Sushant S. Mahajan1, David H. Hathaway2, Andrés Muñoz-Jaramillo3, & Petrus C. Martens1
1. Department of Physics & Astronomy, Georgia State University, Atlanta, GA, USA
2. W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA, USA
3. Southwest Research Institute, Boulder, CO, USA
4. High Altitude Observatory, Boulder, CO, USA
5. National Solar Observatory, Boulder, CO, USA
The Sun’s axisymmetric flows, differential rotation (DR) and meridional flow (MF), play key roles in virtually all models of the Sun’s magnetic dynamo. The differential rotation (variation in rotation rate with both latitude and depth) stretches the radial and latitudinal components of the magnetic field in the longitudinal direction – thereby increasing the field strength and changing the field direction. The meridional flow transports the radial and longitudinal components of the magnetic field in the latitudinal direction – thereby building up the polar fields on the surface and annihilating oppositely directed longitudinal fields across the equator in the solar interior. Thus, measurements of the meridional flow are important for the prediction of the buildup of polar fields that in turn are good predictors for the next solar cycle.
These flows have been measured on the solar photosphere using several different techniques including local helioseismology, tracking small scale magnetic features, tracking granules, and tracking supergranular-scale magnetic or velocity patterns. Local helioseismology and granule tracking are known to be plagued by a center-to-limb error that makes it seem that the Eastern limb of the Sun rotates at a different speed than the western limb[1,2]. This is a systematic error with a hitherto unknown origin. We have discovered a similar center-to-limb error in local correlation tracking of supergranular-scale magnetic patterns using the Sun’s line-of-sight magnetograms. Magnetograms recorded by the Michelson Doppler Imager (MDI) onboard Solar Heliospheric Observatory (SOHO) from 1996 to 2010 and from the Helioseismic Magnetic Imager (HMI) onboard Solar Dynamics Observatory (SDO) from 2010 to 2020 were analyzed in this study.
Local correlation tracking of supergranular-scale magnetic patterns involves masking out active regions and tracking supergranular-size blocks in the magnetograms over time. Each selected block in one magnetogram is correlated with a block chosen from another magnetogram at a certain time-lag and this process is optimized to find the displacement of the block at which it shows the highest correlation.
Figure 1| The first shows the residual rotation rate measured at one hour time-lag obtained after removing the average rotation rate measured at each latitude, while the second column shows the residual rotation rate measured at the equator at several time-lags. The third column shows the residual meridional flow speed measured at one hour time-lag obtained after removing the average meridional flow speed measured at each latitude, while the fourth column shows the latitudinal variation of meridional flow speed at several time-lags.
Figure 1 shows the residual rotation rate of the Sun measured at the equator (after subtracting the mean rotation rate) as well as the meridional flow profile measured at different time-lags. Apart from creating an apparent east-west asymmetry in rotation rate, the center-to-limb effect also makes meridional flow at high latitudes seem faster at lower time-lags. The numerical extent of this effect decreases as the time-lag between successive magnetograms used for tracking increases.
We developed a robust methodology to measure and correct such a center-to-limb error in measurements of velocity. Even though this methodology was developed for tracking supergranular-scale magnetic patterns using local correlation tracking on solar magnetograms, it is scalable and applicable to tracking of other features using other techniques.
The measured displacement at each time-lag (nΔt) was decomposed into three components, all of which are independent of time-lag: a baseline (true) velocity (v), a constant shift (Δ) that is present at all time-lags and a quantity with the dimensions of acceleration (δ) that allows for small changes in the flow profile as a linear function of time-lag (nΔt). Here, Δt was one hour for HMI magnetograms and 96 minutes for MDI magnetograms. Thus, the displacement at a time-lag of nΔt can be written as
In order to solve for the baseline velocity (v), the constant shift (Δ) and the acceleration parameter (δ), we used three sets of Carrington rotation averaged velocity measurements at time-lags of 1, 2 and 4 Δt. The results are shown in Figures 2 & 3.
Figure 2| From left to right: vMF, the baseline meridional flow velocity, constant shift ΔMF and the time-lag dependency parameter δMF derived from combinations of measurements at different time-lags as in Equation 1 and averaged over one year from May 2010 to April 2011 for HMI. Their agreement supports our assumption that Δ, δ and the baseline flow are independent of time-lag.
Most of the east-west asymmetry in rotation rate turned out to be a part of the constant shift (Δ) while the baseline velocity obtained using different combinations of time-lags was found to be nearly identical (see Figure 2). The meridional component of the constant shift made meridional flow at shorter time-lags appear faster than it actually is. The constant shift parameter also accounts for most of the difference between MDI and HMI making velocity measurements during their overlap period in 2010 consistent. This constant shift parameter is nothing but the center-to-limb error.
While there is no clear explanation for the acceleration parameter δ, we speculate that this parameter entails depth dependency. As we track the magnetic pattern at longer time-lags, we are biased towards tracking longer lived, stronger magnetic features that may be anchored deeper into the Sun and thus may move with the speed in deeper layers. Thus, it may be indicative of the weakening of meridional flow and speeding up of rotation with depth.
Figure 3| Latitude-time plots of the components of displacement and the temporal variation of Legendre coefficients of the baseline velocities. The top row shows the baseline meridional flow and torsional oscillation obtained from over 23 years of measurements from MDI and HMI data in units of m/s. The boundaries of active latitudes obtained from Royal Greenwich Observatory records are marked with solid black lines. The corresponding plots for Δ (in km) and δ (in m/s/hr) are shown in the second and the third row respectively. The bottom row shows the scaled sunspot number (in black) and our fitting coefficients of associated Legendre polynomial of order 1, degree 2 to the meridional flow profile and of order 1, degree 1 to the torsional oscillation (translated by 55 m/s) with 2σ errors in blue whereas the fitting coefficients from Ref  are shown with red dots and standard error.
The variation of center-to-limb error with the phase of the solar cycle seen in Figure 3 clearly indicates that the center-to-limb effect cannot be assumed to be constant in time and it has to be calculated synchronously with the flow profiles. If it is assumed to be a constant, as in many previous studies, its temporal variation may contaminate measured flow profiles.
The baseline velocity, Δ and δ measurements as well as the Legendre fitting coefficients to baseline flow profiles shown in Figure 3 are available in an easy-to-use form at https://dataverse.harvard.edu/dataverse/lct-on-solar-magnetograms [5,6].
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 Mahajan, S. S., Hathaway, D. H., Muñoz-Jaramillo, A., & Martens, P. C. 2021, ApJ, 917, 100
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 Mahajan, S. 2021a, Level 3 Photospheric Flow Measurements: Longitudinally averaged f(latitude, time),vV2, Harvard Dataverse, doi:10.7910/DVN/I0OWHG. https://doi.org/10.7910/DVN/I0OWHG
 Mahajan, S. 2021b, Level 4: Fitted Photospheric Flow Profiles for modelers, vV2, Harvard Dataverse, doi:10.7910/DVN/0IZBHY. https://doi.org/10.7910/DVN/0IZBHY