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Virtanen, I. O. I. and Virtanen, I. I. and Pevtsov, A. A. and Yeates, A. and Mursula, K. (2017)'Reconstructing solar magnetic �elds from historical observations.', Astronomy astrophysics., 604 . A8.

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A&A 604, A8 (2017)DOI: 10.1051/0004-6361/201730415c© ESO 2017

Astronomy&Astrophysics

Reconstructing solar magnetic fields from historical observations

II. Testing the surface flux transport model

I. O. I. Virtanen1, I. I. Virtanen1, A. A. Pevtsov2, 1, A. Yeates3, and K. Mursula1

1 ReSoLVE Centre of Excellence, Space Climate research unit, University of Oulu, PO Box 3000, 90014 Oulu, Finlande-mail: iiro.virtanen@oulu.fi

2 National Solar Observatory, Sunspot, NM 88349, USA3 Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK

Received 10 January 2017 / Accepted 13 April 2017

ABSTRACT

Aims. We aim to use the surface flux transport model to simulate the long-term evolution of the photospheric magnetic field fromhistorical observations. In this work we study the accuracy of the model and its sensitivity to uncertainties in its main parameters andthe input data.Methods. We tested the model by running simulations with different values of meridional circulation and supergranular diffusionparameters, and studied how the flux distribution inside active regions and the initial magnetic field affected the simulation. Wecompared the results to assess how sensitive the simulation is to uncertainties in meridional circulation speed, supergranular diffusion,and input data. We also compared the simulated magnetic field with observations.Results. We find that there is generally good agreement between simulations and observations. Although the model is not capable ofreplicating fine details of the magnetic field, the long-term evolution of the polar field is very similar in simulations and observations.Simulations typically yield a smoother evolution of polar fields than observations, which often include artificial variations due toobservational limitations. We also find that the simulated field is fairly insensitive to uncertainties in model parameters or the inputdata. Due to the decay term included in the model the effects of the uncertainties are somewhat minor or temporary, lasting typicallyone solar cycle.

Key words. Sun: magnetic fields – Sun: photosphere – Sun: activity

1. Introduction

Surface flux transport (SFT) models have been studied anddeveloped for several decades, starting from Leighton (1964),and continuing to modern models capable of replicating finedetails of magnetograms (Jiang et al. 2014; Yeates et al. 2015;Lemerle et al. 2015). They have been used for various purposesincluding, for example, studies of long-term solar activity overhundreds of years (Jiang et al. 2011), simulations of magneticfield between observations (Worden & Harvey 2000), and stud-ies on how individual active regions affect poleward flux surges(Yeates et al. 2015). The models usually include meridional cir-culation, differential rotation and diffusion, and a term describ-ing the emerging flux. Sometimes additional terms are includedto describe the decay of the magnetic field. The main prob-lem in SFT modeling is the parameterization of these processes.Meridional circulation and diffusion in particular are difficult toobserve, and hence difficult to model accurately.

We aim to use SFT modeling to simulate the long-term evo-lution of the photospheric magnetic field over the past one hun-dred years from pseudo-magnetograms created from historicalcalcium II K line and sunspot observations (Pevtsov et al. 2016).In particular we are studying the polar fields, which are con-nected to coronal holes. Coronal holes are a source of openmagnetic flux that extends into the heliosphere as the helio-spheric magnetic field. They therefore play an important role inspace weather in general, but are not included in the pseudo-magnetograms. The lack of information about polar areas and

the exact structure of active regions pose further challenges toSFT modeling.

Magnetic flux emergence can be included in an SFT sim-ulation in many different ways, varying from creating artificialbipoles based on the observed sunspot number (Jiang et al. 2011)or the observed sunspot area (Jiang et al. 2010; Baumann et al.2004) to assimilating observed active regions (Yeates et al.2015) or full disk magnetograms (Worden & Harvey 2000) di-rectly. The pseudo-magnetograms include active regions that canbe directly assimilated into the simulation. However, the ex-act flux distributions inside these active regions are not veryaccurate, which leads to an additional source of uncertaintyfor pseudo-magnetograms compared to assimilating observedmagnetograms.

The most widely used SFT model includes only merid-ional circulation, differential rotation, and supergranular dif-fusion (Mackay & Yeates 2012). In long simulations spanningmultiple solar cycles, this model is in some cases unable tocorrectly reverse the polarity of the polar fields (Schrijver et al.2002; Baumann et al. 2006; Lean et al. 2002). It has also beenshown to be insufficient to simulate the decay of the axialdipole during the minimum between cycles 23 and 24 (Yeates2014). Including an additional decay term that slowly weak-ens the polar fields over time can help eliminate these prob-lems. Schrijver et al. (2002) and Yeates (2014) use simple ex-ponential decay terms without a clear physical interpretation.Baumann et al. (2006) derive a similar decay term based on the

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A&A 604, A8 (2017)

diffusion of the radial field in the convection layer. A significantincrease in the speed of meridional circulation can also reducethe build up of flux at the poles and help weaken the unrealis-tically strong polar fields (Wang et al. 2002b; Yeates 2014), butthe circulation speed required to do this is much higher than theobserved circulation speed (Hathaway & Rightmire 2010).

The speed of meridional circulation can be measured, forexample, by tracking magnetic features temporally in magne-tograms or by determining it from helioseismic observations.Although many SFT models assume a constant meridional cir-culation profile, varying speed profiles have also been used andshown to improve the stability of the polar fields (Wang et al.2002a). Upton & Hathaway (2014) use feature tracking to findthe correct meridional flow and differential rotation speeds foreach rotation. This approach leads to more accurate simulations,but cannot be used with historical observations, which do notcontain information about the speed of surface flows.

Supergranular diffusion is a challenging process toparametrize, because it cannot be measured directly. Earlystudies suggested diffusivities as large as 770−1540 km2 s−1(Leighton 1964), but later estimates of the optimal value of su-pergranular diffusivity have been significantly lower. For ex-ample, values of 600 km2 s−1 (Wang et al. 1989), 500 km2 s−1(Wang et al. 2002b), and 350 km2 s−1 (Lemerle et al. 2015) havebeen suggested. The large uncertainties in supergranular diffu-sivity imply that the sensitivity of the model to this parametershould be investigated.

In this paper we use the SFT model by Yeates et al. (2015)extended with the decay term by Baumann et al. (2006) to simu-late the radial photospheric magnetic field of the past three solarcycles using synoptic maps from the National Solar Observa-tory (NSO). The data assimilation technique of the model canbe easily adapted to be compatible with pseudo-magnetograms,and the model uses a meridional circulation profile that is con-stant in time. These properties make it particularly suitable foruse with pseudo-magnetograms. In this paper we show that evenif there are significant uncertainties related to meridional circula-tion, supergranular diffusion, and the structure of active regions,the model is fairly robust, and can reliably simulate the largescale evolution of polar fields. We also demonstrate the neces-sity of the additional decay term. The paper is organized as fol-lows. In Sect. 2 we describe the data used in this paper and theinstrumentation used to obtain the data. In Sect. 3 we discuss theSFT model and the additional decay term in detail. In Sect. 4 werun various tests with the model to assess how sensitive it is touncertainties in meridional circulation, supergranular diffusion,the exact structure of active regions, and the initial polar field.In Sect. 5 the simulated magnetic field is compared to observa-tions, and in Sect. 6 we discuss the results in light of the intendeduse of the model with pseudo-data constructed from historicalobservations.

2. Data

The SFT model uses synoptic Carrington rotation maps of themagnetic field as input. The photospheric longitudinal mag-netic field has been measured in the NSO at Kitt Peak (KP)since early 1970s. In 1970–1974 (Carrington rotations 1558–1611) observations were carried out with a 40-channel pho-toelectric magnetograph using the FeI 523.3 nm spectral line(Livingston & Harvey 1971). From February 1974 (CR 1625)until March 1992 (CR 1853) the instrument was a 512-channeldiode array magnetograph using the Fe I 868.8 nm spectralline (Livingston et al. 1976). The CCD spectromagnetograph

(SPMG) started operating in 1992, first using the Fe I 550.7 nmspectral line from April to October 1992 (CR 1855–1862)and then the same Fe I 868.8 nm line as the previous instru-mentation (Jones et al. 1992). These observations terminated inAugust 2003 (CR 2006), when the Synoptic Optical Long termInvestigations of the Sun (SOLIS) vector spectromagnetograph(VSM) replaced the old instrumentation (Keller et al. 2003). In2006 the VSM modulators were replaced, and in 2010 the orig-inal Rockwell cameras, which had a pixel size of 1.125′′, werereplaced with Sarnoff cameras that have a smaller pixel size of1′′ (Pietarila et al. 2013). The size of the full-disk images of the512-channel magnetograph is 2048 by 2048 pixels and of theSPMG 1788 by 1788 pixels. The size of the SOLIS full-disk im-ages is 2048 by 2048 pixels. The full resolution images wereused to create pseudo-radial synoptic maps with a reduced res-olution of one degree in longitude and 1/90 in sine of latitude,resulting in a synoptic map with a size of 360 by 180 pixels inlongitude-sine-latitude, under the assumption that the magneticfield in the photosphere is radial. These are the maps used in thesimulations of this work.

Thus, the measurements at the highest latitudes are less reli-able due to a combination of projection effects and noise. Also,due to the inclination of the orbit of the Earth relative to thesolar equator, solar poles are not observable during some peri-ods of time, and it is customary to fill the missing pixels withinterpolated data. The polar areas in these maps were filled us-ing cubic spline fitting over the periods when the poles arenot well observed, but the values are known to be partly erro-neous (Harvey 2000; Harvey & Munoz-Jaramillo 2015). Thereare several periods in KP data with known errors, especially dur-ing the 512-channel magnetograph era. The polar fields in late1980s and early 1990s are unreliable. The average strength ofthe southern polar field increases in a step-like manner by al-most four Gauss in 1985 and remains very strong for severalyears. During the maximum of cycle 22 the average strengthsof both polar fields often change by several Gauss betweentwo rotations. Similar behaviour can also be seen during themaximum of cycle 21. However, the problems mainly relateto polar fields (Virtanen & Mursula 2016), not active regions,which means that our simulations are not affected by this prob-lem. Therefore one could expect that the amplitudes of polarfields obtained from an SFT simulation might not agree veryclosely with those of the original synoptic maps. We note thatthere is an ongoing effort to recalibrate Kitt Peak measurements(Harvey & Munoz-Jaramillo 2015), but the recalibrated data isnot yet available. Because of many errors in data we excludedthe period of the 40-channel photoelectric magnetograph, 1970–1974, and the first years of the 512-channel diode array magne-tograph, 1974–1978, from this study.

The SFT simulations are sensitive to the polar fields of thefirst rotation, but after that only observations of active regionsare used as input and can affect the results. We started the sim-ulation from January 1978 (CR 1664), which has no data gapsor obvious errors, meaning that there were no missing longitudestrips or large areas of unrealistically strong magnetic fields.

Figure 1 shows the KP and SOLIS data sets over the time in-terval studied here. Each vertical line was formed by averaginga synoptic map over longitude. We call this presentation a super-synoptic map. The data set started from January 1978 (CR 1664)and ended in May 2016 (CR 2177). The switch from the earlierKitt Peak instrumentation to SOLIS happened in August 2003between rotations 2006 and 2007. On July 21, 2014, SOLISwas relocated from Kitt Peak to Tucson, Arizona, leading tothe longest data gap seen also in Fig. 1. As described above,

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I. O. I. Virtanen et al.: Reconstructing solar magnetic fields from historical observations. II.

1980 1990 2000 2010

Year

-80

-60

-40

-20

0

20

40

60

80

La

titu

de

[d

eg

]

-10

-8

-6

-4

-2

0

2

4

6

8

10

Br [G

]

Fig. 1. Super-synoptic map of the KP and SOLIS photospheric mag-netic field observations from January 1978 (CR 1664) to May 2016(CR 2177). Yellow and blue tunes correspond to positive and negativepolarity. White vertical lines are periods with no observations.

the synoptic maps contain polar field values obtained by filling.However, the filling method is not perfect, as indicated by therapidly alternating colors of the polar-most bins of the super-synoptic map. This does not affect our results because the polarfields were obtained from simulations with the SFT model. Thewhite vertical lines are periods with no observations. The datagaps were not filled, meaning that no active regions were assim-ilated into the simulation during rotations with no observations.

3. The SFT model

We used the SFT model of Yeates et al. (2015) to model the evo-lution of the photospheric magnetic field and compare it withthe observed evolution. The radial magnetic field Br(θ, φ, t) canbe written in a spherical coordinate system as a function of thetwo-component vector potential [Aθ, Aφ]:

Br(θ, φ, t) =1

R sin(θ)

(∂

∂θ(sin(θ)Aφ) −

∂Aθ∂φ

)(1)

where R is the radius of the Sun, θ is the colatitude, φ is theazimuth angle, and t is the time. The vector potential evolves intime as follows (Yeates et al. 2015):

∂Aθ∂t

= ω(θ)R sin(θ)Br(θ, φ, t) −D

R sin(θ)∂Br(θ, φ, t)

∂φ+ S θ(θ, φ, t)

(2)∂Aφ∂t

= −uθ(θ)Br(θ, φ, t) +DR∂Br(θ, φ, t)

∂θ+ S φ(θ, φ, t) (3)

where ω(θ) is the differential angular velocity of rotation, D isthe supergranular diffusivity coefficient, uθ(θ) is the velocity ofthe meridional circulation, and S θ and S φ are the source termsthat represent the emergence of new flux.

We used the angular velocity of differential rotation ω(θ) (inthe Carrington rotation frame) of Yeates et al. (2015):

ω(θ) = (0.521 − 2.396 cos2(θ) − 1.787 cos4(θ)) deg/day. (4)

We did not investigate here the effects of possible variations oruncertainties in differential rotation speed because differentialrotation parameters are better constrained than the parameters

studied here. Moreover, if the parameters are kept within a sen-sible parameter range they do not significantly affect the aver-age strength of polar fields in SFT simulations (Baumann et al.2004).

For meridional circulation, the profile of Schüssler &Baumann (2006) was used. It is a semi-empirical profile thathas been adapted to the profile obtained from helioseismicmeasurements:

uθ(θ) = u0 sin(2(π

2− θ

))exp

(π − 2

∣∣∣∣∣π2 − θ∣∣∣∣∣) (5)

where u0 is the peak velocity. The optimization of these pa-rameters has been studied by, for example, Yeates (2014) andLemerle et al. (2015), and is further discussed in Sect. 4.1 (seealso Whitbread et al., in prep.).

The source terms S θ and S φ represent the emerging flux ofactive regions. It should be noted that they are included in theequations only for the sake of clarity and have no functionalform. In practice the active regions are copied from the obser-vations directly to the simulated radial field Br(θ, φ, t), not viathe vector potential.

The radial magnetic field at the start of the simulation wasdefined by the first observed synoptic map. Thereafter only ac-tive regions of the remaining maps are used as input. The simula-tion computed the vector potential from the radial field by settingAφ = 0 every time the vector potential was solved. Because offreedom of gauge we could make this choice as long as Aθ ischosen so that B = ∇ × A. Equation (1) then gives:

Aθ = −R sin(θ)∫ φ

0Br(φ′, θ)dφ′. (6)

The vector potential was evolved in time according to Eqs. (2)and (3). Due to Eq. (3), Aφ becomes non-zero, even though itwas set to zero at the beginning of each time step. The new radialfield is then computed from Eq. (1). Active regions, the sourceof emerging flux in the simulation, were added to the radial fieldat the time when they crossed the central meridian.

The differential equations were solved using a finite-difference method. To ensure convergence, the time step ofthe numerical integration must be small enough to fulfill theCourant-Friedrichs-Lewy condition (Courant et al. 1967):

∆t 6∆xu

(7)

where ∆t is the time step, ∆x is the spatial step, and u is thevelocity of a process. Therefore the maximum value of ∆t de-pends on the parameters of the model and the spatial resolutionof the data. We computed the maximum value of ∆t that satis-fied Eq. (7) for differential rotation and meridional circulation.For supergranular diffusion D the criteria is:

∆t 6(∆x)2

D· (8)

We set the time step of the simulation to be 20% of the smallestacquired value to ensure convergence. In the simulations pre-sented in this paper the time step is typically between 10 and20 min.

3.1. Additional decay term

The above model, which we initially used, is successful in repro-ducing the properties of the polar fields of cycles 21–23. How-ever, we later found that the model fails to reproduce the polarity

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Table 1. Decay times τl.

l 1 2 3 4 5 6 7 8 9τl [day] 1888 1623 1340 1088 881 718 591 491 413

reversal of cycle 24. This will be shown later in this work. We at-tribute this failure to the weak sunspot activity of cycle 24, whichwas insufficient to cancel out the magnetic flux of the previous,stronger cycle. Thus, we extend the SFT model of Yeates et al.(2015) with the decay term suggested by Baumann et al. (2006).The purpose of this term is to remove the long-term memory ofthe model. Physically, this corresponds to the radial diffusion ofthe magnetic flux into the convection zone. Without it, a weakcycle might not be able to reverse the polarity of strong polarfields generated during a previous stronger cycle. In SFT modelsthe evolution of the magnetic field is governed by supergranulardiffusion, which simply spreads the magnetic field over a largerarea, and the meridional flow, which transports flux to polar re-gions. Without emergence of new flux, these two processes willlead to a preservation of magnetic flux in polar areas. Even ifnew flux emerges, the flux transported to the poles may be insuf-ficient to reverse the polarity of the magnetic field. Additionaldiffusion is necessary to overcome this limitation.

Baumann et al. (2006) assume that the magnetic field B isradial at the surface and that the radial component disappears atthe bottom of the convection layer at 0.7R. The poloidal field canbe written in terms of a scalar function S (r, θ, φ, t):

B = −∇ × (r × ∇S ), (9)

which obeys the diffusion equation:

η∇S − ∂S∂t

= 0. (10)

The solution of the diffusion equation is written as a sum of or-thogonal decay modes:

S (r, θ, φ, t) =∞∑

n=0

∞∑l=1

l∑m=−l

Rln(r)Ylm(θ, φ)Tln(t) (11)

where Ylm(θ, φ) are spherical harmonics. The monopole terml = 0 has been omitted. Equation (10) implies that Tln(t) hasthe following time dependence:

Tln(t) = exp(−ηk2lnt) (12)

where η = 100 km2 s−1 is the volume diffusion coefficient. Thisleads to an exponential decay of each mode characterized by land n, and the decay time τln = 1/(ηk2ln) depends on the valueof kln, which can be solved numerically. We considered only themodes where n = 0 and l 6 9. The decay times τl for these modesare shown in Table 1. The decay times decrease for larger valuesof l. The highest mode used in the computation of the decayterm, l = 9, had a decay time of about a year. Higher modes couldbe omitted because they do not affect the long-term evolution ofthe field, which we are studying. Radial modes higher than n = 0have even shorter decay times and could be omitted for the samereason (Baumann et al. 2006).

The radial surface field can be presented with sphericalharmonics:

Br(θ, φ, t) =∞∑

l=1

m=l∑m=−l

clm(t)Ylm(θ, φ), (13)

where clm are coefficients that are evaluated for the synoptic mapobserved at time t. Combining this with the above decay timesleads to the following decay rate D(θ, φ, t) of the radial surfacefield:

D(θ, φ, t) =l=9∑l=1

1τl(kl)

m=l∑m=−l

clm(t)Ylm(θ, φ). (14)

The decay term was computed and subtracted from the simu-lated field every 100th time step, meaning that this part of thesimulation ran at a lower time resolution than the rest. Becauseone time step is typically 10 to 20 min, the decay was computedroughly once a day. This was done to save computation time anddoing so did not reduce the accuracy of the simulation becausethe decay process is very slow compared to the other processesincluded in the model, as can be seen from the decay times inTable 1.

3.2. Emerging flux

We identified active regions from each synoptic map of the KPand SOLIS data sets with the following method. The overall fluximbalance, meaning the total sum of signed flux of the map, wasfirst removed by subtracting the overall mean value of the synop-tic map from every pixel. The maps were then combined to forma continuous series of observations from the beginning of thefirst map in the data set to the end of the last, effectively creatingone large map. This allowed us to treat active regions that arelocated on longitude zero or 360 degrees, meaning those that re-side on two different maps at the same time, as one active regionrather than two regions. The combined maps were then trans-formed to absolute value maps and smoothed with a Gaussianfilter. The width of the filter was 4 pixels in both latitude andlongitude and the standard deviation was 2 pixels. The purposeof the filtering was to smooth out small gaps between areas ofnegative and positive polarities within an active region. Other-wise the negative and positive parts could be identified as twoseparate active regions, which would then be discarded duringthe flux balancing phase, which will be discussed later.

We used a threshold of 50 G to identify active regions fromthe smoothed map. Any connected group of pixels with a mag-netic field greater than the threshold counted as an active region.The identified regions were then selected from the unsmoothedmap pixel by pixel. The filter width and threshold were selectedwith trial and error. As long as the threshold was clearly abovethe strength of the background field and the filter was not exces-sively wide they had only a minor effect on how the edges of theactive regions were defined and how much flux was containedwithin each active region.

Flux imbalance was removed from each active region by sub-tracting the active region mean value from every pixel inside theregion. If the total imbalance of the region was more than half ofthe total unsigned flux inside it, meaning if the weaker field wasless than one third of the stronger, the region was discarded. Thisprevented areas of strong magnetic fields in the polar regions andinside poleward flux surges from being identified as active re-gions, and left only the newly emerged active regions, which aretypically fairly balanced in negative and positive flux. Prior toapplying the flux balance condition the majority of the selectedregions were not emerging active regions in the activity belts, butsmall areas of strong unipolar magnetic field at higher latitudes.This was especially true for the KP observations, which oftencontained areas of very strong magnetic fields at high latitudes.

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Every remaining active region was added to the simulation atthe time when the center of the region crossed the central merid-ian. The added region replaced the values of the correspondingpixels in the simulation. Any flux imbalance that existed insidethe area in the simulation was kept by adding the average im-balance to every pixel of the substitute region. This was done topreserve the overall flux balance of the simulation.

4. Testing the SFT model

In light of the intended future use of the SFT model on historicaldata, we investigated the sensitivity of the model when varyingthe values of some of the main parameters. In particular, we in-vestigated the outcome of the model for different diffusion coef-ficients, different meridional circulation speeds, different initialpolar fields, and a simplified representation of emerging flux.

4.1. Testing parameters for meridional circulationand diffusion

Figure 2 shows nine simulations that were run using differentpeak velocities u0 of meridional circulation and different super-granular diffusion coefficients D. Peak velocity increases fromleft column, u0 = 9 m s−1, to middle column, u0 = 11 m s−1,and right column, u0 = 13 m s−1. Diffusion coefficient in-creases from top row, D = 300 km2 s−1, to middle row, D =400 km2 s−1, and bottom row, D = 500 km2 s−1. The parameterrange was selected based on earlier studies on SFT parameters(Yeates 2014; Lemerle et al. 2015) and meridional circulation(Hathaway & Rightmire 2010).

Increasing supergranular diffusion coefficient D makes thepoleward surges wider and causes them to merge into a polarfield at a lower latitude. Increasing meridional circulation speedhas the opposite effect, making the flux surges reach the polesfaster, hence giving them less time to diffuse.

In the top right panel, where the meridional circulation speedis high and supergranular diffusion coefficient low, the polarfields exist only at high latitudes and decay fairly quickly. Thisis especially true for the southern polar field of cycle 22, whichis formed by a single strong poleward surge. On the other hand,in the bottom left panel, where supergranular diffusion is strongand meridional circulation slow, the polar fields form at fairlylow latitudes. Both of these situations are inconsistent with theobservations. Based on this visual analysis and comparison ofFigs. 1 and 2 we selected D = 400 km2 s−1 and u0 = 11 m s−1(middle panel) as the reference parameters. This is also consis-tent with Yeates (2014) and Hathaway & Rightmire (2010).

However, we also find that the simulation is not very sen-sitive to the exact values of u0 and D. Even though the rela-tive change in parameter values between the panels of Fig. 2is significant, there are no dramatic changes in the simulatedfield, and several parameter pairs produce almost identical simu-lations. The u0 and D parameters mostly affect how the polewardsurges develop and polar fields spread. The polar field strengthand the times of polarity reversals are not significantly affected.

4.2. Testing the structure of active regions

To study how the shape of the active regions affects the sim-ulation we replaced all active regions identified in input mag-netograms (see Sect. 3.2) with artificially constructed active re-gions. For this exercise, we took into account the location, tiltangle, and total flux of each observed active region. The tilt angle

Fig. 2. Super-synoptic maps from nine SFT simulations from January1978 (CR 1664) to May 2016 (CR 2177) using different values ofmeridional circulation peak velocity u0 and supergranular diffusion co-efficient D. Yellow and blue tones correspond to positive and negativepolarity.

was defined as the angle between the line that connects the cen-troids of opposing polarities and the east-west direction. Figure 3shows one active region from February 1978 (CR 1665) and thegenerated artificial substitute as an example.

It should be noted that the purpose of this exercise is not tocreate artificial regions that are identical to observed regions, butto demonstrate that the simulation is not sensitive to the exactstructure of the active regions. The parameters that define theproperties of the artificial active regions were selected by trialand error and are not based on observations. The artificial sub-stitute regions were created by the following method:

We created a bipolar region consisting of two circles of equalradius r, one containing positive flux and one containing nega-tive flux. In the following, ΦAR is the total flux of the observedactive region. To define the radius r we took the absolute fluxin every bipole pixel i to be Φi = Φ0 = 200 G and the area ofthe active region to be 2πr2. These assumptions were made onlyto define the radius and were not used in the construction of thefinal substitute region. The radius r was then

r =

√ΦAR

400π· (15)

The distance between the centers of the circles was set to bethe radius r, which meant that the circles partly overlap. The tiltangle, meaning the angle between the line connecting the centersof the circles and the east-west direction, was set to be the sameas the tilt angle of the observed region.

We then set the flux distribution of both the positive and neg-ative part to be a two-dimensional normal distribution centered atthe center of the circle. The variance of the distribution in eitherdirection was 1.5r. The distributions were normalized so that theabsolute flux inside the region was the same as the flux insidethe observed active region. In the region where the negative andpositive circles overlapped the distributions were added together.

Finally we added random variation to the artificial active re-gion by adding to each pixel a value that is 25% of the maximumfield intensity of the artificial region times a random numberdrawn from the standard normal distribution. After this the re-gion was again normalized so that the total absolute flux matchedthe observations and was placed on the centroid of the observedregion.

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Fig. 3. An active region in February 1978 (CR 1665) and its artificialsubstitute. The artificial active region consists of two circles, which ap-pear as ellipses due to unequal scaling in x and y axes. Yellow and bluetones correspond to positive and negative polarity.

As seen in Fig. 3, this leads to artificial regions that havea size that is roughly comparable to the size of the real activeregions. The substitute may be smaller or larger than the ob-served region depending on the flux distribution of the observedregion. In the case of the region shown in Fig. 3 the substitute isslightly smaller than the observed region and has a simpler fluxdistribution.

The top panel of Fig. 4 shows the photospheric field from asimulation run using the artificial substitutes of the observed ac-tive regions as input. For comparison, the bottom panel showsthe field from a simulation that uses the observed active re-gions. The observed field of January 1978 (CR 1664) was usedas the starting point. The peak velocity of meridional circulationis 11 m s−1, and the diffusion coefficient is D = 400 km2 s−1.The most notable difference between the two simulation runsis the second strong southward flux surge that starts in January1992 (CR 1851) and is marked with a black arrow in the toppanel of Fig. 4 but is not present in the observations. In this case,the reshaping of the active regions to their artificial substitutesis momentarily producing an intense surge of negative flux thatis transported southward. This is caused by large active regionsduring this period of high activity, which have complex struc-tures that are very different from the simple artificial regions.We note, however, that the simulation with the observed activeregions shows a number of smaller surges of negative flux at thesame time, roughly balancing the effects of the intense surge ofthe artificial run to the overall polar field strength and the timingof the following polarity reversal. The long-term polar field con-tribution depends predominantly on the axial dipole moment ofthe active region, because it is the slowest decaying mode.

As an additional note, the tests of replacing actual active re-gions with simple bipoles also suggest that the procedure of pre-serving the observed flux distribution described in Sect. 3.2 is notcritical for successful modeling of polar fields. Model runs thatreplace actual field distributions with balanced artificial bipo-lar distributions are capable of reproducing the polar fields verywell.

4.3. Testing the effect of the initial polar field

The magnetic field at the start of all the above simulations wasdefined by the observed synoptic map of the first rotation. To

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Fig. 4. Super-synoptic maps of SFT simulations run using artificial ac-tive regions (top panel) and observed active regions (bottom panel). Yel-low and blue tones correspond to positive and negative polarity. Theblack arrow marks a poleward surge that appears in the top panel butnot in the bottom panel.

determine how sensitive the simulation is to the change of thefirst map we ran simulations using artificial synoptic maps asstarting points. Figure 5 shows three super-synoptic maps fromthree simulations that used artificial initial maps with a very sim-ple structure of the overall magnetic field. The artificial mapshave a constant polar field above 60◦ and below −60◦ latitudeand are zero elsewhere. The strength of the polar field is ±2.5 Gin the top panel, ±5.0 G in the middle panel and ±7.5 G in thebottom panel. The field is positive in the north and negative in thesouth. All three simulations start from January 1978 (CR 1664).The average strength of the simulated polar fields is shown inFig. 6.

The differences in polar field strength, which are very clear atthe beginning of the simulations, gradually disappear with time.The simulations start from the ascending phase of cycle 21, andthe polar field strengths have converged to almost identical val-ues by the polarity reversal of cycle 22. This convergence of sim-ulations with different starting field strengths is mostly due to theadditional decay term in the model (see Sect. 3.1). To demon-strate the importance of the decay term, Fig. 6 also includes polarfields from a simulation that was run without the additional de-cay term and with an initial polar field of ±7.5 G. In this case, thefield does not converge with the fields of the other simulations.The strong initial polar field causes the simulation to alternate

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Fig. 5. Super-synoptic maps of three different SFT simulations that useartificial synoptic maps for the first rotation. The simulations start fromJanuary 1978 (CR 1664). Top panel: field strength is ±2.5 G above andbelow ±60◦ latitude and zero elsewhere. Middle panel: ±5.0 G aboveand below ±60◦, zero elsewhere. Bottom panel: ±7.5 G above and below±60◦, zero elsewhere. Yellow and blue tones correspond to positive andnegative polarity.

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Fig. 6. Average field strength in the polar regions (above and below±60◦). Solid lines are from the simulations shown in Fig. 5. Dashedlines are from a simulation without the additional decay term.

between strong and weak polar fields, and this pattern does notfade away within the time frame of the simulation.

Figure 7 shows a simulation that was started in March 1982(CR 1720) when the polar fields were weak and about to reversetheir polarity. The simulation starts with an empty map, mean-ing that the magnetic field is set to zero at CR 1720. In this case,the polar fields remain very weak in the declining phase of cy-cle 21. This is because the strong poleward surges that started inthe ascending phase of the cycle around 1980, as can be seen inFig. 1, are not included in the simulation. The active regions thatcreated the surges appeared before the start of the simulation andthe surges themselves are not included in the first synoptic mapof this run. The polar fields start forming only after the flux fromthe first active regions that appear after CR 1720 reach high lat-itudes. By the descending phase of cycle 22, in the early 1990s,the polar fields are as strong as in Fig. 6, meaning that the uncer-tainty in the initial field affects the simulation for only one solarcycle.

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5. Comparing modeled and observed field evolution

The top panel of Fig. 8 shows a larger version of the middle panelin Fig. 2, meaning the simulation where D = 400 km2 s−1 andu0 = 11 m s−1, and the bottom panel shows the observed fieldfor comparison. The activity belts of the modeled and observedmagnetic fields are almost identical, which is expected becausethe observed active regions were used in the simulation. Mostof the poleward surges produced by the active regions behavesimilarly at mid-latitudes, having similar strengths and slopesand causing the polar fields to change polarity at roughly thesame times in simulations and observations.

There are some differences in how the surges connect tothe polar fields at high latitudes, especially around and afterthe maximum of cycle 22. However, rather than reflecting prob-lems in the simulation, these differences may be due to problemsin the polar fields of the KP maps, which are known to haveerrors (Virtanen & Mursula 2016). This interpretation is evenlikely, because KP data depict uniquely large annual variationsat this time, as seen in the bottom panel of Fig. 8. Also, the sev-eral sudden polarity reversals, especially in the northern polarfield around 1990, are known to be errors in the data (see, e.g.,Virtanen & Mursula 2016), which is why the smooth transitionof the simulation is probably closer to reality.

The simulation starts in the ascending phase of cycle 21,about forty rotations before the polarity reversal of cycle 21.The large active regions in the beginning of the simulation causelarge poleward surges of flux in both hemispheres that eventu-ally turn the polarity of both poles. During the next solar min-imum, the maximum polar field is stronger in the south than inthe north. The northern field is weakened by a large surge of pos-itive flux around December 1982 (CR 1730). This developmentis more clearly presented by simulations than observations, inwhich weakening of the northern field is less systematic, mostlikely due to the vantage point effect and other problems relatedto polar fields. We note also that the observed southern field isintensified in a step-like manner in March 1985 (CR 1760), indi-cating another error in KP data (Virtanen & Mursula 2016) andleading to artificially large fields at at the southern pole for manysubsequent years.

In simulations, the polarity reversal of cycle 22 takes placeten to twenty rotations earlier at the north pole than at the southpole. Both reversals are caused by large clusters of active re-gions and the subsequent poleward surges, the northward surge

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Fig. 8. Super-synoptic map of the simulated photospheric magnetic fieldfrom February 1978 (CR 1665) to May 2016 (CR 2177) (top panel) andthe observed field (bottom panel). Yellow and blue tones correspond topositive and negative polarity. White vertical lines are periods with noobservations.

of positive flux starting in April 1989 (CR 1815), and the south-ward surge of negative flux in June 1990 (CR 1830). The polarfield again grows to be stronger in the south than in the northduring the late declining to minimum phase, but field intensitymaxima in both hemispheres are stronger than during the pre-vious minimum. This hemispheric asymmetry of polar fields isalso seen in the measured KP data (Virtanen & Mursula 2016)and is one demonstration of the "bashful ballerina" phenomenon(Mursula & Hiltula 2003).

The polarity reversal of cycle 23 also takes place earlier inthe north than in the south. There is a continuous stream of weakpoleward surges throughout the cycle. This is most clearly seenin the southern hemisphere. On the other hand, during cycle 22the poleward transportation of flux was more clearly connectedto a few strong surges that were also responsible for the polarityreversal and the formation of the polar fields.

During the deep minimum between cycles 23 and 24 the po-lar fields decay steadily in the absence of sufficiently large ac-tive regions. In 2014 the first poleward surges of cycle 24 reachthe poles and reverse the polarity. However, because the cycleis weak and there are only a few large active regions, the polarfields remain weak until the end of the simulation.

Figure 9 shows the average field strength above 60◦ latitudeand below −60◦ latitude, meaning the mean field strength in the

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Fig. 9. Average field strength above 60◦ and below −60◦ latitude fromthe reference simulation shown in the top panel of Fig. 8 (crosses), asimulation without the decay term (dashed line) and the KP and SOLISdata (open circles). The solid black line is the zero-line.

polar region, for simulations with and without the decay termand for observations. The differences between the simulationsand the observations are clearly visible. The largest differencesoccur in the southern hemisphere in 1985–1989. These are atleast partly caused by the sudden and significant strengtheningof the polar field in the KP data in May 1985 (CR 1762), whichis also clearly visible in the bottom panel of Fig. 8. This is mostlikely an error in the KP data.

The maximum value of the simulated field of cycle 21 ishigher in the south, reaching 3.7 G, whereas the correspond-ing value in the north is −3.0 G. The same asymmetry is true,even more clearly, for cycle 22, where the southern field reaches−5.6 G and the northern 3.6 G. However, during cycle 23 thereis no significant hemispheric asymmetry in the simulated fields.The maximum value is 3.6 G in the south and 3.8 G in the north.

The polarity reversals of cycles 21–23 happen from a fewmonths to a few years earlier in simulations than in observa-tions, but the exact timings of the observed reversals of cycles 21and 22 are difficult to determine because of the very large anderroneous variations of field strength from rotation to rotationthat cause the polar fields to change polarity multiple times overseveral years. The first reversal of the northern field during cy-cle 24 seems to happen earlier in the observations, in 2013,whereas the simulated field does not change polarity until 2015.However, there is large variance in the observed values duringthis time, and some of them are still negative in 2015. The fi-nal reversal in the north is observed in 2015. The reversal of thesouthern field happens within only a few rotations in simulationand observations in 2014.

The necessity of the decay term is clearly demonstrated inFig. 9. Without it the simulation starts to significantly deviatefrom the observations during the deep minimum between cycles23 and 24. Without a sufficient emergence of active regions in theascending phase of cycle 24, and without the additional decayterm, the polar fields remain almost constant for several years ofcycle 24 and fail to reverse polarity by the end of simulations.

The polar fields are on average weaker in the simulationthan in the observations. This is at least partly due to the errorsthat affect the observed polar fields especially in the 1980s andearly 1990s and are discussed above. During cycle 23, which is

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covered by the more reliable SOLIS observations, the simula-tion follows the observed polar fields more closely than duringcycles 21 and 22.

One should also note that the observations used in this workare from three different instruments and their mutual intensityscaling is not known in detail. The KP 512-channel magneto-graph and SPMG operated simultaneously for a short period in1992–1993. The linear scaling factor of 1.46 has been used toscale the KP-512 full-disk observations to SPMG level. How-ever, based on 22 days of observations between November 28,1992 and April 10, 1993, scaling factors in the range of 1.38–1.63 were derived (Wenzler et al. 2006). An alternative method,where KP-512 observations are corrected without comparisonswith other instruments (Arge et al. 2002) gives a lower scalingfactor of 1.242 (Wenzler et al. 2006).

The first version of the synoptic SOLIS data set was cali-brated to match the magnetic flux of the SPMG synoptic maps,but the calibration process was changed after the modulator up-date in 2006. All SOLIS observations prior to 2006 were re-calibrated using the new procedure in 2016, which increasedthe magnetic field intensity of the synoptic maps in 2003–2006.Therefore the SOLIS data set used in this work is not scaled toSPMG level. A comparison of SPMG and SOLIS observationsbased on harmonic expansions (Virtanen & Mursula 2017) indi-cates that the correct SPMG to SOLIS scaling factor would be1.3–1.4.

There is an ongoing recalibration effort that will causechanges to both the KP 512-channel magnetograph and SPMGobservations (private communication with Jack Harvey). Be-cause the scaling and its effects in the simulation are non-linear,we are at the moment unable to estimate in detail how much therecalibration will change the active region flux densities and thesimulated polar fields, but it is clear that polar field strength dur-ing the KP-512 period in 1974–1992 and the SPMG period in1992–2003 will increase. We expect that more detailed informa-tion about the mutual scaling of the NSO instruments will beavailable in near future.

6. Discussion and conclusions

We study here the applicability of SFT models for future mod-eling of historical data, which may have uncertainties in severalmain parameters, including the initial magnetic field, supergran-ular diffusion, speed of meridional flow, and the precise distri-bution of magnetic field in active regions. We tested peak merid-ional flow speeds u0 ranging from 9–13 m s−1 and supergranulardiffusion coefficients D ranging from 300–500 km2 s−1. Parame-ters within this range produce simulations that agree fairly wellwith observations. Selecting an optimal parameter set is some-what difficult, because the two parameters can have similar ef-fects. As a further development, Whitbread et al. (in prep.) hasbeen developing a new automated optimisation technique. It isknown that the meridional circulation may have varied in time(Hathaway & Rightmire 2010). However, we show that the sim-ulations are not very sensitive to the exact values of the D andu0 parameters, and that changes within the parameter range stud-ied here, athough impacting on the strength of poleward surgesand polar fields, lead to a fairly similar solar cycle and long-termevolution.

We show that changing the distribution of magnetic flux in-side active regions while preserving the tilt angles does not causesignificant difference in model outcome (see Fig. 4). The struc-ture of the polar fields remains intact, and significant differences

are observed only during certain short time periods involvinglarge active regions with complex structures. The effect of theseregions is temporary and does not affect the long-term evolutionof the simulated field.

The structure and even the strength of the first synoptic mapaffects the simulation typically for only one solar cycle. Withinthis time period any uncertainties caused by the initial fieldsdisappear and the field becomes completely determined by theemerging flux of the active regions.

We also verify that the additional decay term (Baumann et al.2006) is necessary. Without it the simulations are somewhat sen-sitive to errors in the initial polar fields and the model does notreproduce the polarity reversal of cycle 24 correctly. The addi-tional decay term is necessary for the polar fields to weaken.

Comparing the simulated magnetic field with the observa-tions, we find that the activity belts and poleward surges are verysimilar, but the polar fields are sometimes different. During thetime period from May 1985 (CR 1762) until the polarity reversalof cycle 22 in 1990 there are large differences in field strength,which are at least partly due to errors in the polar fields of theKP maps. Simulations may yield a more realistic view than ob-servations during, for example, the polarity reversals of cycles21 and 22 and in the mid-1980s. Simulations and observationsagree with each other considerably better during the SOLIS pe-riod. We also note that simulations verify the larger maximumfield strength in the southern hemisphere during the decliningto minimum phase of cycles 21 and 22, as well as in the earlydeclining phase of cycle 24, in agreement with the “bashfulballerina” (Mursula & Hiltula 2003; Mursula & Virtanen 2011;Virtanen & Mursula 2016).

We conclude that the SFT model is stable and reliable evenwithout accurate information about the initial polar fields or theexact structure of the active regions. The initial field affects thesimulation for roughly one solar cycle, whereas the structure ofactive regions can affect individual poleward surges but does notchange the long-term evolution of the polar fields. The model isalso not very sensitive to the exact values of parameters of merid-ional circulation or supergranular diffusion. Although the modelis not able to reproduce the finer details of the field, it is robustenough to be used as a tool to study the long-term evolution ofthe polar fields.

Acknowledgements. We acknowledge the financial support by the Academy ofFinland to the ReSoLVE Centre of Excellence (project No. 272157). The Na-tional Solar Observatory (NSO) is operated by the Association of Universitiesfor Research in Astronomy, AURA Inc under cooperative agreement with theNational Science Foundation (NSF).

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