Sensitivity Kernels for Inferring Lorentz Stresses from Normal-mode Frequency Splittings in the Sun

Abstract

Departures from standard spherically symmetric solar models, in the form of perturbations such as global and local-scale flows and structural asphericities, result in the splitting of eigenfrequencies in the observed spectrum of solar oscillations. Drawing from prevalent ideas in normal-mode-coupling theory in geophysical literature, we devise a procedure that enables the computation of sensitivity kernels for general Lorentz-stress fields in the Sun. Mode coupling due to any perturbation requires careful consideration of self- and cross coupling of multiplets. Invoking the isolated-multiplet approximation allows for limiting the treatment to purely self coupling, requiring significantly less computational resources. We identify the presence of such isolated multiplets under the effect of Lorentz stresses in the Sun. Currently, solar missions allow for precise measurements of self coupling of multiplets via “a-coefficients” and the cross-spectral correlation signal that enables the estimation of the “structure coefficients”. We demonstrate the forward problem for both self coupling (a-coefficients) and cross coupling (structure coefficients). In doing so, we plot the self-coupling kernels and estimate a-coefficients arising from a combination of deep-toroidal and surface-dipolar axisymmetric fields. We also compute the structure coefficients for an arbitrary general magnetic field (real and solenoidal) and plot the corresponding “splitting function”, a convenient way to visualize the splitting of multiplets under 3D internal perturbations. The results discussed in this paper pave the way to formally pose an inverse problem and infer solar internal magnetic fields.

Inferring solar differential rotation through normal-mode coupling using Bayesian statistics

“…In this section we discuss the goodness of the isolated multiplet approximation in estimating a n due to Ω(r, θ) for all the HMI-resolved modes shown in Figure 3. A similar result but for ≤ 30 was presented in Appendix G of Das et al (2020). In Figure 9 we color code multiplets to indicate the departure of frequency shifts obtained from QDPT δ n ω Q m as compared to shifts obtained from DPT δ n ω D m .…”

“…We therefore state at the outset that estimates of a-coefficients determined from frequency splitting serve as reliable measures of differential rotation (Chatterjee & Antia 2009). Nevertheless, the estimation of non-axisymmetric perturbations requires a rigorous treatment honoring the cross coupling of multiplets (Hanasoge et al 2017;Das et al 2020). In such cases, measuring changes to the eigenfunctions is far more effective than, for instance, the a-coefficient formalism.…”

Minuscule Corrections to Near-surface Solar Internal Rotation Using Mode Coupling

“…where δ n ω ℓm and δ n ξ ℓm are the corrections to the eigenfrequencies and eigenfunctions, and ω ref is a reference frequency that may be chosen to be close to the unperturbed frequency n ω ℓ of the multiplet n S ℓ whose perturbed eigenstate we are interested in. This results in a new eigenvalue problem for the supermatrix Z (see Appendix B in Das et al 2020),…”

Recipe for inferring sub-surface solar magnetism via local mode-coupling using Slepian basis functions

“…In the absence of sensitivity kernels for global magnetic fields of a general configuration, earlier studies were limited to a simplified toroidal magnetic field (Gough 1990;Antia et al 2000;Dziembowski & Goode 2004;Kiefer et al 2017;Kiefer & Roth 2018). This long-standing problem was addressed in Das et al (2020) (hereafter D20), where sensitivity kernels for Lorentz-stress due to magnetic field of general geometry was presented. The resultant operator that describes the coupling of two modes closely spaced in frequency in the presence of magnetic field, called the coupling matrix, was demonstrated to be Hermitian.…”