A&A 466, 377?388 (2007)
DOI: 10.1051/00046361:20041632
c?ESO 2007
Astronomy
&
Astrophysics
Global oscillations in a magnetic solar model
II. Oblique propagation
star
B. Pint?r
1
,R.Erd?lyi
2
, and M. Goossens
3
1
Solar System Physics Group, Institute of Mathematical and Physical Sciences, The University of Wales, Aberystwyth,
Penglais Campus, Aberystwyth, Ceredigion, SY23 3BZ, Wales, UK
email: b.pinter@aber.ac.uk
2
Solar Physics and Space Plasma Research Centre (SP
2
RC), Department of Applied Mathematics, University of She?eld,
Hicks Building, Hounsfield Road, She?eld, S3 7RH, England, UK
email: robertus@sheffield.ac.uk
3
Centre for PlasmaAstrophysics, Departement Wiskunde, Faculteit Wetenschappen, Katholieke Universiteit Leuven,
Celestijnenlaan 200B, 3001 Heverlee, Belgium
email: Marcel.Goossens@wis.kuleuven.ac.be
Received 9 July 2006/Accepted 29 January 2007
ABSTRACT
The coupling of solar global acoustic oscillations to a magnetised solar atmosphere is studied here. The solar interior ? atmosphere
interface is modelled by a nonmagnetic polytrope interior overlayed by a planar atmosphere embedded in nonuniform horizontal
atmospheric magnetic field. Pint?r & Goossens (1999, A&A, 347, 321) showed that parallel propagating acoustic waves can couple
resonantly to local magnetohydrodynamic (MHD) slow continuum modes only. In general, global acoustic modes can, however,
propagate in arbitrary directions with respect to local atmospheric fields giving rise to an additional e?cient coupling mechanism that
has consequences on mode damping and atmospheric energetics. In this paper we study obliquely propagating global modes that can
couple also to local MHD Alfv?n continuum modes. The atmospheric magnetic e?ects on global mode frequencies are still much of
a debate. In particular, the resulting frequency shifts and damping rates of global modes caused by the resonant interaction with both
local Alfv?n and slow waves are investigated. We found the coupling of global f and p modes and the Lamb mode, that penetrate into
the magnetic solar atmosphere, will strongly depend on the direction of propagation with respect to the solar atmospheric magnetic
field. These frequency shifts, as a function of the propagation direction, give us a further elegant tool and refinement method of local
helioseismology techniques. Finally we briefly discuss the importance of studying obliquely propagating waves and discuss the results
in the context of possible helioseismic observations.
Key words. Sun: helioseismology ? Sun: oscillations ? Sun: atmosphere ? Sun: chromosphere ? Sun: magnetic fields ?
magnetohydrodynamics (MHD)
1. Introduction
Observations reveal complex structures of the solar atmospheric
magnetic field. Beneath the solar surface the magnetic field can
be described by confined, vertical thin flux tubes. When these
flux tubes break through the photosphere, it is observed that
the magnetic field lines incline in most cases from the verti
cal direction. They fan out and create a local magnetic canopy,
i.e. structures with horizontal magnetic field, throughout the
chromosphere.
From the solar atmospheric magnetic structures we are inter
ested in global (or coherent) structures that exist for a relatively
long time, at least when compared to the life time of the global
oscillations. In practical terms this means a few hours. Title &
Schrijver (1998) argue that the life time of the lower solar atmo
spheric magnetic carpet satisfies this condition. Although esti
mates are changing as instrumentation improves, it is now gen
erally believed that the replacement time of the magnetic carpet
is of the order of 10?14 h; well above the life time of the global
oscillations, that is around 7?10 periods. When considering the
star
Appendix A is only available in electronic form at
http://www.aanda.org
global pmodes, their life time is around 2000 to 3000 s. This
observation allows us to work in a framework where the time
dependent variation of the background magnetic carpet can be
neglected.
If the picture of static magnetic carpet holds, this is even
more true for global chromospheric and coronal background
fields on the time scales of the life time of acoustic global os
cillations allowing an investigation of the coupling of these os
cillations to the solar atmosphere in stationary state. Before we
embark on the analysis, let us briefly recall some evidence that
may directly or indirectly indicate the mechanism of such cou
pling. Of course, there is no direct observational proof of the
actual resonant layer since that would require resolution of the
order of few hundreds of meters in the solar plasma. However,
there are studies indicating that oscillations found in the lower
solar atmosphere, also called the lower boundary layer, or even
in the low corona show strong correlation with the periodic mo
tions that characterize the photosphere (Erd?lyi 2006).
The existence of oscillations within the atmospheric mag
netic structures is clear from observations. The presence of os
cillations in sunspots was already known (e.g. Bogdan 2000)
before the generation of high angular and temporal resolution
Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/00046361:20041632
378 B. Pint?r et al.: Obliquely propagating global solar oscillations. II.
measurements. Images from SOHO and TRACE prove unques
tionably that di?erent kinds of oscillations propagate from the
solar interior through the chromosphere, and reach even the
corona (e.g., Schrijver et al. 2002; Aschwanden et al. 2002; and
Roberts 2002).
De Pontieu et al. (2003) found correlation between lower
transition region oscillations and photospheric pmode oscilla
tions. De Pontieu et al. (2004) showed how global modes leak
into the atmosphere along vertical thin flux tubes and can drive
periodic spicules along inclined thin flux tubes. This leakage
can even cause loop oscillations if these photospheric motions
reach the lower corona, as show by De Pontieu et al. (2005) and
De Pontieu & Erd?lyi (2006).
Last but not least, Rutten & Krijger (2003) have studied low
frequency brightness modulation of internetwork regions in the
low chromosphere using image sequences from TRACE, and
have shown that even atmospheric gravity waves are at present
in the low chromosphere. The examples will improve with the
launch of SolarB giving a timely aspect to questions like: what
are the physical details of the coupling of global oscillations to
the lower boundary layer? What is the coupling mechanism?
Could resonant absorption, an already popular mechanism for
plasma heating and damping of loop oscillations, play a role in
the coupling? What could be the manifestation of resonant cou
pling of global modes? In this paper we try to demonstrate the
answers to these questions in a simple theoretical model.
A number of solar models were put forward to investigate the
features of global oscillations in the solar atmosphere. An ap
proximation for the inhomogeneous solar interior and atmo
sphere is a twolayer model, where the lower layer represents the
magnetic fieldfree subphotospheric region while the upper one
is for the atmosphere embedded in a horizontal (canopy) mag
netic field with constant Alfv?n speed. Miles et al. (1992) used
that model with isothermal layers to study magnetoacoustic
gravity surface waves. Miles & Roberts (1992) carried out an
other followup study, in which the upper layer has a uniform
magnetic field. Campbell & Roberts (1989) studied the e?ects of
an atmospheric magnetic field on global solar oscillations, sim
ilarly in a twolayer model, with constant Alfv?n speed in the
layer above the interface. They considered the solar interior as
a polytrope, instead of an isothermal medium. Evans & Roberts
(1990) removed the assumption of constant Alfv?n speed and
considered a uniform chromospheric magnetic field instead.
Vanlommel et al. (2002) also studied the e?ects of a uniform,
horizontal coronal magnetic field on global oscillations. They
focused on coupling of global modes (which propagate paral
lel to the magnetic field lines) to local slow MHD oscillations,
and used an equilibrium model somewhat more realistic than in
Campbell & Roberts (1989) and Evans & Roberts (1990).
Magnetic e?ects on obliquely propagating f and p modes
were investigated by Jain & Roberts (1994) in the twolayer
model, with constant Alfv?n speed in the upper layer. Jain &
Roberts came to the conclusion that the positive frequency shift
of the f mode is reduced as?is increased from zero to 90
?
.They
explained this result as being due to the magnetic tension, which
is reduced when the angle of the wave vector with respect to
the magnetic field becomes larger. They also found that a pos
itive shift in pmode frequencies is reduced by an increase in
the propagation angle. The e?ect is smaller for p modes with
higher radial order since their frequency shifts are thought to
depend strongly on temperature and magnetic pressure in the
chromosphere, and so the orientation of the wave vector is less
important.
A more advanced threelayer model with an intermediate
zone, where the magnetic field, together with the Alfv?n speed,
varies continuously from zero was introduced by Tirry et al.
(1998). The importance of the Alfv?n continuum is that global
modes may interact resonantly to local Alfv?n oscillations at the
height were their frequency matches the frequency of the global
mode, hence the model can be used to investigate the e?ects of
resonant coupling between global modes and local magnetohy
drodynamic (MHD) oscillations. Tirry et al. (1998) gives a thor
ough mathematical description of the threelayer model, and out
lines the basic idea of resonant coupling. However, the physical
results presented in Tirry et al. (1998) are far from complete,
details were left for subsequent studies.
Pint?r & Goossens (1999), hereafter Paper I, exposed the
case of parallel propagation in a followup study to Tirry et al.
(1998). Di?erent types of oscillation modes have been deter
mined for a wide range of the magnetic field strength and for dif
ferent degrees of the spherical harmonic. The emphasis is on the
possible coupling of global solar oscillation modes to localized
continuum eigenmodes of the magnetic atmosphere. For propa
gation parallel to the magnetic field, the global oscillation modes
can couple only to slow continuum modes and this has been
found to occur for a rather large range of parameters. Damping
of parallel global oscillation modes due to resonant absorption
and frequency shifts of global modes due to the magnetic field
have been examined.
The model in the present paper is an enhancement of models
that have been used by those referred to previously (Campbell
& Roberts 1989 to Pint?r & Goossens 1999). We consider the
phenomena of resonant coupling of solar global oscillations to
the inhomogeneous solar atmosphere for oblique propagation.
Figure 1 shows the equilibrium profiles for the plasma den
sity, pressure, sound speed and Alfv?n speed for some typical
values. Notice that the number density, n
0
(z) is plotted in Fig. 1a,
though in the calculations the mass density,?
0
(z) ? m
p
n
0
(z)is
used, where m
p
is the average mass of the plasma particles.
The atmospheric scale height, H
co
,inFig.1aismuchlarger
than the range of the atmosphere shown. This is why the rate
of the exponential decay of n
0
(z)andp
0
(z) in the upper layer is
hardly recogniseble.
Although the model is an e?ective tool to describe crucial
solar phenomena, we have to emphasize that the current ap
proach does not claim to be a perfect representation of the highly
complex and dynamic Sun. The assumption of a steady uni
directional horizontal atmospheric magnetic field is obviously
a crude representation of the threedimensional magnetic struc
tures of the real solar chromosphere and corona. Magnetic fluxes
are continuously emerging at the solar surface and expanding
into the atmosphere. Consequently, the orientation of the atmo
spheric magnetic field changes temporarily and spatially. It is our
aim to further develop the present model by considering stochas
tic magnetic fields, similar to that in Erd?lyi et al. (2004a,b,
2005).
We describe the model in Sect. 2. In Sect. 3 the structure
of the frequency spectrum and the global oscillation modes to
gether with spatial solutions are derived. Section 4 is devoted
to a detailed investigation of the obtained atmospheric e?ects on
the fandpmodes. Section 5 is for the summary and discussion
of the results.
2. The model
The main characteristics of the model are presented in this sec
tion. First, a basic description is given, explaining the main
B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 379
Fig.1. Illustration of the equilibrium of a) plasma density and pressure and b) sound and Alfv?n speeds. (The Alfv?n speed profile is given for
B
L
?90 G. The scale heights shown are defined in Eqs. (11) and (12)).
features of the model, its relevance to the Sun together with
its limitations. Finally, the dispersion relation between the wave
number, k, and the frequency,?, of the eigenoscillations is de
rived.
Model description. A planeparallel, threelayer model in
Cartesian coordinate system is used here, similar to the one in
Paper I. While Paper I investigated the case of parallel propa
gation, the present study focuses on the results for the case of
nonparallel propagation in detail.
The static equilibrium model is derived from the ideal mag
netohydrodynamic (MHD) equations, which are the continuity
equation, the equation of momentum, the equation of energy and
the induction equation:
??
?t
+?.(?u)=0, (1)
?
bracketleftBigg
?
?t
+u.?
bracketrightBigg
u=??p+
1
?
(??B)?B+?g, (2)
bracketleftBigg
?
?t
+u.?
bracketrightBigg
p?
?p
?
bracketleftBigg
?
?t
+u.?
bracketrightBigg
?=0, (3)
?B
?t
=??(u?B), ?.B=0. (4)
The adiabatic index (or ratio of specific heats) is taken?= 5/3
throughout the paper. The three layers are the semiinfinite solar
interior (z< 0) and atmosphere (z>L) ? which is basically the
corona ? with a transitional layer between them (0 ? z ? L).
Note that this transitional layer is not taken just for the transition
region of the Sun. The equilibrium quantities (temperature, T
0
,
density, ?
0
, gas pressure, p
0
, and magnetic induction, B
0
)are
inhomogeneous and vary continuously in the zdirection, which
is oriented towards the solar centre. (The 0 index refers to the
state of equilibrium.)
The interior is a polytrope (i.e., p(z)??
?
(z)), where?=5/3
is the ratio of specific heats or the adiabatic index, with an equi
librium temperature decreasing from central to surface regions.
The top of the interior is the photosphere (z= 0). The temper
ature, T
0
(z), increases linearly in the intermediate transitional
layer from its photospheric minimum to its maximum, which
is the temperature of the isothermal corona.
The plasma density, ?
0
(z), decreases throughout the
three layer of the model with di?erent steepness. Its decay is
exponential in the corona.
A horizontal magnetic field, B
0
(z), is considered in the at
mosphere (z>0) representing a canopylike structure. From the
nonmagnetic photosphere, the field strength increases sharply
in the transitional layer to its top in a way that the square of
the Alfv?n speed, v
2
A
(z) ? B
2
0
(z)/(??
0
(z)), increases linearly.
Above that, the magnetic field strength decreases exponentially
together with the plasma density so that the Alfv?n speed re
mains constant in the corona. Such an equilibrium will have all
the necessary ingredients to investigate the mechanism of reso
nant coupling of solar global oscillations and will also be treat
able analytically.
The equilibrium profile of the plasma pressure, p
0
(z), can be
derived from the temperature, density and magnetic induction
by assuming pressure equilibrium and that the plasma obeys the
perfect gas law.
The equilibrium density, plasma pressure and Alfv?n speed
square, hence, have the following profiles, respectively:
?
0
(z)=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
ph
parenleftBigg
1?
z
H
in
parenrightBigg
1/(??1)
, z?0,
?
ph
parenleftBigg
1+
z
H
tr2
parenrightBigg
?
, 0?z?L,
?
L
exp
parenleftBigg
?
z?L
H
co
parenrightBigg
, L?z,
(5)
p
0
(z)=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
p
ph
parenleftBigg
1?
z
H
in
parenrightBigg
?/(??1)
, z?0,
p
ph
parenleftBigg
1+
z
H
tr1
parenrightBigg
parenleftBigg
1+
z
H
tr2
parenrightBigg
?
, 0?z?L,
p
L
exp
parenleftBigg
?
z?L
H
co
parenrightBigg
, L?z,
(6)
380 B. Pint?r et al.: Obliquely propagating global solar oscillations. II.
Fig.2. Sketch of the threelayer model.
v
2
s
(z)=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
v
2
s,ph
parenleftBigg
1?
z
H
in
parenrightBigg
, z?0,
v
2
s,ph
parenleftBigg
1+
z
H
tr1
parenrightBigg
, 0?z?L,
v
2
s,co
, L?z,
(7)
v
2
A
(z)=
?
?
?
?
?
?
?
?
?
?
?
0 ,z?0,
v
2
A,co
z
L
, 0?z?L,
v
2
A,co
, L?z,
(8)
with
p
ph
=
1
?
?
ph
v
2
s,ph
,v
2
s,co
=
T
co
T
ph
v
2
s,ph
(9)
?
L
=?
0
(z=L), p
L
= p
0
(z=L) (10)
H
in
=
v
2
s,ph
(??1)g
, H
co
=
1+?
?
v
2
s,co
?g
(11)
H
tr1
=
v
2
s,ph
L
v
2
s,co
?v
2
s,ph
,H
tr2
=
2v
2
s,ph
L
2v
2
s,co
?2v
2
s,ph
+?v
2
A,co
(12)
?=1+
2?gL
2v
2
s,co
?2v
2
s,ph
+?v
2
A,co
,?=
2v
2
s,co
?v
2
A,co
? (13)
The indices ph, L and co refer to equilibrium quantities taken
at the photosphere (z = 0), at the top of the transitional layer
(z=L) and in the corona (z?L), respectively.
The equilibrium profiles, Eqs. (5) to (8), can be derived
from the vertical component of the equation of motion, Eq. (2)
(which takes the form dP
0
(z)/dz =?g?
0
(z) for the equilibrium
quantities) and from the definition of the sound speed square,
v
2
s
(z)??p
0
(z)/?
0
(z). Here P
0
is the total pressure of the equilib
rium: P
0
(z)? p
0
(z)+B
2
0
(z)/2?.
Although the model is simple, it mimics a fairly realistic ba
sic solar structure. The atmosphere features a sudden density and
pressure decrease in the transitional layer. The temperature pro
file in the model reflects the observed transitional profile in the
sense that the interior becomes cooler towards the photosphere,
above which the corona becomes hot again at a certain height.
Most importantly, the magnetic profile represents a canopy mag
netic field above the observed magnetic carpet, where the con
densed magnetic flux tubes ? penetrating from the photosphere
into the atmosphere ? fan out and the magnetic field lines form
horizontal, canopy structures. The solar plasma is stratified in the
model by constant gravity,g=?ge
z
, withg= 274 m/s
2
,using
its observed photospheric value. Because the global waves are
expected to be concentrated right beneath the photosphere, they
are most a?ected by the photospheric values of the equilibrium
quantities.
Governing equations. The disturbances in the plasma are de
scribed by the linearized MHD equations, derived from Eqs. (4).
Ohmic heating is included via resistivity. Dissipation has to be
taken into account in regions only where steep gradients occur,
i.e., in the vicinity of the resonant heights. Elsewhere the ideal
MHD equations give an accurate description.
The perturbed quantities take their Fouriertransformed form
f
1
(x,y,z,t)= f(z;?,k
x
,k
y
)e
i(k
x
x+k
y
y??t)
. (14)
The aim is to obtain a dispersion relation, f(?,k
x
,k
y
) = 0. The
linear Fourier transformed ideal MHD equations can be reduced
to two ordinary di?erential equations of the first order for the
vertical component of the Lagrangian displacement, ?
z
(z)and
for the Eulerian perturbation of total pressure, P(z):
D
d?
z
dz
=C
1
?
z
?C
2
P, D
dP
dz
=C
3
?
z
?C
1
P. (15)
The total pressure is the sum of the plasma pressure and the mag
netic pressure, P(z) ? p(z)+B
2
(z)/(2?). The coe?cient func
tions D, C
1
, C
2
and C
3
in (15) are given by
D(z)=?
0
(v
2
s
+v
2
A
)(?
2
??
2
c
)(?
2
??
2
A
),
C
1
(z)=g?
0
?
2
(?
2
??
2
A
),
C
2
(z)=?
4
?k
2
(v
2
s
+v
2
A
)(?
2
??
2
c
),
C
3
(z)=
parenleftBigg
?
0
(?
2
??
2
A
)+g
d?
0
dz
parenrightBigg
D+g
2
?
2
0
(?
2
??
2
A
)
2
.
(16)
The two components of the horizontal wave vector, k,arek
x
=
k cos?and k
y
=k sin?. The angle?measures the inclination of
the propagation direction to the magnetic field lines (see Fig. 3).
The x axis in the model is taken to be parallel to the magnetic
field lines, i.e.
cos??
k.B
kB
? (17)
The local (i.e. zdependent) Alfv?n frequency?
A
(z) and the lo
cal slow or cusp frequency?
c
(z) play a fundamental role in the
coupling of global solar oscillations to continuum oscillations.
The local sound frequency?
s
(z) is, indirectly, also a key param
eter for the frequency spectrum. The squares of these frequencies
are given by
?
2
s
(z)=(k
2
x
+k
2
y
)v
2
s
(z),
?
2
A
(z)=k
2
x
v
2
A
(z),
?
2
c
(z)=k
2
x
v
2
c
(z),
(18)
where the cusp speed square is given by
v
2
c
(z)?
v
2
s
(z)v
2
A
(z)
v
2
s
(z)+v
2
A
(z)
?
Equations (15) describe how linear oscillations of a one
dimensional, inhomogeneous magnetic plasma are governed.
B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 381
Fig.3. The idea of oblique propagation.
The three layers of the model (interior, transitional layer
and corona) are joint by the physical requirements that the
Lagrangian displacement,?
z
(z), and the Lagrangian perturbation
of total pressure, P(z)+g?
0
(z)?
z
(z), must be continuous functions
of height. Equations (15) with these boundary conditions and
boundary conditions at??define an eigenvalue problem for the
global frequency?. The boundary conditions far away from the
transitional layer are that the kinetic and magnetic energy of the
eigenoscillations diminish to zero for z???(toward the solar
interior) and for z ??(toward the outer corona). The aim of
this study is to understand how the frequency spectrum of linear
eigenoscillations is changed when the direction of propagation,
?,varies.
3. Solutions to the governing equations
Interior. The general solutions of Eqs. (15) for?
z
(z)andP(z)in
the polytrope interior are a product of an exponential term and
a linear combination of the Kummer functions M and U. (About
Kummer functions, see, e.g., Abramovitz & Stegun 1965). The
integrational coe?cient of the terms containing function M has
to be zero due to the boundary condition that the kinetic energy
density, E
kin
??
0
(z)?
2
?
2
z
(z)/2, of the perturbation has a finite
asymptotic value as z tends to??. The solutions remain free up
to a constant factor, as the linear wave theory does not predict
the size of the perturbations.
Transitional layer. Once having the values ?
z
(z = 0) and
P(z = 0), from the internal solutions, a numerical integration
of Eqs. (15) can be carried out from z= 0toL, using the equi
librium profile of the transitional layer, in case of no resonance
(?>?
A
). If resonance occurs in the transitional layer, the numer
ical integration evaluates?
z
(z)andP(z) at the lower edge of the
resonant layer; there the connection formulae are used to obtain
?
z
(z)andP(z) at the top of the resonant layer, which are then the
initial values for further numerical integration, up to z = L.If
the global frequency takes place not only in the Afv?n but also
in the slow continuum, both sets of connection formulae have to
be used: around the slow resonant position, z
C
, and then around
the Alfv?n resonant position, z
A
. In this latter case the evalua
tion of ?
z
(z = L)andP(z = L) consists of numerical integra
tions along three sections in the transitional layer (first, below
the Alfv?n resonant height, then between the Alfv?n and slow
resonant heights, finally, above the slow resonant height) and the
use of two connection formulae (first, across the Alfv?n resonant
height, then across the slow resonant height).
Corona. In order to properly investigate the cuto?frequencies
(see the following paragraph), it is useful to write down explic
itly the coronal solutions for?
z
(z)andP(z). The general coronal
solution is
?
z
(z)=A
0
exp
parenleftBiggparenleftBigg
1
2H
??
parenrightBigg
z
parenrightBigg
+A
1
exp
parenleftBiggparenleftBigg
1
2H
+?
parenrightBigg
z
parenrightBigg
,
P(z)=
D
C
2
?
c
?
bracketleftBiggparenleftBigg
C
1
D
?
1
2H
+?
parenrightBigg
A
0
exp
parenleftBigg
?
parenleftBigg
1
2H
+?
parenrightBigg
z
parenrightBigg
+
parenleftBigg
C
1
D
?
1
2H
??
parenrightBigg
A
1
exp
parenleftBiggparenleftBigg
?
1
2H
+?
parenrightBigg
z
parenrightBiggbracketrightBigg
,
(19)
whereA
0
andA
1
are integration constants,?
c
is the equilibrium
density at the top of the transitional layer (at z = L)andthe
parameter?is defined by its square as
?
2
(z)=
parenleftBigg
C
1
D
?
1
2H
parenrightBigg
2
?
C
2
C
3
D
2
? (20)
The boundary condition lim
z??
E
kin
(z)=0 is satisfied for positive
values of?
2
and forA
1
=0 in Eq. (19).
Characteristic frequencies. The structure of the inhomoge
neous model indicates that the frequency spectrum of the
eigenoscillations is divided into di?erent regions by character
istic frequencies. Although such divisions of the solar eigen
spectrum have not been observed clearly yet, it is necessary to
analyse the e?ects of the characteristic frequencies arising in the
present structured model.
It follows from the definitions of the coe?cients D
and C
1
?C
3
, Eqs. (16), that ?
2
can be written as a cubic poly
nomial divided by a quadratic polynomial in?
2
:
?
2
=?
(?
2
??
2
I
)(?
2
??
2
II
)(?
2
??
2
III
)
(v
2
s
+v
2
A
)(?
2
??
2
A
)(?
2
??
2
c
)
? (21)
Here?
2
I
,?
2
II
and?
2
III
denote the three yet unknown roots of the
cubic polynomial in?
2
and play the role of cuto?frequencies.
The parameter?
2
changes sign when? matches one of the
cuto?frequencies or one of the two other characteristic frequen
cies, ?
A
or?
c
. The importance of the sign of?
2
is that pertur
bations in the corona (see Eqs. (19)) propagate for?
2
<0 (leaky
modes) and are evanescent for?
2
>0 (eigenmodes).
The frequency spectrum in an equilibrium which is free of
a magnetic field is characterized only by two cuto?frequencies,
?
I
and?
II
. They can be expressed as
?
2
I

B=0
=
v
2
s
2
parenleftBigg
k
2
+
1
4H
2
parenrightBigg
parenleftBig
1?
?
1??
parenrightBig
,
?
2
II

B=0
=
v
2
s
2
parenleftBigg
k
2
+
1
4H
2
parenrightBigg
parenleftBig
1+
?
1??
parenrightBig
,
(22)
382 B. Pint?r et al.: Obliquely propagating global solar oscillations. II.
where
?=
64k
2
H
2
(v
2
s
?gH)gH
v
4
s
(1+4k
2
H
2
)
2
, (23)
The sound speed,v
s
, and the isothermal density scale height, H?
v
2
s
/?g, are all constant in the corona.
Notice that ?, defined in Eq. (23), is always positive, as
?(=5/3) > 1. It follows from this that it never happens in the
nonmagnetic model that only the upper cuto? frequency ex
ists while the lower cuto? frequency does not exist (i.e., if
?
2
II
(B0)>0then?
2
I
(B0)>0 too).
Equation (20) reduces to a quadratic equation for ?
2
for
an equilibrium with a magnetic field (B nequal 0) and for parallel
propagation (k
y
= 0). The three frequencies characterising the
eigenspectrum are now?
I
,?
II
and?
c
. The two cuto?frequen
cies can be expressed analytically as
?
2
I
vextendsingle
vextendsingle
vextendsingle
k
y
=0
=
1
2
parenleftBigg
k
2
x
+
1
4H
2
co
parenrightBigg
(v
2
s
+v
2
A
)
parenleftBig
1?
?
1??
parenrightBig
,
?
2
II
vextendsingle
vextendsingle
vextendsingle
k
y
=0
=
1
2
parenleftBigg
k
2
x
+
1
4H
2
co
parenrightBigg
(v
2
s
+v
2
A
)
parenleftBig
1+
?
1??
parenrightBig
,
(24)
where
?=
4k
2
x
parenleftBiggparenleftBigg
k
2
x
+
1
4H
2
co
parenrightBigg
v
2
s
v
2
A
+
parenleftBigg
??
1+?
?1
parenrightBigg
g
2
parenrightBigg
parenleftBigg
k
2
x
+
1
4H
2
co
parenrightBigg
2
(v
2
s
+v
2
A
)
2
? (25)
The coronal magnetic density scaleheight, H
co
modified by the
presence of the magnetic canopy, is defined in Eq. (11). The
Alfv?n speed, the sound speed, the plasma?and hence H
co
are
all constant in the corona.
The Alfv?n and slow frequencies tend to zero while the
sound frequency becomes?
s
=k
y
v
s
for k
x
? 0. There are only
two cuto?frequencies for k
x
=0. Their asymptotic values are
?
2
I
vextendsingle
vextendsingle
vextendsingle
k
x
=0
=
1
2
(k
2
y
+
1
4H
2
co
)(v
2
s
+v
2
A
)
parenleftBig
1?
?
1??
parenrightBig
,
?
2
II
vextendsingle
vextendsingle
vextendsingle
k
x
=0
=
1
2
(k
2
y
+
1
4H
2
co
)(v
2
s
+v
2
A
)
parenleftBig
1+
?
1??
parenrightBig
,
(26)
where in this limit
?=
(??1)??1
1+?
k
2
y
parenleftBigg
k
2
y
+
1
4H
2
co
parenrightBigg
2
g
2
(v
2
A
+v
2
s
)
2
? (27)
The transitional layer between the photosphere and the corona
introduces an Alfv?n continuum and a slow continuum in the fre
quency spectrum of oscillations. These two frequency continua
are [min(?
A
(z)), max(?
A
(z))] (i.e. [0, ?
A
]) and [min(?
c
(z)),
max(?
c
(z))] (i.e. [0,?
c
]), respectively. A global mode that has
a matching frequency in the slow and/or Alfv?n continuum cou
ples resonantly to a local slow mode and/or a local Alfv?n mode,
respectively. The coupling also transforms the eigenmode into
a damped mode. The damping rate is the nonzero imaginary
part of the eigenfrequency ? as investigated in detail in the
Appendix.
Global oscillation modes propagating parallel to the atmo
spheric magnetic field lines (?= 0) can interact only with local
slow continuum modes. However, obliquely propagating global
oscillations (? nequal 0) can be coupled resonantly also to local
Alfv?n continuum modes.
Although the frequencies are expressed in terms of ? in
the derivations, henceforth the results are presented in terms of
?(??/2?), which is more commonly used when quoting obser
vations. The imaginary part, Im?, which represents the damping
rate of the mode, is replaced in the discussions by the, more fa
miliar, line width of the modes,?(??2Im?), for similar reason.
The characteristic frequencies are labeled in the plots of the solu
tions of the eigenvalue problem with I, II, III, A and C, referring
to the frequencies of?
I
,?
II
,?
III
,?
A
and?
C
, respectively.
It was discussed earlier that?
2
, defined in Eq. (20), has a pos
itive value for eigenmodes with a frequency right below?
II
.The
parameter?
2
is complex in the Alfv?n continuum and in the slow
continuum. The regions with positive real part of?
2
inthe Alfv?n
and slow continua are for (damped) eigenmodes, while those
with negative real part of?
2
inthe Alfv?n and slow continua are
for leaky modes.
Input parameters. The focus is on the e?ects introduced by
the atmospheric magnetic field strength and the angle of prop
agation on the frequency spectrum. Hence the thickness of the
transitional layer is fixed at L = 2 Mm throughout the numer
ical analysis, representing a transition between the photosphere
and corona. Also, the temperature increase through the transi
tional layer is fixed at T
co
/T
ph
= 200. The sound speed is taken
at the photosphere v
s,ph
= 7.6kms
?1
, which corresponds to
T
ph
? 4170 K photospheric temperature minimum. These im
ply that v
s,co
? 108 km s
?1
and T
co
? 834 000 K. The plasma
density at the photosphere is fixed at ?
ph
= 0.17 g m
?3
.The
avearage molar mass of the plasma particles is approximated in
the model as M
p
(?N
A
m
p
) = 1.3gmol
?1
. From this, the pho
tospheric plasma pressure is p
ph
? 8320 N m
?2
. The plasma
density,?
L
??
0
(z = L), and pressure, p
L
? p
0
(z = L), at the
top of the transitional layer are a function of the magnetic field
strength B
L
, as given by Eqs. (5) and (6).
The dependence of the mode frequencies on the harmonic
degree, l, is basically parabolic,??
?
l, as also obtained in many
other early studies, e.g. in Campbell & Roberts (1989). (The re
lation between the wave number, k, and the harmonic degree, l,
of an oscillation mode is k?
?
l(l+1)/R
circledot
.) The only restriction
to the possible choices of l in the present planar geometry comes
from the requirement that the horizontal wavelength of the per
turbations has to be small compared to the solar radius, which
is R
circledot
= 696 Mm in the model. It can be easily shown that this
condition is fulfilled for l > 6. Since at present we do not wish
to study the l dependence of the results, the harmonic degree is
fixed at 100 throughout the paper.
Structure of spectrum. First, we study the properties of charac
teristic frequencies, that define and border the di?erent regions
of the frequency spectrum. The propagation angle is varied be
tween 0 and 90
?
in Figs. 4a,b for l = 100, B
L
= 10and50G,
respectively. B
L
? B(z = L) is the magnetic field strength at
the top of the transitional layer, characterizing the overall atmo
spheric magnetic field strength. The values 10 and 50 G for B
L
are chosen to represent weak atmospheric magnetic fields.
The propagation angle,?, can obviously take values also be
tween 0 and?90
?
, but all the mathematical expressions related
to the frequency spectra are even functions of ?, and so solu
tions for characteristic frequencies of eigenfrequencies for any
B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 383
Fig.4. Frequency spectrum with varying ? for l = 100 and for weak
atmospheric magnetic fields: a) B
L
=10 G and b) B
L
=50 G.
value of?90
?
???0 are equal to the solution for??,whichis
between 0 and 90
?
.
The characteristic Alfv?n frequency,?
A
, and the third cut
o? frequency, ?
III
, are distinct for ? nequal 0, and ?
A
, ?
c
and ?
III
decrease to zero as ? ? 90
?
. The di?erences between ?
III
, ?
c
and ?
A
in Fig. 4a are so small that the frequencies cannot be
easily distinguished. A small region of the spectrum, around?=
50
?
, is enlarged and inserted in Fig. 4a to show the thin (<1?Hz)
layer between?
A
and?
c
(which is part of the Alfv?n continuum),
and between ?
c
and ?
III
(which is for leaky modes). The lower
cuto?frequencies,?
I
and?
III
, decrease while the upper cuto?
frequency,?
II
, increases with increasing?.
Next, let us embark on how the characteristic frequencies di
vide the frequency spectrum into di?erent regions for weak mag
netic field. Eigenmodes have real frequencies between?
I
and?
II
.
The thin region max(?
III
,?
c
)?
II
, ?
A
1) to the strong field region
(?<1). For a weak field the lower curve belongs to?
III
and the
upper one is ?
I
, while for a strong field the upper curve repre
sents?
III
and the lower one is for?
I
. The boundary between the
two regions,?= 1, corresponds to B
L
? 83 G. This change of
character is an example of coupling or avoided crossing between
characteristic frequencies.
4. Global oscillations
The inhomogeneous (e.g., threelayer) structure of the solar
atmosphere introduces characteristic frequencies. It has been
shown in Sect. 3 how strongly an atmospheric magnetic field af
fects them. Here we turn our attention to the global oscillations
themselves, and investigate how their frequencies are influenced
by the magnetic solar atmosphere.
4.1. Frequencies of eigenmodes in weak atmospheric
magnetic field
For a weak atmospheric magnetic field, the slow and Alfv?n con
tinua are confined at the lowfrequency region of the spectrum
(Fig. 4). Consequently, the eigenmodes, of which the frequen
cies are higher, are not coupled resonantly to either Alfv?n or
slow continuum modes. Figure 4a is the frequency spectrum for
a weak, B
L
= 10 G, field. The eigenmode with the lowest fre
quency in the spectrum is the fundamental mode (or f mode).
Above that the frequencies of the first two pressure modes (or
p modes) can be seen. Besides the f and p modes, the a mode
(also known as Lamb mode) also appears as an eigenmode, with
highest frequencies. The a mode is an acoustic mode, first time
found by Lamb (1932). In a nonmagnetic medium, the Lamb
mode frequency is near the sound frequency. The Lamb mode
usually appears as an eigenmode in isothermal models. The ex
istence of this mode in the present model is most likely due to
the isothermal corona. The Lamb mode can also arise in a model
in which the magnetic field of then isothermal corona is uniform,
i.e.v
A,co
=cst.is replaced by B
co
=cst.(Pint?r 1999). For more
details on this acoustic mode see also Paper I or, e.g., Vanlommel
&
?
Cade? (1998). The frequency of the a mode is right below the
upper cuto?frequency for all values of?. According to this, the
a mode oscillation can propagate parallel as well as obliquely
to the magnetic field lines. All the four modes (f, p
1
, p
2
and a)
in Fig. 4a are hardly influenced by the weak magnetic field, the
shifts of the mode frequencies are of the order of?Hz.
More pronounced magnetic e?ects can be seen in Fig. 4b,
where B
L
=50 G. First of all, there are two examples of charac
ter change between eigenoscillation modes, similar to that seen
between the two lower cuto?frequencies, discussed in the pre
vious subsection and visualised in Fig. 6. One of them happens
between the a and p
3
modes. For increasing?,theamode fre
quency moves away from?
II
at?? 20
?
,andthea mode trans
forms into the p
3
mode. The p
3
mode comes to existence with
increasing propagation angle at?? 28
?
, due to the fact that the
upper cuto?frequency is increased by the increase of?. The fre
quency of p
3
starts to increase sharply at?? 30
?
, showing that
the p
3
mode transforms into the a mode, of which the frequency
stays near below?
II
for further increasing?.
The p modes of radial order one and two exist between ?
I
and ?
II
in the whole range of? without any gap. The variation
of their frequencies with?is not large enough to see in Fig. 4b,
as it is still of the order of few tens of nHz, while the scale here
is?mHz.
The fundamental mode (f mode) is also present in the spec
trum. The frequency of the f mode falls into the Alfv?n and
slow continua for small?, i.e., the mode is resonantly coupled to
an Alfv?n mode and to a slow mode.
For a nonzero ?, the third cuto? frequency is no longer
equal to the Alfv?n frequency, as?
III
??
A
for?=0, and eigen
modes may occur also with frequencies between?
A
and?
III
.We
found a new mode of which the real frequency is within this re
gion, and exists for?>0 only, i.e. it does not propagate along
the magnetic field lines, only obliquely. We call this new mode
b mode. Being in the Alfv?n continuum, the b mode is coupled
resonantly to a local Alfv?n oscillation. The bmode frequency is
always just above the third cuto?frequency such as the amode
frequency is always right below the upper cuto?frequency. The
apparent similarities between the a and b modes allow us to con
sider the b mode another Lamb mode, that exists together with
the a mode, but is coupled to Alfv?n modes, and propagates only
obliquely to the magnetic field lines. We refer to this mode as the
magnetic Lamb mode.
The fmode frequency occurs also between ?
A
and ?
III
for
propagation angles 26
?
40
?
, above the slow and
Alfv?n continua.
4.2. Frequencies of eigenmodes in strong atmospheric
magnetic field
For strong atmospheric magnetic fields the increase of the upper
cuto?frequency allows the presence of more p mode frequen
cies within the Alfv?n and slow continua. Ten of them is plot
ted in Fig. 5a. The frequency spectrum is shown as a function
of?, as in Figs. 4a,b, but here with B
L
fixed at 100 G. The third
cuto?frequency is higher then?
A
for?>0and?
A
decreases
with increasing?slightly faster then?
III
. The gap between them,
which is for leaky mode frequencies, increases with increasing?.
Hence, the gap in the pmode frequencies due to the leaky region
between?
A
and?
III
is more and more recognisable for p modes
of lower order. An important consequence of this phenomenon
B. Pint?r et al.: Obliquely propagating global solar oscillations. II. 385
could be the direct leakage of solar oscillations into the magnetic
solar atmosphere (see e.g. De Pontieu et al. 2005; De Pontieu &
Erd?lyi 2006).
The magnetic Lamb mode can be found also for strong mag
netic fields. Its frequency is near the lowest cuto? frequency,
which is?
I
,as?
III
exceeds?
I
for strong magnetic fields. As the
b mode frequency decreases together with?
I
, the mode changes
character with the p
2
, p
1
and f modes with increasing ?.The
avoided crossings of the b and f modes are enlarged in the
inserted panel in Fig. 5a. The frequencies of both the b and
f modes can be found above the slow and Alfv?n continua for
large propagation angles.
The large region of slow continuum contains the frequencies
of the p
1
, f and b modes until a certain propagation angle.
Panel b in Fig. 5 is the frequency spectrum with the fre
quencies of the f, b and p modes for B
L
= 120 G, a mag
netic field strength possibly more characteristic for active re
gions. The pmode frequencies are displayed up to the p
8
mode.
The slow continuum is not increased much compared to that
for B
L
= 100 G, contrary to the Alfv?n continuum, which al
most fills now the entire spectrum for the given frequency range
(0 ??? 5 mHz). The gaps in the eigenmodes due to the leaky
regions between ?
I
and ?
c
and between ?
A
and ?
III
are hardly
noticeable.
4.3. Frequency shift and linewidth variation due to varying
propagation angle
Although the frequency spectra for di?erent atmospheric mag
netic field strengths show remarkable di?erences, only those due
to the varying characteristic frequencies are recogniseable, as
seen in Figs. 4 and 5. Frequency shifts of eigenmodes caused
by the varying magnetic field strength and propagation angle are
smaller than the order of mHz. However, they are of high impor
tance from observational point of view.
The frequency shift related to the frequencies for parallel
propagation,??(?; B
L
) ??(?; B
L
)??(?= 0; B
L
), and the line
width, ?(?; B
L
), of the f, p
1
, p
2
, p
3
and p
4
modes are given
for propagation angle 0 ??? 90
?
in Figs. 7a,b, respectively,
for B
L
= 100 G. The f, p
1
and p
2
modes have a gap in the
Alfv?n continuum.
Notice that here the frequency shift caused only by vary
ing ? is studied, because frequency shifts due to varying mag
netic fields have already been investigated in Paper I.
The frequency shifts of the f, p
1
and p
2
global modes due to
nonzero propagation angle are basically negative and their mod
ulus is less than 0.2?Hz. The frequencies of the three modes are
in the slow continuum and also in the Alfv?n continuum (which
is the frequency region coloured in green in Fig. 5a) for propa
gation angles smaller than 59.4
?
, 38.8
?
and 13.0
?
, respectively.
The modes, hence, are coupled resonantly to a local slow mode
and to a local Alfv?n mode in the atmosphere. The damping rate
of the global modes caused by resonant absorption can be mea
sured by the nonzero line width of the modes (which is shown
in the lower panel, in Fig. 7b). The three modes modes do not
propagate in certain directions, as their frequency is in a region
for leaky modes for an interval of?.Thep
2
mode terminates at
??13
?
,thep
1
mode at??39
?
and the f mode at??59
?
.The
lack of the three modes appears also in Fig. 5a as gaps in the
??(?) graphs. The frequencies of the f, p
1
and p
2
modes reap
pear for larger values of?. With a frequency right above?
I
,the
magnetic Lamb mode changes character first with p
2
(at??27
?
)
then with p
1
(at??43
?
) and finally with the f mode (at??60
?
)
(see also Fig. 5a). The influence of the magnetic Lamb mode on
Fig.7. a) Frequency shift and b) line width of the f and first
four p modes as functions of propagation angle, ?,forl = 100 and
B
L
=100 G.
?
The actual value for the line width of the f mode can be
obtained by multiplying the values shown in panel b) by a factor of ten.
The frequency shifts in panel a) and line widths in panel b) are plotted
as dashed lines for which?the f, p
1
or p
2
mode shows the characteris
tics of the Lamb mode.
the f, p
1
and p
2
mode frequencies around the given propagation
angles appears as a sharp increase in the frequency shift with
decreasing?. Hence, positive frequency shifts occur only due to
the magnetic Lamb mode. The frequency shifts e?ected by the
interaction with the magnetic Lamb mode is plotted with dashed
lines in Fig. 7a.
The frequencies of p
3
and p
4
, however, are shifted further
from the values obtained for parallel propagation, also to the
negative direction (i.e.??(?)<0). The negative frequency shifts
take their maximal values at?= 90
?
, which shift is of the order
of?Hz.
The f, p
1
and p
2
modes are coupled resonantly to a local
slow and a local Alfv?n mode for the intervals of ? given in
the latter paragraph. The largest line width can be found for the
f mode, for 0