Propagating Linear Waves in Convectively Unstable Stellar Models: A Perturbative Approach

Abstract

Linear time-domain simulations of acoustic oscillations are unstable in the stellar convection zone. To overcome this problem it is customary to compute the oscillations of a stabilized background stellar model. The stabilization, however, affects the result. Here we propose to use a perturbative approach (running the simulation twice) to approximately recover the acoustic wave field, while preserving seismic reciprocity. To test the method we considered a 1D standard solar model. We found that the mode frequencies of the (unstable) standard solar model are well approximated by the perturbative approach within 1 µHz for low-degree modes with frequencies near 3 mHz. We also show that the perturbative approach is appropriate for correcting rotational-frequency kernels. Finally, we comment that the method can be generalized to wave propagation in 3D magnetized stellar interiors because the magnetic fields have stabilizing effects on convection.Keywords: Stellar models · Helioseismology · Magnetic fields · Numerical methods Time-Domain Simulations of Linear OscillationsHelioseismology is used to study complex phenomena in the solar interior and atmosphere, such as flows and magnetic heterogeneities that cover many temporal and spatial scales. Numerical simulations of wave propagation are a crucial tool for modeling and interpreting helioseismic observations. The same simulations should find applications in the study of stellar oscillations as well (low-degree modes).Acoustic waves in the Sun have very low amplitudes compared with those of the background (Christensen-Dalsgaard, 2002) and thus can be treated as weak perturbations with respect to a background reference model.

Simulating acoustic waves in spotted stars

“…The use of the linear approximation is justified since acoustic wave perturbations in the Sun and solar-like stars have much smaller amplitudes compared to the stellar background quantities (e.g., velocity perturbations at the surface are < 20 cm/s, four orders of magnitude smaller than the local sound speed, see Libbrecht 1988). For the stellar model we considered a spherically symmetric static equilibrium described by Model S (Christensen-Dalsgaard et al 1996) stabilized against convection (Papini et al 2014) and including the photosphere up to R = 1.0007 R , R being the solar radius: that is our quiet Sun (QS) background model. We then added the starspot model to the background.…”

Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flows

“…However, this requires a stabilization of the background model by changing the buoyancy frequency (e.g. Schunker et al 2011;Papini et al 2014). Unless this operation is performed first, the linearized equations allow convective modes that grow exponentially.…”

“…Unless this operation is performed first, the linearized equations allow convective modes that grow exponentially. Unfortunately the mode frequencies are seriously affected by the stabilization of the solar model and become too far from the solar observations (Papini et al 2014). …”

Forward Modeling Helioseismic Signatures of One- and Two-cell Meridional Circulation

“…Convective action in the background model is treated separately by isolating the Brunt-Väisälä frequency (N 2 ). In order to avoid growing convective instabilities, we set the slightly negative values of the Brunt-Väisälä frequency in the convection zone (N 2 < 0) to zero -obviating the need to modify background profiles of density, pressure and the adiabatic ratio (Γ 1 ), see Hanasoge et al (2006); Parchevsky & Kosovichev (2007); Papini et al (2014). A full description and validation of the model's algorithm can be found Stejko et al (2021).…”