Alfvén Wave Turbulence as a Coronal Heating Mechanism: Simultaneously Predicting the Heating Rate and the Wave-induced Emission Line Broadening

The Alfvén Wave Solar Model (AWSoM) is a global, wave-driven, extended-MHD numerical model starting from the top of the chromosphere and extending into the heliosphere beyond Earth’s orbit. The waves drive the model by exchanging momentum and energy with the plasma: gradients in the wave pressure accelerate the plasma, while dissipation converts wave energy into thermal energy. Wave dissipation is the only explicit heating mechanism invoked, and no ad hoc or geometric heating functions are included. The time and spatial scales associated with the wave phenomena are much smaller than the characteristic scales of the entire system, making it impractical to resolve the wave motions in a global model. We therefore adopt an approach where the wave energy evolves according to wave kinetic equations under the WKB approximation. Two such equations, for low-frequency Alfvén waves propagating parallel and anti-parallel to the magnetic field, are coupled to the MHD equations.

The model is based on BATS-R-US (Tóth et al. 2012), a versatile, massively parallel MHD code with adaptive mesh refinement. The computational domain used in this work is a non-uniform spherical grid which allows us to treat the sharp gradients in the transition region as well as resolve the heliospheric current sheet. AWSoM is implemented within the Space Weather Modeling Framework (SWMF; Tóth et al. 2012).

The model equations are based on the two-temperature MHD equations derived in Braginskii (1965), with the following simplifications: the Hall effect is neglected, and the electrons and protons are assumed to have the same bulk velocity. This leads to single-fluid continuity and momentum equations, and separate pressure equations for electrons and protons. The latter allow us to incorporate non ideal-MHD processes such as electron heat conduction, radiative cooling, and electron−proton heat exchange. The extended-MHD equations are coupled to wave kinetic equations for parallel and anti-parallel waves, as described in Sokolov et al. (2009), van der Holst et al. (2010), and Sokolov et al. (2013). The governing equations then become

These equations describe the evolution of the mass density, ρ; the bulk flow velocity, ; the magnetic field, ; and the proton and electron thermal pressures, p_p and p_e, respectively. is the energy density of Alfvén waves propagating parallel (+) or anti-parallel (−) to the magnetic field. Next, G is the gravitational constant, is the solar mass, is the magnetic permeability, and γ is the polytropic index set to be constant at 5/3. The Alfvén velocity is given by . The pressure tensor due to Alfvén waves is given by (see, e.g., Jacques 1977).

The wave kinetic equations are given in Equation (6), which represents two separate equations, for the two wave polarities. The wave energy density dissipation rate of the two wave modes is denoted by . The total wave energy density dissipation rate is given by . The pressure equations for protons and electrons are given in Equations (7) and (8), respectively. The dissipated wave energy is partitioned between the two species, where the fraction heating the protons is denoted by f_p, which is set at 0.6 (see Cranmer et al. 2009; Breech et al. 2009). The radiative cooling rate of the electrons, Q_rad, represents the loss of energy due to emission from the plasma at a given location. The emission is assumed to be due to electronic de-excitation, which is important in the cooler chromosphere to the lower corona, and becomes negligible in the  1 MK hot corona. The cooling rates are calculated from the CHIANTI 7.1 atomic database (Dere et al. 1997; Landi et al. 2013). The ion population responsible for the radiation is calculated by assuming coronal elemental abundances as reported in Feldman et al. (1992) and ionization equilibrium (due to the ionization and recombination rates appearing in Landi et al. 2013).

Heat exchange due to Coulomb collisions between electrons and protons enters the energy equations through the second term on the right-hand side of both equations. The collisional heat exchange results in temperature equilibration on a timescale , which is given by (Goedbloed & Poedts 2004)

where m_p and m_e are the proton and electron masses, respectively, e is the elementary charge, k is the Boltzmann constant, T_e is the electron temperature, is the permittivity of free space, n is the plasma number density (under the assumption of quasi-neutrality), and is the Coulomb logarithm, taken to be uniform with . Since the thermal coupling between the two species is proportional to the plasma density, it becomes negligible as the density drops off with distance from the Sun. The electron energy equation, Equation (8), includes the field-aligned thermal conduction tensor, denoted here by and given by the Spitzer form.

The model assumes that a Poynting flux of Alfvén waves propagates through the top of the chromosphere, with a magnitude proportional to the local magnetic field and constrained by observations (see Table 1). The polarity of the wave emitted from each point on the inner boundary is determined by the direction of the local radial magnetic field. The wave energies of the two wave polarities evolve self-consistently with the plasma according to Equations (6)–(8). This allows us to directly account for the effects of wave pressure on the flow. However, a physics-based description of the wave dissipation requires us to consider the evolution of the wave spectrum, an approach that would essentially make a global model four dimensional (as was discussed in Sokolov et al. 2009; Oran et al. 2010). An alternative to this approach is to adopt a Kolmogorov-type dissipation rate, based on the assumption that the wave energy represents an average over a turbulent spectrum, and that it dissipates due to a fully developed turbulent cascade (Matthaeus et al. 1999). This approach, first presented in Hollweg (1986) for open magnetic field lines, allows one to treat the effects of wave dissipation although the detailed microphysical processes are not simulated explicitly. Further, by incorporating a Kolmogorov-type dissipation term into the coupled system of wave-transport and MHD equations, we can achieve a self-consistent description that is based on sound physical arguments while still accounting for the unresolved evolution of the wave field. A self-consistent heating mechanism is a major step forward compared to coronal models that employ geometric or fully empirical heating functions in the MHD equations, because the heating source is itself evolving. The present model employs the wave dissipation term presented in Sokolov et al. (2013), who generalized the Hollweg (1986) approach treating wave dissipation in both open and closed magnetic field lines. The resulting dissipation rate allows the boundary between the open and closed field lines to evolve self-consistently. This mechanism was analyzed in detail in Oran et al. (2013) and shown to be equivalent to the dissipation terms invoked in Matthaeus et al. (1999) and Cranmer et al. (2007). Further, the modeled thermal structure in the lower corona was validated against full-disk EUV images, while the flow properties of the wind at 1–2 au were validated against Ulysses observations. Further validation of the thermal and flow structure at the lower corona was presented in Landi et al. (2014), where the model performance was compared to other coronal models (for an ideal dipole magnetic field case), and in Oran et al. (2015), who showed that the modeled electron density and temperature are consistent with differential emission measure tomography of the lower corona (for another solar minimum case). As for comparing the simulated wave transport to observations, it was shown in Oran et al. (2013) that the modeled wave amplitudes were consistent with observations taken from the lower corona and up to 1 au along open polar magnetic field lines during solar minimum. In this work, we wish to extend this comparison by focusing on the equatorial lower corona, producing synthetic spectra, and comparing them to detailed spectroscopic observations of line broadening. We will thus be able to study how well the turbulent cascade dissipation assumed here reproduces the observations, and to make the first (to our knowledge) comparison of modeled and observed emission line broadening due to Alfvén waves propagating and dissipating in a realistic coronal magnetic field.

Table 1.  Input Parameters and Inner Boundary Values for the AWSoM Steady-state Simulation for CR 1916

Input Parameter	Value
a	25 kma
	0.06
Poynting flux per unit Bb
Base electron temperature, Te	50,000 K
Base proton temperature, Tp	50,000 K
Base electron density, ne
Base proton density, np

Notes.

^aThe correlation length, , in Equation (10) is determined by , where denotes a magnetic field measured in units of Tesla. ^bThis value is based on the Hinode observations reported in De Pontieu et al. (2007), and corresponds to an rms wave velocity amplitude, km s⁻¹, observed at an altitude where the plasma density is cm⁻³.

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The wave dissipation term used in this work is based on a phenomenological description of a fully developed turbulent cascade (Matthaeus et al. 1999). For the sake of brevity, we will not repeat the full derivation and justification of this term and refer the reader to Sokolov et al. (2013) and Oran et al. (2013) for a more detailed analysis. It is worthwhile, however, to describe this term and the adjustable parameters controlling it. The energy density dissipation, , is given by

Note that the dissipation depends on the relative magnitudes of the two wave polarities, leading to different heating rates in open and closed field lines. This unified approach, presented in Sokolov et al. (2013), ensures that the spatial distribution of the coronal heating rates will emerge automatically and self-consistently with the magnetic field topology. A detailed analysis of this approach and its implications can be found in Oran et al. (2013). The total dissipated wave energy heats both protons and electrons, with the fraction of heating going into the protons denoted by the constant f_p = 0.6 (see Breech et al. 2009; Cranmer et al. 2009 for more details). It is important to note that the model version and global simulation used here are those from Oran et al. (2013), and so reflections are not directly simulated by the model; rather, the pseudo-reflection coefficient serves to mimic their effect under the assumption of a fully developed turbulent cascade. In this approximation, any wave energy created by reflections is dissipated locally by the cascade process before it can be carried away by the reflected wave (Matthaeus et al. 1999; Dmitruk & Matthaeus 2003; Cranmer et al. 2007; Chandran & Hollweg 2009). Thus, in practice there is no need to convert the outgoing wave energy into the opposite polarity, as the wave energy is converted into heat. A less restrictive treatment that includes a self-consistent description of wave reflections in a global model was implemented in van der Holst et al. (2014). The dissipation mechanism is controlled by two adjustable parameters: a constant pseudo-reflection coefficient, , and the transverse correlation length for Alfvénic turbulence, L_⊥, which varies with the width of the magnetic flux tube such that (Hollweg 1986).

In the AWSoM model, the wave energy evolves under the WKB approximation. The perturbations due to Alfvén waves propagating parallel and anti-parallel to the background magnetic field can be conveniently described by the Elsässer fluctuation variables (Tu & Marsch 1994), defined as , where and are the velocity and magnetic field perturbations, respectively, and is the permeability of free space. The energy densities of non-compressible Alfvén waves can be expressed as , while the square of the velocity perturbation can be obtained from

On open field lines, only one wave polarity should dominate if the reflection is negligible so that the product will be zero. On closed field lines, opposite wave polarities are injected at the two footpoints of the field line, giving rise to counter-propagating waves. However, in the balanced turbulent regime near the top of the closed field lines, these perturbations are presumed to be uncorrelated: . Thus, the last term on the right-hand side of Equation (11) will drop out in any magnetic topology. The square of the velocity perturbation now becomes

Combining Equations (2) and (12), we can relate the non-thermal speed to the wave energies as

Note that under the WKB approximation, the wave energy density is already an average over timescales much larger than the wave period, and there is no need for averaging.

In order to produce a realistic steady-state solution for the period during which the SUMER observations were taken, we derive the inner boundary conditions of the model using a synoptic line-of-sight magnetogram of the photospheric radial magnetic field, acquired during CR 1916 (lasting from 1996 November 11 to 1996 December 9). The magnetogram was obtained by the Michelson–Doppler Interferometer (MDI) instrument on board the SoHO spacecraft (Scherrer et al. 1995). In order to compensate for the reduced accuracy at the polar regions, we use a polar-interpolated synoptic magnetogram, provided by the Solar Oscillations Investigation (SOI) team (Sun et al. 2011). The resulting radial magnetic field is shown in Figure 1.

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The values used for the model’s adjustable parameters and inner boundary conditions for this simulation are listed in Table 1. The dissipation length and reflection coefficient are not unique, but may vary in a defined range that is empirically motivated and consistent with previous models, as explained in Sokolov et al. (2013) and Oran et al. (2013). Due to the high computational cost of global coronal models, it is not feasible to perform a complete sensitivity study of parameter space. Nonetheless, the specific values used here were chosen in Oran et al. (2013) under the constraint that the solution is capable of reproducing the observations both near the corona and far in the solar wind for a different solar minimum case. That simulation was validated against a myriad of observations from the lower corona to interplanetary space at 2 au in Oran et al. (2013, 2015). The use of the same values for two different Carrington Rotations during solar minimum is reasonable and does not introduce additional uncertainty. Indeed, the global solution reproduced the global structure of the corona well, as we demonstrate in Section 6.1.

The data were reduced using the standard SUMER software made available by the SUMER team through the SolarSoft IDL package (Freeland & Handy 1998); each original frame was flat-fielded, corrected for geometrical distortions, and aligned with all other frames. In order to increase the signal-to-noise ratio, the data were averaged along the slit direction in 30 bins, each 0.01 R_⊙ wide. Spectral line profiles were fitted with a Gaussian curve removing a linear background. The resulting count rates were then calibrated using the standard SUMER calibration also available in SolarSoft. The accuracy of the spectral flux calibration of SUMER detector B before 1998 June is ≈20% (Wilhelm 2006, and references therein).

The micro-roughness of the SUMER optics causes the instrument to scatter the radiation coming from the solar disk into the detector, even when the instrument is pointing outside the limb. The scattered light forms a ghost spectrum of the solar disk at rest wavelength superimposed onto the actual spectrum emitted by the region imaged by the SUMER slit.

This ghost spectrum can provide important, though undesired, contributions to measured line fluxes when the local emission of the Sun is weak; these contributions need to be evaluated and, when necessary, removed. Unfortunately, the strength of the ghost spectrum depends on a number of factors (slit pointing, strength of the disk spectrum etc.), which make it impossible to devise a procedure to automatically remove it from the observations; its estimation needs to be performed on a case-by-case basis.

In the case of the present observations, the almost radial pointing of the SUMER slit allows us to use the rate of decrease of spectral line intensities with distance from the limb in order to determine an upper limit on the contributions of the ghost spectrum. Since emission line intensities depend on the square of the electron density, the rapid decrease of the latter with height causes the coronal line intensities to decrease by almost two orders of magnitude from the closest to the farthest end of the slit in the present observation; in contrast, the scattered light intensity, which is not emitted by the plasma in the observed region, is only reduced by a factor over the same range.

Landi (2007) devised a two-step method to determine an upper limit on the scattered light contribution to any spectral line for off-disk observations stretching over a large range of distances from the limb. First, the rate of decrease of the scattered light intensity with height is determined, based on several lines that are not emitted by the corona and whose off-disk intensity is entirely due to scattering. Second, the rate of decrease of the scattered light intensity is used to get an upper limit on its contribution to a specific coronal line as follows. We measure the intensity of the coronal line at the location farthest from the limb in the instrument’s field of view and assume that this intensity is entirely due to scattered light. The radial rate of decrease of the scattered light intensity is then normalized to match that coronal line intensity at the same height, giving an upper limit to the scattered light contribution at all other heights. Note that this method actually overestimates the scattered light contribution to coronal lines.

To estimate the radial rate of decrease of the scattered light intensity, we have used the intensity of the continuum at 1475 Å and of the following lines: He i 584 Å, C ii 1335 Å, C iii 977 Å, O i 1302 Å, 1304 Å and 1306 Å, O iii 835 Å, and Si iii 1206 Å. These lines and continuum are emitted by the solar chromosphere, so that they are expected to be too weak to be observed at the heights covered by the SUMER field of view: their observed intensity is entirely due to scattered light. The rate of decrease of each of these lines and continuum have been normalized to the value of the intensity at the largest distance from the limb and averaged together to provide the final scattered light intensity versus height curve. This curve appears as the light green curve in Figure 2. The normalized intensity versus height curve for the lines in Table 2 is also shown for comparison. We verified that all of them decreased at a rate much faster than the scattered light intensity: this suggests that the latter is at best a minor contributor to the intensity of each of the lines in Table 2. We also determined the maximum heliocentric distance R_max below which the scattered light contribution to the coronal line intensity is less than 50%. We take this arbitrary limit as an indication of the range of heights where we can safely neglect the scattered light. This height is reported in the third column of Table 2. We note that all of the emission lines considered here possessed a clear Gaussian line shape that could be separated from the background up to distances larger than R_max.

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The total line emission in a plasma volume, dV, having electron temperature T_e and density N_e is given by

where is the contribution function for a spectral line associated with an electronic transition from an upper level j to a lower level i, defined as

where G_ji is measured in units of photons cm³ s⁻¹. X^+m denotes the ion of the element X at ionization state . The contribution function also depends on the following quantities:

• 1.  
  is the relative level population of ions at level j, and depends on the electron density and temperature;
• 2.  
  is the abundance of the ion relative to the abundance of the element X, and depends on the electron temperature;
• 3.  
  is the abundance of the element X relative to hydrogen;
• 4.  
  is the hydrogen abundance relative to the electron density (∼0.83 for fully ionized plasmas); and
• 5.  
  A_ji is the Einstein coefficient for the spontaneous emission for the transition .

As T_e and N_e are known from the model solution, the contribution function in any computational volume element can be calculated. In this work, we used coronal element abundances as given in Feldman et al. (1992) and the latest ionization equilibrium computation available in CHIANTI (Landi et al. 2013).

Once the contribution function is calculated at every point along the line of sight, the total observed flux in the optically thin limit is given by integrating the emissivity along the line of sight:

where d is the distance of the instrument from the emitting volume dV. F_tot is measured in units of photons cm⁻² s⁻¹. This volume integral can be replaced by a line integral by observing that dV = Adl, where A is the area observed by the instrument and dl is the path length along the line of sight. In the case of the present observations, the area covered by the instrument is 4″ × 1″. In order to calculate the line-of-sight integral from the 3D model results, we interpolate G_ji and N_e from the AWSoM non-uniform spherical computational grid onto a uniformly spaced set of points along each observed line of sight. The spacing used for the interpolation was set to match the finest grid resolution of the model. This procedure ensures that the integration is second-order accurate.

Knowledge of the magnitude of thermal and non-thermal ion motions allows us to calculate a synthetic spectrum, which explicitly includes their effects on the line profile. Thus, instead of merely predicting the total flux of an emission line, we can predict the full spectral line profile to be compared with the observed spectrum.

For each location along the line of sight, the local spectral flux can be calculated by imposing a Gaussian line profile characterized by the predicted total flux, F_tot, the rest wavelength , and line width, , determined from the ion temperature and the magnitude of non-thermal motions. The spectral flux, measured in units of photons cm⁻² s⁻¹ Å⁻¹, can be written as

where is the normalized line profile. In the case of a Gaussian line profile, is given by

and the line width, in accordance with Equation (1), can be written as

The non-thermal speed, v_nt, can be calculated from the AWSoM model through Equation (13). The emitting region in our case is a three-dimensional non-uniform plasma, where each plasma element along the line of sight gives rise to different values of the total flux and the line width. In order to synthesize the line profile from the model, we must perform the line-of-sight integration for each wavelength separately, i.e., we must calculate the spectral flux at the instrument, , given by

The spectral flux is calculated over a wavelength grid identical to the SUMER spectral bins. In order to compare the synthetic spectra with observations, we must also take into account the SUMER instrumental broadening. For this purpose, we convolve the line-of-sight-integrated spectral flux with the wavelength-dependent instrumental broadening for SUMER detector B, as given by the standard SUMER reduction software available through the SolarSoft package.

Atomic data uncertainties directly affect the line fluxes calculated from the AWSoM simulation results. It is therefore necessary to discuss the accuracy of the data available for the emission lines for which we wish to produce synthetic spectra. Table 2 lists the five spectral lines that were used for detailed line profile calculations. They were chosen mainly because they are bright and clearly isolated from neighboring lines, so that their profile could be resolved accurately to as large a height as possible.

5.3.1. Ne viii 770.4 Å and Na ix 681.7 Å

5.3.2. Mg ix 706.0 Å 

5.3.3. S x 1196.2 Å

5.3.4. Fe xii 1242 Å

Comparing observed full-disk images to those synthesized from model results allows us to test how well the global, three-dimensional solution, and specifically the temperature and density distributions, can reproduce the observations. Such a comparison also tests the model’s prediction of the location and shape of the boundaries between open and closed magnetic field regions, as the coronal holes appear much darker than closed field regions in EUV images. In the most general case, creating synthetic images requires solving the full radiative transfer through the entire line of sight. However, EUV emission lines from the corona and transition region can be treated within the optically thin approximation. This assumption becomes less accurate at the limb, where the optically thin approximation may break down due to the large density along the line of sight. The procedure used to calculate the synthetic images in this work is identical to that presented in Downs et al. (2010), Sokolov et al. (2013), and Oran et al. (2013), and its details will not be repeated here.

We compare our model results for CR 1916 to images recorded by the EUV Imaging Telescope (EIT; Delaboudinière et al. 1995) on board SoHO. In preparing instrument-specific response tables, as well as observed images from the raw data, including calibration, noise reduction, and normalization of the photon flux by the exposure time, we used the SolarSoft IDL package.

Figure 3 shows the observed versus synthesized images of the 284 Å band, which is dominated by the Fe xv ion, corresponding to an electron temperature of ∼2.2 MK. We present images taken at two different times: the top image shows the solar disk as viewed by SoHO at the time of the SUMER observations, while the bottom figure shows the emission from the solar disk a week earlier, so that the region containing the plane of the sky during the SUMER observation can be viewed close to disk center. As can be seen, the large-scale features of the corona, such as coronal hole boundaries and active region locations, are reproduced by the simulation.

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In order to perform 3D line-of-sight analysis, we begin by extracting model results, such as the electron and proton densities, and temperatures, as well as the Alfvén wave energy density, along the line of sight to the SUMER observational slit. The geometry of the problem is illustrated in Figure 4, where the SUMER line of sight for the entire slit width is traced within the three-dimensional space of the model solution. The figure shows the solar surface, colored by the radial magnetic field magnitude, the horizontal plane containing the SUMER slit colored by the electron density, and the plane of the sky for the time of SUMER observations.

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Using the model results and the CHIANTI database, we calculated the spectral flux line-of-sight integral according to Equation (20) for each of the lines in Table 2 at each of the 30 radial sections of the SUMER slit. The resulting spectra are compared to the observed spectra in Figures 5–9. The left panel in each figure shows a contour plot of the synthetic and observed line spectra at all heights covered by the SUMER slit. The middle panel compares the line profile in absolute units at two different distances above the limb: 1.04 R_⊙ and 1.14 R_⊙. The blue symbols and error bars show the observed flux and the associated uncertainty, which takes into account a calibration error of 20% for SUMER detector B (Wilhelm 2006), and the statistical error in the photon count. The blue curve shows the fit to a Gaussian of the measured flux. The red curve shows the model result. On the right, we show the normalized line profile in each of these heights, using the same color coding as before. The normalized line profile allows us to examine the accuracy of the model prediction of the line width, independent of the absolute value of the predicted total flux. The first thing to notice is that for all lines, the observed and predicted line widths are in good agreement at both heights. These results imply that the combination of thermal and non-thermal motions predicted by the AWSoM model is accurate. The predicted and observed spectral line fluxes are in good agreement for the Mg ix and Na ix ions, while the model underpredicts their magnitude in the S x, Fe xii, and Ne viii ions. We discuss possible causes of these discrepancies in Section 6.3.

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The total flux predicted by the model depends on the distribution of electron density and temperature along the line of sight. In turn, the radial profiles of the electron density and temperature depend on the heating rate, which in our case is a result of turbulent dissipation of Alfvén waves. Thus, comparing the radial profiles of the total flux to the observations allows us to verify that the large-scale distribution of heating rates predicted by the model gives realistic results.

Figure 10 shows a comparison of the radial profiles of the total flux for all of the lines listed in Table 2. The left panels display the predicted and observed total flux, F_tot, at all heights covered by the SUMER slit. The panels on the right side of Figure 10 display the ratio between the observed and predicted total line fluxes as a measure to determine the agreement or disagreement between model and observations. The discrepancies between the model and the observations seem to decrease with radial distance, as all ions show agreement above 1.2 solar radii. However, this decrease is due in part to the increase with height of the uncertainties of the observed fluxes. The regions shaded by an orange color correspond to the height where the error in the measured flux is larger than the measured value itself. For these cases, the ratio between predicted and observed total flux becomes meaningless, and these points are excluded from the ratio calculation. The regions shaded in blue correspond to the height above the limb where the scattered light contribution may reach up to 50% of the observed line flux, as discussed in Section 4.2. These heights are summarized in the third column of Table 2. We next discuss the results for the separate lines in more detail.

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Mg ix and Na ix—The successful comparison for Mg ix and Na ix is very important. Since no atomic physics problems were expected for these lines (see Section 5.2), the agreement indicates that the overall temperature and density distributions predicted by the AWSoM model along the line of sight are realistic, although line-of-sight effects might compensate for local inaccuracies.

Fe xii—The total flux of the Fe xii line is underestimated, but it is important to note that the factor of 2 to 3 discrepancy we find is similar to the underestimation we expected from this line (see Section 5.2) so that the disagreement could be largely due to atomic data inaccuracies.

Ne viii and S x—The synthetic fluxes for Ne viii and S x are underestimated by a factor of ≈1.5 and ≈2, respectively. The S x line is density sensitive, and its contribution function , defined by Equation (15), decreases as the electron density increases beyond  cm⁻³: the underestimation might be stemming from an overestimation of the AWSoM-predicted electron density along the line of sight. However, no such effect is expected for Ne viii; thus, the underestimation of the flux must stem from a different reason and is most likely due to the uncertainties in the abundances of these two elements. In fact, the coronal abundances of S and Ne are uncertain, as they are affected by the still not-understood fractionation processes known as the “FIP effect” (Feldman & Laming 2000 and references therein). This effect consists in an enhancement of the coronal abundance ratio of elements with low (<10 eV) First Ionization Potential (FIP) to those with high (>10 eV) FIP over their photospheric values. Ne is a very high-FIP element (FIP = 21.6 eV) while S (FIP = 10.4 eV) sits at the threshold between the two classes of elements. The FIP bias of S is ill-determined: for example, the bias reported in Feldman et al. (1992), used in the present work, set it at 1.15, while Feldman et al. (1998) found a value between 1.2 and 2.0, which would solve this discrepancy. Furthermore, models of the FIP effect predict that S behaves as a high-FIP element in closed field lines in the corona, but exhibits enhancement in coronal holes and the fast solar wind (Laming 2015). The abundance of neon is even more uncertain as its absolute value cannot be measured in the photosphere and is usually inferred through the Ne/O abundance ratio. In turn, the coronal value of the Ne/O ratio itself is disputed: it is usually assumed to be  0.17 (Young 2005) but measurements carried out in the solar wind and in coronal holes have found that the coronal Ne/O is  0.1 (von Steiger et al. 2000; Landi & Testa 2015), is dependent on the solar cycle (Shearer et al. 2014; Landi & Testa 2015), and is much smaller than in the photosphere (Drake & Testa 2005). An additional complication comes from the fact that the value of the oxygen abundance in the photosphere and the corona is disputed as well. Thus, while we note that the coronal hole measurements of Landi & Testa (2015) would decrease the predicted intensities further and deepen the disagreement, we feel that the uncertainties in the neon coronal abundance are sufficiently large to account for the discrepancy until better determinations of that value are provided.

The comparison of the radial variation of the line width in the synthetic spectra to that found in the observations allows us to determine how well the predicted plasma and wave properties are able to account for the observed line broadening in the inner (1.04–1.34 R_⊙) part of the equatorial solar atmosphere.

Figure 11 compares the radial profiles of the synthetic and observed line widths for each of the spectral lines in Table 2. The regions where the scattered light flux may contribute up to 50% to the line flux are shaded in blue. These radial distances are reported in Table 2. The panels on the left-hand side show the model and observed width cast in units of speed using Width (km s⁻¹) = , where c is the speed of light in km s⁻¹. This quantity is often referred to as the effective speed. The blue curve with error bars shows the observations, while the red dashed line shows the model results. In order to examine the relative contribution from the thermal and non-thermal speeds, we repeated the calculation of the line widths while ignoring the non-thermal speed as a line-broadening mechanism. The results are shown as the green curves on the left panels. The panels on the right-hand side show the ratio of the observed to synthetic line width (blue curve). The solid black line denotes a ratio of 1, i.e., a perfect agreement. The first thing we note is that the ratios for all lines are all very close to unity, with a discrepancy of less than 10% at most heights. This implies that the combination of ion temperatures and non-thermal speeds predicted by AWSoM can produce synthetic line widths whose magnitudes are very close to the observed ones, at least in the case of the lower equatorial corona. As in the case of the total flux comparisons, we note that line-of-sight effects may compensate for any local inaccuracy in the AWSoM prediction. The removal of the non-thermal speed from the calculation of the synthetic profiles greatly reduces the agreement between the model and the observations. This implies that the non-thermal motions induced by the waves are necessary for predicting line widths that are consistent with observations. While the line width due to thermal motions alone does not change considerably with radial distance, the total line width which includes the wave-induced motions shows a clear radial dependence. This dependence is due in part to the effects of the magnetic topology, as we will discuss in Section 6.6. This comparison also sheds some light on the validity of our assumption that all the ions have the same temperature. The reasoning behind this assumption is that in the lower corona, where the density is still very high, collisions will allow the different ions to achieve thermal equilibrium. Since the spectral lines considered here are emitted by different elements, the thermal contribution to the line width is different for each of them, while the non-thermal contribution is the same. The simultaneous agreement of the predicted and observed widths for several ions make it less probable that their temperatures do in fact differ from one another. We note that the agreement between the synthetic and observed line widths decreases as the height above the limb increases for the case of Fe xii. This discrepancy may be due to the uncertainty in the observations, but it is also possible that our assumption that this ion, which has the largest mass of all of the ions considered here, also has the same temperature as the protons breaks down at higher altitudes, where the density has already fallen off considerably and the plasma becomes collisionless.

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In the previous sections, we showed that the modeled wave amplitude is consistent with the observed line widths of several different ions, suggesting that the model correctly predicts the amount of wave energy propagating in the corona. To complete this discussion, we wish to verify that the observed coronal heating rate, which depends on the wave dissipation rate, is also reproduced. Since the heating rate impacts the electron density and temperature, comparing the modeled and measured electron properties along the SUMER slit serves as an independent check of the dissipation mechanism assumed in the model. Oran et al. (2013) found that the AWSoM model’s prediction of the electron properties in a polar coronal hole during solar minimum was in good agreement with measurements. The simple geometry of the coronal hole allowed the authors to compare the line-of-sight measurements to model results extracted along the coronal hole axis. However, in the present case of observations of the equatorial quiet corona, which exhibits a more complex magnetic topology, it becomes less clear which region along the line of sight should be compared to the measurements. We therefore adopt a more detailed approach, one that takes into account the variable emission from the different magnetic structures crossing the line of sight.

6.5.1. Overcoming Line-of-sight Effects: 3D Emission Analysis

6.5.2. Region of Maximum Emission for Electron Density Measurements

The electron density along the SUMER slit was obtained from the line flux ratio of the S x 1196 Å and S x 1212 Å lines. Figure 12 shows the relative contribution of each location along the line of sight to the total emission, calculated using the AWSoM results and the CHIANTI database. The left and right panels correspond to 1196 and 1212 lines, respectively. In each of the panels, the horizontal axes correspond to the radial distance for the 30 bins along the slit. The vertical axes represent the distance from the plane of the sky, measured in units of solar radii and ranging from −1 to 1. The regions outside this range have no clear emission pattern and are omitted for clarity. The top row shows the fractional contribution to the total emission along each point along the lines of sight. The bottom panels show the cumulative contribution to the normalized line-of-sight integral of the emission for these lines. The color scale ranges from 0 to 1 (corresponding to the two edges of the line of sight, i.e., the normalized integral equals 1 at the far end of the line of sight). It can be seen from the top panels of both lines that the largest relative contribution comes from a narrow region around the plane of the sky. Below r = 1.15 R_⊙, there is a significant contribution coming from an additional region behind the central region. We have found that ∼24% of the total emission of both lines comes from a region that is less than 0.2 R_⊙ wide, marked by the black and purple curves. The black curves show the bounds of the 24% region for S x 1196 Å, while the purple curves show the same for S x 1212 Å . Since the two regions more or less overlap, the line flux ratio calculated by integrating the line fluxes only over this region will be equal to the line flux ratio when the integration is over the entire line of sight. Therefore, the density modeled in this region is suitable for comparison with the density measurement using the S x 1196/1212 line flux ratio. It should be emphasized that this comparison is consistent, even though most of the total emission comes from outside this region. This is due to the line-ratio diagnostics. If another diagnostic method is used, the analysis should be different. We also note that the region we highlighted does not account for most of the total line-of-sight emission, but its relative contribution per unit length is much higher than other regions, and it possesses the most distinct emission pattern along the line of sight, while the relative contribution from the other regions is much smaller and structureless.

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We next locate this region in the model’s three-dimensional magnetic topology. Figure 13 shows the MHD solution in an equatorial plane. The color contours show the radial speed while the black curves show the magnetic field. (Both the velocity and magnetic field are readily simulated by the model, which is based on the MHD equations). The boundaries of the 24% region for S x 1196 Å are marked by the white squares (corresponding to the purple curves in Figure 12). Interestingly enough, we see that a large part of the emission is coming from a distinct magnetic structure of a pseudo-streamer, i.e., a loop structure topped by open field lines of a single polarity. The flow speed above the streamer is slower than that in the surrounding regions.

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6.5.3. Region of Maximum Emission for Electron Temperature Measurements

The electron temperature along the SUMER slit was obtained from the line flux ratio of Mg ix 706 Å and Mg ix 749 Å. As for the S x line pair, we wish to verify that both lines give a similar relative contribution to the line-of-sight emission in the same region. The cumulative contribution along the line of sight is shown in Figure 14. The overlaid curves represent the region where the relative contributions of the two lines are similar, and account for 36% of the total line-of-sight emission. The black and purple curves correspond to the 706 Å and the 749 Å lines, respectively. As can be seen, these regions almost entirely overlap. Calculating the temperature from the observed line flux ratio also requires us to know the electron density, which we take from the measurement discussed in the previous section. We therefore wish to compare the location of the region of equal contribution of the Mg ix lines to the region of equal contribution of the S x lines, i.e., the pseudo-streamer region selected in the previous section. The comparison is shown in Figure 15. The panels show the fractional contribution for Mg ix 749 Å (left) and for S x 1212 Å (right). The purple curves represent the region of equal contribution of the S x line pair (as in Figures 12 and 13), while the black curves represent the region of equal contribution of the Mg ix line pair (as in Figure 14). As can be seen, the spatial distributions of the emission are quite different, mostly at low altitudes. The regions of equal contribution more or less overlap above a heliocentric distance of 1.15 R_⊙. We therefore restrict the comparison of the measured and predicted electron temperatures to these altitudes only, where we can safely assume that the density and temperature observations apply to the same region. Examining Figure 13, we can see that this altitude corresponds to the purely open field line region of the pseudo-streamer, while at lower altitudes the lines of sight intersect both open and closed field line structures.

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6.5.4. Electron Density and Temperature in a Pseudo-streamer

We located a distinct and narrow region that accounts for equal parts of the total fluxes of the lines used in the electron density and temperature measurements. For each line of sight, we average the predicted quantity over the segment bounded by the white squares in Figure 13 to obtain a radial profile along the SUMER slit.

Figure 16 shows the comparison of the predicted electron density in the pseudo-streamer with the SUMER measurement. The blue curve with the error bars shows the measured electron density while the dashed red line shows the model results. The error bars in the measured density is a result of the error in the measurement of the line fluxes participating in the line-ratio calculation. The line flux errors were discussed in Section 6.2. The error bars in the model indicate the minimum and maximum electron densities found along the line-of-sight segments over which we take the average. The shaded region represents the altitude where the observed flux of the lines used for this measurement has decreased to below twice the scattered light flux, making the measurement less reliable at these heights. As can be seen, the modeled density falls inside the measured ranges at all altitudes, although the uncertainty in the electron density measurement is quite large.

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The predicted electron temperature along the SUMER slit and its comparison to observations is shown in Figure 17. The color coding, as well as the role of the error bars, is the same as in Figure 16. The comparison starts at r = 1.15 R_⊙ since below that height the lines used in the temperature measurement are not emitted from the same region as the lines used for the density measurement. The shaded region corresponds to the altitudes where the observed flux of the lines used for this measurement has decreased to below twice the scattered light flux, making the measurement less reliable at these heights. The measured temperature exhibits large uncertainties and variations with height, with no clear radial trend. The predicted electron temperature falls within the range of observed values, suggesting that the heating supplied by the heating mechanism is sufficient to achieve the observed coronal temperatures in the quiet corona.

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The three-dimensional, magnetogram-driven solution allows us to study not only the synthetic line-of-sight line width, but also the variation of the wave amplitude along selected field lines. We recall that the line width observed from a particular direction depends on both the wave energy and the magnetic topology, as is clear from Equation (2). Examining Figure 13, we can see that the magnetic field in the region of largest emission is composed of a closed loop structure up to a radial distance of ∼1.1 R_⊙, above which all field lines are open. In the closed loop region, the magnetic field direction changes from approximately perpendicular to parallel to the line of sight. Thus, while the wave amplitude is expected to increase with height in this region (due to the sharp decrease in the plasma density), its fraction that lies along the SUMER line of sight will decrease. Above the closed loop structure, the magnetic field direction is very close to perpendicular to the line of sight, and thus a larger share of the wave-induced motions will contribute to the line width. This dependence on the line of sight and magnetic field geometry is illustrated in the radial variation of the line widths in Figure 11, where an evident change in the synthetic line widths of all ions occurs around r = 1.1 R_⊙.

In order to study the actual variation of the wave amplitude, we extracted the model results along three open field lines inside the region of largest emission. This will enable us to remove the effects of the line-of-sight geometry and directly study the wave dissipation taking place in this region. We calculate the rms of the wave velocity amplitude, , using Equation (12). Hassler et al. (1990) and Moran (2001) have shown that if no wave damping is taking place, the rms wave amplitude would vary as as a result of energy conservation along a magnetic flux tube. Thus, we would expect the rms wave amplitude predicted by the model to be lower than the undamped values. The results are shown in Figure 18. The location of the selected field lines is shown in the inset. Line 1, colored in blue, is an open field line on the edge of the pseudo-streamer, while line 3, colored in red, is the first open field line straddling the closed loop structure. Line 2, colored in green, lies in between the other two lines. The solid curves show the wave amplitude as a function of the path length S along each of the field lines, while the dashed curves show the hypothetical curves for the wave amplitude assuming no dissipation, normalized to the value of the corresponding modeled curve at S = 0.05 R_⊙. As expected, the rms wave amplitude sharply increases close to the inner boundary due to the sharp drop in density. Departures of the simulated wave amplitude from the curve of undissipated waves become prominent above R_⊙, although each of the field lines exhibit a different dissipation rate. It is interesting to compare the curves for dissipated and undissipated waves to a similar analysis presented in Hahn et al. (2012) for a polar coronal hole. In Figure 5 therein, the observed effective speeds of several emission lines are compared to undamped values. Departures from the undamped wave amplitude curves start above heights of 0.1 R_⊙ and 0.2 R_⊙ above the limb, depending on the ion. Of the three field lines in Figure 18, line 1 most resembles a coronal hole field line, as it is an open line with minimal bending around the closed loop structure. The wave amplitude along line 1 shows very similar behavior to the one reported in Hahn et al. (2012). For line 2 and line 3, the wave amplitude of the dissipated waves shows larger departures from the non-dissipated case, with more wave energy being converted into heat. This is most prominent for line 3, where the wave amplitude is significantly reduced near the tip of the loop structure. In this location, higher dissipation is expected to occur due to the presence of counter-propagating waves, and the first term under the square root in Equation (10) will be taken into account. Above that point, the rms wave amplitude increases at a rate similar to that of line 1, consistent with the fact that the dissipation rate is now dominated by reflections, i.e., the second term under the square root in Equation (10). Line 2 also exhibits a signature of this behavior, although it is less pronounced.

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