Numerical simulations of oscillations for axisymmetric solar backgrounds with differential rotation and gravity

TL;DR

The paper develops a forward solver for axisymmetric solar oscillations in backgrounds that include gravity and differential rotation, using a Hybridizable Discontinuous Galerkin discretization. By recasting the oscillation equations through Liouville-type changes of variables and performing a cylindrical modal decomposition, the authors obtain stable, mode-wise 2.5D formulations that can reproduce observed rotational effects in helioseismic data. They validate the approach by comparing HDG-based power spectra and cross-covariances with solar observations and with Gyre, demonstrating accurate representation of rotation-induced frequency shifts and gravity-driven wave behavior. The framework supports synthetic observables and sets a solid foundation for subsequent inversions, enabling more realistic modeling of solar interior dynamics and potentially inertial modes like Rossby waves.

Abstract

Local helioseismology comprises of imaging and inversion techniques employed to reconstruct the dynamic and interior of the Sun from correlations of oscillations observed on the surface, all of which require modeling solar oscillations and computing Green's kernels. In this context, we implement and investigate the robustness of the Hybridizable Discontinuous Galerkin (HDG) method in solving the equation modeling stellar oscillations for realistic solar backgrounds containing gravity and differential rotation. While a common choice for modeling stellar oscillations is the Galbrun's equation, our working equations are derived from an equivalent variant, involving less regularity in its coefficients, working with Lagrangian displacement and pressure perturbation as unknowns. Under differential rotation and axisymmetric assumption, the system is solved in azimuthal decomposition with the HDG method. Compared to no-gravity approximations, the mathematical nature of the wave operator is now linked to the profile of the solar buoyancy frequency N which encodes gravity, and leads to distinction into regions of elliptic or hyperbolic behavior of the wave operator at zero attenuation. While small attenuation is systematically included to guarantee theoretical well-posedness, the above phenomenon affects the numerical solutions in terms of amplitude and oscillation pattern, and requires a judicious choice of stabilization. We investigate the stabilization of the HDG discretization scheme, and demonstrate its importance to ensure the accuracy of numerical results, which is shown to depend on frequencies relative to N, and on the position of the Dirac source. As validations, the numerical power spectra reproduce accurately the observed effects of the solar rotation on acoustic waves.

Figures (16)

• Figure 1: Representation of the reference flow $\boldsymbol{\chi}_0$ and perturbed flow $\boldsymbol{\chi}$ and displacement $\Delta^L_{\boldsymbol{\chi}}$ defined in \ref{['Ldisp::def']}.
• Figure 2: Illustration of the numerical domain for axisymmetry which corresponds to the meridional half-disk. The boundary is separated between $\partial D^a$ on axis $\mathbf{e}_z$, and $\partial D^b$ for the exterior.
• Figure 3: Radial solar background models for the density $\rho_0$ (top), wave-speed $c_0$ (bottom left) and gravity potential $\phi_0'$ (bottom right) corresponding to the standard model-S, christensen1996currentAtmoI2020 (all units in SI). The plot of the density also highlights the different layers of the Sun. The position $r=1$ corresponds to the solar radius $r_{\odot}$, numerical experiments are carried out up to scaled radius $r_{\max}=1.001$ which corresponds to 700 above solar radius.
• Figure 4: The buoyancy frequency (or Brunt--Väisälä frequency) is represented with a solid line, Lamb frequencies $S_\ell$ are represented with dotted line for different harmonic degree $\ell$. The horizontal dashed lines indicate the three representative frequencies that we use in the numerical experiments, 0.2, 3.0, and 6.0m. The right panel shows $N^2$ crossing with $\omega^2$ near the surface at 0.2m (in $r=$0.999949) and at 3m (in $r=$0.9999995).
• Figure 5: Comparison of the solutions $\mathrm{Re}(w_\bullet)$ for the different formulations for a source positioned in $(1,0)$, at mode $m=0$ and frequency $6$m. The zoom near the source corresponds to the zone $(0.99,1.001) \times (-0.005,0.02)$, and uses a different scaling to improve visualization.

Theorems & Definitions (33)

• Remark 1: Centrifugal potential
• Remark 2: Effective potential
• Remark 3: No-resonance assumption
• Remark 4
• proof
• Definition 1
• Remark 5
• Remark 6
• Definition 2: Differential flow backgrounds
• proof