:spec_full']}). The constant offset $\pi/4r_0$ is added to account for the initial phase shift caused by a dipole spatial structure of the driver Eq. (\ref{['eq:driver']}) being an odd function of $r$. The vertical dashed and dotted lines show the lower ($r=0$) and upper ($r/a=1$) boundaries of the cavity and the position of the wave source ($r=r_0$), respectively. The dashed oscillatory signals in both panels illustrate the corresponding solutions with no reflection at the lower boundary, i.e. no interference effect. The normalisation parameters $\nu_0$ and $a$ are the same as in Fig. \ref{['fig:cutoff_profile']}.
• Figure 4: Time-distance plots illustrating the full spatio-temporal evolution of the high-frequency acoustic perturbation function $\psi(t,r)$ obtained numerically from Eq. (\ref{['eq:KG']}) and the same model setup as described in the caption to Fig. \ref{['fig:snapshots']}. The top and bottom panels illustrate the regimes of constructive and destructive intereference of acoustic waves, forming pseudomode peaks and troughs in the high-frequency part ($\omega>\alpha$) of the Fourier spectrum shown in Fig. \ref{['fig:spec_full']}. In both panels, the colour bars show $\psi(t,r)$ normalised to its maximum; the time and distance are normalised to $\tau_\mathrm{A}=20$ min and $a=0.1 R_\odot$.
• Figure 5: Left column: pseudomode oscillation spectra given by Eq. (\ref{['eq:spec_pseudo']}) for the acoustic cut-off frequency $\alpha = 38$ (blue boxes), 30 (green diamonds), 10 (red circles) and fixed source height $r_0=0.8$ (top); and the source height $r_0 = 0.85$ (blue boxes), 0.86 (red circles) and fixed cut-off frequency $\alpha=30$ (bottom). Right column: the corresponding pseudomode peak frequency shifts caused by the effect of $\alpha$ varying from 10 to 38 (top) and $r_0$ varying from 0.86 to 0.85 (bottom). The normalisation of all parameters is the same as in Fig. \ref{['fig:cutoff_profile']}.