Application of Synoptic Magnetograms to Global Solar Activity Forecast
aa r X i v : . [ a s t r o - ph . S R ] O c t Application of Synoptic Magnetograms to Global Solar ActivityForecast
I. N. Kitiashvili , NASA Ames Research Center, Moffett Field, Mountain View, CA 94035, USA Bay Area Research Enviromental Institute, Moffett Field, Mountain View, CA 94035, USA
ABSTRACT
Forecasting solar activity is a long-standing problem, mainly due to limita-tions of available observational data and model inaccuracies. Synoptic magne-tograms, available for the last four solar cycles, provide us with knowledge aboutthe evolution of magnetic fields on the solar surface and present important infor-mation for forecasting future solar activity. In this work, poloidal and toroidalmagnetic field components derived from synoptic magnetograms are assimilated,using the Ensemble Kalman Filter method, into a mean-field dynamo modelbased on Parker’s migratory dynamo theory complemented by magnetic helicityconservation. Tests performed by predicting previous solar cycles 23 and 24 showgood potential for this approach for prediction of the upcoming solar cycle andsupport future developments of this methodology. It was found that the pre-dicted toroidal field is in good agreement with observations for almost the entirefollowing solar cycle. However, poloidal field predictions agree with observationsonly for the first 2 – 3 years of the predicted cycle. The results, based on the syn-optic magnetograms obtained during Cycles 21 – 24, indicate that the upcomingSolar Maximum of Cycle 25 (SC25) is expected to be weaker than the currentCycle 24 (which nearing its end). The model results show that a deep extendedsolar activity minimum is expected during 2019 – 2021, and that the next solarmaximum will occur in 2024 – 2025. The sunspot number at the maximum willbe about 50 (for the v2.0 sunspot number series) with an error estimate of 15 –30 %. The maximum will likely have a double peak or show extended periods (for2 – 2.5 years) of high activity. According to the hemispheric prediction results,SC25 will start in 2020 in the Southern hemisphere, and will have a maximumin 2024 with a sunspot number of about 28. In the Northern hemisphere thecycle will be delayed for about 1 year (with an error of ± . e-mail: [email protected] Subject headings:
Sun: solar activity, magnetic fields; Methods: numerical; dataassimilation
1. Introduction
Growing interest in solar magnetic activity, and, in particular, in its prediction, isdriven by the upcoming Solar Cycle 25 (SC25). As we approach the solar minimum of thecurrent cycle, more predictions of the upcoming activity cycle are released. Predictions ofthe previous Solar Cycle 24 (SC24) were obtained with a wide range of methods and resultedin a wide range of predicted activity strengths, e.g., the predicted maximum sunspot numberranged from 42 to 197 (Pesnell 2012). Such diversity of the predicted cycle properties reflectslimitations in our understanding of global processes in the Sun.The most successful empirical predictions of Solar Cycle 24 (e.g, Schatten 2005) weremade by assuming that the polar magnetic field strength during the preceding solar minimumrepresents the poloidal magnetic field which is later converted into the toroidal field bydifferential rotation. However, a dynamo model which is qualitatively capable of reproducingthis phenomenon and predict solar-cycle evolution of the global magnetic field has not beendeveloped.The limitations of available models and poor knowledge of the current and past statesof global magnetic field can be partially resolved by driving a model solution with ob-servational data. For instance, Choudhuri et al. (2007) used the correlation between thestrength of the polar fields during a solar minimum and the amplitude of the following ac-tivity cycle (Svalgaard et al. 2005) to drive a 2D flux-transport model using observationsof dipole moment parameters that indirectly characterize the polar field strength. The re-sulting forecast of SC24 was correct in the cycle strength but gave an incorrect time of thecycle maximum. Nevertheless, this indicates the potential of the data-driven methodologyto improve the model predictions by including information from observations. Recently,Lemerle & Charbonneau (2017) suggested driving a model solution by an ‘emergence func-tion’, which describes emergence of magnetic flux on the solar surface. They predict thatSC25 will be about about 20% weaker than SC24 and that the mean sunspot number duringthe cycle maximum will be about 89 (Labonville et al. 2019). Surface flux-transport modelshave been used to advect magnetic fields obtained from synoptic magnetograms to predictthe polar field strength and then use the empirical relation to estimate the next solar maxi-mum (Upton & Hathaway 2014). According to this approach, the upcoming SC25 maximumwill be similar to SC24 with an uncertainty of 15% (Upton & Hathaway 2018). 3 –Recently, machine-learning techniques, in particular neutral network methods, havebeen employed to predict solar activity. For instance, statistics-based analysis of the sunspotarea’s butterfly diagram, trained on historical data sets, showed reasonable ability to recon-struct the evolution of the sunspot area for one solar cycle (Covas 2017). However, thereconstructed longitudinal distribution is noisy, and the value of the sunspot area is underes-timated. This approach suggests that SC25 will have a maximum sunspot number of 57 ± ∼
2. Parker-Kleeorin-Ruzmaikin dynamo model
We employ the mean-field dynamo model previously formulated by Kitiashvili & Kosovichev(2008, 2009). It combines Parker’s migratory α -dynamo model (Parker 1955) with the equa-tion of the magnetic helicity balance (Kleeorin & Ruzmaikin 1982; Kleeorin et al. 1995) and 5 –describes the coupling between turbulence and magnetic fields: ∂A∂t = αB + η ∇ A∂B∂t = G ∂A∂x + η ∇ B, (1) ∂α m ∂t = − α m T + Q πρ (cid:20) h B i · ( ∇ × h B i ) − αη h B i (cid:21) , where A and B are the vector-potential and the toroidal field component of magnetic field B , α = α h + α m is total helicity, which includes the hydrodynamic ( α h ), and magnetic ( α m )parts, η = η t + η m (where usually η m ≪ η t ) is total magnetic diffusion, η t is the turbulentand η m is the molecular magnetic diffusivity, G describes the rotational shear, ρ is density, Q and T are coefficients which describe characteristic scales and times, and t is time.Following the suggestion of Weiss et al. (1984), the dynamo model equations (Eq. 1) areconverted into a non-dimensional dynamical system by averaging the model properties in theradial direction. Taking into account a single Fourier mode propagating in the equatorwarddirection (Kitiashvili & Kosovichev 2009) the initial set of equations has the following formd A d t = DB − A, d B d t = i A − B, (2)d α m d t = − να m − D (cid:2) B − λA (cid:3) , where A , B , α m , and t are non-dimensional variables, D = D α is a non-dimensional dynamonumber, D = α Gr /η , α = 2 Qkυ A /G , υ A is the Alfv´en speed, λ = ( k η/G ) = Rm − , k is a characteristic wavelength, and ν is the ratio of characteristic turbulence time-scales.The sign of the dynamo number gives the direction of the dynamo waves migration inlatitude: from higher to lower latitude if the dynamo number is negative, and the oppositefor positive D . We consider only negative dynamo number to make the model consistent withthe butterfly diagram. The reduced dynamo model describes the evolution of three basicproperties: the mean global toroidal and poloidal field components and magnetic helicity.The dynamical system (Eg. 2) can be decomposed into real and imaginary parts as A = a + ia , B = b + ib , α m = c + ic , where the dynamo number D = αD , total helicityis α = α h + α m with quenching α h = α k / (1 + q ( b + b )). Then, the dynamical system canbe rewritten in the following form˙ a = − a + D ( α h b + b c − b c )˙ a = − a + D ( α h b + b c + b c ) 6 –˙ b = − a − b (3)˙ b = a − b ˙ c = − νc + D ( α h + c )[ λ ( a − a ) − ( b − b )] − D c ( λa a − b b )˙ c = − νc + 2 D ( α h + c )( λa a − b b ) + D c [ λ ( a − a ) − ( b − b )] . As was demonstrated before (e.g. Kitiashvili & Kosovichev 2008, 2009), the PKR modelperiodic solution qualitatively reproduces the basic properties of solar activity cycles, suchas the Waldmeier rules, the mean shape of the cycles, etc. In this work, we apply this modelto available synoptic magnetograms and investigate the model’s predictive capabilities forboth the magnetic field components and the sunspot number in each hemisphere of the Sun.The model parameters: D = − . α k = 1 . ν = 0 . q = 5 .
87, and λ = 3 . × − ,as well as the initial conditions, were chosen by performing trial runs to obtain a periodicsolution, which qualitatively reproduces the observed cyclic evolution of the global magneticfield.
3. Observations: Synoptic magnetograms
In the current work, we use synoptic magnetograms for the last four solar cycles (Fig. 1a),which were obtained for the period from 1976 (Carrington rotation 1645) to 2019 (Carringtonrotation 2216) from Kitt Peak Observatory (Harvey et al. 1980; Worden & Harvey 2000),the SOLIS instrument (Keller et al. 2003), and SOHO/MDI and SDO/HMI (Scherrer et al.1995, 2012). The synoptic magnetograms have been reduced to the original KPO spatialresolution of 360 ×
180 pixels and to only the radial magnetic field component. Magnetic fieldmeasurements unavailable in the polar regions were copied from lower latitudes. Synopticmagnetograms for Carrington rotations 1854, 2015, 2016, 2040, and 2041 are not available,and to avoid gaps in the data, the closest magnetograms have been used.Decomposition of the synoptic magnetograms into toroidal and poloidal field compo-nents is a challenging task due to the difficulty of finding a unique solution. To simplifythe magnetic field decomposition problem we assume that the high-latitude magnetic field(above the active latitudes) characterizes the poloidal field component and that the unsignedflux in the active latitudes corresponds to the toroidal field. This assumption is acceptablefor the 1D model with some level of uncertainty, because it requires estimates of the relativebehavior of the field components.To account for toroidal field reversals, the sign of the estimated toroidal field is prescribedaccording to the Hale polarity law. Figure 1b shows variations of magnitude of the estimatedtoroidal field with time for each hemisphere. The time-series of the estimated toroidal and 7 –poloidal fields are averaged over 1-year intervals and are shown by circles for each hemispherein Figure 2. Thin curves show unsmoothed variations of the fields for reference.The resulting annual observations have been normalized to match the model periodicsolutions for the toroidal and poloidal fields (Fig. 3) for each hemisphere. Normalization forthe poloidal field was chosen for a best agreement for the field amplitude. For the toroidalfield, the normalization is performed relative to the last observed solar cycle, following theapproach of Kitiashvili & Kosovichev (2008). Figure 3b shows an example of the toroidalcomponent of the magnetic field calibration in the model solutions for the prediction of SC25.Traditionally, solar activity cycles are characterized by the sunspot number; the toroidal fieldcan be converted to the sunspot number with a corresponding normalization. Comparisonof the observed hemispheric sunspot number and that estimated from the synoptic magne-tograms is shown in Figure 4.
4. Data assimilation methodology: Ensemble Kalman Filter
The problem of forecasting the behavior of a physical system with multiple interactingand evolving non-linear processes is very common in many fields, such as climate change,atmosphere and ocean dynamics, and others. Discrepancies between model solutions and ob-servations significantly restrict or even prevent building reliable forecasts. The origin of thesediscrepancies is incompleteness of the models, as well as shortage and uncertainties of obser-vations. In such situations, observations and models can be linked through a cross-analysisof measurements and model solutions together with estimation of errors and uncertainties.This mathematical procedure, called
Data Assimilation , represents a wide class of methods(see e.g., Evensen 1997; Kalnay 2002).In this paper, we use the Ensamble Kalman Filter method (EnKF, Evensen 1997),which is an extension of the Kalman Filter method (Kalman 1960) for non-linear systems.Because of errors and uncertainties, observations are described in terms of the true state ofsolar global activity, ψ t , which in our case is represented by the evolution of the magneticfield components and magnetic helicity. Measurements, m , can be described by the relation m = M ψ t + ǫ , where M is the measurement functional, describes the relationship betweenthe model properties and observational errors, ǫ . Since the true state of global solar activityis unknown, observational data is considered as an ensemble of possible measurements, m j : m j = M ψ t + ǫ j , (4)where j = 1 , ..., N , and N is the number of ensemble members. The ensemble error covariancematrix of observations is then C eǫǫ = ǫǫ T . In this work, all analysis is performed for N = 300. 8 –The physical model can be presented in the form (Evensen 1997):d ψ = G ( ψ ) dt + h ( ψ ) dq, (5)where G ( ψ ) is a model operator, the term h ( ψ ) dq describes the model errors, in which q represents random variations, h ( ψ ) is the uncertainty of the model state, and t is time. Inthis context, the model solution can be considered as a predicted state ( ψ f ) of the global solaractivity for each moment of time (thick black curve, Fig. 5), described as a combination of the‘true’ state and errors, ψ f = ψ t + e f , with an error covariance C fψψ = ( ψ f − ψ t )( ψ f − ψ t ) T .Because the ‘true’ state is not known, the error covariance for the ensemble of states becomes( C eψψ ) f = ( ψ f − ψ f )( ψ f − ψ f ) T , and allows us to compute the Kalman gain as K e = ( C eψψ ) f M T M ( C eψψ ) f M T + C eǫǫ (6)to obtain a first guess for the corrected model solution (blue curve, Fig. 5). Such estimatesare obtained sequentially for each observation, which is considered as an ensemble of possiblestates for the unknown true state. This analysis has been used to compute a best estimatefor the likelihood solution (green curve in Fig. 5) in the following form ψ aj = ψ fj + K e ( d j − M ψ fj ) . (7)The practical implementation of the EnKF method to the solar activity prediction prob-lem has been performed in three steps, previously described in detail by Kitiashvili & Kosovichev(2011): 1) preparation of the observational data, 2) assimilation for past activity states, and3) forecast.Step 1 includes analysis of the synoptic magnetograms collected from different spaceand ground instruments, by performing decomposition of the magnetograms into toroidaland poloidal field components, and recalibration of the observational data as described inSection 3. To separate analysis for the past and future activity, two phases are identified:‘analysis’ (corresponding to Step 2) and ‘prediction’ (Step 3). Each of these phases is indi-cated in all figures showing the results of this study.
5. Test predictions of past and current solar activity5.1. Application of the toroidal and poloidal fields to solar cycle forecasts
In this section we describe testing of the predictive capabilities of assimilation of toroidaland poloidal field observations into the PKR dynamo model using the EnKF method. The 9 –goal is to reproduce the evolution of the magnetic field components for each hemisphere, aswell as the sunspot number, during SC23 and SC24 based on the data for previous cycles.
Reconstruction of global solar activity during SC23 has been tested using synoptic mag-netograms obtained during two previous cycles, SC21 and SC22 (section 3) from 1977.5 to1996.5. Because the available observations start from the rising phase of SC21, to assimi-late the global field evolution corresponding to the solar minimum between SC20 and SC21,two synthetic observations for 1976.5 and 1975.5 have been added to the observational time-series. Figure 6a shows the annual observational data (circles) and the model solution for thetoroidal field (dashed curves) normalized to the SC22 maximum data for each hemisphere.Also, the model solution phase is chosen to match the phase of the toroidal field of SC22.The start of the prediction phase is indicated by the vertical dashed line (Fig.6a). Usingthe EnKF procedure, the periodic model solution (dashed curves) is corrected accordingto the annual observations for the toroidal (thin red and blue curves, Fig.6a) and poloidalfields (thin black curves, Fig.7a,b) for the corresponding hemispheres. The additional modelvariables, e.g. magnetic helicity, for which observations are not available, are generated fromthe model solution with imposed noise of 10%.Comparison of the model prediction with the actual toroidal field variations shows goodagreement for both hemispheres up to the SC23 maximum (Fig. 6b). After the maximum,the predicted toroidal field in the Northern hemisphere quickly deviates from the observedevolution. In the Southern hemisphere, deviations of the predicted toroidal field becomesignificant two years after the SC23 maximum. Figure 7a,b shows results of analysis for themean poloidal field for each hemisphere. The predicted evolution of the poloidal field inboth hemispheres quickly deviates from the actual data after the first year after predictionstart. The predicted time of the field reversals for both hemispheres is one year earlierthan the actual one. Nevertheless, the maximum strength of the poloidal fields is estimatedcorrectly for both hemispheres. The predicted time of the strongest poloidal field strength istwo years earlier for the Northern hemisphere and about four years earlier for the Southernhemisphere. The origin of the prediction discrepancies is likely due to a significant phasedeviations between the model solution (dashed curves in Fig. 7a,b) and the observations(empty circles).Because the toroidal magnetic field correlates with the sunspot number, the resultscan be presented in terms of sunspot number variations (Fig. 8). The sunspot numberprediction for SC23 is in good agreement with the actual hemispheric sunspot number data 10 –for the whole cycle in the Northern hemisphere (Fig. 8a). For the Southern hemisphere,the prediction results are in agreement with the amplitude and time of the solar activitymaximum. Deviations between the predicted and actual observations gradually increase inthe declining phase of the cycle (Fig. 8b). The total sunspot number (panel c) is correctlyreconstructed for most of the cycle.
Test prediction for Solar Cycle 24 is performed by using assimilation of observationaldata for the previous two and three solar cycles. Utilization of only two cycles can beconsidered as an additional test to investigate the limitations of very short observationaltime-series.
Case 1: Assimilation of data for two solar cycles
In this case, we use the available synoptic magnetograms from 1987.5 to 2008.5 for predictionof SC24. The phase of the periodic dynamo solution fits the activity phase very well, whichallows us to make a good prediction for the toroidal field for the whole solar cycle in theNorthern hemisphere (Fig. 9a,c). In the Southern hemisphere, despite growing discrepanciesbetween the estimated and actual toroidal field variations, the cycle duration is predictedcorrectly. The poloidal field prediction has good agreement with the observations for thefirst two years in both hemispheres, and then it significantly deviates from the observations(Fig. 9b,d). The forecast of the sunspot number (Fig. 10) is in agreement with observations.Most of the discrepancies are related to the complicated shape of the cycle near its maximumin the Northern hemisphere (panel a). Such relatively short-time variations are not describedin the current model formulation. The estimates of the total sunspot number correctlypredict the rise and decay rate, but the strength of the activity cycle is underestimated by20% (Fig. 10c).
Case 2: Assimilation for three solar cycles
In this test case, we use the available magnetic field measurements from 1977.5 to 2008.5for prediction of SC24. Results of the analysis are presented in Figures 11 and 12. Thepredicted evolution of the toroidal field is in good agreement for both hemispheres, althoughthere are some discrepancies during the solar maximum in the Northern hemisphere. In-creasing discrepancies between the predicted and observed toroidal fields during the decayphase of solar activity in the Southern hemisphere are expected because of the step-like 11 –variations of the predicted toroidal field evolution (red thin curve, Fig. 11a) and the sunspotnumber (red thin curve, Fig. 12b). This behavior indicates accumulation of errors duringthe analysis and, in general, gives us a warning that the forecast quality is potentially low.This effect previously was discussed by Kitiashvili & Kosovichev (2008) for assimilation ofsunspot number time-series.Including the additional solar cycle in the assimilation procedure improves the forecastfor the poloidal fields in both hemispheres (Fig. 11b,d). Accuracy of the prediction is goodfor up to 3 years and provides a correct prediction for the time of the polar field reversals. Af-ter this, the predicted and observed field components quickly diverge. The sunspot numberestimates (Fig. 12) show good agreement with the actual data for both hemispheres. Somedeviations in the shape of the predicted activity cycles are expected, and this reflects re-strictions of the dynamo model formulation. The total sunspot number maximum is slightlyoverestimated, but in general the prediction results show a good agreement for the wholesolar cycle.
Our previous predictions of solar activity cycles (Kitiashvili & Kosovichev 2008; Kitiashvili2016) have been performed using sunspot number data and assuming that the sunspot num-ber correlates with the toroidal field (Bracewell 1988). In those studies, the poloidal field wasgenerated from the model solution and perturbed with noise. The synoptic magnetogramsavailable for the last four cycles of solar activity allow us to more accurately estimate globalvariations of the toroidal and poloidal fields and their uncertainties (section 3). For testinghow the poloidal field contribution improves the forecasts of SC23 and SC24, we replacethe poloidal field observational data with synthetic observations generated by the dynamomodel. Figure 13 shows comparison of the predicted toroidal field variations and the sunspotnumber for both hemispheres when only the toroidal field data were used (dashed curves) andfor the case when both toroidal and poloidal field data were assimilated (solid curves). Forboth hemispheres the strongest deviations takes place at the solar maximum (Fig. 13). Thedifference in the sunspot cycle maximum predictions reaches 10.5% for SC24 in the South-ern hemisphere using the observations of 3 cycles and –8.6% in the Northern hemisphere(Table 1). Other tests showed significantly weaker discrepancies. Nevertheless, these testresults indicate that the poloidal field measurements can be very important for improvingprediction accuracy. 12 –
6. Solar Cycle 25 prediction
To perform prediction of the upcoming Solar Cycle 25, we use four solar cycles of synop-tic magnetic field data from 1977 to 2019. Following the procedure described in Section 3, weobtained estimates of the annual variation of the toroidal and poloidal fields. The magneticfield observations were calibrated to approximate the periodic model solution (Fig. 3). In ad-dition, the sunspot number, calculated from the toroidal field measurements, was normalizedto the sunspot number amplitude of Solar Cycle 24 to link the magnetograms and sunspotnumber data (Fig. 4). Figure 14 shows the results of prediction for the toroidal (panel a)and poloidal (panel b) fields. The sunspot number forecast is shown in Figure 15. As ex-pected, the forecast for the toroidal fields (and the sunspot number) is more accurate forthe Northern hemisphere than for the Southern hemisphere because of smaller discrepanciesbetween the model solution and observations at the end of Cycle 24.The model solutions show strong variation in the toroidal fields near and after 2026.5(red curves, Figs 14, 15a,b). These strong variations indicate that prediction uncertaintiessignificantly increase after 2026.5. Thus, the sunspot number prediction for SC25 in theNorthern hemisphere is about 30 (that is ∼
50% weaker than SC24) with an error of 15– 20% and about 25 for Southern hemisphere ( ∼
65% weaker than SC24) with error 25 –30%. The solar maximum is expected during 2024 – 2026 in the Northern hemisphere, andduring 2024 – 2025 in the Southern hemisphere. The total sunspot number predicted for theSC25 maximum is about 50 (for the v2.0 sunspot number definition) with an error estimateof ∼ −
7. Discussion and conclusions
Building accurate forecasts of solar activity requires knowledge of the non-linear multi-scale interactions of turbulent and large-scale flows and magnetic fields, which are describedby global MHD models. At the present time, long-term synoptic observations provide evo-lution of magnetic fields and sunspots on the solar surface, while very limited information isavailable about long-term subsurface dynamics. In addition, the absence of realistic globalmodels of the Sun requires finding a way to estimate solar activity evolution using simplifiedmodels. In such a state of limited knowledge of the global dynamics, shortage of observa-tional data, and incomplete models, the data assimilation approach provides an efficient wayto combine the data and models while taking into account uncertainties of both the modelsand observations. Currently, data assimilation includes a wide range of mathematical meth-ods applicable to diverse problems in weather prediction, planetary sciences, fluid dynamics, 13 –etc. (e.g. Evensen 1997; Kalnay 2002).Previously, we applied the Ensemble Kalman Filter method (Evensen 1997) to pre-dict SC24 by using the sunspot data time-series and a simplified Parker-Kleeorin-Ruzmaikin(PKR, Parker 1955; Kleeorin & Ruzmaikin 1982) mean-field dynamo model (Kitiashvili & Kosovichev2009). This approach allowed us to create a reliable forecast for the whole activity cycle(Kitiashvili & Kosovichev 2008). However, it is important to develop capabilities for pre-dicting not only the general properties of solar activity in terms of the sunspot number, butalso to forecast the evolution of solar magnetic fields including hemispheric activity. In thiscase, we have to deal with relatively short series of available observational data.Recent studies based on the sunspot number data series (Kitiashvili 2019) show thepossibility of obtaining a reasonable solar cycle prediction by assimilating the data for onlytwo previous solar cycles. This encouraged us to apply the previously developed methodologyto synoptic magnetogram data, available for four solar cycles. For this work, we combined allavailable synoptic observations from the National Solar Observatory and the SOHO and SDOspace mission archives. The data were used to estimate hemispheric variations of the toroidaland poloidal field components and assimilate them into the dynamo model for predictionof the hemispheric solar activity. The dynamo model includes an additional parameter, theglobal magnetic helicity, for which there is no observational data, and which is recovered inthe data assimilation procedure.First, we performed three test predictions of SC23 and SC24 using different numbersof the available cycles (two and three). The goal of these tests was to evaluate the predic-tive capabilities of the method. For instance, the test prediction of SC23 was done usingthe synoptic magnetograms only for the period 1977.5 to 1996.5. However, it is found thatan accurate prediction can be made only by including the data for 1975.5 and 1976.5 corre-sponding to the solar minimum period. This testing was performed by adding synthetic datafor this period. As we have known from the previous studies (Kitiashvili & Kosovichev 2008;Kitiashvili 2019), the discrepancy between the model initial conditions and the observationsis less important for assimilation of longer time series. In this case, it is critical to have aclose phase match between the observed variations and the model solution.It is important to note that despite a good qualitative correlation between the meantoroidal field and the sunspot number, the exact quantitative relationship is not known. Weused the three-halves law,
SSN ∼ B / t (Bracewell 1988), but this relation leads to somenoticeable deviations (Fig. 4) that introduce additional uncertainties in prediction of thesunspot number cycle. In addition, the total sunspot number variations are obtained bycombining results for each hemisphere, which already include some uncertainties. It some-times causes cancellation of the uncertainties and thus improves the forecast, but sometimes 14 –amplifies these uncertainties. For instance, the sunspot number predicted for SC23 in theSouthern hemisphere is greater than the actual value, but the predicted total sunspot numbermatches the observations quite well. Contrarily, in the case of SC24 prediction based on 2 cy-cles, the sunspot number forecast for both hemispheres was in agreement with observations,but the total sunspot number variations were overestimated (Fig. 10).To summarize, we can identify the following primary results of the data assimilationapproach: • Using two cycles of the synoptic magnetograms can provide a reasonable forecast ofthe solar activity for the following solar cycle. • Including additional observations improves the predictive capability. • Taking into account poloidal field observations can noticeably improve the forecast,particularly in the case when the data of three preceding cycles are assimilated in themodel. • Forecasted hemispheric toroidal field variations are in good agreement with obser-vations, at least up to the following solar maximum, and often make a reasonableprediction for the whole activity cycle. • Forecasted poloidal fields are in good agreement with observations for up to two yearsin the case of assimilation of data for two preceding activity cycles, and for about threeyears if data for three cycles is assimilated. • According to the presented analysis, the next Solar Cycle 25 will be weaker than thecurrent cycle and will start after an extended solar minimum during 2019 – 2021. Themaximum of activity will occur in 2024 – 2025 with a sunspot number at the maximumof about 50 ±
15 (for the v2.0 sunspot number series) with an error estimate of 30%. • SC25 will start in the Southern hemisphere in 2020 and reach maximum in 2024 witha sunspot number of ∼
28 ( ± ± . ∼ ± ± Acknowledgment.
The work is supported by NSF grant AGS-1622341. 15 – year year s i n ( l a t i t ude ) ab s ( t o r o i da l f i e l d ) , G North HemisphereSouth Hemisphere a)b)
SC21 SC22 SC23 SC24
Fig. 1.— Panel a: Synoptic magnetogram covering four solar cycles from 1976 to 2019. Thegrey color-scale is saturated at range ±
10 G. Panel b: Temporal variations of the meanunsigned toroidal magnetic field component in the Northern (blue curve) and Southern (red)hemispheres. 16 – T o r o i da l f i e l d , G P o l o i da l f i e l d , G a)b) North HemisphereSouth Hemisphere1980 years
North HemisphereSouth Hemisphere
Fig. 2.— Temporal variations of the toroidal (panel a) and poloidal fields (panel b) in theNorthern (blue curves) and Southern hemispheres (red curves). Circles indicate magneticfield values used in the analysis.
REFERENCES
Babcock, H. W. 1961, ApJ, 133, 572Bracewell, R. N. 1988, MNRAS, 230, 535Cameron, R., & Sch¨ussler, M. 2007, ApJ, 659, 801Choudhuri, A. R., Chatterjee, P., & Jiang, J. 2007, Physical Review Letters, 98, 131103Covas, E. 2017, A&A, 605, A44Covas, E., Peixinho, N., & Fernandes, J. 2019, Sol. Phys., 294, 24Dikpati, M., & Gilman, P. A. 2007, New Journal of Physics, 9, 297Evensen, G. 1994, J. Geophys. Res., 99, 10—. 1997, Data Assimilation: The Ensemble Kalman Filter (Springer) 17 –
4 6
8 10 time
4 6
8 10 time North Helisphere South Helispheretoroidal fieldpoloidal fieldmodelsolution observations toroidal fieldpoloidal fieldmodelsolution observations m agne t i c f i e l d s t r eng t h m agne t i c f i e l d s t r eng t h a) b) Fig. 3.— Time-series of the annual toroidal (red dots) and poloidal (blue) field obser-vations calibrated to the corresponding periodic dynamo solutions (thick curves) for theNorthern (panel a) and Southern hemispheres (b). The magnetic fields and time units arenon-dimensional.Harvey, J., Gillespie, B., Miedaner, P., & Slaughter, C. 1980, NASA STI/Recon TechnicalReport N, 81Jiang, J., Cameron, R. H., Schmitt, D., & I¸sık, E. 2013, A&A, 553, A128Jiang, J., & Cao, J. 2018, Journal of Atmospheric and Solar-Terrestrial Physics, 176, 34Jiang, J., Wang, J.-X., Jiao, Q.-R., & Cao, J.-B. 2018, ApJ, 863, 159Jouve, L., Brun, A. S., & Talagrand, O. 2011, ApJ, 735, 31 predicted
23 2 3 .
96% 0 . . − . − .
6% 10 . North hemisphere
Obs. sunspot number(Uccle station)Sunspot number frommagnetogramscalibrated to observedObs. sunspot number(WDC network) s un s po t nu m be r year South hemisphere
Obs. sunspot number(Uccle station)Sunspot number frommagnetogramscalibrated to observedObs. sunspot number(WDC network) s un s po t nu m be r year a) b) time T o r o i da l f i e l d Fig. 5.— Illustration of the EnKF analysis procedure of a model solution correction (thickblack curve) according to observations (circles) to build a forecast (green dashed curve). Bluecurve shows an initial guess of the field variations, and green curve shows a best estimate ofthe field, obtained by the EnKF analysis.Kitiashvili, I. N., & Kosovichev, A. G. 2011, in Lecture Notes in Physics, Berlin SpringerVerlag, Vol. 832, Lecture Notes in Physics, Berlin Springer Verlag, ed. J.-P. Rozelot& C. Neiner, 121Kleeorin, N., Rogachevskii, I., & Ruzmaikin, A. 1995, A&A, 297, 159Kleeorin, N. I., & Ruzmaikin, A. A. 1982, Magnetohydrodynamics, 18, 116Labonville, F., Charbonneau, P., & Lemerle, A. 2019, Sol. Phys., 294, 82Leighton, R. B. 1969, ApJ, 156, 1Lemerle, A., & Charbonneau, P. 2017, ApJ, 834, 133Macario-Rojas, A., Smith, K. L., & Roberts, P. C. E. 2018, MNRAS, 479, 3791Parker, E. N. 1955, ApJ, 122, 293Pesnell, W. D. 2012, Sol. Phys., 281, 507Pipin, V. V., & Kosovichev, A. G. 2015, ApJ, 813, 134Schatten, K. 2005, Geophys. Res. Lett., 32, 21106Scherrer, P. H., Bogart, R. S., Bush, R. I., et al. 1995, Sol. Phys., 162, 129Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207 20 –Svalgaard, L., Cliver, E. W., & Kamide, Y. 2005, in Astronomical Society of the PacificConference Series, Vol. 346, Large-scale Structures and their Role in Solar Activity,ed. K. Sankarasubramanian, M. Penn, & A. Pevtsov, 401Upton, L., & Hathaway, D. H. 2014, ApJ, 780, 5Upton, L. A., & Hathaway, D. H. 2018, Geophys. Res. Lett., 45, 8091Wang, Y.-M. 2016, Space Sci. Rev., doi:10.1007/s11214-016-0257-0Wang, Y.-M., & Sheeley, Jr., N. R. 1991, ApJ, 375, 761Weiss, N. O., Cattaneo, F., & Jones, C. A. 1984, Geophysical and Astrophysical FluidDynamics, 30, 305Worden, J., & Harvey, J. 2000, Sol. Phys., 195, 247
This preprint was prepared with the AAS L A TEX macros v5.2.
21 – T o r o i da l f i e l d , G year a)b) Analysis Prediction year
Analysis Prediction
SC21 SC22 SC23 ( B t e s t - B t r e f ) ! , G North South Hemispheres
Initial model solutionCorrected model solutionInitial prediction of the fieldPrediction of the field with included uncertaintiesEstimated annual observations of the fieldSynthetic observations of the toroidal fieldObservations NorthSouth
Fig. 6.— Panel a: Evolution of the mean toroidal field in the Northern and Southernhemispheres based on field observations for SC21 and SC22, prediction of the mean toroidalfield component variation during SC23, and comparison of the prediction with the toroidalfield observations. Panel b shows errors of the initial periodic solutions relative to the actualobservations during data assimilation analysis and prediction steps. Blue lines show errorsfor the Northern and red lines for Southern hemispheres. Vertical dashed lines indicate theprediction start time. 22 –
Analysis Prediction
SC21 SC22 SC23 year P o l o i da l f i e l d , G Analysis Prediction
SC21 SC22 SC23 year P o l o i da l f i e l d , G Analysis Prediction year ( B po l e s t - B po l r e f ) ! , G Initial model solutionCorrected model solutionInitial predictionPrediction with uncertaintiesEstimated annual observationsSynthetic observationsObservations
North HemisphereSouth Hemisphere a)b)c)
Fig. 7.— Evolution of the mean poloidal field in the Northern (panel a) and Southern(panel b) hemispheres based on field observations of SC21 and SC22, prediction of the meanpoloidal field variation during SC23, and comparison of the prediction with the poloidal fieldobservations. Panel c shows errors of the initial periodic solutions (dashed curves) relativeto the actual observations (empty circles) during data assimilation analysis and predictionsteps. Blue lines show errors for the Northern and red lines for Southern hemispheres.Vertical dashed lines indicate the prediction start time. Blue lines correspond to errors forNorthern and red for Southern hemispheres. Vertical dashed lines indicate prediction starttime. 23 –
SC21 SC22 SC23SC21 SC22 SC23 SC21 SC22 SC23 year year year he m i s phe r i c s un s po t nu m be r t o t a l s un s po t nu m be r North he m i s phe r i c s un s po t nu m be r South a) b)
Initial model solutionCorrected model solutionInitial prediction of the fieldPrediction of the field with included uncertaintiesPredicted annual sunspot number
ObservationsModel
Sunspot number from synoptic magnetogrammsHemispheric sunspot number (Uccle station)Hemispheric sunspot number (WDC station)Monthly sunspot number (WDC station)Total annual sunspot number c) Fig. 8.— Reconstruction of sunspot number variations during SC23 for the Northern hemi-sphere (panel a), Southern hemisphere (panel b), and the total sunspot number (panel c).Solid black curves show an initial periodic model solution, blue curves correspond to thecorrected solution according to the observed fields, thick brown curves represent the initialforecast of the sunspot number variations, and red thin curves show the prediction estimatestaking into account data uncertainties and model errors. Black squares show estimates forthe annual sunspot number, other symbols show the actual observational data. Green dotsshow the monthly sunspot number and are given for reference. 24 – T o r o i da l f i e l d , G year a)c) Analysis Prediction
SC22 SC23 SC24 P o l o i da l f i e l d , G ( B t e s t - B t r e f ) ! , G ( A e s t - A r e f ) ! , G year b)d) Analysis Prediction year
Analysis Prediction year
Analysis Prediction
SC22 SC23 SC24
NorthSouth NorthSouth
North South Hemispheres
Initial model solutionCorrected model solution Initial prediction of the fieldPrediction of the field with included uncertaintiesEstimated annual observations of the fieldObservations
North South Hemispheres
Fig. 9.— Evolution of the mean toroidal (panel a) and poloidal (b) fields in the Northernand Southern hemispheres based on field observations for SC22 and SC23 (case 1), andprediction of the mean toroidal and poloidal field components during SC24. Panels c) andd) show deviations of the model solutions for the magnetic field components from the actualobservational data. Blue curves correspond to the errors for the Northern hemisphere, andthe red curves for the Southern hemisphere. Vertical dashed lines indicate the predictionstart time. 25 –
SC22 SC23 SC24 North yearyear SC22 SC23 SC24 South he m i s phe r i c s un s po t nu m be r he m i s phe r i c s un s po t nu m be r a) b) SC22 SC23 SC24year t o t a l s un s po t nu m be r c) Initial model solutionCorrected model solutionInitial prediction of the fieldPrediction of the field with included uncertaintiesPredicted annual sunspot number
ObservationsModel
Sunspot number from synoptic magnetogrammsHemispheric sunspot number (WDC station)Monthly sunspot number (WDC station)Total annual sunspot number
Fig. 10.— Prediction of the sunspot number variations during SC23 for the Northern (panela), Southern hemisphere (b), and total sunspot number (panel c) assuming availability ofobservations only for two previous cycles (case 1). 26 – T o r o i da l f i e l d , G year a)c) Analysis Prediction
SC21 SC22 SC23 SC24
Analysis P o l o i da l f i e l d , G year b)d) Prediction
SC21 SC22 SC23 SC24 year
Analysis Prediction Analysis year
Prediction ( B t e s t - B t r e f ) ! , G ( A e s t - A r e f ) ! , G North South Hemispheres
Initial model solutionCorrected model solution Initial prediction of the fieldPrediction of the field with included uncertaintiesEstimated annual observations of the fieldObservations
North South Hemispheres
NorthSouth North South
Fig. 11.— Evolution of the mean toroidal (panel a) and poloidal (b) fields in the Northernand Southern hemispheres based on the field observations for three solar cycles (case 2),and prediction of the mean toroidal and poloidal field components variation during SC24.Panels c) and d) show deviations of the model solutions for the magnetic field componentsfrom the actual observational data. Blue curves correspond to the errors for the Northernhemisphere, and the red curves for the Southern hemisphere. Vertical dashed lines indicatethe prediction start time. 27 – yearSC22 SC23 SC24 North yearSC22 SC23 SC24 South he m i s phe r i c s un s po t nu m be r he m i s phe r i c s un s po t nu m be r a) b) yearSC22 SC23 SC24 t o t a l s un s po t nu m be r c) Initial model solutionCorrected model solutionInitial prediction of the fieldPrediction of the field with included uncertaintiesPredicted annual sunspot number
ObservationsModel
Sunspot number from synoptic magnetogrammsHemispheric sunspot number (Uccle station)Hemispheric sunspot number (WDC station)Monthly sunspot number (WDC station)Total annual sunspot number
Fig. 12.— Prediction of the sunspot number variations in SC24 for the Northern (panel a),Southern hemisphere (b), and the total sunspot number (panel c) assuming availability ofobservations for three previous cycles (case 2). 28 –
SC23, 2 cycles T o r o i da l f i e l d , G
1 2 year a) s un s po t nu m be r
1 2 year b) North hem.South hem. SC24, 2 cyclesNorth hem.South hem. SC24, 3 cyclesNorth hem.South hem.
Fig. 13.— Comparison of the first guess predictions for the toroidal field (panel a) andsunspot number (panel b) for two cases: (1) when only toroidal field observations are as-similated (dashed curves), and (2) when both the toroidal and poloidal fields are used (solidcurves). The tests are performed for reconstruction of SC23 and SC24. Time t = 0 corre-spondts to the prediction start time. T o r o i da l f i e l d , G year a) b) Analysis Prediction
SC21 SC22 SC23 SC24 SC25 P o l o i da l f i e l d , G year Analysis Prediction
SC21 SC22 SC23 SC24 SC25
North South Hemispheres
Initial model solutionCorrected model solution Initial prediction of the fieldPrediction of the field with included uncertaintiesEstimated annual observations of the fieldObservations
North South Hemispheres
Fig. 14.— Prediction for the mean toroidal (panel a) and poloidal (b) fields in the Northernand Southern hemispheres based on field observations for three solar cycles. Vertical dashedlines indicate the prediction start time. 29 – yearSC23 SC24 SC25 North he m i s phe r i c s un s po t nu m be r yearSC23 SC24 SC25 South he m i s phe r i c s un s po t nu m be r a) b) yearSC23 SC24 SC25 t o t a l s un s po t nu m be r c) Initial model solutionCorrected model solutionInitial prediction of the fieldPrediction of the field with included uncertaintiesPredicted annual sunspot number
ObservationsModel