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Inertial wave modes in the Sun are of interest owing to their potential to reveal new insight into the solar interior. These predominantly retrograde-propagating modes in the solar subsurface appear to deviate from the thin-shell Rossby-Haurwitz model at high azimuthal orders. We present new measurements of sectoral equatorial inertial modes at $m>15$ where the modes appear to become progressively less retrograde compared to the canonical Rossby-Haurwitz dispersion relation in a co-rotating frame. We use a spectral eigenvalue solver to compute the spectrum of solar inertial modes in the presence of differential rotation. Focussing specifically on equatorial Rossby modes, we find that the numerically obtained mode frequencies lie along distinct ridges, one of which lies strikingly close to the observed mode frequencies in the Sun. We also find that the $n=0$ ridge is deflected strongly in the retrograde direction. This suggests that the solar measurements may not correspond to the fundamental $n=0$ Rossby-Haurwitz solutions as was initially suspected, but to a those for a higher $n$. The numerically obtained eigenfunctions also appear to sit deep within the convection zone -- unlike those for the $n=0$ modes -- which differs substantially from solar measurements and complicates inference., Comment: 16 pages, 12 figures

Inertial waves, which are dominantly driven by the Coriolis force, likely play an important role in solar dynamics, and additionally, provide a window into the solar subsurface. The latter allows us to infer properties that are inaccessible to the traditional technique of acoustic-wave helioseismology. Thus, a full characterization of these normal modes holds promise in enabling the investigation of solar subsurface dynamics. In this work, we develop a spectral eigenvalue solver to model the spectrum of inertial waves in the Sun. We model the solar convection zone as an anelastic medium, and solve for the normal modes of the momentum and energy equations. We demonstrate that the solver can reproduce the observed mode frequencies and line-widths well, not only of sectoral Rossby modes, but also the recently observed high-frequency inertial modes. In addition, we believe that the spectral solver is a useful contribution to the numerical methods on modeling inertial modes on the Sun., Comment: 6 Figures, accepted for publication in ApJS

Context: Helioseismic analysis of large-scale flows and structural inhomogeneities in the Sun requires the computation of sensitivity kernels that account for the spherical geometry of the Sun, as well as systematic effects such as line-of-sight projection. Aim: I aim to develop a code to evaluate helioseismic sensitivity kernels for flows using line-of-sight projected measurements. Methods: I decomposed the velocity field in a basis of vector spherical harmonics and computed the kernel components corresponding to the coefficients of velocity in this basis. The kernels thus computed are radial functions that set up a 1.5D inverse problem to infer the flow from surface measurements. I demonstrate that using the angular momentum addition formalism lets us express the angular dependence of the kernels as bipolar spherical harmonics, which may be evaluated accurately and efficiently. Results: Kernels for line-of-sight projected measurements may differ significantly from those that don't account for projection. Including projection in our analysis does not increase the computational time significantly. We demonstrate that it is possible to evaluate kernels for pairs of points that are related through a rotation by linearly transforming the terms that enter the expression of the kernel, and that this result holds even for line-of-sight projected kernels. Conclusion: I developed a Julia code that may be used to evaluate sensitivity kernels for seismic wave travel times computed using line-of-sight projected measurements, which is made freely available under the MIT license., Comment: 7 pages, 3 figures, accepted for publication in Astronomy and Astrophysics

Helioseismic inferences of large-scale flows in the solar interior necessitate accounting for the curvature of the Sun, both in interpreting systematic trends introduced in measurements as well as the sensitivity kernel that relates photospheric measurements to subsurface flow velocities. Additionally the inverse problem that relates measurements to model parameters needs to be well-posed to obtain accurate inferences, which necessitates a sparse set of parameters. Further, the sensitivity functions need to be computationally easy to evaluate. In this work we address these issues by demonstrating that the sensitivity kernels for flow velocities may be computed efficiently in a basis of vector spherical harmonics. We are also able to account for line-of-sight projections in Doppler measurements, as well as center-to-limb differences in line-formation heights. We show that given the assumed spherical symmetry of the background model, it is often cheap to simultaneously compute the kernels for pairs of observation points that are related by a rotation. Such an approach is therefore particularly well-suited to inverse problems for large-scale flows in the Sun, such as meridional circulation., Comment: 32 pages, 9 Figures, Accepted for publication in ApJ

As helioseismology matures and turns into a precision science, modeling finite-frequency, geometric and systematical effects is becoming increasingly important. Here we introduce a general formulation for treating perturbations of arbitrary tensor rank in spherical geometry using fundamental ideas of quantum mechanics and their extensions in geophysics. We include line-of-sight projections and center-to-limb differences in line-formation heights in our analysis. We demonstrate the technique by computing a travel-time sensitivity kernel for sound-speed perturbations. The analysis produces the spherical harmonic coefficients of the sensitivity kernels, which leads to better-posed and computationally efficient inverse problems.

Supergranules are divergent 30-Mm sized cellular flows observed everywhere at the solar photosphere. Their place in the hierarchy of convective structures and their origin remain poorly understood (Rincon et al., 2018). Estimating supergranular depth is of particular interest since this may help point to the underlying physics. However, their subsurface velocity profiles have proven difficult to ascertain. Birch et al. (2006) had suggested that helioseismic inferences would benefit from an ensemble average over multiple realizations of supergranules due to the reduction in realization noise. Bhattacharya et al. (2017) used synthetic forward-modelled seismic wave travel times and demonstrated the potential of helioseismic inversions at recovering the flow profile of an average supergranule that is separable in the horizontal and vertical directions, although the premise of this calculation has since been challenged by Ferret (2019). In this work we avoid this assumption and carry out a validation test of helioseismic travel-time inversions starting from plausible synthetic non-separable profiles of an average supergranule. We compute seismic wave travel times and sensitivity kernels by simulating wave propagation through this background. We find that, while the ability to recover the exact profile degrades based on the number of parameters involved, we are nevertheless able to recover the peak depth of our models in a few iterations where the measurements are presumably above the noise cutoff. This represents an important step towards unraveling the physics behind supergranules, as we start appreciating the parameters that we may reliably infer from a time-distance helioseismic inversion., Comment: 20 pages, 7 figures

We develop a helioseismic inversion algorithm that can be used to recover sub-surface vertical profiles of 2-dimensional supergranular flows from surface measurements of synthetic wave travel times. We carry out seismic wave-propagation simulations through a 2-dimensional section of a flow profile that resembles an averaged supergranule, and a starting model that has flows only at the surface. We assume that the wave measurements are entirely without realization noise for the purpose of our test. We expand the vertical profile of the supergranule stream function on a basis of B-splines. We iteratively update the B-spline coefficients of the supergranule model to reduce the travel-times differences observed between the two simulations. We carry out the exercise for four different vertical profiles peaking at different depths below the solar surface. We are able to accurately recover depth profiles of four supergranule models at depths up to $8-10\,\text{Mm}$ below the solar surface using $f-p_4$ modes, under the assumption that there is no realization noise. We are able to obtain the peak depth and the depth of the return flow for each model. A basis-resolved inversion performs significantly better than one where the flow field is inverted for at each point in the radial grid. This is an encouraging result and might act as a guide in developing more realistic inversion strategies that can be applied to supergranular flows in the Sun., Comment: Accepted for publication in in A&A, 11 pages, 6 figures

The inference of internal properties of the Sun from surface measurements of wave travel times is the goal of time-distance helioseismology. A critical step toward the accurate interpretation of travel-time shifts is the computation of sensitivity functions linking seismic measurements to internal structure. Here we calculate finite-frequency sensitivity kernels in spherical geometry for two-point travel-time measurements. We numerically build Green's function by solving for it at each frequency and spherical-harmonic degree and summing over all these pieces. These computations are performed in parallel ("embarrassingly"), thereby achieving significant speedup in wall-clock time. Kernels are calculated by invoking the first-order Born approximation connecting deviations in the wavefield to perturbations in the operator. Validated flow kernels are shown to produce travel times within 0.47% of the true value for uniform flows up to 750 m/s. We find that travel-time can be obtained with errors of 1 millisecond or less for flows having magnitudes similar to meridional circulation. Alongside flows, we also compute and validate sensitivity kernel for sound-speed perturbations. These accurate sensitivity kernels might improve the current inferences of sub-surface flows significantly., Comment: 32 pages, 10 figures, accepted for publication in ApJ

Observations of the solar surface reveal the presence of flows with length scales of around $35$ Mm, commonly referred to as supergranules. Inferring the sub-surface flow profile of supergranules from measurements of the surface and photospheric wavefield is an important challenge faced by helioseismology. Traditionally, the inverse problem has been approached by studying the linear response of seismic waves in a horizontally translationally invariant background to the presence of the supergranule; following an iterative approach that does not depend on horizontal translational invariance might perform better, since the misfit can be analyzed post iterations. In this work, we construct synthetic observations using a reference supergranule, and invert for the flow profile using surface measurements of travel-times of waves belonging to modal ridges $f$ (surface-gravity) and $p_{1}$ through $p_{7}$ (acoustic). We study the extent to which individual modes and their combinations contribute to infer the flow. We show that this method of non-linear iterative inversion tends to underestimate the flow velocities as well as inferring a shallower flow profile, with significant deviations from the reference supergranule near the surface. We carry out a similar analysis for a sound-speed perturbation and find that analogous near-surface deviations persist, although the iterations converge faster and more accurately. We conclude that a better approach to inversion would be to expand the supergranule profile in an appropriate basis, thereby reducing the number of parameters being inverted for and appropriately regularizing them., Comment: Accepted in ApJ, 12 pages, 10 figures, 1 table

The advent of space-based observatories such as CoRoT and Kepler has enabled the testing of our understanding of stellar evolution on thousands of stars. Evolutionary models typically require five input parameters, the mass, initial Helium abundance, initial metallicity, mixing- length (assumed to be constant over time), and the age to which the star must be evolved. Some of these parameters are also very useful in characterizing the associated planets and in studying galactic archaeology. How to obtain these parameters from observations rapidly and accurately, specifically in the context of surveys of thousands of stars, is an outstanding ques- tion, one that has eluded straightforward resolution. For a given star, we typically measure the effective temperature and surface metallicity spectroscopically and low-degree oscillation frequencies through space observatories. Here we demonstrate that statistical learning, using artificial neural networks, is successful in determining the evolutionary parameters based on spectroscopic and seismic measurements. Our trained networks show robustness over a broad range of parameter space, and critically, are entirely computationally inexpensive and fully automated. We analyze the observations of a few stars using this method and the results com- pare well to inferences obtained using other techniques. This method is both computationally cheap and inferentially accurate, paving the way for analyzing the vast quantities of stellar observations from past, current, and future missions., Comment: Submitted. 9 pages, 7 Figures, 1 Table, MNRAS

Measurements of oscillation frequencies of the Sun and stars can provide important independent constraints on their internal structure and dynamics. Seismic models of these oscillations are used to connect structure and rotation of the star to its resonant frequencies, which are then compared with observations, the goal being that of minimizing the difference between the two. Even in the case of the Sun, for which structure models are highly tuned, observed frequencies show systematic deviations from modeled frequencies, a phenomenon referred to as the "surface term." The dominant source of this systematic effect is thought to be vigorous near-surface convection, which is not well accounted for in both stellar modeling and mode-oscillation physics. Here we bring to bear the method of homogenization, applicable in the asymptotic limit of large wavelengths (in comparison to the correlation scale of convection), to characterize the effect of small-scale surface convection on resonant-mode frequencies in the Sun. We show that the full oscillation equations, in the presence of temporally stationary 3-D flows, can be reduced to an effective "quiet-Sun" wave equation with altered sound speed, Br\"{u}nt--V\"{a}is\"{a}la frequency and Lamb frequency. We derive the modified equation and relations for the appropriate averaging of three dimensional flows and thermal quantities to obtain the properties of this effective medium. Using flows obtained from three dimensional numerical simulations of near-surface convection, we quantify their effect on solar oscillation frequencies, and find that they are shifted systematically and substantially. We argue therefore that consistent interpretations of resonant frequencies must include modifications to the wave equation that effectively capture the impact of vigorous hydrodynamic convection., Comment: 9 pages, 4 figures