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Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
On the theory of MAG waves and a comparison with sunspot
observations from CDS/SoHO
D. Banerjee 1 , E. O'Shea 2 , M. Goossens 1 , J.G.Doyle 3 , and S. Poedts 1
1 Centre for Plasma Astrophysics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium
2 Instituto de Astrosica de Canarias, C/ V  ia Lactea s/n, 38200 La Laguna, Tenerife, The Canary Islands, Spain
3 Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
Received date, accepted date
Abstract. We examine the in uence of non-adiabatic eects on the modes of an isothermal stratied magnetic
atmosphere. We present new solutions for magneto-acoustic-gravity(or mag) waves in the presence of a radiative
heat exchange based on Newton's law of cooling. An analytic expression for the dispersion relation is derived,
which allows the eect of a weak magnetic eld on the modes to be studied. The insight so gained proves useful
in extending the computations to the moderate{high eld case. In the second part we present observations of
two sunspots obtained in the euv wavelength range with the Coronal Diagnostic Spectrometer (cds) on SoHO.
We examine the time series for the line intensities and relative velocities and calculate their power spectra using
wavelet transforms. We nd oscillations in the chromosphere and transition region above the sunspots in the
temperature range log T = 4.6-5.4 K. Most of the spectral power above the umbra is contained in the 5-7 mHz
frequency range. When the cds slit crosses the sunspot umbra a clear 3 min oscillation is observed. The observed
oscillation frequencies are compared with the computed frequencies and the observations are interpreted in terms
of the slow magneto-acoustic waves.
1. Introduction
In the past thirty years, observations of oscillations with
periods around 3 minutes have been widely reported
in the atmosphere of sunspots (See reviews by Lites,
1992; Bogdan 2000; Fludra 1999, 2001; Brynildsen et al.
1999a,b, 2000, 2002; Maltby et al. 1999, 2001; Tziotziou
et al. 2002; O'Shea et al. 2002). It is widely believed that
these oscillations are the signatures of waves propagat-
ing in the sunspot atmosphere. A study of these oscilla-
tions can therefore be used to reveal information about
the form of the waves and the structure and nature of
the sunspot. The aim of the present study is to con-
tribute towards developing a theory of such wave mo-
tions and to compare them with observations performed
by the Coronal Diagnostic Spectrometer (cds) on SoHO.
The waves that we consider here are magneto-acoustic-
gravity (mag for short) waves. We study the physical na-
ture of the mag oscillations and try to understand the
cause for the existence of dierent types of elementary
wave modes in a magnetized radiative isothermal atmo-
sphere, subject to dierent sets of boundary conditions.
The present investigation is a continuation of earlier work
by Hasan & Christensen-Dalsgaard (1992) and Banerjee
Send oprint requests to: D. Banerjee, e-mail:
dipu@wis.kuleuven.ac.be
et al. (1995), who examined the eects of a weak verti-
cal magnetic eld on the normal adiabatic modes of an
isothermal atmosphere by combining a semi-analytic ap-
proach, based on asymptotic dispersion relations, with nu-
merical solutions. However, oscillations in a realistic solar
atmosphere are aected by radiative dissipation and en-
ergy losses at the boundaries. Thus the modes are damped
and have complex frequencies. In this paper, we exam-
ine the in uence of non-adiabatic eects on the modes of
an isothermal stratied magnetized atmosphere. The in-
clusion of radiative dissipation based on Newton's law of
cooling demonstrates the importance of this eect in the
study of magneto-atmospheric waves. It was pointed out
by Bunte & Bogdan (1994) that Newtonian cooling can
be incorporated in the solution of the isothermal magneto-
atmospheric wave problem by replacing , the ratio of spe-
cic heats, by a complex frequency-dependent quantity.
This procedure permits one to generalize easily the pre-
vious calculations to include radiative dissipation. Bunte
& Bogdan treated a planar, isothermal and stratied at-
mosphere in the presence of a horizontal magnetic eld,
whereas in this study we consider a vertical magnetic eld.
Babaev et al. 1995 derived an exact solution of the mag
waves in the case of oblique propagation with respect to
the magnetic eld. The solutions are expressed in terms
of the generalized Meijer's hypergeometric G functions.

2 D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations
Their solutions are very similar to our analytical results.
We further perform a normal mode analysis subject to
dierent sets of boundary conditions and compare our an-
alytical results with full numerical solutions.
We consider wave damping by radiative energy ex-
change, which is likely to be eôcient in the solar pho-
tosphere, where the radiative relaxation times are very
short compared to the typical wave periods. By compari-
son, damping of hydro-magnetic waves due to viscous dis-
sipation and particle conduction is entirely negligible in
those layers of the solar atmosphere where small ampli-
tude disturbances are likely to occur. As we will see later,
the condition for the propagation of gravity waves, which
depends on the existence of the buoyancy force, is more
stringent in the presence of radiative damping than in its
absence (Bray & Loughhead, 1974). The radiative damp-
ing of oscillatory modes in a optically thin, isothermal, un-
magnetized medium was studied by several authors (Stix,
1970; Sourin, 1972; Mihalas & Mihalas, 1984). Bogdan &
Knolker (1989) obtained the dispersion relation for linear
compressive plane waves in a homogeneous, unstratied,
uniformly magnetized radiating uid. Here we consider
the propagation of optically thin mag waves in a strati-
ed, uniformly magnetized medium in which the radiative
energy exchange occurs through Newton's law of cooling.
The main eect of radiation is to damp the waves. The
plan of the paper is as follows: in Sect. 2, the basic wave
equations are presented, including Newton's law of cool-
ing. In Sects. 3 and 4 we present the dispersion relation
for a weak eld followed by numerical results in the form
of a
K
diagram. In Sect. 5 we treat the strong eld
case which will be more relevant for comparison with our
observations. This is followed by an observational report
of sunspot oscillation studies by cds in Sects. 6 and 7. A
discussion of the observational results and a comparison
with the theoretical results are taken up in Sect. 8 and
nally the conclusions are drawn in Sect. 9.
2. The wave equation with Newtonian cooling
We conne our attention to an isothermal atmosphere
with a vertical magnetic eld which is unbounded in the
horizontal direction. Using a uid description and assum-
ing an ideal plasma, we write the energy equation as,
@ t
ôT
T + ôv z
1
T
dT
dz + ( 1)r
ôv = 1
 R
ôT
T ; (1)
where ôv = @ t  is the velocity perturbation, T is the tem-
perature of the gas and ôT the temperature perturbation.
We assume the Lagrangian displacement  varies as 
e i(!t kx) , where ! is the angular frequency and k is the
horizontal wave number. We allow for radiative losses, ap-
proximating them by Newton's law of cooling (eg. Spiegel,
1957; Mihalas & Mihalas, 1984), which assumes that the
temperature uctuations are damped radiatively on a time
scale  R , given by
 R = c v
16T 3
; (2)
Fig. 1. The complex parameter  as a function
of
~
R .
Panels (a) represents the variation of the real part and (b)
represents the variation of the imaginary part.
where  is the mean absorption coeôcient per unit length,
c v the specic heat per unit volume, and  the Stefan-
Boltzmann constant. For simplicity, we assume that  R is
constant over the atmosphere. If the vertical dimension
of the perturbation is small compared to the local scale
height, the relation between the Lagrangian perturbations
in pressure p and density  is then approximately given
by ôp=p '  ô=, where
 (!) = 1 + i! R
1 + i! R
: (3)
With these assumptions, the linearized equations for mag
waves are given by a system of two coupled dierential
equations (Banerjee et al. 1997),
[v 2
A
d 2
dz 2 (~ c 2
S +v 2
A )k 2 +! 2 ] x ik(~ c 2
S
d
dz g) z = 0 ;(4)
[~ c 2
S
d 2
dz 2 ~ g d
dz +! 2 ] z ik[~ c 2
S
d
dz (~ 1)g ] x = 0 ;(5)
where  z and  x are the amplitudes of the vertical and
horizontal components of the displacement, g is the accel-
eration due to gravity, and ~ =  = . The adiabatic sound
speed and the Alfven speed are given, respectively, by
c S =
r
p
 and v A = B
p 4 : (6)

D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations 3
We should point out here that the equation governing the
propagation of the purely transverse Alfven waves is de-
coupled from Equations (4) and (5) and will not be con-
sidered in the present investigation. We have implicitly as-
sumed that the propagation and the motions of the mag
modes are conned to the x z plane. This involves no
loss of generality. Equations (4) and (5) have the same
structure as the linearized wave equation for adiabatic
perturbations (see Eqs. (1){(2) of Hasan & Christensen-
Dalsgaard 1992), apart from the appearance of the pa-
rameter ~
, which incorporates radiative cooling. In non-
dimensional form equation (3) can be written as
 = 1
+i
~
 R
1
+i
~
 R
; (7)
where the dimensionless relaxation time scale is given by
~  R = (c S =H)R , where H = p=g is the scale height of the
atmosphere, which is constant for an isothermal medium
and the dimensionless frequency,
is given by expression
(9).
As
~
 R varies from 0 to 1 ,  describes a semi-
circle in the complex plane (see Bunte & Bogdan 1994).
Figures 1a & b show the variation of the real and imag-
Fig. 2. The variation of the real part (solid line) and the imagi-
nary part (dashed line) of the eective Brunt-Vaisala frequency
~
BV , (in dimensionless units) as a function
of
~ R .
inary parts of  respectively as a function
of
~
 R . We
nd that
for
~  R < 0:1,  approaches the isothermal
limit, and  = 1.
For
~
 R > 10; Re(  ) = 5/3 = .
Thus in the limit  R ; !1 i.e. in the limit of adiabatic
perturbations,  = and ~ = 1. Letting ~ = 1 in equa-
tions (4) and (5) we recover the linearized equations given
by Hasan & Christensen-Dalsgaard (1992). The imaginary
contribution to  is maximal
for
~
 R  1. Thus to study
the maximal eect of radiative heat exchange we choose
our parametric values such
that
~
 R becomes close to 1.
In order to obtain a dimensionless wave equation we in-
troduce three dimensionless parameters,
K = kH ; (8)

= !H
c S
; (9)
and the dimensionless vertical coordinate
 = !H
vA
= c S
v
A;0

e z=(2H) ; (10)
where v A;0 is the Alfven speed at z = 0. In terms of the
variables dened by equations (8) { (10), equations (4)
and (5) can be combined into a fourth-order dierential
equation for  x ,

 4 d 4
d 4
+ 4 3 d 3
d 3
+

1 +
4(

2
~ K 2 ) + 4 2

 2 d 2
d 2

1
4(

2
~
+K 2 ) 12 2

 d
d
+ 16

2
~
+K 2 (
~
2
BV
2
1)



2

2 K 2
~
 
 x = 0 ; (11)
where
~
2
BV = (  1)=  is the square of the eective
Brunt-Vaisala frequency (in dimensionless units). This
Brunt-Vaisala frequency is a function of frequency in the
non-adiabatic case.
Figure 2 shows the dependence of
~
BV on the radia-
tive relaxation time ~  R . The solid line depicts the real part
whereas the dashed line represents the imaginary part of
~
BV . Note that
for
~  R > 10,
~
BV reaches a constant
value of 0.5 (corresponding to the adiabatic limit). On the
other hand,
as
~  R ! 0 , Re(
~
BV ) < 0:1. Thus, in the
isothermal limit,
~
BV , which is the higher cuto frequency
for the g modes, is very low. The consequences of this
will be taken up again when we discuss the properties of
g modes in detail. Figure 2 also reveals that the imagi-
nary part of
~
BV is signicant only
for
~  R  1.
The general solution of equation (11) can be expressed
in terms of Meijer functions (Zhugzhda 1979) as follows:
 (h)
x = G 12
2;4

 h ; a 1 ; a 2
 1 ;


;  i ;


;  4
j  2

; (12)
where (i; h = 1; : : : ; 4; i 6= h) and
 1;2 = (1  i)
2 ;  3;4 = K ; (13)
a 1;2 = (1  )
2 ; (14)
 =
s
4

2
~

1 ; (15)
 =
q
 2 + 4K 2 (1
~
2
BV
=
2 ) : (16)
These solutions are very similar in nature to the purely
adiabatic case (see Banerjee et al. 1995). Once  (h)
x is
known, it is fairly straightforward to determine the corre-
sponding solutions  (h)
z from either of equations (4) or (5).
The complete solutions satisfying the required boundary
conditions can be constructed as linear combinations of
 (h)
x and  (h)
z .

4 D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations
Fig. 3. Diagnostic diagram for non adiabatic modes. Panel (a)
for g1 mode and (b) for p1 mode. Dierent line styles cor-
respond to dierent radiative relaxation time ~ R as labelled.
3. Normal modes in a weak magnetic eld
We now examine the asymptotic properties of waves and
normal modes of a stratied atmosphere with a weak mag-
netic eld (corresponding to the limit of small , where
 = v A;0 =c S ). The analytical results are used for the inter-
pretation of the numerical solutions presented in Sect. 4.
In order to get a physical picture of the solution, we con-
sider the upward propagation of a wave, excited from be-
low at z=0, in an isothermal atmosphere. It is well known
that acoustic modes are easily re ected if the temperature
of the medium changes with height (for a good discus-
sion see Leibacher & Stein 1981). The slow mode can be
re ected due to the increasing Alfven speed with height
from layers where vA  c S , through conversion into a fast
mode (eg. Zhugzhda et al. 1984). We implicitly assume
that the properties of the atmosphere change abruptly at
the top boundary, resulting in downward re ection of the
waves. The lower boundary condition is chosen to simu-
late a forcing layer. This permits standing wave solutions.
It should, however, be kept in mind that an isothermal at-
mosphere by itself does not trap modes, rather we use this
assumption to understand the physical properties of the
modes in a stratied atmosphere with a vertical eld. Let
us now derive approximate dispersion relations for various
boundary conditions.
3.1. Rigid boundary condition
Let us rst consider rigid boundary conditions, viz.
 x =  z = 0 at z = 0 and z = d : (17)
here d is the height of the upper boundary, and D = d=H
is the dimensionless height. The asymptotic properties of
Fig. 4. Variation of the imaginary part
of
with K for ~ R =
0:5. The dierent line styles corresponds to dierent  values
(which is a measure of magnetic eld strength) as labelled.
the solution in the purely adiabatic limit are presented
in Hasan & Christensen-Dalsgaard (1992). Following the
same line of treatment using equations (12){(16) and ap-
plying the boundary conditions given by equation (17) one
can derive the following dispersion relation in the weak
eld limit
(

2
~

K 2 ) sin ~  sin(K z D) =
2
p
~


e D=4

K z K 2 
cosh(D=4) cos ~
 cos(K z D) 1

+ sinh(D=4) cos ~
 sin(K z D)

M(

2
~
K 2 )
K 2 ( 1
~
1
2 )
 
+O 
2
2


;(18)
where
 0 = (0) ;  D = (D) ; ~
 = 2( 0  D ) ; (19)
and K 2
z is given by
K 2
z
=

2
~
K 2 (1
~
2
BV
2
) 1
4 ; (20)
and
M = K 2
~
2
BV

2
1
16
: (21)

D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations 5
For  <<
the dispersion relation to lowest order in
=
becomes
(

2
~ K 2 ) sin ~
 sin(K z D) = 0 : (22)
Equation (22) admits the following solutions
sin(K z D) = 0 ; (23)
sin ~  = 0 ; (24)

=
p
~ K: (25)
We rst consider the solution given by equation (23) which
implies that K z D = n, where n is an integer and denotes
the order of the mode. Using this condition equation (20)
yields,

4
i
~


2
i (K 2
t + 1
4 ) +K 2
~
2
BV = 0 (i = p; g) ; (26)
where K 2
t = K 2
z +K 2 . Note that equation (26) looks very
similar to the usual relation for p and g modes (see
Banerjee et al. 1995) apart from the factor ~
and modied
~
BV . Because of the presence of these two factors, the
properties of these modes change drastically. Figures 3a,b
give an overview of the behaviour of the g and p type
modes respectively. Their properties are re ected in these
two
K
diagrams produced by solving equation (26) for
dierent values of the relaxation time ~
 R . Figure 3a clearly
shows that the g 1 mode has been pushed down to the low
frequency part of the diagnostic diagram with decreasing
value of ~  R . As ~  R ! 0, the g 1 mode tends to disappear,
because
~
BV ! 0. On the other hand, Figure 3b shows
that the inclusion of radiative exchange has not greatly
changed the behaviour of the p 1 mode, apart from a
decrease of the acoustic cuto frequency.
The solution corresponding to equation (25) can be
recognized as a modied Lamb mode (compare with

= K, for pure Lamb mode). Thus we expect a fre-
quency shift of the adiabatic Lamb mode. Turning our
attention to the solution given by equation (24), we nd
that these modes are the same magnetic modes present
in the adiabatic conditions, which arise solely due to the
presence of the magnetic eld. The magnetic modes, here-
after referred to as m modes, have frequencies

m = l
2s ; (l = 1; 2; : : :) ; (27)
where s = (1 e D=2 ). These modes are approximately
transverse ( it has be shown by Banerjee et al. 1995
that  (1;2)
x = (1;2)
z  O() ). Physically, these modes can
be interpreted as gravity-modied slow modes in a weak
magnetic eld. Thus these slow modes are not aected by
the inclusion of radiative losses in the weak eld limit.
This result complements the result of Bogdan & Knolker
(1989), where it was conjectured that the uniform mag-
netic eld reduces the temperature perturbations associ-
ated with these waves and therefore suppresses the radia-
tive damping of these disturbances. This aspect will be
taken up later when we discuss the numerical solutions.
3.2. Zero-gradient boundary condition
If we use zero-gradient boundary conditions at the top and
bottom of the layer,
d x
dz = d z
dz = 0 at z = 0 and z = d : (28)
In addition to the modes discussed previously, we nd
another wave mode, namely the gravity-Lamb mode. The
dispersion relation for this mode is given by
K 2
z + 1
4 = 0 : (29)
Combining equation (29) with (20) yields

4 ~

2 K 2 + ~
K
2
2
BV = 0 : (30)
This equation has the solution

2 = ~ K 2
2
2
4 1 

1 4
~
2
BV
K 2 ~

! 1=2
3
5 : (31)
The solution resembles a modied gravity mode on the
lower branch and a Lamb mode on the upper branch. In
order to see this, consider the limit K !1. The smaller
solution in equation (31) has the
limit
'
~
BV , which
is the dispersion relation for a modied g mode for large
K; the larger solution has the
limit
'
p
~ K for large K,
which shows that the mode behaves like a modied Lamb
wave.
Thus the separate modes have changed their behav-
ior in the diagnostic diagram in the non-adiabatic case.
It is important to know how these modied modes inter-
act with one another in the presence of radiative losses.
Mode coupling in the non adiabatic case will be dierent
as compared to the adiabatic case studied by Banerjee et
al (1995) (the right hand side of equation (18) contributes
to the coupling).
4. Numerical results
The behavior of the mag waves is re ected in their prop-
erties in the
K
diagram namely the variation of the
real and imaginary part of the complex frequency with the
horizontal wave number K. The solutions were obtained
by solving equation (11) numerically, using a complex ver-
sion of the Newton-Raphson-Kantorovich scheme (Cash
& Moore, 1980) subject to a dierent sets of boundary
conditions. Banerjee et al (1997) presented the numerical
solutions for the weak eld case subjected to rigid bound-
ary conditions. In this paper we would like to compare our
theoretical results with some observational results so we
concentrate here on higher magnetic eld strengths.
First we show the eect of the strength of the magnetic
eld on the damping of these waves. To delineate the in u-
ence of the magnetic eld, we choose the m 2 mode, which
is predominantly magnetic in nature. Figure 4 shows the
variation of the imaginary part of the frequency (which is a
measure of the damping) with K for xed ~
 R = 0:5; D = 1;

6 D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations
Table 1. Eigenfrequencies of dierent order p modes for a model atmosphere with D = 10; B = 2kG; ~ R = 0:5 and K = 0:1.
Adiabatic case
(~ R = 100)
Radiative case
(~ R = 0:5)
Isothermal case
(~ R = 0:05)
Mode
Re(
Im(
P(S) Re(
Im(
P(S) Re(
Im(
P(S)
p1 0.5903 0.0007 164 0.5203 0.0658 186 0.458 0.0125 211
p2 0.803 0.0009 120 0.7075 0.0895 136 0.624 0.017 155
p3 1.07 0.0013 90 0.94 0.119 103 0.828 0.0227 117
and = 5=3 (for rigid boundary conditions). The dier-
ent line styles correspond to dierent  values. It clearly
reveals that as we increase the value of  (increasing mag-
netic eld strength) the imaginary part reduces, indicating
less damping of these wave modes. This result is in agree-
ment with the conclusions drawn by Bogdan & Knolker
(1989), that the magnetic eld suppresses radiative damp-
ing. For horizontal magnetic eld Bunte & Bogdan (1994)
also reported a \stiening" of the atmosphere with in-
creasing  values.
Fig. 5. Region in the diagnostic diagram for moderate eld
strength ( = 0:1) and ~
R = 0:5, where the modied
Lamb mode, magnetic modes and the gravity-Lamb mode are
present. These results are for zero gradient boundary condi-
tions.
Let us now consider a moderate magnetic eld
strength. Figure 5 shows a region in the diagnostic dia-
gram for  = 0:1 (B  240 G) and ~  R = 0:5, subject
to the zero-gradient boundary conditions. The mode cou-
pling in this case is much more complicated because we
have three mode interaction regions as indicated. As K in-
creases, the m 1 mode begins to acquire the character of
a modied Lamb mode. Figure 6a, which shows the varia-
tion of imaginary part of the frequency of the m 1 mode
with K also reveals that, there is an enhancement as it ap-
proaches an avoided crossing (near K = 0:8) followed by
a suppression due to mode transformation. Up to K = 0:8
Fig. 6. Variation of the imaginary part
of
with K for the
same set of parameters as in Fig. 5. Panel (a) shows the m1
mode and panel (b) the m2 mode.
this mode behaves as a magnetic Lamb mode and after
the mode transformation it becomes a magnetic type. This
process is repeated at higher frequency (around K = 2).
Note the large drop due to magnetic eld suppression.
Figure 6b shows the variation of the imaginary part of the
frequency for the m 2 mode (Figure 5) with K. The two
peaks correspond to modied Lamb and m mode cou-
pling and modied gravity-Lamb (gL ) and m mode
coupling respectively. Note the steep rise of the imaginary
part after the avoided crossing which indicates the eect of
the gravity mode. Figure 5 also shows the lower branch of
the modied gravity-Lamb mode (indicated as
~
gL ) which
was absent in the purely adiabatic case.
5. High magnetic eld case
We now consider a situation which is more realistic as
far as the solar atmosphere is concerned. We consider
an isothermal atmosphere extending over several scale
heights for which vA  c S over most of the atmosphere.
This situation is somewhat similar to the atmosphere in
sunspots. We consider the solution for small K. There are
three types of wave modes present in this situation, the
slow, fast and magneto-gravity-Lamb (MgL) modes. The
gL mode acquires a more pronounced magnetic behavior
because of the higher magnetic eld strength and so we
call them MgL mode (see Banerjee et al. 1995 for fur-
ther details). From a study of the energy density variation
of these modes we nd that the fast and MgL modes are
essentially conned to photospheric regions (i.e. the lower
part of the atmosphere), whereas the wave energy density
of the slow waves is spread over the entire extension of the
cavity. As far as the wave heating is concerned the slow

D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations 7
modes appear to be a more promising candidate than the
other two type of modes.
The frequencies of the slow magneto-acoustic modes or
p modes can be found from equation (26) with K = 0,
i.e.

p;n =
s
~

n 2  2
D 2
+ 1
4

; (32)
where n denotes the order. Following Scheuer & Thomas
Table 2. Eigenfrequencies (corresponding to a sunspot with
radius 5000 km) of dierent p modes for a model atmosphere
with D = 20; B = 2kG; ~ R = 0:5.
Mode
Re(
Im(
P(S) (mHz) D(S) D/P
p1 0.520 0.0659 186 5.4 233 1.25
p2 0.605 0.0767 160 6.2 200 1.25
p3 0.706 0.089 137 7.3 173 1.26
p4 0.820 0.104 118 8.5 148 1.25
(1981) let us treat the sunspot umbra as a cylinder of ra-
dius a. It can easily be shown that our analysis for a plane
can be carried over in a straightforward way to cylindrical
geometry, by treating axisymmetric modes and regarding
 x and k as the radial displacement and wave number
respectively. Assuming that the radial component of the
displacement vanishes at r = a, we nd that k takes dis-
crete values given by ka = j 1; , where j 1; denotes the
zero of the Bessel function J 1 of order . We consider the
lowest-order mode (where order refers to the horizontal
direction) corresponding to =1. This provides us with a
relation between the horizontal wave number and the ra-
dius of the spot. Table 1 represents the eigenfrequencies
of dierent order p modes from our model atmosphere
with D = 10, ~
 R = 0:5 and  = 0:84 (B  2KG). Table 1
reveals that radiative cooling shifts the eigenfrequencies
away from the real axis. Note that the computed frequen-
cies match very well with the ones calculated from expres-
sion (32). Radiative cooling leads to a temporal decay of
oscillations of the form exp( t= D ). The frequency eigen-
values of the four modes are listed in Table 2 for a typical
sunspot with radius of 5000 km (K = 0:06) together with
the ratio of characteristic decay time  D =Im(!) 1 and os-
cillation period = 2=Re(!). In the presence of Newtonian
cooling all four modes are damped by a factor e 1 within
two oscillation periods.
6. Observations and data reduction
For these observations we have used the normal inci-
dence spectrometer (nis) (Harrison et al. 1995), which
is one of the components of the Coronal Diagnostic
Spectrometer (cds) on-board the Solar Heliospheric
Observatory (SoHO). The data discussed here were se-
lected from the observing period 14th April and 19-20
Fig. 7. mdi intensity-gram showing the location of the slit
of the s19332r00 dataset, relative to the sunspot umbra and
penumbra. The over-plotted black rectangles are the locations
of the slit at the start (right) and at the end time (left) of the
observations. Pixel number 67 is marked with a white box.
April 2000. The observations were performed for two
dierent active regions. The details of the observations
including pointing and start times are summarized in
Table 3. Two dierent cds sequences were run, one tempo-
ral series sequence called CHROM N6 and another raster
sequence called CHROM N5. Temporal series datasets of
85 min. duration were obtained for the three lines of
He i 584  A(log T=4.6), O iii 599  A(log T= 5.0) and
O v 629  A(log T=5.4) using exposure times of 25 sec
and the 2  240 arcsec 2 slit. The cds pixels in the y di-
rection ( i.e. spatial resolution) are of size 1.68 arcsec. In
the raster sequence the 2  240 slit was moved 30 times in
steps of 2 arcsec so as to build up 60  240 arcsec 2 raster
images within a duration of 24 minutes. For this sequence
the lines used were: He i 584  A(log T=4.6), O iii 599  A(log
T= 5.0), O v 629  A(log T=5.4), Ca x 574  A(log T=5.8),
and Mg ix 368  A(log T=6.0).
In order to get good time resolution the rotational
compensation was switched o (sit-and-stare mode) and
so it becomes important to calculate the lowest possible
frequency we can detect from this long time sequence af-
ter taking the solar rotation into account (see Doyle et al.
1998 for details). We estimate that the maximum eect
of the sit and stare mode on the resulting power, for all
datasets would be a spreading of the frequencies by around
1.5 mHz, depending on the size of the source. For all our
sunspots, whose sizes are several arc sec and which have
(well-dened) primary oscillating frequencies much above
3 mHz, the eect of the sit and stare is not considered to
be important.

8 D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations
Fig. 8. cds raster images for the s19331r00 dataset, in dierent temperature lines (as labeled), and the Kitt peak magnetogram.
The over-plotted white rectangles are the locations of the slit for the s19332r00 dataset at the start (right) and the end time
(left). Pixel 67 is marked as a black box on the images. The contours indicate the location of the umbra and penumbra.
Table 3. A log of the datasets used in this paper obtained during April 2000.
Active Date Dataset Type of Pointing Starting time Lines used
region observation (X, Y) UT
AR 8951 14 April 2000 s19331r00 Raster (116,277) 04:25 O iii, O v, He i, Mg ix, Ca x
s19332r00 Temporal (122,277) 04:48 O iii, O v, He i
s19333r00 Raster (135,277) 06:14 O iii, O v, He i, Mg ix, Ca x
s19334r00 Temporal (136,277) 06:37 O iii, O v, He i
s19335r00 Raster (152,276) 08:02 O iii, O v, He i, Mg ix, Ca x
s19336r00 Temporal (153,275) 08:26 O iii, O v, He i
AR 8963 19 April 2000 s19377r00 Raster (-1,418) 18:21 O iii, O v, He i, Mg ix, Ca x
s19378r00 Temporal (4,412) 18:45 O iii, O v, He i
s19379r00 Raster (18,416) 20:10 O iii, O v, He i, Mg ix, Ca x
s19380r00 Temporal (20,415) 20:34 O iii, O v, He i
s19381r00 Raster (33,416) 21:59 O iii, O v, He i, Mg ix, Ca x
s19382r00 Temporal (32,414) 22:23 O iii, O v, He i
AR 8963 20 April 2000 s19387r00 Raster (204,402) 18:00 O iii, O v, He i, Mg ix, Ca x
s19388r00 Temporal (204,402) 18:24 O iii, O v, He i
s19389r00 Raster (217,401) 19:49 O iii, O v, He i, Mg ix, Ca x
s19390r00 Temporal (218,401) 20:13 O iii, O v, He i
s19391r00 Raster (232,401) 21:38 O iii, O v, He i, Mg ix, Ca x
s19392r00 Temporal (233,401) 22:02 O iii, O v, He i

D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations 9
Fig. 9. Frequencies measured as a function of spatial position
along the slit (X-F slice) for the O v 629  A line (left panels) and
the s19332r00 dataset. The right panels show the total number
of counts in a pixel (summed counts) over the observation time.
The tting of the dierent CDS lines was done using
a single Gaussian as the lines were found to be generally
symmetric. Details on the cds reduction procedure, plus
the wavelet analysis, may be found in O'Shea et al. (2001).
Before applying the wavelet analysis we rst removed the
trend of the data (i.e. the very lowest frequency oscilla-
tions) using a 30 point running average. By dividing the
results of this running average (or trend) into the original
data and subtracting a value of one we obtained the re-
sulting detrended data used in the analysis. Fludra (2001)
have shown that this method is very eôcient in removing
the low frequency background oscillation. The statistical
signicance of the observed oscillations was estimated us-
ing a Monte Carlo or randomization method. The advan-
tage of using a randomization test is that it is distribution
free or nonparametric, i.e. it is not limited or constrained
by any specic noise models, such as Poisson, Gaussian
etc. We follow the method of Fisher randomization as out-
lined in Nemec & Nemec (1985), performing 250 random
permutations to calculate the probability levels. The lev-
els displayed here are the values of (1 p)100, where p is
the proportion of the permutations that show a null test
result (see O'Shea et al. 2001). We choose a value of 95%
as the lowest acceptable probability level. Occasionally the
estimated p value can have a value of zero, i.e. there being
an almost zero chance that the observed time series oscil-
lations could have occurred by chance. In this case, and
following Nemec & Nemec (1985), the 95% condence in-
terval can be obtained using the binomial distribution,
and is given by 0.0 < p < 0.01, that is, the probability
((1 p)  100) in this case is between 99{100%. The ve-
locity values presented in this paper are relative velocities,
that is, they are calculated relative to an averaged prole,
obtained by summing over all pixels along the slit and all
time frames.
7. Observational Results
We rst present results from the sunspot in active region
8951 as observed on 14th April 2000. In Figure 7, using
an MDI intensity-gram image, we show an enlarged region
around the sunspot together with an overlay of a por-
tion of the slit from the temporal series dataset s19932r00
showing its location at the beginning and end times of
the observation. The mdi intensity-gram used was ob-
tained from the le fd Ic 01h.63844.0048.ts. The MDI
intensity-gram observations were performed at 04:47:33,
the same time approximately as the start time of the
cds temporal series observations of dataset s19332r00. In
Figure 8 we show cds intensity rasters of size 60  240
arcsec 2 for dierent temperature lines along with a (low
resolution) Kitt peak magnetogram. The cds rasters were
obtained from dataset s19331r00, with a starting time of
04:25 on the 14 April 2000. The low resolution Kitt Peak
magnetogram (resolution4 arcsec) with a start-time of
14:26:31 on the same day was thus obtained about 10
hours after the CDS observations. The thin rectangles
over-plotted on these images show the location of the slit
(for the s19332r00 dataset) at the beginning (the right
one) and end of the temporal sequence (the left one). Pixel
67 is marked as a black box in all the images. The CDS
rasters shown are the square-root images (i.e. the square
root of the intensities has been taken to reduce the con-
trast between the most bright and dark values). A com-
parison of the raster images and the magnetogram clearly
reveals that all the intensity enhancements are closely re-
lated with concentrated magnetic eld regions.
The contours for the umbra and penumbra (in Figs. 7
& 8) are plotted using the average value for the whole
mdi intensity-gram as a guide. The penumbra is dened
as the parts of the mdi intensity where the intensity falls
below a factor of 1.5 that of the average, i.e. it is the
average/1.5. The outer contour around the sunspot shows
the contour of the average value, while the inner contour
shows the contour of the average/1.5 values. The umbra
is then dened as anything that is contained within this
average/1.5 contour.
In order to show the spatial variation of the observed
oscillation frequencies across the sunspot region we se-
lect the strongest line, i.e. O v 629  A. In Figure 9 we plot
the variation of the frequencies over a section of the ob-
serving slit. This section includes the umbra, penumbra
and the adjacent regions. The top panel shows a contrast
enhanced intensity map (X-T slice), obtained by remov-
ing the low frequency trend of the oscillations for each
of the positions along the slit. The lower two left panels
show the measured frequencies as a function of position
along the slit (X-F slice) for velocity and intensity respec-

10 D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations
Fig. 10. Wavelet results corresponding to the He i 584  A line in the s19332r00 dataset at pixel 67. Panels (a) and (b) represent
intensity and velocity results respectively. The middle row left panels show the time frequency phase plot corresponding to the
variations shown in the top panels. The middle row right hand panels show the average of the wavelet power spectrum over
time, i.e. the global wavelet spectrum. The continuous dashed horizontal lines in the wavelet spectra indicate the lower cut o
frequency. The lowest panels show the variation of the probability with time from the randomization test, with the dot-dash
line indicating the 95% signicance level.
tively. The crosses correspond to the primary maxima in
the global wavelet spectra. The total number of counts in
a pixel (summed counts) during the observation is shown
in the right columns and is useful in identifying the loca-
tion of the umbra of the sunspot. Note that pixel 67 is the
brightest pixel across our slit (see right panel of Figure 9).
This pixel may correspond to a plume region. According
to Maltby et al. (1999), locations that show I > 5 
I , where

I is the average intensity in the sunspot area being in-
vestigated, can be considered as plumes in the sunspot
umbra, if they also coincide with the location of an um-
bral region seen in, for example, a MDI intensity-gram. In
the X-T slice, for portions of the image, roughly from 50-
60 and then between 80-90 pixels along the slit there are
brightenings and darkenings, representing longer period
oscillations, which correspond to the penumbra and adja-
cent regions. The frequencies in these regions are typically
in the 2-4 mHz range, whereas for the central part of the
slit, roughly from pixels 60-75, there are many alternate
dark and bright ridges, corresponding to the umbral oscil-
lations, with frequencies of oscillation in the range 5.5-7
mHz, with most peaks at 6.2 mHz. Another point to note
in the X-T image is that there is a drift in the ridges,
a slanting from top to bottom, which is due to the solar
rotational drift (i.e. a sit and stare eect), that is, the os-
cillating umbra source moves down along the slit as the
Sun rotates under the slit. As mentioned before, the peak
counts occur at pixel 67. Below we investigate this pixel
in all three spectral lines, He i, O iii and Ov.
In Figure 10 we show as a representative umbral os-
cillation, the power spectra analysis corresponding to the
He i 584  A line at pixel location 67 (marked as a box
in Figure 8). In the wavelet spectrum, the dark contour
regions show the locations of the highest powers. Only
locations that have a probability greater than 95% are re-
garded as being real, i.e. not due to noise. Cross-hatched
regions, on either side of the wavelet spectrum, indicate
the `cone of in uence' (COI), where edge eects become
important (see Torrence & Compo, 1998). The dashed hor-
izontal lines in the wavelet spectra indicate the lower fre-
quency cut-o, in this instance 1.5 mHz. The results from
the phase plots show that the He i 584  A intensity and
velocity both show signicant power in the 6.0-7.0 mHz
range, for the periods between the 20-30th and 40-65th
minutes of the observing sequence. From the overlay of
the mdi intensity-gram and the slit location (Figure 7)

D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations 11
Fig. 11. Wavelet results corresponding to the O v 629  A line in the s19332r00 dataset at pixel 67. Representations are same as
Fig. 10.
one can clearly see that this particular pixel was over the
sunspot umbra between the time interval 20-65th minute
of the observing sequence (for a total of 45 minutes). We
should point out that the wavelet analysis has been car-
ried out on relative intensity and velocity values and hence
there is a lack of low frequency power in the wavelet spec-
trum plots. The global wavelet spectra (on the right of
Figures 10(a) & (b), which are the average of the wavelet
power spectrum over the entire observing period, show the
strongest intensity and velocity power at 6.2 mHz (161
seconds). This is printed out in Figure 10 above the global
wavelet plots, together with the probability estimate for
the global wavelet power spectrum. In the lowest panels
we show the variation of the probability level as estimated
from the randomization test. Note that the statistical sig-
nicance is calculated only for the maximum powers in the
wavelet spectrum marked by the dotted white line in the
dark patches. From these panels we can clearly see that
the oscillations were signicant in the period between the
20-30 and 40-65 minutes of the time sequence.
The O iii 599  A line formed in the low-to-mid transi-
tion region, is rather faint and to increase the signal to
noise ratio we binned over two pixels (67-68). The inten-
sity wavelet shows power around 5.6 mHz and also some
strong power around 3 mHz for the rst 20 minutes. The
slit was not positioned over the sunspot umbra for the rst
20 minute of the observing sequence and thus the rst 20
minute power corresponds to the umbra boundary. In the
global wavelet the main power peak is at 5.6 mHz, but
there is also a strong peak around 3 mHz, which corre-
sponds to the rst 20 minutes during which the penumbra
rotates under the slit. The most signicant oscillations
take place during the time intervals between 20-25 and
45-50 minutes. For the O iii 599  A line the velocity signal
is too weak and hence the oscillations for this component
are not reliable and so are not included in this analysis.
Now we turn our attention to the transition region
O v 629  A line. Figure 11 shows the wavelet results for the
same pixel location, 67, and dataset, s19332r00. Intensity
and velocity both shows strong power around 6.7 mHz in
the phase and global wavelet spectra. The Ov oscillation
is strong for the same time interval (as in He i and O iii),
namely between the 20{65 minutes, with a drop in sig-
nicance for the time interval between 30-40 minutes of
the sequence. From this one pixel located in the sunspot
umbra we thus nd that the average frequency of oscil-
lation over the entire observing time for the O v line is
at 6.7 mHz (165 s), as estimated from the global wavelet
spectrum.
Using the same techniques as before, we also exam-
ine the same sunspot 30 minutes later using dataset
s19334r00. The results from this temporal series dataset
are summarized in Table 4. For the central portion of the
umbra we nd that the global peaks show frequencies in

12 D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations
Fig. 12. Wavelet results corresponding to the O v 629  A line in the s19336r00 dataset at pixel 67. Representations are the same
as Fig. 10.
the range 5.2-5.7 mHz for all the three lines observed and
the lifetimes of the oscillations are between 10-20 minutes,
very similar to the previous case of s19332r00. We concen-
trate on the same single pixel as before, namely pixel 67,
in the umbra of the sunspot (plume). From Figure 12 it is
clear that the main oscillations in intensity and velocity,
take place in a 5-60 minute interval in the observing se-
quence, with most signicant oscillations occurring in the
5-15 and 25-65 minute intervals for intensity and the 20-55
minute interval for the velocity (this can be conrmed by
looking at the variation of probability in the lowest pan-
els). The global peak for the intensity and velocity both
appear at 6.2 mHz with a very high probability level. The
results from the other lines from this dataset are given in
Table 4. In Figure 13, we show the overall spatial varia-
tions of the oscillations in O v. In the X-T slice we can see
faint light and dark ridges in the interval between pixels
60-75, where the central part, roughly between pixels 66-
68 corresponds to the plume (where I > 5 
I, see rightmost
panels). Once again a slow downward drift of the bright
ridges may be seen in the X-T slice, which is a rotational
eect due to the movement of the oscillating source down
relative to the slit with time. The frequency distribution
shows that the umbra oscillates in intensity and velocity
in the 5-6.5 mHz range, with most oscillations occurring
at 6.2 mHz.
Now we turn our attention to the study of the other
active region, AR8963 over the period 19-20 April 2000
(see Table 3 for details). To save space, we do not show
here detailed wavelet plots for individual dataset, rather
we choose some selective pixel locations in each dataset
(corresponding to the umbra of a sunspot) and summarize
our results in the form of Table 4. We also list the dura-
tion of the oscillations, estimated as the periods of time
dierent oscillation packets showed signicant oscillations
above the 95% signicance level. This can be easily mea-
sured from a comparison of the wavelet phase plots and
the variation of the probability level in the wavelet analy-
sis (e.g in Fig 12). We just point out here some of the other
additional features which we noticed for this active region.
Firstly we should point out that this active region was
much larger compared to the previous one with 12 beta
type spots. In certain cases we found that the oscillation
frequency changed during the observation which might in-
dicate that a new oscillating region was rotating into the
eld of view of the cds slit. For 20 April 2000 dataset,
we also encountered a small aring event which interfered
with the measurement of the underlying higher frequency
oscillations. It was noted that active region AR8963 had
slightly evolved in comparison with the previous day (the
are was a result of that). In all the datasets correspond-
ing to 20th April we nd a wider distribution of frequency
measured from the peak of the global wavelet spectrum,

D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations 13
Fig. 13. Frequencies measured as a function of spatial position
along the slit (X-F slice) for the O v 629  A line (left panels)
and the s19336r00 dataset.
so we have listed the range of frequency over which the
oscillation was most signicant.
8. Discussions
Using UVSP data obtained in emission lines formed at
temperature of 7  10 4 K to 1:3  10 5 K, Gurman et al.
(1982) observed transition region oscillations in sunspots
with frequencies in the range of 5.8-7.8 mHz. Their in-
phase intensity and velocity oscillations lead them to in-
terpret the oscillations in terms of upward propagating
acoustic waves. For the rst time Thomas et al. (1987)
made simultaneous detection of umbral oscillation at dif-
ferent heights, starting from the chromosphere to the tran-
sition region. Their power spectra of intensity and velocity
both show multiple peaks at the 3 min band.
With the launch of SoHO there has been renewed in-
terest in the study of umbral oscillations. Fludra (1999,
2001) investigated 3 min intensity oscillations with cds
by observing the chromospheric line He i and several
transition region lines. He concluded that the 3 min um-
bral oscillations can occur both in the so called sunspot
plumes (bright features seen in the transition region above
sunspot) or in the lower intensity plasma closely adjacent
to the plumes. He found the spectral power to be con-
tained in the 5.55-6.25 mHz range. No oscillations were
detected by him in the Mg ix 368  A line, suggesting that
the 3 min oscillation does not propagate into the corona.
Tziotziou et al. (2002) have presented two-dimensional
intensity and Doppler shift images computed at dierent
wavelengths within the Ca ii 8542  A line. Their power
spectrum analysis shows a 6 mHz frequency, for the stand-
ing umbral oscillations only for the upper half part of the
umbra. For the penumbra they report a 3 mHz frequency.
They also conclude that the umbral oscillations are a local-
ized phenomena. SUMER observations (in both intensity
and velocity) have conrmed that the sunspot oscillations
are prominent in transition region lines above the umbra
(Maltby et al. 2001). They also state that the umbral os-
cillations are a localized phenomenon and that the 3 min
oscillations ll the sunspot umbra in the transition re-
gion and tends to stop at the umbral rim. Support for the
acoustic wave hypothesis was presented by Brynildsen et
al. (1999a,b). They observed oscillations in intensity and
velocity to test the hypothesis and found the oscillations
to be compatible with upwardly propagating waves.
More recently O'Shea et al. (2002) and Brynildsen et
al. (2002) have both presented joint observations of the
3 min umbral oscillations with trace and cds. O'Shea
et al. (2002) nd oscillations at all temperatures from the
temperature minimum, as observed by trace 1700  A up
to the upper corona, as measured by the Fe xvi 335  A
line with cds. Both these authors report that the oscil-
lation amplitude above the umbra increases with increas-
ing temperature, reaches a maximum in the transition re-
gion and decreases for higher temperature lines, though
O'Shea et al. (2002) nds evidence for another increase
in amplitude for lines formed above 1 MK. O'Shea et al.
interpreted their observations in terms of slow magneto-
acoustic waves propagating upwards (as conrmed from
their time delays) along magnetic eld lines. In a recent
theoretical paper, Zhukov (2002) calculated the spectrum
of eigenmodes of umbral oscillations. It was shown that
the 3 min umbral oscillations are the p- modes modied
by the magnetic eld.
The salient feature of our observation is that we have
detected both intensity and velocity oscillations in chro-
mospheric and transition region lines as observed by cds.
We should point out that the velocity resolution of cds
is, at best, 5 km s 1 , and generally it is quite diôcult to
detect velocity oscillations with any condence from noisy
data. But with inclusion of a reliable probability test and
wavelet technique we were able to extract velocity infor-
mation in most of the cases with a 95% condence level
or higher. Most of the earlier work on sunspot oscillations
with cds (Fludra et al. 1999,2001; Brynildsen et al. 2002;
O'Shea et al. 2002) presented only intensity results. Our
results clearly show that the 3 minute intensity and ve-
locity oscillations are a property of the umbra, and not
just the sunspot plume (Figs 9 & 13, shows that power
peak around 6 mHz is present over several pixels in the
umbra). We also detect 3 mHz oscillations corresponding
to the penumbra, which supports the recent observation
by themis, Tziotziou et al. (2002). We should point out
that the He i line is thought to have a complex formation
history and its emission may not correspond to that which
one might expect from a chromospheric line. It is believed
that there are two main mechanisms by which He i can
form (Andretta & Jones 1997); either by collisional exci-
tation in the lower transition region from electrons with

14 D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations
Table 4. Summary of the oscillation frequencies observed corresponding to the dierent dataset.
Dataset Lines Pixel Intensity results Velocity results
Freq. maxima Prob. level Duration Freq. maxima Prob. level Duration
s19332r00 He i 67 6.2 mHz 99-100% 20-30, 40-65 6.2 mHz 99-100% 20-25, 40-65
O iii 67-68 5.6 mHz 99.2% 20-25, 45-50
O v 67 6.7 mHz 99-100% 20-30, 40-65 6.7 mHz 99-100% 20-30, 40-50, 60-65
s19334r00 He i 66 5.7 mHz 99-100% 10-30, 40-50 5.2 mHz 98.8% 15-30, 40-50
O iii 66-67 5.2 mHz 95.6% 15-20, 45-55
O v 66 5.7 mHz 99-100% 15-30, 40-55 5.7 mHz 99-100% 15-30, 40-50
s19336r00 He i 67 6.2 mHz 99-100% 5-10, 20-30, 40-60
O iii 66-67 5.7 mHz 98.4% 5-10, 40-50
O v 67 6.2 mHz 99-100% 5-15, 25-60 6.2 mHz 99-100% 20-55
s19378r00 He i 20 4.8 mHz 99-100% 5-20, 25-45
O iii 21-22 4.8 mHz 99-100% 10-20, 25-50 4.8 mHz 98.8% 10-20, 40-50
O v 21 4.8 mHz 99-100% 10-20, 25-45 4.8 mHz 99-100% 0-20
s19380r00 He i 21 4.8 mHz 99-100% 5-20, 25-30, 45-55
O iii 21-22 5.2 mHz 99-100% 5-20, 45-55
O v 21 4.8 mHz 99-100% 5-20, 45-55 4.0 mHz 99.8% 5-20
s19382r00 He i 21 5.2 mHz 99-100% 45-70
O iii 21-22 5.6 mHz 99-100% 45-60
O v 21 5.2 mHz 99-100% 10-20, 25-35, 45-65 4.0 mHz 99-100% 25-35
s19388r00 He i 33 3.5-5.0 mHz 99-100% 0-35 3.5-5.0 mHz 99-100% 5-35
O iii 32-33 4.0-5.0 mHz 99-100% 0-45 5.7 mHz 95.0% 5-10, 35-45
O v 33 4.0-5.0 mHz 99-100% 5-20, 35-55 5.2 mHz 99-100% 40-55
s19390r00 He i 33 3.5-5.0 mHz 99-100% 5-35 4.5-5.0 mHz 95.0% 0-20
O iii 32-33 3.5-5.0 mHz 95.0% 5-10, 25-30
O v 33 3.0-5.0 mHz 99-100% 10-45
s19392r00 He i 32 3.0-6.0 mHz 99-100% 30-75 4.4 mHz 99.2% 10-20, 40-45
O iii 31-32 4.8 mHz 98.4% 30-40
O v 32 3.0-5.0 mHz 99-100% 0-10, 35-50, 65-70 3.5-5.0 mHz 99-100% 0-20, 40-45
kinetic temperature higher than the local temperature of
the helium atom or by a process in which coronal photons
penetrate into the chromosphere and photoionize helium
atoms which then recombine to form He i. Thus the emis-
sion from He i can re ect conditions at temperatures above
that of its putative formation temperature. Furthermore,
the denition of where the chromosphere ends and tran-
sition region begins is a bit arbitrary. Thus there exists
an uncertainty and controversy over the He i formation
height. Moreover, the O iii and O v lines are formed close
to each other in temperature. This does not allow us to
make a time-delay analysis for the calculation of the wave
propagation speed using this data.
9. Conclusion
There exists several observational reports of umbral os-
cillations in the literature and there have been several
theoretical attempts to explain them. However, no gener-
ally accepted model exists for the understanding of these
mechanical structures, their physical mechanisms and en-
ergy transport to the surroundings. In this paper we pre-
sented new solutions for magneto-atmospheric waves in
an isothermal atmosphere with a vertical magnetic eld in
the presence of radiative heat exchange based on Newton's
law of cooling. Radiation can radically alter the dynamical
properties of wave modes in a uid. This radiative heat ex-
change gives rise to a temporal decay of oscillations with a
characteristic dimensionless decay time ~  D =
1=
I , where

I is the imaginary part of
Depending on the value of
the radiative relaxation time ~  R , the modes are eectively
damped by the radiative dissipation in as short a time
as two oscillation periods; however, in the limits of very
large or very small ~  R , corresponding to nearly adiabatic
or nearly isothermal oscillations, the modes are essentially
undamped. We would also like to point out the merits and
demerits of using Newton's law to model heat exchange.
At suôciently low frequencies, the wavelength of a distur-
bance is so long, that it becomes optically thick (no matter
how transparent the material is), and the Newtonian cool-
ing approximation no longer holds. Conversely, at high
frequencies the wavelength of a disturbance becomes so
small that it is optically thin (no matter how opaque the
material) and the Newtonian approximation holds good.
Bunte & Bogdan (1994) have already pointed out that ra-
diative eects on oscillations in photospheric and higher
layers are clearly important. Radiative dissipation based
upon Newton's cooling law is clearly an oversimplication
of the problem; nevertheless it allows us to assess the ef-
fects of radiative damping on the modal structure. It also
enables us to look at the full frequency spectrum and the
interaction amongst various modes. Our treatment of the

D. Banerjee et al.: On the theory of MAG waves and a comparison with sunspot observations 15
weak eld limit has permitted an analysis of the
K
diagram in terms of asymptotic approximations; this has
allowed us to understand the nature of the modes in a ver-
tical magnetic eld in the presence of radiative exchange.
The insight so gained has proved useful in extending the
computation to the moderate to strong eld case. The
transition region lines as observed by cds on SoHO are ca-
pable of diagnosing, Alfven, slow and fast magnetoacoustic
waves. The Alfvenic oscillations are essentially velocity os-
cillations and do not cause any density uctuations. The
compressional modes may however reveal themselves in
the form of intensity oscillations through a variation in
the emission measure. This fact, together with the oscil-
lations in intensity, allows us to interpret the waves as
slow magneto-acoustic in nature. We have computed the
frequencies of the modes from the full mag equation (see
Eq. 11) and found out that for our model atmosphere they
correspond to the slow magneto-acoustic modes. The p 1
and p 2 mode frequencies fall very well within the observed
range (compare Tables 2 & 4). Our observational results
very much complement earlier results and provide addi-
tional input for the study of the characteristics of the
wave modes. Our observations reveal that umbral oscil-
lations are a localized phenomenon, where intensity and
velocity both shows a clear peak around 6 mHz. In all the
wavelet plots, we also notice a smaller peak in the global
wavelet spectra and some power in the phase plot around
3 mHz, for part of the time sequence, which corresponds
to the penumbra. In the theory part of this paper we have
shown that the life time of the oscillations are dependent
on the relaxation time scale and in some cases these oscil-
lations could be damped within a few oscillations periods
as well. Our observations also indicate that the oscilla-
tions seems to come in packets with life times of  10-20
mins, which matches fairly well with the damping behav-
ior of our mag waves. We should also point out that the
envelope of these packets do not show exponential decay,
as one would expect from the theory, rather the intensity
amplitude usually remain sinusoidal. An alternative ex-
planation for the appearance of the packets could be due
to the rotation of the sun under the slit than the actual
length of the oscillations. We see oscillations for only 20
minutes as that may be the time necessary for a source
of say, 2 arcsec wide in the 2 arcsec wide slit, to rotate
out of the eld of view, if the sun is rotating at say, 6 arc-
sec/hour. In general we nd good agreement between the
model and observations as far as the duration of oscillation
and range of frequency is concerned.
Acknowledgements. DB expresses his gratitude to Profs.
S.S.Hasan and Joergen Christensen-Dalsgaard for many valu-
able discussions which has enabled to develop the theory of
the mag waves. DB wishes to thank the FWO for a fellow-
ship (G.0344.98). EOS is a member of the European PLATON
Network. We would like to thank the cds and eit teams at
Goddard Space Flight Center for their help in obtaining the
present data. cds and eit are part of SoHO, the Solar and
Heliospheric Observatory, which is a mission of international
cooperation between ESA and NASA. Research at Armagh
Observatory is grant-aided by the N. Ireland Dept. of Culture,
Arts and Leisure. This work was supported by PPARC grant
PPA/G/S/1999/00055. The original wavelet software was pro-
vided by C. Torrence and G. Compo, and is available at URL:
http://paos.colorado.edu/research/wavelets/.
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