Beyond MHD: modeling and observation of partially ionized solar plasma processes
Highlights of Spanish Astrophysics VIII, Proceedings of the XI Scientific Meeting of the Spanish Astronomical Society held on September 8 – 12, 2014, in Teruel, Spain. A. J. Cenarro, F. Figueras, C. Hernández-‐Monteagudo, J. Trujillo, and L. Valdivielso (eds.)
Beyond MHD: modeling and observation ofpartially ionized solar plasma processes
E. Khomenko , Instituto de Astrof´ısica de Canarias, 38205 La Laguna, Tenerife, Spain Dpto de Astrof´ısica, Universidad de La Laguna, 38205, La Laguna, Tenerife, Spain
Abstract
The temperature and density conditions in the magnetized photosphere and chromosphereof the Sun lead to a very small degree of atomic ionization. In addition, at particular height,the magnetic field may be strong enough to give rise to a cyclotron frequency larger thanthe collisional frequency for some species, while for others the opposite may happen. Thesecircumstances influence the collective behavior of the particles and some of the hypothesesof magnetohydrodynamics may be relaxed, giving rise to non-ideal MHD effects. In thispaper we discuss our recent developments in modeling non-ideal plasma effects derived fromthe presence of a large amount of neutrals in the solar photosphere and the chromosphere,as well as observational consequences of these effects.
The plasma in the photosphere and chromosphere of the Sun is often treated in the approx-imation of magnetohydrodynamics (MHD). MHD modeling has been very successful both inconstructing idealized models of basic processes, and in providing realistic models of complexsolar phenomena such as magneto-convection, formation of magnetic structures, flux emer-gence, wave propagation, reconnection, etc. [2, 5, 20, 29]. While in ideal MHD the plasma issupposed to be completely ionized, the solar plasma is only weakly ionized, and the ioniza-tion degree can drop as low as 10 − in the upper photosphere. Together with the decrease ofthe collisional coupling with height, this may lead to the break of magnetohydrodynamic as-sumptions (depending on temporal and spatial scales of the phenomena under consideration)and lead to a series of non-ideal plasma effects. Since the degree of magnetization (the ratioof cyclotron to collision frequency of the different species) depends on the magnetic field, itis expected that non-ideal effects may play role in strongly magnetized photospheric regions(sunspots, flux tubes) and all over the chromosphere. a r X i v : . [ a s t r o - ph . S R ] A p r In order to take into account the non-ideal effects derived from the presence of neutrals,a fluid-like description has often been used in the photosphere and the chromosphere. Thisis possible because the collisions are still strong enough to partially couple different plasmacomponents. Many of single-fluid and multi-fluid models include only hydrogen or hydrogen-helium plasma, and a simplified treatment of radiative energy exchange (or even withouttaking radiative transfer aspects into account), which greatly simplifies the complexity of theproblem. Such an approach has been frequently used in studies of different types of waves inthe solar atmosphere [13, 19, 38, 42, 43] and studies of instabilities [7, 9, 37]. Idealized modelsof reconnection have been constructed in single-fluid and two-fluid approaches [27, 34, 46].More complex models include those of magnetic flux emergence [1, 26] and of chromosphericthermal structure [21, 24, 28]. The presence of neutrals, via collisions, provides a source ofadditional dissipation and energy supply. A weakly ionized plasma is able to diffuse throughthe magnetic field, partially breaking the frozen-in condition. The influence of neutralsresults in creation or removal of cut-off frequencies for waves and instabilities, additionalheating by dissipation of waves and currents, anisotropic dissipation of currents perpendicularto magnetic field facilitating the generation of potential structures, and other phenomena.Therefore, the influence of neutrals on the dynamics and energetics of solar plasma is provedto be important.With a constant increase of resolution capabilities of solar observations, both fromspace and from the ground, it is essential that new realistic models are built including non-ideal effects derived from the presence of neutrals. In this paper, we describe the effortsundertaken by our group in the development of an analytical theory and a numerical codefor such modeling. We attempt to provide a self-consistent description of a multi-componentmulti-species solar plasma and its interaction with the radiation field. Direct observationaldetection of ion-neutral effects has also been attempted and its first results are described.
We consider a plasma composed of a mixture of atoms of different atomic species. Theseatoms can be excited to excitation levels and/or to different ionization stages. Withoutloss of generality we limit the ionization stages to only two, as the number of multiplyionized ions is small in the regions of interest of the solar atmosphere. The interaction withradiation is included via excitation/deexcitation and ionization/recombination processes andphotons are treated as another type of particles interacting with the rest of the mixture. Thedetailed derivation of the macroscopic equations of continuity, momentum and energy foreach ionization-excitation level of species composing solar plasma components is given in [22].After summing up the motion equations over excitation states, a two-fluid system of equationsis derived that describes the behavior of neutral and ionized species. The electron and ioncontributions are added up together to get a single equation for the charged component. Thisway, the following system of equations is obtained: ∂ρ n ∂t + (cid:126) ∇ ( ρ n (cid:126)u n ) = S n (1) . Khomenko ∂ρ c ∂t + (cid:126) ∇ ( ρ c (cid:126)u c ) = − S n (2) ∂ ( ρ n (cid:126)u n ) ∂t + (cid:126) ∇ ( ρ n (cid:126)u n ⊗ (cid:126)u n ) = ρ n (cid:126)g − (cid:126) ∇ ˆp n + (cid:126)R n (3) ∂ ( ρ c (cid:126)u c ) ∂t + (cid:126) ∇ ( ρ c (cid:126)u c ⊗ (cid:126)u c ) = [ (cid:126)J × (cid:126)B ] + ρ c (cid:126)g − (cid:126) ∇ ˆp ie − (cid:126)R n (4) ∂∂t (cid:18) e n + 12 ρ n u n (cid:19) + (cid:126) ∇ (cid:18) (cid:126)u n ( e n + 12 ρ n u n ) + ˆp n (cid:126)u n + (cid:126)q (cid:48) n (cid:19) = ρ n (cid:126)u n (cid:126)g + M n − (cid:126) ∇ (cid:126)F nR (5) ∂∂t (cid:18) e ei + 12 ρ c u c (cid:19) + (cid:126) ∇ (cid:18) (cid:126)u c ( e ei + 12 ρ c u c ) + ˆp ie (cid:126)u c + (cid:126)q (cid:48) ie (cid:19) = ρ c (cid:126)u c (cid:126)g + (cid:126)J (cid:126)E ∗ − M n − (cid:126) ∇ (cid:126)F cR (6)In the above equations ρ n = (cid:80) α n αn m αn and ρ c = (cid:80) α n αi m αi + n e m e are the total massdensity of neutrals and charges; (cid:126)u n and are (cid:126)u c their center of mass velocities; (cid:126)J is the totalcurrent density; e n = 3 p n / (cid:80) α χ αn and e ei = 3 p ei / (cid:80) α χ αi are the internal energies ofneutrals and ions, where χ αn and χ αi are the ionization energies of species α . The summationin α goes over N chemical components. The definition of neutral, ˆp n , and electron-ion, ˆp ie ,pressure tensors and heat flux vectors, (cid:126)q (cid:48) n and (cid:126)q (cid:48) ie , is done using a common system of referencefor neutral and charged velocities, see [22]. The heat flow vectors are then corrected for thepotential ionization-recombination energy flux, e.g. (cid:126)q (cid:48) n = (cid:126)q n + (cid:80) α χ αn (cid:126)w αn . In this expression,the drift velocities of a neutral/ion of species α (e.g., (cid:126)w αn ) are taken with respect to thecentral of mass velocity of all neutrals/ions, e.g. (cid:126)w αn = (cid:126)u αn − (cid:126)u n . For the closure, one needsto provide the for collisional mass, momentum and energy exchange, S n , (cid:126)R n and M n . Themass source term S n is due to imbalance between the ionization/recombination processesand depends on the number densities and on the collisional rate coefficients. The term M n accounts for the energy exchange of neutrals through the elastic collisions with charges. Dueto their complexity in a general case, we do not provide specific expressions for these terms.The collisional momentum exchange can be simplified to give [22]: (cid:126)R n ≈ − ρ e ( (cid:126)u n − (cid:126)u e ) N (cid:88) β =1 ν en β − ρ i ( (cid:126)u n − (cid:126)u c ) N (cid:88) α =1 N (cid:88) β =1 ν i α n β (7)where ν en β are the collisional frequencies between electrons and neutrals of species β and ν i α n β are those between ions and neutrals of different species.The energy exchange by photons in Eqs. 5 and 6 is split into the contributions due toneutrals and charges to the radiative energy flux as follows: (cid:126)F R = (cid:126)F nR + (cid:126)F cR (8)where (cid:126) ∇ (cid:126)F nR = (cid:90) ∞ (cid:73) ( j nν − k nν I ν ) d Ω dν ; (cid:126) ∇ (cid:126)F cR = (cid:90) ∞ (cid:73) ( j cν − k cν I ν ) d Ω dν (9)In these definitions we have separated the contributions in the absorption k ν and emission j ν coefficients related to neutrals and ions. The intensities I ν are obtained from the completeradiative transfer equation, with total coefficients j ν and k ν . d I ν d s = j ν − k ν I ν (10)To close the system (1–6), we use the generalized Ohm’s law providing the electric field, (cid:126)E : (cid:126)E ∗ = [ (cid:126)E + (cid:126)u c × B ] = ρ e ( en e ) (cid:88) α ν ei α + (cid:88) β ν en β (cid:126)J + 1 en e [ (cid:126)J × (cid:126)B ] − (cid:126) ∇ ˆp e en e (11) − ρ e en e ( (cid:126)u c − (cid:126)u n ) (cid:88) β ν en β − (cid:88) α (cid:88) β ν i α n β This equation is obtained assuming stationary currents and neglecting second-order terms.From left to right the right-hand side terms in the Ohm’s law are: Ohmic, Hall, battery, andambipolar term. The Ohm equation has similar form as the one for hydrogen plasma derivedin e.g., Zaqarashvili et al. [45] except that we use (cid:126)u c instead of (cid:126)u i and include more completeexpressions for the coefficients that depend on the collisional frequencies.The two-fluid formulation (1–6) is applied when the difference in behavior betweenneutrals and ions is larger than between the neutrals/ions of different kind themselves. Theforces acting on neutrals and charges are different since the magnetic Lorentz force onlyaffects charges, and the neutrals feel its action exclusively via collisions. Thus, the two-fluidapproach is based on a stronger coupling between charged particles than between charged andneutral particles [44, 45]. According to estimations of the magnetization factor of the solaratmosphere, the two-fluid approach is best valid at heights above 1000 km. Otherwise, whenthe collisional coupling of the plasma is strong enough, it is more convenient to use a single-fluid quasi-MHD approach i.e, including resistive terms that are not taken into account bythe ideal MHD approximation. The conservation equations for the multi-species solar plasmabecome in this case: ∂ρ∂t + (cid:126) ∇ ( ρ(cid:126)u ) = 0 (12) ∂ ( ρ(cid:126)u ) ∂t + (cid:126) ∇ ( ρ(cid:126)u ⊗ (cid:126)u ) = (cid:126)J × (cid:126)B + ρ(cid:126)g − (cid:126) ∇ ˆp (13) ∂∂t (cid:18) e + 12 ρu (cid:19) + (cid:126) ∇ (cid:18) (cid:126)u ( e + 12 ρu ) + ˆp (cid:126)u + (cid:126)q (cid:48) (cid:19) + (cid:126) ∇ (cid:126)F R = (cid:126)J (cid:126)E ∗ + ρ(cid:126)u(cid:126)g (14)where we use a similar notation as for the two-fluid formulation. This system of equations isclosed by the generalized Ohm’s law formulated for the single-fluid case, that reads as: (cid:126)E ∗ = [ (cid:126)E + (cid:126)u × (cid:126)B ] = ρ e ( en e ) (cid:88) α ν ei α + (cid:88) β ν en β (cid:126)J + 1 en e [ (cid:126)J × (cid:126)B ] − en e (cid:126) ∇ ˆp e (15) − ξ n α n [( (cid:126)J × (cid:126)B ) × (cid:126)B ] + ξ n α n [ (cid:126)G × (cid:126)B ] . Khomenko (cid:126)G and α n are given by (cid:126)G = ξ n (cid:126) ∇ ˆp ie − ξ i (cid:126) ∇ ˆp n (16) α n = N (cid:88) β =1 ρ e ν en β + N (cid:88) α =1 N (cid:88) β =1 ρ i ν i α n β (17)and ξ n = ρ n /ρ , ξ i = 1 − ξ n are neutral and ion fractions.After applying Faraday’s and Ampere’s laws (neglecting Maxwell’s displacement cur-rent) to the Ohm’s equation formulated above, the induction equation is obtained for theevolution of the magnetic field: ∂ (cid:126)B∂t = (cid:126) ∇ × ( (cid:126)u × (cid:126)B ) − ρ e ( en e ) (cid:88) α ν ei α + (cid:88) β ν en β (cid:126)J − en e [ (cid:126)J × (cid:126)B ] + 1 en e (cid:126) ∇ ˆp e ++ ξ n α n [[ (cid:126)J × (cid:126)B ] × (cid:126)B ] − ξ n α n [ (cid:126)G × (cid:126)B ] (cid:35) (18)This equation closes the system of single-fluid equations. The single-fluid equations 12–14 and 18 are solved by the extended version of the codeMancha3D. This is a non-ideal magnetohydrodynamic code with hyper diffusion algorithmsand Cartesian grid written in Fortran 90. It solves the non-linear equations for perturbation in2D, 2.5D and 3D, and requires an arbitrary magneto-hydrostatic equilibrium to be explicitlyremoved from the equations. Spatial discretization is based on a six-order centre-differencescheme. The numerical solution of the system is advanced in time using an explicit fourth-order Runge-Kutta scheme. The code is fully MPI-parallelized using the distributed memoryconcept, and allowing full arbitrary 3D domain decomposition. A description of an earlypurely MHD version of this code can be found in [12]. Compared to the version described inthat paper, the new code includes the non-ideal terms derived from the generalized Ohm’slaw (Eq. 15), and the module for the calculation of non-ideal equation of state and the RTmodule that solves the Radiative Transfer equation (Eq. 10).Order of magnitude estimates of the importance of the different terms in the generatedsingle-fluid Ohm’s law (Eq.15) done in [22] for various atmospheric models show that onlyambipolar, Hall and (potentially) battery terms can reach significant values in the photo-sphere and chromosphere of the Sun. Therefore, in the current version of the code we onlyretain these three terms. This feature is partially described in [21, 23].The native set of variables used in the code is { ρ, ρ(cid:126)u, e, (cid:126)B } . The pressure, p , is assumedto be a scalar and is obtained from e through the implementation of realistic Equation ofState (EOS) for the solar mixture that takes into account the effects on ionization. This isdone by assuming a non-ideal gas in thermodynamical equilibrium in the convection zone,Figure 1: From left to right: initial distribution of the magnetic field; ion fraction ξ i ; am-bipolar diffusion coefficient η A = ξ n | B | /α n µ (in m s − ); and the quantity J η A in a fluxtube model.and an ideal partially ionized gas of a mixture including molecules in the photosphere, and isimplemented as interpolations of lookup tables. An ideal gas with parametrized thermody-namical quantities is used as the first approximation elsewhere. We compute additionally thetables for electron pressure, so that the ionization fraction is obtained self-consistently withthe rest of thermodynamical variables. We assume instantaneous ionization balance. TheEOS implemented in that way provides the required precision while enormously reducing thecomputing time.The newly implemented RT module solves the non-gray Radiative Transfer equation(Eq. 10) assuming Local Thermodynamic Equilibrium (LTE). The wavelength dependenceof the emission and absorption coefficients j ν and k ν is discretized by the opacity binningmethod [30]. The angle discretization is set prior to the code execution. Both discretizationsare performed by sets of routines accompanying the code. The formal solver used in the RTmodule is based on the short-characteristics method [31]. The module is fully parallelizedusing MPI. The LTE approximation limits the application of the RT module to simulationsof the top of the convective zone and the photosphere. In the optically thin corona, the codeuses Newtonian cooling approximation.The code has passed standard tests on numerical performance and speed, with scalingtests in up to several thousand cores, showing its reasonable efficiency. The code is mod-ular, includes conditional compilation and a modern input/output file format (HDF5). Anextended 2/2.5D version of the code uses Adaptive Mesh Refinement (AMR) implementedvia the PARAMESH library. Dissipation of currents due to the action of the ambipolar term provides an efficient source ofheat. We performed 2.5D simulation of the chromospheric heating in an idealized atmosphericmodel composed of a set of flux tubes. We used a 2nd-order thin magnetic flux tube modelconstructed after [32], see Figure 1. The model represents a horizontally infinite series of fluxtubes that merge at some height in the chromosphere. This magnetic field configuration isnon force-free. In order to evaluate the efficiency of the heating mechanism by ambipolardiffusion, we intentionally selected the temperature stratification of the flux tubes to be given . Khomenko − , even inthe chromosphere. The ambipolar diffusion coefficient, defined as η A = ξ n | B | /α n µ is ordersof magnitude larger than the Ohmic one, reaching values of 10 m s − at a height of 2 Mm.Such values of η A imply important current dissipation on very short time scales.The initial model is in MHS equilibrium and does not evolve without external per-turbation. As a perturbation we use ambipolar term in the induction (18) and energy (14)equations. Thanks to the Joule heating in the energy equation, (cid:126)J (cid:126)E , the magnetic energyis efficiently converted into thermal energy, producing heat. This heat is balanced by theradiative cooling term, (cid:126) ∇ (cid:126)F R . The left panel of Figure 2 shows the simulation snapshot 800s after the introduction of the perturbation. The right panel of this figure shows the heightdependence of the temperature at a fixed horizontal position inside the flux tube for differenttime moments. The most important heating is achieved at the upper part of the domain,close to the tube borders, since the term responsible for the heating is orders of magnitudelarger at these locations (right panel of Fig. 1).At the first few seconds of simulation, the chromospheric temperature significantlyincreases reaching up to 8 kK at the upper part of the domain, i.e. about 4 kK above itsinitial value. The time scales associated to the Joule heating are very short, of the order ofseconds, similar to those of radiative cooling. Due to the balance between these two processes,after the initial heating, we observe damped temperature oscillations while converging tosome constant value about 6 kK, forming hot chromosphere. Thus, we conclude that theJoule heating and radiative cooling terms in the energy equation can balance each other andFigure 3: Time evolution of density in the simulations of Rayleigh-Taylor instability including ambipolarterm (top) and without this term (bottom). The size of each snapshot is 1 × lead to the chromospheric temperature rise. In a more complex situation, magneto convectionsimulations performed in [28] show that the action of the ambipolar diffusion allows to removeartificially cool bubbles produced by adiabatic plasma expansion in the chromosphere andto provide a more realistic model of its temperature structure. Therefore it is important toinclude ambipolar diffusion in further models of the chromosphere. Plasma of solar prominences is yet another environment where the manifestation of partialionization effects can be important. Typical values of temperature, derived from the Dopplerwidth of spectral lines observed in prominences, are of the order of 8-9 kK [25, 40]. Theionization fraction is less known, but the values provided in the literature vary in the range ξ i =0.2–0.7 [25]. The prominence-corona transition region (PCTR) is thought to be subjectto tangential instabilities, as the Kelvin-Helmholtz instability (KHI) or the Rayleigh-Taylorinstability (RTI) [4, 15, 33, 35]. The linear theory has demonstrated that both instabilitiesare affected by the presence of neutrals in the prominence plasma [7, 9, 37].We performed 2.5D simulations of the non-linear phase of the RTI at the PCTR con-sidering partial ionization of the prominence plasma in the single-fluid formulation. As initialsetup we used a purely hydrostatic stratification of pressure and density for a given constant . Khomenko θ = 90 ◦ , top) and skewed from theplain at θ = 89 ◦ (middle) and θ = 88 ◦ (bottom). In the last two cases, the vertical dotted linesmark the cut-off wavelength of the instability, λ c = 38 and 155 km.temperature in the prominence (5 kK) and coronal (400 kK) parts of the domain. A constanthorizontal magnetic field of 10 G was set over the whole domain, and its orientation withrespect to the perturbation plane varied from one simulation to another. For these param-eters, the neutral fraction in the prominence makes ξ n = 0 . η A = 2 . × m s − . The instability is initiated by a multi-mode perturbationof the interface position. For each orientation of the magnetic field, we compare simula-tions with and without ambipolar term, see [23]. Figure 3 shows a time series of simulationsnapshots illustrating the development of the instability for the case with ambipolar termon (top) and off (bottom). The flows start the same, but then develop differences on smallscales, since the ambipolar term introduces additional diffusion acting perpendicular to themagnetic field. These slight differences in the particular form of the turbulent flows can beobserved at the end of the simulation. The ambipolar term is larger in the regions withlow density, i.e. regions with, generally, low momentum ρ(cid:126)u . Therefore, the impact of suchregions into the overall flow dynamics is not large. A similar behavior is observed in thesimulations of multi-fluid turbulence in molecular clouds [10, 11]. However, statistically, theflows in both cases are different, as is shown in Figure 4. In agreement with the linear theory,non-linear simulations demonstrate that the introduction of the ambipolar diffusion removesthe cut-off wavelength for the growth rate of the instability and allows the small scales todevelop. As the non-linear development of the instability is such that small scales merge andgive rise to larger scales [16, 18, 41], the larger growth rate of small scales leads to largerextreme velocities in the ambipolar case [23]. We also find up to 30% larger temperatures atthe PCTR in the simulations with ambipolar term, as a result of the Joule dissipation. Asignificant drift momentum, defined as (cid:126)p D = (cid:112) ( ρ i ρ n )( (cid:126)u i − (cid:126)u n ) is present at the PCTR [23]. Direct observational confirmation of the uncoupled behavior of neutral and ionized speciesin the solar atmosphere is still missing. Order of magnitude calculations show that thescales at which ion-neutral effects become important are rather small. At maximum, spatialscales reach few kilometers, and temporal scales reach fraction of seconds, depending on0Figure 5: Slit-jaw image in H α of the prominence observed on 11th of September, 2012. Blacknearly horizontal line shows the location of the slit in one of the scans.Figure 6: Top: amplitudes (left), widths (middle) and displacements (right) of the Ca ii − .Bottom: same for the He i . Khomenko ii line (red) and from He i line (blue) as a functionof time at a fixed location. Notice the oscillatory behavior and slightly larger amplitudes inCa at the velocity extremes. Right: velocities in both lines along the slit for a fixed time.Notice the lag between both velocities at the locations with larger gradients. The observations were done on 11th of September, 2012, using the Vacuum Tower Telescope(VTT) at the Observatorio del Teide on Tenerife. We targeted a prominence located atthe limb close to the active region NOAA 11564 that was undergoing evolution during theobservations. A slit-jaw image of the observed prominence is shown in Figure 5. The datarepresent a time series of 10-steps scans (0.14 (cid:48)(cid:48) per step) of a small part of the prominence.It took 1.5 sec per scan position, a fixed position was repeatedly measured every 15 sec. Weobserved Ca ii i r parameter staying around 10 cm. Standard reduction of the datawas performed including the correction for the differential refraction for spatial alignment ofthe spectra taken at the two different wavelengths. The detailed description of the data andtheir analysis is given in [8].The scientific analysis included a gaussian fit to the observed emission spectral lines,from which we extracted their amplitudes, widths and displacements [8]. The results of thisfit are given in Figure 6. The upper raw of this figure gives the results for the Ca ii linewhile the bottom one is for the He i line. Due to the low sensitivity of the CCD at thewavelength of Ca ii line the data for this line are much noisier than for He i . Therefore, thereare locations where we could not perform any fit of the Ca ii line, or the quality of the fit wasnot adequate. In most of the locations both lines show simple shapes that can be interpretedas due to a single-component atmosphere along the line of sight. However, at some pointsthe profiles show a clear two-component structure due to the appearance of the surges in theline of sight. In the preliminary analysis detailed here, we did not give any special treatment2Table 1: Parameters of the spectral lines in sunspot observations. Atomic data are takenfrom [14]. Formation heights are calculated in the penumbra models C and D from [36].Elm Ion λ (˚A ) EPL (eV) log gf g eff H C (km) H D (km) IFe ii − .
976 1.18 449 293 0.69Fe i − .
500 1.90 280 276 0.48Fe ii − .
643 1.67 162 171 0.41Fe i − .
900 1.40 277 274 0.41Fe i − .
493 1.09 336 350 0.69Fe ii − .
348 0.67 539 328 0.72Fe i − .
640 0.36 216 194 0.39Fe i − .
135 1.50 398 494 0.79Fe ii − .
279 0.87 553 331 0.72Fe i − .
497 0.39 235 244 0.38Fe ii − .
432 1.07 281 250 0.46Fe i − .
460 1.15 327 351 0.51to these profiles. Figure 6 shows that, outside of the locations where the Ca ii line couldnot be fitted, the distribution of the amplitudes and widths of both lines in time and alongthe slit is rather similar. To first order, we take this as a confirmation that both lines format the same location in the prominence and therefore measure the dynamics of the sameplasma. It can be observed at the right panels of Fig. 6 that indeed, the velocities derivedfrom both lines are very similar. A more detailed look at the velocities is given in Figure 7.It shows two selected examples of the time dependence at a fixed location (left) and spatialdependence at a fixed time (right). It can be observed that, while the velocities in Ca ii andHe i are very similar, differences appear at the locations of extreme velocities (left panel) or atthe locations of more rapid changes in space where the velocity gradients are stronger (rightpanel). Having in mind the preliminary nature of our analysis, it may indicate that velocitiesof neutral and ionised atoms in prominences shows differences, partially attributing this toion-neutral effects. Observations were taken at the VTT during the morning of 7th of October, 2012. As a targetwe used an isolated regular-shaped sunspot belonging to the NOAA 11582 group located atcoordinates (914 (cid:48)(cid:48) , -216 (cid:48)(cid:48) ) off solar disc center. By means of the spectrograph of the VTTwe scanned the sunspot using several carefully selected spectral intervals. Each intervalwas selected from the the list of unblended spectral lines with a clean continuum providedin [14] under a criterium of containing at least one Fe i and one Fe ii spectral line closein wavelength to fit on the same CCD. We selected pairs of lines with as similar formationheights as possible. The lines are listed in Table 1. The scan step was 0.18 (cid:48)(cid:48) and 250 positionswere taken. Different spectral intervals were scanned successively starting from 9:26 UT and . Khomenko i ii λ -meter technique at the position around the core of each line.The circles mark the distances in the penumbra where we performed the fit to the velocities.Figure 9: λ -meter velocities as a function of height at the half radius of the visible penumbra,measured at different spectral intervals.it took about 5 min to complete each scan. The seeing conditions were exceptional duringthe morning, with the r parameter varying around 15-20 cm.The data were reduced using a standard procedure and velocities were extracted bymeans of λ -meter technique [39]. No absolute wavelength calibration was possible due to theabsence of telluric spectral lines in the observed spectral intervals. We performed wavelengthcalibration by fitting the average quiet Sun spectrum to the FTS atlas. The 2D images of λ -meter velocities and continuum intensity were rotated and corrected for the limb perspective.An example of the continuum intensity and velocities of the Fe i ii v LOS ( r, φ ) = v r ( r ) sin θ cos φ + v z ( r ) cos θ , where r represents the radial distance along penumbra, θ is the heliocentric angle, φ is the azimuthal angle around the spot center, and v r and v z are the vertical and radialcomponents of the velocity in the local reference frame. The fit was performed for thevelocities at six radial distances in the penumbra located at various ranges of r , see Figure 8.The λ -meter technique allows to obtain the velocity at several positions along the lineprofile that correspond to different heights in the solar atmosphere. In order to compare Fe i and Fe ii velocities, we need these heights to be reliably calculated. The formation height ofspectral lines can strongly vary at the locations of bright and dark penumbra filaments. We4calculated the formation heights of the observed spectral lines in two semi-empirical modelsof sunspot penumbra, corresponding to a dark filament (model C) and a bright filament(model D) as provided in [36] at the observed heliocentric angle of θ = 77 ◦ (Shchukina,private communication). The height where the optical depth is equal to one at a givenwavelength along the line profile was taken to be its formation height. Each λ -meter velocitywas assigned a height calculated this way in the models C and D and then averaged for agiven r and λ -meter position. For a reference, Table 1 provides heights corresponding to theline minimum.Figure 9 shows the amplitude of the Evershed flow velocity v = (cid:112) ( v r + v z ) as a functionof height in the model D for all five spectral intervals separately for the Fe i lines (red) andFe ii lines (black). The results are qualitatively similar for the heights from model C with adifferent temperature structure. In all the cases the outward velocity decreases with heightaccording to the well known behavior of the Evershed flow. The Fe i lines show systematicallylarger velocities than Fe ii ones at all spectral intervals. Figure 9 shows the velocities at thehalf radius of the visible penumbra. The difference between the Fe i and Fe ii velocities ismaximum at radial distances close to the umbra-penumbra boundary and becomes almostnegligible at distances close to the visible end of the penumbra. While the results of ouranalysis are subject to uncertainties due to the height of formation of spectral lines, it isintriguing that all spectral intervals independently show similar behavior with slightly largervelocity of neutral iron at the same photospheric heights. More extensive analysis of thesedata will be presented in our future publications. This paper summarizes our recent effort in the theoretical and observational investigation ofthe effects of partial ionization of the solar plasma into its dynamics and energy balance. Wehave developed a self-consistent mathematical formulation for the two-fluid and single-fluiddescription of the partially ionized multi-species solar plasma and its coupling to the radiationfield. The single-fluid equations are solved by the extended version of the 3D non-ideal MHDcode Mancha. First results of our simulations show the high importance of the ion-neutraleffects (ambipolar diffusion) for the heating of the magnetized solar chromosphere. Thepresence of neutrals is also shown to modify the stability criterium of the Rayleigh-Taylorinstability at the border of solar prominences. Observational detection of ion-neutral effectshas been attempted. Our preliminary analysis indicates possible differences in the velocitiesof ions and neutrals in solar prominences and in sunspot penumbra.
Acknowledgments
This work is partially supported by the Spanish Ministry of Science through projects AYA2010-18029 and AYA2011-24808 and by the Leverhulme Trust through project IN-2014-016. Thiswork contributes to the deliverables identified in FP7 European Research Council grant agree-ment 277829, “Magnetic connectivity through the Solar Partially Ionized Atmosphere”. The . Khomenko
References [1] Arber, T. D., Haynes, M., Leake, J. E. 2007, ApJ, 666, 541[2] Asplund, M., Nordlund, ˚A., Trampedach, R., Allende Prieto, C., Stein, R. F. 2000, A&A,359, 729[3] Bellot Rubio, L. R., Balthasar, H., Collados, M., Schlichenmaier, R. 2003, A&A, 403, L47[4] Berger, T. E., Shine, R. A., Slater, G. L., et al. 2008, ApJ, 676, L89[5] Cheung, M. C. M., Sch¨ussler, M., Moreno-Insertis, F. 2007, A&A, 467, 703[6] Collados, M., Lagg, A., D´ıaz Garc´ıa, J. J., et al. 2007, in P. Heinzel, I. Dorotoviˇc, R. J.Rutten (eds.), The Physics of Chromospheric Plasmas, Vol. 368 of