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

TL;DR

The paper investigates how self-consistently generated large-scale magnetic fields beneath the solar surface affect the surface gravity ($f$) mode. Using a 3D two-layer Cartesian model with an $oldsymbol{α}^2$ dynamo in the lower layer, the authors analyze $k$–$ ext{ω}$ diagrams from the vertical velocity at the interface and fit Lorentzians to extract $f$-mode properties. They find that during the kinematic dynamo phase the $f$-mode resembles the non-magnetic case, but once the magnetic field saturates near equipartition, the $f$-mode strengthens, shifts to higher frequencies, and broadens—particularly at larger horizontal wavenumbers $k_x$. This supports observational indications of $f$-mode strengthening by subsurface magnetic fields and highlights a potential diagnostic for mapping such fields, while acknowledging limitations from the idealized isothermal bulk and high-$ ext{ℓ}$ relevance.

Abstract

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.

Figures (5)

• Figure 1: Top: Schematic of the the two-layer simulation domain. Bottom: Equilibrium profiles of (a) density, (b) pressure, and (c) temperature as a function of $z$ for $L_z/L_0 = \pi$. The red dotted line marks the interface at $z=0$.
• Figure 2: $k$-$\omega$ diagram for the non-magnetic run h1. The dotted and dot-dashed lines show $\tilde{\omega}=c_\mathrm{su}\tilde{k}_x$ and $\tilde{\omega}=c_\mathrm{sd}\tilde{k}_x$ respectively. The dashed and solid curves show $\tilde{\omega}_\mathrm{f0}$ and $\tilde{\omega}_\mathrm{f}$ respectively.
• Figure 3: (a) Time evolution of the kinetic and magnetic energies for run d1. Three phases, (I) kinematic, (II) weak nonlinear, and (III) saturated, of the $\alpha^2$ dynamo are highlighted. (b) Kinetic and magnetic energy spectra are shown from the plane at $z=-0.1L_0$, during the kinematic and saturated phases. Black dashed line shows the forcing wavenumber $k_\mathrm{f}$. (c) Space-time diagram of the mean magnetic field components. Magnetic fields are first averaged over the $x$-$z$ plane to define $\langle B_x\rangle_{xz}$ and $\langle B_z\rangle_{xz}$. These quantities are then normalized by their corresponding rms values $\sqrt{\langle\langle B_x\rangle_{xz}^2\rangle_y}$ and $\sqrt{\langle\langle B_z\rangle_{xz}^2\rangle_y}$, respectively. (d) Vertical profile of horizontally averaged plasma beta $\beta(z)$, in the saturated phase. Black circles show the eigenfunction of the $f$-mode near the free surface, scaled by a factor of $10^{10}$ for clarity. The red dashed line marks $z=-0.1L_0$.
• Figure 4: Left: Same as Figure \ref{['fig:2']} but for the magnetic run d1. A noticeable broadening of the $f$-mode is seen with increasing $\tilde{k}_x$. Right: Line profiles of the $f$, $p_0$, and $p_1$-mode at $\tilde{k}_x=2$ for the kinematic phase (panel a) and for the saturated phase (panel b); dotted curves represent the data. To ensure a consistent comparison, the background has been removed. Blue dashed lines show the locations of the $f$-mode.
• Figure 5: Variation of different $f$-mode characteristics with $\tilde{k}_x$ for different runs. (a) Mode strength ($\mu_f$), (b) relative frequency shift, and (c) full width at half maximum (FWHM) of the $f$-mode, all as functions of $\tilde{k}_x$. Tick marks at the top indicate the corresponding spherical harmonic degree $\ell$.