THE DYNAMICS OF THE SOLAR MAGNETIC FIELD: POLARITY REVERSALS, BUTTERFLY DIAGRAM, AND QUASI-BIENNIAL OSCILLATIONS

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1 C The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi: / x/749/1/27 THE DYNAMICS OF THE SOLAR MAGNETIC FIELD: POLARITY REVERSALS, BUTTERFLY DIAGRAM, AND QUASI-BIENNIAL OSCILLATIONS A. Vecchio 1,2, M. Laurenza 3, D. Meduri 1,4, V. Carbone 1,5, and M. Storini 3 1 Dipartimento di Fisica, Università della Calabria, Rende (CS), Italy; vecchio@fis.unical.it 2 Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia (CNISM), unità di ricerca di Cosenza-Ponte P. Bucci, Cubo 31C, Rende (CS), Italy 3 INAF/IFSI-Roma, Via del Fosso del Cavaliere 100, Roma, Italy 4 Max-Planck-Institut für Sonnensystemforschung, Katlenburg-Lindau, Germany 5 Liquid Crystal Laboratory (CNR), Rende (CS), Italy Received 2010 June 1; accepted 2012 January 31; published 2012 March 19 ABSTRACT The spatio-temporal dynamics of the solar magnetic field has been investigated by using NSO/Kitt Peak magnetic synoptic maps covering the period 1976 August 2003 September. The field radial component, for each heliographic latitude, has been decomposed in intrinsic mode functions through the Empirical Mode Decomposition in order to investigate the time evolution of the various characteristic oscillating modes at different latitudes. The same technique has also been applied on synoptic maps of the meridional and east west components, which were derived from the observed line-of-sight projection of the field by using the differential rotation. Results obtained for the 22 yr cycle, related to the polarity inversions of the large-scale dipolar field, show an antisymmetric behavior with respect to the equator in all the field components and a marked poleward flux migration in the radial and meridional components (from about 35 and +35 in the southern and northern hemispheres, respectively). The quasi-biennial oscillations (QBOs) are also identified as a fundamental timescale of variability of the magnetic field and associated with poleward magnetic flux migration from low latitudes around the maximum and descending phase of the solar cycle. Moreover, signs of an equatorward drift, at a 2 yr rate, seem to appear in the radial and toroidal components. Hence, the QBO patterns suggest a link to a dynamo action. Finally, the high-frequency component of the magnetic field, at timescales less than 1 yr, provides the most energetic contribution and it is associated with the outbreaks of the bipolar regions on the solar surface. Key words: methods: data analysis Sun: activity Sun: surface magnetism Online-only material: color figures 1. INTRODUCTION The 11 yr cyclic behavior (Schwabe cycle) of the solar activity, mainly characterized by a large poloidal field during activity minima and by the toroidal field in the maxima, represents one of the most intriguing and open aspects of solar physics. The polarity of the magnetic field reverses every 11 yr, producing the 22 yr heliomagnetic variation (Hale cycle). Solar activity is driven by dynamo action taking place in the solar interior, and it is closely related to the emergence and evolution of the magnetic field in the solar atmosphere. In particular, the toroidal field mainly manifests itself as bipolar active regions with sunspots migrating toward the equator during the Schwabe cycle, giving rise to the well-known butterfly diagram (Spörer law). Two main classes of dynamos have been developed to explain the main features of the heliomagnetic activity. Both consider the toroidal field as generated by the ω-effect, i.e., the shearing of a preexisting poloidal field by differential rotation. In the model of Babcock (1961), the regeneration and inversion of poloidal field is due to the decay of bipolar active regions, at the solar surface, releasing a net surface dipole moment that, transported across high latitudes, reverses the polar fields (Babcock Leighton type α-effect). This picture has been subsequently developed through quantitative models (e.g., Leighton 1964; Wang et al. 1989) of the poleward migration involving meridional circulation and supergranular diffusion. On the other hand, in the mean field α ω dynamo models, referring to the Parker (1955) theoretical investigations, the poloidal field is generated by an average electromotive force (α-effect). The α ω dynamo equations support traveling wave solutions whose direction of propagation is constrained by the Parker Yoshimurasignrule (Parker1955; Yoshimura 1975).In particular, equatorward propagating wave solutions are found when the product Γ = α Ω/ r (where α is the dynamo parameter related to the helicity of turbulent fluctuations and Ω is the Sun s angular velocity profile varying with the radial distance r) is negative (positive) in the northern (southern) hemisphere. The internal rotation profile, inferred through helioseismic measurements, shows that Ω/ r < 0 at high latitudes (θ 30 ) and Ω/ r > 0 at low latitudes (Howe et al. 2000). Thus, in order to obtain the observed equatorward migration of the active regions, the sign of α must be negative to satisfy the Parker Yoshimura sign rule (Charbonneau 2010; Dikpati & Gilman 2001). The different behaviors of Ω/ r at high and low latitudes suggests the presence of two activity belts in which magnetic tracers migrate along in opposite directions. Such an approach agrees with the observations of latitude migration of several indicators of solar activity during the Schwabe cycle (Benevolenskaya 1998). In fact, sunspots and faculae, observed in the activity belts, migrate toward the equator while polar prominences (e.g., Benevolenskaya et al. 2001) and coronal emissions (see Rusin et al. 1998) migrate poleward, likely within a 2 3 yr time period. The most recent α ω dynamo models have been upgraded to also include advective flux transport by meridional circulation, which allows us to overcome the Parker Yoshimura sign requirement (Charbonneau 2010 and references therein). The mean field 1

2 dynamo models (see, e.g., Krause & Rädler 1980; Rüdiger & Hollerbach 2003; Ossendrijver 2003; Hoyng 2003), based on the α ω equations, and including a steady meridional circulation (van Ballegooijen & Choudhuri 1988), are able to fit the observed surface magnetic field evolution to yield a robust setting of the cycle period and to reproduce reasonably solarlike time latitude butterfly diagrams with a suitable choice of the model parameters, such as differential rotation, meridional flow, magnetic diffusivity, and α-effect (Rüdiger & Elstner 2002; Bonanno et al. 2003; Käpylä et al. 2006). In particular, the antisymmetric behavior with respect to the equator, found in the photospheric field, can be reproduced by an α-effect localized near the bottom of the convection zone (Ossendrijver 2003). The most advanced dynamo models include the contribution of two types of α-effects, i.e., a Babcock Leighton poloidal source term in a thin layer near the surface and a deep seated α-effect (e.g., Dikpati & Gilman 2001; Dikpati et al. 2004). Apart from the main 11 yr cycle, variations at lower timescales have been recognized in several solar cycle indicators. The most prominently recognized periods are the quasi-biennial oscillations (QBOs) at timescales around 2 yr. This component of the cycle, although weaker than the main component, has been identified in many activity indices (Rao 1973; Valdés- Galicia et al. 1996; Bazilevskaya et al. 2000; Kudela et al. 2002; Bai 2003; Knaack & Stenflo 2005; Vecchio & Carbone 2008, 2009; Laurenza et al. 2009). The QBO origin is still unknown even if it could be related to the dynamo action in the inner solar layers (Benevolenskaya 1998), being also detected in phenomena directly connected with the solar interior. In fact, the equatorial rotation rate close to the tachocline varies with a 1.3 yr period (Howe et al. 2000), as detected from GONG and MDI observations (although it is not confirmed after 2001; Howe 2009) as well as the solar angular momentum (Komm et al. 2003); the solar neutrino flux shows a significant modulation at the QBO rate ( 2010); a 2 yr signal has been detected for the natural p-mode frequencies of the Sun (Fletcher et al. 2010). In this paper we focus on the solar magnetic field variability, observed on the surface, through the Empirical Mode Decomposition (EMD) technique, for a better understanding of the spatio-temporal dynamics of the solar cycle and the underlying physical processes. The data used and analysis are given in Sections 2 and 3, respectively. In Sections 4 and 5, the results obtained for three magnetic field components (radial, meridional, and east west) are discussed. Constraints for dynamo models are derived and emphasized in Section DATA USED Magnetic synoptic maps are obtained from direct measurements of the magnetic field of the solar surface. These maps have been used to investigate the main large-scale features of the solar magnetic field, its variability, and periodic behavior (e.g., Knaack & Stenflo 2005; Cadavid et al. 2005; Laurenza & Storini 2009). The classic procedure to derive magnetic synoptic maps (one per Carrington Rotation, CR), from full-disk photospheric magnetograms, is as follows (Gaizauskas et al. 1983; Harvey & Worden 1998). The coordinates of each magnetic measurement are first transformed into the Carrington coordinate system which rigidly rotates with the mean solar rotational rate ( days). In particular, the Carrington longitude is calculated by summing up the heliographic longitude L (also called the central meridian angle, i.e., the angular distance measured from the central meridian) with the Carrington longitude of the central meridian L 0 (see, e.g., Thompson 2006). The next step is to merge the arrays of transformed elements from all the magnetograms, which belong to a full solar rotation, into a single array representing the entire solar surface. Note that the measured line-of-sight (LOS) component of the magnetic field (B LOS ) is converted to an approximate flux density under the assumption that the fields are radial in the solar photosphere. Then, the magnetic fluxes, corresponding to a given pair of sine latitude and Carrington longitude values, are combined into a weighted average, the weighting function varying as cos 4 (L). This ensures that, in each synoptic map, the magnetic configuration of an active region dominates at its central meridian passage. The assumption of a radial field is fairly good for weak active regions and network structures but it is not necessarily true in strong active regions (see, e.g., Harvey & Worden 1998), which are the surface manifestations of the toroidal field. Thus, these kinds of synoptic maps could not accurately describe the toroidal component of the solar magnetic field. In order to obtain complete information on the magnetic field vector, synoptic maps of the meridional and east west components can be derived from B LOS, combined with the differential solar rotation, by applying the method described in Ulrich et al. (2002). Briefly, this method works as follows. First, the coordinates of each magnetogram are converted into a rotating system by taking into account the differential rotation. This allows us to identify the set of points having the same position on the solar surface to be combined to form a single synoptic chart point. Then, the magnetic field components B merid and B EW are obtained by weighting the observed B LOS values with both sin(l) and cos(l) (Ulrich et al. 2002; Ulrich & Boyden 2005). For each point i on the solar surface, the LOS, the meridional, and the EW components are related as B LOS(i) = cos(l i ) B merid(i) +sin(l i ) B EW(i). (1) The components B merid and B EW are resolved by inverting Equation (1) and by making use of weighted averages of the observed fields: with sb = i cb = i sin(l i ) B LOS(i) = sc B merid + ss B EW (2) cos(l i ) B LOS(i) = cc B merid + sc B EW (3) ss = i sc = i sin 2 (L i ) (4) sin(l i ) cos(l i ) (5) cc = cos 2 (L i ). (6) i Note that the sum is restricted to the number of times the point i can be observed during each solar rotation. It follows that the fields components are B merid = ss cb sc cb ss cc sc 2,B EW = cc sb sc sb ss cc sc 2. (7) Following Ulrich et al. (2002), L is taken negative to the west in a way that the field projected in the east west direction is positive toward the west. 2

3 Figure 1. Butterfly diagram of the net radial magnetic flux averaged over longitude for each CR. The black horizontal line marks the latitude θ = 0. (a) (b) Figure 3. Butterfly diagram of the meridional component B merid (upper panel) and east west component B EW (lower panel) averaged over longitude for each CR. The black horizontal line marks the latitude θ = 0. The vertical white stripes indicate data gaps. (c) Figure 2. Time history of the magnetic field B r (θ,t) at three different latitudes θ. In this paper we analyze both the B r synoptic maps, derived by the classic procedure, and the B merid and B EW components as obtained by following the above approach. In detail, we analyze the NSO/Kitt Peak synoptic maps of the radial magnetic fields, with a resolution of 1 in longitude and 0.01 in sine latitude, respectively, consisting of 363 maps covering CRs from 1976 August to 2003 September. 6 The latitude time diagrams were calculated, for the whole considered time period, by longitudinally averaging the field over each CR. The latitude band investigated was limited to the range [ 65, +65 ] because the measurements at higher latitudes are less reliable due to annual periodic variations caused by the inclination of the 6 Data available at ftp://nsokp.nso.edu/kpvt/synoptic/mag/ Earth s orbit. Figure 1, illustrating the latitude time evolution of the magnetic field B r (θ,t), shows the typical 11 yr butterfly diagram of equatorward magnetic flux with occasional strong poleward surges (Wang et al. 1989). No well-defined periodicity has been directly associated with these surges, although Knaack & Stenflo (2005), through a Fourier analysis, detected power enhancements, in the QBO range, at the solar latitudes where these magnetic flux concentrations seem to emanate. The time evolution of the magnetic field strongly depends on the latitude. Figure 2 shows three examples of the corresponding time series at three different fixed latitudes θ. At high latitudes, the fields are strongly modulated by the 22 yr variation (panel (c)), while at the equator no evident periodicity can be recognized and the signals are dominated by small-scale fluctuations (panel (a)). In addition, at latitudes of about θ 25, noticeable high fluxes during the solar maxima are observed opposed to very low values during the minima (Figure 2, panel (b)). The meridional (B merid ) and east west (B EW ) components were calculated from the daily full-disk NSO/Kitt Peak magnetograms. 7 The differential rotation profile, ω(θ), was chosen according to Ulrich & Boyden (2005) and L 0 was retrieved from the data header. We remark that the L 0 values for the data during 1979 and from 1992 May 30 to 1992 June 6 were retrieved from astronomical ephemeris/almanac series (Astronomical Ephemeris 1978; Astronomical Almanac 1991) since the corresponding values in the data headers were missing. The meridional and EW fields were derived for 321 synoptic maps, from 1976 August to 2000 July, i.e., the period for which the full-disk magnetograms are available. Also in this case, latitude time maps were obtained through longitudinal averages over each CR and limited to the latitudinal range [ 65, +65 ]. In Figure 3, the latitude time evolution of B merid (θ,t) and B EW (θ,t) is depicted. The histograms of the EW (black line) and meridional (red line) component amplitudes are shown in 7 Data available at ftp://nsokp.nso.edu/kpvt/daily/ 3

4 Figure 4. Histograms N(B) oftheb EW (black line) and B merid (red line) amplitude. Figure 4. This allows us to quantitatively estimate that both components have very similar amplitudes, in agreement with results from Ulrich & Boyden (2005). Moreover, a comparison of our Figure 3 with Figure 2 of Ulrich & Boyden (2005)shows that very similar features of the field components can be derived. In detail, the meridional field (top panel of Figure 3) is characterized by strong fluxes from mid latitudes toward the polar regions and it is almost unipolar at high latitudes (changing polarity in each cycle maximum). On the other hand, the B EW (θ,t) component (bottom panel of Figure 3) shows welldefined, opposite field polarity for each wing of the butterfly and it is mainly detected in the region ±35, consistently with the location of the sunspot formation zone. We remark that the EW and meridional magnetic fields have been calculated from magnetograms having a daily temporal resolution, while in Ulrich & Boyden (2005) multiple magnetograms per day are sampled. Hence, some small-scale features could be different. B merid and B EW are supposed to be representative of the superficial poloidal and toroidal field, respectively, and provide a distant reflection of the dynamo-generated poloidal and toroidal components (Hoeksema & Scherrer 2010). Thus, they are the only possibility to have traces of the internal field of the Sun. 3. DATA ANALYSIS The time variability of all the field data sets has been investigated through the EMD analysis. This technique was developed to process non-stationary data (Huang et al. 1998) and was successfully applied in several different contexts (Cummings et al. 2004; Terradas et al. 2004; 2010; Barnhart & Eichinger 2011). It describes a signal in terms of Figure 5. Time evolution of the IMFs b j for the radial field component at latitude θ 25 (see the text for details). 4

5 (a) (b) Figure 6. Energy density (E j ) vs. period (P j, j = 0 8) for the EMD significance test applied to the IMFs of Figure 5. Dashed line indicates the 99th percentile (see the text for details). time-dependent amplitude and phase functions, thus overcoming one of the major limitations of Fourier analysis, namely, a correct description of nonlinearity and non-stationarities (Huang et al. 1998). Through the EMD technique, a time series B i (t), at a fixed latitude θ i, is decomposed into a finite number m of intrinsic mode functions (IMFs) as Figure 7. Reconstruction through partial sums of modes b 3, b 4, b 5, with typical average periods between 1 and 4.5 yr, for the latitude θ 25 (a) and corresponding Fourier power spectrum (b). m 1 B i (t) = b ji (t)+r mi (t). (8) j=0 The IMFs b ji (t) represent a set of basis functions obtained from the data set under analysis by following the sifting procedure described by Huang et al. (1998). This process starts by identifying local minima and maxima of the raw signal B i (t). The envelopes of maxima and minima are then obtained through cubic splines, and the mean between them, namely, m 1 (t), is then calculated. The difference between the raw time series and the mean series, h 1 (t) = B i (t) m 1 (t), represents an IMF only if it satisfies two criteria: (1) the number of extrema and zero crossing does not differ by more than one; (2) at any point the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. If h 1 (t) does not support the criteria, the previous steps are repeated by using h 1 (t) as raw series and h 11 (t) = h 1 (t) m 11 (t), where m 11 (t) is the mean of the envelopes in this case, is generated. This procedure is repeated k times until h 1k (t) satisfies the IMF s properties. Thus, b 1i (t) = h 1k (t) is the first IMF, containing the shortest timescale of the process and having a zero local mean. To guarantee that the IMF components contain enough physical sense with respect to both amplitude and frequency modulations, a criterion to stop the sifting process has been introduced (Huang et al. 1998). A kind of standard deviation, calculated using two consecutive siftings, is defined, σ = [ ] N h 1(k 1) (t) h 1k (t) t=0 h 2 1(k 1) (t) and the iterative process is stopped when σ is smaller than a threshold value, in our case chosen as 0.3 (Huang et al. 1998). The function r 1i (t) = B i (t) b 1i (t) is called the first residue, and it is analyzed in the same way as just described, thus obtaining a new IMF b 2i (t) and a second residue r 2i (t). The process continues until b ji or r ji are almost zero (9) Figure 8. EMD power histogram of B r (θ,t), as a function of latitude, in five bands of periods T k as explained in the text. The x-axis indicates the center of each band T k. The color table has been normalized to the maximum power for each latitude. everywhere or when the residue r ji (t) becomes a monotonic function from which no more IMF can be extracted. At the end of the procedure the original time series is decomposed into m empirical modes ordered with increasing characteristic timescale, and a residue r mi (t) which describes the mean trend if any. Each b ji (t) has its own timescale and represents a zero mean oscillation experiencing amplitude and frequency modulations (Huang et al. 1998). In other words, each IMF is not restricted to a particular frequency, but it experiences both amplitude and frequency modulations, i.e., it can be written as b ji (t) = A ji (t) cos(ω ji (t) t), where A ji (t) and ω ji (t) represent, respectively, the instantaneous amplitude and frequency of the j th mode. This kind of decomposition is local, complete, and orthogonal. Since each b ji (t) has its own timescale, we can relate a typical average period P j to each IMF identified by j. We remark that the average period calculated for each EMD mode is not to be intended as the Fourier one. It gives just an indication of the timescale characterizing the EMD mode for which it is computed, although many modes with different average periods may contribute to the variability of the actual signal at a particular timescale. The EMD has been applied to B r, B merid, and B EW for each latitude θ. An example of the IMFs obtained for B r,atlatitude θ 25, is shown in Figure 5. Nine IMFs have been extracted 5

6 The Astrophysical Journal, 749:27 (10pp), 2012 April 10 Figure 10. EMD power histogram of BEW (θ, t) and Bmerid (θ, t) (panels (a) and (b), respectively), as a function of latitude, in five bands of periods Tk as reported in the text. The x-axis indicates the center of each band Tk. The color table has been normalized to the maximum power for each latitude. the spread line at 99th percentile and bullets correspond to the energy density of the IMFs. For this latitude, significant modes j = 3, 4, 5 have QBO timescales and j = 7 describes the 22 yr mode. The orthogonality property of EMD modes can be exploited to reconstruct the signal through partial sums of Equation (8) (Huang et al. 1998; Terradas et al. 2004; 2010; Barnhart & Eichinger 2011). Figure 7 (panel (a)) illustrates the obtained QBOs, at θ 25, reconstructed by summing up the modes b3, b4, and b5. Note that by performing the Fourier spectral analysis over the obtained signal, the power spectrum (panel (b)) is peaked at about 2 yr. We stress that EMD is a very powerful tool to identify the typical periodicities of the magnetic field signals. In fact, this technique overcomes some difficulties of the Fourier analysis when it is applied to real data. Two major disadvantages are the a priori definition of the Fourier modes, which often are far from being eigenfunctions of the phenomenon at hand, and the non-periodicity of boundary conditions. In the latter case, Fourier modes are mixed together in order to build up a solution that corresponds to the fictitious periodic boundary conditions imposed by the analysis. Figure 9. Reconstruction through partial sums of IMFs bj (θ, t), extracted from Br, for the three different base periods Pj (see the text), namely, Pj 1 yr (panel (a)), 1 yr < Pj 4.5 yr (panel (b)), and 18 yr < Pj (panel (c)). In panel (a), the black horizontal line marks the latitude θ = 0. In panel (b), black horizontal lines mark the latitudes θ = 25, θ = 0, and θ = +25. In panel (c), black horizontal lines mark the latitudes θ = 35, θ = 0, and θ = +35. from Br ( 25, t) with average periods ranging from 2.5 months to 22 yr. The statistical significance of information content for the chosen IMFs, with respect to a white noise, can be checked by applying the test developed by Wu & Huang (2004) based on the following considerations. When the EMD is applied to a white noise series, the constancy of the product between the energy density Ej of each IMF (Ej = bj (t) 2, where, denotes time averages) and its corresponding averaged period can be deduced. This relation can be used to derive the analytical energy density spread function of each IMF as a function of different confidence levels. Thus, by comparing the energy density of the IMFs extracted from the actual data with the theoretical spread function, IMFs containing information at the selected confidence level can be distinguished from purely noisy IMFs. As an example, the case for the decomposition of Br ( 25, t) is shown in Figure 6. The dashed line indicates 4. RESULTS FOR THE RADIAL COMPONENT Let us discuss the amplitude of the modes as detected by the EMD. For each latitude θ, we compute the energy density Ej and the average period Pj of the j th mode. Then, we identify five characteristic period ranges: T0 [Pj < 1] yr, T1 [1 Pj < 4.5] yr, T2 [4.5 Pj < 8] yr, T3 [8 Pj < 18] yr, and T4 [Pj 18] yr. Generally, higher amplitudes are found for IMFs having periods shorter than 1 yr. They are associated with the outbreaks of the active regions at the solar surface on such timescales, representing the high-frequency fluctuating component of the magnetic field. 6

7 The Astrophysical Journal, 749:27 (10pp), 2012 April 10 Figure 11. Reconstruction through partial sums of IMFs bj (θ, t), extracted from Bmerid, for the three different base periods Pj (see the text), namely, Pj 1 yr (panel (a)), 1 yr < Pj 4.5 yr (panel (b)), and 18 yr < Pj (panel (c)). In panel (a), the black horizontal line marks the latitude θ = 0. In panel (b), black horizontal lines mark the latitudes θ = 25, θ = 0, and θ = +25. In panel (c), black horizontal lines mark the latitudes θ = 35, θ = 0, and θ = +35. Figure 12. Reconstruction through partial sums of IMFs bj (θ, t), extracted from BEW, for the three different base periods Pj (see the text), namely, Pj 1 yr (panel (a)), 1 yr < Pj 4.5 yr (panel (b)), and 18 yr < Pj (panel (c)). In panel (a), the black horizontal line marks the latitude θ = 0. In panel (b), black horizontal lines mark the latitudes θ = 25, θ = 0, and θ = +25. In panel (c), black horizontal lines mark the latitudes θ = 35, θ = 0, and θ = +35. Nevertheless, as we are interested to study the QBO and higher periods, the power level in the bin T0 is set to 0 to obtain Figure 8 where the power histogram as a function of the bin Tk (k = 0 4) is shown. For each latitude the power is normalized to the maximum value to emphasize the amplitudes at latitudes where the magnetic field level is low. Figure 8 indicates that the magnetic field power, calculated through the EMD, at timescales greater than 1 yr is mainly concentrated in the yr range and equally distributed over all latitudes. This range of average periods refers to IMFs relevant for the QBOs. The interval is wider than the commonly accepted QBO range of yr (e.g., Bazilevskaya et al. 2000), considering that the IMF period is not constant and it is calculated through an average operation (see Section 3). In fact, the limits 1 and 4.5 yr reconcile the intention of isolating the QBOs and the need that the selection of the EMD modes is clear and unambiguous. In particular, the upper limit at 4.5 yr is applied since timescales between 3 yr and 4 yr have generally been found in EMD decomposition of the considered data sets; nevertheless, there could be other principal components with periods close to, but just above 4 yr. In addition, the QBOs are known to be variable in both amplitude and period (up to 4 yr) between different solar cycles and/or within the same cycle. Hence, all the mid-term variations possibly contributing to the QBOs should be considered in the reconstruction. On the other hand, we remark that by selecting only average periods in the restricted range yr, results of the analysis do not show substantial differences. A power excess at 22 yr is also detected at polar latitudes. We note that the Fourier analysis would be able to capture mainly the amplitude excess at 22 yr, while the occurrence of significant power in the QBO range might be underestimated, considering that the QBO amplitude, as found in solar activity indices, is high only during the maxima (Bazilevskaya et al. 7

8 Figure 13. Spatio-temporal behavior of the QBO contribution for radial (a), meridional (b), and EW (c) components of the magnetic field as obtained after smoothing over about 0.1 in sine latitude. Black horizontal lines mark the latitudes θ = 25, θ = 0,andθ = ; Valdés-Galicia & Velasco 2008; 2010) and their frequency is not constant from one cycle to another (Rybák et al. 2000; Vecchio & Carbone 2009; Laurenza et al. 2009). It is well known that in the presence of non-stationary signals, although characterized by a well-defined frequency, the Fourier power spectrum, and to a lesser extent the global wavelet spectrum, detects broader and lowered peaks. Let us discuss the spatial properties of the reconstructions obtained through partial sums of IMF, in Equation (8), at different latitudes. By following the criterion described in Section 3 we consider, for the reconstruction, only the modes above the 99% confidence level. Independent spatio-temporal patterns (Figure 9) have been obtained by selectively choosing IMFs, passing the significance test, with timescales P j in three different ranges. The high-frequency pattern (panel (a); i.e., timescales 1 yr) traces out the butterfly diagram, thus indicating the magnetic flux emergence progressively toward the equator during the Schwabe cycle. In fact, it is apparent that the flux emerges at lower latitudes as time goes on and, after 11 yr, it starts re-emerging at relatively higher latitudes. The magnetic butterfly found in the spatio-temporal pattern corresponding to the temporal dynamics of the magnetic field less than 1 yr is consistent with the lifetime typical of sunspots (of the order of some months) and should be related to the emergence of magnetic flux concentration at active region scales. The reconstruction for periods 1 yr P j < 4.5 yr corresponds to variations in the QBO range. The main feature of this spatio-temporal pattern (panel (b)) is the poleward migration at latitudes greater than +25 in the northern hemisphere and less than 25 in the southern hemisphere. The poleward migration is clearly seen starting at much lower latitudes, and even from the equator, in cycle 21. The ±25 limits have been indicated because beyond this range the poleward migration is clearly observed as a general feature in both hemispheres and for all the considered solar cycles. This general behavior can be interpreted as a quasi-periodic poleward migration of the radial magnetic field at about 2 yr rate, around the sunspot maximum, even if just some few strong poleward surges can be directly identified in the raw data. In the latitude range [ 25, +25 ] no well-defined drift directions can be identified. Nevertheless, during cycle 22 and 23, traces of equatorward drift are apparent. This could represent evidence that the polar and equatorial activity belts, in which the magnetic tracers migrate along in opposite directions, do exist and are associated with the QBO timescales. The EMD reconstruction at periods longer than 18 yr (panel (c) of Figure 9) points out, with respect to the raw data, 8

9 the spatio-temporal properties of the periodic polarity reversals of the Sun and clearly shows some well-known features of the Hale cycle. The reconstruction highlights that the polarity reversal at the polar regions takes place around the maximum of the solar activity. On the other hand, an antisymmetry with respect to the equator is apparent in the belt ±35, where each polarity is constant for 11 yr with inversion at the activity minima. Moreover, butterfly-like structures can be recognized in the active latitude band. At intermediate latitudes (35 < θ < 80 ), the inversion of the poloidal field seems to happen shifted in time with increasing latitude, with an 11 yr periodicity. Finally, we remark that the spatio-temporal pattern associated with intermediate periods, namely, in the range 4.5 yr P j < 18 yr, has very low amplitude with respect to the 22 yr and QBO ones. 5. RESULTS FOR MERIDIONAL AND EAST WEST COMPONENTS The EMD has also been used to analyze B merid and B EW synoptic maps computed as described in Section 2. For both components the magnetic field power, at timescales greater than 1 yr, is enhanced in the range yr and distributed almost over all latitudes between 65 and +65 as for the radial case (Figure 10, panels (a) and (b)). Both fields show a power excess, around the 22 yr period, in the butterfly regions between [ 35, +35 ]. The meridional field power is increased, analogous to the radial case, at high latitudes. The reconstructions at three different temporal scales are derived as in Section 4 and shown in Figures 11 and 12 for B merid and B EW, respectively. The spatio-temporal patterns (panel (a) of Figures 11 and 12) at high frequencies (P j 1 yr) are similar to each other as well as to the corresponding one derived for the radial component (panel (a) of Figure 9). The patterns at 22 yr show that, being the EW field concentrated at active latitudes, the poleward flux migration is only characteristic of B merid (compare panel (c) of Figures 11 and 12). For this component, the magnetic inversion takes place at later times at increasing latitudes, starting from +35 and 35 in the northern and southern hemispheres, respectively. A similar behavior is observed in panel (b) of Figure 11 for the pattern at 2 yr scales. Poleward migrations from low latitudes are detected in the meridional component during the maxima and descending phases of each solar cycle, whereas the equatorward migrations are not clear. On the other hand, B EW mainly shows traces of equatorward drift (particularly during solar cycle 22). The migration trends in the three components at the QBO timescale are highlighted by performing data smoothing in latitude (over about 0.1 in sine latitude). This approach allows us to catch the large-scale features without altering their time behavior. The smoothed QBO patterns for the radial, meridional, and EW components are depicted in Figure 13. Because of the smoothing procedure, values are averaged out and the color bar range has been saturated at ±3 G. Panels (a) and (b) show a very well defined poleward migration. As can be seen in panel (c), the migration toward the equator in the toroidal QBO pattern is present in all cycles during the maximum and descending phase: in cycle 21, in cycle 22, and in cycle 23. We note that the characteristics derived from the B r synoptic maps are emphasized in those obtained from B merid and B EW.In particular, the 22 yr spatio-temporal patterns are characterized by an antisymmetric behavior with respect to the equator. Moreover, the equatorward (poleward) migration is mainly related to the superficial toroidal (poloidal) field. 6. CONCLUSIONS The spatio-temporal dynamics of the heliomagnetic field for about three solar cycles has been investigated. The EMD has been applied to the radial, meridional, and east west components of the magnetic field. We underline that B merid and B EW can provide information on the dynamo-generated poloidal and toroidal components, although they are just a distant reflection and that there is no way to further resolve these two components without a true vector magnetograph (Ulrich et al. 2002; Hoeksema & Scherrer 2010). In particular, the EW component (superficial toroidal field) is found to be confined primarily in the range [ 35, +35 ] and shows an antisymmetric behavior with respect to the equator, as also happens for the theoretical toroidal field as derived from dynamo models (e.g., Dikpati et al. 2004) at the tachocline level ( 0.7 R ). The highest amplitudes correspond to modes at timescales shorter than 1 yr, which can be associated with the outbreaks of the active regions at the solar surface. When focusing on longer periods, two main scales of variability have been identified in the three components of the magnetic field, namely, the wellknown 22 yr period and QBOs. The spatio-temporal patterns associated with both variations present common features such as a marked flux migration from mid latitudes toward the poles and traces of migration toward the equator. Some features of the main solar cycle are clearly detected when the results obtained for B r, B merid, and B EW are compared: the equatorward migration during each 11 yr solar cycle for the three components and a periodic flux migration in the meridional and radial components from mid latitudes to the poles during each 11 yr cycle. These results are useful to test theoretical dynamo models and to set their parameters for a better understanding of the spatio-temporal evolution of the solar magnetic field. In fact, some models, in which a deep seated α-effect and meridional circulation play a fundamental role, produce representative solutions showing the equatorward migration of the deep toroidal field and the poleward migration and intensification of the surface poloidal field as a direct consequence of advection by meridional circulation (Charbonneau 2010). Moreover, an α-effect localized near the bottom of the convection zone selects an antisymmetric toroidal field (Ossendrijver 2003). On the other hand, pure Babcock Leighton models tend to produce symmetric-parity solutions and a strong (often dominant) polar branch in the toroidal field butterfly diagram, which we do not observe in the EW field. For instance, our analysis underlines, for the three components of the surface magnetic field, a net separation (see the horizontal black lines in Figures 9, 11, and 12 at ±35 ) between the poleward and equatorward branches. This particular feature can be reproduced by deriving the α-effect source from the global hydrodynamic instability of latitudinal differential rotation in the tachocline, as calculated using a shallow-water model (Dikpati & Gilman 2001). The same observed features could be obtained by a dynamo model distributed in the bulk of the convective zone, including the region of subsurface rotational shear, by using suitable surface boundary conditions (Pipin & Kosovichev 2011). At lower timescales, we found significant variations in the QBO range almost uniformly distributed over all latitudes with high amplitudes during the maximum and descending phase of each solar cycle. We characterized for the first time the spatio-temporal dynamics of the surface magnetic field 9

10 at 2 yr timescales. The QBOs can be associated with flux migration in the poleward direction for both meridional and radial components (more accentuated in the former) around the maximum and descending phase of each solar cycle. Weaker signs of equatorward drift, confined in the belt ±25, appear in the QBO patterns of the radial and toroidal components (mainly during cycle 22). This behavior, reflecting the main features of the 22 yr cycle, could be interpreted in terms of an additional dynamo action responsible for QBOs. The possibility of having a dynamo working at timescales of 2 yr is reasonable from a theoretical point of view. As a simple order of magnitude estimate, the frequency associated with wave solutions of the linear α ω equation can be expressed as ω 1 α 0 Δu 2 l Δr, (10) where α cm s 1 (Charbonneau 2010) and l 10 9 cm are, respectively, a characteristic value for the α-effect coefficient and the length scale of the dominant turbulent eddies, as estimated, e.g., from mixing length theory. The quantity Δu/Δr, representing the radial shear, has been estimated as the mean angular rotation ΔΩ 157 nhz (Howe et al. 2000) across the tachocline, where the radial shear is stronger. This yields an estimate of ω s 1, corresponding to a period of about 2π/ω 2.24 yr, a value consistent with the QBO period. Nevertheless, as this estimation depends on the values of the turbulent parameters of the convection zone, which are supposed to be stable, the dynamo period should be almost constant as well. Our results show that the period itself can change on short timescales. Thus, more advanced dynamo models are required in order to explore the link between the QBOs and the dynamo mechanism. The possible source for the 2 yr periodicity could be seated at the tachocline level, also considering that the same variation is found in several processes taking place in the solar interior involved in the generation of the solar magnetic field (solar rotation rate and angular momentum close to the tachocline). In addition, Fletcher et al. (2010), by analyzing the p-mode frequency shifts, found a clear sign of the QBOs in the profound layers of the Sun and suggested it to be a signature of a second dynamo. We remark that two or more dynamo waves at different periods can be obtained, for instance, by coupling a tachocline and a Babcock Leighton type α-effect (Dikpati & Gilman 2001) as well as in nonlinear mean field α ω models (Hoyng 1990). 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