152. Probing the Solar Meridional Circulation using Fourier Legendre Decomposition

Contributed by Doug Braun. Posted on March 16, 2021

D. C. Braun1, A. C. Birch2, and Y. Fan3

1. NorthWest Research Associates, Boulder, CO 80301, USA
2. Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
3. National Center for Atmospheric Research, HAO Division, Boulder, CO 80301, USA

The solar meridional circulation is a crucial, but poorly constrained, component of magnetic flux transport and dynamo models. In a recent publication[1], we apply the helioseismic methodology of Fourier Legendre Decomposition (hereafter FLD) to 88 months of HMI Dopplergrams as the basis of inferring the depth variation of the mean meridional flow. The FLD method, first suggested over twenty years ago[2], has been substantially updated and includes control procedures designed to assess and remove the well-known center-to-limb artifacts[3] among other improvements. Similar to ring-diagram analysis, FLD infers subsurface flows by measuring and modeling the Doppler distortion in the power spectra of advected f and p modes, but accounts for spherical geometry using combinations of Legendre functions of the first and second kind to characterize poleward or equatorward propagating waves. Figure 1 shows the annular ranges over which the nominal (north and south hemisphere) measurements, as well as the control measurements (to correct for center-to-limb artifacts), are made.

Figure 1| The regions of the Sun for which nominal and control FLD measurements are performed. The solid black lines in the left (right) panel shows the nominal latitude ranges employed in the northern (southern) hemisphere. The boundaries span a latitude range of +20 to +60 degrees in the north (left panel) and -60 to -20 degrees in the south (right panel). For the control measurements, used to assess center-to-limb artifacts, we carry out identical analyses with the geometry rotated 90 degrees from the true north (or south) poles to analogous locations at the east and west limbs. The solid (dashed) red lines indicate the start (end) of an 8-hour interval over which the annular control regions are tracked with a Carrington rotation rate. The green circle indicates a heliocentric angle of 60 degrees, beyond which Dopplergram pixels are excluded from the analysis.

Shifts between poleward and equatorward propagating waves are obtained from the power spectra using a multi-ridge fitting procedure. Although characterized as “frequency shifts”, they actually result from a redistribution of power, through mode coupling produced by advection, between modes of nearby degree and which leak into the observed spectra[4]. Figure 2 shows the raw frequency shifts for the northern hemisphere, control frequency shifts as determined from the east and west oriented geometries, and the resulting corrected frequencies.

Figure 2| Raw frequency shifts (top left) as a function of phase speed (frequency divided by wavenumber), in the northern hemisphere. Different ridges (radial orders) have colors and symbols as indicated. The center-to-limb effect is shown in the top right, determined from the east and west control measurements, with the bottom-right panel showing its dependence with frequency. Corrected frequency shifts are shown in the bottom left with vertical lines indicating lower turning-point depths.

Forward modeling is carried out, using sensitivity functions proportional to the mode kinetic energy density, to infer the depth variation of the meridional circulation in the top half of the convection zone. The results, shown in Figure 3 imply substantial differences between the meridional circulation in the northern and southern hemisphere.

Figure 3| Binned averages of the corrected frequency shifts are shown as black circles with errors (left panels), with forward-model predictions shown by blue (north) and red (south) dots. For each hemisphere, two models were constructed to roughly bracket the error bars. The upper right panel overlays both hemisphere measurements to highlight the hemispheric asymmetry of the corrected frequency shifts. The bottom right panel shows the meridional flow models, as a function of depth below the surface, which give rise to the predictions shown on the left.

While real hemispheric differences in the meridional circulation may very well exist, it is worth considering the possibility that the asymmetry results from systematic errors. The inferred presence of a return (equator-ward propagating) flow at a depth of approximately 40 Mm below the photosphere in the northern hemisphere (Figure 3) is surprising and appears to be inconsistent with many other helioseismic analyses. Analysis of the time variation of the hemispheric frequency-shift asymmetry shows that it persists during both high and low solar activity and is not simply related to hemispheric differences in solar activity. The results appear to be at least qualitative consistent with recent findings[5] which point to an anomaly in HMI data that is not present in MDI or GONG data. One possibility is that the center-to-limb effect is not axisymmetric as assumed for the correction, but instead varies (for example, due to instrumental causes) with the position around the limb. Variations of only 5 to 10% in the center-to-limb artifact would be needed to produce the observed asymmetries. The FLD method offers a frequency-wavenumber perspective on the problem which complements time-distance methods and may shed some light on the systematics which currently hinder a reliable determination of the deep properties of the meridional circulation.

For more details, see reference [1].


[1] Braun, D.C., Birch, A.C., and Fan, Y. 2021, ApJ, in press, http://arxiv.org/abs/2103.02499
[2] Braun, D.C. and Fan, Y. 1998, ApJL, 508, L105.
[3] Zhao, J., Nagashima, K., Bogart, R.S., Kosovichev, A.G., and Duvall, T.L. Jr. 2012, ApJL, 749, L5.
[4] Roth, M.. Doerr, H.-P., and Hartlep, T. 2016, A&A, 592, A106.
[5] Gizon, L., Cameron, R.H., Pourabdian, M., Liang, Z.-C., et al. 2020 , Science, 368, 1469.

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