Twist, Writhe & Helicity in the inner penumbra of a sunspot
aa r X i v : . [ a s t r o - ph . S R ] N ov Draft version November 27, 2017
Preprint typeset using L A TEX style emulateapj v. 2/16/10
TWIST, WRITHE, AND HELICITY IN THE INNER PENUMBRA OF A SUNSPOT
B. Ruiz Cobo and K. G. Puschmann (1) Instituto de Astrof´ısica de Canarias (IAC), E-38200 La Laguna, Tenerife, Spain(2) National Solar Observatory (NSO), 950 North Cherry Avenue, Tucson, AZ 85719, USA(3) Departamento de Astrof´ısica, Universidad de La Laguna (ULL), E-38205 La Laguna, Tenerife, Spain(4) Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), D-14482, Potsdam, Germany
Draft version November 27, 2017
ABSTRACTThe aim of this work is the determination of the twist, writhe, and self magnetic helicity of penumbralfilaments located in an inner Sunspot penumbra. To this extent, we inverted data taken with thespectropolarimeter (SP) aboard
Hinode with the SIR (Stokes Inversion based on Response function)code. For the construction of a 3D geometrical model we applied a genetic algorithm minimizing thedivergence of ~B and the net magnetohydrodynamic force, consequently a force-free solution would bereached if possible. We estimated two proxies to the magnetic helicity frequently used in literature:the force-free parameter α z and the current helicity term h c z . We show that both proxies are onlyqualitative indicators of the local twist as the magnetic field in the area under study significantlydepartures from a force-free configuration. The local twist shows significant values only at the bordersof bright penumbral filaments with opposite signs on each side. These locations are precisely correlatedto large electric currents. The average twist (and writhe) of penumbral structures is very small. Thespines (dark filaments in the background) show a nearly zero writhe. The writhe per unit length ofthe intraspines diminishes with increasing length of the tube axes. Thus, the axes of tubes related tointraspines are less wrung when the tubes are more horizontal. As the writhe of the spines is verysmall, we can conclude that the writhe reaches only significant values when the tube includes theborder of a intraspine. Subject headings: methods: observational - methods: numerical - Sun: magnetic topology - sunspots- techniques: polarimetric INTRODUCTIONInvestigating the physical nature and the dynamics ofpenumbral filaments is essential in order to understandthe structure and the evolution of sunspots and their sur-rounding moat regions. Many important observationalaspects of penumbral filaments are well settled down, al-though their interpretation is still a source of debate,e.g., the brightness of penumbral filaments, the inwardmotion of bright penumbral grains, the Evershed flow,the Net Circular Polarization (NCP), as well as mov-ing magnetic features in the sunspot moat. During thelast few years, new observational discoveries, e.g., darkcored penumbral filaments (Scharmer et al. 2002), strongdownflow patches in the mid and outer penumbra (Ichi-moto et al. 2007a), penumbral micro-jets (Katsukawa etal. 2007; Jurˇ c ´ak & Katsukawa 2008), or twisting motionsof penumbral filaments (Scharmer et al. 2002; Rimmele& Marino 2006; Ichimoto et al. 2007b; Ning et al. 2009)broadened the number of unknowns and gave new im-pulse to the investigation of sunspots. For recent reviewssee Borrero (2011), Bellot Rubio (2010), Borrero (2009),Schlichenmaier (2009), or Tritschler (2009). An espe-cially controversial issue is the study of the twisting mo-tions of penumbral filaments. On the one hand, Ichimotoet al. (2007b) consider twisting motions as an apparentphenomenon, produced by lateral motions of intensityfluctuations associated with overturning convection. Onthe other hand, Ryutova et al. (2008) propose that theobserved twist is an intrinsic property of penumbral fila- [email protected], [email protected] ments and is produced as a consequence of reconnectionprocesses which take place in the penumbra. Su et al.(2008, 2010) conclude that the twist of penumbral fila-ments changes with time caused by an unwinding pro-cess.In the present paper we study the twist of filaments ofthe inner penumbra of a sunspot by means of the mag-netic helicity. We take advantage of the 3D geometricalmodel of a section of the inner penumbra of a sunspot de-scribed in Puschmann et al. (2010a) [hereafter, Paper I].We use observations of the active region AR 10953 nearsolar disk center obtained on 1 st of May 2007 with theHinode/SP. The inner, center side, penumbral area un-der study was located at an heliocentric angle θ = 4.63 ◦ .To derive the physical parameters of the solar atmo-sphere as a function of continuum optical depth, the SIR(Stokes Inversion based on Response function) inversioncode (Ruiz Cobo & del Toro Iniesta 1992) was applied tothe data set. The 3D geometrical model was derived bymeans of a genetic algorithm that minimized the diver-gence of the magnetic field vector and the deviations fromstatic equilibrium considering pressure gradients, gravityand the Lorentz force. We can not assess the unicity ofthe resulting model: the found solution just minimizesthe divergency of the magnetic field and the modulus ofthe net force, neglecting the contribution of the aceler-ation terms. For a detailed description we refer to Pa-per I. In Puschmann et al. (2010b) [hereafter, Paper II],we calculated the electrical current density vector ~J inthe above mentioned area and found the horizontal com-ponent of the electrical currents ∼ Fig. 1.—
Left panel: Length ℓ of the field lines of ~B integratedfrom the top ( z = 200 km) to the bottom layer ( z = 0 km) or until x = 4 . ℓ define 14 areas (flux tubes)related to spines (black) and intraspines (white), respectively. Thinisolines denote different thresholds for selecting the flux tube area.Right panel: Vertical component of ~B at z = 200 km. Contour linescorrespond to horizontal cuts through the 14 flux tubes with thelargest area at z = 200 , , , , the vertical component J z (thus confirming the results ofPevtsov & Peregud 1990; Georgoulis & LaBonte 2004).In addition, we concluded that the magnetic field at theborders of bright penumbral filaments departs from aforce-free configuration (see also Zhang 2010). These re-sults are strongly significant considering that we haveimposed that our solution minimizes the net force, in-cluding the Lorentz force, and consequently, a force freesolution should be found if it were possible.We can evaluate the magnetic field lines by the inte-gration of ~B starting at each pixel of the top layer ofour volume of the inner penumbra (of 4.2 Mm × × ℓ of each field line. As our sunspothas negative polarity, ~B points downwards and we thusintegrate the field lines from the top layer to the bottomlayer. The majority of field lines end up at the bottomlayer, except the lines starting at larger X coordinatesin our FOV. In the whole analized volume the magneticfield has the same polarity, and consequently, the fieldlines do not present maxima nor minima in our region,i.e., the field lines always travel downwards. This fact, aswe will see later, simplifies the evaluation of the magnetichelicity. Areas with larger ℓ correspond to intraspines,since ~B is more horizontal, the field lines thus traverselarger distances inside our volume.In the left panel of Fig. 1, we selected 14 areas, 7 cor-responding to intraspines (thick white contours) and 7 tospines (thick black contours), according to the length ℓ of the magnetic field lines. For each of the selected zoneswe define a volume delimited by the field lines setting offfrom each pixel of the closed curve of the top layer. Inthe right panel of Fig. 1 we show the vertical component of ~B at z = 200 km together with horizontal cuts througheach of these volumes at z = 200 , , , , & 0 km.Since the umbra is placed at the right hand side in theFOV, the cuts at deeper layers are displaced to the right.Finally we checked that the magnetic flux traversing eachof the cuts is approximately constant: the standard de-viation of the relative variation of the magnetic flux be-tween the top and the bottom layer is 4.5%. Thus, eachvolume can be approximately considered as a flux tube.However, the 14 areas were selected quit arbitrarily.In order to study the dependence of twist, writhe andmagnetic helicity on the flux tube area, 40 additionallysmaller tubes have been defined inside the larger ones(thin isolines in the left panel of Fig. 1). Thus, for mostof the 14 zones we have several tubes of different size,many of them being the internal part of the larger one.Consequently, we have 54 tubes, 27 of them related tointraspines and 27 to spines. MAGNETIC HELICITY, TWIST, AND WRITHEThe study of helicity of solar magnetic features hasbeen a hot topic during at least the last 25 years. Mag-netic helicity has been investigated in solar structures atdifferent spatial scales in the photosphere and chromo-sphere, as well as in the solar wind (see e.g. the reviewsof Brown et al. 1999; Rust 2002; Pevtsov & Balasubra-maniam 2003; D´emoulin 2007; D´emoulin & Pariat 2009,and references therein). The helicity in penumbral fila-ments has been analyzed by means of some proxies by,e.g., Ryutova et al. (2008), Tiwari et al. (2009), Su et al.(2010), and Zhang (2010).The magnetic helicity, H m , quantifies how the mag-netic field is twisted, writhed, and linked. H m plays akey role in magneto-hydrodynamics because it is almostconserved in a plasma having a high magnetic Reynoldsnumber (see e.g., Berger 1984). The magnetic helicityof a vector field ~B , fully contained within a volume V and bounded by a surface S (i.e., the normal component B n = ~B · ~n vanishes at any point of S ), is (Elsasser 1956): H m = Z V ~A · ~B d x, (1)where the vector potential ~A satisfies ~B = ∇× ~A . Berger& Field (1984) showed that Eq. 1, is not gauge-invariantif the volume of interest is not bounded by a magneticsurface, i.e., if ~B crosses S (as in the case of the volumeof the penumbra retrieved from our observations). Inthis case the relative magnetic helicity (Finn & Antonsen1985) should be used: H relm = Z V ( ~A + ~A p ) · ( ~B − ~B p ) d x, (2)where ~B p is a potential field having the same normalcomponent B n on S , and ~A p is its vector potential.The relative helicity reflects twist, writhe, and linkagewith respect to a current-free (potential) field, i.e., itsminimum-energy state for the given B n -condition on S . The relative magnetic helicity so defined is gauge-invariant and has the same conservation properties andamount of topological information as the magnetic helic-ity. Throughout this article, the term magnetic helicitywist, Writhe, and Helicity 3 Fig. 2.—
From left to right: local twist ( T loc ), α z / π , current helicity density ( h c ), and 4 h c z evaluated at z = 200 km. The isolines arethe same as in the left panel of Fig. 1. refers to the relative magnetic helicity. For an isolatedmagnetic flux rope, H relm is proportional to the sum ofits twist T and writhe W (Berger & Field 1984; T¨or¨oket al. 2010): H relm = ( T + W )Φ (3)where Φ is the magnetic flux of the rope. The twistis the turning angle of a bundle of magnetic field linesaround its central axis, whereas the writhe quantifies thehelical deformation of the axis itself. Following Berger &Prior (2006), the twist of an infinitesimal rope is givenby T = R T loc d l , l being the arc length along the centralfield line of the rope and T loc the local twist: T loc = d T d l = µ J || πB || , (4)being J || and B || the components of the current and mag-netic field parallel to the central field line of the rope.With this definition, T = 1 when the field lines twistaround the axis by an angle of 2 π and T loc is the localtwist per unit length evaluated at a given geometricalheight at each pixel. If the rope has a non-infinitesimalcross section Σ, the local twist of the rope T loc i is given bythe average of the infinitesimal local twist, T loc over Σ.Berger & Prior (2006, see also T¨or¨ok et al. (2010)) giveexpressions for the writhe of specific geometrical con-figurations. Provided that the magnetic field lines inthe inner penumbral region studied in this paper alwaystravel downwards in z , i.e. without showing maxima norminima between the z =200 km and z =0 km heightlayers, we can evaluate the writhe using a very simplifiedformula: W = 12 π Z z z
11 + | τ z | ( ~τ × ~τ ′ ) z d z , (5)obtained as a particular case of the more general for-mula of Berger & Prior (2006). In Eq. 5, ~τ stands for the tangent vector to the tube axis; τ z for the verticalcomponent of ~τ ; and ~τ ′ = d ~τ d z .Equations 3, 4, & 5 allow the evaluation of the twist,the writhe and the magnetic helicity of an isolated tube.What happens in the case of a non-isolated tube, as isclearly the case of the penumbral tubes? The writhe ofa magnetic rope, isolated or not, is a measure of the he-lical deformation of the axis of the rope, while the twistquantifies the winding of the magnetic field lines of therope around its axis. The magnetic helicity measures thelinking number of the field lines, averaged over all pairsof lines, and weighted by the flux (Berger & Prior 2006;Moffatt 1969). We can simplify the case of a non-isolatedtube to a scenario in which we have just two adjoiningtubes. It is clear that we can define the writhe and twistfor each individual tube, and consequently its magnetichelictiy, but the helicity of the whole configuration is notjust the sum of both contributions. We would need toinclude an extra term taking into account the linking be-tween both tubes: the mutual helicity. Consequently theapplication of equations 3, 4, & 5 to our tubes retrievesonly the contribution of the local values of the twist,writhe and the self helicity. The contribution of the sur-rounding tubes to the helicity of each flux tube is notconsidered. This contribution, the mutual helicity, couldbe larger than the self helicity (see e.g., R´egnier & Priest2007). The mutual helicity can be calculated using theprocedure described in Berger & Prior (2006). However,for the scope of this paper we limit the calculation to selfhelicity. PROXIES OF THE MAGNETIC HELICITYThe current helicity density is defined (see e.g., See-hafer 1990) as h c = ~B · ∇ × ~B . If we take as mag-netic ropes the tubes defined by the field lines start-ing in each pixel, (being then B || = B ), Eq. 4 becomes T loc = h c / πB . The parameter α is usually defined in aforce-free configuration, i.e., when ~B is parallel to its curl, B. Ruiz Cobo & K. G. Puschmannby ∇ × ~B = α ~B . The α parameter can be defined for nonforce-free fields: α = ~B ·∇× ~B/B . This is the definition we will use throughout the paper. In order to see howthis re-defined α differs from the force-free definition, letus decompose ∇ × ~B = ( ∇ × ~B ) || + ( ∇ × ~B ) ⊥ . Fromits definition, α becomes equal to α = ±| ( ∇ × ~B ) || | /B which is equal to the classical definition for a force-freefield. The ± is needed to consider the case when ( ∇× ~B ) || and ~B point in opposite direction, i.e., when α is nega-tive. For a no force free-field, the parameter α is then theratio between the parallel component of the curl of themagnetic field and its modulus. Evidently, in a generalcase, we will have α = h c /B = 4 π T loc .Given the difficulty of empirically obtaining H relm and ~J , one finds many works where different proxies wereused. Before evaluating the magnetic helicity we cancalculate, from our data, some of the most usual proxiesof the magnetic helicity. Among them the most commonproxies, with several different but more or less equiva-lent definitions, are the α z = ( ∇ × ~B ) z /B z parameter(see e.g., Su et al. 2009, 2010; Pevtsov et al. 2008, andreferences therein), and the parameter h c z = B z ( ∇× ~B ) z (see e.g., Zhang 2010). It is evident that for a force-freefield α z = α and h c z = αB z = h c B z /B . That meansthat the h c z parameter could be meaningless for nearlyhorizontal magnetic fields, such as those found in sunspotpenumbrae.Many authors suppose that the sign of the integral of α z (or h c z ) over the volume of a magnetic structure coin-cides with the sign of H relm , although this fact has not yetbeen demonstrated (D´emoulin 2007). On the other hand,Hagyard & Pevtsov (1999) point out that h c z only con-siders the vertical component J z of the electric currentdensity vector, h c z can strongly differ from h c , providedthat J z is much smaller than the horizontal components,at least in the inner penumbra (see Paper II). Besides,Pariat et al. (2005) comment that, since the magnetic he-licity is a global quantity, it is not obvious that a helicitydensity has any physical meaning.We evaluated these proxies in the inner penumbralregion under study. In the left panels of Fig. 2 wepresent T loc (evaluated from Eq. 4) and α z / π evalu-ated at each pixel at z = 200 km. As in the innerpenumbra the magnetic field is not force-free at the bor-ders of bright penumbral filaments (see Paper II), α z / π (2 nd panel) is only qualitatively similar to T loc . In the3 rd and 4 th panel we present h c = ~B · ∇ × ~B and h c z evaluated at z = 200 km. h c z is multiplied by a fac-tor 4 just to make easier its comparison with h c . Aswe have seen before, T loc = h c / πB , and thus thegeneral aspect of h c is very similar to T loc . However, h c z resembles h c only marginally (the standard devia-tions are σ ( h c ) = 2 . G m − and σ ( h c z ) = 0 . G m − ).The values of α z / π and h c z at z = 200 km obtainedhere are very similar to the results found in the lit-erature: Tiwari et al. (2009) found that α z / π variesaround ± . M m − along azimuthal paths in the mid-dle penumbra; Su et al. (2010) found a fluctuation of Yeates et al. (2008) denominate current helicity to this gener-alized α parameter. α z / π larger than ± . M m − over an inner penumbralregion; Su et al. (2009) found that h c z fluctuates along anazimuthal path in the inner penumbra with an amplitudelarger than 1 G m − while Balthasar & G¨om¨ory (2008)found penumbral mean values of about 0.04 G m − . Inthe four panels of Fig. 2, the outlined areas were selectedby different thresholds of ℓ , the length of the magneticfield lines between the layers z = 200 km and z = 0 km(see also Section 1 and Fig. 1). The sign of the inte-gral of T loc and α z / π over the above mentioned areasonly coincides in 39% of the 54 tubes (if we consider onlythe areas related to the intraspines this value decreasesto 15% of the 27 tubes). The same figures are obtainedfor the percentage of coincidence between the signs of theintegrals of h c and h c z , approximately. This weak coinci-dence demonstrates that, at least in the inner penumbraof a sunspot, α z / π and h c z are not good estimates of T loc and h c , respectively. As already shown in Paper II,the magnetic field in the area under study significantlydepartures from a force-free configuration.Note that T loc reaches significant values only at theborders of the intraspines: these are exactly the areaswhere the electric current density is large (see Fig. 1of Paper II). Furthermore, often T loc changes its sign atboth sides of bright filaments, i.e., for the majority ofthe intraspines, T loc shows negative values at the upper(larger Y-coordinate) borders of the filaments and posi-tive values at the lower borders. The alternation of thesign in the twist is also observed (although less evident)in the maps of α z and h c z and it is clearly visible in Ti-wari et al. (2009), Su et al. (2009), Su et al. (2010), andZhang (2010).This phenomenon can be explained if we consider thatthe field lines of the magnetic background componentwrap around the intraspines (Borrero et al. 2008) andtend to meet above the intraspines, thus generating acurvature of different sign in the field lines at both sidesof the intraspines. Thus T loc could be measuring thetwist of the background field wrapping around the in-traspines rather than the twist of the field lines of theintraspines themselves. However, the alternation of signsof the twist at both sides of a penumbral filament is com-patible with the magnetohydrostatic equilibrium modelof a magnetic flux tube built by Borrero (2007). Thismodel includes a transverse component of ~B having op-posite twist at both sides of a plane longitudinally cuttingthe flux tube. This model is able of explaining both thedark cored penumbral filaments and the net circular po-larization observed in penumbral filaments (Borrero etal. 2007). Magara (2010) suggests the existence of anintermediate region where the magnetic field has a tran-sitional configuration between a penumbral flux tube andthe background field: in such areas, coinciding with thelargest electrical current density (see Paper II), penum-bral micro-jets are produced as observed by Katsukawaet al. (2007). NUMERICAL TESTTo check its correctness, the procedure used for theevaluation of twist, writhe, and magnetic helicity wasapplied to two different analytical cases. In the first casewe consider that the magnetic field lines follow a helixaround a vertical straight line. The magnetic field vec-wist, Writhe, and Helicity 5tor is defined by ~B = B ˆ z + B r ˆ θ with B and B being constant. ~B could easily be decomposed in a po-tential ~B p = B ˆ z and a close (toroidal) field ~B c = B r ˆ θ .Obviously, the potential component ~B p fulfills the con-ditions required in Eq. 2 in the case of a cylindricaltube: ~B and ~B p have the same normal component onthe external surface of the tube. The determinationof the vector potential, in this case, is straightforward: ~A = B r/ θ − B r / z . The vector potential of thepotential component will be ~A p = B r/ θ . Using Eq. 2,the magnetic helicity of a cylindrical tube of height L andradius R becomes H m = π B B R L . The magneticflux of this tube is Φ = π B R . This tube has a zerowrithe because its axis is a straight line. From Eq. 3, thetwist follows as T = B L π B . (6)This is obviously the expected result, provided that thepitch (of screw-step) of our helix is π B B and the twist isa measure of the number of turns done by the magneticfield lines along a longitude L .In the first four rows of Table 1 we present the analyti-cal (i.e., using Eq. 6) and numerical results (using Eqs. 4and 5) for tubes with B = − . B = 0 .
04 T Mm − , L = 0 .
225 Mm and a radius R equal to 0 . R h turning an angle Ψ through a length L = 0 .
22 Mm. Then, the magnetic field at the axis willbe ~B = B ˆ z + B R h ˆ θ with B = B Ψ /L . FollowingBerger & Prior (2006), the writhe of a magnetic flux tubewhose axis is a helix can be easily evaluated in terms ofits polar writhe (i.e., area/2 π of the section of the unitysphere limited by the tantrix curve and the north pole.The tantrix curve is the path, the tip of the tangent vec-tor takes on the unit sphere). In our case, the writhebecomes: W = Ψ2 π (1 − B | B | ) . (7)Once we have the axis, we can easily build a tube witha given twist around such an axis. We chose the radiusof the tube as R = 0 . R h = 0 . B = 0 . − . The angleΨ takes a value of 9 .
124 degrees in order to have ananalytical writhe (i.e., using Eq. 7) of 0 . .
0, 0 . .
01, 0 .
1, and − . RESULTSIn Table 2 we present the resulting values of the axislength, magnetic flux, twist, writhe and self magnetic he-licity for the flux tubes of the 14 largest zones. As the
TABLE 1Test results: Writhe ( W ), twist ( T ), and magnetichelicity ( H m ), for a helical magnetic field (first fourrows) and a helical magnetic field winding around ahelical axis. R [Mm] W T H m [Mx ]0.2 analytical 0.00 -7.16e-3 -4.52e+34numerical -9.e-28 -7.16e-3 -4.50e+341.0 analytical 0.00 -7.16e-3 -2.83e+37numerical -9.e-28 -7.16e-3 -2.81e+370.2 analytical 1.00e-3 0.00 8.16e+34numerical 0.99e-3 1.90e-6 8.07e+340.2 analytical 1.00e-3 1.00e-3 1.63e+35numerical 0.99e-3 1.07e-3 1.67e+350.2 analytical 1.00e-3 1.00e-2 8.97e+35numerical 0.98e-3 1.06e-2 9.47e+350.2 analytical 1.00e-3 1.00e-1 7.01e+36numerical 0.97e-3 0.99e-1 6.91e+360.2 analytical 1.00e-3 -1.00e-3 0.00numerical 0.99e-3 -1.06e-3 -6.14e+33 Fig. 3.—
Panel (a): Length of the axis of each flux tube. Thevalues corresponding to tubes of the same zone are connected bylines. Cross symbols correspond to intraspines and small circles tospines. Panels (b), (c), and (d): writhe, twist, sum of writhe andtwist. magnetic helicity depends on the square of the magneticflux, and our selected areas are very different in area,the resulting magnetic helicity varies over a wide rangeof several orders of magnitude. To study the dependenceof the precedent quantities on the length of the respec-tive axis, in Fig. 3, we plot the length of the axis of eachflux tube ζ , the writhe, the twist, and the sum of twistand writhe for the 54 selected tubes related to intraspines(index ranging form 0 to 26) and spines (index rangingform 27 to 53). The values corresponding to tubes ofthe same zone (see left panel of Fig. 1) are connected bystraight lines. As the magnetic field in the spines is morevertical than in the intraspines, the length of the axis ofthe tubes related to the spines is clearly shorter. For eachintraspine/spine zone the length of the axis grows withthe index because each of the related tubes was chosenin the interior of the preceding one. The writhe of the in-traspines does not follow a clear pattern, but most of the B. Ruiz Cobo & K. G. Puschmann Fig. 4.—
Panels (a) and (b): normalized twist (
T/ζ ) and writhe(
W/ζ ) versus the length of the axis of each flux tube ζ . The ab-solute value | W/ζ | has been overplotted in panel (b) with trianglesymbols. The dashed line is the linear fit of | W/ζ | . Panel (c):wavenumber. Panel (d): T/ζ versus α/ π averaged over the sec-tion of each flux tube at z = 200 km. Panels (e) and (f): T versusthe average of α z / π and h c z respectively. In panels (d), (e) and(f) the straight line with slope 1 has been overplotted. Cross sym-bols correspond to intraspines and small circles to spines. TABLE 2Axis length ζ , magnetic flux Φ , twist, writhe andmagnetic helicity for the 14 largest zones. The 7 first(last) rows are related to intraspines (spines). index ζ [Mm] Φ [Mx] T W H m [Mx ]0 0.79 -3.05e+18 -0.0138 0.0258 1.11e+354 0.80 -1.56e+18 -0.0272 0.0182 -2.19e+349 1.06 -4.71e+18 -0.0210 0.0024 -4.12e+3515 0.92 -2.55e+18 -0.0224 -0.0163 -2.51e+3520 0.80 -5.69e+18 0.0111 0.0041 4.94e+3523 1.02 -2.64e+17 -0.0054 0.0166 7.80e+3225 1.00 -5.13e+17 0.0096 0.0079 4.59e+3328 0.32 -4.43e+18 -0.0023 0.0000 -4.42e+3429 0.33 -3.41e+18 -0.0028 0.0021 -8.34e+3330 0.34 -6.29e+18 -0.0041 0.0011 -1.20e+3536 0.34 -7.73e+18 -0.0065 0.0010 -3.30e+3540 0.37 -6.30e+18 -0.0041 0.0015 -1.01e+3541 0.36 -1.13e+19 -0.0001 0.0007 7.56e+3446 0.35 -2.04e+19 -0.0010 -0.0017 -1.15e+36 tubes have a positive writhe. The spines show a nearlyzero writhe. The twist of the intraspines, however, takesnearly always negative values. Consequently, the twist ispartially canceled by the writhe, thus the absolute valueof the self magnetic helicity is, nearly always, lower thanthe absolute value of the twist.In panels (a), and (b) of Fig. 4 we plot the twist, and writhe per unit length versus the length of the axis ofeach flux tube ζ . The normalized twist does not show aclear dependence with the length of the axis, but the ab-solute value of the normalized writhe in the intraspinesclearly decreases with increasing ζ . The axes of tubesrelated to intraspines are less wrung when the tubesare more horizontal (i.e., in the central part of the in-traspines). As the writhe of the spines is very small, wecan conclude that the writhe reaches only significant val-ues when the tube includes the border of a intraspine.In panel (c) we plot the wavenumber (1/ pitch ), i.e., thenumber of turns done by the magnetic field per lengthunit as a function of ζ . Flux tubes related to spines showslightly smaller values but any clear dependence is notobserved.In panel (d) we plot the normalized twist versus α/ π (i.e., the local twist T loc at each pixel) evaluated at z = 200 km and averaged over each structure. Giventhe good correlation between both magnitudes we canuse the average of the local twist as a good proxy forthe normalized twist. Consequently, we can explain thesmall obtained twist values in terms of the local twist: aswe have seen in Fig. 2, T loc reveals significant values withopposite sign at the borders of the penumbral filaments,leading to a cancellation of the twist when integratingover the filament.In order to assess the reliability of the most used prox-ies we plot, in panels (e) and (f), the average twist of eachstructure versus the average value of α z / π and h c z , re-spectively. Both proxies are very bad indicators of theaverage twist of intraspines but they give a qualitativelygood agreement (better in the case of h c z ) for the spines.This asymmetry could be explained by the fact that both α z and h c z are only related with the vertical component J z of the electric current density vector, which, as weshow in Paper II, is much smaller than the horizontalcomponents, mainly in and around the intraspines. Onthe other hand, in a force-free configuration α z = α and h c z = h c , and then, the discrepancy between these pa-rameters is a clear result of the non-validity of the force-free approximation in the inner penumbra: In Paper IIwe have already shown that, at the borders of brightpenumbral filaments, the magnetic field strongly depar-tures from a force-free configuration. CONCLUSIONSIn the present work we calculated the parameter α andits proxy α z , the current helicity density h c and its proxy h c z , the twist, the writhe, and the magnetic helicity ofdifferent structures of the inner penumbra of a sunspot.The parameters are evaluated from a three-dimensionalgeometrical model obtained after the application of a ge-netic algorithm on inversions of spectropolarimetric dataobserved with Hinode (see Puschmann et al. 2010a, Pa-per I). We demonstrate, that in the inner penumbra thefrequently used proxies α z and h c z are only qualitativeindicators of the local twist (twist per unit length, evalu-ated under the assumption that the axis of a flux tube isparallel to the magnetic field) of penumbral structures.As shown in (Puschmann et al. 2010b, Paper II), themagnetic field in the area under study many times de-parts significantly from a force-free configuration and thehorizontal component of the electrical current density issignificantly larger than the vertical one.wist, Writhe, and Helicity 7The local twist shows only significant values at the bor-ders of bright penumbral filaments and reveals oppositesign at each side of the bright filaments. The oppositesign might be the reason for a cancellation of the twistwhen integrating over the filament, thus the twist of thepenumbral structures is very small. Significant valuesof the local twist are exactly related to areas where theelectric current density is large. The local twist couldbe measuring the twist of the background field wrappingaround the intraspines and/or the twist of the field linesof the intraspines themselves; in the latter case the in-ternal structure of the tube would consist in two ”cotyle-dons” (at both sides of a vertical plane longitudinallycutting the tube), harboring each one a magnetic fieldof opposite twist, compatible with the MHS model ofBorrero (2007).The writhe per unit length diminishes with increasinglength (decreasing inclination) of the axis of flux tubesrelated the intraspines. The small amount of twist andwrithe shown by the spines indicates that the backgroundfield lines, in these zones, are nearly straight.A future study should clarify if the helicity apparentin the intensity maps of penumbral filaments in the midand outer penumbra of sunspots is produced by helicalflux tubes with a strong writhe or just by spurious effectsproduced by lateral intensity fluctuations. In any case,it is clear that a twisted tube does not per se generateany intensity fluctuation similar to the observations by Ryutova et al. (2008). Rather, there is the necessity ofa writhe of the tube, in such a way that different lon-gitudinal portions of the tube were at different opticaldepths producing changes in the observed intensity. Wewill extend the present work (and necessarily the workpresented in Paper I and II) on the entire sunspot.We thank the referee for fruitful comments. Hin-ode is a Japanese mission developed and launched byISAS/JAXA, collaborating with NAOJ as a domesticpartner, NASA and STFC (UK) as international part-ners. Scientific operation of the Hinode mission isconducted by the Hinode science team organized atISAS/JAXA. This team mainly consists of scientistsfrom institutes in the partner countries. Support for thepost-launch operation is provided by JAXA and NAOJ(Japan), STFC (U.K.), NASA, ESA, and NSC (Norway).Financial support by the Spanish Ministry of Science andInnovation through projects AYA2010–18029, ESP 2006-13030-C06-01, AYA2007-65602, and the European Com-mission through the SOLAIRE Network (MTRN-CT-2006-035484) is gratefully acknowledged. The NationalSolar Observatory (NSO) is operated by the Associationof Universities for research in Astronomy (AURA), Inc.,under a cooperative agreement with the National ScienceFoundation. We thank V. Mart´ınez Pillet, C. Beck, andH. Balthasar for fruitful discussions. REFERENCESBalthasar, H. & G¨om¨ory, P. 2008, A&A, 488, 1085Bellot Rubio, L. R., in Magnetic coupling between the Interiorand the Atmosphere of the Sun, S.S. Hassan and Rutten (eds.),(Springer-Verlag: Berlin) ASP Ser., 2010, 193Berger, M.A. 1984, Geophys. Astrophys. Fluid Dyn., 30, 79Berger, M.A. & Field, G. B. 1984, J. Fluid. 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