S.C. Tripathy et al. Probing the Sun’s Near Surface Shear Layer 16 2024
10.1017/xxxxx Proceedings IAU Symposium

# Probing the Sun’s Near Surface Shear Layer using HMI Spherical Harmonic
Coefficients

S. C. Tripathy1 K. Jain1 S. Kholikov1,2 and R. Komm1 1 National Solar
Observatory, 3665 Discovery Dr., Boulder, CO 80303, USA  2 Institute of
Fundamental and Applied Research, National Research University, TIIAME,
Uzbekistan

###### Abstract

We have measured zonal and meridional components of subsurface flows up to a
depth of 30 Mm below the solar surface by applying the technique of ring
diagram on Dopplergrams which are constructed from the spherical harmonic (SH)
coefficients. The SH coefficients are obtained from the Helioseismic and
Magnetic Imager (HMI) full-disk Dopplergrams. We find a good agreement and
some differences between the flows obtained in this study with those from the
traditional methods using direct Dopplergrams.

###### keywords:

Solar physics, Helioseismology, Solar interior, Solar oscillations, Solar
activity, Solar convection zone, Solar rotation, Solar meridional circulation

††editors: A. Getling & L. Kitchatinov, eds.

##  1 Introduction

Helioseismic studies have illustrated that the most significant changes with
the solar cycle occur in the near-surface shear layer (NSSL) occupying outer
5% or about 35 Mm of the Sun by radius. Within this relatively thin layer, the
density changes by several orders of magnitude and the rotation rate shows a
local maxima. It is also believed that a nonlinear $\alpha\Omega$ dynamo could
be operating in this layer where the velocity shear converts a part of the
poloidal magnetic field into the toroidal field. The existence of this
secondary dynamo is in addition to the global dynamo operating in the
tachocline region and may be responsible for the small-scale field of the
quiet Sun (Brandenburg, 2013). However, the dynamics of NSSL is poorly
understood mainly because of the complex interactions between rotation,
magnetism and convection. The advent of high-resolution Doppler observations
from both space-borne and ground-based instruments, in conjunction with local
helioseismic techniques such as ring-diagram (Hill, 1988) and time-distance
(Duvall et al., 1993), have made it possible to infer the subsurface flows in
NSSL and their temporal evolution (Zhao & Kosovichev, 2004; Basu & Antia,
2010; Komm et al., 2018; Lin & Chou, 2018). But local helioseismology
techniques are limited in the depth range that can be probed. In particular
most of the studies using ring-diagram analysis is confined to a depth of 15
Mm. Only a handful of studies have used large-aperture tiles to analyse flows
up to a depth of 30 Mm (González Hernández _et al._ , 2006; Komm, 2021).

In this context, we measure zonal and meridional components of the horizontal
flows up to a depth of 30 Mm below the surface during the first six and half
years of the Helioseismic and Magnetic Imager (HMI) observations on board
Solar Dynamics Observatory (SDO). We use a non-traditional method comprising
of spherical harmonic (SH) coefficients derived from the full-disk
Dopplergrams and the ring-diagram (RD) technique. This method has the
advantage to infer flows from low-resolution as well as high-resolution
Dopplergrams.

##  2 Data and Analysis Technique

We use HMI SH coefficient time series111series name: hmi.ap_V_sht_gf_72d for
$\ell$ = 0 to 400 obtained from the Joint Science Operation Center (JSOC)
covering the period 2010 May 1 to 2016 December 31. Each time series is
organized as series of individual ($n$, $\ell$, $m$) coefficients and are 72
days long. We also download the window function of each series to track the
duty cycle. Use of this SH time series has several advantages as these have
been generated from the velocity images which are corrected for known
systematics, e.g., image scale, cubic distortion, CCD misalignment,
inclination error and CCD tilt etc. The Dopplergrams were further detrended
and gap filled. A description of all the corrections applied to the time
series along with other details are provided in Larson & Schou (2018).

Figure 1: A typical $\ell$-$\nu$ diagram showing the distribution of the
fitted modes at the disk center using HMI spherical harmonic coefficients (red
circles), HMI Dopplergrams (green diamonds) and the GONG Dopplergrams (blue
plus). Vertical line at $\ell$ = 400 defines the boundary of the fitted modes
using spherical harmonic coefficients.

Reconstruction of Dopplergrams using the SH coefficient time series (inverse
SH decomposition) is carried out following the procedure described in Kholikov
et al. (2014)

$V(\theta,\phi)=\sum_{\ell=0}^{\ell=400}\sum_{m=0}^{\ell}C_{l}^{m}P_{l}^{m}(\theta)e^{im\phi+\delta\phi}\;,$
(1)

where $C_{\ell}^{m}$ are SH coefficients, $P_{\ell}^{m}$ is the associated
Legendre polynomial of spherical degree $\ell$ and order $m,\theta$ and $\phi$
are latitude and longitude, respectively. We also use the differential
rotation profile ($\delta\phi$) of Libbrecht and Morrow (1991) to remove the
surface differential rotation where tracking is carried out by introducing the
surface differential rotation profile into the argument of the complex
exponential in the right hand side of Eq. (1). The reconstructed images were
then remapped to 187 patches of the size 30∘ $\times$ 30∘ from disk center to
$\pm$60∘ in latitude and $\pm 37.5^{\circ}$ in longitude spaced by 7.5∘. All
the patches were tracked for 1664 min (the time series are divided in to 1664
min time interval to produce the data cubes). A three-dimensional fast Fourier
transform (FFT) is applied to each cube in both spatial and temporal
directions to obtain a three-dimensional power spectrum which was subsequently
processed through the RD pipeline (Corbard et al., 2003). The fitting
algorithm in the RD pipeline uses the maximum-likelihood procedure described
in Anderson, Duvall, and Jefferies (1990) and fits about 270 modes for
$n$-values between 0 – 9 and $\ell$ values between 110 – 400 in the frequency
range of 1400 – 5200 $\mu$Hz. Figure 1 displays a typical $\ell$-$\nu$ diagram
comparing modes fitted at the disk center patch for three different data sets.
It is evident that we recover most of the input modes in the $\ell$ range of
approximately 100 to 400 (the RD technique does not fit modes below
approximately $\ell$ of 100). We also note that we fit many more modes in the
low $\ell$ and higher frequency ranges compared to methods using direct
Dopplergrams.

Figure 2: Temporal variation in zonal flow at three different depths: 5.2
$\pm$ 2.2 Mm (top), 15 $\pm$ 7.2 Mm (middle) and 25.6 $\pm$ 7.2 Mm (bottom).
The temporal mean has been subtracted at every latitude and depth. Black
contours in each panel indicate the magnetic activity.

The subsurface zonal and meridional flows are finally derived using the
Regularized Least Square (RLS) inversion method (Haber et al., 2002). The
depth dependence in the inverted flows is known to have significant influence
from the mode kernels used in the inversion. The kernels peak sharply near the
surface owing to the availability of significant number of modes but are less
localized with increasing depth. Thus the depth resolution turns out to be
poor resulting in relatively larger errors in deeper layers (Jain et al.,
2017). However, in this study we have a limited set of high-degree modes and
thus we find that the errors associated with the flows measured closer to the
surface are larger compared to deeper depths. We also correct the flows for
the annual variation with the solar inclination angle towards Earth (B0 angle)
and other systematic effects following the procedure described in Komm et al.
(2015).

##  3 Results

Figure 2 shows the zonal flow at three depths (5.2 $\pm$ 2.2 Mm, 15 $\pm$ 7.2
Mm, and 25.6 $\pm$ 7.2 Mm) as a function of time and latitude where the mean
is subtracted at each latitude and depth. The contours on top of the flows are
plotted to depict the evolution of magnetic activity. The figure illustrates
the bands of slower and faster rotation relative to the mean flow, known as
torsional oscillations, of few m/s at all depths. These are comparable with
those obtained in the traditional methods (e.g., Komm et al., 2018; Zhao et
al., 2014, for ring-diagram and time-distance analyses, respectively). As is
well known, we find that the bands of faster rotation at mid latitudes are
well aligned with the migration of magnetic activity towards the equator but
appears about a year earlier before the activity appears at the surface.
Although the signature of the poleward branch is visible starting around 2012
during the ascending phase of the cycle, we do not clearly see the branch
probably due to the weak cycle 24. Similar results are reported in torsional
oscillations derived from the global modes (Howe et al., 2018). Also, the
connection between the mid-latitude branch in this solar cycle with the polar
branch of the previous cycle is not apparent in the figure due to the temporal
coverage. Nevertheless, a close comparison of the zonal flows obtained in Komm
et al. (2018) displays similarities as well as some differences in finer
details. These differences probably arise due to the different patch sizes
used in these studies covering different areas at each grid point and the
different smoothing parameters. However the overall patterns are similar in
both the studies. Further, we find a discontinuity in the mid-latitude branch
in the Northern hemisphere around 2012 in the top and bottom panels of Figure
2 corresponding to 5 and 25 Mm (the band is barely visible in 5 Mm). While the
traditional methods provide continuous branches in both hemispheres, our
analysis shows the continuous branch to be present in the stronger hemisphere
only. It is not clear if this effect is due to smaller temporal coverage which
provides different mean values over the two hemispheres. It is also plausible
that the corrections that is applied at deeper depths to remove the systematic
variations are not optimized.

Figure 3: Same as Figure 2 but for meridional flows Figure 4: The equatorial
meridional flow averaged over the depth range of 0.9 – 3.4 Mm. Positive values
indicate northward flows, while negative values indicate southward flows. The
blue line represents the strength of the magnetic activity where the MAI of
7.5∘ S tile is subtracted from the 7.5∘ N tile. For clarity, the error bars
for flow measurements are shown only for few points.

It is well recognized that the meridional flow is poleward on the surface and
in shallow depths and increases gradually with depth. We also find similar
results with a peak amplitude of about 15 m/s near the surface and 30 m/s at a
depth of 29 Mm. Figure 3 displays the temporal evolution of the meridional
flow relative to the mean flow at each latitude and depth for the same depths
as the zonal flows. We obtain bands of faster and slower meridional
circulation residuals with time but with opposite signs in both hemispheres
implying poleward flows as positive values indicate flows to the north and
negative values for the flows to the south. At the shallow depths, the band of
poleward meridional flow appears at about 60∘ latitude around the mid-2010
(beginning of our data) and slowly migrates towards the equator reaching there
in about 2013. A similar result is reported in Komm et al. (2018). But at
deeper depths, the flow pattern is reversed; the fast band has changed to the
slow band and vice versa and differs from the result of Komm (2021) where
results from HMI 30∘ tiles were used. However, we should note that the HMI RD
result has a coarser resolution in latitude and longitude since the flow
measurement are available only at a grid spacing of 15∘.

The amplitude and direction of the meridional flow at the equator is an
important topic in the context of cross-equatorial cancellation of magnetic
flux and formation of polar flux. Early investigations that found flows
crossing the equator believed it to be an error in the alignment of the
telescope since its magnitude is one order smaller than the maximum amplitude
of the meridional flow. But recent advances in the surface flux transport
models indicate that it plays a crucial role in transporting magnetic flux
across the equator from the dominant hemisphere to the other one and thus
cancels the excess flux. We further investigate, as shown in Figure 4, the
meridional flows at the equator at a depth closer to the surface (averaged
over 0.9 – 3.4 Mm). It is evident that the meridional flows at the equator are
small but non-zero. It is mainly positive at the beginning of cycle and
oscillates between positive and negative values as the solar cycle progresses
which is believed to be a consequence of the imbalance of activity between the
two hemispheres. To quantify this flux imbalance, we subtract the magnetic
activity index (MAI) of 7.5∘ S tiles from 7.5∘ N tiles. The MAI values for
each tile is calculated by tracking and re-mapping line-of-sight magnetograms
in the same fashion as Dopplergrams for the same length of time. The absolute
values of all pixels higher than a defined threshold are then averaged to
compute the unsigned proxy for magnetic field (MAI). It is evident from Figure
4 that the cross equatorial flows follow the hemisphere with dominant
activity. The Pearson’s linear cross correlation is found to be 75% confirming
the relation between the cross-equatorial flow and strength of the magnetic
activity. These findings are in agreement with those reported in Komm (2022)
where subsurface flow measurements were derived from 15∘ tiles.

##  4 Summary

Reconstructing Dopplergrams from the HMI SH coefficients and subsequently
applying the ring-diagram technique, we have derived subsurface flows up to a
depth of 30 Mm from mid-2010 to the end of 2016. Our preliminary results of
zonal and meridional flow profiles mostly agree near the surface and
intermediate depth ranges with other studies using observed Dopplergrams.
However, the measured meridional flows at greater depths show a flow pattern
which is reversed compared to the flows observed near the surface. Although
there are limited helioseismic studies which probes the flows at depths
greater than 15 Mm using RD technique, our result appears to differ from these
findings. It is plausible that the applied empirical correction for the
systematic variations has not removed all artifacts. We plan to revisit the
correction scheme when flow measurements for whole cycle 24 is computed.

The data used here are courtesy of NASA/SDO and the HMI Science Team. This
work was supported by NASA grants 80NSSC20K0194, 80NSSC21K0735, and
80NSSC23K0404 to the National Solar Observatory. SCT, KJ and RK acknowledge
the partial financial support from NASA Cooperative Agreement 80NSSC22M0162 to
Stanford University for the COFFIES Drive Science Center.

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