Solar Magnetic Field Signatures in Helioseismic Splitting Coefficients
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Preprint typeset using L A TEX style emulateapj v. 08/22/09
SOLAR MAGNETIC FIELD SIGNATURES IN HELIOSEISMIC SPLITTING COEFFICIENTS
Charles S. Baldner
Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT, 06520-8101
H. M. Antia
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Sarbani Basu
Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT, 06520-8101 andTimothy P. Larson
Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085
Draft version June 16, 2018
ABSTRACTNormal modes of oscillation of the Sun are useful probes of the solar interior. In this work, we usethe even-order splitting coefficients to study the evolution of magnetic fields in the convection zoneover solar cycle 23, assuming that the frequency splitting is only due to rotation and a large scalemagnetic field. We find that the data are best fit by a combination of a poloidal field and a double-peaked near-surface toroidal field. The toroidal fields are centered at r = 0 . R ⊙ and r = 0 . R ⊙ and are confined to the near-surface layers. The poloidal field is a dipole field. The peak strengthof the poloidal field is 124 ±
17 G. The toroidal field peaks at 380 ±
30 G and 1 . ± . Subject headings:
Sun: activity, Sun: helioseismology, Sun: magnetic fields INTRODUCTION
Understanding the nature of the Sun’s magnetic fields— their structure and variability, their generation mech-anisms, and their effects on the heliosphere — is one ofthe key aims of current research in solar physics. It isgenerally believed that the magnetic fields are generatedby a cyclic dynamo that operates somewhere in the solarinterior. In this paper, we use helioseismology to studythe global scale internal magnetic fields over the courseof solar cycle 23.Helioseismology is the most powerful tool available tosolar physicists to study the interior of the Sun. The os-cillation frequencies have been used to study the struc-ture and dynamics of the solar interior with great preci-sion. Magnetic fields, however, have proved to be muchmore challenging. There are a number of important dif-ficulties in dealing with magnetic fields in a helioseis-mic context. The magnitudes of the signatures in thedata are quite small, making statistically significant mea-surements challenging. Secondly, the interpretation ofdata is very difficult. The physics of wave propagationin the presence of magnetic fields is far more complexthan in the non-magnetic case. Further, the geometryof the underlying field strongly affects the signatures inhelioseismic global mode frequencies, meaning differentfield configurations and strengths can be difficult to dis-tinguish from their helioseismic signatures. Even worse,Zweibel & Gough (1995) showed that because magnetic
Electronic address: [email protected] fields act on mode frequencies both by perturbing thethermal structure of the Sun and by changing the wavepropagation speeds directly, there is a degeneracy be-tween magnetic field effects and other thermal perturba-tions which cannot be distinguished a priori from helio-seismic data.Although helioseismic determinations of magneticfields are difficult, there have been many attempts to doso. Isaak (1982) suggested that the then observed fre-quency splittings in the solar acoustic spectrum couldbe caused by a large scale magnetic field situated inthe core. Dziembowski & Goode (1984) used an asymp-totic approximation to study the effects of magnetic fieldon the splitting coefficients, and Dziembowski & Goode(1988) argued that a 1 MegaGauss field at the base of theconvection zone was necessary to explain the observedsplitting coefficients. However, Basu (1997); Antia et al.(2000) placed a limit of 0.3 MG on the field at the baseof the convection zone; thus the situation was unclear. Amega-Gauss magnetic field is also inconsistent with dy-namo theories and constraints from other observations(e.g., D’Silva & Choudhuri 1993).Gough & Thompson (1990) developed a formalism tocompute the effects of rotation and axisymmetric mag-netic fields on the frequency splittings (discussed in thefollowing section), which Antia et al. (2000) used to an-alyze the first year of Michelson Doppler Imager (MDI)data. They placed limits on the strengths of internaltoroidal fields, finding a limit of 20 kG at a depth of30 Mm, and a limit of 300 kG at the base of the con- Baldner et al.vection zone ( r = 0 . R ⊙ ). Dziembowski et al. (2000)inverted the mean frequencies and splitting coefficientsfor changes in temperature, and found that the result-ing temperature perturbation could be explained by achange in magnetic field of 60 kG at a depth of 45 Mm( r ∼ . R ⊙ ).Dziembowski et al. (2001) found that changes in f -mode frequencies from solar minimum to solar maximumimplied a decrease in solar radius with activity, whichthey associated with a change between 4 and 8 Mm indepth. In explaining this result with changing magneticfields, they assumed a tangled field, but even so the mag-nitude of the change in field strength was strongly depen-dent on the radial distribution of the field. The changethey required was 7 kG for a uniform field, or substan-tially less (1 kG at 8 Mm) for an inwardly increasingfield. Chou & Serebryanskiy (2002, 2005) looked for sig-natures of a change at the base of the convection zonefrom low activity to high activity, and found signs ofa small change, which they proposed could be due to achange in magnetic field of 170 – 290 kG. Baldner & Basu(2008), working with an entire solar cycle’s worth of he-lioseismic data, found a change in sound speed betweensolar maximum and solar minimum at the base of theconvection zone, which, if due to a change in magneticfield, could indicate a change in field strength of 290 kGat that depth.In this work, we exploit the fact that we have muchmore helioseismic data than previous investigators hadaccess to, and try to get a coherent picture of sub-surfacesolar magnetic fields and their temporal evolution. Weextend the work of Antia et al. (2000), who consideredtoroidal magnetic fields, to include poloidal fields. Thismeans that we can, in principle, consider any axisym-metric magnetic field configuration. We compute the ef-fects of a wide variety of magnetic field configurations onthe a splitting coefficients, and compare them to a so-lar cycle’s worth of MDI data. It is not clear if the solarmagnetic field has large scale structure of the form we as-sume or whether it is in tangled state due to turbulencein the convection zone. Since the effect of magnetic fieldmanifests through a quadratic term in magnetic field,our estimate may also be applicable to tangled field withsome degree of approximation. PERTURBATIONS TO SOLAR OSCILLATIONFREQUENCIES
The frequencies of normal modes of oscillation ν nℓm are degenerate in m in the case of a spherically symmet-ric star. Departures from spherical symmetry lift thisdegeneracy. When the departures from spherical sym-metry are small, as they are in the case of the Sun, thedifferences in frequency for different values of m will besmall, and it is natural therefore to express the normalmode frequencies in terms of the mean frequency of themultiplet ν nℓ and splitting coefficients a j : ν nℓm = ν nℓ + j max X j =1 a j ( n, ℓ ) P ( ℓ ) j ( m ) . (1)As is common in the current literature, the polynomi-als P ( ℓ ) j ( m ) are the Ritzwoller-Lavely formulation of theClebsch-Gordan expansion (Ritzwoller & Lavely 1991). The odd-order splitting coefficients are caused by the ro-tation of the Sun, and will not be directly consideredin this work. The even-order coefficients are caused bysecond order effects of rotation, and by the effects of mag-netic fields or any other departure from spherical symme-try in the solar structure. In this work, we treat rotationand magnetic fields as perturbations on the sphericallysymmetric case, which allows us to avoid explicitly con-structing a model of a rotating, magnetized star. Theformalism was developed by Gough & Thompson (1990)and Antia et al. (2000) extended the formalism to in-clude the perturbation to the gravitational potential (i.e.,to relax the Cowling approximation) and to include dif-ferential rotation.The first order correction to the mode frequencies dueto rotation affects only the odd-order splitting coeffi-cients. These effects are due to the perturbation of themode frequencies by advection of the waves. The secondorder correction affects only the even-order splitting co-efficients, and is caused by the perturbation to the eigen-functions and the centrifugal force. The odd-order coeffi-cients can be used to determine the rotation profile Ω( r )(Thompson et al. 1996; Schou et al. 1998), which can inturn be used to compute the second order rotation cor-rection (Antia et al. 2000) to the even-order coefficients.This correction needs to be made if the magnetic per-turbation is comparable in size to second order rotationeffect, which appears to be the case (Gough & Thompson1990; Antia et al. 2000).In this work, we consider two different axisymmetricmagnetic field configurations: toroidal and poloidal. Fol-lowing Gough & Thompson (1990), the toroidal field isexpressed in the form B = (cid:20) , , a ( r ) ddθ P k (cos θ ) (cid:21) , (2)where P k are the Legendre polynomials of degree k and a ( r ) describes the radial profile of the magnetic field. Weconsider only even values of k to ensure antisymmetryabout the equator, consistent with the observed field atthe surface. The poloidal field is assumed to be of theform B = (cid:20) k ( k + 1) b ( r ) r P k (cos θ ) , r dbdr ddθ P k (cos θ ) , (cid:21) , (3)where b ( r ) describes the radial profile of the magneticfield. In this case we use only odd values of k to ensurethat the field is antisymmetric about the equator. Withappropriate combinations of these two fields we can, inprincipal, represent any axisymmetric magnetic field.The effect of these magnetic field configurations on thefrequency splittings of p -modes is calculated using theformulation of Gough & Thompson (1990); Antia et al.(2000). There are two ways in which the magnetic fieldcan affect the frequencies, one is the so-called direct effectdue to the additional force, and the second is the distor-tion effect due to the equilibrium state being distortedfrom the original spherically symmetric one. Both theseeffects are included in all calculations. These formula-tions treat the effect of these magnetic fields separately.Unfortunately, the effect of magnetic fields is not linearand hence strictly the contributions from two differentconfigurations cannot be added. In principle, there willagnetic Field Signatures 3be some cross-terms when the combination of toroidaland poloidal fields have a region of overlap in the solarinterior. In this work, we neglect these terms and addthe contributions from toroidal and poloidal fields to getthe total effect. We expect the cross terms to be small. DATA
The data we use for comparison are 72-day mode pa-rameter sets from the Michelson Doppler Imager (MDI)on the SOlar and Heliospheric Observatory (SOHO). Weuse mode parameter sets from the corrected pipeline de-scribed by Larson & Schou (2008). The original MDIanalysis pipeline (Schou 1999) did not take in to accounta number of instrumental effects which introduced secu-lar trends in the mode parameter sets. In particular, theplate scale of the MDI instrument has changed somewhatover SOHO’s mission, and this results in an apparentchange in the solar radius if not properly corrected inthe analysis. Baldner & Basu (2008) found a signaturein the mean frequencies which became increasingly sig-nificant over the course of the solar cycle. A repetition ofthat work with reanalyzed mode parameter sets removedthis effect completely (Baldner et al. 2008). The splittingcoefficients, which we focus on in this work, suffer fromsimilar instrumental effects as the mean frequencies, andhence we use the reanalyzed data in this work.We include 56 mode sets which cover solar cycle 23.The mode sets are identified by the MDI start day, begin-ning with set a splitting coefficients, as the higherorder splitting coefficients have larger errors, and as suchdid not distinguish well between different field configu-rations. The rotation profile determined from the odd-order splitting coefficients (Antia et al. 2008) was usedto calculate the second-order contribution to the even-order coefficients, and this contribution was subtractedfrom the data. RESULTS
Models
In Fig. 1, we show the second splitting coefficient forfour different poloidal field configurations. The actualquantities plotted are ℓa , both as a function of frequency ν and as a function of the lower turning radius of themodes, r t . The radial profile in this case is taken to be b ( r ) = B r − k , (4)where B is a constant which determines the peak fieldstrength and r is the radial distance measured in units ofsolar radius. The models shown in Fig. 1 all have a peakstrength of B = 1 G (note that they do not all have thesame value of B ). The most obvious difference betweendifferent order poloidal fields is that for the k = 1 fieldthe splitting coefficients are all positive, whereas for thehigher order fields they are largely negative, although theshallow modes have positive a .The toroidal field we employ is similar to that used by Fig. 1.—
Splitting coefficients ℓa due to poloidal magnetic fields.The left hand panels are shown as a function of frequency ν , theright hand panels are shown as a function of lower turning radius r t .The four configurations shown have peak field strengths of 1 G atthe surface. The fields have four different values of k . To facilitatedirect comparison with later figures, only modes measured in theMDI data (specifically, the high activity set Antia et al. (2000), with a radial profile given by a ( r ) = ( √ πpβ (cid:16) − ( r − r d ) (cid:17) if | r − r | ≤ d p is the gas pressure, β is the ratio of the mag-netic to gas pressure at r , and r and d are positionand width of the field. As is the case for the poloidalfields, the toroidal field corrections are linear in magneticfield strength squared. Excepting field strength, there-fore, our toroidal fields are described by three quantities:the order of the Legendre polynomial k , which deter-mines the latitudinal distribution of the field, the cen-tral radius r , which determines the location, and thewidth d . Figure 2 shows the splitting coefficients dueto toroidal fields with different values of k but the sameradial profile (in this case, β = 10 − , r = 0 . R ⊙ and d = 0 . R ⊙ ). For the a coefficient, the order k of thefield makes very little difference except to the scale ofthe perturbation — increasing k for the same β effec-tively increases the total amount of flux, but except forthis effect, the a coefficients are not sensitive to differentlatitudinal distributions. For the remainder of the work,therefore, we restrict ourselves to k = 2 fields.Figure 3 shows the splitting coefficients ℓa for near-surface toroidal fields with different central radii r andwidths d as a function of frequency. Figure 4 showsthe same, but as a function of the lower turning ra-dius, r t . The behavior of the splitting coefficients isnot surprising. In general, the fields which penetratebelow the surface show oscillatory behavior as a func-tion of frequency similar to that seen in mode frequency(Gough 1990; Gough & Thompson 1990) and used byRoxburgh & Vorontsov (1994); Basu et al. (1994) andothers to study the convection zone base. The period ofthese oscillations is related to the acoustic depth of theperturbation in the structure. Decreasing the depth of Baldner et al. Fig. 2.—
Splitting coefficients ℓa due to toroidal field with different latitudinal distributions. The upper panels show the coefficients asa function of frequency ν , the lower panels show the coefficients as a function of lower turning radius r t . All the results are with β = 10 − , r = 0 . R ⊙ and d = 0 . R ⊙ . Only modes present in the MDI data have been plotted. the perturbation lengthens the period of the oscillatorybehavior. Fields which are confined near the surface, onthe other hand, do not exhibit oscillatory behavior, butinstead resemble the ‘surface term’ correction which isremoved in structure inversions (e.g., Dziembowski et al.1990; Antia & Basu 1994). Increasing the width of theperturbation smears out the oscillatory signature, as seenin Fig. 3. Because all the modes sampled have lower turn-ing radii below the magnetic fields considered here, thereare no obvious signatures in the splitting coefficients asa function of r t .In addition to fields near the surface, in Fig. 5, weshow the splitting coefficients due to some toroidal fieldslocated at the base of the convection zone. The fieldsshown differ only in the width d of the fields. Unlikethe surface fields shown in previous figures, the deep fieldsignatures show both positive and negative splitting coef-ficients. These models are most interesting as a functionof lower turning radius r t . The splitting coefficients arepositive above the center of the magnetic field, and neg-ative below the center of the magnetic field. Further,as the width of the field is increased, the width of theperturbations to the splitting coefficients (in r t figure)increases as well.The a splitting coefficients due to various poloidal andtoroidal fields are shown in Fig. 6, which shows the resultsfor two poloidal fields, the k = 3 and k = 7 fields, as wellas two toroidal fields with different values of k ( k = 2 and k = 8), each with r = 0 . R ⊙ and d = 0 . R ⊙ . The k = 1 field has essentially no effect on the a splittingcoefficients. Fits to observed data
In order to choose the fields which best match the ac-tual data, we have computed the splitting coefficients fora large grid of field configurations, with fields through-out the convection zone. For poloidal fields we varied k — the form of the radial profile was found not to mat-ter very much for the splitting coefficients, so long as thefield penetrated below the surface. For the toroidal fields, we varied the location r , the width of the field d , andthe latitudinal distribution with k . The range in r wasbetween 0 . R ⊙ and 1 . R ⊙ . The values for d rangedfrom 10 − R ⊙ to 0 . R ⊙ . In order to judge goodness-of-fit, we use the χ statistic. For both the poloidal andthe toroidal fields, the perturbations vary linearly withthe square of the field strength, so to fit the field, weallowed the field strength to vary freely, and chose thestrength that minimized the χ . We have computed the χ for all the field configurations in our grid, as well asfor many combinations of two and three different fields.The results we present below represent the best fits fromthe entire grid of computed models.The largest signal-to-noise ratio in a is found at peakactivity, and so the highest activity set ought to be theeasiest to fit. Comparison of different field configura-tions with the splitting coefficients at high activity areshown in Fig. 7, and the fits are shown both as a functionof frequency and as a function of lower turning radius.The residuals, normalized by the errors in the data, arealso shown. A fit to a k = 1 poloidal field is shownin panel (a). The reduced χ for this fit is 16, and itis evident that the field does a poor job of reproducingthe observed splitting coefficients. Higher order poloidalfields are considerably worse, as an examination of Fig. 1will show — these fields perturb all the splitting coeffi-cients negatively, whereas the observed splittings are allpositive. Panel (b) shows the effect of a toroidal fieldsituated near the surface. Although we attempted to fittoroidal fields throughout the convection zone, fields notlocated very near the surface were extremely poor fitsto the data. The field shown in panel (b) is the bestfit for a single toroidal field, with r = 0 . R ⊙ and d = 0 . R ⊙ . The reduced χ is 5. The residuals aremostly without structure in r t , but are oscillatory in fre-quency, a hint that there could be a second, somewhatdeeper field. The splitting coefficients at peak solar ac-tivity cannot be well fit by either a toroidal field or apoloidal field of the form considered by us.The third field configuration shown (panel c) is a com-agnetic Field Signatures 5 Fig. 3.—
Splitting coefficients ℓa due to near-surface toroidal magnetic fields, as a function of frequency ν . The results are shownfor k = 2 with five different values of central radius r (from 0 . R ⊙ to R ⊙ ), and three different values of the width of the field d (0 . , . , and 0 . R ⊙ . Only modes present in the MDI data have been plotted. Fig. 4.—
Same as Fig. 3, but plotted as a function of the lower turning radius r t . Baldner et al.
Fig. 5.—
Effects of toroidal magnetic fields at the base of the con-vection zone on the ℓa splitting coefficients. Results for two mag-netic field configuration with k = 2, r = 0 . R ⊙ and β = 10 − are shown. The fields have widths of d = 0 . R ⊙ and d = 0 . R ⊙ for the top and bottom panels, respectively. Left hand panels showthe splitting coefficients as a function of frequency ν , right handpanels show the splitting coefficients as a function of lower turningradius r t . Only observed modes have been plotted. Fig. 6.—
Effects of various magnetic fields on the ℓa splittingcoefficients, as a function of both frequency ν (left hand panels) andlower turning radius r t (right hand panels). The top two panelsshow the results for poloidal fields with k = 3 (panel a) and k = 7(panel b). The bottom two panels show the results for toroidalfields, both with β = 10 − , r = 0 . R ⊙ and d = 0 . R ⊙ .Panel (c) is for a k = 2 field and panel (d) for a k = 8 field. Onlyobserved modes have been plotted. bination of a k = 1 poloidal field and a near-surfacetoroidal field ( r = 0 . R ⊙ , d = 0 . R ⊙ — the samefield from panel (b)). This combination of fields yieldsa much better fit to this data set, with a reduced χ of2.8. Using a surface toroidal field instead of a poloidalfield does not fit the data as well, although it is an im-provement over the fit in (b), with a χ value of 3.5.Like the toroidal-only fit, the residuals are more or lesswithout structure in r t , but show oscillatory behavior infrequency. The peak field strengths of the two fields are133 G and 368 G for the poloidal and toroidal fields, re-spectively. The residuals from this fit can be fit by atoroidal field centered at r = 0 . R ⊙ , so in panel (d)we show the best fit to the data: a k = 1 poloidal fieldwith two toroidal fields, one centered at r = 0 . R ⊙ and another centered at r = 0 . R ⊙ . Both toroidalfields are k = 2 fields and have widths d = 0 . R ⊙ .The poloidal field has a peak field strength at the sur-face of 124 ±
18 G, while the toroidal fields have peakfield strengths of 380 ±
30 G and 1 . ± . χ of this fit is 1.7. Attempts tofit the data with a single toroidal field which occupiesthe same region as the two field in this fit did not yielda good fit — the data seem to require a double peakedfield.Figure 8 shows the a splitting coefficients for two dif-ferent data sets ( a coefficients,but some other field configurations also fit the a coeffi-cients equally well, so we do not use them to constrainthe field configurations or determine the field strengths.Having fit the high activity set, we repeat the fits for all56 sets in our study. Figure 9 shows fits of the same k = 1poloidal plus toroidal field combination to six representa-tive mode sets, covering the rise and fall of solar cycle 23.The first set, F . = 126 .
4) and a set from the currentminimum ( k = 2 and d = 0 . R ⊙ , we find that a fieldsof up to 300 kG can be fit to the data, although this isan upper limit, not a detection, since the models of thatfield strength or lower give the same χ as a zero fieldstrength model. At high activity, the dominant signalis from the surface, which we have attempted to explainwith magnetic fields located in those layers.The field strengths of the poloidal and shallow toroidalfield fits to all 56 sets used in this study are shown inFig. 10. Also shown in this figure is the ratio of thepoloidal field strength to the r = 0 . R ⊙ toroidal fieldstrength. With the exception of the low activity setsat the beginning and end of the solar cycle, where theuncertainty in the fits is relatively large, the ratio be-tween the poloidal and toroidal field strengths is roughlyconstant. The field strengths from Fig. 10 are corre-lated with global activity indices from solar cycle 23.The correlation coefficients are 0.90, 0.93, and 0.92 forthe poloidal and two toroidal field components, respec-tively. In Fig. 11, we plot the toroidal and poloidal fieldstrengths as a function of one such global index, the10.7 cm radio flux. The field strengths prove to be highlycorrelated with activity, although there is a hysteresis-like effect evident in the toroidal field strengths — therising phase (shown in blue) is weaker than the declin-ing phase fields. The same effect may also be present atlow activity in the poloidal field strengths. The poloidalagnetic Field Signatures 7 Fig. 7.—
Fits to observed splitting coefficients ℓa for different magnetic field configurations. Four different fits are shown, both as afunction of frequency ν and as a function of lower turning radius r t . The data are shown in black, and the modeled points in red. Theresiduals, scaled by the errors in the data, are also shown below the comparisons. The data are the ℓa splitting coefficients from an MDI72 day mode parameter set, taken at the peak of solar cycle 23 (MDI set k = 1poloidal field. Panel (b) shows the fit from a near-surface toroidal field. Panel (c) shows the best fit field combination with two fields tothis data set — a combination of a dipole poloidal field with a toroidal field located just below the surface ( r = 0 . R ⊙ , d = 0 . R ⊙ ).Panel (d) shows the best fit with three fields — the same poloidal and toroidal field as in panel (c) (though with slightly different fieldstrengths) and another toroidal field at r = 0 . R ⊙ and a width of d = 0 . R ⊙ . field strengths do seem to saturate at high activity. Thestrengths of the two toroidal fields are extremely wellcorrelated. DISCUSSIONS AND CONCLUSIONS
We have attempted to use the first even order splittingcoefficient ( a ) to infer the configuration and strength ofthe Sun’s internal magnetic fields over the course of solarcycle 23, assuming that the entire signature in a aftercorrection for rotation effects is magnetic and that thefields are axisymmetric. The field that we have found isa combination of poloidal field and a double-peaked near-surface toroidal field. The strengths of the poloidal andtoroidal components, at least for high activity period, arewell correlated. The relative strengths of the two toroidalfields are also extremely well correlated.Although the fits we have shown are the best fit to thedata from the grid of models that we have computed, wecan say nothing about the uniqueness of these fits overthe set of all possible magnetic field configurations in thesolar interior. In particular, the choice of radial profile ofthe toroidal fields is virtually limitless, and by restrict- ing our work to profiles of the form (2), we have limitedour search to a restricted class of fields. It is possiblethat there are fields we did not consider with quite dif-ferent radial and latitudinal distributions which fit thedata as well as the fields we have presented as best fits.In addition, as noted above, it is not strictly correct toadd the splitting coefficient perturbations together as wehave done without explicitly accounting for the pertur-bations arising from the cross terms. We do not expect,however, these corrections to be significant, and a fulltreatment of these corrections would be considered in afuture work.Our inferred magnetic field does not change its latitu-dinal distribution over the course of the solar cycle. Thisis in part due to the fact that we are only fitting the a coefficient (as noted above, the higher order splittingshad large errors), so our sampling of the interior is notreally latitudinally sensitive. Thus, we do not see a but-terfly diagram in our magnetic fields. Ulrich & Boyden(2005) measured the surface toroidal component of thesolar magnetic field over almost an entire 22-year cycle.The field they measure is roughly a tenth of the peak Baldner et al. Fig. 8.—
Comparisons of data to models for ℓa splitting coefficients from two different data sets. The field configurations are the sameas from Fig. 7 (a). The left hand panels are from set ν and as a function of lower turning radius r t . The residuals are shown below the data,and are normalized by the errors in the data. The model is obtained by fitting only the a coefficients. Fig. 9.—
Fits to measured splitting coefficients ℓa for six different sets throughout solar cycle 23. The data (shown in black) are fromMDI 72 day mode parameter sets. The magnetic field configuration is the same as panel d in Figure 7: a dipole poloidal field and twotoroidal fields at r = 0 . R ⊙ and r = 0 . R ⊙ , with d = 0 . R ⊙ and k = 2. The fits are shown both as a function of frequency ν and as a function of lower turning radius r t . The residuals scaled by the errors in the data are also shown. The toroidal field strengthsat r = 0 . R ⊙ correspond to β = 10 − , 7 . × − , 2 × − , 2 . × − , 1 . × − , and 5 × − for the six sets, respectively. Thetoroidal field strengths at r = 0 . R ⊙ correspond to β = 1 . × − , 2 × − , 5 . × − , 6 . × − , 2 . × − , and 1 . × − . Thepoloidal field strengths at the surface are B = 0 G, 68 G, 115 G, 125 G, 94 G, and 58 G. agnetic Field Signatures 9 | B | ( G ) a) r =0.999R (cid:0) poloidal fieldtoroidal field | B | ( G ) b) r =0.996R (cid:1) | B p o l / B t o r | c) r =0.999R (cid:2) Fig. 10.—
The strength of the inferred magnetic fields as a func-tion of time over solar cycle 23. Each MDI 72 day mode parameterset is fitted by with the same magnetic field configuration as Fig. 9.The strengths (in Gauss) of the poloidal field at r = 0 . R ⊙ (solidblack line) and the toroidal field at r = 0 . R ⊙ (dashed line) areshown in the upper panel (a). The middle panel (b) shows thesame quantities as in the upper panel, but this time at a radius of r = 0 . R ⊙ . The lower panel (c) shows the ratio of the poloidalfield strength to the toroidal field strength at r = 0 . R ⊙ . Theratio of poloidal to toroidal at r = 0 . R ⊙ looks very similar. | B | ( G ) toroidal, r =0.999R (cid:3) rising phasefalling phase | B | ( G ) toroidal, r =0.996R (cid:4)
60 80 100 120 140 160 180 200 220 24010.7 cm flux (SFU)04080120 | B | ( G ) poloidal Fig. 11.—
The strength of the inferred magnetic fields as a func-tion of 10.7 cm radio flux. The top panel shows the r = 0 . R ⊙ toroidal field strength, the middle panel shows the r = 0 . R ⊙ toroidal field strength, and the bottom panel shows the poloidalfield strength at r = 0 . R ⊙ . Rising and declining phase are dis-tinguished with blue circles for the rising phase and red trianglesfor the declining phase. The toroidal field shows a hysteresis effect.The poloidal field shows some hysteresis at low activity, as well asa hint of saturation at high activity. strength of our toroidal field. Strictly speaking, how-ever, we see no toroidal field at all at the surface, sincein our inferred field, the field strength becomes zero pre-cisely at r = 1 R ⊙ . The peak strength that we measure,however, is only 700 km below the surface, and the fieldcould penetrate the surface somewhat. Ulrich & Boyden(2005) find a field which gives a β ∼ × − at the sur-face at high activity, and drops to nothing at low activity,while we find a field that changes from β ∼ − at low activity to β = 2 × − at high activity at a radius of r = 0 . R ⊙ (a depth of approximately 700 km).Recently, attention has been focused on the strengthand configuration of the quiet Sun surface magnetic field.Harvey et al. (2007) reported the presence of a ‘seething’horizontal magnetic field with an rms field strength of1.7 G. With the launch of Hinode (Solar B), the highspatial resolution of the onboard spectropolarimeter hasbeen used to study the horizontal fields of the solar pho-tosphere. Lites et al. (2007, 2008) have measured thehorizontal flux, which they find to be 55 G, comparedto the average vertical flux of 11 G. Petrie & Patrikeeva(2009) found that the zonal component (component inthe East-West direction) was much smaller than the ra-dial component, reporting an inclination angle of lessthan 12 ◦ from vertical in the East-West direction.The fields being studied in the aforementioned worksare generally very tangled fields which thread throughthe intergranular lanes and so they are not axisymmet-ric fields. It is worthwhile to compare our results withtheirs, since tangled fields on local scales can organizeinto roughly axisymmetric fields on global scales. How-ever, the contribution to splittings are more sensitive to h B i rather than h B i and hence tangled field may alsocontribute to it, even when the average h B i is very small.Further, considering the general behavior of the pertur-bation to the mode frequencies, our inference about thelocation of required magnetic field is more robust as adifferent location will yield a very different behavior ofsplitting coefficients. The exact magnitude of the fieldmay depend on the assumption of geometry and on itbeing tangled or large scale. Nevertheless, we believethat our estimate is of the right order, though the statis-tical errorbars obtained by us may not be realistic. Thesystematic errors in these estimates would be certainlylarger. The dominance of poloidal field orientation atthe surface found by Petrie & Patrikeeva (2009) is foundin our own results — at the surface, the toroidal fieldis weak or vanishing, but the poloidal field remains. Inthe period analyzed by Lites et al. (2007, 2008), we finda poloidal field strength of 40 G, and a toroidal field of90 G at a depth of 700 km. The vertical flux they find(11 G) is weaker than what we detect, but their 55 G hor-izontal flux may be roughly consistent with our toroidalfield.Schrijver & Liu (2008) found that the dipole momentof the surface magnetic field, measured from MDI mag-netograms, was half the strength in 2008 that it was in1997, during the last solar minimum. We do not see sucha difference from the beginning of our period to the end— in fact, we find the poloidal field strength is slightlyhigher during the current minimum, although the levelof the difference is within the errors, and our data setsend in 2007, so the comparison is not contemporaneous.Hysteresis in the relations between activity indiceshas been observed before, for example in the rela-tion between low degree (Anguera Gubau et al. 1992;Jimenez-Reyes et al. 1998) and intermediate degree(Tripathy et al. 2000, 2001) acoustic modes and globalmagnetic indices. It should be noted that an analysis ofa full solar cycle’s worth of intermediate degree p -modesdata does not show any hysteresis in mean frequenciesas a function of 10.7 cm flux (Baldner & Basu 2008).Tripathy et al. (2001) noted that, among the global mode0 Baldner et al.indices, the relation between global line-of-sight mag-netic flux and 10.7 cm radio flux showed a hysteresiseffect, but the relation between the radiative indices and10.7 cm flux did not. Moreno-Insertis & Solanki (2000)argued that the observed hysteresis could be almost en-tirely due to the latitudinal distribution of magnetic fluxon the surface of the Sun. We believe that this is acompelling explanation for the hysteresis that we find.The 10.7 cm flux is the integrated flux received at theEarth and does not contain any information about thelatitudinal variation, while the a splitting coefficient isassociated with definite latitudinal variation, given by P (cos θ ) (Antia et al. 2001), and hence the two wouldnot be the same. More importantly, we expect the ac-tual magnetic fields in the near surface layers to driftequatorward — as the surface fields do.Few conclusions can really be drawn from this workwith respect to dynamo theory since the fields we haveinferred are predominantly shallow fields, whereas mostdynamo models operate much deeper down, in the shearlayer at and below the base of the convection zone.(some useful recent reviews include Ossendrijver 2003;Charbonneau 2005; Miesch & Toomre 2009). The up-per limits that we place on fields at that depth areconsistent with earlier helioseismic results (e.g., Basu1997; Antia et al. 2000; Chou & Serebryanskiy 2002, 2005; Baldner & Basu 2008). Many deep-seated dy-namo mechanisms predict an anticorrelation betweenthe poloidal and toroidal field components, as the dy-namo converts poloidal to toroidal field and toroidal fieldback to poloidal. We do not see any evidence of suchconversion. Some dynamo mechanisms, however, op-erate in the near-surface shear layer (e.g. Brandenburg2005). Although the fields generated in these modelsare generally extremely tangled, on global scales thesefields can have toroidal and poloidal components (e.g.Brown et al. 2007, 2009). In particular, although theywere considering a more rapidly rotating star than theSun, Brown et al. (2007) noted that their field containedboth a poloidal and a toroidal component, and that thetoroidal component was much the stronger of the two.This work utilizes data from the Solar Oscillations In-vestigation/ Michelson Doppler Imager (SOI/MDI) onthe Solar and Heliospheric Observatory (SOHO). SOHOis a project of international cooperation between ESAand NASA. MDI is supported by NASA grants NAG5-8878 and NAG5-10483 to Stanford University. This workwas partially supported by NSF grants ATM 0348837and ATM 0737770 to SB. CB is supported by a NASAEarth and Space Sciences Fellowship NNX08AY41H.) (Antia et al. 2001), and hence the two wouldnot be the same. More importantly, we expect the ac-tual magnetic fields in the near surface layers to driftequatorward — as the surface fields do.Few conclusions can really be drawn from this workwith respect to dynamo theory since the fields we haveinferred are predominantly shallow fields, whereas mostdynamo models operate much deeper down, in the shearlayer at and below the base of the convection zone.(some useful recent reviews include Ossendrijver 2003;Charbonneau 2005; Miesch & Toomre 2009). The up-per limits that we place on fields at that depth areconsistent with earlier helioseismic results (e.g., Basu1997; Antia et al. 2000; Chou & Serebryanskiy 2002, 2005; Baldner & Basu 2008). Many deep-seated dy-namo mechanisms predict an anticorrelation betweenthe poloidal and toroidal field components, as the dy-namo converts poloidal to toroidal field and toroidal fieldback to poloidal. We do not see any evidence of suchconversion. Some dynamo mechanisms, however, op-erate in the near-surface shear layer (e.g. Brandenburg2005). Although the fields generated in these modelsare generally extremely tangled, on global scales thesefields can have toroidal and poloidal components (e.g.Brown et al. 2007, 2009). In particular, although theywere considering a more rapidly rotating star than theSun, Brown et al. (2007) noted that their field containedboth a poloidal and a toroidal component, and that thetoroidal component was much the stronger of the two.This work utilizes data from the Solar Oscillations In-vestigation/ Michelson Doppler Imager (SOI/MDI) onthe Solar and Heliospheric Observatory (SOHO). SOHOis a project of international cooperation between ESAand NASA. MDI is supported by NASA grants NAG5-8878 and NAG5-10483 to Stanford University. This workwas partially supported by NSF grants ATM 0348837and ATM 0737770 to SB. CB is supported by a NASAEarth and Space Sciences Fellowship NNX08AY41H.