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Meridional flow in the Sun’s convection zone is a single cell in each hemisphere

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Science  26 Jun 2020:
Vol. 368, Issue 6498, pp. 1469-1472
DOI: 10.1126/science.aaz7119

The flow of plasma inside the Sun

The activity of the Sun, including the occurrence of sunspots, is driven by magnetic fields that originate from the motion of charged plasma beneath the surface. Helioseismology uses acoustic oscillations to probe the Sun's interior, analogous to seismology's use of earthquakes to investigate Earth's interior. Gizon et al. analyzed helioseismology data from 1996 to 2019, covering two 11-year solar cycles. They measured the latitudinal and radial flow of plasma as a function of depth within the Sun and how it varies with time. The results support magnetic flux-transport dynamo models, which can explain the distribution of sunspots over each solar cycle.

Science, this issue p. 1469

Abstract

The Sun’s magnetic field is generated by subsurface motions of the convecting plasma. The latitude at which the magnetic field emerges through the solar surface (as sunspots) drifts toward the equator over the course of the 11-year solar cycle. We use helioseismology to infer the meridional flow (in the latitudinal and radial directions) over two solar cycles covering 1996–2019. Two data sources are used, which agree during their overlap period of 2001–2011. The time-averaged meridional flow is shown to be a single cell in each hemisphere, carrying plasma toward the equator at the base of the convection zone with a speed of ~4 meters per second at 45° latitude. Our results support the flux-transport dynamo model, which explains the drift of sunspot-emergence latitudes through the meridional flow.

Heat is transported by convective motions of the plasma in the outer 29% of the Sun (the solar convection zone). In this layer, convection interacts with rotation to drive global-scale axisymmetric flows (1). The longitudinal component of these flows is the solar differential rotation: The equator rotates once every 25 days, the poles once every 34 days. The latitudinal and radial components are the Sun’s meridional flow. The differential rotation and the meridional flow both play a role in the solar dynamo (2). Differential rotation acts on latitudinal magnetic field to generate a longitudinal (toroidal) magnetic field. At the surface, the meridional flow transports the radial magnetic field from the equator toward the poles. The role of the deep meridional flow is less certain. In the class of models known as flux-transport dynamos (3), the meridional flow near the base of the convection zone is assumed to be equatorward and to transport the toroidal magnetic field to match the drift of 2° to 3° per year in the latitudes at which sunspots appear (2). Thus, these models provide testable predictions about the amplitude and sign of the deep meridional flow.

Testing these predictions is challenging. The geometry of the meridional flow is difficult to compute theoretically from first principles, as it results from a small imbalance between two large terms (4). Mass conservation implies that the plasma is carried around closed loops (cells). A poleward flow at the surface must return equatorward at some depth. There may be additional closed cells stacked on top of each other in the radial direction. Both one-cell and two-cell meridional flow geometries have been proposed on the basis of theory and numerical simulations (5). Observationally, the meridional flow can be constrained using helioseismology (6, 7). This technique relies on measurements of the times taken by solar acoustic waves to travel between points on the surface through the interior. Suitable data are available for 1996–2011 from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO) spacecraft and for 2010 onward from the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) spacecraft. Studies of the HMI data have reached differing conclusions on the geometry of the meridional flow: either one or two cells in the radial direction (811). The MDI data and the HMI data also show different flow structures (12). These differences may result from instrumental systematic errors, the calibration of the observations, and/or different assumptions made during the data analysis.

To confirm the validity of a helioseismic inference, it is necessary to compare results from two independent datasets that have an extended overlap period. For example, the Sun’s internal rotation has been validated (13) by comparing observations from MDI to those of the ground stations operated by the Global Oscillation Network Group (GONG) (14).

Using multiple datasets, we studied the structure and time variability of the meridional flow in the convection zone. We used maps of the line-of-sight velocity at the Sun’s surface (dopplergrams) at reduced spatial resolution, which provide information about sound waves propagating in the solar interior with spherical harmonic degrees up to 300. The reduced-resolution data are known to be less prone to instrumental errors (15). We considered all three main datasets: MDI, HMI, and GONG. The MDI observations consist of dopplergrams with 192 pixels by 192 pixels per frame for the period 1 May 1996 to 11 April 2011 (16). After April 2003, the SOHO spacecraft was rotated 180° every 3 months; we only used the observations when MDI was in the orientation it had before 2003 to ensure consistency of the data (17). The HMI data (18) that we used cover the period from 1 May 2011 to 30 April 2019; they were processed by the HMI team to have a format (204 pixels by 204 pixels) similar to that of the MDI data. We constructed the third dataset from the merged and calibrated high-resolution GONG dopplergrams (19) for 1 August 2001 to 30 April 2019 (839 pixels by 839 pixels) by applying Gaussian smoothing and downsampling to 200 pixels by 200 pixels.

We produced time series of MDI, HMI, and GONG images—remapped onto a heliographic coordinate system (scale of 0.6° per pixel)—by tracking areas on the solar surface at the Carrington rotation rate of 456.03 nHz (20). The MDI data were corrected to account for a misalignment of the instrument with respect to the spacecraft, corresponding to a 0.20° error in the solar P angle (one of the two angles describing the direction of the Sun’s rotation axis with respect to the plane of the sky). This error was determined using HMI images as reference during the 2010 overlap period (21). The orientation of the HMI images is known to within 0.01°, based on the analysis of the transit of Venus from 5 to 6 June 2012 (22). The MDI, HMI, and GONG images were all corrected for a 0.08° difference between the inclination of the solar rotation axis to the ecliptic and the value of 7.25° measured by Carrington in 1863 (20).

The data were analyzed using time-distance helioseismology (23). Solar subsurface flows affect the time it takes for acoustic waves to travel between two points on the surface, A and B. The travel times were measured in both directions by cross-correlating the Doppler velocity data at A and those at B. The difference between the two travel times is sensitive to flows near the ray path that connects A and B through the interior. To determine the meridional flow, we considered points separated in latitude, using a quadrant geometry with arcs of 30° [(12) and fig. S1]. Points in locations of strong magnetic fields were excluded from the averages (17). By fitting a one-parameter model (24) to the cross-correlation functions computed daily, we measured the south-north travel-time differences τ(Δ, λ), where 6° ≤ Δ ≤ 42° is the angular distance between the arcs and −54° ≤ λ ≤ 54° is the latitude of the midpoint between the arcs. We applied a center-to-limb correction using travel-time differences measured in the east-west direction (25) (fig. S2).

Figure 1, A to C, shows the measured travel times averaged over three ranges of travel distances at low latitudes in the north and south. The travel times are averaged in bins of 3 years starting from 1 May 1996 to aid in the comparison of the three datasets. For the period before 1 May 2002, we only show MDI data. For the periods thereafter, we included in the time averages only those days when travel times are available for two instruments: MDI (original orientation) and GONG until 30 April 2011 and HMI and GONG thereafter. The signs of the measured travel times are consistent with a poleward meridional flow near the surface. The time variations of the travel times are caused by surface inflows into magnetic regions (26) and therefore are related to the sunspot number (Fig. 1D). The MDI and GONG observations during the period from 2002 to 2011 differ by ~0.1 s (Fig. 1, A to C, and fig. S3), which is much smaller than the 1σ uncertainties caused by the random excitation of the acoustic waves. Taking noise correlations into account, MDI and GONG travel times are consistent for distances ≤30° [(20) and table S1]. However, the HMI travel times differ from the GONG travel times, with smaller values for HMI times in the northern hemisphere. We have been unable to identify the source of this discrepancy, so we do not use the HMI data for the rest of the analysis. We also considered data from the GONG network from 1 May 1996 to 31 May 2001, when it operated with lower-resolution cameras. The corresponding travel times are consistent with the MDI data for the same period but noisier [(20) and fig. S4]. We therefore combine the MDI data from 1 May 1996 to 30 April 2003 (before the first change in orientation of the SOHO spacecraft) and the GONG data from 1 May 2003 to 30 April 2019.

Fig. 1 Averages of the travel-time differences caused by the meridional flow.

(A to C) The travel times were averaged in 3-year periods and over latitudes 10° ≤ λ ≤ 25° (North) and −25° ≤ λ ≤ −10° (South). Between 1 May 2002 and 30 April 2011, only the days when travel times for both MDI (original orientation) and GONG are available were included in the time averages. From 1 May 2011 onward, only the days when travel times for both HMI and GONG are available were included in the averages. Travel times were further averaged over angular distances from 6° to 18° (A), from 18° to 30° (B), and from 30° to 42° (C). The error bars indicate ±1σ. (D) Monthly sunspot number versus time, displaying solar cycles 23 and 24.

The south-north travel-time differences τ are linearly related to the meridional flow through functions of position known as sensitivity kernels. We computed the kernels using a finite element solver in the frequency domain (27). Denoting the colatitude by θ = 90° − λ, we represented the radial and colatitudinal components of the meridional flow, Ur and Uθ, as linear combinations of basis functions (20). The coefficients in these expansions are the parameters describing the flow, which were determined by inverting the travel times. We solved the linear inverse problem under the physical constraints of mass conservation and that the flow does not cross the convection zone boundaries (20). Inversions were validated using synthetic data (20). Eleven years of data allow us to distinguish between one- and two-cell flow profiles, with a noise level of ~1.5 m s−1 at the base of the convection zone. For 3 years of data, the noise increases to 2.5 m s−1. These uncertainties are consistent with previous estimates (28).

Figure 2A shows the inferred Uθ in the convection zone, averaged over each solar cycle. The stream functions in Fig. 2B show that the flow takes the form of a single cell in each hemisphere. The flow is poleward at the surface (Fig. 3C) and equatorward at the base of the convection zone (Fig. 3A). The flow profile at the base of the convection zone approximately follows Uθ = Ub sin 2θ, with Ub = 4.8 ± 1.0 m s−1 for cycle 23 and Ub = 3.6 ± 1.0 m s−1 for cycle 24 (the cycles are identified in Fig. 1D). The flow switches sign near 0.79 R, where R is the solar radius (Fig. 3B). This is consistent with previous inversions that used the constraint of mass conservation (9, 11).

Fig. 2 Inferred meridional flow in the solar convection zone.

(A) Flow Uθ for solar cycles 23 and 24 (Ur is shown in fig. S5). (B) Stream functions ψ for each solar cycle, defined by ρU=×[φ^ψ/(rsinθ)], where ρ is the density and φ^ is the unit vector in the longitudinal direction. The stream functions are normalized to their maximum absolute value. (C and D) Temporal variations with a cadence of 3 years.

Fig. 3 Cuts through the solutions of the meridional flow.

(A) Latitudinal dependence of Uθ at the base of the convection zone. The purple and light blue solid curves show Uθ for solar cycles 23 and 24, respectively, with the shaded regions indicating ±1σ. (B) Radial dependence of Uθ at latitudes of ±30° for each cycle. (C) Latitudinal dependence of Uθ at the surface for each cycle. (D) Time dependence of Uθ at latitudes of ±30° at the base of the convection zone. The purple and light blue horizontal lines show the averages over cycles 23 and 24 (Fig. 1D), respectively, with the shaded regions indicating ±1σ. The dark blue and gray lines connect the 3-year averages from MDI and GONG, respectively. Error bars indicate ±1σ. (E) Time dependence of Uθ at latitudes of ±30° at the surface.

The 3-year averages are shown in Fig. 2, C and D, and Fig. 3, D and E. The noise is higher than that for the 11-year averages. Except for the period between the two cycles, a single cell is evident in each hemisphere. At the surface, the variations of Uθ at 30° latitude are substantial and anticorrelated with the sunspot number (Fig. 3E). Figure 4A shows the surface Uθ smoothed in time with a low-pass filter (5 years) and its relation to the latitudes where sunspots emerge. The changes with time and latitude can be understood as local inflows into magnetic regions, which are likely driven by horizontal pressure gradients caused by the surface magnetic field (26). In the middle of the convection zone (fig. S6), the meridional flow averaged over cycle 24 is poleward in the north and very weak in the south. The 3-year averages show a decrease of the amplitude of Uθ from about 2004 during the decaying phase of solar cycle 23. This decrease is seen in both the MDI and GONG data.

Fig. 4 Sunspot-emergence locations in relation to the meridional flow.

(A) Line-of-sight magnetic field contours at the surface (black curves), overlain on Uθ averaged over the top 2% of the Sun by radius. The flow inversion used all travel distances, and variations on time scales faster than 5 years were filtered out. Inflows into regions where the surface magnetic field is strong dominate the time dependence of Uθ. The dashed curve shows where Uθ = 0. (B) The same contours of the magnetic field at the surface, overlain on Uθ averaged between the base of the convection zone (0.713 R) and 0.8 R. (C) Surface longitudinal (east-west) magnetic field Bϕ from Wilcox Solar Observatory observations (30) and, superimposed with black lines, latitudes at which the subsurface longitudinal field from the flux transport model—based on the measured Uθ—is maximum. The solid lines are the median and the dashed lines are the 16th and 84th percentiles, obtained from 300 realizations of the flow.

The agreement between the MDI (original orientation) and GONG data is better for angular distances of Δ ≤ 30° than for Δ > 30° (Fig. 1). Travel distances ≤30° are capable of distinguishing between single- and double-cell models (12), so we also performed inversions for Δ ≤ 30° only. The results (figs. S7 to S9) confirm the single-cell solution for each cycle for the MDI and GONG data. Additionally, the inversions for MDI and GONG restricted to the days in common and for separation distances Δ ≤ 30° are almost indistinguishable (fig. S10).

In flux-transport dynamo models, the amount of longitudinal magnetic field created in the convection zone is determined by the distribution of the radial magnetic field at the surface (29). The longitudinal magnetic field is then transported by Uθ near the base of the convection zone to produce the drift of latitudes at which sunspots emerge. To test this hypothesis, we used the one-dimensional mean-field equation governing the evolution of the longitudinal magnetic field (20), with Uθ from each solar cycle averaged from the bottom of the convection zone to 0.8 R. Using a cycle average for Uθ is appropriate because the temporal variations of Uθ deep in the convection zone (Fig. 4B) are consistent with noise (Fig. 3D). The latitude of the peak subsurface longitudinal magnetic field from the model shows an equatorial propagation that is consistent with the equatorward drift of locations of strong surface field (Fig. 4C). The one-cell meridional flow in each hemisphere that we observed is thus consistent with the equatorial migration of the sunspots under a simple flux-transport model.

Supplementary Materials

science.sciencemag.org/content/368/6498/1469/suppl/DC1

Materials and Methods

Figs. S1 to S14

Tables S1 and S2

References (3143)

References and Notes

  1. Materials and methods are available as supplementary materials.
Acknowledgments: We thank R. Burston for help with the netDRMS data management system. We thank H. Barucq and the Magique 3D team at Inria Bordeaux Sud-Ouest and Université de Pau et des Pays de l’Adour (UPPA) for making the finite element wave solver Montjoie available to us. SOHO is a project of international cooperation between the European Space Agency (ESA) and NASA. The MDI data are courtesy of the SOHO/MDI consortium. This work utilizes GONG and SOLIS (Synoptic Optical Long-term Investigations of the Sun) data obtained by the National Solar Observatory (NSO) Integrated Synoptic Program (NISP), managed by the NSO, the Association of Universities for Research in Astronomy (AURA), Inc. under a cooperative agreement with the National Science Foundation. The HMI data used are courtesy of NASA (SDO) and the HMI science team. The sunspot numbers are from WDC-SILSO (World Data Center Sunspot Index and Long-term Solar Observations), Royal Observatory of Belgium, Brussels. Funding: The data were processed at the German Data Center for SDO, funded by the German Aerospace Center under grant DLR 50OL1701. L.G. acknowledges support from European Research Council Synergy grant WHOLE SUN 810218. The Center for Space Science at New York University Abu Dhabi (NYUAD) is funded by the NYUAD Institute under grant G1502. M.P. was funded in part by the International Max Planck Research School (IMPRS) for Solar System Science at the University of Göttingen. Author contributions: L.G. and Z.-C.L. designed the research. Z.-C.L. measured the travel times. D.F. and C.S.H. computed the sensitivity kernels for flows. M.P. and D.F. inverted the travel times. R.H.C. developed the flux-transport dynamo model. L.G. and R.H.C. wrote the draft paper. All authors discussed the results and contributed to the final version of the paper. Competing interests: The authors declare no competing interests. Data and materials availability: The MDI data were taken from the Joint Science Operations Center (JSOC) export tool at http://jsoc.stanford.edu/ajax/lookdata.html?ds=mdi.vw_v for the period May 1996 to April 2011. The HMI data were taken from the JSOC export tool at http://jsoc.stanford.edu/ajax/lookdata.html?ds=hmi.vw_v_45s for the period May 2010 to April 2019. The GONG merged velocity data (“mrvzi”) were taken from https://gong2.nso.edu/archive/patch.pl?menutype=g for GONG dates 960501 to 190501. Our analysis software and the data necessary to reproduce our results are available on the Open Research Data Repository of the Max Planck Society (Edmond) at https://edmond.mpdl.mpg.de/imeji/collection/0MJjNql7GfpEl5Mb.

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