Time-Dependence of Subsurface Solar Convection Using the Time-Distance Deep-Focus Method

TL;DR

The study re-evaluates the deep-focus time-distance helioseismology approach for measuring subsurface solar convection near $ ilde{0.96 R_\odot$ ($\\sim 30$ Mm) by validating with GALE and EULAG simulations and applying the method to Solar Cycle 24 data. It demonstrates that the inferred convective power spectrum is reliable up to $\\ell \\approx 15$–$20$ before diverging from ground truth at higher angular degrees, and finds modest cycle-related variations with an average spectrum about $0.5$ dex higher than earlier estimates. Despite the improved calibration, the results do not resolve the Convective Conundrum, indicating persistent gaps between observations and global convection simulations. The work also outlines a plan to cross-validate with ring-diagram analyses to better constrain the subsurface convection power spectrum and guide future modeling efforts.

Abstract

We re-examine the deep-focus methodology of time-distance helioseismology previously used to estimate the power spectrum of the solar convection at a depth of about 30 Mm, which was found to be significantly weaker than predicted by theory and simulations. The Global Acoustic, Linearized Euler (GALE) and Eulerian Lagrangian (EULAG) codes are used to generate ground-truth simulations to evaluate the accuracy of the inferred convective power spectrum. This validation process shows that the power spectrum derived using the time-distance methodology diverges significantly from ground truth beyond spatial scales corresponding to the spherical harmonic degree $\ell=15$--$30$ because of the limited resolution of helioseismic measurements at that depth. However, the power estimated at larger spatial scales ($\ell<15$) is sufficiently accurate. We then apply the methodology to solar data selected from throughout Solar Cycle 24 and find some evidence that the magnitude of the convective power changes throughout the Cycle. An average of the convective power across the Solar Cycle reveals a spectrum that is qualitatively similar to previous estimates, though about half an order of magnitude greater. The disagreement between observations of solar convection and the magnitudes predicted by simulations persists.

Figures (7)

• Figure 1: Acoustic waves that intersect the focal point at radius 0.96R$_\odot$. We discard wave paths that intersect the focal point at an angle greater than 45$^\circ$ (gray lines) and keep only those that intersect at an angle less than 45$^\circ$ (black lines). The focal depth is indicated by the red dashed line.
• Figure 2: Results of the calibration procedure. Panel a shows the input flow map with the 500 randomly placed "delta" flows. Panel b shows the travel time deviations measured from the GALE simulation with the background flow map shown in panel a. Panel c shows the "clean" travel time deviations, after the contribution from the random sources has been removed. Panel d shows the travel time response based on a linear fit of the SHCs from panels a and c; by definition, the error for the fit at $\ell=0$ and $\ell=1$ must be zero.
• Figure 3: Panel a shows the fit (dashed line) of the travel time variance of CR 2184 segment #12 to the time-dependent noise and time-independent signal model. The threshold applied to the travel time maps (black star) is chosen as the one that minimizes the cumulative relative error (panel c) between the residuals (red), signal uncertainty (blue), and noise uncertainty (green). Panels c and d show the same quantities for the noise-only model of the empty GALE simulation.
• Figure 4: Effects of data reduction on the computed power spectrum. Panels a and b correspond to the input background flow field. Panels c and d correspond to the travel time measurements and derived power spectrum, respectively. In the third row, we replicate the construction of a pseudo-synoptic map from two independent simulations, with the power spectrum from panel d shown with the blue dashed line. In the fourth row, the pseudo-synoptic map is truncated at the poles as in the solar data; the blue dashed line in panel h corresponds to the power spectrum from panel f.
• Figure 5: The convective power at each $\ell$ is shown throughout Solar Cycle 24, with the corresponding angular degree labeled above each panel. Note that the scales for each panel are different in order to highlight the time-dependence. The individual measurements are shown as symbols with corresponding error bars, the black line shows the 13-month rolling average, and the gray line shows the monthly sunspot number. The black diamonds indicate an average of the central and preceding Carrington rotations, while the orange symbols denote a single-rotation measurement. Orange stars indicate that only the central Carrington rotation was available, while orange diamonds indicate that only the preceding Carrington rotation was available.