Astrophysics and Space Science Proceedings Phần 3 pptx

92 D. Nandy 4.3 Origin of Grand Minima Small, but significant variations in solar cycle amplitude is commonly observed from one cycle to another, and models based on either stochastic fluctuations, or nonlinear feedback, or time-delay dynamics exist to explain such variability in cy- cle amplitude (for overviews, see Charbonneau 2005; Wilmot-Smith et al. 2006). However, most models find it difficult to switch off the sunspot cycle completely for an extended period of time – such as that observed during the Maunder minimum – and subsequently recover back to normal activity. Two important and unresolved questions in this context are what physical mecha- nism stops active region creation completely and how does the dynamo recover from this quiescent state. The first question is the more vexing one and still eludes a co- herent and widely accepted explanation. The second question is less challenging in my opinion; the answer possibly lies in the continuing presence of another ˛-effect (could be the traditional dynamo ˛-effect suggested by Parker), which can work on weaker, sub-equipartition toroidal fields – to slowly build up the dynamo amplitude to eventually recover the sunspot cycle from a Maunder-like grand minima. These are speculative ideas and one thing that can be said with confidence at this writing is that we are just scratching the surface as far as the physics of grand minima like episodes is concerned. 4.4 Parametrization of Turbulent Diffusivity Typically, in many dynamo models published in the literature, the coefficient of tur- bulent diffusivity employed is much lower than that suggested by mixing-length theory (about 10 13 cm 2 s 1 ; Christensen-Dalsgaard et al. 1996). This is done to ensure that the flux transport in the SCZ in advection dominated (i.e., meridional circulation is the primary flux transport process). There are many disadvantages to using a higher diffusivity value in these dynamo models. Usage of higher diffusivity values makes the flux transport process diffusion dominated, reducing the dynamo period to values somewhat lower than the observed solar cycle period. It also makes flux storage and amplification difficult and shortens cycle memory; the latter is the basis for solar cycle predictions. Nevertheless, this inconsistency between mixing- length theory and parametrization of turbulent diffusivity in dynamo models is, in my opinion, a vexing problem. In the absence of any observational constraints on the depth-dependence of the diffusivity profile in the solar interior, this problem can be addressed only theoreti- cally. One possible solution to resolving this inconsistency is by invoking magnetic quenching of the mixing-lengththeory suggested diffusivity profile. The idea is sim- ple enough; as magnetic fields have an inhibiting effect on turbulent convection, strong magnetic fields should quench and thereby be subject to less diffusive mix- ing. The magnetic quenching of turbulent diffusivity is challenging to implement numerically, but seems to me to be the best bet towards reconciling this inconsis- tency within the framework of the current modeling approach. Outstanding Issues in Solar Dynamo Theory 93 4.5 Role of Downward Flux Pumping An important physical mechanism for magnetic flux transport has been identified recently from full MHD simulations of the solar interior. This mechanism, often re- ferred to as turbulent flux pumping, pumps magnetic field preferentially downwards, in the presence of rotating, stratified convection such as that in the SCZ (see, e.g., Tobias et al. 2001). Typical estimates yield a downward pumping speed, which can be as high as 10 ms 1 ; this would make flux pumping the dominant downward flux transport mechanism in the SCZ, short-circuiting the transport by meridional circu- lation and turbulent diffusion. However, turbulent flux pumping is usually ignored in kinematic dynamo models of the solar cycle. If indeed the downward pumping speed is as high as indicated, then turbulent flux pumping may influence the solar cycle period, crucially impact flux storage and am- plification, and also affect solar cycle memory. Therefore, turbulent flux pumping must be properly accounted for in kinematic dynamo models and its effects com- pletely explored; this remains an issue to be addressed adequately. 5 Concluding Remarks Now let us elaborate on and examine some of the consequences of the outstanding issues highlighted in the earlier section. 5.1 A Story of Communication Timescales To put a broader perspective on some of these issues facing dynamo theory, specifi- cally in the context of the interplay between various flux-transport processes, it will be instructive here to consider the various timescales involved within the dynamo mechanism. Let us, for the sake of argument, consider that the BL mechanism is the predominant mechanism for poloidal field regeneration. Because this poloidal field generation happens at surface layers, but toroidal field is stored and amplified deeper down near the base of the SCZ, for the dynamo to work, these two spatially segre- gated layers must communicate with each other. In this context, magnetic buoyancy plays an important role in transporting toroidal field from the base of the SCZ to the surface layers – where the poloidal field is produced. The timescale of buoy- ant transport is quite short, on the order of 0:1 year and this process dominates the upward transport of toroidal field. Now, to complete the dynamo chain, the poloidal field must be brought back down to deeper layers of the SCZ where the toroidal field is produced and stored. There are multiple processes that compete for this downward transport, namely meridional circulation, diffusion, and turbulent flux pumping. 94 D. Nandy Considering the typical meridional flow loop from mid-latitudes at the surface to mid-latitudes at the base of the SCZ, and a peak flow speed of 20 ms 1 , one gets a typical circulation timescale  v D 10 years. Most modelers use low values of diffu- sivity on the order of 10 11 cm 2 s 1 , which makes the diffusivity timescale (L 2 SCZ =Á, assuming vertical transport over the depth of the SCZ)  Á D 140 years; that is, much more than  v , therefore making the circulation dominate the flux transport. However, if one assumes diffusivity values close to that suggested by mixing length theory (say, 5  10 12 cm 2 s 1 ), then the diffusivity timescale becomes  Á D 2:8 years; that is, shorter than the circulation timescale – making diffusive dispersal dominate the flux transport process. If we now consider the usually ignored process of turbulent pumping, the situ- ation changes again. Assuming a typical turbulent pumping speed on the order of 10 ms 1 over the depth of the SCZ gives a timescale  pumping D 0:67 years, shorter than both the diffusion and meridional flow timescales. This would make turbulent pumping the most dominant flux transport mechanism for downward transport of poloidal field into the layers where the toroidal field is produced and stored. 5.2 Solar Cycle Predictions As outlined in Yeates et al. (2008), the length of solar cycle memory (defined as over how many cycles the poloidal field of a given cycle would contribute to toroidal field generation) determines the input for predicting the strength of future solar cycles. The relative timescales of different flux transport mechanisms within the dynamo chain of events and their interplay, based on which process (or pro- cesses) dominate, determine this memory. For example, if the dynamo is advection (circulation)-dominated, then the memory tends to be long, lasting over multiple cycles. However, if the dynamo is diffusion (or turbulent pumping) dominated, then this memory would be much shorter. Now, within the scope of the current framework of dynamo models, I have ar- gued that significant confusion exists regarding the role of various flux transport processes. So much so that we do not yet have a consensus on which of these pro- cesses dominate; therefore, we do not have a so-called standard-model of the solar cycle yet. Should solar cycle predictions be trusted then? Taking into account this uncertainty in the current state of our understanding of the solar dynamo mechanism, I believe that any solar cycle predictions – that does not adequately address these outstanding issues – should be carefully evaluated. In fact, under the circumstances, it is fair to say that if any solar cycle predictions match reality, it would be more fortuitous than a vindication of the model used for the prediction. This is not to say that modelers should not explore the physical processes that contribute to solar cycle predictability; indeed that is where most of our efforts should be. My concern is that we do not yet understand all the physical processes that constitute the dynamo mechanism and their interplay well enough to Outstanding Issues in Solar Dynamo Theory 95 begin making predictions. Prediction is the ultimate test of any model, but there are many issues that need to be sorted out before the current day dynamo models are ready for that ultimate test. Acknowledgement This work has been supported by the Ramanujan Fellowship of the Depart- ment of Science and Technology, Government of India and a NASA Living with a Star Grant NNX08AW53G to the Smithsonian Astrophysical Observatory at Harvard University. I gratefully acknowledge many useful interactions with colleagues at the solar physics groups at Montana State University (Bozeman) and the Harvard Smithsonian Center for Astrophysics (Boston). I am indebted to my friends at Bozeman, Montana, from where I recently moved back to India, for contributing to a very enriching experience during the 7 years I spent there. References Babcock, H. W. 1961, ApJ, 133, 572 Charbonneau, P. 2005, Living Reviews in Solar Physics, 2, 2 Christensen-Dalsgaard, J., et al. 1996, Science, 272, 1286 Dikpati, M., Charbonneau, P. 1999, ApJ, 518, 508 D’Silva, S., Choudhuri, A. R. 1993, A&A, 272, 621 Fan, Y., Fisher, G. H., Deluca, E. E. 1993, ApJ, 405, 390 Leighton, R. B. 1969, ApJ, 156, 1 Nandy, D. 2002, Ap&SS, 282, 209 Nandy, D., Choudhuri, A. R. 2001, ApJ, 551, 576 Nandy, D., Choudhuri, A. R. 2002, Science, 296, 1671 Parker, E. N. 1955a, ApJ, 121, 491 Parker, E. N. 1955b, ApJ, 122, 293 Schrijver, C. J., Liu, Y. 2008, Solar Phys., 252, 19 Tobias, S. M., Brunnell, N. H., Clune, T. L., Toomre, J. 2001, ApJ, 549, 1183 Wilmot-Smith, A. L., Nandy, D., Hornig, G., Martens, P. C. H. 2006, ApJ, 652, 696 Yeates, A. R., Nandy, D., Mackay, D. H. 2008, ApJ, 673, 544 Status of 3D MHD Models of Solar Global Internal Dynamics A.S. Brun Abstract This is a brief report on the decade-long effort by our group to model the Sun’s internal magnetohydrodynamics in 3D with the ASH code. 1 Introduction: Solar Global MHD The Sun is a complex magnetohydrodynamic object that requires state-of-the-art observations and numerical simulations in order to pin down the physical processes at the origin of such diverse activity and dynamics. We here give a brief summary of recent advances made with the Anelastic Spherical Harmonic (ASH) code (Clune et al. 1999; Brun et al. 2004) in modeling global solar magnetohydrodynamics. 2 Global Convection A series of papers has been published on this important topic (Miesch et al. 2000; Elliott et al. 2000; Brun and Toomre 2002), most recently by Miesch et al. (2008). In this paper, for the first time, a global model of solar convection with a density contrast of 150 from top to bottom and a resolution equivalent to 1,500 3 has been achieved. This has lead to significant results regardingthe turbulent convection spec- tra from large-scale (like giant cells) down to supergranular-like convection patterns and their correlation with the temperature fluctuations, leading to large (150% L ˇ ) convective luminosity. A.S. Brun (  ) CEA/CNRS/Universit´e Paris 7, France S.S. Hasan and R.J. Rutten (eds.), Magnetic Coupling between the Interior and Atmosphere of the Sun, Astrophysics and Space Science Proceedings, DOI 10.1007/978-3-642-02859-5 7, c  Springer-Verlag Berlin Heidelberg 2010 96 Status of 3D MHD Models of Solar Global Internal Dynamics 97 3 Differential Rotation and Meridional Circulation A recent review by Brun and Rempel (2008) discussed the respective role of Reynolds stresses, latitudinal heat transport, and baroclinic effect in setting the pe- culiar conical differential rotation profile observed in the Sun. Indeed, basic rotating fluid dynamic considerations imply that the differential rotation should be invariant along the rotation axis, yielding a cylindrical rotation profile. As this is not observed, it is necessary to find the source of the breaking of the so-called Taylor–Proudman constraint. In particular, in a recent paper by Miesch et al. (2006), we have been able to show that baroclinic effects are associated with latitudinal variation of the tem- perature and that convection by transporting heat poleward contributes a significant part of that variation, but not all. A temperature contrast of about 10 K is compatible with helioseismic inferences for the inner solar angular velocity profile. Meridional flows in most cases are found to be multicellular, and fluctuate significantly over a solar rotation. These flows contribute little to the heat transport and to the kinetic energy budget (accounting for only 0.5% of the total kinetic energy). However, it plays a pivotal role in the angular momentum redistribution by opposing and bal- ancing the equatorward transport by Reynolds stresses (Brun and Toomre 2002; Brun and Rempel 2008). 4 Global Dynamo In continuation of Gilman (1983) and Glatzmaier (1985), we have studied, at much higher resolution, dynamo action in turbulent convective shells (Brun 2004; Brun et al. 2004). We have found that dynamo action is reached above a critical magnetic Reynolds number and that the magnetic field is mostly intermittent and small-scale (Fig. 1), with the large-scale axisymmetric field only contributing for about 3% of the total magnetic energy. Reversals of the field occur on a time scale of about 1.5 year, as opposed to the observed 11 year cycle of solar activity. This is partly due to the absence of a tachocline at the base of the convective envelope. In an attempt to resolve this issue, we have in Browning et al. (2006) computed the first 3D MHD model of a convection zone with an imposed stable tachocline. In that layer, the field that has been transported or pumped down from the turbulent convection zone above it, is found to be organized in strong axisymmetric toroidal ribbons with dominant antisymmetry with respect to the equator. The poloidal field in the convection zone is stabilized by the presence of that layer with much less frequent, if any, reversals. The magnetic energy reaches in both cases about 10% of the total kinetic energy. We also find that the differential rotation is reduced in amplitude due to the nonlinear feedback of the field on the flow via the Lorentz force. In a recent study by Jouve and Brun (2007, 2009), we have also studied flux emergence in isentropic and turbulent rotating convection zone. We confirmed that a certain amount of field concentration and twist is required for the structure to emerge at the Status of 3D MHD Models of Solar Global Internal Dynamics 99 Fig. 2 First 3D integrated solar model coupling nonlinearly the convective envelope to the radiative interior. Shown is a 3D rendering of the density perturbations, with red corresponding to positive fluctuations. We have omitted an octant in order to be able to see the equatorial and meridional planes within the domain. We note the clear presence of internal waves in the radiative zone 6 Towards a 3D Integrated Model of the Sun Coupling nonlinearly the convection zone with the radiative interior is the key to understand the solar global dynamo and inner dynamics. Brun (in preparation) has developed the first 3D solar integrated model from r D 0:07 R ˇ up to 0.97R ˇ .We show in Fig. 2 a 3D rendering of the density fluctuations over the whole computa- tional domain. The presence of internal waves is obvious in the radiative interior. The penetrative convection is at the origin of these gravito-inertial waves. We are currently studying in detail the source function at every depth in the model and the resulting power spectrum at different locations in the radiative interior and find that a large spectrum near the base of the convection zone is excited. The tachocline is kept thin in this model by using a step function at the base of the convection zone for the various diffusion parameters, making the thermal and viscous spread of the latitudinal shear imposed by the convective envelope slow with respect to the con- vective overturning time. We intend in the near future to redo the study of Brun and Zahn (2006) by introducing in the integrated model a fossil field, taking advantage of the more realistic boundary conditions realized in this new class of models. Acknowledgement I am thankful to my friends and colleagues J. Toomre, J P. Zahn, M. Miesch, M. Derosa, M. Browning, and L. Jouve without whom the results reported in this paper would not have been obtained. I also thank the IFAN network for partial funding during my visit to India. Finally, I am grateful to Profs. S. Hasan, K. Chitre, and H.M. Antia for the wonderful time I spent in Bangalore and Mumbai. Measuring the Hidden Aspects of Solar Magnetism J.O. Stenflo Abstract 2008 marks the 100th anniversary of the discovery of astrophysical magnetic fields, when George Ellery Hale recorded the Zeeman splitting of spectral lines in sunspots. With the introduction of Babcock’s photoelectric magnetograph, it soon became clear that the Sun’s magnetic field outside sunspots is extremely structured. The field strengths that were measured were found to get larger when the spatial resolution was improved.It was therefore necessary to come up with methods to go beyond the spatial resolution limit and diagnose the intrinsic magnetic-field properties without dependence on the quality of the telescope used. The line-ratio technique that was developed in the early 1970s revealed a picture where most flux that we see in magnetograms originates in highly bundled, kG fields with a tiny volume filling factor. This led to interpretations in terms of discrete, strong-field magnetic flux tubes embedded in a rather field-free medium, and a whole indus- try of flux tube models at increasing levels of sophistication. This magnetic-field paradigm has now been shattered with the advent of high-precision imaging po- larimeters that allow us to apply the so-called “Second Solar Spectrum” to diagnose aspects of solar magnetism that have been hidden to Zeeman diagnostics. It is found that the bulk of the photospheric volume is seething with intermediately strong, tan- gled fields. In the new paradigm, the field behaves like a fractal with a high degree of self-similarity, spanning about 8 orders of magnitude in scale size, down to scales of order 10 m. 1 The Zeeman Effect as a Window to Cosmic Magnetism 2008 marks the 100th anniversary of the discovery of magnetic fields outside the Earth (cf. Fig.1). George Ellery Hale had suspected that the Sun might be a mag- netized sphere from the appearance of the solar corona seen at total solar eclipses, and from the structure of H˛ fibrils around sunspots, which was reminiscent of iron files in a magnetic field. The proof came when Hale placed the spectrograph slit in J.O. Stenflo (  ) Institute of Astronomy, ETH Zurich, Zurich, Switzerland S.S. Hasan and R.J. Rutten (eds.), Magnetic Coupling between the Interior and Atmosphere of the Sun, Astrophysics and Space Science Proceedings, DOI 10.1007/978-3-642-02859-5 8, c  Springer-Verlag Berlin Heidelberg 2010 101 Measuring the Hidden Aspects of Solar Magnetism 103 stars and galaxies elsewhere in the universe. Our increasing empirical knowledge about the Sun’s magnetism has helped guide the development and understanding of various theoretical tools, like plasma physics and magnetohydrodynamics. The experimental tool is spectro-polarimetry, which needs the Zeeman effect (and more recently also the Hanle effect, see below) as an interpretational tool to connect theory and observation. Outside sunspots, the polarization signals of the transverse Zeeman effect are much smaller than those of the longitudinal Zeeman effect. For weak fields, the linear polarization from the transverse Zeeman effect is approximately proportional to the square of the transverse field strength rather than in linear proportion, and it is limited by a 180 ı ambiguity.In contrast, the circular polarization is easy to measure, and to first order it is proportional to the line-of-sight component of the field, with sign. Therefore, magnetic-field measurements have been dominated by recordings of the circular polarization due to the longitudinal Zeeman effect. The breakthrough in these measurements came with the introduction by Babcock of the photoelectric magnetograph (Babcock 1953). Soon afterwards, full-disk magnetograms (maps of the circular polarization) were being produced on a regular basis, forming a unique data base for the understanding of stellar magnetism and dynamos. 2 Emergence of the Flux Tube Paradigm When directly resolved magnetic-field observations are not available, like for mag- netic Ap-type stars, one usually makes models assuming that the star has a dipole or low-degree multipolar field. The solar magnetograms, however, showed the Sun’s field to be highly structured. It was found that the measured field strength increases with the angular resolution of the instrument used (Stenflo 1966). As the measured field strength also depended on the spectral line used, many believed that this was a calibration problem that could be solved by a coordinated campaign, organized by an IAU committee, to record the same regions on the Sun with different instruments. It was only with the introduction of the line-ratio technique (Stenflo 1973) that the cause for this apparent “calibration problem” could be found. The magnetic flux is highly intermittent, with most of the flux concentrated in elements that were far smaller than the available spatial resolution. The magnetograph calibration (con- version of measured polarization to field strength) was based on the shape of the spatially averaged line profile and the assumption of weak fields (linear relation between polarization and field strength). The average line profile is, however, not representative of the line formation conditions within the flux concentrations, and also the weak-field approximation is not valid there (we have “Zeeman saturation”), as the concentrated fields are intrinsically strong. Inside the strong-field regions, the thermodynamic conditions are very different from the rest of the atmosphere, which leads to temperature-induced line weakenings. The magnitude of the line-weakening and Zeeman saturation effects vary from line to line, which leads to the noticed dependence of the field-strength values on 104 J.O. Stenflo the spectral line used. This effect cannot be calibrated away, as the line-formation properties in the flux concentrations are not accessible to direct observations when they are not resolved. A further effect is that different lines are formed at different atmospheric heights, and the field expands and weakens with height. All these ef- fects contribute jointly in an entangled way to the “calibration error.” The line-ratio technique was introduced to untangle them, and it is described in Fig. 2. The trick is to use a combination of lines, for which all the various entangled factors are identical, except one. Thus it was possible to isolate the Zeeman satu- ration (nonlinearity) effect from all the thermodynamic and line formation effects by choosing the line pair Fe I 5250.22 and 5247.06 ˚ A. Both these lines belong to multiplet No. 1 of iron, have the same line strength and excitation potential, and therefore have identical thermodynamicresponse and line-formation properties. The only significant difference between them is their Land´e factors, which are 3.0 and 2.0, respectively. No other line combination has since been found, which can so cleanly isolate the Zeeman saturation effect from the other effects. 1.0 200 Slope gives intrinsic field strength Stokes V (%) V 5250 / (1.5 V 5247 ) 5250 / 5247 line ratio technique 100 –100 –200 –200 –100 B 5250 (G) B 5247 (G) 100 2000 5 1.5 1.0 0.5 20 40 60 80 04080 5247 (x1.5) 5247 (x1.5) 5250 weal plage strong plage 5250 –Δλ (mÅ) –Δλ (mÅ) 120 Line ratio vs. Δλ (verifies physical validity of the model) 4 3 2 1 0 0 4 1 5247.0585IFE 1 5250.2171IFE 2 0 5246 5248 WAVELENGTH (Å) 5250 5252 –2 –4 0.5 INTENSITY STOKES V [%] 18 5247.5737 66 5250.6527 FE I G = 2.5 G = 3 G EFF = 1.5 G EFF = 2 CR 0.0 I Fig. 2 Illustration of the various aspects of the 5,250/5,247 line ratio technique (Stenflo 1973). The linear slope in the diagram to upper left (from Frazier and Stenflo 1978) determines the differential Zeeman saturation, from which the intrinsic field strength can be found. The portion of the FTS Stokes V spectrum to upper right, from Stenflo et al. (1984), shows that the amplitudes of the 5,250 and 5,247 iron lines are not in proportion to their Land´e factors, but are closer to 1:1. In the bottom diagram, from Stenflo and Harvey (1985), the Stokes V profiles and line ratios are plotted as functions of wavelength distance from line center. This profile behavior verifies that the line difference is really due to differential Zeeman saturation [...]... transitions it is only necessary to change the sign of the Q=I and U=I expressions given in this paper To understand the reason for this, see Sect 6.8 of Landi Degl’Innocenti and Landolfi (2004) Chromospheric and Coronal Polarization Diagnostics Q I 3 h p 1 8 2 U I 2 / .3 cos2 ÂB 1/ C 1C 121 2 /.cos2 ÂB i 1/ 3 cos2 ÂB 1/ F ; (3) 3 p p 1 2 2 2 sin ÂB cos ÂB 3 cos2 ÂB 1/ F ; (4) 2 2 where F D W 0 Ju / Z 0 Jl /... the local solar vertical (e.g., Landi Degl’Innocenti and Bommier 19 93) In fact, the information provided by (3) and (4) is contained in the following single formula (cf., Trujillo Bueno 2003a) Q I 3 p sin2 4 2 3 cos2 ÂB 1/ F ; (6) where is the angle between the magnetic field vector and the LOS, ÂB the angle between the magnetic field vector and the local solar vertical, and the reference direction for... structures themselves, such as those of the He I 10, 830 A and ˚ (D3 ) multiplets, and to interpret observations of their intensity and polar5,876 A ization within the framework of the multiterm atom model of the quantum theory of spectral line formation (see Sect 7.6 in Landi Degl’Innocenti and Landolfi 2004) ˚ ˚ The spectral lines of the He I 10, 830 A and 5,876 A multiplets result from transitions between... helium (the singlet level 1 S0 ) The lower term (2s3 S1 ) of the ˚ He I 10, 830 A multiplet is the ground level of ortho-helium, while its upper term 3 ˚ (2p P2;1;0 ) is the lower one of 5,876 A (whose upper term is 3d3 D3;2;1 ) ˚ ˚ The Stokes profiles of the He I 10, 830 A and 5,876 A multiplets depend on the strengths and wavelength positions of the and transitions, which can be calculated correctly... and disadvantages of the Hanle and Zeeman effects as diagnostic tools, and continuing with a more detailed discussion of selected J Trujillo Bueno ( ) Instituto de Astrof´sica de Canarias, La Laguna, Tenerife, Spain ı and Consejo Superior de Investigaciones Cient´ficas, Spain ı S.S Hasan and R.J Rutten (eds.), Magnetic Coupling between the Interior and Atmosphere of the Sun, Astrophysics and Space Science. .. a coexistence of weak and strong fields over a wide dynamic range The PDF for the vertical field-strength component is nearly scale invariant and can be well represented by a Voigt function with a narrow Gaussian core and “damping wings” extending to kG values (Stenflo and Holzreuter 2002; Stenflo and Holzreuter 20 03) A fractal dimension of 1.4 has been found from both observations and numerical simulations... examples of detailed numerical calculations of the emergent Q=I and U=I amplitudes for a variety of ÂB and values can be seen in Fig 9 of Asensio Ramos et al (2008), which the reader will ˚ find useful to inspect Such results for the lines of the He I 10 830 A multiplet can be easily understood via (3) and (4) It is easy to generalize (3) and (4) for any magnetic field azimuth B Such general equations... exploited for removing the 180ı azimuth ambiguity present in vector magnetograms (Landi Degl’Innocenti and Bommier 19 93; see also Fig 2 below) For the case of a magnetic field with a fixed inclination ÂB and a random azimuth below the spatial scale of the mean free path of the line photons we have U=I D 0, while Q I 3 p 1 8 2 2 / 3 cos2 ÂB 12 F : (5) This expression shows that under such circumstances there... Polarization Workshops (Stenflo and Nagendra 1996; Nagendra and Stenflo 1999; Trujillo-Bueno and Sanchez Almeida 20 03; Casini and Lites 2006) 112 J.O Stenflo The structuring in the Second Solar Spectrum is governed by previously unfamiliar physical processes, like quantum interference between atomic levels, hyperfine structure and isotope effects, optical pumping, molecular scattering, and enigmatic, as yet unexplained... Astrophysics and Space Science Proceedings, DOI 10.1007/978 -3- 642-02859-5 9, c Springer-Verlag Berlin Heidelberg 2010 118 Chromospheric and Coronal Polarization Diagnostics 119 developments Recent reviews where the reader finds complementary information are Harvey (2006); Stenflo (2006); Lagg (2007); L´ pez Ariste and Aulanier (2007); o Casini and Landi Degl’Innocenti (2007); and Trujillo Bueno (2009a) 2 . Zurich, Switzerland S.S. Hasan and R.J. Rutten (eds.), Magnetic Coupling between the Interior and Atmosphere of the Sun, Astrophysics and Space Science Proceedings, DOI 10.1007/978 -3- 642-02859-5 8, c . 7, France S.S. Hasan and R.J. Rutten (eds.), Magnetic Coupling between the Interior and Atmosphere of the Sun, Astrophysics and Space Science Proceedings, DOI 10.1007/978 -3- 642-02859-5 7, c . Choudhuri, A. R. 19 93, A&A, 272, 621 Fan, Y., Fisher, G. H., Deluca, E. E. 19 93, ApJ, 405, 39 0 Leighton, R. B. 1969, ApJ, 156, 1 Nandy, D. 2002, Ap&SS, 282, 209 Nandy, D., Choudhuri,