Two-dimensional MHD models of solar magnetogranulation. Testing of the models and methods of Stokes diagnostics
aa r X i v : . [ a s t r o - ph . S R ] O c t Two-dimensional MHD models of solarmagnetogranulation. Testing of the modelsand methods of Stokes diagnostics
V.A. Sheminova
Main Astronomical Observatory, National Academy of Sciences of UkraineZabolotnoho 27, 03689 Kyiv, Ukraine
Abstract
We carried out the Stokes diagnostics of new two-dimensional magnetohydrody-namic models with a continuous evolution of magnetogranulation in the course oftwo hours of the hydrodynamic (solar) time. Our results agree satisfactorily withthe results of Stokes diagnostics of the solar small-scale flux tubes observed in quietnetwork elements and active plages. The straightforward methods often used in theStokes diagnostics of solar small-scale magnetic elements were tested by means ofthe magnetohydrodynamic models. We conclude that the most reliable methodsare the determination of magnetic field strength from the separation of the peaksin the Stokes V profiles of the infrared Fe I line 1564.8 nm and the determinationof the magnetic inclination angle from the ratio tan γ ≈ ( Q + U ) / /V . Thelower limits for such determinations are about 20 mT and 10 ◦ , respectively. We alsoconclude that the 2D MHD models of solar magnetogranulation are in accord withobservations and may be successfully used to study magnetoconvection in the solarphotosphere. Solar small-scale flux tubes still remain spatially unresolved. In this case the Stokes diag-nostics is the only available technique for the investigation of their structure and dynamics.The straightforward methods of Stokes diagnostics [17, 20, 21] allow the necessary informa-tion to be obtained after simple calculations from direct measurements of the observableStokes profile parameters. The inversion methods [2, 14] are used to construct flux tubemodels by fitting synthetic Stokes profiles to the observed ones. The methods of numeri-cal calculation of magnetoconvection in the photospheric layers [1, 5, 7] and constructionof self-consistent nonstationary magnetohydrodynamic models have also been developed.The MHD models are very useful in studying physical processes in solar magnetic features,but they cannot be directly compared to observations.We attempted to make such comparison, using the Stokes diagnostics. In this paper wematch new two-dimensional MHD models [6] to observations and examine the reliabilityof the Stokes diagnostics based on the MHD models.1
Calculations
The MHD models used in our study are described in detail in [6]. The model sequencestarts with a convective model with an initial average magnetic field of about 5 mT andterminates at the moment 120 min with an average field of 50 mT. The sequence contains94 2-D models with a 1-min interval and 52 models with a 0.5-min interval between them.The simulation region is a rectangle of length x = 3920 km and height h = 1820 km,it contains 112 vertical columns (rays) spaced at 35-km intervals. All the atmosphericthermodynamic parameters necessary for calculating the equations of radiative transferin spectral lines for every such column in the presence of magnetic fields were put at ourdisposal by A. S. Gadun.The transfer equations for polarized radiation in each of 112 columns were solved inthe LTE approximation for a plane-parallel atmosphere, and the Stokes profiles obtainedin the solution were averaged over space. The equations were solved numerically by amodified method [11] which is described in detail by Sheminova [16]. The Stokes profileswere calculated for three Fe I lines in the visible ( λλ λ .
85 nm).
The validity of any numerical simulation can be confirmed only by observation data. TheMHD models used here may be quite adequate in their parameters to solar small-scalemagnetic elements. We can calculate the Stokes profile parameters for these models andcompare them to the data of spectropolarimetric observations. For this purpose we usethe Stokes profiles observed in quiet network elements and active plages with a Fourierspectrograph at the McMath telescope in 1979 [24]. The chief value of these data is that alarge number of spectral lines were observed simultaneously in the wavelength range 445–557 nm with a high spectral resolution (420000). Unfortunately, the spatial and temporalresolutions were not high (10 ′′ and 35–52 min). Out of 402 iron lines, we selected 170unblended lines with known laboratory wavelengths. We calculated absolute shifts for theI and V profiles of these lines, using the strong Mg I 517.2 nm line as a reference line andthe laboratory wavelength system by Pierce and Breckinridge [13]. The magnetic fieldstrength was calculated for every line by the method of the center of masses [15]. Wealso determined the asymmetry of amplitudes and areas in the observed V profiles. Thuswe acquired the necessary observational data which could be used to check the results oftheoretical simulations of magnetoconvection in the photospheric layers.The time series of MHD models selected for comparison with observations extendedover the simulation period from 95 min to 120 min. We calculated the Stokes profiles offour spectral lines for each column in the simulation region in each of 52 models (at 0.5-min intervals); we obtained about 5800 profiles for every line. This procedure is analogousto an observation of the Stokes profiles by scanning a very narrow region 3920 km longon the solar surface with a spatial resolution of 35 km at 0.5-min intervals in the courseof 25 min with instantaneous exposures. For each calculated profile we determined thesame parameters as for the observed profiles. A preliminary analysis revealed that theprofiles of the weak IR line 1564.8 nm were the best suited for the comparison. This lineforms deep in the photosphere (log τ R from 0 to -1), it is the least sensitive to temperaturefluctuations [20] and is virtually unaffected by saturation. The other three lines, in thevisible range, are formed high in the photosphere (log τ R from -2 to -4), their central in-2ig. 1. Comparison of theresults of Stokes diagnosticsof MHD models (solid line)and observations in the quietnetwork features (squares)and active plages (crosses):a) field intensity, b) absoluteshift, c) asymmetry of V-profile amplitudes, d) asym-metry of V-profile areas.tensities are temperature-sensitive, they are also sensitive to saturation and NLTE effects.The data for the uppermost layers, which depend on boundary conditions to a greaterextent, are less reliable in our MHD models. The visible lines formed in the upper layersare supersaturated, as a rule, and their diagnostic capabilities are drastically impaired.Figure 1 displays the magnetic field strength B , absolute shift v , asymmetry of am-plitudes and areas, δa and δA , as functions of equivalent width; the quantities obtainedfrom the observed V profiles are shown by squares and crosses, and the calculated pa-rameters are represented by the second-order curves (average values from 5800 IR lineprofiles). We did not expect a complete agreement between the observed and calculatedparameters, since there were some distinctions in resolution, line sampling, and selectionof observation regions. The observed Stokes profiles were measured in magnetic featuresonly, while the calculated profiles referred to the entire simulation region. Only 30 percentof calculated scans, in the best case, may be referred to magnetic elements. It is seen fromFig. 1 that the calculated values of B and v fall within the observed intervals, while theasymmetries of amplitudes and areas are smaller, on the average, than in the observedprofiles. This might be attributed to the well-known observation facts [12] . In the quietnetwork elements the measured asymmetries δa and δA for 1564.8 nm are close to zero (bigdiamonds in Fig. 1). An interpretation for these facts can be found in [7]. Despite somedifferences, we may conclude that realistic Stokes profile parameters are derived withinthe framework of the MHD models, they are close to the parameters observed in the quietnetwork elements and plages. 3 Reliability of Stokes diagnostics
To test the methods of Stokes diagnostics, we selected one 2-D model (snapshot) from theMHD model sequence [6], namely, the model corresponding to the 120th min of simulationtime.Figure 2 demonstrates the distribution of various parameters in the simulation region:isotherms, isobars, velocity field, and field strength, together with the field lines andpolarity. Two flux tubes of various strengths and polarities with clearly defined stronglongitudinal fields stand out in the simulation region; an area at the granule center, wherea new flux tube begins to form, can be also clearly seen. With these data, we calculatedthe Stokes profiles of four spectral lines for each of 112 model columns (scans) and appliedvarious methods of Stokes diagnostics with the aim to determine the flux tube parameters.We examine the agreement between the quantities thus obtained and the thermodynamicmodel parameters and the observation data for magnetic elements.
There are three methods for the determination of this parameter. The first method isbased on the ratio of the amplitudes a V of the V profiles of two lines in the visible rangewith various Zeeman splittings and similar other parameters [22, 23]:MLR = a V (525 . g eff (524 . a V (524 . g eff (525 . ≈ · a V (525 . a V (524 . . Traditionally, the observed magnetic line ratio (MLR) is calibrated with the use ofsimple models with homogeneous magnetic fields; MLR = 1 for weak fields, and MLR < λ ± = R [ I c − ( I c ± V )] λdλ R [ I c − ( I c ± V )] dλ The field strength B (in tesla) is readily found from the well-known expression for theZeeman splitting: B rb = ( λ − − λ + ) / (2 · . · − λ g eff ) , (1) λ being the line wavelength (in nanometers). This method is quite reliable for the solardisk center, but it gives only the lower limit of B , as it specifies the longitudinal fieldcomponent only.The third method is the simplest one — it is based on the measurement of the distance∆ λ br between the peaks in the red (r) and blue (b) wings of the V profile of the IR linewith the Land´e factor equal to 3. This is a very efficient method [9, 20]. With expression(1), we can derive the field strength B br as the upper limit. As the quantity ∆ λ br consists4ig. 2. Snapshots from 2D MHD models [6] at simulation moment 120 min.Isotherms for every 1000 K: dotted line) 4000 K and thick line) 6000 K. The latterroughly indicates the level log τ R = 0. Magnetic field lines are shown in three lowerpanels. Magnetic field intensity: hatching density is proportional to intensities of 0,80, 110, and 140 mT. Magnetic field polarity: hatching density corresponds to fieldintensities of 1, 40, 80, and 120 mT in the positive polarity field.5ig. 3. Magnetic field intensity B along the simulation region: a)from a MHD snapshot at the lev-els log τ R = 0 (solid line) andlog τ R = − λ λ λ λ λ D and the Zeeman splitting ∆ λ H , the distance between the V-profile peaks in weak fields, when the first component is prevailing, is determined mainlyby the nonmagnetic line width, i.e, it is practically independent of magnetic field. Asthe field increases, the Zeeman splitting becomes prevailing. That is the reason why thismethod fails in the case of very weak photospheric fields.One can see in Fig. 3a the variations in the model field strength at the photospherelevels log τ R = 0 and log τ R = −
2. The maximum field strength in the model flux tubesat log τ R = 0 is as large as 200–250 mT in a very narrow interval (35–70 km). It is knownfrom observations (their resolution is no better than 200 km) that the field strength atlog τ R = 0 is 100–150 mT in magnetic elements and 150–200 mT in pores, less than in themodel flux tubes.Figures 3b-d show the field strength determined by three methods. The best fit to themodel values of B is provided by the third method (distance between the V-profile peaks)with the use of the IR line Fe I 1564.8 nm (Fig. 3d). The lower limit for the measurementsby this method is about 20 mT. The line ratio method (Fig. 3b) becomes less reliable atabout 0.1 T, MLR being equal to 0.65–0.7 in this case. The method of the center of masses(Fig. 3c) allows longitudinal magnetic fields to be measured, and their strength obviouslydepends on the inclination of field lines in flux tubes. A greater inclination in the secondtube as compared to the first tube results in an underestimate of about 70 mT at the levellog τ R = 0. 6 .2 Magnetic field inclination The angle γ of the field vector and the line of sight can be found directly from the relationsfor the amplitudes of σ -components in the V, Q, and U profiles [3, 17]: V ≈ cos γ, Q ≈ sin γ cos χ, U ≈ sin γ sin χ, √ Q + U V ≈ sin γ cos γ , UQ = tan χ. Here χ is the field vector azimuth. These relations are valid when the observed line isweak and the angles γ , χ do not change along the line of sight. The ratio ( Q + U ) / /V depends on wavelength, field strength, line saturation, etc. [20].In strong magnetic fields the amplitudes of V , Q , and U profiles do not depend on B due to magnetic saturation of lines, and therefore γ can be reliably determined. Inweak fields, V ≈ B and Q and U are of the order of B , and the ratio depends notonly on γ but on B as well. In actual practice this ratio is calibrated with the use ofmodel calculations. In our two-dimensional MHD models the magnetic vector azimuth isequal to zero or 180 ◦ and U ≈
0. Although the U profile does not vanish altogether dueto magnetooptical effects, it is very small, and it may be ignored. The angle γ can bederived directly from the amplitude ratio when the ratio ( Q + U ) / /V is divided by V and U = 0 is substituted: √ Q + U V ≈ sin γV cos γ = tan γ, QV ≈ tan γ, √ QV ≈ tan γ. In this version the ratio is independent of B under the conditions of weak fields as wellas strong ones. The equality of the amplitude ratio and tan γ is only approximate —the angle being determined is affected by velocities and other parameters. We tested theaccuracy of this approximation by calculating γ from the ratios √ Q/V for four lines andcomparing the calculated values with the model ones.Figures 4a, b show the variations of γ along the simulation region, and Figs 4c, d show γ as a function of B . The inclination is 5–10 ◦ within a range of 300 km in the centralpart of the first flux tube and 10–20 ◦ in the second flux tube. This is in accord with thedata of [19], where an analysis of the V, Q profiles in plages gave an inclination of fluxtubes no less than 10 ◦ . As judged from the distribution γ ( B ) found with the use of theStokes diagnostics (Fig. 4d), there are no fields weaker than 20 mT, but such fields cannotbe detected by the method based on the measurements of the V-profile peaks. It is alsoobvious from Fig. 4d that there are no longitudinal fields with B <
160 mT and γ < ◦ .This is likely to be a result of the limitedness of the method used to determine γ . Besides,the angles γ found with the use of the Fe I 1564.8 nm line are greater than the anglesdetermined from the lines in the visible range. Hence we may conclude that the ratio √ Q/V for the visible lines gives angles overestimated by 10 ◦ , and the overestimation iseven greater for the IR line. The method we used here is simple and quite reliable, as thedistribution γ ( B ) obtained by it is in agreement with the model distribution. In reality,when χ = 0, we have to use the relation ( Q + U ) / /V ≈ tan γ . As of now, there are no observations which would prove the existence of stationary verticalflows with velocities higher than 250 m/s (average from numerous observations) inside7ig. 4(a, b). Magnetic field in-clination along the simulation re-gion: a) from a MHD snapshot atthe levels log τ R = 0 (solid line)and log τ R = − √ Q/V forfour lines. Note the data derivedfrom λ λ γ vs. magnetic field intensity: c)from a MHD snapshot at four log τ R levels — 0, -1, -2, and -3; d) γ from the amplitude ratio √ Q/V for four lines (crosses correspond to λ B was de-rived from the distance between theV peaks of the λ v I , v Q , v V , which were found from the I, V, and Q profiles calculated forfour lines, were compared with the model velocities (Fig. 5). The best agreement was foundfor the velocities measured as shifts of the π -components of Q profiles (Fig 5d), except thecases when γ = 0, Q = 0 (flux tube center), and shifts of the zero crossing of V profiles,except the cases of multicomponent V profiles (Fig 5c). The velocities obtained fromthe shifts of the I-profile center for the IR line ((Fig 5b)) deviate strongly from the true(model) velocities at sites with strong magnetic fields. As the I profile splits completelyinto σ -components of various amplitudes, the risk of mistaking the wavelength of a strong σ -component for the wavelength of the line core is run when the central wavelength of theline is determined automatically from the central intensity minimum.8ig. 5. Vertical velocity along thesimulation region: a) from the MHDmodel at the levels log τ R = 0 (solidline) and log τ R = − The temperature diagnostics in small-scale magnetic features is carried out with the use ofthe temperature line ratio (TLR). This quantity was used for the first time when the trueweakening of the I profile of the line Fe I 525.02 nm with respect to the line Fe I 523.30 nminside unresolved magnetic elements was studied [10]. Ideally, the lines in the TLR musthave different temperate sensitivities and close central depths d c , wavelengths, and effectiveLand´e factors g eff . With the ratio between the amplitudes a V of their V profiles, wecan estimate the line weakening associated with temperature variations in a region withmagnetic fields, i.e., in a flux tube. At elevated temperatures the sensitive line is weakenedto a greater degree and TLR is less than unity, and vice versa. To obtain temperaturefrom this ratio, it should be calibrated by model calculations, as it is done for the magneticline ratio MLR.For lines in the visible spectrum, the ratioTLR = a V (524 . g eff (525 . a V (525 . g eff (524 .
71) = 1 . · a V (524 . a V (525 . a V ≈ d c . We calculated TLR (with a coefficient of 1.11) for our MHDmodel (Fig. 6). It depends not only on temperature, but on the vertical temperature gra-dients and the magnetic and temperature saturation in the lines as well, and this distortsthe results of the diagnostics. In the central regions of flux tubes the magnetic saturationof the lines Fe I 524.7 and 525.06 nm is very strong. The amplitude a V is temperature-insensitive there, and the method fails (TLR ≈ > I c / . The intensity I c was calculated for each model column in thecontinuum of two lines Fe I 525.0 and 1564.8 nm, being the intensity averagedin the horizontal direction over the whole simulation region. Observations suggest thatthe contrast in small magnetic elements heavily depends on spatial resolution. Magneticelements less than 300 km in size have a small brightness (1.1–1.4 in the network), and theelements larger than 300 km have a darker continuum (0.7–0.9 in plages). For IR lines,no brightening is observed even in small magnetic elements, while darkening is typical oflarger elements. This occurs because the IR lines near λ = 1650 . τ R = 0 changed from 5700 to6400 K at the center of the same flux tube (Fig. 6a). Our calculations are in accord withthe brightness variations observed in the continuum in magnetic features. It is well known that not only the amplitudes and asymmetries of the measured StokesQ, U, and V profiles but the areas of the profile wings and shifts as well depend onthe spatial, temporal, and spectral resolution. The reason is the spatial and temporalaveraging as well as the instrumental and atmospheric distortions of the signal measured.Simulations of the spatial smearing made with the use of the modulation transfer function(which allows for the atmospheric and telescopic distortions) reveal that the contrast inthe continuum of a flux tube 100 km in diameter may be by a factor of 7–8 higher thanthe contrast measured with a 0.3 ′′ resolution. At present the spatial resolution is notbetter than 0.25–0.3 ′′ (180–220km), and the reliability of the parameters measured in thinflux tubes still remains a problem. Calculating the V profiles for various intervals alongthe simulation region in the MHD models, we may study the spatial averaging effect onthese profiles. The averaged profiles may have several components with widely differentparameters depending on horizontal temperature gradients, vertical velocities, magneticfield intensities, inclinations, and polarities. The well-known shape of V profiles may be10ig. 6. Temperature distributionalong the simulation region: a)from the MHD model at the levelslog τ R = 0 (thick line), log τ R = − τ R = − τ R = − λ λ a V and theparameters B rb , v V , v Q derived from the V and Q profiles of the line Fe I 1564.8 nmin spectral scans with varying spatial resolution: about 35, 70, 200, 600, 1000, 1300,and 4000 km. The spatial variations in the velocity and the magnetic field intensity andpolarity demonstrated in Fig. 2 can be compared with the diagnostics results plotted inFig. 7. The V profiles will be affected by spatial resolution until very high resolutionsare attained. At a 35-km resolution the derived values of B rb , v V , v Q coincide quitewell with the model values. When the V profile consists of two or more components,the distortions of the profile at low resolutions markedly affect the determination of theamplitude a V = ( a b + a r ) / ≈ . ′′ ), especially in the regionswith steep horizontal gradients. As an example, we give the velocities derived from theshifts of the profiles calculated for the spectral scans with different resolutions from 0 to1300 km in the simulation region. We obtained 0.4, -1, 2.4, 0.5, -0.5, and 0 km/s for sixscans with a 200-km resolution, 0.6, -0.2 km/s for two scans with a 600-km resolution,and 0 km/s for one scan with a 1300-km resolution. The highest velocity obtained witha resolution of 0.3 ′′ (200 km) was four times greater than the velocity obtained with aresolution of 1 ′′ (600 km). Hence it follows that the choice of spatial resolution stronglyaffects the results of Stokes diagnostics with the use of V profiles. We demonstrated with the use of the Stokes diagnostics that new 2-D MHD models [6]are in accord with observations and may be successfully used to study magnetoconvectionin the solar photosphere. We also tested the methods of Stokes diagnostics of small-scalemagnetic elements. The results are as follows.The IR line λ a V ; b) field intensity B rb ;c, d) shifts of V and Q profiles. Hor-izontal averaging of profiles: 35 km(solid line), 70 km (dotted line),200 km (plusses), 600 km (crosses),1000 km (triangles), 1300 km (dia-monds); the entire region (big cross). λλ < . γ ≈ ( Q + U ) / /V with an accuracy of 10 ◦ . Longitudinal fields ( γ < ◦ , B <
160 mT) cannot be measured by this method.The velocities of vertical motions inside magnetic elements are reliably determined fromthe shifts of the π -components in the Q profiles and the zero crossings of V profiles. Whenthe V profiles are of anomalous shape because of the presence of regions with oppositepolarities, the shift of the π -components of Q profiles should be used.The temperature in flux tubes with high spatial resolution cannot be reliably estimatedfrom the temperature line ratio for two lines in the visible spectrum due to strong magneticand temperature saturation of lines in flux tubes.Spatial averaging significantly affects the Stokes diagnostics of flux tubes. Resolu-tions worse than 200 km may produce erroneous results. The reason has to do with thehorizontal gradients of parameters inside flux tubes. Acknowledgements.
We are indebted to A. S. Gadun for furnishing the MHD models12or the Stokes profile calculations and S. K. Solanki for furnishing the observations as wellas for useful discussion of the results and their comments.The study was partially financedby the Swiss National Science Foundation (Grant No. 7UKPJ 48440).
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