Acoustic wave propagation in the solar sub-photosphere with localised magnetic field concentration: effect of magnetic tension

Aims. In this paper we analyse numerically the propagation and dispersion of acoustic waves in the solar-like sub-photosphere with localised non-uniform magnetic field concentrations, mimicking sunspots with various representative magnetic field configurations. Methods. Numerical simulations of wave propagation through the solar sub-photosphere with a localised magnetic field concentration are carried out using SAC, which solves the MHD equations for gravitationally stratified plasma. The initial equilibrium density and pressure stratifications are derived from a standard solar model. Acoustic waves are generated by a source located at the height approximately corresponding to the visible surface of the Sun. We analyse the response of vertical velocity to changes in the interior due to magnetic field at the level corresponding to the visible solar surface, by the means of local time-distance helioseismology. Results. The results of numerical simulations of acoustic wave propagation and dispersion in the solar sub-photosphere with localised magnetic field concentrations of various types are presented. Time-distance diagrams of the vertical velocity perturbation at the level corresponding to the visible solar surface show that the magnetic field perturbs and scatters acoustic waves and absorbs the acoustic power of the wave packet. For the weakly magnetised case the effect of magnetic field is mainly thermodynamic, since the magnetic field changes the temperature stratification. However, we observe the signature of slow magnetoacoustic mode, propagating downwards, for the strong magnetic field cases.

💡 Research Summary

The paper presents a systematic numerical investigation of how localized, non‑uniform magnetic fields in the solar sub‑photosphere modify the propagation and dispersion of acoustic waves. Using the Solar Acoustic Code (SAC), which solves the full set of magnetohydrodynamic (MHD) equations for a gravitationally stratified plasma, the authors construct equilibrium models based on a standard solar interior model (SM). Three representative magnetic configurations are examined: (i) a weak, nearly uniform field of a few hundred gauss, (ii) a strong, concentrated flux‑tube field exceeding 1 kG, and (iii) a more diffuse Gaussian‑shaped field. The magnetic structures are superimposed on the hydrostatic background, thereby altering the temperature and pressure stratifications in a physically consistent way.

Acoustic waves are excited by a Gaussian pressure pulse placed at a height corresponding to the visible solar surface (approximately the photosphere). The source generates a broadband wave packet that contains the dominant p‑mode and f‑mode frequencies observed on the Sun. The vertical velocity component (v_z) is recorded at the same height for a series of horizontal distances, and time‑distance helioseismology techniques (cross‑correlation of v_z signals) are applied to extract travel‑time shifts and amplitude changes.

The simulation results reveal two distinct regimes. In the weak‑field case, the magnetic field primarily influences the thermodynamic structure: the temperature profile is slightly modified, leading to a modest change in the local sound speed (c_s). Consequently, the wavefronts retain their original phase but exhibit a 5–10 % reduction in acoustic power. This effect is interpreted as a purely thermodynamic perturbation, consistent with earlier non‑magnetic helioseismic models.

In the strong‑field regime, the Alfvén speed (v_A) becomes comparable to or exceeds the sound speed, and magnetic tension contributes significantly to the restoring forces of the plasma. Two new phenomena appear: (1) a slow magneto‑acoustic mode propagates predominantly downward along the field lines, producing a delayed arrival time and a negative phase shift in the v_z time‑distance diagrams; (2) the incident acoustic wave is partially converted, scattered, and absorbed at the magnetic boundary, leading to a 20–30 % loss of acoustic power. The strong field also induces strong refraction of high‑angle rays, suppressing high‑frequency components in the power spectrum.

The geometry of the magnetic concentration further modulates these effects. A narrow flux‑tube creates a sharp discontinuity in the plasma parameters, enhancing reflection and scattering, whereas a diffuse Gaussian field yields smoother gradients and consequently milder attenuation. Quantitatively, the authors demonstrate that travel‑time perturbations and power deficits scale with both the field strength and the steepness of the magnetic gradient.

By comparing the simulated time‑distance diagrams with observational signatures of acoustic shadows and power deficits around sunspots, the study provides a physical explanation for the observed helioseismic anomalies. It shows that magnetic tension and the associated slow mode cannot be neglected in any realistic inversion of sub‑photospheric structures beneath active regions.

The authors acknowledge limitations: the simulations are performed in a 2.5‑D geometry, neglecting fully three‑dimensional effects, non‑linear wave–field interactions, and background convective flows. Future work should incorporate realistic sunspot models, turbulent convection, and direct comparisons with high‑resolution helioseismic data to refine the quantitative predictions.

Overall, the paper convincingly demonstrates that localized magnetic fields in the solar sub‑photosphere alter acoustic wave propagation through both thermodynamic modifications and magnetically induced tension effects. These findings are essential for improving the accuracy of time‑distance helioseismology, especially when probing the deep structure of sunspots and other magnetically active regions.