The effect of the solar corona on the attenuation of small-amplitude
  prominence oscillations. I. Longitudinal magnetic field
Astronomy & Astrophysics manuscript no. attIarxiv c© ESO 2018
October 31, 2018
The effect of the solar corona on the attenuation of
small-amplitude prominence oscillations
I. Longitudinal magnetic field
R. Soler, R. Oliver, and J. L. Ballester
Departament de Fı́sica, Universitat de les Illes Balears, E-07122, Palma de Mallorca, Spain
e-mail: [roberto.soler;ramon.oliver;dfsjlb0]@uib.es
Received xxxx; accepted xxxx
ABSTRACT
Context. One of the typical features shown by observations of solar prominence oscillations is that they are damped in time and
that the values of the damping times are usually between one and three times the corresponding oscillatory period. However, the
mechanism responsible for the attenuation is still not well-known.
Aims. Thermal conduction, optically thin or thick radiation and heating are taken into account in the energy equation, and their role
on the attenuation of prominence oscillations is evaluated.
Methods. The dispersion relation for linear non-adiabatic magnetoacoustic waves is derived considering an equilibrium made of
a prominence plasma slab embedded in an unbounded corona. The magnetic field is orientated along the direction parallel to the
slab axis and has the same strength in all regions. By solving the dispersion relation for a fixed wavenumber, a complex oscillatory
frequency is obtained, and the period and the damping time are computed.
Results. The effect of conduction and radiation losses is different for each magnetoacoustic mode and depends on the wavenumber. In
the observed range of wavelengths the internal slow mode is attenuated by radiation from the prominence plasma, the fast mode by the
combination of prominence radiation and coronal conduction and the external slow mode by coronal conduction. The consideration of
the external corona is of paramount importance in the case of the fast and external slow modes, whereas it does not affect the internal
slow modes at all. When a thinner slab representing a filament thread is considered the fast mode is less attenuatted whereas both
internal and external slow modes are not affected.
Conclusions. Non-adiabatic effects are efficient damping mechanisms for magnetoacoustic modes, and the values of the obtained
damping times are compatible with those observed.
Key words. Sun: oscillations – Sun: magnetic fields – Sun: corona – Sun: prominences
1. Introduction
Prominences are dense coronal structures which appear as thin,
dark filaments on the solar disc when observed in Hα. On the
contrary, they show up as bright objects above the solar limb.
The coronal magnetic field is responsible for the support of
prominences against gravity, and it also plays a fundamental role
in the thermal confinement of the cool prominence plasma em-
bedded in the much hotter coronal environment. Nevertheless,
the structure, orientation and strength of the magnetic field in
prominences and the surrounding corona is still enigmatic and
not well-known. High resolution observations reveal that promi-
nences are composed by numerous very thin, thread-like struc-
tures, called fibrils, piled up to form the body of the prominence
(Lin et al. 2003; Lin et al. 2005, Lin et al. 2007) and measures
also indicate that magnetic field lines are orientated along these
thin threads.
The observational evidence of small-amplitude oscillations
in quiescent solar prominences goes back to 40 years ago
(Harvey 1969). The amplitude of these oscillations typically
goes from less than 0.1 km s−1 to 2–3 km s−1, and have been
historically classified, according to their periods, in short- (P <
10 min), intermediate- (10 min < P < 40 min) and long-period
Send offprint requests to: R. Soler
oscillations (P > 40 min), although very short-periods of less
than 1 min (Balthasar et al. 1993) and extreme ultra-long-periods
of more than 8 hours (Foullon et al. 2004) have been reported.
Nevertheless, the value of the period seems not to be related with
the nature or the source of the trigger and probably is linked to
the prominence eigenmode that is excited. There are also a few
determinations of the wavelength and phase speed of standing
oscillations and propagating waves in large regions of promi-
nences (Molowny-Horas et al. 1997; Terradas et al. 2002) and
in single filament threads (Lin et al. 2007). On the other hand,
several observations (Molowny-Horas et al. 1999; Terradas et al.
2002) have informed about the evidence of the attenuation of the
oscillations in Doppler velocity time series, which is a common
feature observed in large areas. By fitting a sinusoidal function
multiplied by a factor exp(−t/τD) to the Doppler series, these
authors have obtained values of the damping time, τD, which
are usually between 1 and 3 times the corresponding oscilla-
tory period. The reader is referred to some recent reviews for
more information about the observational background (Oliver &
Ballester 2002, Wiehr 2004, Engvold 2004, Ballester 2006).
From the theoretical point of view, small-amplitude promi-
nence oscillations can be interpreted in terms of linear magne-
tohydrodynamic (MHD) waves. Although, there is a wide bib-
liography of works that investigate the ideal MHD wave modes
http://arxiv.org/abs/0704.1566v2
2 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
supported by prominence models (see Oliver & Ballester 2002
for an extensive review of theoretical studies), the investigation
of the wave damping has been broached in few papers. By re-
moving the ideal assumption and including dissipative terms in
the basic MHD equations, several works have studied the atten-
uation of prominence oscillations considering radiative losses
based on the Newtonian law of cooling with a constant relax-
ation time (Terradas et al 2001), or performing a more complete
treatment of non-adiabatic effects, assuming optically thin radi-
ation, heating and thermal conduction (Carbonell et al. 2004;
Terradas et al. 2005). The main conclusion of these previous
studies is that only the slow wave is attenuated by thermal
effects, radiation being the dominant damping mechanism in
the range of typically observed wavelengths in prominences,
but the fast wave remains practically undamped. On the other
hand, Forteza et al. (2007) proposed ion-neutral collisions as a
damping mechanism on the basis that prominences are partially
ionised plasmas, but they found that this mechanism is only ef-
ficient in attenuating the fast mode in quasi-neutral plasmas, the
slow mode being almost unaffected.
In the light of these referred studies, it is likely that non-
adiabatic effects are the best candidates for the damping of
small-amplitude oscillations, at least for slow modes. However,
previous results do not asses the influence of the corona. The
main aim of the present work is to perform a step forward in the
investigation of the effect of non-adiabatic mechanisms (radia-
tion losses, thermal conduction and heating) on the time damp-
ing of prominence oscillations. We consider a slab model with
a longitudinal magnetic field and take into account the external
coronal medium. So, we explore for the first time the joint ef-
fect of prominence and coronal mechanisms on the attenuation
of oscillations. The magnetoacoustic normal modes of this equi-
librium have been previously investigated by Edwin & Roberts
(1982) and Joarder & Roberts (1992) in the adiabatic case. Later,
a revision of these works has been done in Soler et al. (2007),
hereafter Paper I, and the normal modes have been studied and
reclassified according to their magnetoacoustic properties.
This paper is organised as follows. The description of the
equilibrium model and the linear non-adiabatic wave equations
are given in Sect. 2, whereas the dispersion relation for the mag-
netoacoustic modes is derived in Sect. 3. Then, the results are
plotted and investigated in Sect. 4. Finally, Sect. 5 contains the
conclusions of this work.
2. Equilibrium and basic equations
Our equilibrium configuration (Fig. 1) is made of a homoge-
neous plasma layer with prominence conditions (density ρp and
temperature Tp) embedded in an unbounded corona (density ρc
and temperature Tc). The coronal density is computed by fix-
ing the coronal temperature and imposing pressure continuity
across the interfaces. The magnetic field is B0 = B0êx, with B0
a constant everywhere. Both media are unlimited in the x- and
y-directions. The half-width of the prominence slab is zp.
The basic magnetohydrodynamic equations for the discus-
sion of non-adiabatic processes are:
+ ρ∇ · v = 0, (1)
= −∇p +
(∇ × B) × B, (2)
+ (γ − 1)[ρL(ρ, T ) − ∇ · (κ · ∇T )] = 0, (3)
Fig. 1. Sketch of the equilibrium.
Table 1. Parameter values of the radiative loss function
corresponding to the considered regimes. The three promi-
nence regimes represent different plasma optical thicknesses.
Prominence (1) regime corresponds to an optically thin plasma
whereas Prominence (2) and Prominence (3) regimes represent
greater optical thicknesses. All quantities are expressed in MKS
units.
Regime χ∗ α Reference
Prominence (1) 1.76 × 10−13 7.4 Hildner (1974)
Prominence (2) 1.76 × 10−53 17.4 Milne et al. (1979)
Prominence (3) 7.01 × 10−104 30 Rosner et al. (1978)
Corona 1.97 × 1024 −1 Hildner (1974)
= ∇ × (v × B), (4)
∇ · B = 0, (5)
, (6)
where D
+ v · ∇ is the material derivative for time vari-
ations following the motion and all quantities have their usual
meaning. Equation (3) is the energy equation, which in the
present form takes into account non-adiabatic effects (radiation
losses, thermal conduction and heating) and whose terms are ex-
plained in detail in Carbonell et al. (2004) and Terradas et al.
(2005). Following these works, only thermal conduction par-
allel to the magnetic field is assumed and we use the typical
value for the parallel conductivity in prominence and coronal
applications, κ‖ = 10
−11T 5/2 W m−1 K−1. Radiative losses and
heating are evaluated together through the heat-loss function,
L(ρ, T ) = χ∗ρTα − hρaT b, where radiation is parametrised with
χ∗ and α (see Table 1) and the heating scenario is given by ex-
ponents a and b. The heating mechanisms taken into account in
this work are (Rosner et al. 1978; Dahlburg & Mariska 1988):
– constant heating per unit volume (a = b = 0);
– constant heating per unit mass (a = 1, b = 0);
– heating by coronal current dissipation (a = b = 1);
– heating by Alfvén mode/mode conversion (a = b = 7/6);
– heating by Alfvén mode/anomalous conduction damping
(a = 1/2, b = −1/2).
Following the same process as in Carbonell et al. (2004),
we consider small perturbations from the equilibrium state, lin-
earise the basic Eqs. (1)–(6) and obtain their Eqs. (9)–(14). Since
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 3
our model is unlimited in the x- and y-directions, we assume all
perturbations are in the form f1(z) exp i(ωt+ kxx+ kyy), and con-
sidering only motions and propagation in the xz-plane (vy = 0,
ky = 0), which excludes Alfvén waves, the linearised equations
become
iωρ1 + ρ0
ikxvx +
= 0, (7)
iωρ0vx = −ikx p1, (8)
iωρ0vz = −
ikxB1z −
, (9)
p1 − c
= −(γ − 1)
k2xκ‖T1 +
ωρρ1 +
ωT T1
, (10)
iωB1x = −B0
, (11)
iωB1z = B0ikxvz, (12)
where c2s =
is the adiabatic sound speed squared and
L + ρ0Lρ
, ωT ≡
T0LT ,
Lρ, LT being the partial derivatives of the heat-loss function with
respect to density and temperature, respectively,
, LT ≡
Now, it is possible to eliminate all perturbations in favour of
vz to obtain a single differential equation
+ k2z vz = 0, (13)
in which
k2z =
ω2 − k2xv
ω2 − k2xΛ
v2A + Λ
ω2 − k2xc̃
) , (14)
where v2A =
is the Alfvén speed squared. Λ2 and c̃2T are the
modified sound and cusp (or tube) speed squared, respectively,
(γ − 1)
x + ωT − ωρ
+ iγω
(γ − 1)
κ‖k2x + ωT
, (15)
c̃2T ≡
v2A + Λ
. (16)
Expressions for the perturbations in terms of vz are given in
App. A. In all the following formulae, subscripts p or c denote
quantities computed using prominence or coronal values, respec-
tively.
3. Dispersion relation
We impose some restrictions on the solutions of Eq. (13) in or-
der to obtain the dispersion relation for the linear non-adiabatic
magnetoacoustic waves. We restrict this analysis to body waves
which are evanescent in the corona, since we are looking for
solutions which are essentially confined to the slab. For such so-
lutions, vz(z) is of the form
vz(z) =
A1 exp
z + zp
, if z ≤ −zp,
A2 cos
+ A3 sin
, if −zp ≤ z ≤ zp,
A4 exp
z − zp
, if z ≥ zp.
withℜ(kzp) > 0 andℜ(kzc) > 0.
Imposing continuity of vz and the total (gas plus magnetic)
pressure perturbation across the interfaces, we find four alge-
braic relations between the constants A1, A2, A3 and A4. The
non-trivial solution of this system gives us the dispersion rela-
Ac − ω
kzpzp
Ap − ω
kzc = 0, (18)
where cot/tan terms and ± signs are related with the symmetry of
the perturbations. The cot term and the + sign correspond to kink
modes (A3 = 0), whereas the tan term and the − sign correspond
to sausage modes (A2 = 0).
The dispersion relation for the magnetoacoustic waves pre-
sented in Eq. (18) is equivalent to the relation investigated in
Edwin & Roberts (1982) and Joarder & Roberts (1992), and re-
vised Paper I, in the case of adiabatic perturbations, since all
non-adiabatic terms are now enclosed in kzp and kzc through
Eq. (14).
4. Results
Now, we assume Prominence (1) conditions inside the slab (i.e.
an optically thin prominence) and a heating mechanism given
by a = b = 0. Unless otherwise stated, the following equilibrium
parameters are considered in all computations: Tp = 8000 K,
ρp = 5 × 10
−11 kg m−3, Tc = 10
6 K, ρc = 2.5 × 10
−13 kg m−3,
B0 = 5 G and zp = 3000 km. The solution of the dispersion
relation (Eq. [18]) for a fixed real kx gives us a complex fre-
quency ω = ωR + iωI. We then compute the oscillatory period,
the damping time and the ratio of the damping time to the pe-
riod because this is an important quantity from the observational
point of view,
, τD =
Since we are interested in studying the behaviour of the most
relevant solutions of the dispersion relation, we only compute the
results for the fundamental modes, which are labelled, accord-
ing to the classification of Paper I, as internal and external slow
modes and fast modes. The band structure described in Paper I is
slightly modified when non-adiabatic terms are considered (see
Fig. 2). The phase speed of the internal slow modes is now en-
closed in the bandℜ(c̃Tp) < ωR/kx < ℜ(Λp). The adiabatic fast
modes exist in two separated bands in the phase speed diagram
due to the presence of a forbidden region (cTc < ωR/kx < csc),
but now the forbidden band is avoided and a continuous fast
mode is found with vAp < ωR/kx < vAc. Finally, and like in
the adiabatic case, among the external slow modes only the fun-
damental kink one exists as a non-leaky solution in a restricted
4 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
wavenumber range and couples with the fundamental fast kink
mode. Its phase speed is ωR/kx ≈ ℜ(Λc). Therefore, we see that
in the non-adiabatic case Λ plays the role of cs in the adiabatic
case.
Fig. 2. Phase speed versus the dimensionless wavenumber for
the three fundamental oscillatory modes. Solid lines denote kink
modes whereas dotted lines correspond to sausage modes. The
shaded zones are projections of the forbidden (or leaky) regions
on the plane of this diagram. Note that the vertical axis is not
drawn to scale.
In Fig. 3 P, τD and τD/P are represented for the fundamental
modes and for a range of the longitudinal wavenumber between
10−10 m−1 and 103 m−1. The shaded zones correspond to wave-
lengths between 5 × 103 km and 105 km, the typically observed
values. It turns out that the values of the period are very sim-
ilar to those obtained in the adiabatic case (Joarder & Roberts
1992; Paper I). The damping time presents a strong dependence
with the wavenumber and its behaviour is very different from
one mode to another. This fact suggests that the non-adiabatic
mechanisms can affect each mode in a different way (Carbonell
et al. 2004). This is studied in detail in Sect 4.1. Observations
show that prominence oscillations are typically attenuated in a
few periods (Terradas et al. 2002), so a damping time of the order
of the period is expected. In our results, the fundamental modes
present values of τD/P in the range 1 to 10 in the observed wave-
length region, which is in agreement with observations.
4.1. Regions of dominance of the damping mechanisms
The importance of the different non-adiabatic terms included in
the energy equation (Eq. [3]) depends on the wavenumber. In
order to know which is the range of dominance of each mech-
anism, we compare the damping time obtained when consider-
ing all non-adiabatic terms (displayed in the middle column of
Fig. 3) with the results obtained when a specific mechanism is
removed from the energy equation. With this analysis, we are
able to know where the omitted mechanism has an appreciable
effect on the damping. The results of these computations for the
fundamental kink modes (Fig. 4) are summarised as follows:
– The fundamental internal slow kink mode is not affected by
the mechanisms related with the corona. This is a conse-
quence of the nature of this mode, which propagates strictly
along the prominence without disturbing the corona (see Fig.
4, top row, of Paper I). For this reason, in the adiabatic case
it is also independent of the coronal conditions. On the other
hand, the prominence-related mechanisms show different ef-
fects in two different ranges of kx. For kx . 10
−3 m−1 promi-
nence radiation dominates, while for kx & 10
−3 m−1 promi-
nence conduction is the dominant mechanism. Beginning
from small values of the wavenumber, prominence radiation
becomes more efficient as kx grows and the damping time
falls following a power law until kx ≈ 10
−5 m−1, where τD
saturates in a plateau between kx ≈ 10
−5 and kx ≈ 10
−3 m−1.
Then, prominence conduction becomes the dominant mech-
anism and the damping time falls again until kx ≈ 10
−1 m−1
where a new plateau begins. This last part of the curve cor-
responds to the isothermal or superconductive regime, in
which the amplitude of the temperature perturbation drops
dramatically (Carbonell et al. 2006). Prominence radiation
is responsible for the attenuation of the slow mode in the
observed wavelength range. An approximate dispersion re-
lation for the internal slow modes is included in App. B.
– The fundamental fast kink mode is affected by the four
mechanisms. For kx . 3 × 10
−9 m−1 coronal radiation dom-
inates but for 3 × 10−9 m−1 . kx . 5 × 10
−7 m−1 the effect
of coronal conduction grows and becomes the main damping
mechanism. Then, for kx & 5×10
−7 m−1 the corona loses dra-
matically its influence and prominence mechanisms become
responsible for the attenuation of this mode. First, promi-
nence radiation is dominant in the range 5 × 10−7 m−1 .
kx . 10
−3 m−1, then prominence conduction governs the
wave damping for kx & 10
−3 m−1 and finally the isother-
mal regime begins for kx ≈ 10
0 m−1. The minimum of τD
occurs into the coronal conduction regime, for the value of
kx which corresponds to the coupling with the external slow
mode. The transition between the coronal conduction regime
and the prominence radiation regime occurs in the observed
wavelength range. The reason for the sensitivity of the fast
mode damping time on prominence and coronal conditions
is that this wave has a considerable amplitude both inside the
prominence and in the corona, the later becoming more im-
portant for long wavelengths (see the second and third rows
of Fig. 4 of Paper I).
– The behaviour of the damping time of the fundamental exter-
nal slow kink mode is entirely dominated by coronal mech-
anisms whereas the prominence mechanisms do not affect it
at all. This behaviour is a result of the negligible amplitude
of this wave in the prominence (see the fourth and fifth rows
of Fig. 4 of Paper I). For kx . 3× 10
−9 m−1 coronal radiation
dominates, but for shorter wavelengths coronal conduction
becomes more relevant and is responsible for the damping
in the observed wavelength range until the frequency cut-off
is reached. At the cut-off, τD has a value of the order of the
period.
Regarding the fundamental sausage modes, the behaviour of
the internal slow sausage mode is exactly that of the slow kink
mode, so no additional comments are needed. The fundamen-
tal fast sausage mode (Fig. 5) presents the same scheme as the
fundamental fast kink mode for kx & 10
−8 m−1. The main dif-
ference between the fast kink and sausage modes happens in the
observed wavelength range, where the effect of coronal conduc-
tion on the sausage mode is less relevant. If coronal conduction
is omitted, the fundamental fast sausage mode is not able to tra-
verse the forbidden region in the dispersion diagram and then
shows frequency cut-offs as in the adiabatic case. This means
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 5
Fig. 3. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus the longitudinal wavenumber
for the fundamental oscillatory modes. Upper panels: internal slow kink (solid line), fast kink (dotted line) and external slow kink
(dashed line). Lower panels: internal slow sausage (solid line) and fast sausage (dotted line). Shaded zones correspond to those
wavelengths typically observed. Note the cut-offs of the external slow kink mode and the fast sausage mode. Prominence (1)
radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0.
Fig. 4. Damping time versus the longitudinal wavenumber for the three fundamental kink oscillatory modes: internal slow (left),
fast (centre) and external slow (right). Different linestyles represent the omitted mechanism: all mechanisms considered (solid line),
prominence conduction eliminated (dotted line), prominence radiation eliminated (dashed line), coronal conduction eliminated (dot-
dashed line) and coronal radiation eliminated (three dot-dashed line). Prominence (1) radiation conditions have been taken for the
prominence plasma and the heating scenario is given by a = b = 0.
that coronal conduction causes the fast mode to cross the forbid-
den region in the dispersion diagram in the non-adiabatic case.
Approximate values of kx for which the transitions between
regimes take place can be computed by following a process simi-
lar to that in Carbonell et al. (2006). The thermal ratio, d, and the
radiation ratio, r, quantify the importance of thermal conduction
and radiation, respectively (De Moortel & Hood 2004),
(γ − 1)κ‖T0ρ0
γ2 p20τs
τcond
, (19)
(γ − 1)τsρ
, (20)
where τs is the sound travel time and τcond and τrad are character-
istic conductive and radiative time scales. Taking τs = 2π/k
the value of k∗ for which the condition d = r is satisfied is
k∗ = 2πρ0
χ∗Tα−10
. (21)
Now, we use prominence values to compute k∗ for the promi-
nence radiation–prominence conduction transition (k∗p), and
6 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
Fig. 6. Damping time versus the longitudinal wavenumber for the fundamental internal slow kink mode (left), the fundamental fast
kink mode (centre) and the fundamental external slow kink mode (right). The different linestyles represent different values of the
prominence temperature: Tp = 8000 K (solid line), Tp = 5000 K (dotted line) and Tp = 13000 K (dashed line). The heating scenario
is given by a = b = 0 and the optical thickness for the prominence plasma is Prominence (1).
Fig. 7. Same as Fig. 6 with ρp = 5 × 10−11 kg m−3 (solid line), ρp = 2 × 10−11 kg m−3 (dotted line) and ρp = 10−10 kg m−3 (dashed
line).
Fig. 8. Same as Fig. 6 with B0 = 5 G (solid line), B0 = 2 G (dotted line) and B0 = 10 G (dashed line).
coronal values for the coronal radiation–coronal conduction
transition (k∗c). This gives the values k
p ≈ 1.7 × 10
−3 m−1, and
k∗c ≈ 2.2 × 10
−8 m−1. For the transition of the fast kink mode
between the coronal conduction and the prominence radiation
regimes, the boundary wavenumber k∗p↔c can be roughly calcu-
lated by imposing dc = rp, that gives
k∗p↔c = 2πρp
cscχ∗pT
cspκ‖cTc
, (22)
and whose numerical value is k∗p↔c ≈ 1.4 × 10
−6 m−1. All these
wavenumbers for the transitions between different regimes are
independent of the wave type, be it fast or slow, internal or ex-
ternal (this agrees with Figs. 4 and 5). On the other hand, the be-
ginning of the isothermal regime can be estimated by following
Porter et al. (1994). Considering c2sp/v
Ap ≪ 1 and the approxi-
mations ωR ≈ kxcsp for the slow wave and ωR ≈ kxvAp for the
fast wave, the critical wavenumber is
kcrit−slow =
2ρpkBcsp
κ‖pmp cos θ
, (23)
for the internal slow mode, and
kcrit−fast =
2ρpkBvAp
κ‖pmp cos2 θ
, (24)
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 7
Fig. 5. Same as Fig. 4 for the fundamental fast sausage mode.
for the fast mode, where mp is the proton mass, kB is the
Boltzmann constant and θ is the angle between B and k. Taking
cos θ = 1 for simplicity, the approximate critical values are
kcrit−slow ≈ 1.7 × 10
−1 m−1 and kcrit−fast ≈ 9.1 × 10
−1 m−1. We
note that all these approximate values describe correctly the tran-
sitions between the diverse regimes shown in Figs. 4 and 5, but
their numerical values overestimate by almost an order of mag-
nitude the actual critical wavenumbers.
4.2. Exploring the parameter space
4.2.1. Dependence on the equilibrium physical conditions
In this section, we compute the solutions for different values
of the equilibrium physical conditions. We only present the re-
sults for the fundamental kink modes since they are equivalent to
those of sausage modes. Figures 6, 7 and 8 display the damping
time as function of kx for some selected values of the prominence
temperature, the prominence density and the magnetic field, re-
spectively.
For the internal slow mode, a decrease of the prominence
temperature or the prominence density raises the position of
the radiative plateau and increases its length. The opposite be-
haviour is seen when the density or the temperature are in-
creased. However, the value of the magnetic field does not in-
fluence the attenuation of this mode, such as expected for a slow
wave.
Increasing the value of the prominence temperature causes
a vertical displacement of τD of the fast mode in those regions
in which prominence mechanisms dominate. The value of the
prominence density has a smaller effect and its main influence
is in changing the coupling point with the external slow mode,
which moves to higher kx for greater values of the density. The
magnetic field strength has a more complex effect on τD and also
modifies the coupling point.
Finally, the external slow mode is only slightly affected by
a modification of the prominence physical parameters since it
is mainly dominated by coronal conditions, and the influence of
the magnetic field is very small due to the slow-like magnetoa-
coustic character of this solution.
4.2.2. Dependence on the prominence optical thickness
The optically thin radiation assumption is a reasonable approxi-
mation in a plasma with coronal conditions but prominence plas-
mas often are optically thick. In this section we compare the
results obtained considering different optical thicknesses for the
prominence plasma (see Fig. 9 for the fundamental kink modes).
The results corresponding to the slow sausage mode have not
been plotted since they are equivalent to those obtained for slow
kink mode; those for the fundamental fast sausage mode, how-
ever, are displayed in Fig. 10.
The variation of the prominence optical thickness modi-
fies the prominence conduction–prominence radiation critical
wavenumber, k∗p (see analytical approximation of Eq. [21]). For
the internal slow mode, an increase in the optical thickness
raises the position of the radiative plateau and shifts it to smaller
wavenumbers. This fact causes an a priori surprising result in the
observed wavelength range, since τD has a smaller value for opti-
cally thick radiation, Prominence (3), than for optically thin radi-
ation, Prominence (1). Regarding fast modes, the damping time
increases when the optical thickness is increased, but only in
the region in which prominence radiation dominates. The value
of τD inside the observed wavelength range is partially affected
and raises an order of magnitude for Prominence (3) conditions
in comparison with the results for Prominence (1) conditions.
Finally, the damping time of the external slow mode is not af-
fected by the prominence optical thickness since it is entirely
dominated by the corona, as it has been noticed in Sect. 4.1.
Fig. 10. Same as Fig. 9 for the fundamental fast sausage mode.
4.2.3. Dependence on the heating scenario
Now, we compute the damping time for the five possible heating
scenarios. For simplicity, we only consider the fundamental kink
modes (Fig. 11). Carbonell et al. (2004) showed that in a plasma
with prominence conditions the different heating scenarios have
no significant influence on the damping time. Nevertheless, in
coronal conditions wave instabilities can appear depending on
the heating mechanism. In our results, we see that the heating
scenario affects the value of τD only in the ranges of kx in which
radiation is the dominant damping mechanism. The heating sce-
nario has a negligible effect when prominence radiation domi-
nates, since τD is only slightly modified. On the contrary, wave
instabilities appear in those regions in which coronal radiation
dominates. Thermal destabilisation occurs when the imaginary
part of the frequency becomes negative, so oscillations are not
attenuated but amplified in time. Instabilities only occur in the
fundamental fast kink and the external slow modes for very small
values of kx, outside the observed wavelength range.
8 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
Fig. 9. Same as Fig. 6 with the prominence optical thickness given by Prominence (1) (solid line), Prominence (2) (dotted line) and
Prominence (3) (dashed line) conditions.
Fig. 11. Same as Fig. 6 with the heating scenario given by a = b = 0 (solid line); a = 1, b = 0 (dotted line); a = b = 1 (dashed line);
a = b = 7/6 (dot-dashed line); a = 1/2, b = −1/2 (three dot-dashed line).
4.3. Comparison with the solution for an isolated slab
In order to assess the effects arising from the presence of two
different media in the equilibrium, a comparison between the
previous results and those corresponding to a single medium is
suitable. So, we consider a simpler equilibrium made of an iso-
lated prominence slab with the magnetic field parallel to its axis.
The external medium is not taken into account. Magnetoacoustic
non-adiabatic perturbations are governed by Eq. (13), and rigid
boundary conditions for vz are imposed at the edges of the promi-
nence slab,
vz(−zp) = vz(zp) = 0. (25)
Then, the solution is of the form
vz(z) = C1 cos
+C2 sin
, (26)
and after imposing boundary conditions (Eq. [25]), we deduce
the dispersion relation for the magnetoacoustic slow and fast
non-adiabatic waves,
kzpzp =
π, (n = 0, 1, 2, . . .), (27)
for the kink modes, and
kzpzp = nπ, (n = 1, 2, 3, . . .), (28)
for the sausage modes. Inserting expressions (14) and (15) for
kzp and Λp respectively, one can rewrite the dispersion relations
(27) and (28) as polynomial equations in ω. See App. C for the
details.
Next, considering only the fundamental kink modes for sim-
plicity, we compute the period and the damping time and com-
pare with those obtained when the surrounding corona is taken
into account (Fig. 12). We see that there is a perfect agree-
ment between both results in the case of the internal slow mode,
whereas the solutions for the fast mode only coincide for inter-
mediate and large wavenumbers, and show an absolutely differ-
ent behaviour in the observed wavelength range and for smaller
wavenumbers. Additionally, one must bear in mind that the ex-
ternal slow mode exists because of the presence of the coronal
medium, hence it is not supported by an isolated slab.
In Paper I we proved that the internal slow mode is essen-
tially confined within the prominence slab and that the effect of
the corona on its oscillatory period can be neglected. Now, we
see that the corona has no influence on the damping time either.
On the other hand, the confinement of the fast mode is poor for
small wavenumbers, the isolated slab approximation not being
valid. As it has been noted in Section 4.1, the corona has an es-
sential effect on the attenuation of the fast mode in the observed
wavelength range.
4.4. Application to a prominence fibril
Since magnetic field lines are orientated along fibrils, our model
can also be applied to study the oscillatory modes supported by
a single prominence fibril. In order to perform this investigation,
we reduce the slab half-width, zp, to a value according to the
typical observed size of filament threads, which is between 0.2
to 0.6 arcsec (Lin et al. 2005). Since these values are close to the
resolution limit of present-day telescopes, it is likely that thinner
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 9
Fig. 12. Comparison between the solutions for a prominence
plus corona system and for an isolated slab with prominence
conditions. The upper panels correspond to the fundamental in-
ternal slow kink mode and the lower panels to the fundamental
fast kink mode. The solid lines are the solutions for a prominence
plus corona equilibrium whereas the dotted lines with diamonds
represent the solutions for an isolated slab. Prominence (1) pa-
rameters and a = b = 0 have been used in the computations.
threads could exist. So, assuming now zp = 30 km, we compute
P, τD and τD/P for the fundamental kink modes and compare
these results with those obtained for zp = 3000 km.
Such as displayed in Fig. 13, both internal and external slow
modes are not affected by the width of the prominence slab since
they are essentially polarised along the x-direction and so they
are not influenced by the equilibrium structure in the z-direction.
Nevertheless, the location of the cut-off of the external slow
mode and the coupling point with the fast mode are shifted to
larger values of kx when the slab width is reduced. On the other
hand, the fast mode, which is responsible for transverse motions,
is highly influenced by the value of zp. The τD curve for the
fast mode is displaced to larger values of kx when smaller zp is
considered. This causes that higher values of τD/P are obtained
in the observed wavelength range. Hence, these results suggest
that local prominence oscillations related with transverse fast
modes supported by a single fibril could be less affected by non-
adiabatic mechanisms than global fast modes supported by the
whole or large regions of the prominence. However, according to
the results pointed out by Dı́az et al. (2005) and Dı́az & Roberts
(2006), large groups of fibrils tend to oscillate together since the
separation between individual fibrils is of the order of their thick-
ness. In a very rough approximation one can consider that a thick
prominence slab could represent many near threads which oscil-
late together and that the larger the slab width, the more threads
fit inside it. So, our results show that the slab size (i.e. the number
of threads which oscillate together in this rough approximation)
has important repercussions on the damping time of collective
transverse oscillations, hence the oscillations could be more at-
tenuated when the number of oscillating threads is larger. This
affirmation should be verified by investigating the damping in
multifibril models.
5. Conclusions
In this paper, we have studied the time damping of magne-
toacoustic waves in a prominence-corona system considering
non-adiabatic terms (thermal conduction, radiation losses and
heating) in the energy equation. Small amplitude perturbations
have been assumed, so the linearised non-adiabatic MHD equa-
tions have been considered and the dispersion relation for the
slow and fast magnetoacoustic modes has been found assuming
evanescent-like perturbations in the coronal medium. Finally, the
damping time of the fundamental oscillatory modes has been
computed and the relevance of each non-adiabatic mechanism
on the attenuation has been assessed. Next, we summarise the
main conclusions of this work:
1. Non-adiabatic effects are an efficient mechanism to obtain
small ratios of the damping time to the period in the range of
typically observed wavelengths of small-amplitude promi-
nence oscillations.
2. The mechanism responsible for the attenuation of oscilla-
tions is different for each magnetoacoustic mode and de-
pends on the wavenumber.
3. The damping of the internal slow mode is dominated by
prominence-related mechanisms, prominence radiation be-
ing responsible for the attenuation in the observed wave-
length range. Such as happens in the adiabatic case (see
Paper I) the corona does not affect the slow mode at all, and
these results are in perfect agreement with those for an iso-
lated prominence slab.
4. The attenuation of the fast mode in the observed wavelength
range is governed by a combined effect of prominence radia-
tion and coronal conduction. The presence of the corona is of
paramount importance to explain the behaviour of the damp-
ing time for small wavenumbers within the observed range of
wavelengths. Non-adiabatic mechanisms in both the promi-
nence and the corona are significant because the fast mode
achieves large amplitudes in both regions.
5. Since the external slow mode is principally supported by
the corona, its damping time is entirely governed by coro-
nal mechanisms, coronal conduction being the dominant one
in the observed wavelength range.
6. The consideration of different optical thicknesses for the
prominence plasma causes an important variation of the
damping time of the internal slow and fast modes in the ob-
served wavelength range. Hence a precise knowledge of the
radiative processes of prominence plasmas is needed to ob-
tain more realistic theoretical results.
7. The heating scenario has a negligible effect on the damp-
ing time of all solutions in the observed wavelength range.
Depending on the scenario considered, thermal instabilities
can appear for small values of the wavenumber, in which
coronal radiation dominates.
8. The width of the prominence slab does not affect the results
for both internal and external slow modes. However, fast
modes are less attenuated in the range of observed wave-
lengths when thinner slabs or filaments threads are consid-
ered.
Taking into account the results in the observed range of
wavelengths, one can conclude that radiative effects of the
prominence plasma are responsible for the attenuation of the in-
ternal slow modes, which can be connected with intermediate-
and long-period prominence oscillations, whereas a combined
effect of prominence radiation and coronal thermal conduction
10 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
Fig. 13. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus kx for the fundamental
kink oscillatory modes: internal slow (top panels), fast (mid panels) and external slow (bottom panels). Solid lines correspond
to zp = 3000 km whereas dotted lines correspond to zp = 30 km. Prominence (1) radiation conditions have been taken for the
prominence plasma and the heating scenario is given by a = b = 0.
governs the damping of fast modes, whose periods are compati-
ble with those of short-period oscillations.
Acknowledgements. The authors acknowledge the financial support received
from the Spanish Ministerio de Ciencia y Tecnologı́a under grant AYA2006-
07637. R. Soler thanks the Conselleria d’Economia, Hisenda i Innovació for a
fellowship.
Appendix A: Expressions for the perturbations
Combining Eqs. (7)–(12), one can obtain the expressions for the
perturbed quantities as functions of vz and its derivative
−ikxΛ
ω2 − k2xΛ
, (A.1)
ω2 − k2xΛ
, (A.2)
iωρ0Λ
ω2 − k2xΛ
, (A.3)
ω2 − k2xΛ
, (A.4)
B1x =
, (A.5)
B1z =
vz. (A.6)
Now, we write the expressions for the perturbations to the mag-
netic pressure, p1m, and the total pressure, p1T,
p1m =
B1x =
, (A.7)
p1T = p1 + p1m =
ω2 − k2xv
. (A.8)
In the limit Λ → cs (i.e. in the absence of conduction, radiation
losses and heating), all the expressions reduce to those corre-
sponding to the adiabatic case.
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 11
Appendix B: Approximate dispersion relation for
the internal slow modes
Internal slow modes are almost non-dispersive and for adia-
batic perturbations a good approximation for the frequency is
ω ≈ cspkx, csp being the prominence sound speed. In the non-
adiabatic case, we can consider the equivalence between cs and
Λ to propose ω ≈ Λpkx as an approximate dispersion relation.
Taking into account Eq. (15) for Λ, the approximate dispersion
relation for the internal slow modes is a third order polynomial
in ω,
− iBω2 − k2xc
spω + i
Ak2x = 0, (B.1)
A = (γ − 1)
κ̂‖pk
x + ωTp − ωρp
, (B.2)
B = (γ − 1)
κ̂‖pk
x + ωTp
, (B.3)
κ̂‖p = κ‖p
In Fig. B.1 a comparison between the exact and approximate
solutions is displayed and a perfect agreement is seen.
Fig. B.1. Period (left) and damping time (right) versus the lon-
gitudinal wavenumber for the fundamental internal slow kink
mode. The solid line corresponds to the exact solution and the di-
amonds correspond to the approximate solution. Prominence (1)
parameters and a = b = 0 have been used in the computations.
Appendix C: Dispersion relation for an isolated slab
We here deduce a polynomial dispersion relation for the magne-
toacoustic normal modes of a slab with a longitudinal magnetic
field. Taking Eqs. (27) and (28) as the dispersion relations for
the kink and sausage modes, respectively, one can replace kz and
Λ with their correspondent expressions (Eqs. [14]–[15]), and the
following fifth order polynomial equation is found,
− iBω4 −
v2A + c
v2AB +A
+ v2Ac
iAv2Ac
= 0, (C.1)
= k2x +
(n + 1/2)2 π2
, (n = 0, 1, 2, . . .),
for the kink modes, and
= k2x +
, (n = 1, 2, 3, . . .),
for the sausage modes. Quantities A and B are given by
Eqs. (B.2) and (B.3), respectively.
References
Ballester, J. L. 2006, Phil. Trans. R. Soc. A, 364, 405
Balthasar, H., Wiehr, E., Schleicher, H. & Wöhl, H. 1993, A&A, 277, 635
Carbonell, M., Oliver, R. & Ballester, J. L. 2004, A&A, 415, 739
Carbonell, M., Terradas, J., Oliver, R. & Ballester, J. L. 2006, A&A, 460, 573
Dahlburg, R. B. & Mariska, J. T. 1988, Sol. Phys., 117, 51
De Moortel, I. & Hood, A. W. 2004, A&A, 415, 705
Dı́az, A J., Oliver R. & Ballester, J. L. 2005, ApJ, 440, 1167
Dı́az, A. J. & Roberts, B. 2006, Sol. Phys., 236, 111
Edwin, P. M. & Roberts, B. 1982, Sol. Phys., 76, 239
Engvold, O. 2004, Proc. IAU Collq. on Multiwavelength investigations of solar
activity (eds. A. V. Stepanov, E. E. Benevolenskaya & A. G. Kosovichev),
Forteza, P., Oliver, R., Ballester, J. L. & Khodachenko, M. L. 2007, A&A, 461,
Foullon, C., Verwichte, E. & Nakariakov, V. M. 2004, A&A, 427, L5
Harvey, J. 1969, Ph.D. thesis, University of Colorado, USA
Hildner, E. 1974, Sol. Phys., 35, 123
Joarder, P. S. & Roberts, B. 1992, A&A, 256, 264
Lin, Y., Engvold, O. & Wiik, J. E. 2003, Sol. Phys., 216, 109
Lin, Y. et al. 2005, Sol. Phys., 226, 239
Lin, Y., Engvold, O., Rouppe van der Voort, L. H. M. & van Noort, M. 2007,
Sol. Phys., in press
Milne, A. M., Priest, E. R. & Roberts, B. 1979, ApJ, 232, 304
Molowny-Horas, R., Oliver, R., Ballester, J. L. & Baudin, F. 1997, Sol. Phys.,
172, 181
Molowny-Horas, R., Heinzel, P., Mein, P. & Mein, N. 1999, A&A, 345, 618
Oliver, R. & Ballester, J. L. 2002, Sol. Phys., 206, 45
Porter, L. J., Klimchuk, J. A. & Sturrock, P. A. 1994, ApJ, 435, 482
Rosner, R., Tucker, W. H. & Vaiana, G. S. 1978, ApJ, 220, 643
Soler, R., Oliver, R. & Ballester, J. L. 2007, Sol. Phys., submitted (Paper I)
Terradas, J., Oliver, R. & Ballester, J. L. 2001, A&A, 378, 635
Terradas, J., Molowny-Horas, R., Wiehr, E. et al. 2002, A&A, 393, 637
Terradas, J., Carbonell, M., Oliver, R. & Ballester, J. L. 2005, A&A, 434, 741
Wiehr, E. 2004, Proc. SOHO 13, ESA SP-547, 185
	Introduction
	Equilibrium and basic equations 
	Dispersion relation 
	Results
	Regions of dominance of the damping mechanisms
	Exploring the parameter space
	Dependence on the equilibrium physical conditions
	Dependence on the prominence optical thickness
	Dependence on the heating scenario
	Comparison with the solution for an isolated slab
	Application to a prominence fibril
	Conclusions
	Expressions for the perturbations
	Approximate dispersion relation for the internal slow modes
	Dispersion relation for an isolated slab
