Effects of Dynamo-Generated Large-Scale Magnetic Fields on the Surface Gravity ($f$) Mode

By modelling the upper layers of the Sun in terms of a two-layer setup where a free-surface exists within the computational domain, we numerically study the interaction between the surface gravity, or the fundamental ($f$) mode, and the magnetic fields. Earlier such works were idealized in the sense that the static magnetic fields were imposed below the photosphere, i.e., the free-surface, to detect signatures of sub-surface magnetic fields and flows on the $f$-mode. In this work, we perform three-dimensional (3D) numerical simulations where the interior fluid below the photosphere is stirred helically at small scales, thus facilitating an $α^2$-dynamo. This allows us to investigate how these self-consistently generated large-scale magnetic fields influence the properties of the $f$-mode. We find that when the magnetic fields saturate near the equipartition values with the turbulent kinetic energy of the flow, the $f$-mode is significantly perturbed. Compared to the non-magnetic case, or the kinematic phase of the dynamo when fields are too weak, we note that the frequencies and the strengths of the $f$-mode are enhanced in presence of saturated magnetic fields, with these effects being larger at larger wavenumbers. This qualitatively confirms the earlier findings from observational and numerical works which reported the $f$-mode strengthening due to strong sub-surface magnetic fields.

💡 Research Summary

This paper investigates how large‑scale magnetic fields generated by an α² dynamo influence the solar surface gravity (fundamental, or f) mode. The authors construct a simplified two‑layer Cartesian model that mimics a small patch of the solar surface: a deep, isothermal lower layer representing the upper convection zone and a thin, hotter upper layer representing the atmosphere. The interface at z = 0 acts as a free surface where the f‑mode is naturally trapped.

Instead of imposing a static magnetic field below the surface—as done in many earlier numerical studies—the authors drive helical turbulence in the lower layer with a stochastic forcing function. This helicity breaks mirror symmetry and enables a self‑consistent α² dynamo to develop. The magnetic field therefore grows from a weak seed, first in a kinematic phase (magnetic energy ≪ kinetic energy) and later saturates at roughly equipartition with the turbulent kinetic energy.

The governing equations are the compressible magnetohydrodynamic (MHD) set: continuity, momentum, entropy, and induction equations, solved with the high‑order Pencil Code. Viscosity (ν) and magnetic diffusivity (η) are kept small (10⁻³–10⁻⁴) to minimise artificial damping, while a temperature‑relaxation term is added in both layers to prevent secular heating from viscous and Ohmic dissipation. Periodic boundary conditions are used horizontally; vertically, free‑slip velocity, fixed temperature, zero‑gradient density, and perfectly conducting magnetic boundaries are imposed.

Two simulations are performed at a resolution of 1024 × 256 × 320: (h1) a purely hydrodynamic run, and (d1) an otherwise identical run that allows magnetic field growth. The vertical velocity at the interface, u_z(t, x, y, z = 0), is sampled at regular intervals, forming a three‑dimensional data cube analogous to solar Dopplergrams. A three‑dimensional Fourier transform yields the power spectrum P(ω, k_x, k_y). By setting k_y = 0, the authors construct k‑ω diagrams for each horizontal wavenumber k_x.

For each k_x, the spectrum is fitted with a Lorentzian representing the f‑mode plus a linear background. The fit provides the central frequency ω_c, amplitude A, and linewidth Γ. From these quantities three diagnostics are derived: (i) the relative frequency shift δ = (ω_c² − ω_f²)/ω_f², where ω_f is the theoretical f‑mode frequency in the absence of magnetic fields; (ii) the mode strength μ_f, defined as the integrated excess power over the background; and (iii) the dimensionless linewidth Γ_f = Δω_FWHM/ω_c, the inverse of the quality factor.

Results show that during the kinematic phase of the dynamo the f‑mode properties are indistinguishable from the hydrodynamic case. Once the magnetic field reaches equipartition and saturates, the f‑mode exhibits a clear upward shift in frequency and a substantial increase in amplitude. Both effects become more pronounced at larger k_x (higher‑degree modes). The linewidth slightly narrows, indicating reduced damping. These findings confirm earlier observational reports of f‑mode strengthening prior to active‑region emergence (e.g., Thompson 2006; Singh et al. 2016) and previous numerical experiments that imposed strong static fields beneath the surface (e.g., Kishore et al. 2024).

The novelty of this work lies in using a self‑generated magnetic field rather than an imposed one, thereby providing a more realistic representation of solar interior dynamics. Despite the simplifications (isothermal lower layer, sharp density jump), the study robustly demonstrates that equipartition‑level magnetic fields can measurably modify the f‑mode. The authors suggest future extensions: incorporating a weakly super‑adiabatic convection zone, exploring a broader range of wavenumbers, and performing quantitative comparisons with helioseismic observations. Such advances could establish the f‑mode as a sensitive diagnostic of subsurface magnetic structures, potentially improving forecasts of sunspot and active‑region emergence.