$\displaystyle+{1\over\rho_{0}}\left[\nabla\times\\{\nabla\times\left(\vec{v}_{1}\times\vec{B}_{0}\right)\\}\right]\vec{B}_{1}/\left(4\pi\right)\,.$
(4.37)

##### Wave modes

With Eq. 4.37 we make the ansatz

$\vec{v}_{1}(\vec{r},t)=\vec{v}_{1}\exp\left[i\left(\,\vec{k}\cdot\vec{r}-\omega
t\right)\right]\,,$ (4.38)

with wavevector $\vec{k}$.

For $\vec{B}_{0}=0$ and $\vec{g}=0$ one gets pure sound waves with phase
velocity $v_{ph}=\omega/k=c_{s}$. With $\vec{B}_{0}=0$ and $g>0$ one obtains
atmospheric waves (Bray and Loughhead, 1974).

When the gas pressure is negligible, $p=0$, and with $g=0$, but
$|\vec{B}_{0}|>0$, the dispersion relation results

$\omega^{2}\vec{v}_{1}/v_{A}^{2}=k^{2}\cos^{2}\alpha\,\vec{v}_{1}-(\vec{k}\cdot\vec{v}_{1})\,k\cos\alpha\,\vec{\hat{B}}_{0}+\left[(\vec{k}\cdot\vec{v}_{1})-k\cos\alpha\,(\vec{\hat{B}}_{0}\cdot\vec{v}_{1})\right]\vec{k}\,.$
(4.39)

Here, $\vec{\hat{B}}_{0}$ is a unity vector parallel to $\vec{{B}}_{0}$,
$\alpha$ is the angle between the wavevector $\vec{k}$ and $\vec{B}_{0}$, and
$v_{A}$ is the Alfvén velocity with

$v_{A}^{2}={B_{0}^{2}\over 4\pi\rho_{0}}=2{P_{m,0}\over\rho_{0}}\,,$ (4.40)

where the magnetic pressure is $P_{m}=B^{2}/(8\pi)$. Scalar multiplication of
Eq. 4.39 with $\vec{\hat{B}}_{0}$ shows that
$\vec{\hat{B}}_{0}\cdot\vec{v}_{1}=0$. This means that the (perturbed)
velocity is perpendicular to $\vec{B}_{0}$ (since the Lorentz force on the
perturbed gas is perpendicular to $\vec{B}_{0}$).

Scalar multiplication of Eq. 4.39 with $\vec{k}$ yields

$\left(\omega^{2}-k^{2}v_{A}^{2}\right)\left(\vec{k}\cdot\vec{v}_{1}\right)=0\,.$
(4.41)

From this equation one can derive two magnetic wave modes:

  * (1)

Assuming $\nabla\cdot\vec{v}_{1}=0$ gives the so-called incompressible mode
and from the ansatz Eq. 4.38 one gets $\vec{k}\cdot\vec{v}_{1}=0$. Thus, this
mode is a transversal mode with the velocity perpendicular to the direction of
propagation. From Eq. 4.39 we have $\omega=\pm\cos\alpha\,v_{A}$. The waves
are also called shear Alfvén waves. For $\alpha=0\degr$ one derives that
$\vec{B}_{1}$ and $\vec{v}_{1}$ are parallel and the propagation is along
$\vec{B}_{0}$.

  * (2)

Another solution of Eq. 4.41 is $\omega=kv_{A}$, independent of $\alpha$.
These waves are compressional Alfvén waves, and for $\alpha=90\degr$ the
velocity $\vec{v}_{1}$ is parallel to $\vec{k}$, i.e. we have longitudinal
waves.

Finally, admitting that the gas pressure is not negligible, $p>0$, the phase
velocity comes out as

$v_{ph}={\omega\over k}=\left[{1\over
2}\left(c-s^{2}+v_{A}^{2}\right)\pm\left(c_{s}^{4}+v_{A}^{4}-2c_{s}v_{a}^{2}\cos
2\alpha\right)^{1/2}\right]^{1/2}\,.$ (4.42)

The ‘+’ sign above gives the so-called ‘fast magnetoacoustic waves’ and the
‘$-$’ sign the ‘slow magnetoacoustic waves’. Their phase speeds depend on
$\alpha$ (see the hodographs in Ferraro and Plumpton, 1966; Kippenhahn and
Möllenhoff, 1975; Priest, 1984).

##### Observational results

Figure 4.10: Average temporal power spectra of velocity from AOI C in Figs.
4.4 and 4.5 before filtering (dashed) and after pass-band filtering (solid).

The LOS velocities of the structures contain variations on long time scales of
10 min and longer as well as fluctuations with shorter time scales. To distill
the latter, among them possibly magnetoacoustic waves, we applied a high-pass
temporal filter and removed some high-frequency noise at the same time. The
quantities then fluctuate about zero. Figure 4.10 depicts the average power
spectra of the LOS velocities in AOI C in Fig. 4.5 before and after filtering.
We note that the 5-min oscillations are filtered out, while some oscillations
at the acoustic cutoff (corresponding to periods of approximately 200 s) are
partially retained. Yet the unfiltered and filtered power spectra in Fig. 4.10
do not show any predominant period.

Figures 4.11–4.14 show examples of space-time slices from AOIs C and D. The
fluctuations of several quantities are shown:

  1. 1.

LOS velocities determined from differences of H$\alpha$ intensities at
$\pm$0.5 Å off line center, henceforth referred to as Doppler-gram slices
(bright indicates velocity towards observer);

  2. 2.

H$\alpha$ line center intensities, henceforth LC slices;

  3. 3.

in Figs. 4.11–4.13 differences of intensities at +0.5 Å off line center
$I_{0.5}(t_{i+1})-I_{0.5}(t_{i})$ with cadence $\Delta t=t_{i+1}-t_{i}$ = 22
s, henceforth referred to as $\Delta I_{0.5}$ slices;

  4. 4.

in Fig. 4.14 differences of intensities at line center
$I_{LC}(t_{i+1})-I_{LC}(t_{i})$, henceforth referred to as $\Delta I_{LC}$
slices.

Time runs from bottom to top with $t=0$ at the start of the series. The
interruptions/interpolations at $t\approx$ 18.0–19.6 min and 37.5–38.5 min are
obvious.

Figure 4.11: Example of space-time slices, of 11 width, from AOI C in Figs.
4.4 and 4.5. From left to right: LOS velocity, H$\alpha$ line center
intensity, and intensity differences at H$\alpha$ +0.5Å off line center:
$I_{0.5}(t_{i+1})-I_{0.5}(t_{i})$ with cadence of $\Delta t=t_{i+1}-t_{i}$ =
22 s. The intensity differences in the right column are shifted up by 11 s.
They are referred to as $\Delta I_{0.5}$ slices in the text. Figure 4.12:
Example of space-time slices, of 11 width, from AOI D. Same ordering as in
Fig. 4.11. Figure 4.13: Selected part of space-time slices from AOI D with
slice width of 22. Same order as in Fig. 4.11. Figure 4.14: Selected part of
space-time slices from AOI D with slice width of 22. Left column: H$\alpha$
line center intensity fluctuations; right column: intensity differences at
H$\alpha$ line center: $I_{LC}(t_{i+1})-I_{LC}(t_{i})$ with cadence of $\Delta
t=t_{i+1}-t_{i}$ = 22 s, referred to as $\Delta I_{LC}$ slices in the text.

We focus attention to the oblique stripes in Figs. 4.11–4.14. These are the
signatures of magnetoacoustic waves. From their slopes we can measure phase
velocities projected on the plane perpendicular to the LOS. In Fig. 4.11 from
AOI C, the waves appear to originate near the right edge of the AOI. This is
one side at which the fibrils are rooted. Presumably, the waves are excited by
the buffeting of motions at the photospheric foot points of the magnetic
fields. As seen especially well in the Doppler-gram slices of Fig. 4.11, but
also in the LC slices, steep stripes originate from both sides of this region.
The projected phase speeds are of the order of 8 km s-1.

The stripes are often bent in the course of the temporal evolution, e.g. the
wave parallel to the dashed line ‘1’ in the $\Delta I_{0.5}$ slices of Fig.
4.11. This wave starts off with a phase velocity of 14 km s-1 and speeds up to
approximately 40 km s-1, one of the highest velocities measured.

A prominent period is not detected. Sometimes, the waves appear to be
repetitive, with two or three, at most, wave trains in sequence with periods
between 90 s and 180 s. An example of consecutive wave trains is indicated by
the three dashed lines ‘2’ in the LC slices of Fig. 4.11. Yet most time, the
waves are solitary, with one single wave package traveling across the FoV.
Many of the waves appear to spread out along the direction of propagation and
to fade after having traveled a distance of 5–10 Mm.

The amplitudes of the LOS velocities in the Doppler-gram slices are measured
to approximately 1 km s-1, be it in the waves with low phase speeds or in
those with high phase speeds. With the calibration discussed above in the
context of the cloud model (see Sect. 4.2.2) these amplitudes have to be
multiplied with a factor of approximately 3. The resulting amplitudes are thus
of the order of 3 km s-1, which is not a small perturbation compared with the
sound speed (c.f. below the discussion on the magnetoacoustic waves).

Figure 4.12 from AOI D shows similar space-time slices as those from AOI C in
Fig. 4.11. Yet here, the waves are excited at both sides and travel into the
AOI, sometimes crossing from left and right and possibly colliding as in the
example parallel to the dashed lines ‘1’ in the Doppler-gram. The long lasting
(more than 7 min), solitary wave train (parallel to dashed line ‘2’ in Fig.
4.12) has a phase velocity of approximately 13 km s-1, a typical speed of the
‘slow’ waves in this AOI. A correction for foreshortening, i.e. that we see
only the projection of the phase speed on the plane perpendicular to the LOS
is to be excluded since the fibrils in AOI D, as well as those in AOI C, are
oriented almost perpendicularly to the direction to the limb (see Fig. 4.5),
thus perpendicularly to the LOS. The wave parallel to the dashed line ‘3’ in
the $\Delta I_{0.5}$ slices of Fig. 4.12 gives a phase speed of 30 km s-1,
again not to be corrected for foreshortening. This is a typical phase speed of
the fast waves.

Figure 4.13 gives another example of space-time slices from AOI D, with wider
slices of 22 width and shorter time span of 13 min duration than in Figs. 4.11
and 4.12. The long lasting wave train in the Doppler-gram slices (parallel to
dashed line ‘1’) gives again the typical phase speed of 13.3 km s-1 with
(calibrated) LOS velocity amplitudes of approximately 2 km s-1. These LOS
velocities are transversal, in the sense that they are perpendicular to the
propagation and to the H$\alpha$ fibrils. The ‘fast’ wave in the $\Delta
I_{0.5}$ slices (parallel to dashed line ‘2’) exhibits also the typical phase
speed of 32 km s-1 with calibrated LOS velocities of approximately 1.5 km s-1.

Figure 4.14 gives a 7.25 min long section of the temporal development of
fluctuations in AOI C with slice widths of 22, but this time the LC slices and
the $\Delta I_{LC}$ slices only. Note that dark and bright features in
H$\alpha$ LC indicate increased and decreased absorption, respectively, not
enhanced and reduced temperature (see Al et al., 2004; van Noort and Rouppe
van der Voort, 2006). The two solitary waves between the pairs of horizontal
dashed lines (a, a) and (b, b) have phase speeds of approximately 25 km s-1.
Inspection of the LC slices shows that the waves consist of elongated, thin
blobs with length of 1–2 and width of approximately 05. Apparently, the waves
do not travel in the spatial direction along straight lines, but along sinuous
lines with deviations from straight lines of approximately 05 in amplitude.
This suggests that on these small scales the magnetic field is not straight
and homogeneous but entangled.

The presentation of the difference slices $\Delta I_{LC}$ in Fig. 4.14 is
prepared to study temporal displacements of absorption features. These are
only seen if the displacements have a strong component perpendicular to the
LOS. The bright and dark small-scale features, lying parallel and next to each
other, in the upper part of the $\Delta I_{LC}$ slices, between the dashed
line pair (b, b) are suggestive of such displacements perpendicular to the
direction of propagation.

We summarize in short the observational findings on magnetoacoustic waves:

  1. 1.

Generally, we find two kinds of waves: slow waves with phase velocities of
12–14 km s-1 and fast waves with phase velocities of 25–33 km s-1 (maximum
velocity found 42 km s-1). The waves appear to develop from low phase speed to
high phase speed waves and vanish after having traveled a distance of 5–10 Mm.

  2. 2.

Irrespectively of the wave mode, the LOS gas velocities are of the order of
2–4 km s-1.

  3. 3.

The waves are mainly solitary. They consist of short (1–2) and thin ($\approx
0\farcs 5)$ blobs of compressed gas.

  4. 4.

The waves appear to follow wiggly, entangled magnetic field lines with
possible lateral displacements.

##### Interpretation – waves in thin magnetic flux tubes

For the interpretation of the observations from AOI C and D, we adopt the
picture of waves in thin magnetic flux tubes, whose radius is small compared
to the pressure scale height. The propagation of waves in magnetic flux tubes
were treated by, among others, Defouw (1976), Wentzel (1979), Spruit (1982),
and recently by Musielak et al. (2007). Spruit assumes a thin, cylindrical
magnetic flux tube parallel to the $z$ axis, with radius $R$, magnetic field
along the tube of strength $B$, gas pressure $p$, mass density $\rho$, and
temperature $T$. The gravity is neglected. The tube is embedded in an external
medium with properties $B_{e}$, $p_{e}$, $\rho_{e}$, and $T_{e}$. Inside and
outside the tube the magnetic and atmospheric parameters are constant. In
Spruit’s (1982) work, the MHD equations are linearized and a mode analysis is
performed, with proper conditions at the interface between flux tube and
surrounding medium.

Incompressible Alfvén waves ($\nabla\cdot\vec{v}_{1}=0$, with small velocity
perturbation $v_{1}$) are also possible in flux tubes. They are torsional
Alfvén waves. The compressive solutions lead to

$\nabla\cdot\vec{v}_{1}=A\,{\cal B}_{m}(nr)\exp[i\,(\omega t+m\phi+kz)]\,,$
(4.43)

with amplitude $A$, ${\cal B}_{m}(nr)$ Bessel functions of order $m$, $r$ the
distance from the axis of the tube, and $\phi$ the azimuthal angle. Inside the
tube, the waves propagate along the $z$ direction. For $n$ the relation holds

$n^{2}=(\omega^{2}-v_{A}^{2}k^{2})\,(\omega^{2}-c_{s}^{2}k^{2})/[(\omega^{2}-c_{t}^{2}k^{2})\,(v_{A}^{2}+c_{s}^{2})]\,.$
(4.44)

Here, the tube speed $c_{t}$ is introduced with

$c_{t}^{2}={v_{A}^{2}c_{s}^{2}\over v_{A}^{2}+c_{s}^{2}}\,,$ (4.45)

which shows that the tube speed is smaller than both the Alfvén and the sound
velocity.

Spruit (1982) showed that in the limit $k\,R\rightarrow 0$ the mode with $m=0$
is a longitudinal mode with $v_{ph}=c_{t}$, which is approximately the sound
speed $c_{s}$ for $v_{A}\gg c_{s}$. This mode is often referred to as the
‘sausage mode’, with velocity inside the tube parallel to the magnetic field.

In the same limit and for $m>0$ one obtains the so-called ‘kink waves’, with
phase speeds related to the magnetic fields and densities through

$v_{ph}^{2}={\rho v_{A}^{2}+\rho_{e}v_{A,e}^{2}\over\rho+\rho_{e}}={1\over
4\pi}\cdot{B^{2}+B_{e}^{2}\over\rho+\rho_{e}}\,.$ (4.46)

These waves are transversal waves, and Spruit’s (1982) analysis takes into
account the dragging by the ambient medium. The phase speeds are obviously
$v_{ph}^{2}=v_{A}^{2}$ for $\rho_{e}=\rho,\,\,B_{e}=B$;
$v_{ph}^{2}=v_{A}^{2}/2$ for $\rho_{e}=\rho,\,\,B_{e}=0$, and
$v_{ph}^{2}=2\cdot v_{A}^{2}$ for $\rho_{e}=0,\,\,B_{e}=B$.

We now compare the observations of waves with the expectation from this linear
wave theory. We adopt that the waves propagate along the magnetic field and
that the influence of gravity on the wave properties is negligible. The period
at the acoustic cutoff of 200 s is longer than the periods, actually seen only
rarely, in our data. Likewise, the period for the cutoff of kink waves
(Spruit, 1981; Choudhuri et al., 1993) is approximately 400 s, for small
plasma $\beta$, which is the ratio of gas pressure to magnetic pressure,
$\beta=(8\pi p)/B^{2}$.

With the parameters in Table 4.1 for the surge discussed above in Sect. 4.2.3,
i.e. with gas pressure $p=0.38$ dyn cm-2 and mass density $\rho=2.3\times
10^{-13}$ g cm-3, the sound velocity is $c_{s}=16.6$ km s-1. From the
determination of parameters in a wide range of chromospheric H$\alpha$
structures by Tsiropoula and Schmieder (1997), Tsiropoula (2000), and
Tsiropoula and Tziotziou (2004) we obtain values of the sound speed in the
range of 13.5–16.7 km s-1. The widely found temperatures of $T=10^{4}$ K and
the mean molar mass of 0.8 from the ionization equilibrium of hydrogen found
from Table 4.1 and from the above works give a sound velocity of 14.4 km s-1.
The phase velocities of the slow waves observed here are compatible with these
values, if one accounts for possible small projection effects and for a small
reduction for the velocity of tube waves (cf. Eq. 4.45).

De Pontieu et al. (2004) adopted magnetic field strengths of the order of 100
Gauss in the chromosphere of active regions. With this value and the commonly
found mass densities of 0.8–2.3$\times 10^{-13}$ g cm-3, the Alfvén velocity
is $v_{A}=1\,000\dots 600$ km s-1, much higher than the velocities of the fast
waves in the present observations. We believe, that 100 Gauss is an upper
limit of the field strengths in the chromosphere of AOIs C and D. From high
spatial resolution (approximately 035) data from a plage region by Bello
González and Kneer (2008) we find an average field strength in the photosphere
of 60–90 Gauss. This may possibly be reduced by a factor of 2 in chromospheric
fibrils as in AOIs C and D by spreading out of the field lines over areas
which possess little field in the photosphere. Otherwise the fibrils would not
be so elongated. Yet still this yields to Alfvén velocities of $v_{A}\approx
200$ km s-1, as a minimum value.

Giovanelli (1975) has measured velocities of 70 km s-1 in chromospheric
H$\alpha$ structures. With a magnetic field strength of 10 Gauss and with
reasonable particle densities he arrived at the Alfvén velocity in agreement
with these measured phase velocities. In the present work, one would need
field strengths as low as 5 Gauss for an Alfvén velocity of 32 km s-1 as
observed. We note that even with 5 Gauss the motions are still dominated by
the magnetic field, i.e. $\beta\ll 1$ holds.

We estimate the maximum phase speed measurable from our data to 250–300 km
s-1. Such velocities would still be detectable. The phase speeds found here
are in the range 25–35 km s-1. (The highest measured speed amounts to 42 km
s-1). These are obviously incompatible with Alfvén waves in a homogenous
magnetic field with 30–100 Gauss. We mention several possibilities to
reconcile our measurements with the picture of fast mode magnetoacoustic waves
along the magnetic field, i.e. of Alfvén waves.

  1. 1.

The magnetic field strength in the fibrils of AOIs C and D is indeed as low as
5 Gauss, which is not very probable considering the very high activity in the
whole area observed. AOIs C and D are not especially located at the outskirts
of this activity.

  2. 2.

Propagation of a fast mode wave in a flux tube surrounded by a medium with low
or zero field strength but with high gas density would reduce the phase speed
(cf. Eq. 4.46).

  3. 3.

Apparently, the waves start as slow mode waves with phase velocities of the
order of 10–14 km s-1 and then are transformed into fast mode waves
propagating with Alfvén velocity. Yet the transformation does not occur
immediately. Examples are seen in Fig. 4.11. While the solitary waves evolve
into fast mode waves their wave packages get dispersed and they decay by
spreading out along the direction of propagation.

  4. 4.

We do not measure phase velocities but group velocities of solitary wave
packages. We have calculated for the slow mode the cusped surface of the wave
front according to Ferraro and Plumpton (1966, cf. their Fig. 13), which is
rotationally symmetric about the direction of the magnetic field. The adopted
Alfvén and sound velocities were 200 km s-1 and 16 km s-1, respectively. The
maximum velocity of this surface is only marginally larger than the sound
speed by 3.2%, and the maximum deviation from the direction of the magnetic
field is 001. Thus, the propagation of such slow mode pulses is practically
along the magnetic field with the sound velocity.

  5. 5.

The picture is actually more complicated: The waves with low phase speed seen
here are not pure longitudinal waves. The gas velocities of the waves have a
strong transversal component of the order of 3 km s-1. Furthermore, the
propagation of the fast waves deviates from straight lines, their motion
appears more wiggly, possibly because the magnetic fields are entangled. Under
the aspect of these observations the linear theory of small perturbations of
straight flux tubes appears to be not sufficient.

####  4.2.5 Summary on some properties of the active chromosphere

Thanks to the good resolution, we could follow the evolution of small-scale
chromospheric structures of an active region. From the rich dynamical
processes in the observed, very active, flaring region some areas were
selected for detailed investigation in the present work:

  1. 1.

A small surge: It showed repetitive occurrence with a rate of some 10 minutes.
The surge developed from initial small active fibrils to a straight, thin
stucture of approximately 15 Mm length, then retreated back to its mouth to
reappear again two times. The gas velocities reach approximately 100 km s-1.
The rebound shock model by Sterling and Hollweg (1989) seems to be a viable
explanation.

  2. 2.

Two small-scale, synchronous, possibly sympathetic flashes, or mini-flares: In
a pair of small areas, two brightenings occurred simultaneously and
disappeared during two H$\alpha$ scans with total duration of 45 s.
Presumably, the evolutionary time scale is much shorter, few to 10 s. Yet we
could follow the evolution with a temporal resolution of 2 s by analysing
H$\alpha$ filtergrams at different wavelengths. One of the two flashes showed
an apparent proper motion with a speed up to 200 km s-1, while it was
developping a high emission, above the continuum intensity, in the red part of
the H$\alpha$ profile. However, the cadence of the scanning was too slow to
decide whether the temporal evolution consisted in a rapid horizontal proper
motion with a final fast down flow or in a rapid change of emission at fixed
local postitions.

  3. 3.

Magnetoacoustic waves in long fibrils: In two areas with long fibrils, the
structures exhibited many magnetoacoustic waves running parallel to the
fibrils, thus presumably also parallel to the magnetic field. The waves are
mostly solitary. Few times, two or three repetitive wave trains could be seen
with periods of 100–180 s. The waves start at the footings of the fibrils with
a speed of 12–14 km s-1, which is not much lower than the sound speed
estimated for such structures and similar to the tube speed. Most of the waves
get accelerated to reach phase speeds of approximately 30 km s-1. Then they
spread out along the fibrils and fade. The final phase speed is much lower
than the Alfvén speed of $\geq 200$ km s-1, estimated from reasonable magnetic
field strengths in the active region chromosphere of 30–100 Gauss and
reasonable mass densities in the fibrils of 2$\times 10^{-13}$ g cm-3.
Furthermore, we observe that the slow waves have strong transversal (LOS)
velocity components with $\sim$3 km s-1, i.e. are not purely longitudinal, and
that the fast waves consist of short (1–2), thin ($\sim$05) blobs and
apparently move along sinuous lines. We conlude from these findings that a
linear theory of wave propagation in straight magnetic flux tubes is not
sufficient.

###  4.3 Comparison between speckle interferometry and blind deconvolution

In Sec. 3.1 we introduced the image degradation problem due to atmospheric
distortions for all ground based solar observations and astronomical
observations on general. In Sec. 3.2.1 we explained the adaptive optics
approach used at the VTT to reduce the image distortions in real time.
Finally, in Sec. 3.3.3 we described the basis of two different _post factum_
image reconstruction techniques, the Speckle Imaging (SI) and one type of
Blind Deconvolution (BD) with simultaneous Multiple Objects and Multiple
Frames (MOMFBD).

In the case of data from the G-FPI, the SI approach reconstructs separately
the broadband images and uses both the original recorded frames (after dark
subtraction and flat fielding) and the reconstructed image from this channel
to obtain the reconstructed narrow-band images at the various positions along
the spectral line.

The MOMFBD code applied to our G-FPI data uses at the same time, for each
spectral position, the various (15) pairs of simultaneously recorded broadband
and narrow-band frames. Thus, at 21 wavelength positions along the spectral
line, two different objects were observed and their images were reconstructed
with 15 frames per object.

In this thesis work we used the SI method for fields of view on the solar
disc. The last version of the code takes into account the field dependence of
the PSF around the AO lockpoint, so we will refer to this version as SI+AO.
However, data frames on and off the limb cannot be reconstructed with the
current code. Near the limb the contrast is lower than near disc center, which
makes any reconstruction more difficult. Moreover, KAOS can lock on the low-
contrast structures near the limb only under very good seeing conditions.
Also, the limb darkening at large heliocentric angles makes it difficult to
determine the STF on the rings around the lockpoint for the SI+AO. Beyond the
problems inside the limb, off-the-limb emission features seen in the narrow-
band images lack broadband counterparts, and therefore there exist no
simultaneous data from which we could apply the second part of the SI to
reconstruct the off-limb parts of the images.

Dataset _limb_ (results in Sec. 5.2) was recorded under very good seeing
conditions, with KAOS locked on a nearby facula correcting 27 (Zernike
polynomial) modes most of the time. For the post factum image reconstruction
we used the MOMFDB, for which the limb darkening presents no problems.
Spicules above the limb do not posses a simultaneously observed broadband
object, so it is expected that their spatial resolution is lower, since there
are no multiple objects, just the multiple frames for each spectral position
in the narrow-band channel.

In order to compare results from both approaches we have reconstructed with
both methods the same field of view on the disc, a frame of the _sigmoid_ data
set. In this Section we present the comparison. As we show, for the BD case we
made two different reconstructions with different limits of the expansion of
the aberration in Karhunen-Loeve (K-L) modes. In the first case the expansion
of the wavefront aberration is done until de $17^{th}$ K-L polynomial, while
on the second run we expanded to the first 100 modes. The running time of the
code is highly sensitive to the number of modes, being slower with more modes.

Figure 4.15: Results for different image reconstruction techniques. Axes are
in arcseconds. White rectangles enclose the region where the RMS contrast is
calculated. _Upper left_ : Best speckle raw image, contrast is 5.9%. _Upper
right_ : SI+AO result, contrast is 11.1%. _Lower left_ : MOMFBD running 17
modes, contrast is 6.7%. _Lower right_ : MOMFBD running 100 modes, contrast is
10.0%.

Figure 4.16: Comparison of the power spectra (in arbitrary units) of the same
image for different image reconstruction techniques.

Figure 4.17: Results for different image reconstruction techniques for the
line center narrow-band filtergram. Scales on the axes are in arcseconds.
_Upper left_ : Best speckle raw image.  _Upper right_ : SI+AO result. _Lower
left_ : MOMFBD running 17 modes.  _Lower right_ : Image difference. In this
case the differences reach 45% of the fluctuations in the reconstructed
frames.

Figure 4.18: Comparison of the power spectra of the same image for different
image reconstruction techniques.

##### Broadband

Figure 4.16 compares the full FoV image, while Fig. 4.16 shows the
corresponding power spectra. The speckle frame corresponds to the one with
highest rms contrast (5.9% inside the white rectangle). The SI+AO
reconstruction shows a much higher resolution, with more power at all
frequencies than the speckle frame for angular scales larger than $\sim
0\farcs 32$. The reconstructed image has also less noise than the speckle
frame (small-scale end of the power spectra). The rms contrast of granulation
for this image is 11.1%. In the case of the MOMFBD with 17 modes the power of
the reconstructed image is significantly lower than for SI+AO, albeit having a
lower noise level (comparable even with the burst average). We have to run up
to 100 modes to arrive at a similar contrast as for SI+AO. The noise threshold
for the last run, coincides with that of the SI+AO approach. The rms contrast
of granulation for these images are 6.7% with 17 modes and 10.0 % with 100
modes.

Figure 4.16 shows also the power spectrum of the difference between SI+AO and
MOMFBD100 (green line), which is many orders magnitude lower than one of the
power spectra themselves. Only at scales smaller than $\sim 0\farcs 4$, the
difference becomes of the same order as the power spectra. Taking the
differences of the reconstructed images shows that $99.8\%$ of the pixels in
the FoV have intensity fluctuations lower than $15\%$ of the intensities in
the images themselves.

##### Narrow-band

Figure 4.19: Close-up subfield from the narrow-band spectrogram at the
H$\alpha$ core. Axes are in arcseconds.

The narrow-band images have lower intensities than the broadband images,
especially at the core of the H$\alpha$ line. Much less images are used for
reconstruction, so a lower resolution is expected. Figure 4.18 compares the
images at the H$\alpha$ line center, while Fig. 4.18 shows the corresponding
power spectra. The SI + AO reconstruction shows a higher resolution than the
speckle frame, with more power at scales larger than $\sim 0\farcs 5$. At
smaller scales, the speckle frame is dominated by noise. The MOMFBD with only
17 modes gives already a similar resolution than the SI+AO and better
treatment of the noise. Fig 4.19 shows a close-up region where the better
noise treatment of the MOMFBD is clearly visible.

The difference between the methods is bigger than in the broadband case, as
expected since the intensity and resolution are lower. Nonetheless the
agreement is very high, 92.8% of the pixels in the difference image have
amplitudes smaller than $0.15$ of the average intensity in the reconstructed
images (similar results are found for other spectral position, reaching 99% in
the wings, at wavelengths $\pm 1$ Å off the line center).

##### Conclusions

In this Section we have shown the good convergence of both post-processing
approaches. Using different techniques we arrive at similar results and
spectral profiles. The amplitudes of the difference images are lower than
$0.15$ of the average image intensity in more than 99% of the broadband and
above $\sim 90$% for the narrow-band images. In the case of the broadband
reconstruction it was necessary to use 100 modes for the MOMFBD method to
reach similar results as for SI+AO, while, in the narrow-band case, already
with 17 modes the MOMFBD gives better images than SI+AO.

The main disadvantage of BD methods is the computational load. The
reconstruction of the single data set from broad and narrow-band and only 17
modes takes $\sim 7$ hours to process with 20 CPU cores of $3.2$GHz. For the
100 modes run, given the limited resources, we only used the broadband frames
(Multi Frame BD). If the data set “sigmoid” were reconstructed with BD
methods, even with only 17 modes, it would have taken around 130 days on our
computing resources.

The main advantage of the BD is its ability to reconstruct an image even with
only few frames. This is of special importance when observing fast evolving
targets. The SI needs much more frames. The property of reconstructing
_simultaneously_ recorded images from different “objects” (e.g. broadband and
the H$\alpha$ narrow-band) leads to a perfect sub-alignment of the results,
which avoids spurious signals in derived quantities. Note however that, not
simultaneously observed objects, like in the several consecutive scans with
the G-FPI, are not aligned since they are not recorded under identical
_seeing_ conditions.

The SI+AO method is considerably much faster, around 10 and 15 minutes for the
broadband and narrow-band images, respectively, with the same computers used
for the MOMFBD reconstruction and gives better results for the broadband
reconstruction, even using 100 modes in the latter method. However, with
MOMFBD, the resolution and treatment of the noise is better in the narrow-band
case. The main current advantage of the BD methods for our work and data is
the possibility of reducing narrow-band limb and off-the-limb data scans.

Anisoplanatism is an issue common to both approaches. In both cases the large
FoV is divided into smaller subfields, where the assumption of isoplanatism is
valid. It is therefore important to address this point for both cases. The
image difference does not show any subfield pattern. However, there can still
be some small effects. For this reason we have used the integrated contrast
profile of the difference, defined as

$\mathbb{CI}=\sum_{\lambda}\Big{|}\frac{I_{SI+AO}(\lambda)-I_{MOMFBD_{17}}(\lambda)}{I_{SI+AO}(\lambda)}\Big{|}\,,$
(4.47)

where $I_{SI+AO}(\lambda)$ and $I_{MOMFBD_{17}}(\lambda)$ correspond to the
reconstructed images using the SI+AO method, and to the images using MOMFBD
with 17 modes, respectively.

$\mathbb{CI}$ qualitatively measures the total difference between the
profiles. If they were equal, then $\mathbb{CI}$ would be 0, while an
increasing difference in the profiles increases the value of $\mathbb{CI}$.
Since the subfield locations are the same for all the spectral positions, this
information is added along the scan, while the intensities of the structures
at each position are essentially subtracted out. The subfield pattern does not
disappear with the subtraction of images reconstructed with different methods
since they do not necessarily coincide.

Figure 4.20 shows the calculated $\mathbb{CI}$. The weak subfield pattern is
revealed, especially in regions where the difference is low (dark background).
The mean edge length of the squares is approximately around 32 pixels.

The amplitude of the grid pattern is very low, only revealed after the
calculation of $\mathbb{CI}$. Presumably this comes from the apodization. When
joining common regions on overlapping subfields, the common parts are overlaid
in the final image. This, while preserving the structures, reduces the noise,
which leads to slightly smaller noise levels in these overlapping lanes. This
grid is common for all wavelength positions. The difference between the
methods is low enough to reveal this small decrease of the noise level
(leading to darker areas in Fig. 4.20) when the total effect is calculated, by
using the $\mathbb{CI}$ parameter. Therefore, regions with more contrast,
where the difference between SI and BD is bigger, the presence of this pattern
is masked, as shown in the figure.

Figure 4.20: Isoplanatic subfield array pattern when calculating
$\mathbb{CI}$. The mean edge length of the squares is approximately 32 pixels.
Axis scale is in arcseconds.

##  Chapter 5 Spicules at the limb 111Contents from this Chapter have been
partially published as Sánchez-Andrade Nuño et al. (2007a)

Spicules, known for more than 130 years (see the hand drawings by Secchi,
1877), represent a prominent example of the dynamic chromosphere. We refer the
reader to reviews by Beckers (1968, 1972) and to the paper by Wilhelm (2000)
on UV properties. According to these works, spicules are seen at and outside
the limb of the Sun as thin, elongated features. They develop speeds, measured
from both proper motion and Doppler shifts, of 10–30 km s-1 and reach heights
of 5–9 Mm on average, during their lifetimes of 3–15 minutes. Recent
observations from HINODE (Kosugi et al., 2007) and own results presented below
in Sec. 5.2 have changed the traditional picture. Some spicules live for only
few seconds, and spicules may be much more inclined with respect to the
vertical than adopted hitherto.

As pointed out by Sterling (2000), a key impediment to develop a satisfactory
understanding has been the lack of reliable observational data. Many
theoretical models have been proposed to understand the nature of spicules,
using a wide variety of motion triggers and driving mechanisms. In this
Chapter we focus on the He i 10830 Å triplet emission line (see Sec. 2.3),
using recent technical improvements in observational facilities, and on the
results from the limb observations in H$\alpha$.

###  5.1 Spicule emission profiles observed in He i 10830 Å

The energy levels that take part in the He i 10830 Å triplet are basically
populated via an ionization-recombination process (Avrett et al., 1994). The
EUV coronal irradiation (CI) at wavelengths $\lambda<504$ Å ionizes the
neutral helium, and subsequent recombinations of singly ionized helium with
free electrons lead to an overpopulation of all ortho-helium levels.
Alternative theories suggest other mechanisms that might also contribute to
the formation of the helium lines relying on the collisional excitation of the
electrons in regions with higher temperature (e.g., Andretta and Jones, 1997).
We are able to provide observational evidence of the link between the corona
and the infrared emission of this line, in the frame of the current
theoretical models of the solar atmosphere.

Centeno (2006) modelled the ionisation and recombination processes using
various amounts of CI, non-LTE radiative transfer, and different atmospheric
models (see also Centeno et al., 2007). They have simulated limb observations
for different heights, obtaining synthetic emission profiles in spherically
symmetric models of the solar atmosphere. One conclusion of their study is
that the ratio of intensities $({\cal R}=I_{\rm blue}/I_{\rm red})$ of the
‘blue’ to the ‘red’ components of the He i 10830 Å emission is a very good
candidate for diagnosing the CI. The population of the metastable level
depends on optical thickness, whose variation with height governs the change
in the ratio $\cal R$ as a function of the distance to the limb.

Trujillo Bueno et al. (2005) measured the four Stokes parameters of quiet-Sun
chromospheric spicules and could show evidence of the Hanle effect by the
action of inclined magnetic fields with an average strength of the order of 10
G. They modelled the He i 10830 Å profiles assuming the medium along the
integrated line of sight as a slab of constant properties and with its optical
thickness as a free parameter. Trujillo Bueno et al. (2005) showed that the
observed intensity profiles and their ensuing $\cal R$ values can be
reproduced by choosing an optical thickness significantly larger than unity.
Centeno (2006) demonstrated that this optical thickness is related to the
coronal irradiance (through the ratio $\cal R$), thus providing a physical
meaning to the free parameter in the slab model (see also Centeno et al.
2007).

Figure 5.1: Measured He i 10830 Å emission profiles for increasing distances
to the solar limb, scanning a broad range of the height extension of the
spicules. Each profile is the average of the 312 pixels along the slit (which
was always kept parallel to the limb).

####  5.1.1 Observational intensity profiles and intensity ratio

We present novel observations showing the spectral emission of He i 10830 Å
and its dependence on the height of the spicules above a quiet region. We
compare the deduced observational $\cal R$ with that obtained from detailed
non-LTE numerical calculations using available atmospheric profiles.

These data correspond to the data set described in Table 3.2. After the
standard reduction process (Sec. 3.4.3 ) we obtain 21 intensity profiles above
the infrared limb, with a step size of $0\farcs 35$. Figure 5.1 shows the
emission profiles of the He i 10830 Å (after the reduction process) for
different heights above the limb. Figure 5.2 illustrates this in three
dimensions, as a function of wavelength and the distance to the solar limb,
clearly showing a change in the intensity ratio of the blue and red components
of the multiplet $({\cal R}=I_{\rm blue}/I_{\rm red})$ with height.

Figure 5.2: 3D representation of the measured He i 10830 Å emission profiles
for increasing distances to the solar visible limb. Note that the x-axis is
wavelength, the y-axis the height above solar limb and the z-axis the
intensity normalised to the maximum emission in the line center of the red
component.

For the calculation of $\cal{R}$ we need to determine the amplitudes of the
blue and red components of the emission profile (as shown in Fig. 5.3).

To determine the core wavelength of the red component of the triplet we fitted
a Gaussian profile to its core, in a 1.3 Å range around the maximum. After
symmetrising the observed profile around this maximum, using the values on the
red side of the red component, we fitted another Gaussian function to the
resulting symmetric profile. Subtraction of the fitted symmetric profile from
the data leaves the emission profile of the blue component, which was also
approximated by a Gaussian to determine its central wavelength. Our tests
trying to fit directly both profiles using two Gaussians failed in a number of
cases, probably due to the following reasons: (a) the red component is in fact
the result of two blended lines, (b) the much weaker blue component was almost
hidden in the broadened red component, and (c) the presence of noise. Our
technique determines first the red component and then, after subtraction of
the fitted profile, the blue one.

We have thus separated the helium emissions into their red and blue components
assuming only that both are present and that they are both symmetric. We can
now measure their widths and intensities and also check that the line core
positions coincide with the theoretical ones. After the fitting process the
residuals between measured and observed profiles were small, the largest
errors occurring in the determination of the core intensities of the red line.
This happens because the red component consists of two blended lines (with a
separation of 0.09 Å), a fact that flattens the emission profile near the core
as opposed to a more peaked Gaussian function. Nevertheless, the differences
between fitted profiles and data are only significant in the red core and are
always lower than +0.08 of the maximum normalised intensity, with a mean
difference of $\sim$0.02. To avoid systematic errors, we used the
observational values for the center of the red component when calculating
$\cal R$.

Figure 5.3: Determination of the blue and red components of the He i 10830 Å
triplet from the observed emission profiles. In this example the slit was
placed at 14 off the solar visible limb. See text for details. The solid line
represents the average emission profile. The dotted line is the Gaussian fit
to the symmetrised red component. Subtraction of this from the observed
profile leaves the blue component, which is also fitted by a Gaussian profile
(thin solid line). The sum of both Gaussians (dashed line) gives the fit to
the observed profile.

####  5.1.2 Results

The chromospheric temperature and density are too low to populate the ortho-
helium levels via collisions (Avrett et al., 1994). The EUV irradiation from
the corona (CI) ionises the para-helium, and the subsequent recombinations
lead to an overpopulation of all the ortho-helium levels, in particular those
involved in the 10830 Å transitions. Centeno (2006) and Centeno et al. (2007)
have modelled the off-the-limb emission profiles and concluded that the ratio
$\cal{R}$ = $I_{blue}/I_{red}$ is a function of the height and a direct tracer
of the amount of CI. Here we compare the results from the theoretical
modelling with observations.

Trujillo Bueno et al. (2005) modelled their spectropolarimetric observations
assuming a slab with constant physical properties with a given optical
thickness. In the optically thin regime $\cal R$ $\sim$ 0.12, which is the
ratio of the relative oscillator strengths of the triplet. As the optical
thickness (at the line-center of the red blended component) grows, this ratio
also increases until it reaches a saturation value slightly larger than 1 for
$\tau\sim 10$. (This type of calculation can be done and improved as explained
in Trujillo Bueno & Asensio Ramos 2007). To reproduce the observed emission
profile Trujillo Bueno et al. (2005) had to choose $\tau\sim 3$.
Interestingly, the values of $\tau$ yielded by this modelling strategy are
consistent with the more realistic approach of Centeno (2006), where non-LTE
radiative transfer calculations in semi-empirical models of the solar
atmosphere are presented, using spherical geometry and taking into account the
ionising coronal irradiation. With our data we are able to test such
theoretical calculations by comparing the measured values of $\cal R$ with
those resulting from various chromospheric models. This way we may eventually
trace the amount of CI inciding on the spicules. The analysis described above
yielded the values of $\cal R$ for the observed profiles. The resulting
dependence on the distance to the solar limb, for each pixel along the slit
and each position of the slit above the limb, are presented in Fig. 5.4. The
solid black line gives the average value of $\cal R$.

Figure 5.4: Measured ratio $\cal R$ = $I_{blue}/I_{red}$ as a function of
distance to the solar limb. Thin lines come from each pixel along the slit.
The thick solid line represents the average and the dashed line the value of
the optically thin regime.

The dependence of $\cal R$ with height can be understood in a qualitative way
as follows: In the outer layers of the chromosphere the density is so low that
the transitions occur in the optically thin regime. With decreasing altitude
the ratio $\cal R$ increases (proportionally with density) until a maximum
optical thickness is reached. At even lower layers, although the density still
continues to rise, the extinction of the coronal irradiance leads to a
reduction in the number of ionizations, which results in a decrease of the
optical thickness in the core wavelength of the red component, and thus in a
decrease of $\cal R$.

For a quantitative comparison with theoretical modelling we have used the
results from Centeno (2006) and Centeno et al. (2007) where they calculated
the ratios $\cal R$ for different standard model atmospheres: FAL-C and FAL-P
(Fontenla et al., 1991) and FAL-X (Avrett, 1995). The FAL-C and FAL-X models
may be considered as illustrative of the thermal conditions in the quiet Sun,
while the FAL-P model of a plage region. The FAL-X model has a relatively cool
atmosphere in order to explain the molecular CO absorption at 4.6 $\mu$.

The comparison is shown in Fig. 5.5. We notice that the modelled height
variations of $\cal{R}$ agree only in a qualitative manner with what is found
in our observations. However, the calculations from different models of the
solar atmosphere are unable to reproduce the measured ratio. Higher values of
the coronal irradiance lead to an increase of the optical thickness (at the
line centers of the He i multiplet) and an upward shift in the run of
$\cal{R}$ vs. height. Yet the shape of the height dependence is mainly given
by the atmospheric density profile and the attenuation of the ionising
radiation as it reaches the lower layers of the chromosphere. It is also clear
from Fig. 5.5 that the models do not extend high enough.

Figure 5.5: Observed (average) vs. theoretical variation of the ratio $\cal
R$$=I_{blue}/I_{red}$ with height.

####  5.1.3 Conclusions

The theoretical behaviour of the ratio $\cal R$ agrees qualitatively with
observations. Yet, a quantitative comparison shows poor agreement. Also, the
simulated ratios are highly model dependent. As already explained, the failure
to reproduce the observed profiles is very likely due to the density
stratification not being adequate for spicule modelling and to the limited
vertical extension of the atmospheric models. Modelling of the intensity ratio
$\cal R$ in the He i infrared triplet should account for the fact that the
solar chromosphere is inhomogeneous on small scales and that the spicules are
small-scale intrusions of chromospheric matter into the hot corona.

###  5.2 High resolution imaging of spicules

De Pontieu et al. (2007) recently published high resolution observations of
spicules with the Solar Optical Telescope on board Hinode (Kosugi et al.,
2007) in the Ca II H line at 3968 Å. They find at least two types of spicules
that dominate the structure of the magnetic solar chromosphere: Type I with
3-7 minute timescales that correspond to the hitherto known spicules, and the
new Type II spicules, developing in $\sim 10$ s, and lasting 10-150 s. These
are also very thin, with widths down to the spatial resolution (120 km). Also,
Trujillo Bueno et al. (2005) used spectropolarimetric observations and a
theoretical modeling accounting for radiative transfer effects. They find that
the magnetic field in spicules is aligned with the visible structures and
measure field strengths of up to 40 G with an inclination of 35∘ with respect
to he local vertical.

This Section compares these observational and theoretical properties with our
high spatial resolution observations with the G-FPI. We use the dataset
“limb”. It consists of several consecutive H$\alpha$ scans with a field of
view that includes the limb and a region outside the disc up to a height where
no emission from spicules in the H$\alpha$ core is observed. The _seeing_
conditions were extremely good during the observations and the AO system could
lock on a nearby facula. After usual dark subtraction and flat fielding we
have used the BD method (see Sec. 3.3.3) to achieve highest spatial
resolution. We were observing the limb near both poles. In Figures 5.6 and 5.7
we present some examples of the reduced data.

Our time series of four minutes duration already shows a wide range of
dynamics. We observe both long lasting spicules and fast evolving phenomena.
Measuring the inclination of the projected spicules to the local vertical we
find angles up to 30 for the north pole, as it has been reported (Beckers,
1968) . The projected height above the limb varies from the wings to the core,
from 2770 to 3750 km at $\pm 0.5$ and $\pm 1$Å respectively. Near the south
pole we find, however, much stronger emission and higher inclinations. The
maximum angle is close to 70 from the local vertical, while the maximum height
reaches up to 8250 km. We also find one horizontal fibril/spicule, as well as
the presence of kinks or bends in some spicules. The width of single resolved
spicules varies from a maximum width of 1 000 km at the spicule footing to a
minimum size of 250 km, almost down to the resolution limit of the images,
both in faint spicules and in others with strong emission. We also can
retrieve the spectral profile at each pixel.

Figure 5.6 demonstrates an important contribution to the understanding of
spicules. It solves the long-standing question about the counterparts of
spicules on the disc (Grossmann-Doerth and Schmidt, 1992). The first and last
four filtergrams of the scan across H$\alpha$ in this figure show that
spicules outside the limb continue as dark fibrils inside the disc.

In Fig. 5.8 we show the mean variation from the disc to the limb of the
intensity in H$\alpha$ around the north pole. Further, Fig. 5.9 presents mean
intensity variations from the disc to the limb for several wavelengths around
the H$\alpha$ line center. The emission at the line center is almost constant
from the disc up to a height of around 5 above the limb, where the intensity
starts to decrease.

Figure 5.6: Reconstructed narrow-band scan observed near the solar north pole.
The size of each image is $56\farcs 1\times 19\farcs 1$. The wavelength of the
filtergrams increases by $0.1$Å from top left to bottom right row by row. The
third image in the third row is closest to the center of the mean line
profile. The images have been rotated to have the limb parallel to the
horizontal axis.

Figure 5.7: Limb H$\alpha$ 2D filtergram at $\lambda_{0}+1.1$ Å near the south
pole, where a coronal hole was present. This region shows much stronger
emission and more variation of spicule width, height and inclination as Fig.
5.6. A background of thin vertical spicules can be seen overlaid with wider
and more inclined spicules, including nearly horizontal jets. Some of the
spicules appear to be bent and show internal structure such as splitting into
parallel jets. The maximum projected height above the limb is $\approx 8\,250$
km, while the mean height at this wavelength is $\approx$3700 km. The image
has been rotated to show a horizontal limb in the presentation.

Figure 5.8: Image representation of the mean measured spicular profiles from
image 5.6. the x-axis is the height above limb, while the y-axis is the
wavelength around H$\alpha$ line center (black horizontal line). Horizontal
cuts at $\lambda-\lambda_{0}=[0,\pm 0.5,\pm 1]$Å are shown in Fig. 5.9. Figure
5.9: Average over $11\farcs 2$ of H$\alpha$ intensity profiles inside and
outside the limb, for several line positions, observed near the solar north
pole. _Dotted line_ : broadband intensity at 6300 Å, the inflection point
defines the position of the solar limb; _green thick line_ : intensity at
H$\alpha$ line center, which is nearly constant till $\approx$5 above the limb
and then decreases outwards; _dashed lines_ : intensities at $-0.5$ Å (blue)
and $+0.5$ Å (red) off line center, with height of spicular visibility
decreasing at $\approx 3\farcs 7$; _solid lines_ : intensities at $\pm 1$Å off
line center. It is seen that the H$\alpha$ line turns from absorption inside
the limb into an emission line (line intensities higher than the continuum
intensity) above the limb.

##  Chapter 6 Conclusions and outlook

We have studied the dynamics of the solar chromosphere, both on the disc and
above the limb, using two spectral regions (H$\alpha$ in visible light and the
infrared He i 10830 Å multiplet). By means of real-time correction and
different post-processing techniques we have reduced the image degradation
induced by the Earth’s atmosphere achieving resolutions in H$\alpha$ up to
$0\farcs 5$ and better. This Chapter summarize the main conclusions of this
work.

##### Observations and analysis

The basic results from the observations taken from the disc are:

  * •

Data taken in the combination with the “Göttingen”-Fabry Perot Interferometer
(G-FPI), adaptive optics and speckle interferometry have high quality. We have
obtained a time sequence in H$\alpha$ of 55 min from the active region AR
10875 at heliocentric angle $\vartheta\approx 36\degr$. The time cadence is 22
seconds, and its field of view $77\arcsec\times 94\arcsec$. For each
interpolated time step we can retrieve 23 filtergrams along the H$\alpha$
spectral line with 45 mÅ FWHM and spatial resolution better than $0\farcs 5$ .
Simultaneous broadband images at 6300 Å were also obtained, with spatial
resolution of $0\farcs 25$, close to the telescopic diffraction limit.

  * •

We have observed the dynamics of a small surge in detail: It showed repetitive
occurrence with a rate of some 10 min. The surge developed from initial small
active fibrils to a straight, thin structure of approximately 15 Mm length,
then retreated back to its mouth to reappear again two times. The gas
velocities reach approximately 100 km s-1. The rebound shock model by Sterling
and Hollweg (1989) seems to be a viable explanation.

  * •

The region was very active during the observations. We studied two small-
scale, synchronous, possibly related flashes, or mini-flares. The simultaneity
is within seconds, while their total evolution time was $\sim 45$s. The
brightenings were separated by $\sim 14$ Mm. The used scanning parameters of
the G-FPI were slow for this fast evolution, yet we could follow it with a
temporal resolution of 2 s by analysing filtergrams taken consecutively at
different wavelengths across the H$\alpha$ line. One of the two flashes showed
an apparent proper motion with a speed up to 200 km s-1, while developing a
high emission in H$\alpha$, above the continuum intensity.

  * •

For the observations of waves we restricted our study to two areas exhibiting
long fibrils. Yet the results likely represent the typical behavior of
chromospheric magnetoacoustic waves within this active region. By means of
high-pass frequency filtering, we observe waves running parallel to the
fibrils, thus presumably also parallel to the magnetic field. They were mostly
solitary waves, although sometimes repetitive wave trains could be seen with
periods of 100–180 s. Most pulses start with velocities on the order of 12–14
km s-1 and get accelerated to reach phase speeds of approximately 30 km s-1.
Furthermore, we observe that the slow waves have strong transversal (LOS)
velocity components with $\sim$3 km s-1, i.e. are not purely longitudinal, and
that the fast waves consist of short (1–2), thin ($\sim$05) blobs and
apparently move along sinuous lines.

Further, we have analyzed observations of spicules inside the disc and above
the limb with the G-FPI data. Given the properties for this kind of
observations we could not use the speckle interferometry method to reduce the
atmospheric distortions. Instead we have used the blind deconvolution
approach, in particular the version developed at the Swedish Institute for
Solar Physics for multiple simultaneous objects with multiple frames per
object (van Noort et al., 2005). The observations and analysis yielded the
following main results:

  * •

It is possible to successfully use multi-object multi-frame blind
deconvolution methods with the G-FPI to reduce atmospheric distortions. This
is specially important for on-limb observations, where the current speckle
interferometry method is not applicable.

  * •

We have observed spicules in H$\alpha$ at both solar polar caps. Compared with
the solar north pole, we find much stronger spicular emission at the south
pole that could be related to the presence of a coronal hole. The maximum
projected height reaches 8250 km, while we see inclinations of the spicules up
to 70 form the local vertical. We can resolve the detailed structure of the
spicules as well as the presence of kinks or bends in some cases. The width of
a single spicule ranges from 1 000 km down to the resolution limit of around
250 km.

  * •

Using the retrieved spectral profile we can observe that spicules outside the
limb continue as dark fibrils inside the disc, as shown in Fig. 5.6. This
answers a long standing question, e.g. cited by Grossmann-Doerth and Schmidt
(1992).

Thus, we have used two different post-processing approaches to reduce the
image degradation with the H$\alpha$ spectral line data. Since both methods
are based on different approaches, we have reduced the same observational data
with both techniques and compared the results. These are the main conclusions:

  * •

The agreement of the images from both approaches is high. The achieved
resolution comes close to the diffraction limit in both cases. Even though
both methods split the image into isoplanatic subfields for individual
reconstruction, there is no difference of subfield re-composition when
comparing the results.

  * •

In general, the biggest advantage of speckle interferometry over blind
deconvolution is the small computational time required. A complete restoration
of one full H$\alpha$ scan like the ones used in this work is roughly $\sim
400$ times faster with speckle interferometry than with blind deconvolution
methods.

  * •

The main advantage of blind deconvolution methods is their versatility. It can
be applied with only few frames, with one or few simultaneous objects, or both
at the same time. This method is highly advisable when aiming for e.g., fast
evolving targets, or limb observations. The perfect sub-alignment of
simultaneous objects avoids spurious signals on deduced quantities, like
magnetograms.

  * •

For the broadband channel we find that the speckle interferometry gives images
with high contrast. Only when forcing the blind deconvolution method to
reconstruct up to high ($100$) Karhunen-Loeve modes, the results are similar.
We note that the speed of the reconstruction is proportional to the number of
modes.

  * •

The narrow-band images are clearly better reconstructed with the blind
deconvolution, even with only 17 Karhunen-Loeve modes. The noise treatment
gives smoother images with details at smaller scales (of the order of $\approx
0\farcs 3$).

Further, we have obtained and analysed spectroscopic measurements in the
infrared. We have centered our studies here on the intensity profiles of the
He i 10830 Å multiplet above the limb.

  * •

Recent work, like e.g. by Trujillo Bueno et al. (2005); Centeno (2006), has
demonstrated the importance of the intensity ratio between the blue and red
component of this triplet as tracer of the coronal irradiance. In this work we
present novel observations showing the variation of this parameter with
distance to the limb with a resolution of $0\farcs 35$ up to 7 above the solar
visible limb (See Fig. 5.4).

##### Interpretation of observations

For the interpretation of the observed data we have used several models and
previous theoretical results to compare with the presented data. The main
results from this analysis are:

  * •

From the intensity profiles of the H$\alpha$ spectral line inside the disc we
can infer many physical parameters. We have applied the lambdameter method as
a fast and easy way to retrieve qualitative velocity maps. Also we have used
Beckers’s 1964 cloud model. Our simple non-LTE inversion code provides the
possibility to infer many physical parameters, e.g. hydrogen and electron
density, mass density, temperature, …The results are in agreement with the
data given in the current literature.

  * •

From the linearization of the MHD problem, we discuss the interpretation of
the observed waves as magnetoacoustic waves. We assume estimates with
reasonable magnetic field strengths in the chromosphere of the active region
of 30–100 Gauss and reasonable mass densities in the fibrils of 2$\times
10^{-13}$ g cm-3. The observed wave speeds are much lower than the expected
Alfvén velocities. We conclude from these findings that a linear theory of
wave propagation in straight magnetic flux tubes is not sufficient.

  * •

From the infrared observations we have calculated the ratio of amplitudes in
the two main components of the He i 10830 Å multiplet. Centeno (2006) has
modelled synthetic limb observations according to the current theories of
formation of this triplet and chromospheric models. The agreement is only
qualitative. The failure to reproduce the observed profiles is very likely due
to the density stratification not being adequate for spicule modelling used
and to the limited vertical extension of the atmospheric models. Modelling of
the intensity ratio should account for the fact that the solar chromosphere is
inhomogeneous on small scales and that the spicules are small-scale intrusions
of chromospheric matter into the hot corona. Future models of the solar
chromosphere should be constrained by the observational evidences presented
within this work.

##### Outlook

The solar chromosphere represents a lively and exciting field of research. The
wealth of structures, its dynamics and the wide range of evolution timescales
are the consequence of the peculiar properties of this atmospheric layer.
Current instruments like the ones used here, are able to observe and study in
great detail new phenomena, that test current models and, as a consequence,
helps our understanding of the solar atmosphere. This thesis aimed at
contributing to the understanding. Yet, work to extend this research is
already in progress. Here we shortly describe some of this work and give an
outlook to further steps to be undertaken next.

  * •

The blind convolution method provides a practical way to study the spicules in
H$\alpha$ near and above the limb. Data from a short time sequence, taken
under very good seeing conditions, are currently under reconstruction with
phase diversity methods. The analysis will shed light onto the dynamic
phenomena in spicules.

  * •

We have learned that the sequential scanning, with the G-FPI, with cadence of
22 s is not fast enough in some cases. For future observations, we can design
scanning modes of 2–3 second resolution taking images at fewer wavelength
positions in a spectral line, like H$\alpha$.

  * •

New infrared data of spicules near the solar poles and the equator, below
coronal holes or in coronal active regions will help us to understand the
detailed formation process of the He i 10830 Å lines.

  * •

Full Stokes spectropolarimetric data of the He i 10830 Å multiplet are
available from an earlier observing campaign. Scans above the limb were
performed under very good seeing conditions. We can therefore extend our
present analysis and study the polarization. We aim to investigate the Hanle
effect as suggested by Trujillo Bueno et al. (2005) and make use of available
inversion techniques like e.g. from Lagg et al. (2004).

  * •

The new Gregor telescope (Balthasar et al., 2007) will host the G-FPI from the
coming year on. The combination of this new facility with other instruments
like Hinode (Kosugi et al., 2007), will provide new exciting resources for
further research.

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## Publications

  1. _Refereed papers_

  2. 1.

B. Sánchez-Andrade Nuño, R. Centeno, K. G. Puschmann, J. Trujillo Bueno, J.
Blanco Rodríguez, and F. Kneer. Spicule emission profiles observed in He i
10830 Å. A&A, 472:L51–L54, Sept. 2007.

  3. 2.

B. Sánchez-Andrade Nuño, N. Bello González, J. Blanco Rodríguez, F. Kneer, and
K. G. Puschmann. Fast events and waves in an active region of the Sun observed
in H$\alpha$ with high spatial resolution. A&A, submitted, Dec. 2007.

  4. 3.

J. Blanco Rodríguez, O. V. Okunev, K. G. Puschmann, F. Kneer, and B. Sánchez-
Andrade Nuño. On the properties of faculae at the poles of the Sun. A&A,
474:251–259, Oct. 2007.

  5. _Conference contributions_

  6. 4.

B. Sánchez-Andrade Nuño, R. Centeno, K. G. Puschmann, J. Trujillo Bueno, and
F. Kneer. Off-limb spectroscopy of the He I 10830 Å multiplet: observations
vs. modelling. In F. Kneer, K. G. Puschmann, and A. D. Wittmann, editors,
Modern solar facilities - advanced solar science, pages 177–180, 2007.

  7. 5.

B. Sánchez-Andrade Nuño, K. G. Puschmann, and F. Kneer. Observations of a
flaring active region in H$\alpha$. In F. Kneer, K. G. Puschmann, and A. D.
Wittmann, editors, Modern solar facilities - advanced solar science, pages
273–276, 2007.

  8. 6.

B. Sánchez-Andrade Nuño, K. G. Puschmann, M. Sánchez Cuberes, J. Blanco
Rodríguez, and F. Kneer. Analysis of a Wide Chromospheric Active Region. In D.
E. Innes, A. Lagg, S. A. Solanki, and D. Danesy, editors, Chromospheric and
Coronal Magnetic Fields, volume 596 of ESA Special Publication, Nov. 2005.

  9. 7.

B. Sánchez-Andrade Nuño, K. G. Puschmann, M. Sánchez Cuberes, J. Blanco
Rodríguez, and F. Kneer. Chromospheric Dynamics of a Solar Active Region. In
The Dynamic Sun: Challenges for Theory and Observations, volume 600 of ESA
Special Publication, Dec. 2005.

  10. 8.

B. Sánchez-Andrade Nuño. Study case: Solar Science Communication. In L.
Lindberg Christensen, and M. Zoulias, editors, Communicating Astronomy with
the Public 2007. An IAU/Nat. Obs. of Athens/Eugenides Foundation Conference,
Oct. 2007.

  11. 9.

J. Blanco Rodríguez, B. Sánchez-Andrade Nuño, K. G. Puschmann, and F. Kneer.
Study of Polar Faculae. In The Dynamic Sun: Challenges for Theory and
Observations, volume 600 of ESA Special Publication, Dec. 2005.

  12. 10.

J. Blanco Rodríguez, B. Sánchez-Andrade Nuño, K. G. Puschmann, and F. Kneer.
Study of Polar Faculae. In D. E. Innes, A. Lagg, S. A. Solanki, and D. Danesy,
editors, Chromospheric and Coronal Magnetic Fields, volume 596 of ESA Special
Publication, Nov. 2005.

  13. 11.

F. Kneer, K. G. Puschmann, J. Blanco Rodríguez, B. Sánchez-Andrade Nuño, and
A. D. Wittmann. Magnetic Structures on the Sun: Observations with the New
”GÖTTINGEN” Two-Dimensional Spectrometer on Tenerife. In D. E. Innes, A. Lagg,
and S. A. Solanki, editors, Chromospheric and Coronal Magnetic Fields, volume
596 of ESA Special Publication, Nov. 2005.

  14. 12.

L. Valdivielso Casas, N. Bello González, K. G. Puschmann, B. Sánchez-Andrade
Nuño, and F. Kneer. Analysis of Polarimetric Sunspot Data from
Tesos/vtt/tenerife. In D. E. Innes, A. Lagg, S. A. Solanki, and D. Danesy,
editors, Chromospheric and Coronal Magnetic Fields, volume 596 of ESA Special
Publication, Nov. 2005.

  15. 13.

C. Denker, A.P. Verdoni, F. Wöger, A. Tritschler, T.R. Rimmele, F. Kneer, K.
Reardon, B. Sánchez-Andrade Nuño, I. Domínguez Cerdeña and K.G. Puschmann
Speckle Interferometry of Solar Adaptive Optics Imagery DFG-NSF Astrophysics
Research Conference “Advanced Photonics in Application to Astrophysical
Problems”, June 2007

## Acknowledgements

I would like to acknowledge here the contributions from many people to the
successful completion of this work. I want to express my gratitude to all
those who helped me, from my supervisor this three last years to my teachers
at my first university who encouraged me to start this adventure in
astrophysics. During all this time I have been also surrounded and supported
by many people to whom I want to thank within these lines. They all made this
experience of doing a PhD, professionally and personally, one of the best
times of my life.

My supervisor of this research work was Prof. Dr. Franz Kneer, or better
_franz_. He was the first person I saw when I came to the train station back
three years ago (along with Julian). He was the guide to all my work, the
observations, the interpretation, …basically, my formation as solar physicist.
The amount of knowledge I learnt from him cannot be measured by any means, and
for that I am professionally extremely grateful. In a personal way he was also
very kind and helped me always with good advices whenever I needed it. He
understood me perfectly when I had problems and also encouraged me to travel
as much as I wanted (which was not a little). Working with him this time made
the whole experience of PhD leaps better. Ich kann nicht diese Arbeit ohne Ihn
zu vorstellen. Danke Franz.

Above all, I am and will be always grateful to my reference in life, my
parents Conchita Nuño López and Julio Sánchez-Andrade Fernández, and my sister
Deva Sánchez Nuño. They always supported me, although a bit far in physical
distance during this last years. When I was 5 years old I was going home with
my mother from the playground when I saw a poster announcing a conference
about starts. I asked my mother to read it loud for me, and to her surprise I
had to explained her: “Don’t you yet know that I want to be a _researcher of
stars_?”. The day after she gave me a children book about stars and explained
me that if I wanted to be so I should know that they are called _astronomers_.
And that’s how it all began.

My work was supported by two institutions: the _Max Planck Institute für
Sonnensystemforschung_ (MPS) granted me the fellowship and the _Institut für
Astrophysik Göttingen_ (IAG) provided me the facilities to work with Franz at
Göttingen and support for the observations at Tenerife. Also, being part of
the _International Max Planck Research School on Physical Processes in the
Solar System and Beyond at the Universities of Braunschweig and Göttingen_
(IMPRS) I could attend seminars, retreats and courses on various astrophysical
topics. I am very grateful to these institutions for providing such a broad
curricula. I would like also to thank the coordinator of the MPS and IMPRS,
Dr. Dieter Schmitt.

The Vacuum Tower Telescope used for the observations is operated by the
Kiepenheuer-Institut für Sonnenphysik, Freiburg, at the Spanish Observatorio
del Teide of the Instituto de Astrofísica de Canarias.

Here at the solar group of IAG we had a really stimulating environment with
long discussions: Franz, Klaus, Markus, Nazaret, Julian and the various
Erasmus people that went by (like Luisa, Manu, …). We worked in the beginning
at the beautiful historical _Sternwarte_ and then at the Faculty of Physics. I
would like to thank specially Klaus for all the help and scientific
discussions during all the time he was here.

Many other professional colleagues contributed to this work. Their input was
extremely helpful. A short list with few names would include: (from the MPS)
Andreas Lagg, Vasili Zakarov, (from the IAG) the solar group, Axel Wittman,
Volker Bothmer, (from the IAC) Basilio Ruiz Cobos, Manolo Collados, Javier
Trujillo, Rebecca Centeno. Thanks to Nurol Al and H. Schleicher for their
computing codes. For the Blind Deconvolution section I got much help from
Jaime and Mats Löfdahl at the Institute for Solar Physics, Stockholm.

From Asturias to here there is long way, specially passing through Canary
Island. Without the advices and guidance from many people I would have been
lost. In these sense I specially thank Cristina, Juanjo, Basilio and Fernando.

Throughout all these years I had the luck to be surrounded by the best
companion of friends. I would like to mention lastly some few friends that
helped me to keep my feet on the ground. Here in Göttingen Thorsten, Gonzaga,
Klaus, Miguel, Cristiano, Vladi, Cathi, Niko, Lorne, Diego, Iria, Teresa,
Nora, Carlos, Olga, Julian, Nazaret, Benoit, Markus, Manu and Luisa. Thorsten
in particular was not only a good friend but also my flat-mate, climbing-mate,
snowboard-mate and of course party-mate. My new flat-mate, Richard, had to
stand me this last months with my thesis mood, though. Thanks a lot!. Mark,
Lucas, Clementina, Pedro, Emre or Martin, all the guys from Lindau, always
warmly welcomed his lost member of the family emigrated to the civilization.

During my short intense period in Tenerife I made good friends: Onti, Borja,
Bendinat, Adriana, Manu, Miguel. Some time later I met Rebecca, with whom I
also worked these years.

Whenever a crazy trip came up I had AEGEE people coming along. They would join
me to any place in Europe. With them I had some of my best times: Luis, David,
Neila, Saioa, Javiero, Marti, Marta and recently of course Carol. All those
friends with crazy names: Annia, Annamary, Bir, Konstantina, Marios or Lutzn.
I am sure I’ll meet you again, somewhere.

Since I left Asturias, I have always missed the green and astounding
landscapes, but also my old friends: Roberto, Raul, Estela, Flaci, Nacho,
Cova, Lisa, David and of course my big and warm family.

Por ultimo me gustaría terminar esta sección, y con ello el fin de este
trabajo, con la memoria de tres personas a quienes siempre echaré de menos. En
mi primer año en Alemania tuve que despedir a mi abuela materna y el segundo a
mi abuela paterna. Aunque sea ley de vida no deja de ser doloroso. Este último
año, a mi primo Abraham, a quien tuve la suerte de ver una última vez. Con
ellos he aprendido la lección más importante:

_Disfruten de la vida._

Göttingen, January 2008

## Lebenslauf

Name: | Bruno Sánchez-Andrade Nuño
---|---
Geburtsdatum: | 6\. Mai 1981
Geburstort: | Oviedo, Asturias, Spanien
Familienstand: | Ledig
Eltern: | Julio Miguel Sánchez-Andrade Fernández
| Concepción Nuño López
Staatsangehörigkeit: | Spanisch
Schulbildung: | September 1987 - Juni 1995 |  Grundschule am _Colegio Público “Río Piles”_ in Gijón, Asturias
| September 1995 - Juni 1999 |  Weiter führende Schule am _I. B. “Río Piles”_ in Gijón, Asturias
Studium: | September 1999 - September 2003 |  Physikalische Fakultät der Universität Oviedo, Asturias
| September 2003 - September 2004 |  Physikalische Fakultät der Universität La Laguna, Teneriffa (Fachrichtung Astrophysik)
Promotion: | Januar 2005 - Januar 2008 |  Promotion an der Universitäts-Sternwarte Göttingen (seit Juni 2005 Institut für Astrophysik Göttingen, IAG)
| Januar 2005 - Januar 2008 |  Stipendium des Max-Planck-Instituts für Sonnensystemforschung