Spectropolarimetrically accurate magnetohydrostatic sunspot model for forward modelling in helioseismology
aa r X i v : . [ a s t r o - ph . S R ] A p r Draft version March 27, 2019
Preprint typeset using L A TEX style emulateapj v. 5/2/11
SPECTROPOLARIMETRICALLY ACCURATE MAGNETOHYDROSTATIC SUNSPOT MODEL FORFORWARD MODELLING IN HELIOSEISMOLOGY
D. Przybylski, S. Shelyag, P.S. Cally
Monash Centre for Astrophysics, School of Mathematical Sciences, Monash University, Clayton, Victoria, Australia, 3800.
Draft version March 27, 2019
ABSTRACTWe present a technique to construct a spectropolarimetrically accurate magneto-hydrostatic modelof a large-scale solar magnetic field concentration, mimicking a sunspot. Using the constructed modelwe perform a simulation of acoustic wave propagation, conversion and absorption in the solar interiorand photosphere with the sunspot embedded into it. With the 6173˚A magnetically sensitive photo-spheric absorption line of neutral iron, we calculate observable quantities such as continuum intensities,Doppler velocities, as well as full Stokes vector for the simulation at various positions at the solar disk,and analyse the influence of non-locality of radiative transport in the solar photosphere on helioseismicmeasurements. Bisector shapes were used to perform multi-height observations. The differences inacoustic power at different heights within the line formation region at different positions at the solardisk were simulated and characterised. An increase in acoustic power in the simulated observations ofthe sunspot umbra away from the solar disk centre was confirmed as the slow magneto-acoustic wave.
Subject headings:
Sun: magnetic fields - Sun: oscillations - Sun: helioseismology - sunspots INTRODUCTION
Techniques of local helioseismology are currently un-able to unambiguously determine sub-surface structureof the flows and sound speed perturbations in and aroundlarge-scale solar magnetic field concentrations, such assunspots and pores (Shelyag et al. 2007a; Gizon et al.2009; Moradi et al. 2010). Due to the complexity of mag-netohydrodynamic processes involved, our understand-ing of the behaviour of magnetoacoustic waves as theyare absorbed, reflected and refracted by sunspots is farfrom complete.Recently, it was demonstrated that solar magneticfields and the process of magneto-acoustic wave modeconversion associated with them lead to significantchanges in the wave travel times used in helioseis-mic inversions (Moradi & Cally 2014; Hansen & Cally2014). Four processes associated with strong magnetismdominate wave behaviour in sunspots (Cally et al.2015): fast/slow mode conversion at the Alfv´en/acousticequipartition level v a = c s , allowing acoustic (slow)waves to transmit into the upper atmosphere if the ‘at-tack angle’ α between wave vector and magnetic field issmall but converting them to magnetic (fast) waves oth-erwise (Cally 2006; Schunker & Cally 2006); the “rampeffect” that reduces the effective acoustic cutoff frequency ω c to ω c cos θ , where θ is the magnetic field inclinationfrom the vertical (Bel & Leroy 1977); fast wave reflec-tion around the height where the Alfv´en speed matchesthe wave’s horizontal phase speed; and fast/Alfv´enmode conversion that typically occurs over several scaleheights near the fast wave reflection level, generat-ing both upward and downward propagating Alfv´enwaves (Cally & Hansen 2011). Fast/slow conversion isfound to produce large negative travel time shifts, whilefast/Alfv´en conversion generates countervailing positiveshifts provided the vertical plane containing the wavevector is nearly perpendicular to the vertical plane con-taining the magnetic field lines (Cally & Moradi 2013). The ramp effect allows field-guided acoustic waves toenter the atmosphere in inclined field where frequency ω > ω c cos θ while normally the acoustic cutoff wouldprevent their propagation ( ω < ω c ). The complexity andsensitivity to magnetic field direction of wave motionsabove the equipartition level makes interpretation of ob-servations difficult but potentially rewarding. Moradiet. al. (2015, submitted) studied the effects of direc-tional time-distance helioseismology on the travel timemeasurements in the sunspot model.There are also possible significant discrepancies intravel time measurements originating from the effectsof non-locality of radiative transport in the solar at-mosphere. Changes in spectral line formation heightsdue to magnetic field presence (see e.g. Shelyag et al.2007b), systematic centre-to-limb variations in absorp-tion line formation (Shelyag & Przybylski 2014), as wellas instrumental effects, such as stray light, and otherprocesses involved in formation and measurement of ra-diation intensities and Doppler shifts result in our in-ability to unambiguously measure the travel time pertur-bations and, therefore, infer solar sub-surface structure(Rajaguru 2011).Rapid improvements in computational power alreadymake it possible to perform forward modelling of mag-netohydrodynamic wave propagation and mode con-version in “realistic” solar magnetic field structures(Shelyag et al. 2009; Moradi et al. 2009; Felipe et al.2010; Cameron et al. 2011; Khomenko & Cally 2012;Felipe 2012; Zharkov et al. 2013; Felipe et al. 2014).Spectral line synthesis codes and radiative diagnosticstools also allow computations of mock observables fromthe simulated plasma parameters, allowing for directcomparison between simulations and observations incomputational helioseismology.Creating a sunspot that is both spectropolarimetri-cally accurate and magnetohydrostatically stable is in-herently difficult, as the sound speed and temperaturecan change significantly with small changes in the den- Przybylski et al. sity and pressure stratification. The sunspot model ofKhomenko & Collados (2008) was created to allow em-pirical quiet and umbral solar models to be used inthe near-surface layers in combination with a Schl¨uter-Temesvary flux tube model (Schl¨uter & Temesv´ary 1958)in the interior. However, the model created this wayis still not convectively stable. Convective instabilityis fatal to linear MHD simulations, but these codes areless expensive than full non-linear simulations, and idealfor the long time series required in helioseismology; forthe study of fast and slow magneto-acoustic waves; andfor simulating fast-slow and fast-Alfv´en mode conver-sion in the photosphere and lower chromosphere. Theeffects of convective stabilisation on the eigenmodes ofsolar models for helioseismic simulations were studied bySchunker et al. (2011). A technique for stabilising theatmosphere is discussed in Sec. 2.In this paper, we present a model of a magneto-hydrostatic and spectropolarimetrically accuratesunspot. Our model is based on the sunspot-likemodel of Khomenko & Collados (2008). The modelwas adjusted to provide a more accurate replicationof photospheric sunspot properties taken from semi-empirical models, while still maintaining a smoothtransition of physical properties between the magneticand non-magnetic regions required for stable numericalsimulation. This technique makes it possible to obtainaccurate photospheric absorption line formation heightsas well as allowing the study of observational signaturesof acoustic wave propagation in the simulated modelat different positions on the solar disk. We perform amagnetohydrodynamic simulation of the propagation ofa wave through this sunspot-like model and investigatethe behaviour of acoustic waves in the simulated modelusing the synthesised radiation, as if it were observed.We also investigate effects of the centre-to-limb variationeffects on Doppler velocity measurements and studythe line bisector shapes to allow for a multi-heightview in the line formation region, which can be usedto observationally disentangle wave mode conversionprocess in the solar atmosphere.The structure of the paper is as follows. In Section 2 wedescribe the background model. In Section 3 we explainthe magnetohydrodynamic simulation and the spectralsynthesis methods used to provide artificial observables.Section 4 provides results and description of the radiativeeffects on acoustic wave measurements in observations.In Section 6, we discuss our findings. MODEL
To provide a convectively stable quiet Sunbackground model, the method described byParchevsky & Kosovichev (2007) is used. The Brunt-V¨ais¨al¨a frequency N = gγp ∂p∂z − gρ ∂ρ∂z must be positive forconvective stability. Rearranging this equation in termsof ∂ρ/∂z , combining with the equation for hydrostaticstability (Equation 1) and introducing a free parameter α gives, ∂p∂z = − ρg (1) ∂ρ∂z = − gρ γp − α ρN g , (2) where the gravity acceleration g , Brunt-V¨ais¨al¨a fre-quency N , and the adiabatic index γ are functions ofdepth. The equations are solved using a fourth-orderRunge-Kutta method on a one-dimensional grid. Theequispaced grid covers the height range from −
50 to+2 .
48 Mm and is resolved with the vertical step ∆ z =0 . z is modified by setting the neg-ative values in the convectively unstable solar interiorto small positive ones. The free parameter α must begreater than zero to enforce convective stability and isincreased so to match the pressure in the model with theStandard Solar Model S (Christensen-Dalsgaard et al.1996) pressure at a point in the interior z = − a , η and B which change the sunspotradius, magnetic field inclination and strength, re-spectively. A full description of the effects of theseparameters on the magnetic field configuration is givenby Khomenko & Collados. The model is defined on atwo-dimensional r - z plane discretised into a domainfrom −
10 to 2 Mm in height, with a radius of 100 Mmand resolution of ∆ z = 0 . r = 0 . − z > − u ( r, z ) (Pizzo 1986) ∂ u∂r − r ∂u∂r + ∂ u∂z = − πr ∂p ( u, z ) ∂u , (3)where p ( u, z ) is the gas pressure along the field lines.The Pizzo method boundary conditions require both aquiet Sun (denoted with index q ) and umbral (denotedwith index um ) pressure, density, temperature, and pres-sure scale height ( h = pρg ) and temperature distributionsas functions of depth. The quiet Sun model ( p q , ρ q , h q )generated above was used for the outer boundary condi-tion. For the inner boundary the Avrett semi-empiricalmodel (Avrett 1981) is used, which is then joined to thepressure and density profiles at the axis of the self-similarflux tube using log-linear interpolation. This is then con-vectively stabilised using Equations (2) and (1) as de-scribed for the quiet Sun above. The Wilson depressionadiatively accurate magnetohydrostatic sunspot model 3can be prescribed by shifting the log( τ ) = 0 of theumbral model ( p um , ρ um , h um ). The pressure and scaleheight are then distributed throughout the domain usingthe following: p ( u, z ) = p q ( z ) − ( p q ( z ) − p um ( z )) (cid:18) − u ( r, z ) u ( n r , (cid:19) , (4) h ( u, z ) = h q ( z ) − ( h q ( z ) − h um ( z )) (cid:18) − u ( r, z ) u ( n r , (cid:19) . (5)The potential solution given by Equation (3) is used asan initial guess. The pressure distribution given by Equa-tions (4) and (5) is iterated together with Equation (3)using a Gauss-Seidel method. Thus, the complete forcebalance is calculated with a specified precision, giving afinal distribution of the potential and pressure.The Pizzo and Low type flux tubes are then joinedat z = − B r , B z are calculated accordingto: ρ ( r, z ) = p ( r, z ) g ( z ) h ( r, z ) (6) B r ( r, z ) = − r ∂u∂z (7) B z ( r, z ) = 1 r ∂u∂r . (8)To extend this model below z = −
10 Mm a verticalflux tube with a constant B z and zero B r is used, andthe pressure and density profiles are continued smoothlydownwards.Finally, the FreeEOS equation of state (Irwin 2012)is applied to find the adiabatic index, temperature andsound speed at each grid cell in the model. The model isthen converted to Cartesian geometry, giving the full setof physical parameters required for the MHD simulationsand radiative transfer calculations.Using the procedure explained above the magnetic fieldstructure pictured in Figure 1 was constructed. Thebackground image in the figure shows the modulus ofmagnetic field B . The field lines are nearly vertical inthe “umbral” region ( r <
10 Mm), and show inclinationof about 60 ◦ in the “penumbral” region, r >
10 Mm, ofthe sunspot model.In the figure, the dashed line shows the log( τ ) =0 layer, while the dotted contours represent c s /v A =0 . , , and 10 levels. As is evident from the figure,in the umbral region at the axis of the sunspot, thelog( τ ) = 0 layer is positioned higher than c s /v A = 1layer, suggesting formation of the continuum radiationin the magnetically-dominated sunspot atmosphere.The 6173˚A photospheric absorption line of neutral ironis used for observations of the full solar disk with theHelioseismic Magnetic Imager (HMI) onboard the So-lar Dynamic Observatory (SDO). Therefore, this linewas chosen to carry out radiative diagnostics of thesunspot model using the SPINOR code (Solanki 1987;Shelyag et al. 2007b). For each one-dimensional col-umn of the model, continuum intensity and spectral lineprofile calculations are performed by solving the Unno-Rachovsky (Unno 1956) radiative transfer equation for Fig. 1.—
Magnetic field structure of the sunspot model. Themagnetic field strength is shown with the magnetic field lines over-plotted (solid). Also shown are the c s /v A = 1 (middle dotted), 0.1(upper dotted) and 10 (lower dotted) contours. The dashed line isthe log( τ ) = 0 contour, representing the visible photosphere.Note that the aspect ratio is severely stretched. the Stokes vector I = [ I, V, Q, U ]. Off-disk centre ob-servations are simulated by inclining the numerical do-main and interpolating the density, temperature, mag-netic field and velocities onto the new line of sight ( los ).The slanting is performed around the z = 0 km heightand in the direction of positive y (Figure 2). The ve-locity and magnetic field vectors are then projected intothe new reference frame. The calculation uses 500 wave-length points with a δλ = 0 . los velocity is given by v los = v z cos θ + v x sin θ . The magnetic field is recalcu-lated using a similar relation.Figure 2 shows the continuum images of the sunspotmodel calculated for 0 ◦ , 30 ◦ and 60 ◦ angles between thesurface and the los , which correspond to viewing cosine µ = 1 ., .
866 and 0 .
5, respectively. We find that themodel produces a realistic limb darkening dependencewith a continuum value of 79% of the disk centre intensityat µ = 0 .
5. This is only slightly higher than the 75%of the limb darkening curve determined by Foukal et al.(2004).The velocity response functions of the 6173˚A spectralline are shown in Figure 3 for the quiet Sun, two penum-bral regions at ±
10 Mm, and in the centre of the sunspotumbra for the chosen positions at the solar disk. Theselocations have been marked with crosses in Figure 2.Since, for observations away from solar disk centre, twopoints at the same distance from the sunspot axis arenot equivalent, the penumbral models have been chosenso that los of P1 crosses the umbral region, while the los of P2 inclines further into the penumbra. The responsefunctions were calculated by computing a perturbed pro-file with a small positive (directed towards the observer) los velocity perturbation and subtracting from it an un-perturbed profile for the same location.The top row of Figure 3 shows the response functionsof the four points in the model for the disk centre cal-culation. The perturbation is directed towards the ob-server, causing the line to be blue-shifted. The lobes ofthe response function tilt inwards towards the line core, Przybylski et al.
Fig. 2.— ◦ , 30 ◦ and 60 ◦ to thevertical. The figures have been normalised to the quiet Sun valueat 0 ◦ inclination. Inclination is performed towards an observerdisplaced in the negative y direction. The crosses show the twopenumbral, and one umbral point used in Figure 3 marked as 0 . x -axis) are formedhigher in the atmosphere.The top-right figure shows a fully Zeeman-split pro-file in a strong umbral magnetic field. Notably, whilethe line formation height range is narrower compared tothe quiet Sun, the response function lobes are wider inwavelength, suggesting higher sensitivity of the line tovelocity perturbations.The two penumbral points are identical in the so-lar disk centre simulation due to the symmetry of thesunspot. As the penumbral magnetic field is weaker, theline is not completely split. In the line core, the responsefunction shows two smaller regions of sensitivity to thevelocity perturbation.Figure 3 demonstrates that the line formation heightrange increases with the inclination angle. In the case ofthe quiet Sun (left column of the figure), it increases from ∼
400 km at the disk centre to ∼
800 km at µ = 0 .
5. Asthe line width does not change significantly, the wave-length range of the response function does not changewith the inclination.For the cases of magnetised penumbral and umbral at-mospheres, the observed visible sunspot surface increaseswith the inclination angle. Between µ = 1 and 0.5, thelog τ = 0 level for the penumbral points P1 and P2is shifted downwards by ∼
100 km, and by ∼
400 kmfor the umbra. The line sensitivity height range also in-creases further away from the disk centre, similarly tothe quiet Sun.Notably, the line profiles and the response function shapes for P1 and P2 are very different. The far-side um-bra (second column of Figure 3, P1 in Figure 2) will havea formation range that extends into the highly magneticumbra. This can be observed as an increasingly splitprofile as inclination increases in the 2nd and 3rd rows.The near-side penumbral pixel (third column, P2) willsimilarly form in a region of lower magnetic field. Dueto the inclination of the magnetic field, P1 will measurea higher magnetic field strength along the los while P2will incline against the direction of field line inclination.The angle of the two ridges is seen to be larger in theumbral distributions than the quiet sun. For a smallwavelength range in the quiet sun up to 500 km of theatmosphere will be measured. Comparitively, in theumbral distribution a similar filter would only samplearound 100 km.The large range of responses seen in the differentpenumbral and umbral positions will lead to larger un-certainty in the observation height of velocity measure-ments. The impact of Zeeman-split profiles on velocitymeasurements is only amplified at higher inclinations. NUMERICAL SIMULATIONS
We perform a simulation of acoustic wave propagationin the simulated model with the SPARC code. The codewas designed to solve the linearised ideal MHD equa-tions for wave propagation in a stratified solar environ-ment (Hanasoge 2011). The version of the code we em-ploy for the simulations uses Message Passing Interface(MPI) to parallelise the computation and reduce compu-tation time. It uses an implicit compact 6th order finitedifference scheme applied to the horizontal and verticalderivatives. An explicit filter is used to prevent numeri-cal instabilities in the solution. The boundary conditionsused in the simulation include a Perfectly Matched Layer(PML) (Hanasoge et al. 2010) at the top and bottomboundaries, allowing for efficient absorption of the out-going waves. A 12 . v A = 125 km s − , which is sufficiently high to al-low fast MHD waves of interest, which have horizontalphase speed less than this, to propagate and refract cor-rectly while still allowing us to use a manageable timestep. Our cap is large enough for this to not be an oner-ous constraint. The implications of the limiter on helio-seismic travel time shifts have been studied in detail byMoradi & Cally (2014).The numerical grid has horizontal extent of n x = n y =256 grid points, with a physical size of 140 Mm, givinga resolution of ∆ x = ∆ y = 0 .
55 Mm in the horizontaldirections. To deal with the large variation in physicalparameters over the domain from the convective zoneto chromosphere, the code uses a vertical grid spacingbased on the sound speed. The grid has n z = 300 pointsbetween 1 . −
25 Mm and is distributed such thatthe acoustic travel times between each cell are the samefor the quiet Sun, ∆ z ∝ /c s . This gives a resolution ofaround 50 km near the photosphere, and around 1 Mmadiatively accurate magnetohydrostatic sunspot model 5 Fig. 3.— los velocity response functions of Stokes I profile of 6173˚A Fe i line computed for the models of quiet Sun (first column), near-sideand far-side penumbrae (second and third columns, respectively), and umbra (fourth column) for µ = 1 ., .
866 and 0 . los at which the perturbation is placed, where 0 Mm represents the log τ = 0layer for the quiet Sun photosphere. The 5000˚A optical depth axis is also shown, with a dashed line showing the observed depression ofthe photosphere. The corresponding Stokes- I (top right) and Stokes- V (top-left) profile shapes were over-plotted in white in each panelover the wavelength range shown in the figure. in the lower solar interior. This means we do not resolveslow waves in the large β regime, but these are effectivelydecoupled from the system anyway, so their neglect is notimportant. The following acoustic source, similar to thatdescribed by Shelyag et al. (2009), was used: v z = A sin( 2 πtT o ) exp − ( t − T ) σ t exp − ( r − r ) σ r × (9) × exp − ( z − z ) σ z , where T = 300 s, T = 600s, σ t = 100 s, σ xy = 1 Mm, σ z = 0 .
25 Mm. The position of the pulse is r ( x, y ) =(45 ,
70) Mm, z = − .
65 Mm.The SPARC code solves the MHD equations for theperturbations around the MHS background model. Amaster-slave Open-MPI code has been written to takethese perturbations, combine them with the backgroundmodel and incline them as required. The SPINOR rou-tines are then applied to each pixel to generate the fullStokes vector for each pixel. Using the generated Stokes- I profiles, the los centre-of-gravity Doppler velocity iscalculated by computing the position of the centre ofgravity of the line profile and determining its shift fromthe unperturbed counterpart, computed for the back-ground model, according to: ∆ λ = λ cog − λ = R ( I c − I ) λdλ R ( I c − I ) dλ − λ . (10)To calculate bisector Doppler velocities from the spec-tral line the relative intensity I rel was determined bynormalising the measured Stokes I between 0 and 1. Bi-sectors of the spectral line were calculated at 100 evenlyspaced values between 0 . − .
95 of I rel . The bisectorswere calculated for the background model and for eachoutput snapshot. A Doppler velocity was then deter-mined for each snapshot using the shift from the unper-turbed background value, according to: v bsr = ( λ − λ bsr ) cλ . (11) RESULTS
Using the model described in Section 2 and methodol-ogy given in Section 3, a 2 . los Doppler velocity mea-sured using Equation (10). The first three wave bouncescan be easily seen. A shift in the wave arrival time canbe observed as a flattening of the wavefront as it passesthrough the sunspot umbra at y = 0 Mm. The middle Przybylski et al. Fig. 4.—
Response of the model to the acoustic source. Toppanel - simulated Doppler velocities at 0 ◦ inclination, measured at x = 0. Middle panel - simulated vertical component of velocity ata geometric height z = 0 Mm, measured at x = 0. The differencesbetween these two can be seen to be small. Bottom pannel - √ ρv y through spot centre (0,0) Mm showing the propagation of slowmodes down through the box. Two dashed horizontal lines in thetop plots mark the position of the sunspot umbra. panel of Figure 4 shows a time-distance plot of the ver-tical component of velocity at the z = 0 Mm level of thesimulation domain. A comparison between the top twopanels shows that the vertical velocity in the domain andthe los Doppler velocity are visually identical. Some re-flection can be seen from the top and bottom PMLs, andfrom the side boundaries.The bottom panel of Figure 4 shows the horizontalcomponent of velocity, scaled by √ ρ to provide a viewof the slow magnetoacoustic wave in the strong magneticfield. A slice is taken through the centre of the simu-lated sunspot ( x = 0 , y = 0). The fast wave can be seento propagate through the sunspot in the lower interiorwhere plasma- β is high. At around z = − .
400 Mm inthe umbra, the incoming fast wave hits c s /v A = 1 layer(See Figure 1) and undergoes partial transmission as aslow mode (effectively acoustic in c s < v A ). The slowmagnetoacoustic wave (now magnetic in c s > v A ) canbe seen to propagate back down into the sunspot as aflattening banding in the time distance plot. The waveamplitude in the atmosphere is low due to scaling bythe very low densities, however, it still can be seen tocontinue to travel upwards above the photosphere andescapes through the absorbing upper boundary.Figure 5 shows a power spectrum plotted with az-imuthally averaged wavenumber and frequency. Thespectrum has been calculated from the 2 . . l . Similar gaps are found in the simu-lations of Parchevsky & Kosovichev (2007) with high top Fig. 5.— A ν − l power spectrum for the sunspot box calculatedfrom synthetic Doppler velocities. boundary (1.75 Mm; their Fig. 6c), and attributed totrapping of acoustic modes. We do not understand howacoustic trapping explains this phenomenon. The gapsare also present in quiet sun simulations (no magneticfield), but are largely removed when the top boundaryis lowered to 500 km above the solar surface (their Fig.6b). This suggests that there is some numerical dissi-pation mechanism operating in our model chromosphere( z >
500 km) that we are yet to identify.Using the Fe i x, y ) pixelsin the model. The acoustic power is binned into differ-ent frequency bins by applying a Gaussian filter with aFWHM of 0 . x = 45 Mm, y = 70 Mm has been masked in the power maps. Fig-ures 6 and 7 show the acoustic power calculated fromthe Doppler shifts measured at the bisector positions of0 . I rel and 0 . I rel , respectively. The acoustic powerwas binned into frequency 1 mHz bands centred at 3, 4,5, 6 and 7 mHz (left to right in the figures), and the diskpositions of − ◦ , 0 ◦ and 60 ◦ were used (top to bottom). los velocity measurements off the disk centre are affectedby a larger line formation region and the presence of hor-adiatively accurate magnetohydrostatic sunspot model 7izontal velocity components in the los velocities. Imme-diately obvious in the figure is a series of concentric ringsof power travelling out from the source. These rings canbe thought of as the spatial analogue of the ridges seenin Figure 5 and occur at discrete values because a singlepoint-like source was used in the simulation.The differences between the acoustic power maps inFigure 6 and 7 represent changes in the wave behaviourover the formation height of the spectral line. This for-mation height can span almost a megameter close to thesolar limb (see Figure 3), and the differences are substan-tially more pronounced in the magnetic regions.In the acoustic power maps (Figure 6 and 7) a solid linerepresenting the sunspot umbra is over-plotted. Outsidethe sunspot umbra the sunspot shadow is observed atall frequencies. In the sunspot shadow, the power fromthe pulse has been absorbed or reflected by the magneticfield structure. This is most obvious at lower frequencies(left two columns). As frequency increases, the ridgesof increased acoustic power can still be seen behind thesunspot. Interestingly, the magnetic field perturbs theconcentric rings seen at 7 mHz, as the fast wave prop-agation speed and turning height change. Behind thesunspot in the 7 mHz band (right column), a small regionof increased power can be seen between the two outer-most rings of the power ridges (around y = 30 Mm). Asthis is seen in high frequencies and as a ring around thesunspot, rather than around the source, it appears to bea far-side acoustic halo. Acoustic halos are seen aroundactive regions as an excess of acoustic power comparedto the quiet sun. Comparison of the weakly-magnetic regions of each col-umn in Figure 6 shows little variation with inclination,regardless of frequency. In these regions the propagat-ing fast wave dominates the simulated observations asthere is little magnetic field to alter the fast-wave turn-ing point or to allow for mode-conversion process to takeplace. Inside the umbral region, the acoustic power mapsfor the disk centre case show very little power in 3 and 4mHz, and the power in the vertically-aligned oscillationsis seen to be almost completely absorbed. As the incli-nation increases, the 3 and 4 mHz power in the umbraremains low, while the 5, 6 and 7 mHz power bands showa significant power enhancement.From the response function (first column in Figure 3)we expect there to be larger differences in power betweenthe line core (Fig. 6) and line wings (Fig. 7) as we ob-serve further away from the solar disk centre. There islittle difference found in the disk centre cases (top row)between the two figures. However, at 60 ◦ inclination sig-nificant differences can be seen between the power mapsat the line core and line wings. Particularly, (1) the um-bral power increase is only seen at the line core (Fig. 6);(2) the ring-like structure (marked by the dashed circlein the left panel of Fig. 6) found at around y = −
20 Mmin the bottom left two panels is somewhat wider at theline core than in the line wings (Fig. 7). This struc-ture is most apparent in the 3 − A more comprehensive look at physics of acoustic halos, basedon three-dimensional simulations of this sunspot model can befound in Rijs et al. (2015), which expands on the previous worksof Hanasoge (2008); Khomenko & Collados (2009). They are at-tributed to fast MHD waves reflected in active region atmospheresby the steep Alfvn speed gradients there. (first two columns), and the power in it decreases withincreasing frequency. While the inclination of the mag-netic field at the surface at the radius of 20 Mm is 60 ◦ de-grees from vertical the magnetic field strength is low, andthe equipartition layer c s = v A is located above the lineformation region. Therefore, the ring is of acoustic na-ture and cannot be related to the slow magneto-acousticmode, as it is found at the source side in both − ◦ and60 ◦ inclinations corresponding to the los direction whichis either parallel or highly inclined to the magnetic field.As demonstrated, the umbral and penumbral acousticpower structures are mostly seen near the line core (Fig-ure 6). In the los velocities measured from bisectors inthe line wings (Figure 7) only a faint structure can beseen at high inclinations, again more obvious in the 7mHz power band (bottom right).To better understand the three-dimensional structureof the umbral power increase, in accordance with the re-sponse functions shown in Figure 3, a multi-height obser-vation is made by computing the Doppler velocities frombisector shifts measured at different line depths withinthe line formation region. In Figure 8, a Doppler veloc-ity map for a slice marked by a dashed line in Figure 7 isplotted for each bisector depth in the range 0 . − . I rel .For the 0 ◦ and 30 ◦ inclinations (top and middle rows ofFigure 8), the changes in the structure and magnitude ofacoustic power over the line formation range are limitedto an increase in power in the high frequency bands forobservations made closer to the line core. This matchesthe observations of acoustic halos by Rajaguru et al.(2013), where the acoustic power was weaker using filter-grams close to the line-wings. At 60 ◦ inclination (bottomrow, left two columns) the faint x = ±
20 Mm radius lowfrequency ring can be made out, increasing in radius atlarger heights.The vertical extent of the umbral power structure, seenat high inclinations at 6–7 mHz (bottom right two pan-els in Figure 6–8) shows a significant increase in powerhigher in the formation region. The formation of thisacoustic power structure seems to start mid-way up theline formation region, suggesting a highly localised phe-nomenon. The inclination of the field lines at the centreof this power increase ( y = 4 .
11 Mm) is approximately20 ◦ from the vertical. Taking into account the observa-tion angle of ± ◦ , the field line is almost perpendicularto the line-of-sight. Comparison with Fig. 1 shows thecontinuum formation height of the spectral line and the c s /v A = 1 layer cross at z = 0 Mm and 4 Mm radiusfrom the sunspot centre, allowing for direct observationof conversion of fast (parallel to the magnetic field line,perpendicular to the los ) waves to slow (perpendicularto the magnetic field line, and parallel to the los ) waves.The increased power in this region corresponds to theslow magneto-acoustic wave in the region where the mag-netic field is close to perpendicular to the los . This resultconfirms previous findings by Zharkov et al. (2013) thatthe observed acoustic power increase in the sunspots is asignature of slow magneto-acoustic waves. DISCUSSION AND CONCLUSION
In this paper we have: (1) described a modificationof the Khomenko sunspot model we developed to pro-vide a more accurate line formation region, allowing foraccurate spectral synthesis to be performed; (2) anal- Przybylski et al.
Fig. 6.—
Acoustic power map calculated from the shifts in the bisector of the Fe i . I rel . The columns,from left to right, show power in the 3, 4, 5, 6 and 7 mHz bands. The rows, from top to bottom, show measurements made at − ◦ , 0 ◦ and 60 ◦ inclination from the vertical, where the field of view has been inclined in the y -direction. The power at each inclination anglehas been normalised by its average in each frequency band. This represents a velocity measurement made near the line-core, showing thebehaviour higher in the atmosphere. The sunspot umbra has been marked with a solid circle, while the dashed line in the bottom rightpanel represents the slice taken in Figure 8. The low frequency ring has been marked in the top left panel. Fig. 7.—
Acoustic power map calculated from the shifts in the bisector of the Fe i I rel . The layoutof the columns and rows is as seen in Figure 6. This figure represents measurements made near the line wings. adiatively accurate magnetohydrostatic sunspot model 9 Fig. 8.—
Bisector power map at x = 0 Mm. The layout of the columns is as in Figure 6. The rows, from top to bottom, show measurementsmade at 0 ◦ , 30 ◦ and 60 ◦ inclination from the vertical. The Y axis in this figure represents the line depth where the bisector wavelengthand Doppler velocity are measured and covers a large part of the line formation region of 400 −
800 km in length. The formation regionwill depend on the magnetic field strength and inclination (Figure 3). et al. ysed response functions in our model to understandour synthesised centre-to-limb observations of the Fe i ◦ inclination with faint traces at lower inclina-tions. This geometry suggests that it is driven by slowmagneto-acoustic waves, as it is seen when the hori-zontal velocity component dominates the los velocity(bottom rows Figure 6 and 7). It can be seen as acrescent-like structure, which is most dominant towardsthe line core (higher in the atmosphere, Figure 6) andvery faint in the spectral line wings (lower in the at-mosphere, Figure 7). Umbral power enhancements areseen in the space-based (HMI, Zharkov et al. (2013))and ground based (Balthasar et al. 1998) observationsof sunspots. Notably, no power excess was observed inthe G-band (Nagashima et al. 2007). The appearanceof this power increase in a simulated sunspot suggests amagneto-acoustic phenomenon, rather than photon noise(Donea & Lindsey 2015). There are many differences inboth sunspot properties and the radiative effects on HMI measurements that could explain the lack of the umbralpower increase in all acoustic observations of sunspots.These include changes in the Wilson depression, the widerange of the velocity-response function in a magneticstructure (Figure 3), low resolution and high-noise mea-surements off the disk-centre or issues with using discretefilters on highly split profiles.Current measurements of acoustic travel times in com-putational helioseismology are largely performed usingmeasurements at the geometric heights in the simulationdomain (Moradi & Cally 2013), or on a surface roughlyrepresenting the continuum formation height determinedby the log( τ ) = 1 layer in the simulation domain(Khomenko & Cally 2012; Moradi et al. 2015). Despitethe fact that the physical velocity in the simulationmatches reasonably well to the los velocities calculatedfrom the simulated spectral lines, this method misses alot of information that can otherwise be gained from therange of formation of the spectral lines. As we show,this range also changes substantially if the simulationis performed for positions at the solar disk away fromthe centre. Using the model we described, artificial ob-servables mimicking the HMI and MDI pipelines can bemade (Scherrer et al. 2012; Fleck et al. 2011), as well ascomparisons to ground based observations.The multi-height Doppler measurements made byNagashima et al. (2014), using the HMI filter-grams pro-vide a similar approach to multi-height measurements asthe bisectors used in this study. Rather than velocitymeasurements made using shifts in Stokes- I , HMI usesmeasurements of both Stokes I + V and Stokes I − V REFERENCESAvrett, E. H. 1981, The Physics of Sunspots, 235Balthasar, H., Mart´ınez Pillet, V., Schleicher, H., W¨ohl, H. 1998,Sol. Phys., 182, 65Bel, N., & Leroy, B. 1977, ˚a, 55, 239.Cally, P. S. 2006, Royal Society of London PhilosophicalTransactions Series A, 364, 333Cally, P. S., & Hansen, S. C. 2014, ApJ, 738, 119Cally, P. S., & Moradi, H. 2013, MNRAS, 435, 2589Cally, P. S., Moradi, H., & Rajaguru, S. P. 2015, in
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