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The energy of waves in the photosphere and lower chromosphere: IV. Inversion results of Ca II H spectra

Most static 1D atmosphere models in the quiet Sun predict a rise of the gas temperature at chromospheric layers, but numerical simulations only yield an increase in the brightness temperature. We investigate the thermal structure in the solar chromosphere as derived from an LTE inversion of Ca II H spectra in QS and active regions. We investigate the temperature stratifications on differences between magnetic and field-free regions in the QS, and between QS and ARs. We determine the energy content of individual calcium bright grains (BGs). The rms temperature fluctuations are below 100 K in the photosphere and 200-300 K in the chromosphere. The average temperature stratification in the QS does not exhibit a clear chromospheric temperature rise, opposite to the AR case. We find an energy content of about 7*10E18 J for BGs that repeat with a cadence of about 160 secs. The precursors of BGs have a vertical extent of about 200 km and a horizontal extent of about 1 Mm. The comparison of observed with synthetic NLTE profiles confirms that the solar chromosphere in the QS oscillates between an atmosphere in radiative equilibrium and one with a moderate chromospheric temperature rise. Two-dimensional x-z temperature maps exhibit nearly horizontal canopy-like structures with a few Mm extent around photospheric magnetic field concentrations at a height of about 600 km. The large difference between QS regions and ARs, and the better match of AR and non-LTE reference spectra suggest that magnetic heating processes are more important than commonly assumed. The temperature fluctuations in QS derived by the LTE inversion do not suffice on average to maintain a stationary chromospheric temperature rise. The spatially and vertically resolved information on the temperature structure allows one to investigate in detail the topology and evolution of the thermal structure in the lower solar atmosphere.


Introduction
The thermal structure of the solar chromosphere remains enigmatic and to the greatest extent unexplained more than a century after the first description of the existence of the chromosphere itself (Secchi 1860;Lockyer 1868). Nowadays, two competing views on the thermal structure exist, i.e., a static/stationary and a dynamical view.
The static view is expressed by a long series of onedimensional (1D) thermal stratifications that reproduce observed spectra (e.g., Gingerich et al. 1971;Holweger & Müller 1974;Vernazza et al. 1981;Fontenla et al. 1993Fontenla et al. , 2006, see Rutten & Uitenbroek (2012) and  for additional references). To produce emission in the line cores of chromospheric spectral lines, all these 1D models require a reversal of the temperature gradient with height at some point. Their common feature therefore is the existence of a temperature minimum at a height of about 500 km above the photosphere, where the gas temperature starts to rise towards higher layers (e.g., Vernazza et al. 1976;Ulmschneider et al. 1978;Leenaarts et al. 2011). Such 1D models with a temperature reversal faithfully reproduce observed (average) spectra to the most minute level, but all lack something: they provide a temperature stratification with height in the solar atmosphere, but do not explain the reason for its shape.
The dynamical view of the solar chromosphere is comparably newer than the static 1D models and resulted thanks to the increase in the available computing power during the last two decades. Opposite to the static models that were derived from observations, the dynamical approach usually goes the opposite way and tries to model the solar chromosphere from basic principles, i.e., numerical simulations that encompass the differential equations that govern the temporal evolution of material in the solar atmosphere (Nordlund 1982). One of the advantages of such a forward modeling is that one only needs to decide on what are the relevant physical processes that must be included in the set of differential equations. Whereas numerical simulations of the solar photosphere abound (e.g., Nordlund & Stein 1990;Steiner et al. 1998;Stein & Nordlund 1998;Steiner et al. 2008;Fabbian et al. 2010;Beeck et al. 2012), there are relatively few chromospheric simulations (Rammacher & Ulmschneider 1992;Carlsson & Stein 1997;Wedemeyer et al. 2004;Schaffenberger et al. 2006;Cuntz et al. 2007; Leenaarts et al. 2009).
One reason is the spatial, and especially the vertical extent of the simulation box that has to reach up to a height of about 2 Mm with a sufficiently fine sampling on the order of a few tens of km to encompass the solar chromosphere. A second limitation is caused by the physical conditions in the upper solar atmosphere. The gas density in the solar photosphere is high enough that the assumption of local thermal equilibrium (LTE) is justified. In LTE, the available total energy is distributed over all available degrees of freedom by frequent collisions between particles such that each degree of freedom obtains a fully determined fraction of the total energy. The exact distribution solely depends on the physical laws holding in the material of interest. Assuming LTE, the values of several physical quantities such as population states, thermal motions, etc. are determined instantly as soon as a temperature value is given. Because of the decrease of the gas density with height in the gravitationally stratified solar atmosphere, the LTE condition breaks down when a sufficient number of collisions takes longer than the dynamical evolution of the physical quantities that can change by, e.g., radiative processes (e.g., Carlsson 2007;Wedemeyer-Böhm & Carlsson 2011). A complete treatment of this regime in a non-LTE (NLTE) formalism still exceeds the computing power of even the most recent computer clusters (Auer & Mihalas 1969;Ayres & Wiedemann 1989;Rammacher & Cuntz 1991;Carlsson & Stein 1992Uitenbroek 2000;Asensio Ramos et al. 2003;Leenaarts et al. 2007;Wedemeyer-Böhm & Carlsson 2011;Carlsson & Leenaarts 2012;Martínez-Sykora et al. 2012).
The dynamical numerical simulations of the solar chromosphere revealed a quite distinct thermal structure than the static 1D models. The brightness temperature at chromospheric layers tended to fluctuate strongly, leading to intermittent emission in the synthesized chromospheric spectra. The gas temperature, however, did not show any temperature reversal on a temporal average, contrary to most of the static 1D models (cf. Carlsson & Stein 1995Kalkofen 2001Kalkofen , 2012. Thus, the average brightness and gas temperatures in such simulations differed significantly, while the average synthesized chromospheric line spectra corresponded to neither the average gas nor the average brightness temperature. This puzzling behaviour even raised the question how to define a "mean" temperature at all under this conditions (Rammacher & Cuntz 2005;Uitenbroek & Criscuoli 2011). This point is actually more important than it seems at a first glance because the static 1D models are based on the relation between brightness temperatures (i.e., observed spectra) and gas temperature (i.e., model output). Questioning this relation between gas and brightness temperature, or the definition of temperature therefore converts directly into questioning the results of the static 1D models that provide an excellent fit to observed average spectra (Kalkofen 2012).
There are some common problems for both the static and the dynamical view of the solar chromosphere. One is the width of spectral lines synthesized from the simulations or models, respectively. Observed chromospheric spectral lines are usually found to be significantly broader than the synthetic lines, (far) in excess of the thermal line width for any reasonable temperature range. This behaviour shows up most prominently in offlimb observations (e.g., Beckers 1968;Beck & Rezaei 2011).
Part of this excess in line width can be explained by the spatial resolution of the observations that represent a spatial average over solar structures with different physical properties, and especially different macroscopic mass flows with their related Doppler shifts. But also the average of synthetic spectra from simulations with spatial grid sizes down to a few ten kms lacks the observed line width, which indicates that some physical ingredient is missing in the simulations, or that their spatial sampling -despite being below the photon mean free-path lengthstill is insufficient to capture all of the dynamical evolution of, e.g., small-scale mass flows. The line width in both static and dynamical models is therefore often adjusted ad-hoc by some additional broadening parameters (cf. Allende Asplund et al. 2004, and references therein), usually labeled as macroturbulent and microturbulent velocities that represent spatially unresolved line-broadening mechanisms of unknown origin (e.g., Cram 1981;. A second problem is how to reconcile the temperature stratifications with the observations of molecular lines at chromospheric heights (Ayres & Testerman 1981). This problem affects more the static than the dynamical models, because in the former the location of the temperature reversal at low heights prevents any significant formation of molecules at layers above it (Ayres 1981;Kalkofen et al. 1999;Ayres 2002). A "cool" chromospheric part could partly be included also in static models by "postponing" the chromospheric temperature rise to higher layers (Ayres 2002;Ayres et al. 2006;Fontenla et al. 2007;Rezaei et al. 2008), but it was imposed in some sense as a post-facto crutch. In most numerical simulations, phases with a temperature that is sufficiently low to allow the formation of molecules were usually automatically present (Wedemeyer et al. 2004;Leenaarts et al. 2011).
A third problem is a discrepancy between observations in different spectral lines that yield seemingly contradictory results. Observations with the Solar Ultraviolet Measurement of Emitted Radiation (SUMER) instrument on-board the SOlar and Heliospheric Observatory (SOHO) satellite in ultraviolet (UV) lines revealed that all these lines are in permanent and ubiquitous emission with little intrinsic variation in quiet Sun regions of the solar surface . All these lines are classified as "chromospheric". Contrary to this, chromospheric Ca II lines, especially H and K in the near-UV, show large spatio-temporal fluctuations from reversal-free absorption profiles to moderate emission in the line cores (Liu & Smith 1972;Cram & Dame 1983;Rezaei et al. 2008;Beck et al. 2008). While the temporal evolution is naturally explained in dynamical simulations, persistent emission is only reproduced in static 1D models with a permanent chromospheric temperature rise.
The way to resolve this contradictory behaviour of observed spectral lines and to reconcile the conflicting views of the static and dynamical solar chromospheric models is not clear at present. There is one additional information to be considered: the temperature in the solar photosphere is about 6000 K, while the ion and electron temperature in the solar corona exceeds one million K, where both results are undoubted. Thus, there has to exist a reversal of the temperature gradient at some height, but where exactly is unclear.
Magnetic fields could be a possible mediator between the two conflicting scenarios of the solar chromosphere. Firstly, magnetic fields have not been considered neither in all of the static nor many of the dynamical models (but see also Martínez-Sykora et al. 2012;Khomenko & Collados 2012). Secondly, in quiet Sun (QS) conditions, concentrations of magnetic flux are theoretically expected to spread out from their con-fined diameter in the low photosphere to fill the full volume in the chromosphere. This structure of several expanding flux tubes creates automatically a dividing layer at some chromospheric height by the so-called magnetic canopy that separates field-free from magnetized plasma (Giovanelli 1980;Steiner et al. 1986;Solanki et al. 1991;Štěpán & Trujillo Bueno 2010;Nutto et al. 2012;Holzreuter & Solanki 2012), i.e., that separates a dynamical chromospheric regime below from a stationary one above the canopy (as sketched for instance in Beck et al. 2008, their Fig. 22).
In this contribution, we investigate the spatio-temporal behaviour of the thermal structure in the solar chromosphere as derived from an inversion of Ca ii H spectra assuming LTE (Beck et al. 2013, BE13). This study follows the approach of the static 1D models by deriving a gas temperature from an observed brightness temperature, but uses spatially resolved spectra that trace the dynamical properties of the solar chromosphere -as in the dynamical view of the chromosphere (see also Henriques 2012). We use observations of both QS and active regions (ARs) to investigate differences between these types of solar structures. The observations and the inversion scheme are described in detail in BE13, we thus refer the reader to the latter publication. We will only highlight some observational limitations that are relevant for the current contribution in Sect. 2. We then concentrate on the thermal structure of the chromosphere in QS and ARs in Sect. 3, both on average and spatially resolved. The findings are discussed in Sect. 4. Section 5 provides our conclusions. Appendix A discusses the possible amount of spectral broadening by different mechanisms acting on the observed spectra.

Relevant observational details
We use in total five different observations taken with the POlarimetric LIttrow Spectrograph (POLIS, Beck et al. 2005). POLIS delivered 1 intensity spectra in Ca ii H at 396.85 nm together with spectropolarimetric data of the two Fe i lines at 630 nm. Two data sets were taken in QS regions, i.e., a time series of about one-hour duration and a large-area map (cf. Beck et al. 2009, 2012, BE09 andBE12 in the following). The other three data sets are large-area maps of the active region NOAA 10978 (cf. Beck & Rammacher 2010;Bethge et al. 2012). The parameters most relevant in the present context are the slit width of 0. ′′ 5, the step width in the case of spatial scanning of also 0. ′′ 5, the temporal cadence of the time series of 21 secs, and the spatial sampling along the slit of 0. ′′ 292 in the Ca ii H spectra. These values provide a pixel scale of 0. ′′ 5 × 0. ′′ 292 in the Ca spectra. The data were taken with real-time seeing correction by the Kiepenheuer-Institut adaptive optics system (von der Lühe et al. 2003) and have a seeing-limited spatial resolution of about 1 ′′ , which roughly coincides with the sampling-limited resolution in the scanning direction.
Besides from the study of individual events in the time series (Sect. 3.5), always the average of the statistics in all individual maps is used to expand the statistical base.

Results of LTE inversion
The inversion setup for Ca ii H described in BE13 yields temperature stratifications for spatially resolved spectra. With the inversion results, we now can investigate the statistics of the temperature distributions at different optical depth levels log τ, analogously to BE12, where the intensity distributions at different λ 1 POLIS was officially de-commissioned end of 2010. temperature versus (vs.) optical depth for the full FOV. The relative occurrence of a given temperature is indicated by the colour with the corresponding colour bar at top. Thick black line: original HSRA model. Dashed: average temperature. The vertical dashed lines denote the layers of log τ for which the temperature histograms are shown in the rest of the panels. In the histograms, black/red/blue lines denote the full FOV/magnetic/field-free locations. and the characteristic properties of different spatial samples in the observed field of view (FOV), i.e., field-free or magnetic regions, and the full FOV, were analyzed. The distinction between field-free and magnetic sample is made using the simultaneous polarimetric observations in the 630 nm channel of POLIS by setting a threshold in the observed polarization signal (cf. BE12).

Average temperature and temperature statistics in QS
Following the approach of BE12, we calculated the characteristics of the temperature distributions up to second order (mean, root-mean-squared (rms) fluctuation, skewness). Figure 1 shows some of the statistics of temperature versus (vs.) optical depth for the maps taken in the QS. The upper left panel shows the relative occurrence (denoted by the colour bar) of temperatures in the full FOV for each optical depth level. The three samples in the FOV (field-free, magnetic, full FOV) differed only slightly in this way of presentation, therefore only the latter is shown. The remaining panels show the histograms of temperature for all spatial samples at three optical depth levels of log τ = −0.6, −2.6, and −4.6, respectively. The magnetic locations (blue lines) show a weak preference for slightly higher temperatures compared to the full FOV or the field-free locations. The average temperatures of all three samples do not show an indication of a chromospheric temperature rise as predicted, e.g., by the original Harvard Smithsonian Reference Atmosphere model (HSRA, Gingerich et al. 1971), e.g., compare the black solid line (HSRA) and black dotted line (< T > in the full FOV) in the upper left panel of Fig. 1.
The lower panel of Fig. 2 shows the variation of the rms fluctuations of temperature with optical depth and geometrical height. For the conversion from one to the other we used the tabulated relation given in the HSRA atmosphere model, opposite to a derivation of the height scale from the inversion results themselves as in Puschmann et al. (2005Puschmann et al. ( , 2010. We note that in the derivation of the temperature statistics always the individual temperature stratifications vs. optical depth were used, and that the rms value will (slightly) underestimate the characteristic range of fluctuations in case of asymmetric distributions (cf. Fig. 1). The geometrical heights in Fig. 2 only indicate a probable value for the average location of the formation heights. As in the analysis of the intensity statistics of Ca ii H and Ca ii IR spectra in BE12, we find that the rms fluctuations increase nearly monotonically in geometrical height, with a minimum of fluctuations near z ∼ 100 km (corresponding to log τ ∼ −1) caused by the temperature inversion of granules and inter-granular lanes (cf. Puschmann et al. 2003Puschmann et al. , 2005. The fluctuations stay below 100 K up to 500 km height, and reach only 200 -300 K at 1000 km. The magnetic regions show slightly higher rms fluctuations at all heights and pass through the minimum of rms fluctuations lower in the atmosphere than the other two samples. The latter behaviour is presumably caused by the shift of the optical depth scale on magnetic locations. The maximal temperature variations (dotted lines in the lower panel of Fig. 2) reach about 1000 K at log τ ∼ −4 (z ∼ 700 km).
BE12 found a pronounced difference in the skewness of intensity distributions between field-free and magnetic locations, i.e., an increased skewness on magnetic locations at intermediate wavelengths between the line wing and the core, corresponding to formation heights of about z = 135 -150 km. The upper panel of Fig. 2 shows that this also happens at about the same height in the case of the temperature retrieved by the inversion. A positive skewness indicates that temperature excursions towards values above the average are more frequent than those towards values below it. The skewness on magnetic locations increases strongly already at lower optical depths of about log τ ∼ −4 when compared with the other two samples. This could again be due to a shift of the optical depth scale on magnetic locations, but could also indicate that propagating waves turn into shock fronts at lower heights in the rarefied medium of magnetic flux concentrations than in the field-free surroundings.

Average temperature and temperature statistics in AR
One could argue that with the list of assumptions used in the inversion (LTE, unmodified HSRA density, lack of hydrostatic equilibrium, complete frequency redistribution), it would be impossible to obtain a chromospheric temperature rise in the inversion of spectra because the (partial) decoupling between temperature and emitted intensity that exists in NLTE would be needed. Figure 3 shows that this is not the case, but that a chromospheric temperature rise results also in LTE if input spectra different to those in QS are analyzed. Both the average temperature and the distribution of temperatures in the AR data cover the temperature rise in the original HSRA model (Fig. 3). The offset of the temperature stratifications in the optical depth scale -the HSRA temperature starts to rise at a larger height (log τ = −4) than the average AR temperature retrieved from the inversion (log τ ∼ −3) -is presumably caused by the off-centre location of the AR (θ = 50 • ) and the related shift of the optical depth scale because of the inclined line of sight (LOS). We remark that the drop of temperature at the uppermost optical depth levels (log τ < −5.5) is purely artificial. It results from the need of having a reduction of temperature at the uppermost atmosphere layers for generating double reversals in the Ca line core when using the LTE approximation. The rms fluctuation of the temperature in one of the AR maps is over-plotted in the bottom panel of Fig. 2. It reflects the intrinsically larger spatial variation of temperature in an AR compared to that in QS, but we note that the range goes up to beyond 1000 K rms, implying even larger maximal variations. Temperature fluctuations of this size are large enough to rise the temperature from an atmosphere in radiative equilibrium with about 4000 K in the chromosphere to the temperature values of commonly used atmospheric models such as the HSRA.

Correlation between intensity and temperature
In our inversion approach, each wavelength λ is attributed a corresponding value in τ, i.e., to adjust the intensity I(λ), the corresponding temperature T (τ) should be varied. The iterative inversion process, however, does not only modify the temperature at the given value of τ, but uses a fourth order polynomial for

Fig. 4. Correlation coefficient between intensity at a given wavelength I(λ) and temperature T (log τ).
modifying the full temperature stratification. In addition, when the radiative transfer equation through the model atmosphere is solved in the spectral synthesis, the temperature values of some extended optical depth range contribute to the intensity at one given wavelength. Therefore, the one-to-one correlation between a given wavelength and its attributed optical depth is partially broken up. Additionally, some solar surface structures such as (vertical) magnetic fields leave a signature in intensities, and hence temperature over some extended height/optical depth range.
To determine which layer in log τ corresponds to the intensity at a given wavelength, we calculated the linear correlation coefficient C (cf. Beck & Rammacher 2010) between intensity vs. wavelength and temperature vs. log τ. Figure 4 shows the result for the large-area scan on disc centre. The correlation coefficient between intensity and temperature is similar to the intensity contribution function (e.g., Fig. 1 of BE09) in the line wing up to about 396.65 nm, with a single maximum of correlation between log τ ≈ −0.5 to −1.5 that slowly moves to upper layers for wavelengths closer to the line core. Between 396.65 nm and 396.8 nm, the correlation coefficient shows a second, slightly weaker local maximum of correlation at about log τ ≈ −3.5. In the line-core region from 396.81 nm to 396.89 nm, two clear local maxima of correlation exist, where the upper one above log τ ≈ −4 corresponds to the intensity contribution function, whereas the origin of the lower one at log τ ≈ −1 is not clear at once. Figure 5 reveals that the maximum of correlation between line-core intensity and temperature at low atmosphere layers is most likely caused by locations with large photospheric magnetic flux. A comparison between the line-core intensity I core (lower left panel in Fig. 5) as the most prominent tracer of photospheric magnetic flux and the temperature at optical depths of log τ = −0.9, −2.5, and −5.4 shows that the photospheric magnetic network can be identified at both log τ = −0.9 and −5.4 (correlation coefficient with I core of 0.50 and 0.85, respectively), but is completely inconspicuous at log τ = −2.5 (C = 0.08). This implies that vertical or only weakly inclined magnetic fields -characteristic of the magnetic network and isolated magnetic flux concentrations because of the buoyancy force -leave prominent signatures in temperature in at least two height ranges, i.e., at upper atmospheric layers above log τ = −4, and at lower atmospheric layers at about log τ ≈ −1. This is thus most likely the reason for the increased skewness in the magnetic sample at log τ ≈ −1.2 (z ∼ 150 km) found in the previous section and in BE12: magnetic flux concentrations and their immediate surroundings show up brighter than the quiet inter-network (IN) at intermediate wavelengths, and hence have a slightly enhanced temperature at the corresponding atmospheric layers (cf. Sheminova et al. 2005;Sheminova 2012). This, however, does not explain why the intensity (temperature) is enhanced only at these wavelengths (heights), but not in between. To estimate the vertical energy flux, we determined the evolution of the energy content with height assuming that the energy balance is effected in the vertical direction in a gravitationally stratified atmosphere with radial energy transport at least up to the photosphere with its convective granulation. To separate the dynamical part of the energy balance from any possible static background caused by radiative equilibrium with the incoming photospheric continuum radiation, we converted the rms fluctuations of the temperature ∆T rms (τ) -instead of using the modulus of the temperature T (τ) -to the equivalent increase of internal energy using where R = 8.31 J mol −1 K −1 , µ = 1.3 g mol −1 , and γ = 5/3 are the gas constant, the specific mass, and the adiabatic coefficient, respectively (cf. Eq. (4) of BE09). The tabulated values of the gas density ρ gas in the HSRA atmosphere model were used. With the relation between τ and geometrical height z tabulated in the HSRA model, this yields the (stationary) average variation of energy with height (top panel of Fig. 6). The resulting volume energy density E/V can be compared with the result of the first two papers of this series. The energy density derived from the direct conversion of intensity fluctuations to temperature (BE12) lies slightly above the one derived from the temperature fluctuations in the inversion. The reason is that in the inversion the variation of intensity is implicitly converted to a temperature fluctuation over some optical depth range, whereas in the direct conversion from intensity to temperature all variation is attributed to a single optical depth point. The latter approach therefore leads to larger rms fluctuations than in the case of considering the radiative transport through the atmosphere. The energy density corresponding to the velocity rms (BE09) lies below that derived from intensity fluctuations by direct conversion or by inversion up to z ∼ 1 Mm, but exceeds the others above that height. We note that all temperature fluctuations -when using ∆T without applying Eq. (1)from either velocity, intensity, or the LTE inversion fall short of the energy required to lift an atmosphere in radiative equilibrium to the HSRA or any similar atmospheric model (∼ 1000 K needed at z ∼ 1 Mm, but ∆T is about 500 K at maximum, e.g., Fig. 2). If one assumes that the energy related to the temperature fluctuations is transported vertically by (acoustic) waves moving with a speed v ac of about 6 km s −1 (cf. Sect. 3.5.5 later on), the generic chromospheric energy requirement of about 4 kW m 4 of Vernazza et al. (1976) corresponds to about 0.67 J m −3 (horizontal dotted line in the top panel of Fig.6). The atmospheric energy content from all three approaches (velocity, intensity, LTE inversion) intersects with this requirement at a height of about 750 km, but drops rapidly below this value at higher atmospheric layers. The ratio of the generic energy flux of 4 kW m −2 and the energy density multiplied with a constant sound speed of 6 km s −1 is about ten at z ∼ 1000 km and increases rapidly to a factor of several hundreds at layers above.
The rapid drop of the energy density with height is caused by the exponentially decreasing gas density ρ gas that is overplotted as well in the top panel of Fig. 6. The increase in rms fluctuations with height is not able to compensate the decrease in ρ gas . We used two different gas densities for test purposes, on the one hand the tabulated values of the HSRA, ρ ′ gas , and on the other hand the gas density ρ in the modified HSRA model that uses a linear extrapolation above log τ = −4. The differences between these two gas densities, however, do not significantly change the resulting energy density (compare the black solid and black dashed curve in the top panel of Fig. 6). There is, however, a height range from about z = 100 km to 250 km, where the energy of the temperature fluctuations from the LTE inversion stays roughly constant while the density reduces by a factor of two in the same height interval, similar to the height range of constant energy found for individual bright grains (see Fig. 10 later on).

(Radiative) energy losses
The derivative of the energy content, again multiplied with v ac , gives the energy losses with height (middle panel of Fig. 6). The region of roughly constant energy density can again be identified at about z ∼ 200 km. Integrating the area of the energy gradient for all heights above 750 km that can be considered to contribute to the radiative losses in the Ca line core, i.e., the "chromosphere", yields a value of about 3.5 kW m −2 , which would match the generic chromospheric energy requirement. This would, however, imply that no energy is left anymore for heating the layers above such as the corona since this amount of energy is assumed to be radiated away in all of the chromospheric spectral lines.
For comparison with the gradient of energy losses with height, we used the value of chromospheric radiative losses derived from a set of numerical simulations by Carlsson & Leenaarts (2012, CL12). Their Figs. 15 and 16 yield an average cooling rate of about 100 W g −1 in the chromosphere that can be converted to losses in energy density by a multiplication with the gas density as a function of height. The resulting energy gradient with height using a constant value of 100 W g −1 (red line in the middle panel of Fig. 6) is larger than our estimate (black line in the middle panel of Fig. 6) by a factor of about 2 -3 between z ∼ 200 -300 km, and nearly identical above z ∼ 800 km. This difference might be caused to some extent by the fact that we only considered rms fluctuations of temperature in the energy estimate, whereas the value of CL12 refers to all radiative losses, i.e., losses related to the modulus of temperature itself, not its variation. For a final comparison with CL12, we determined the energy loss rate per mass by dividing our energy gradient with height by the mass density (bottom panel in Fig. 6, compare with Fig. 16 of CL12). The global trend of an increase in radiative losses from the photosphere towards the chromosphere is identical in both estimates with the prediction of a maximum at about 1 Mm. Less pronounced local maxima at z ∼ 800 and 1300 km appear in both estimates, albeit in the case of our estimate their significance is doubtful. In total, a roughly constant value of radiative losses of about 100 -200 W g −1 in the lower chromosphere fits to both our estimate and the one of CL12.  Figure 7 shows the residual energy left at a given height when some amount of energy is dumped into the atmosphere at a height of z = 240 km. To clearly visualize the effect of the energy losses with height, we provide both a linear and a logarithmic plot. The black curve corresponds to our result for the height variation of the internal energy density shown in the upper left panel of Fig. 6, but normalized to the energy density at z = 240 km. With the strong gradient of energy with height, only 5 % of the initial energy are left at a height of 500 km, similar to the reduction of wave energy by 80 % at 400 km found in Cuntz et al. (2007). For comparison, we also overplotted the relative energy fluxes in the numerical simulations of acoustic wave propagation of Fawzy et al. (2012, FA12, first and last column of their Table 3). Their wave energy at 600 km was normalized to the corresponding fraction of our curve at that height. We also read off the value of the energy fluxes for acoustic waves and gravity waves from Straus et al. (2008, ST08;their Fig. 3), both normalized to 100 % at z = 250 km. As last reference, the acoustic energy flux in a wave simulation inside a sunspot umbra from Felipe et al. (2011, their Fig. 16) is also overplotted.

Residual energy content
A comparison of the various curves shows that our result on the variation of the energy density with height, which corresponds to the rms fluctuations of brightness temperature in the QS, matches best with all those derived for acoustic waves in an only weakly magnetized plasma (case a of ST08 and case b of FA12). This implies that any gravity waves present in the atmosphere should have no signature in intensity spectra, and hence temperature, otherwise the inversion would have had to retrieve a significantly different slope of the energy density with height as given by case b of ST08.

Individual bright grains in the quiet Sun
Bright grains (BGs, cf. Rutten & Uitenbroek 1991) are one of the most prominent features in observations of the chromospheric Ca ii lines, regardless whether two-dimensional (2D) imaging data or spectroscopic data are used. They are transient, strong intensity brightenings in chromospheric layers that appear with a random spatial pattern, but a characteristic period of about three minutes. They are clearly related to some sort of energy transport towards and energy increase in the solar chromosphere (e.g., Carlsson & Stein 1997, CS97 in the following; BE08). CS97 studied the purely hydrodynamical case of BGs in numerical simulations and were able to show the photospheric origin of these features.
Following the general approach used throughout our series of papers, we will treat all BG events in our data in the same way without trying to distinguish between a possible acoustic or magnetic origin, which would be possible using the spectropolarimetric data in the photospheric Fe i lines at 630 nm. The simultaneous polarisation measurements (cf. BE08, their Fig. 4) showed a stable magnetic environment without clear indications of neither magnetic flux emergence (e.g., Gömöry et al. 2010) nor flux cancellation events (e.g., Bellot Rubio . To estimate the energy transport related to BGs, we studied the occurrence rate, spatial extension, and energy content of individual BGs for an extrapolation of their total energy contribution to the chromospheric energy balance. We used the data of the time series taken at the disc centre (observation No. 2 in BE12) for that purpose. We define BGs as any temporally isolated temperature enhancement in the lower solar atmosphere, i.e., also below chromospheric layers.

Temporal frequency
To derive the characteristic occurrence frequency of BGs, we analyzed the temporal evolution of the temperature on the existence of local extrema, i.e., local temperature maxima. The top panel of Fig. 8 shows the temporal evolution of the temperature excursions around the mean value at log τ = −3.6 for one pixel along the slit as an example. The temporal evolution of the temperature is similar to the observed intensity at λ = 396.83 nm, which roughly forms at that optical depth (red dash-dotted line). We then applied a method developed for finding polarisation lobes or local intensity reversals in line profiles (e.g., Beck 2006;Rezaei et al. 2008) to all pixels along the slit and all layers in log τ to find the locations of local maxima in the temperature excursions. To be counted as a local maxima, the value at a given instant of time has to be larger than the values on the two time steps before and after, i.e., it has to be the largest value in a range of ±40 secs. With this approach, the influence of noise on the determination is reduced. Given the typical duration and cadence of BGs (cf. Figs. 10 and 13 later on), ±40 secs is a characteristic time span in which usually only a single BG happens. The temperature maps were filtered beforehand for all variations with periods above 7.5 mins to suppress the long-term variation caused by the granular evolution. We implicitly identify in the following local temperature maxima with BGs.
Along the slit, the number of maxima was comparable for all pixels, regardless if they were related to photospheric magnetic fields or not (middle panel of Fig. 8). We find only a weak depen- The lack of variation with optical depth may seem puzzling at a first glance, given that usually solar oscillations are separated into photospheric 5-min and chromospheric 3-min oscillations, but ascribing any period to not necessarily repeated events can also be misleading. For instance, the three BGs traced in Figs. 10 and 13 later on stay three events throughout all optical depth layers (see also Kalkofen et al. 2010;Stangalini & Moretti 2012).
The number of 20 events in 3150 secs yields an occurrence rate of about 0.006 sec −1 , which is comparable to the values given in Tritschler et al. (2007, their Fig. 7, 0.0075 sec −1 at maximum). We note that our rate is an upper limit for the occurrence of BGs because the simple identification method by local temperature maxima presumably has also counted some peaks that could result rather from observational effects such as residual image jitter than from physical processes in the solar atmo-time → sphere (cf. the top panel of Fig. 8 at t ∼ 40 min). We repeated the determination of locations of local temperature maxima with a width of ±4 time steps (≡ ±80 secs) or after applying a running mean of five time steps width, which both yielded only about 15 maxima on average. The dependence of the number of maxima on optical depth did not change by the additional requirement/smoothing. In case of analysing the intensity variation with time for local maxima in a ±40 secs range (top panel of Fig. 8), three additional local maxima are found in comparison to the temperature curve. Thus, the most probable range for the number of temporally local intensity or temperature maxima, and hence BGs is from 15 to 22 in 3150 secs (0.005 -0.007 sec −1 ).
The area fraction of BGs, i.e., the number of pixels along the slit that exhibit a temperature maximum at a given moment of time relative to the total number of pixels, is about 10 %. A similar estimate can be made from Fig. 9 that shows about one BG of 1 Mm diameter in each 2D FOV of about 1 Mm × 10 Mm.

Horizontal spatial extent
To estimate the spatial extent of individual BGs, we used the fact that the time series on disc centre (observation No. 2 in BE12) actually consists of 150 repeated scans of 4 steps with 0. ′′ 5 step width each, even if only the single step of each scan that was cospatial in 396 nm and 630 nm was investigated in detail. Figure  9 shows the temporal evolution of the H 2V intensity in a 2D FOV of 1.45 Mm ×14.8 Mm over 651 seconds (≡ 31 repetitions of scanning the same area). The three consecutive brightenings of, e.g., Fig. 10 are located at (x, y) ≈ (0 − 32, 6) Mm. Several BGs can be identified throughout the FOV, but their full extent in most cases is only covered along the slit in y whereas they are truncated in x when they were not almost centred inside the scanned area of 2 ′′ extent (e.g., the BG in the fourth repetition at y ∼ 6 Mm in Fig. 9).
From the section of the FOV shown in Fig. 9, a roundish shape with an average diameter of about 1 Mm can be attributed to BGs. To improve the statistics, we manually identified about 100 BGs in the full FOV of all the time series. The BGs were required to be located outside of regions with strong polarisation signal and to be visible in at least three repetitions of scanning the same area. We then selected the BGs at the time of their largest brightness and took intensity cuts along the slit through their centres. We fit Gaussians to the intensity cuts to obtain the diameters of the BGs. The diameters ranged between 0.50 and 1.83 Mm, with an average diameter of 1.17 Mm. A set of about 30 BGs that happened on locations with significant polarisation signal showed a slightly larger average diameter of 1.36 Mm. In the following we will thus assume a typical radius of 0.5 Mm for a single BG, which agrees well with the characteristic diameter of BGs of 1 -2 ′′ found in Ca ii imaging data (e.g., Wöger et al. 2006;Wöger 2007;Tritschler et al. 2007).

Fig. 10. Estimate of the energy content of individual BG events.
Top panel: temperature enhancements (left column) and internal energy enhancements (right column) for a pixel with BG activity (bottom row) and a quiet reference pixel (top). Bottom panel: average internal energy density for the active (black) and the quiet pixel (red) vs. geometrical height. The blue line shows their difference.

Energy content
To determine the energy content of individual BGs, one needs to derive the temperature enhancement related to them. One additional problem here is to separate the temperature contribution of BGs from, e.g., the background (brightness) temperature that results from an atmosphere that is in radiative equilibrium with the incoming photospheric radiation. This background radiation also needs not be constant throughout the FOV because photospheric magnetic fields or variations in the granular evolution can modify it. We therefore first determined a minimumtemperature stratification T min (τ, y) for each pixel along the slit defined by the lowest temperature at each layer of log τ attained on the pixel y during the one-hour time series. A subtraction of this minimal temperature from the temperature stratification on the pixel y at a given moment of time then yields only positive temperature fluctuations. These fluctuations represent the minimum amount of additional energy above the lowest energy state that the atmosphere exhibited during nearly one hour, without excluding that already the lowermost energetic state was energetically above radiative equilibrium.
To isolate the BG contribution to the energy and to remove any additional contribution from the background radiation, we first selected two different locations for a comparison, i.e., one pixel with a significant activity in form of a sequence of three prominent and well-defined BGs that appear consecutively, and one pixel that showed no clear BG or other activity (marked in   Fig. 10 shows the temperature fluctuations relative to T min (τ, y) for the two pixels over a period of 651 seconds, with the "active" pixel in the bottom row and the "quiet" reference pixel in the top row. The left column shows the temperature fluctuations ∆T and the right column the corresponding variations of the internal energy density ∆E that were calculated using Eq. (1). For the conversion to geometrical height, we assumed the tabulated relation between log τ and geometrical height z of the HRSA to be valid as first order estimate also for individual spatially resolved temperature stratifications.
To determine the characteristic energy content of a BG, we averaged the value of ∆E for the two pixels over 651 secs (bottom panel of Fig. 10). The difference of the two average values of ∆E reaches a maximum of 22.6 J m −3 at a height of about 250 km. There are several arguments to use the difference at about this height as the actual energy input of one BG. Around this height, the average energy density is nearly constant for the active pixel, whereas it drops steeply at higher atmosphere layers, indicating that the energy is inserted at low altitudes in the atmosphere. This is also evident from the temporal evolution of the temperature fluctuations, where the precursors of the BG appear first low in the atmosphere in each case and then propagate upwards. The temporal evolution of the temperature (upper panel of Fig. 10) also shows clearly that three BGs happen inside the time used for the average, which are caused by three short-lived (< 100 secs) events of energy input into the low atmosphere. The temporal fill factor -as estimated from the time intervals when the BGs can be clearly identified in the temperature variation in Fig. 10 -of all three BGs together in the time span considered is about 50 %. The energy enhancement of a single BG thus should correspond to about twice the temporally average value, i.e., about 45 J m −3 .
To improve the statistical base of the result and to ensure that it is not (strongly) biased by exactly the train of BGs and the reference pixel that we selected, we also calculated the statistics of BGs relative to non-BGs for the full FOV. To that extent, we used the mask of locations with local temperature maxima determined in Sect. 3.5.1. The internal energy density of all locations, where local temperature maxima at three different optical depths occurred, is shown in the middle panel of Fig. 11, whereas the top panel shows the complementary map of the energy density. The bottom panel shows the intensity at the wavelengths that were attributed to these optical depths. A comparison of the middle and upper panel reveals that the BGs with their apparent diameter of 1 Mm are usually only a part of a wave front that extends over up to ten Mm (cf. von Uexküll & Kneer 1995) and induces temperature maxima along its full length. The temperature maxima mark the spines of intensity brightenings, with a similar rise and decay time of the BG events (top panel of Fig. 11). The temporal extent appears strongly compressed in the spatio-temporal map in comparison to the spatial extent.
To determine the maximal energy enhancement caused by a BG, we calculated the average value of the energy density for all temperature maxima and for the complementary sample, i.e., all points in time and space that did not correspond to a local temperature maximum. For the optical depth of log τ = −1.2 (≡ 148 km), BGs have an average energy density of 135 J m −3 , whereas the corresponding value at log τ = −2.2 (≡ 244 km) is 124 J m −3 . For the non-BG sample, the average energy density is 89 J m −3 and 80 J m −3 , respectively. We excluded not only the location of the temperature maximum, but also always the directly preceding and following time step in the calculation of the average non-BG energy. The energy difference between BGs and non-BGs then is 46 J m −3 and 44 J m −3 at 150 km and 250 km height, respectively, matching the estimate above from individual BGs. We will use a characteristic enhancement of the internal energy density of 45 J m −3 for one BG in the following.

Vertical extent
To determine the vertical extent of BGs, in principle the curve of the energy difference between the quiet and the active pixel could be used (lower panel of Fig. 10), but we decided to use the temperature stratifications instead because of the larger amplitude of the temperature fluctuations. We converted the temperature stratifications at the onset of two BGs (indicated by the white vertical dotted lines in the lower left panel of Fig. 10) from optical depth to geometrical height (Fig. 12). The BGs at this stage of their evolution show up as Gaussian-like temperature bumps centred at about z ∼ 300 km. The full-width at half-maximum (FWHM) of the temperature enhancements is about 200 km in height, which would correspond to half a period of a sinusoidal wave with 400 km wavelength (blue line in Fig. 12). Identifying BGs with acoustic waves with a period of 3 or 5 min and assuming propagation with sound speed (about 6 kms −1 ) would predict significantly larger wavelengths of 1 and 1.8 Mm instead. The propagation of BGs in the atmosphere is not purely vertical, but has also a horizontal component (cf. Figs. 9 and Fig. 11, some of the BGs move along the slit with time), which modifies the apparent vertical wavelength. The angle between local vertical and the wave propagation directions seems, however, to be small in most cases.
With the information on the vertical extent and the effective enhancement of the internal energy density, one can derive an estimate of the total energy of a BG by E = V · ∆E. For a single pixel of 0. ′′ 5×0. ′′ 292 extent, assuming a vertical extent of 200 km, and using ∆E = 45 Jm −3 , one obtains a total energy of the BG of 6.91 × 10 17 J. Assuming a roundish structure with a radius of 0.5 Mm, the total energy of a BG is about 7.1 × 10 18 J.

Vertical propagation speed
For estimating the vertical energy flux, the characteristic vertical propagation speed of BGs is needed. The LTE inversion provides the necessary information in form of temperature variations in optical depth. We used again the tabulated relation between τ and z of the HSRA model to convert from optical depth to a geometrical height scale. Figure 13 shows the resulting temperature fluctuations at the active pixel vs. geometrical height. The precursors of the BGs leave inclined traces in the lower part of the atmosphere up to about 600 km height, whereas above the temperature enhancements are nearly vertical, i.e., seemingly affecting all higher layers at the same time. The latter presumably does not indicate that the propagation speed increases strongly, but rather that the relation between τ and z is not valid inside a shock front. CS97 concluded from their 1D NLTE simulations that the emission in Ca ii H at maximal brightness of a BG stems from a height range of about z ∼ 1 to 1.4 Mm, where the strongest contributions to the emergent intensity originate in shocks. For the lower part and the onset of the BGs, the propagation speed of the temperature perturbations can, however, well be measured (inclined dotted lines in Fig. 13). The corresponding slopes denote vertical propagation velocities of about 5.0, 6.3, and 6.7 km s −1 , respectively, comparable to the sound speed (see also von Uexküll et al. 1983). We note that an estimate of the propagation speed and direction in all three spatial dimensions can be derived from an inversion of all steps of each repeated scan, dropping the restriction of using only the step that was cospatial in the Ca ii H and 630 nm spectra.
The magnified view of Fig. 13 on the BG events allows one to estimate an additional characteristic, the typical duration. Above a height of 600 km, none of the BG events lasts longer than about 60 secs from its onset to its subsequent complete disappearance.

Energy flux related to BGs
The temporal evolution of the intensity or temperature shows that the energy of BGs is removed by both propagation and radiation in a short time. There are, however, different ways to estimate the vertical energy flux per unit area. Using the energy content of a single BG of 6.91 × 10 17 J per pixel of 0. ′′ 5×0. ′′ 292 as estimated above and assuming that all this energy disappears again before the onset of the next BG 160 secs later yields an energy flux of 56.3 kW m −2 , but without specifying into which spatial direction (or into which degree of freedom) the energy is dissipated or disappears. The average energy enhancement of 22.6 J m −3 of the active relative to the quiet pixel (Fig. 10) yields an energy flux of 41.4 kW m −2 from both the influx and dissipation of the energy of three BGs in a time span of 651 secs. Excluding a downward energy transport and otherwise equal probability for an energy transport to the remaining five possible spatial directions, the upwards vertical energy flux would be about (41.4 -56.3)/5 ∼ (8.3 -11.3) kW m −2 . This estimate uses the characteristic temporal evolution in the derivation, i.e., about 160 secs cadence between BGs or three events in 651 seconds, respectively, and refers to the energy flux between about 150 to 350 km height.
However, even if individual BGs induce a vertical energy flux of up to a few tens of kW m −2 in the low solar atmosphere (z < 400 km), it seems that at z ∼ 1 Mm far less than 5 % of the energy flux are left over (Fig. 7). We remark that in our case the underlying curve that predicts this behaviour is directly proportional to the temperature rms with height and the gas density (Eq. 1). Modifications in the stratification of either of the two quantities therefore will change the residual energy as well.

Discussion
The reliability of our analysis results depends to some extent on the LTE assumption taken. We therefore first return to a discussion of observed and theoretical profiles that is partly independent of the (N)LTE question.  Figure 14 shows the average observed and best-fit profile in QS for our time series and for one of the AR maps, an average umbral and plage profile from the same AR map, the profile synthesized in LTE from the modified HSRA model without a temperature rise, and two profiles synthesized in NLTE from the FAL-A and FAL-C temperature stratifications (Fontenla et al. 2006) using the code of Uitenbroek (2001). The spectral sampling of the FAL NLTE profiles in the synthesis was 0.75 pm without any additional spectral degradation (cf. the next section). To facilitate an easier comparison of the variations in the very line core in this figure, all profiles from the AR map were normalized to the same intensity at 396.9 nm. Both NLTE FAL profiles exceed the observed emission peaks in the QS on disc centre (lower panel of Fig. 14). The FAL-C profile exceeds even the emission in the QS region of the AR map (top panel). To exclude a possible influence of the largescale averaging that could reduce the amplitude of the emission peaks through the presence of spatially/temporally varying Doppler shifts, we calculated a QS profile that was averaged only over 112 spectra, i.e., one slit position in the time series. This small-scale average QS profile showed a slightly increased amplitude of the H 2V emission peak, but the change stayed minor and the intensity still fell far short of that in the NLTE FAL profiles. Both the observed and average best-fit profile, however, clearly exceed the profile corresponding to the modified HSRA atmosphere. Average profiles in the QS therefore fall between an profile from an atmosphere in radiative equilibrium and an atmosphere with a chromospheric temperature rise such as in FAL-C or FAL-A. For the case of the AR map, the profiles change significantly. The observed average QS profile has two clear emission peaks, and the average best-fit profile in the QS starts to resemble the NLTE FAL profiles. We remark that inside the AR map, the "QS" regions contain far more magnetic flux than found in the QS at disc centre, and furthermore that the AR observations were taken at a heliocentric angle of about 50 • . The emission in average plage and umbra exceeds the FAL NLTE profiles by a factor of about two.

Comparison of average profiles and corresponding temperature stratifications
The corresponding temperature stratifications in Fig. 15 reflect closely the differences in the profiles themselves. The modified HSRA lies below the average temperature of the QS on disc centre retrieved by the inversion, and the two FAL models above it, whereas the stratifications obtained by the inversion for the AR region plage and umbra follow to some extent the chromospheric temperature rise in the FAL models or the original HSRA model (top panel in Fig. 15).
The LTE assumption for sure has a tremendous effect on the retrieved temperature stratifications, but the main problem consists already in reconciling the observed average spectra in QS with those predicted by the NLTE FAL profiles (lower panel of Fig. 14). Appendix A discusses possible spectral broadening mechanisms that could reduce the amplitude of the emission peaks in observed spectra. We find that any reasonable instrumental broadening, or broadening by unresolved micro-and macroturbulent velocities with reasonable velocities (< 10 kms −1 ) are insufficient to reduce the emission peaks from the NLTE FAL models enough to fully match the observed spectra, leaving as only option a clear reduction of the chromospheric temperature (cf. Rezaei et al. 2008).

Difference between temperature stratifications in quiet Sun and active regions
Even if it is well known that the emission in the Ca ii H line core increases in the presence of (photospheric) magnetic fields (e.g., Simon & Leighton 1964;Linsky & Avrett 1970;Liu & Smith 1972;Dunn & Zirker 1973;Mattig & Kneer 1978;Schrijver et al. 1989;Rezaei et al. 2007), the difference between temperatures in QS and AR retrieved by the inversion is surprisingly large. The LTE approach used in the inversion retrieves temperature stratifications with a clear chromospheric rise in the AR, even though it supposedly underestimates the temperature. That the underestimation comes with a caveat is caused by a dichotomy in the (N)LTE problem. In the LTE assumption, (gas) temperature and intensity are directly coupled. In case that an increase in emission is not caused by an increase in the gas temperature but by NLTE effects on the emitted radiation field, the LTE inversion will attribute it still to a temperature change and will yield a too high gas temperature. In case that the emission is caused by an increase in the gas temperature, the LTE approach will yield a too low temperature, because it assumes that the corresponding energy will be evenly distributed over all degrees of freedom, whereas in NLTE a significant fraction can go into additional ionization or other energy sinks without increasing the emitted radiation. For active regions, the temperature structure in the solar atmosphere seems to be more stable than in the QS. While, e.g., umbral profiles show clear indications for upwards propagating wave patterns producing shock waves in the chromosphere like in the QS (e.g., Lites 1986;Centeno et al. 2006;Felipe et al. 2010), Ca ii H spectra in umbrae never loose their emission peaks (Mattig & Kneer 1978;Kneer et al. 1981b,a;Rouppe van der Voort et al. 2003;Felipe et al. 2010), in contrast to QS profiles that oscillate between low or no emission and intensity reversals (e.g, Liu 1974;Cram & Dame 1983;Rezaei et al. 2008, BE12). The strong variability in QS was also found in numerical simulations (CS97, Wedemeyer et al. 2004;Leenaarts et al. 2011), but the corresponding prediction for active regions or the interior of sunspots is still pending. Even if we cannot provide any conclusive proofs, the results of our LTE inversion support a more stable gas temperature structure in active regions with a clear chromospheric temperature rise. This does not exclude that the line shape is caused by NLTE effects, but suggests that the underlying temperature stratification seems to be less influenced by temporal variations than in the QS.
The modulus of the difference between QS and active regions even in our LTE approach indicates that heating processes related to strong magnetic fields are up to five times stronger than those related to acoustic waves in the QS: the (spatial) temperature rms in the AR is about five times larger than the spatial or temporal rms in QS (Fig. 2) and the enhancement of the QS temperature stratification above the modified HSRA model (∆T < 1000 K, Fig. 15) is roughly five times smaller than for the average plage (∆T ∼ 4000 K). Because in most of the stud-ies of low chromospheric lines such as Ca ii H the presence of magnetic fields was neglected, their inclusion could be the turning point to reconcile the high variability of, e.g., Ca ii H with the low variability of chromospheric lines observed by SUMER . We point out that the latter authors also found a permanent presence of emission in the SUMER lines in QS, in conflict to Ca ii H spectra in QS, but in full agreement with the behaviour of Ca ii H spectra in the umbra. This would fit all together with the assumption that the SUMER lines form in a magnetized atmosphere (like Ca ii H in the umbra) in QS, e.g., above a magnetic canopy with a strong damping of impinging acoustic oscillation, whereas Ca ii H forms below the canopy in QS (cf. Beck et al. 2008, their Fig. 22). We note also that the same effect is suggested by the numerical simulations of FA12. The average temperature stratifications shown in their Fig. 6 do not exhibit a clear temperature reversal below a height of 1 Mm, and additionally show a strong increase of the temperature when the magnetic fill factor is increased by an order of magnitude, caused by the related reduction of the gas density in a partially evacuated magnetized plasma.

Energy content of bright grains and related energy flux
We estimated the energy content of the precursors of chromospheric bright grains. We find a spatial extent of about 1 Mm by 0.2 Mm in height, with an increase of the internal energy by about 45 J m −3 and a total energy of about 7 × 10 18 J per event. The temperature enhancements propagate upwards with about 6 km s −1 , comparable to the local sound speed (see also Worrall 2012). With the typical temporal cadence of 160 s for individual events this yields a vertical energy flux of about 10 kW m −2 at a height of about 250 km. Some insecurity in this calculation is related to the characteristic cadence because it is sensitive to how a local maxima is identified and counted. Our determination yields from 15 to 22 maxima in 3150 secs, which gives roughly a 20 % error on the energy flux. Our specific data is less suited for the determination of the occurrence rate of BGs than time series of intrinsically two-dimensional data. The number of about 10 kW m −2 falls in line with the energy fluxes between 3 to 12 kW m −2 at about 250 km height determined from velocity oscillations in, e.g., Bello González et al. (2009González et al. ( , 2010 or Malherbe et al. (2012, MA12). The latter authors also determined the total energy contained in what they call "acoustic events", i.e., a phase of upwards directed energy transport that can cover a few wave trains and can last up to 10 min (cf. Fig. 9 of MA12). They found total energies of about 1.9 × 10 19 J per acoustic event, which fits to the example used in our Sect. 3.5 with three BGs of in total about 2.1 × 10 19 J in 651 seconds.
MA12 found an occurence rate of acoustic events of about 5.7 × 10 −16 s −1 m −2 . With 20 temperature extrema in 3150 secs on each pixel of 0. ′′ 3 × 0. ′′ 5, and a spatial fill fraction of BGs of about 10 % along the slit, we obtain a rate of 8.3 × 10 −15 s −1 m −2 , or about 0.5 min −1 Mm −2 , respectively. The difference between the two rates is about a factor of 15, where our rate should be larger by a factor of 3 -5 because of the existence of multiple temperature maxima during an acoustic event as defined by MA12. The remaining difference could be caused by the lower spatial resolution of our data, where the area of BGs will be slightly overestimated because of the spatial smearing. The recently inaugurated 1.5-m GREGOR telescope  will offer the possibility to obtain highresolution, high-cadence spectra of the Ca ii IR lines around 850 nm with the Gregor Fabry-Perót Interferometer (GFPI, e.g., Puschmann et al. 2012a,b,c) and of Ca ii H with the BLue Imaging Solar Spectrometer (BLISS, Puschmann et al. 2012c) in the near future.
Photospheric energy fluxes in the range of a few kW m −2 would suffice to replenish the generic chromospheric energy requirement of 4 kW m −2 , if -and only if -this amount of energy would finally reach the chromosphere. The estimates of the energy losses with height, however, predict that less than 5 % of an energy input at 250 km height reaches layers above 400 km. This suggests that the chromospheric radiative losses cannot be balanced by the energy of propagating waves or temperature perturbations despite a sufficiently large reservoir of mechanical energy in the photosphere (but see also Petrukhin et al. 2012). We note that the temperature stratifications and the rms temperature fluctuations retrieved by our inversion only predict on average a drastic loss of energy with height, but do not directly provide information about where the energy is going to. To determine the exact fraction of the energy dissipated by radiation, thermal conduction, or changes of the ionization fraction requires a detailed study of individual events.

Visibility and extent of magnetic canopies in temperature maps
Up to here, we discussed the properties of chromospheric brightenings and the chromospheric thermal structure in the quiet Sun as a purely hydrodynamical phenomenon without considering the presence of magnetic fields. The simultaneous polarimetric observations of the 630 nm channel of POLIS, however, also provide information on polarisation signals, hence photospheric magnetic fields. We can thus investigate the influence of the magnetic topology on the thermal structure in the quiet Sun as well.
The typical behaviour of the Ca ii H line with its repeated change from reversal-free absorption to emission and vice versa in "field-free" regions, i.e., on locations that are polarisationsignal-free down to amplitudes of about 10 −3 of I c , and with a persistent stationary emission on locations with photospheric polarisation signal fits to the idea of magnetic field lines as a mediator between a stationary and transient emission pattern in Ca ii H. For photospheric magnetic flux concentrations, geometrical models can be constructed by setting field strength and total magnetic flux in the photosphere, and requiring (magneto-)hydrostatic pressure balance with their surroundings. This yields the classical "expanding flux tubes" whose diameter increases with height in the atmosphere (e.g., Solanki et al. 1991, SO91). Now, if the magnetic fields have a profound influence on the chromospheric temperature structure, they have to show up in the temperature stratifications retrieved by the inversion, regardless of the LTE -NLTE problem. Figure 16 shows that this is (clearly) the case. Panel a shows the temporally averaged temperature stratifications along the slit vs. optical depth. The also temporally averaged unsigned Stokes V signal is overplotted as a black line as a reference of photospheric magnetic fields that were roughly persistent over the duration of the time series of about one hour. Already in the temporally averaged temperature, flux-tube-like features are found co-spatial to the largest values of |V|. To increase the visibility, and because the main effect of the magnetic fields is an increase of temperature relative to field-free surroundings, we subtracted the temporally and spatially averaged temperature stratification from the temporally averaged temperatures (panel b). In that case, temperature enhancements of +50, +100, and +150 K over the average temperature almost exclusively occur on locations of photospheric magnetic fields. For the two strong magnetic flux concentrations at x ∼ 14.5 Mm and 35 Mm, the temperature enhancements at chromospheric layers above 500 km given by the +50 K contour form exactly the "wine-glass" shape predicted and used in many photospheric flux tube models (SO91, Pizzo et al. 1993;Gadun et al. 2001;Khomenko & Collados 2012, their Fig. 10). In both cases, the temperature enhancements yield a nearly horizontal (temperature) canopy at a height of about 620 km that extends between 1 and 5 Mm horizontally away from the central axis (see also de la Cruz Rodríguez et al. 2013). The temperature enhancements delineate vertical structures at the end of the canopies.
The restricted horizontal extent of the temperature enhancements is better seen in panel c that shows the temperature stratifications for one step of the time series. Close to photospheric polarisation signals, the nearly horizontal canopies in temperature appear again prominently, especially from x ∼ 6 to 16 Mm where three neighboring flux tubes create a continuous temperature canopy, but in-between separated flux tubes areas of several Mm extent with a strongly reduced temperature show up (x ∼ 0 −7 Mm, x ∼ 23 −33 Mm). In similar-sized areas, reversalfree Ca ii H spectra were found before . In total, this implies that on the one hand magnetic canopies have a direct effect on the temperature stratification even in a single snapshot of the solar atmosphere, and on the other hand there are no clear indications for volume-filling magnetic fields in the formation height of Ca ii H in the temperature map of Fig. 16. The appearance changes a bit in the temperature map for the largearea scan at high layers (observation no. 1 of BE09, cf. Fig. 5). Figure 16 clearly demonstrates that photospheric magnetic fields are a second acting agent for the chromospheric temperature structure in addition to the transient BGs. A more detailed study of the modification of the thermal structure at or close to the locations of photospheric magnetic fields is, however, beyond the scope of the present paper.

Applicability of LTE inversion
Because of the known problems of assuming LTE in an inversion of any chromospheric line, it might be worth to note a few examples for which purposes the LTE inversion still serves without any, or with only weak restrictions. We remark that mainly the modulus of the temperature derived from the LTE inversion has a high inaccuracy because of the missing instantaneous coupling between gas temperature and radiation field in NLTE conditions, and that the conversion from optical depth to geometrical height suffers from the lack of hydrostatical equilibrium. The latter could, however, be obtained post-facto with a presumably minor impact on the resulting synthetic spectra.
The detectability of canopies in temperature maps -that are implicitly assumed to correspond to the theoretically expected magnetic canopies -does depend only slightly on the modulus of temperature. Our results clearly show that the signature of the canopies can be detected in the surroundings of photospheric magnetic flux concentrations up to a few Mm (cf. Fig. 16). This is in contrast to the results of Judge et al. (2010) who found a strong "confinement" of intensity enhancements, and hence magnetic flux in the chromosphere. The mismatch of these two results shows that a quantitative in-depth analysis of spectra, even with the limiting LTE assumptions, is crucial for tracing the topology at chromospheric layers.
For estimating the energy of individual BGs, we used a height range, or equivalently, a wavelength region in the line wing, where NLTE effects can be assumed to play a minor role (z < 500 km). These results are thus rather robust and would presumably not change significantly when a different inversion method would be used. The advantage of the inversion of the Ca ii H spectra now is to provide on the one hand also a quantitative guess for the corresponding properties at the atmosphere layers above, which cannot be accessed by any photospheric line, and on the other hand to provide clear information on the temporal evolution, such as the vertical propagation speed (Fig. 13), or the exact propagation direction from combining horizontal motions in the 2D sections of the FOV (Fig. 9) with the vertical propagation speed.

Summary and conclusions
In a LTE inversion of Ca ii H spectra, we cannot retrieve on average a chromospheric temperature rise in quiet Sun (QS) regions, whereas in active regions (ARs) this is the case. The difference between QS and ARs can be directly traced back to the observed input profiles themselves, independent of the LTE assumption in the inversion. We confirm the findings of BE12 of a layer at about 150 km height with an increased occurence of highintensity, and hence high-temperature events near and in magnetic features. A comparison of line-core and line-wing images, and the correlation coefficient between intensities and temperature vs. optical depth suggests that the increased skewness of the intensity histograms on magnetic locations is caused by a direct signature of magnetic fields in temperature at two different atmosphere layers, i.e., both in the chromosphere and the mid photosphere. The magnetic network is inconspicuous at the layers in-between. The appearance of the network in the mid photosphere could be caused by the "hot wall" effect in combination with the shift of the optical depth scale inside magnetic fields (Spruit 1976), whereas the appearance in the chromosphere indicates an energy deposit instead.
The energy contained in the rms fluctuations of temperature and the large difference between QS and AR profiles and temperature stratifications indicate the importance of chromospheric heating processes related to the presence of magnetic fields. The physical processes that could yield such a heating should be investigated in more detail in the future and included in the analysis of chromospheric lines. The temperature fluctuations in the QS fall short of the requirements needed to maintain a stationary chromospheric temperature rise.
The properties of individual (heating) events related to chromospheric bright grains (BGs) can be determined during all of their existence in the LTE inversion results, from the BG precursors in the photosphere to their culmination in the lower chromosphere. The mechanical energy content and the vertical energy flux in the photosphere would be sufficient to fulfill the generic chromospheric energy requirement of about 4 kW m −2 , but only a fraction well below 5 % of the mechanical energy reaches the chromosphere, falling short by far of the generic requirement. The decrease of the internal energy density with height agrees with that expected for acoustic waves.
The topology of magnetic fields in the chromosphere is seemingly directly reflected in the temperature structure at layers above a height of 500 km also in an LTE analysis, as indicated by the appearance of canopy structures around magnetic field concentrations both in average and in instantaneous x−z temperature maps. With all of the shortcomings of the LTE approach, this quantitative analysis of large-scale data sets in a chromospheric spectral line provides a wealth of interesting features that can be used to address open questions in solar chromospheric physics of which this paper still has not covered all despite its extent. C. Beck, R. Rezaei, K.G. Puschmann: The energy of waves: IV. Inversion results of Ca ii H spectra, Online Material p 1

Appendix A: Effects of spectral broadening
The effective spectral resolution of the POLIS spectrograph can lead to a loss in amplitude of the emission peaks by spectral smearing because of their small spectral extent. The default spectral resolution of POLIS is 220.000@400 nm (≡ 1.8 pm), but because of the on-chip binning by a factor of two, the resolution is sampling-limited to about 3.8 pm. Convolving an FTS (Kurucz et al. 1984) reference spectrum with the method of Allende Cabrera Solana et al. (2007) to match the observed average Ca profile yielded the need for a Gaussian with σ ∼ 1.7 pm. We note that the FTS spectrum contains line broadening by macroturbulent velocities because of being a spatially averaged QS spectrum.
For an estimate of the maximal spectral smearing between the observed average QS profile and the FAL-C NLTE profilesynthesized without a macroturbulent velocity -by instrumental effects, we first convolved the FAL-C NLTE profile as reference by a Gaussian to match the shape and width of the line blends in the Ca wing in the average observed spectrum. The lower left panel of Fig. A.1 shows that this requires a value of σ of about 3.4 pm, which is in rough agreement with the prediction from the sampling limit. A convolution of the full spectrum with this value, as an estimate of the instrumental and velocity broadening in the atmosphere, reduces, however, the amplitude of the emission peaks of the FAL-C NLTE profile only slightly (lower right panel of Fig. A.1). To force the emission peaks in the synthetic profile to a shape roughly resembling the observed profile requires a convolution of FAL-C with a Gaussian with σ = 9.4 pm. The line blends in the wing, however, clearly exclude such a large instrumental broadening because in the FAL-C NLTE profile convolved with σ = 9.4 pm (purple line) the blends are twice as broad and half as deep as in the observed spectrum (lower left panel of Fig. A.1).
We then decided to try a wavelength-dependent width of the Gaussian used as spectral point spread function (PSF). The width was set to 3.4 pm in the line wing up to λ = 396.75 nm, gradually increasing to 9.4 pm at λ = 396.85 nm, and reducing down to 3.4 pm again at λ = 396.94 nm. As it seemed impossible to match the FAL-C NLTE and the observed profile even with a large broadening in the very core, we applied this variable convolution to the FAL-A NLTE profile instead. The temperature stratification of FAL-A has a weaker chromospheric temperature rise than FAL-C (Fig. 15) and the corresponding NLTE profile differs significantly from FAL-C, especially in the residual linecore intensity (Fig. 14). The approach with the variable spectral PSF maintains the match in the line wing (upper left panel of Fig. A.1) and leads to an acceptable agreement between the emission peaks of the convolved FAL-A NLTE profile and the observed spectrum (upper right panel).
The wavelength-dependent Gaussian spectral PSF would correspond to a microturbulent velocity v mic that changes from 2.6 kms −1 in the photosphere to 7.1 kms −1 in the chromosphere. The NLTE synthesis already included a height-dependent microturbulent velocity of more than 5 kms −1 for log τ < −5.5. Thus, the effective microturbulent velocity necessary to match the averaged observed and the FAL-A NLTE profile should be nearly twice as large as the initial value of v mic , and has to be present already at lower layers of log τ ∼ −3 to yield the correct shape of the H 1V and H 1R minima next to the emission peaks. We note that similar values of v mic would be also required to match the FTS atlas and the FAL-A NLTE profile, because the line shape in the FTS is similar to the average QS spectra shown here. If such a high value of v mic is reasonable for spectra at about 1 ′′ spatial resolution needs a further investigation. Additionally, the increase of v mic at comparably low layers of log τ also already affects the Fe i line blend at 396.93 nm strongly, broadening it significantly beyond its width in observed spectra. Doubling v mic therefore seems not to be a valid option to match observed and theoretical NLTE profiles, leaving a reduction of temperature as the only option (cf. Rezaei et al. 2008).