WAVES IN SUNSPOTS: RESONANT TRANSMISSION AND THE
ADIABATIC COEFFICIENT
A. SETTELE1, J. STAUDE1 and Y. D. ZHUGZHDA2∗
1 Astrophysikalisches Institut Potsdam, Telegrafenberg A27, 14482 Potsdam, Germany
(e-mails: [email protected]; [email protected])
2 IZMIRAN, Troitsk, Moscow Region, 142190 Russia (e-mail: [email protected])
(Received 23 March 2000; accepted 21 May 2001)
Abstract. We investigate linear acoustic-gravity waves in three different semi-empirical model
atmospheres of large sunspot umbrae. The sunspot filter theory is applied, that is, the resonant
transmission of vertically propagating waves is modelled. The results are compared with observed
linear sunspot oscillations. For three umbral models we present the transmission coefficients and
the energy density of the oscillations with the maxima of transmission. The height dependence of
the adiabatic coefficient (the ratio of specific heats) γ strongly influences the calculated resonance
frequencies. The variable γ can explain the observed closely spaced resonance period peaks. The first
resonance in the 3 min range is interpreted as a resonance of the upper chromosphere only, while the
higher order peaks are resonances of the whole chromosphere.
1. Introduction
Oscillations in sunspots have been under discussion for almost three decades. Recent reviews of the history and the present knowledge of the topic have been given
by Staude (1999) and Bogdan (2000), the latter covering rather comprehensively
most aspects of observations as well as of theory and modelling. The present paper will rather focus on some special aspects of modelling, without reviewing the
complete literature; references will be limited to selected examples and to our own
work to point out the original ideas the present paper is based on.
Observations of sunspots show several types of oscillations: there are hints at
shock waves (Brynildsen et al., 1999) as well as signs of linear waves. This paper is
devoted to linear waves. Observations of large old sunspots show linear oscillations
in the transition region with periods from 2 to 5 min (Rendtel et al., 1998; Staude
et al., 1999). Due to the sunspot filter theory, first proposed by Žugžda, Locāns, and
Staude (1983, 1987), and Žugžda, Staude, and Locāns (1984), these oscillations
are excited by acoustic noise from outside and below the sunspot and only waves
with specific frequencies are transmitted through the sunspot atmosphere, thus
producing discrete resonance peaks. As the sunspot filter acts similar to an optical
interference filter it is important that the atmospheric structure is relatively stable
for a time scale which is longer than some periods of the resonances. This can be
∗ Note that Zhugzhda and Žugžda are two different transcriptions of the same name.
Solar Physics 202: 281–292, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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A. SETTELE, J. STAUDE, AND Y. D. ZHUGZHDA
realized by the vertical magnetic field of a sunspot which suppresses the convective
instability. The observations by Rendtel et al. (1998) and Staude et al. (1999) show
that this is generally fulfilled. However, some power peaks are slowly changing in
time, which could be produced by long-term changes in the atmosphere.
For many years there has been a discussion about the type and the position
of the resonators which can produce such peaks as they are observed in the power
spectra of sunspots. An alternative approach to the resonant filter theory mentioned
above is that of resonance oscillations (Uchida and Sakurai, 1975; Antia and Chitre,
1979; Scheuer and Thomas, 1981; Thomas and Scheuer, 1982; Wood, 1990, 1997;
Hasan, 1991; Cally and Bogdan, 1993; Cally, Bogdan, and Zweibel, 1994). Thomas
and Scheuer (1982) proposed a photospheric resonator and produced oscillation
frequencies in the region of the observed peaks by shifting the layer of forcing in
the subphotospheric region. But there is concern whether the forcing layer can be
considered as a lower reflecting level of the resonance layer. In this case, the resonance frequency can be tuned on an arbitrary prescribed frequency by a speculation
about the forcing level. The choice of the lower boundary conditions as well as the
depth, where they are imposed, has to be strongly justified. The best choice is the
fitting with asymptotic solutions as it was done by Cally and Bogdan (1993), Cally,
Bogdan, and Zweibel (1994), and in the present paper.
Žugžda, Locāns, and Staude proposed the alternative idea of a resonance filtering by a chromospheric resonator. The resonator can work as a resonance filter
for incident waves, if it is open: the boundaries should be partly transparent for
the waves, which suffer only partial reflections. It was shown that the sunspot
chromosphere acts as an interference filter for slow waves with periods of 3 min.
The effect has been explored in great detail (Žugžda, Locāns, and Staude, 1983,
1985, 1987; Žugžda, Staude, and Locāns, 1984; Locāns et al., 1988) and confirmed
by independent calculations of Gurman and Leibacher (1984). The latter authors
have shown that resonant filtering should in principle work in every atmosphere
with a temperature minimum; Lee and Yun (1987) found similar results. Thomas
(1984) also mentioned that it is possible to explain ‘the closely-spaced multiple
peaks sometimes observed in the power spectra of chromospheric oscillations’ by
the chromospheric resonator of Žugžda, Staude, and Locāns, 1984; which could be
excited at the same time as his photospheric resonator.
It is the basic advantage of the resonance filtering method that no forcing layer
at a certain depth is needed and the boundaries of the resonator are not prescribed
– all this is formed by the used model of the atmosphere and subphotosphere itself;
it is not added artificially.
In this paper we calculate the resonance transmission coefficients of three different umbral model atmospheres and explore the effect of a variable adiabatic
coefficient (the ratio of specific heats) γ on the chromospheric interference filter. This progress became possible due to a new method for treating the filtering
properties of the sunspot atmosphere developed by Settele, Zhugzhda, and Staude
(1999).
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283
2. Transmission Coefficients and Energy Density
2.1. M ETHOD OF CALCULATION
In a recent paper by Settele, Zhugzhda, and Staude (1999) the linear equations
of magneto-atmospheric waves by Ferraro and Plumpton (1958) have been slightly
extended to consider additionally the dependence of the turbulent pressure (a derivative of gravity is generated by including the turbulent pressure in an effective
gravity g(z)) and of the adiabatic coefficient γ on the geometrical depth z. The latter dependence is produced by the z-dependent degree of ionization in the mean, yet
undisturbed semi-empirical model atmosphere, while the additional, time-dependent
variations by the waves have been neglected.
In the present paper we consider a horizontally widely extended magnetized
atmosphere such as that of a huge sunspot: we set the horizontal wave number
k⊥ = 0 and solve the equations for waves propagating only in vertical direction. In
this case the equations of magneto-atmospheric waves by Settele, Zhugzhda, and
Staude (1999) simplify to those of acoustic-gravity waves:
2
CS2 dγ d
dg
2 d
2
) +ω −
ξ = 0,
(1)
CS 2 − (γ g +
dz
γ dz dz
dz
where CS is the sound speed, ω is the frequency, and ξ is the parallel displacement. We model the waves propagating through stratified semi-empirical model
atmospheres, extending from the corona down to the upper convection zone. The
transition region is assumed to be thin and represented by a jump in temperature.
We took the same program as we used for solving the full set of Ferraro and
Plumpton (1958) equations in Settele, Zhugzhda, and Staude (1999), but set the
horizontal wave number k⊥ = 0; see there for more details.
The calculations start in the isothermal corona. There are analytical solutions
(e.g., Zhugzhda and Dzhalilov, 1984; Cally, 2001), but we used the series solutions
given by Cally, Bogdan, and Zweibel (1994), and chose only the outgoing slow
wave. After the numerical integration in between we fitted an acoustic-gravity
wave, based on an isothermal stratified atmosphere, in the upper convection zone.
From this fit we get the amplitudes of the up- (A ) and down-going (B ) waves in
this region and define the transmission coefficient by
T = 1−
B2
A2
.
(2)
The transmission coefficient derived by this method is comparable to a power spectrum produced by a ‘white’ incoming wave flux (constant over all frequencies) in
the transition region.
The atmospheric models were taken from the mentioned papers (see next section), but we recalculated them by means of the code of Staude (1982) in order to
get the depth dependence of γ and to ensure identical thermodynamic assumptions.
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A. SETTELE, J. STAUDE, AND Y. D. ZHUGZHDA
At higher chromospheric levels the resulting optical depths τ5000 in the continuum
at 5000 Å (a quantity of marginal interest in these layers) show some differences
from those in the original models, which is due to varying assumptions about
opacity sources in the different codes.
2.2. R ESULTS
Figure 1 shows the three umbral model atmospheres discussed here: from the top
to the bottom: Staude (1981; Obridko and Staude, 1988), model M of Maltby et al.
(1986), and Lites and Skumanich (1982). The subphotospheric model in each case
is taken from Staude (1981).
Figure 2 gives the transmission coefficients for the different atmospheres. A
transmission of 0.0 is equal to total reflection of the up-going waves, a transmission
of 1.0 means no reflection inside the atmosphere.
Figures 3 and 4 show the kinetic energy density versus the atmospheric depth
in arbitrary units. Figure 3 is for variable γ , Figure 4 for constant γ = 53 . For
each plot all resonances are supposed to have the same upward flux in the upper
convection zone.
3. Discussion and Conclusions
3.1. T HE INFLUENCE OF THE MODEL ATMOSPHERE
The resonant transmission will work in any atmosphere having at least two levels,
where waves undergo reflection. The frequencies of the transmission bands correspond to the wave resonances between these two levels. If the waves which are
reflected from both levels have a phase shift of 180 deg they will cancel out each
other and the transmission will increase. The chromospheric interference filter differs from the filters in optics, because the ‘refraction coefficient’ of the waves is not
constant between the atmospheric mirrors, and these mirrors are not thin, but rather
broadened. Moreover, there is a strong dependence of the reflection coefficient on
the frequency of the reflecting layer near the temperature minimum Tmin because
of the cut-off frequency. This makes the dependence of the transmission on the
parameters of the atmosphere rather complicated and influences the quality of the
resonators: there is nearly no perfect filter of 100% transmission, and also the half
width of the resonance peaks is broadend in most cases. The reflections in the
atmosphere are produced by the chromosphere-corona interface and by the temperature minimum Tmin . Strictly speaking, the second reflection is not connected
to Tmin but to the minimum of the sound speed. This difference is crucial, because
the consideration of the effective γ in our models affects the height profiles of the
sound speed.
WAVES IN SUNSPOTS: RESONANT TRANSMISSION
285
Figure 1. Used atmospheric models. Top scale gives the optical depth in log τ5000 . Solid/dotted lines
present values for variable γ and dashed/dash-dotted for constant γ . The thick lines are related to
the scales on the left-hand axis and represent the sound speeds. The thin lines show the isothermal
cut-off frequency calculated for each depth point, they are described by the right-hand axis. The
two horizontal lines in each plot give the frequency of the first resonance of the atmosphere. All
parameters are plotted versus atmospheric depth z where our zero point is the jump from the corona
to the chromosphere. Larger values of the atmospheric depth are deeper inside the Sun.
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A. SETTELE, J. STAUDE, AND Y. D. ZHUGZHDA
Figure 2. Transmission coefficients of the model atmospheres. A transmission of 0.0 is equal to total
reflection of the up-going waves, a transmission of 1.0 means no reflection inside the atmosphere. The
solid line indicates a variable γ and the dashed line a constant γ = 53 calculation. The frequencies
of oscillations are given by the lower axis while the upper axis shows the periods in minutes.
WAVES IN SUNSPOTS: RESONANT TRANSMISSION
287
Figure 3. Energy density (arbitrary units) of oscillations in the model atmospheres, considering a
variable γ for each atmosphere. Top scale indicates the optical depth in log τ5000 . For each plot all
resonances are supposed to have the same upward flux in the upper convection zone. The resonances
seen in Figure 2 are counted from the left and belong to the following energy density plots: first
resonance – solid, second – dashed, third – dotted, and fourth resonance – thin solid.
288
A. SETTELE, J. STAUDE, AND Y. D. ZHUGZHDA
Figure 4. Energy density (arbitrary units) of oscillations in the model atmospheres, with γ = 53 . Top
scale indicates the optical depth in log τ5000 . For each plot all resonances are supposed to have the
same upward flux in the upper convection zone. The resonances seen in Figure 2 are counted from
the left and belong to the following energy density plots: first resonance – solid, second – dashed,
third – dotted, and fourth resonance – thin solid.
WAVES IN SUNSPOTS: RESONANT TRANSMISSION
289
Compared to a sunspot, the case of the quiet solar atmosphere is very different
due to its strong dynamics and pronounced fine structure in the chromosphere
which make the resonance filtering almost impossible.
3.2. T HE FIRST RESONANCE PERIOD : RESONANCE OF THE UPPER
CHROMOSPHERE
Five-minute oscillations are not seen in any of the published atmospheres. The
reason becomes clear from the cut-off frequency plots in Figure 1: the region where
the waves are evanescent in the atmosphere is too extended, so the waves cannot
pass it by tunneling (for the sunspot filter approach it is required that the waves
from the subphotospheric region reach the corona).
The waves belonging to the first resonance peak in each atmosphere pass a
region where they have to tunnel. This can easily be seen in Figure 1 where the
cut-off frequency of each atmospheric layer is plotted, and the horizontal line of
the first resonance is crossing it. Looking at the energy density plots (Figures 3
and 4) of the first resonance we see, that they are totally different from those of the
higher resonances. They have their highest amplitudes in the upper chromosphere
and do not match the common node below Tmin in the photosphere which occurs
for the higher resonances (see Subsection 3.3). We can associate the first resonance
to a resonator in the upper chromosphere which is located between the transition
region and the increase of the cut-off frequency in the middle of the chromosphere.
The adiabatic coefficient γ exhibits a strong influence on the shape of the energy density of the first resonance, and this is more pronounced in the atmosphere
of Staude than in the others (see Figures 3 and 4). In Staude’s atmosphere the
region of tunnelling in the variable γ case is very small and situated in the lower
chromosphere. In the other atmospheres, but also for the γ = 53 case, the tunnel
path is extended over the whole area from the lower chromosphere down to the
photosphere. The effect of two closely-spaced peaks is best seen in the Staude
atmosphere. In the model of Staude the increase of the cut-off frequency is near the
middle node of the second resonance, therefore these two resonances appear very
close to each other in frequency. This can happen for the first two resonances only.
In principle this resonance shows signs of a coupling of the upper chromospheric
resonator with the resonator which uses the whole atmosphere as the energy density
is again increasing in the lower chromosphere.
The existence of the upper chromospheric resonator can explain the occurrence of the ‘closely-spaced multiple peaks’, in observations and in the Staude
atmosphere in particular, resulting from the use of more complex atmospheric parameters (such as the variable γ ) in the calculation of the resonance transmission.
Other explanations for the multiple peaks could be a spatial averaging over more
than one flux tube with different atmospheres (see Section 3.4) while measuring
the oscillations. The two slight enhancements in the cut-off frequency seen in the
variable γ plots of Figure 1 near the lower chromosphere and in the photosphere
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A. SETTELE, J. STAUDE, AND Y. D. ZHUGZHDA
might give another resonator if they are more emphasized in reality. Apart from the
resonant transmission model, it was possible to explain the closely spaced peaks
only by a statistical effect, which can appear in the case of rather short time series.
For all atmospheres the frequency of the first peak is shifted to lower frequency,
if we take into account the depth variation of the adiabatic coefficient. This results
from the decreased cut-off frequency across the whole atmosphere, while the extent
of the atmosphere is the same. Of course, also waves with frequencies higher than
the cut-off frequency do feel the ‘potential barrier’ caused by the cut-off frequency,
similar to the analogous behavior in quantum mechanics which is called an overbarrier reflection. Nevertheless the first peak in the 3-min band is very sensitive to
the atmospheric parameters and to the properties of the upper chromosphere.
3.3. H IGHER RESONANCE PERIODS : RESONANCE OF THE WHOLE
CHROMOSPHERE
The waves at higher frequencies do not need to tunnel through any region of the
atmosphere; they all have a common node somewhat below Tmin . Each higher resonance gets one more node between the transition region and this common node.
This means that all of them are the result of the resonant transmission through the
chromospheric interference filter for slow MHD waves. The variable γ shifts all
resonances to lower frequencies in comparison with γ = 53 and also the nodes in
between the transition region and the common node are affected and shifted. In
contrast to a simple optical filter the middle node of the second resonance does
not fit the middle of the fourth resonance. This is due to the variation of the sound
speed with height. In most cases the transmission coefficient of the filter is smaller
than 100%, and partly reflected waves appear as a standing wave pattern in the
photosphere, creating a set of nodes. There is an increase of the mean transmission
coefficient with increasing frequency due to a decrease of the reflection from the
Tmin barrier. This mean transmission coefficient corresponds to the reflection from
the corona-chromosphere interface alone.
The largest velocities are found in the upper chromosphere. However, the largest
kinetic energy density of the oscillations is always in the lower chromosphere/photosphere, because the mass density is rapidly increasing with depth. The maximum energy densities are decreasing for each resonance from the lower chromosphere/photosphere to the upper chromosphere, but this is correct with two
exceptions only: in Figure 3 – in the plots of the Maltby and Lites atmospheres
– the energy densities of the third resonance show a different behaviour. In both
cases this concerns the resonance with the largest transmission. They also show
larger energy densities in the chromosphere.
As we cannot influence the extents of the resonators in our approach it is remarkable that all three atmospheres show maximum transmission in the range
of the observed oscillatory frequencies, although they were derived to fit various
observed spectral features such as line profiles.
WAVES IN SUNSPOTS: RESONANT TRANSMISSION
291
3.4. T HE INFLUENCE OF SIMPLIFYING ASSUMPTIONS
We have shown that the effect of resonance transmission is very different for the
different models. It is very probable that a universal chromospheric model for
different sunspots and across a single sunspot does not exist. The resonance filtering, however, is working for structured sunspot chromospheres as well, because
in the so-called limit of strong magnetic fields the slow waves propagate independently along adjacent field lines (Syrovatskii and Zhugzhda, 1967). That means
the resonance can work in a rather thin flux tube in the chromosphere, because the
dispersion relation in that limit does not depend on the horizontal wavenumber.
The chromospheric models for adjacent tubes in the sunspot chromosphere also
can be very different because the pressure balance between them is controlled by
the magnetic pressure, which is large enough in comparison with the gas pressure.
This point is important for the understanding of observations: there are rather small
amplitudes of the 3-min oscillation of the whole sunspot chromosphere but rather
violent, large amplitudes of umbral flashes in thin tubes. These flashes are probably
due to rather perfect chromospheric interference filters, when the greatest part of
the energy of the incident waves is going to the chromosphere and is creating large
amplitude oscillations and even shock waves. The increase of nonlinear dissipation
could start to work. So, such a filter is not stable in time, in agreement with observations. The small amplitudes of global sunspot oscillations could be the result of
an imperfect interference filter.
Moreover, a main shortcoming of the current calculations is the neglect of
nonadiabatic effects in the chromosphere resulting in a damping of the oscillation
amplitudes, which will reduce the calculated amplitudes in the chromosphere.
3.5. F URTHER PROBLEMS FOR A COMPARISON WITH OBSERVATIONS
It will be very difficult to directly compare calculations of the present type with
observations. It is only possible to give qualitative statements. To perform real
sunspot seismology it would be necessary to have observations of one sunspot for
a longer period of time in many spectral lines simultaneously and to calculate a
sunspot atmospheric model for each of the different power spectra measured. Then
it might be possible to calculate the frequencies more precisely for realistic model
atmospheres, taking into account the exact γ (z).
Of course, an improved comparison with observations should include the calculation of time-dependent line profiles formed in an atmosphere with waves. This
is beyond the scope of the present paper, but calculations of this type are in preparation.
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A. SETTELE, J. STAUDE, AND Y. D. ZHUGZHDA
Acknowledgements
The authors gratefully acknowledge support of the present work by the German
Science Foundation (DFG) through the grants Sta 351/5-3 (AS) and 436 RUS 113/81/4
(YDZ).
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