Resolving the Tachocline using Inversion of Rotational Splitting Derived from Fitting Very Long and Long Time Series

TL;DR

This study uses rotational splittings from very long and long solar time series to invert the solar interior rotation and characterize the tachocline. It compares two inversion frameworks (RLS and SART) on multiple radial grids, validating against simulations and applying to observations from MDI, HMI, and GONG, including a 25.2-year series. Fitting a sigmoid to the inferred rotation profiles reveals a latitude-dependent tachocline location with a sharp discontinuity between low and high latitudes and a sub-percent width, with temporal variations remaining inconclusive due to noise and method dependence. The work highlights the importance of grid design, averaging kernels, and informed initial guesses for robust tachocline inferences from long time-series helioseismic data, in agreement with forward-modeling results and recent analyses.

Abstract

We use rotation splittings derived from very long and long time series, namely 25.2, 12.6 and 6.3 year long, computed by Korzennik (2023) independent methodology to characterize the solar tachocline and its variation with latitude and time. We use two different inversion methodologies and a model of the tachocline to derive its position, width and the amplitude of the radial shear. To validate our methodology we present results from simulated rotational splittings, whether including or not random noise commensurable with the current observational precision. We also describe how we leverage the fact that one of our methodologies uses an initial guess that can be chosen to include a priori information. In order to try to resolve the tachocline, we increased the radial density of the inversion grid and showed how it affect the inferences. We also show how the trade off between smoothing and noise magnification affects these, as well as the effectiveness of using an informed initial guess. Results derived from high-precision rotational splittings show clearly that the location of the tachocline at low latitudes is different for its position at high latitudes. The latitudinal variation of its width is not significantly constrained, but our results agree with estimates based on forward modeling. When using splittings derived from somewhat shorter time series, we find temporal variations that are neither definitive nor significant, since we see systematic differences when using different methodologies.

Figures (20)

• Figure 1: Temporal coverage of the time series used in the work presented here (filled boxes), compared the temporal coverage of the observations (open boxes) for the MDI, HMI and GONG instruments, and to the solar activity illustrated by the monthly averaged sunspot number.
• Figure 2: Positions of the target locations, shown in Cartesian coordinates (top panels) and the grids' radial spacing as a function of radius (bottom panels) for the four grids we used. Shown, from left to right, are the uniform radial spacing grid, two progressive radial spacing grids and the so-called aggressive radial spacing grid. The lowest radial resolution, occurring around the tachocline, is indicated as well.
• Figure 3: Three models used for our simulation. The top panels show the simulated rotation rate in Cartesian coordinates, while the bottom panels show the rotation profile at the equator, with a zoomed view around the tachocline in the lower left panel.
• Figure 4: RLS solutions for noiseless simulations (solid lines with dots) superimposed over the true solutions (dashed lines) for the three models of the tachocline, shown using different colors (red for narrow, green for nominal, and blue for wide). Top to bottom panels correspond to different inversion grids ( aggressive, progressive I, and progressive II), while left to right panels correspond to increasing smoothing (as per the trade-off coefficient $\Lambda$).
• Figure 5: SART solutions for noiseless simulations (solid lines with dots) superimposed over the true solutions (dashed lines) for the three models of tachocline, shown using different colors (red for narrow, green for nominal, and blue for wide). Top to bottom panels correspond to different inversion grids ( aggressive, progressive I and progressive II), while leftmost panel shows solutions when initial guess is set to a constant, while the middle and rightmost panels correspond to solutions when setting the initial to the true solutions and varying the smoothing factor $\alpha$ (i.e., smoothing increases with lower $\alpha$ values). The unique combination of the progressive II grid, exact initial solution and low smoothing (in the lowest middle panel) recovers almost perfectly the three models.