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ASTRONOMY
AND
ASTROPHYSICS
28.10.1996
The effect of the cusp resonance on absorption of
magnetoacoustic waves in the solar corona
V. M. Ÿ
CadeŸz, '
A. Cs'ik ? , R. Erd'elyi ?? and M. Goossens
Center for Plasma­Astrophysics, K.U. Leuven, Celestijnenlaan 200 B, 3001 Heverlee, Belgium.
Received ; accepted
Abstract. The absorption of MHD waves due to the cusp
(or the slow) resonance is studied numerically and the
obtained results are discussed with respect to a possi­
ble application to conditions that are typical for the solar
corona.
The considered configuration is composed of two ho­
mogeneous regions separated by an inhomogeneous slab
through which all physical parameters of the medium
change continuously according to an initially prescribed
distribution.
The MHD waves can propagate through one of the re­
gions only and are being partially absorbed by the non
uniform slab due to resonant processes occurring in par­
ticular locations within the slab.
Computational results show that the absorption aris­
ing from the cusp resonance strongly depends on the char­
acteristics of the driving wave. It can be close to 100% for
certain angles of incidence which indicates the possibility
for an efficient local coronal heating at this resonance too.
The considered resonant absorption can occur for the slow
as well as for the fast magneto acoustic mode.
Key words: Magnetohydrodynamics (MHD) -- Methods:
numerical -- Sun: corona -- Sun: oscillations
1. Introduction
The solar atmosphere is a medium with a complex struc­
ture supporting a variety of physical processes taking place
on different time scales. One of the everlasting phenom­
ena are the MHD waves generated either by the convective
motions in the solar interior or by local releases of energy
in the atmosphere itself.
Send offprint requests to: M. Goossens
? On leave from Department of Astronomy, E¨otv¨os Univer­
sity Budapest, Ludovika t'er 2 H­1083 Budapest, Hungary.
?? Armagh Observatory, College Hill, Armagh, BT61 9DG, N.
Ireland.
The propagating MHD waves transport the energy
from the sources away into the ambient plasma while the
wave dissipation results into local energy depositions. Par­
ticularly the resonant wave absorption, occurring when
the wave frequency matches some characteristic frequency
of the medium, can be responsible for large fractions of the
wave energy to be irreversibly transformed into heat.
Since the first suggestion by Ionson (1978), a consid­
erable work has been done to verify that resonant absorp­
tion of MHD waves can be an effective mechanism for the
heating of the solar corona.
Most of analytical and numerical studies, like those by
Kuperus, Ionson and Spicer (1981) until recent works by
Goossens and Hollweg (1993) and Erd'elyi and Goossens
(1995), have been treating the fast magneto acoustic
waves, known also as the compressional Alfv'en waves,
and their absorption due to the resonant excitation of the
local Alfv'en continuum in inhomogeneous media. These
waves propagate almost isotropically with relatively high
frequencies lying above the upper cutoff ! II and are gen­
erally considered as the main link between the subphoto­
spheric turbulence and the coronal heating through the
intense wave energy dissipation around the Alfv'en reso­
nance.
The absorption of fast MHD waves has also been ap­
plied recently to calculations related to other structures
like sunspots (Stenuit, Poeds and Goossens, 1993), coro­
nal loops (Poedts, Beli¨en and Goedbloed, 1994) and flux
tubes with flows (Erd'elyi, Goossens and Ruderman, 1995,
Erd'elyi and Goossens, 1996).
The other magneto acoustic waves, belonging to the
slow MHD mode, are not considered as an important mean
of energy transfer through the corona mostly due to a high
unisotropy of their propagation and due to a limited range
of their spectrum which is located between the cusp (! c )
and the lower cutoff frequency (! I ).
However, these waves can play a role in the case when
they are generated locally in the corona by some eruptive
phenomena or when their propagation is channeled along
magnetic arcades in the corona. In both cases, the slow

2 V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ...
MHD waves can be resonantly absorbed in an nonuniform
medium when they reach the cusp resonance and excite
the local cusp continuum (Cs'ik, Erd'elyi and Ÿ
CadeŸz, 1996).
In addition to magneto acoustic bulk modes, there
are the guided surface modes that propagate along the
nonuniform coronal arcades and that can be resonantly
absorbed too ( Ÿ
CadeŸz and Ballester, 1996). Furthermore,
the resonant absorption can also occur at some configu­
rations when an external bulk wave first excites surface
modes by a tunneling effect which are then resonantly
converted into heat (OkretiŸc and Ÿ
CadeŸz, 1991).
In this paper, we investigate analytically and numeri­
cally the effects of the cusp resonance on the absorption
of both magnetoacoustic bulk modes in a strongly inho­
mogeneous coronal slab that separates two different, oth­
erwise uniform, regions. The waves are assumed to prop­
agate in only one of them only, while the other region
remains opaque. Thus, the incident wave can be either to­
tally reflected from the layer or partially absorbed if the
resonant conditions within the layer are achieved.
We calculate the absorption coefficient for both mag­
netoacoustic modes under conditions when the cusp reso­
nance occurs in the considered configuration. It is shown
that both of them can be absorbed at the cusp resonance
while the fast mode can sometimes suffer an additional
absorption at the Alfv'en resonance too. Since we are pri­
marily interested in effects of the cusp resonance, the pos­
sibility when the fast mode encounters only the Alfv'en
resonance is not investigated.
The absorption coefficient is obtained from numerical
solutions of ideal MHD equations with a special treatment
of singularities according to the method derived by Saku­
rai, Goossens, Hollweg and Ruderman (in what follows,
the SGHR method) during the last couple of years and
described in details in the review article by Goossens and
Ruderman (1995).
The Chapter 2 contains the description of the consid­
ered model of a stationary ideal fluid, the linear equations
that will be used for calculations of the absorption coeffi­
cient are given in Chapter 3, the SGHR method of solving
the equations in the vicinity of the resonances is described
in Chapter 4, the calculation of the absorption coefficient
is performed in Chapter 5 while the obtained numerical
results are discussed in Chapter 6.
2. The initial fluid configuration
We consider a stationary basic state composed of two
uniform regions separated by a nonuniform layer. In the
Cartesian coordinate system, all the basic state variables
are z dependent inside the layer where 1 – z – 0.
Gravitational effects are not taken into account, the
magnetic field is considered homogeneous throughout the
whole space and is oriented along the x\Gammaaxis: B 0 =
(B 0 ; 0; 0) with B 0 = const:
The initial magnetohydrostatic equilibrium of the
medium and the constancy of the magnetic pressure pm j
B 2
0 =2¯ 0 further indicate the constancy of the thermal pres­
sure p 0 and also of the plasma parameter fi = p 0 =pm . Thus
p 0 = 1
fl
ae 0 (z)v 2
s (z) = const: (1)
meaning that we can freely specify either the sound speed
(i.e. the temperature T 0 ) or the density profile too.
In addition to this, B 0 = const: also implies that
ae 0 (z)v 2
A (z) = const: and ae 0 (z)v 2
c (z) = const:
Fig. 1. The characteristic velocity profiles for the considered
equilibrium state with fi = :6 and n = 0:5.
To investigate the characteristics of the wave absorp­
tion at the slow resonance, it is more convenient to pre­
scribe the profile for the square of the cusp speed v c
v 2
c = v 2
A v 2
s
v 2
A + v 2
s
as a function that being monotonously varying inside the
slab L ? z ? 0 and uniform elsewhere:
v 2
c (z) =
8
!
:
v 2
2 = const:; z – L
v 2
1 \Gamma
\Gamma
v 2
1 \Gamma v 2
2
\Delta \Gamma z
L
\Delta n
; L – z – 0
v 2
1 = const: z Ÿ 0
(2)
Finally, we can express the profiles of the important
basic state quantities as follows:
ae 0 = ae 00
v 2
1
v 2
c (z) ;
v 2
A =
i
1 + 2
flfi
j
v 2
c (z)
v 2
s =
i
1 + flfi
2
j
v 2
c (z);
(3)

V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ... 3
where the profile v 2
c (z) is initially prescribed by (2), as
shown in Fig.1
The described system is then used to model an inho­
mogeneous region in the solar corona in order to study
the effect of the cusp resonance on the resonant absorp­
tion of magneto acoustic waves. The waves can propagate
through the homogeneous region z ? L with the inci­
dent waves coming towards the layer from z AE L while
the waves reflected from the layer and the nontransparent
domain z ! 0 propagate in the positive direction of the
z \Gammaaxis.
The incoming wave parameters will be taken in such
a way that the cusp resonance can be encountered within
the layer either alone or together with the Alfv'en reso­
nance.
3. Perturbation equations and solutions
The governing equations describing the propagation and
the absorption of linear waves through the described un­
perturbed medium is the standard set of ideal MHD equa­
tions
@ae 1
@t + r \Delta (ae 0 v 1 ) = 0;
ae 0
@v1
@t = \Gammarp 1 + 1
¯0 (r \Theta B 1 ) \Theta B 0
+ 1
¯0 (r \Theta B 0 ) \Theta B 1 ;
@B1
@t = r \Theta (v 1 \Theta B 0 )
@p1
@t + v 1 \Delta rp 0 = v 2
s ( @ae 1
@t + v 1 \Delta rae 0 )
(4)
where all perturbed quantities f 1 are assumed harmonic in
time t and in both coordinates, x and y. Their amplitudes
remain z \Gammadependent:
f 1 (x; y; z; t) =
f(\Omega ; z)e i(kx x+kyy\Gamma!t)
with\Omega j f!; k x ; k y g.
The equations (4) further reduce to the following two
coupled equations for the perturbations of the normal
component of the Lagrangian displacement ¸ z j i v z =!
and of the total pressure perturbation P :
D d¸ z
dz
= \GammaC 1 P; dP
dz
= C 2 ¸ z (5)
where
D = ae 0 (v 2
s + v 2
A )(! 2 \Gamma ! 2
c )(! 2 \Gamma ! 2
A );
C 1 = (! 2 \Gamma ! 2
A )(! 2 \Gamma ! 2
s ) \Gamma ! 2 v 2
A k 2
y ;
C 2 = ae 0 (! 2 \Gamma ! 2
A )
(6)
and
!A = vAk x ; ! s = v s
q
k 2
x + k 2
y ; ! c = v c k x :
The cusp and the Alfv'en resonance occur at points
z = z c and z = z A respectively, where the coefficient D
vanishes
! = ! c (z c ) or ! = !A (z A ); (7)
which makes the obtained Eqs (5) singular.
As already said, our basic state has two uniform parts:
the region 1 (z Ÿ 0) and the region 2 (z – L) that are
separated by a nonuniform transitional layer (1 – z – 0)
as described by the profiles (2).
The region 1 is taken opaque to the considered waves
meaning that the Eqs (5) have a simple evanescent solu­
tions for P and ¸ z in this domain which can be written
as
P = e Ÿz ; ¸ z = Ÿ
C 2
e Ÿz (8)
with Ÿ 2 = \GammaC 1 C 2 =D ? 0 and all the coefficients evalu­
ated at z = 0. The integration constants are chosen as to
provide vanishing solutions at z ! \Gamma1 and a unit total
pressure perturbation at z = 0.
The solutions in the transitional layer are obtained
numerically starting from initial values for P and ¸ z at
z = 0. These are given by the solutions (8) evaluated at
the boundary of the region 1. as both, P and ¸ z , should
be continuous at z = 0.
The numerical procedure was the Runge Kutta method
while the singularities were treated according to the
SGHR method from Section 4. Once the solutions were
obtained at z = L, say PL and ¸ zL , they were matched
to the analytical solutions from region 2 by the boundary
conditions requiring their continuity at z = L.
The uniform region 2 is transparent and the solution
of (5) represents a superposition of two waves propagating
in the opposite directions along the z \Gammaaxis
P = P (+) e ikz (z\GammaL) + P (\Gamma) e \Gammaik z (z\GammaL)
¸ z = ikz
C2 P (+) e ikz (z\GammaL) \Gamma ikz
C2 P (\Gamma) e \Gammaik z (z\GammaL)
(9)
where all coefficients are constant and evaluated at z = L
and
k 2
z j C 1 C 2
D
= (! 2 \Gamma ! 2
I )(! 2 \Gamma ! 2
II )
(v 2
A + v 2
s )(! 2 \Gamma ! 2
c ) (10)
Here, ! I and ! II are the lower and the upper cutoff fre­
quencies for which k z in (10) vanishes.
The expression (10) is the dispersion equation for wave
modes in the region 2 and it yields the frequency ranges
for propagating waves: ! I – ! – ! c for the slow magneto
acoustic mode and ! – ! II for the fast magneto acoustic
mode.
Also, an important relation between the z \Gamma compo­
nents of the group and the phase velocity, V gz j @!=@k z
and V pz j !=k z respectively, follows from (10):
V gz V pz
= (v 2
A +v 2
s )(! 2 \Gamma! 2
c ) 2
(2! 2 \Gamma! 2
I \Gamma! 2
II )(! 2 \Gamma! 2
c )\Gamma(! 2 \Gamma! 2
I )(! 2 \Gamma! 2
II )
(11)

4 V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ...
A simple analysis of this expression concerning its sign,
shows that
V gz V pz ? 0 for the fast mode,
V gz V pz ! 0 for the slow mode,
(12)
indicating the sense of the wave energy transports for the
two magneto acoustic modes.
Finally, knowing the calculated values PL and ¸ zL we
can relate them to the solutions (9) at z = L which
yields the following expressions for the amplitudes P (+)
and P (\Gamma) :
P (+) = kzPL \GammaiC 2 ¸L
2kz
P (\Gamma) = kz PL+iC2¸L
2kz
(13)
which are related to the waves with V pz ? 0 and V pz ! 0
respectively.
4. Solutions close to singularities
The frequency matching (7) indicates the existence of a
resonant wave transformation due to the excitation of the
local slow and the Alfv'en continuum respectively. It also
requires a special treatment of solving the singular equa­
tions (5) as done in derivations by Sakurai, Goossens and
Hollweg (1991a,b) and Goossens and Ruderman (1995),
referred to as the SGHR method.
This means that we use the ideal MHD equations ev­
erywhere except within a comparatively narrow dissipa­
tive layer around the resonances where the solutions for P
and ¸ z tend to diverge and the dissipative terms cannot be
neglected and should be included in calculations. As the
consequence, the singularities are shifted from the real to
a complex ! with a small but finite, positive, imaginary
part due to the inevitable dissipations in natural fluids.
Thus, the divergency is removed from the real axis and
the integration can be performed continuously through the
dissipative layer. This, however, requires the additional as­
sumptions on the type of the dominant dissipative process
occurring at the vicinity of the resonances and a proper
modification of Eqs.(5). The thickness of the layer, defined
by some ffi c;A , will obviously depend on the importance of
dissipations: if they are less pronounced the layer is thin­
ner and vice versa.
In this paper, we are not particularly interested about
the details regarding the solutions within the dissipative
layer and all we need is to know how to cross the layer
with the singularity during numerical calculations of ¸ z
and P . For this purpose, the proper connection formulae
relating the solutions for both, P and ¸ z , at the end points
of the layer will be used (Goossens and Ruderman, 1995).
Given below are some details on how the jump con­
ditions for both resonances were obtained by the SGHR
method.
The cusp resonance. The ideal equations (6) with coeffi­
cients linearly Taylor expanded around the cusp resonant
point z c , take the following form
s d¸z
ds = ! 4
c
ae 0 \Delta c v 2
A ! 2
A
P;
s dP
ds = 0
(14)
where the new variable s = s r +iffl is complex with an imag­
inary part coming from dissipations and which is negligible
everywhere but in the neighborhood of z c while s r j z \Gammaz c ,
\Delta c = d(! 2 \Gamma ! 2
c )=ds and all coefficients in (14) have their
values taken at s = 0.
A simple treatment of Eqs.(14) immediately yields a
conservation law of constancy of the total pressure per­
turbation within the dissipative layer around z = z c :
P = const: (15)
The obtained Eqs. (14) can now be easily integrated
across the layer along s r by applying the Cauchy prin­
cipal value integral which finally gives the jumps or the
amplitude differences at the end points s r ú s = \Sigmaffi c of
the layer for both the Lagrangian displacement ¸ z and the
total pressure perturbation P :
[¸ z ] c = \Gammaiú ! 4
c
j\Delta c jae 0 v 2
A ! 2
A
P;
[P ] c = 0
(16)
where all quantities in (16) are taken at z = z c .
The considered dissipations, included in the integra­
tion of Eqs.(14) through a small imaginary part ffl of the
complex variable s, do not enter the final result (16) for
the jump conditions explicitly meaning that they remain
unaffected by the type of dissipations.
For this reason we leave the parameter ffl unspecified
and dissipations will be included in calculations implicitly,
through the prescribed value of the thickness parameter
ffi c .
Finally, the thickness of the dissipative layer over
which the jump conditions (16) are applied, is estimated
from the nonideal MHD equations applied to that layer
(Goossens and Ruderman, 1995b). In particular, when the
electric resistivity j is considered as the dominant dissi­
pative process in the fluid, one can take:
ffi c ú 5
`
!j
j\Delta c j
' 1=3
(17)
The Alfv'en resonance. An analogous procedure can now
be applied to the Alfv'en resonance too, which yields the
approximative equations for ¸ z and P close to z = z A in
the following form
s
d¸ z
ds
= k 2
y
ae 0 \Delta A
P; s
dP
ds
= 0 (18)

V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ... 5
where s = z \Gamma z A + iffl, \Delta A = d(! 2 \Gamma ! 2
A )=ds and all
quantities have their values taken at z = z A .
The second of the equations (17) indicates the con­
stancy of the total pressure perturbation across the do­
main of the Alfv'en resonance and the first equation in
(17) can easily be integrated keeping in mind that s has
a small imaginary part coming from dissipations. Finally,
the jump conditions for the Alfv'en resonance are obtained
as:
[¸ z ] A = \Gammaiú k 2
y
j\Delta A jae 0
P; [P ] A = 0: (19)
The jumps (19) are applied between s r ú s = \Sigmaffi A
where ffi A is estimated (Goossens and Ruderman, 1995b)
as
ffi A ú 5
`
!j
j\Delta A j
' 1=3
(20)
The same as in the case of the cusp resonance, the
dissipative term ffl does not show up explicitly in the Alfv'en
resonance and the jump conditions remain independent on
the type of dissipations.
5. The absorption coefficient
To study the effects of the cusp resonance on the absorp­
tion of magneto acoustic modes we derive an expression
for the absorption coefficient A from the incident and the
reflected wave amplitudes at z = L.
By the definition
A j 1 \Gamma jF (r)
z j
jF (i)
z j
(21)
with F (i)
z and F (r)
z being the z \Gamma components of the en­
ergy flux densities of the incident and the reflected wave
respectively.
They can be further expressed in terms of the z \Gamma com­
ponents of the wave group velocity V (i;r)
gz and the total
wave energy densities w (i;r) as
F (i;r)
z = V (i;r)
gz w (i;r) (22)
The terms 'the incident wave' and 'the reflected wave'
are therefore determined by the sign of the z \Gamma component
of the group velocity in sense that incident waves have
V gz j V (i)
gz ! 0 while the reflected waves are those with
V gz j V (r)
gz ? 0. For the considered MHD waves these two
components are equal by their absolute values as they are
related to the same domain in the fluid. Consequently,
they will cancel out in the expressions (21) and (22) for
calculations of the absorption coefficient.
Taking now into account the inequalities (12) we see
that amplitudes (13) P (+) and P (\Gamma) , corresponding to the
positive and the negative z \Gamma component of the phase ve­
locity respectively, will also be related in the same way to
the z \Gamma component of the group velocity in the case of the
fast magneto acoustic mode. This means that P (+) and
P (\Gamma) are the amplitudes of the reflected and the incident
respectively when the considered wave is the fast MHD
mode.
In the case of the slow MHD modes, when V gz V pz ! 0
according to (12), the situation is reversed. The ampli­
tudes P (+) and P (\Gamma) are now related to the incident and
the reflected wave respectively.
Therefore, we can write the total pressure amplitudes
P (i) and P (r) for the incident and the reflected wave as
P (i) = P (\Gamma) ; P (r) = P (+) for the fast mode
P (i) = P (+) ; P (r) = P (\Gamma) for the slow mode (23)
Knowing now the total pressure amplitudes for the in­
cident and the reflected wave for both magneto acous­
tic modes, we can calculate the related energy densities
needed in (22) and the absorption coefficient (21).
The total wave energy density
w (i;r) = w (i;r)
k + w (i;r)
t + w (i;r)
m (24)
is a sum of related kinetic w (i;r)
k , thermal w (i;r)
t and mag­
netic energy w (i;r)
m densities given as
w k = 1
2 ae 0 jvj 2 ; with jvj 2 = jv x j 2 + jv y j 2 + jv z j 2 ;
w t = v 2
s
2ae 0
jaej 2 ;
wm = 1
2 ae 0 v 2
A jbj 2 ; with jbj 2 = jb x j 2 + jb y j 2 + jb z j 2 :
(25)
where b j B=B 0 and the superscripts (r; i) indicating the
reflected and the incident wave respectively, are omitted.
The perturbed quantities in (25) can all be expressed
in terms of the total pressure perturbation by means of
the initial set of Eqs(4) in the following way
v x = 1
ae 0
kx
!
h
1 \Gamma v 2
A (k 2
y +k 2
z )
! 2 \Gamma! 2
A
i
P j a
1(\Omega\Gamma P
v y = 1
ae 0
ky !
! 2 \Gamma! 2
A
P j a
2(\Omega\Gamma P
v z = 1
ae 0
kz !
! 2 \Gamma! 2
A
P j a
3(\Omega\Gamma P
ae = 1
v 2
s
h
1 \Gamma v 2
A (k 2
y +k 2
z )
! 2 \Gamma! 2
A
i
P j a
4(\Omega\Gamma P
b x = 1
ae 0
k 2
y +k 2
z
! 2 \Gamma! 2
A
P j a
5(\Omega\Gamma P
b y = \Gamma 1
ae 0
kxky
! 2 \Gamma! 2
A
P j a
6(\Omega\Gamma P
b z = \Gamma 1
ae 0
kxkz
! 2 \Gamma! 2
A
P j a
7(\Omega\Gamma P
(26)
where\Omega j (!; k x ; k y ; k z ).

6 V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ...
Substituting expressions (26) into (25) we see that the
total energy densities (24) can be written in this form:
w (i;r) =
jA(\Omega\Gamma j 2 jP (i;r) j 2 (27)
with
A(\Omega\Gamma being a lengthy linear combination of coeffi­
cients an (n = 1; :::; 7) from (26) and the square of its
absolute value remains the same for the incident and the
reflected wave in the considered case.
Consequently, the absorption coefficient A in (21), be­
comes simply
A = 1 \Gamma jP (r) j 2
jP (i) j 2
(28)
with P (r) and P (i) given by relations (23) for the two
considered magneto acoustic modes.
6. Results and conclusions
The numerical example we considered was based on the
basic state profile given by (2) with the numerical values
for the parameters calculated from the initially prescribed
fi = 0:6 and the ratio of the Alfv'en speeds v A (L)=vA (0) =
0:2, according to Eqs.(2) and (3). We also take the expo­
nent n = 0:5 in (2). All lengths are scaled to L, velocities
to vA (0) and the scaling time is then Ü j L=vA (0).
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w
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w
Transition layer
w
w
c
I
II
A
Region 1 Region 2
z=0 z=L
fast
slow
Fig. 2. A schematical view of the considered wave propagation
domains for the slow and the fast mode.
Following now the described procedure, the absorption
coefficient A can be calculated for given wave frequency
! and the wave vector k = (k x ; k y ; k z ) provided they also
satisfy the dispersion equation (10) for propagating waves
in the region 2 and the condition Ÿ 2 ? 0 for evanescent
waves in the region 1. We also impose an upper bound
to the frequency range which excludes the possibility of
having the Alfv'en resonance alone. This means that the
considered frequencies do not exceed the value of the cusp
frequency at z = O: ! Ÿ ! c (0) j v 1 k x as shown schemat­
ically in Fig.2.
Fig. 3. a) The dependence of the absorption coefficient on
the incident angle ` and the azimuth angle OE for the slow
MHD mode. b) The top view of the surface A(`; OE) showing
the boundary of the computational domain. The dimensionless
wave frequency is ! = 0:8.
Fig. 4. The location of the cusp resonance depending on angles
` and OE for the slow MHD mode.
For practical reasons of making the geometry of the
wave propagation more clear we introduced two angles as

V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ... 7
Fig. 5. The slow MHD mode with the frequency ! = 0:8. a)
The dependence of the absorption coefficient A on the incident
angle ` at three indicated azimuthal angles OE. b) The depen­
dence of the absorption coefficient A on the azimuthal angle OE
at three indicated incident angles `.
parameters instead of the wave vector components in the
following way:
k x = k sin(`) cos(OE); k y = k sin(`) sin(OE);
k z = k cos(`); where k 2 = k 2
x + k 2
y + k 2
z
(29)
Clearly, ` is the angle of the wave incidence related to
the direction of the inhomogeneity gradient while OE is an
azimuth angle related to the magnetic field orientation.
Substituting expressions (29) into (10), the dispersion
equation (10) can be written as
k = K s (!; `; OE); k = K f (!; `; OE) (30)
for the slow and the fast MHD mode respectively.
Thus we consider the absorption properties due to the
cusp resonance for a wave with a given frequency ! that
Fig. 6. The dependence of the absorption coefficient A on the
wave frequency ! for the slow MHD mode for: a) OE = 0 ffi
and
b) ` = 50 ffi and other parameters as indicated.
propagates along the prescribed direction determined by `
and OE. These three parameters are now sufficient to deter­
mine the absorption coefficient for both magneto acoustic
modes depending on the solution for k used in (30).
The angle of incidence ` varies from 0 ffi to 90 ffi while
the azimuth angle OE ranges between 0 ffi and 180 ffi . In our
model, however, the wave characteristics are symmetric
with respect to OE = 90 ffi because the change in sign of
either k x or k y in (29) causes no changes in the dispersion
equation (10) as they enter it as squares. Thus, 90 ffi – OE –
0 ffi in our calculations.
We first study the slow mode and the obtained ab­
sorption coefficient is shown in Fig.3a as a function of `
and OE for a typical dimensionless wave frequency ! = 0:8.
The absorption rate reaches values close to 60% at OE = 0 ffi
and ` around 50 ffi . For other values of angles it smoothly
decreases to well below 10 ffi at the edge where the surface
A(`; OE) is cut and ends abruptly. The profile of the cut

8 V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ...
Fig. 7. The fast MHD mode with OE = 0 ffi . a) The depen­
dence of the absorption coefficient A on the incident angle `
at three indicated wave frequencies !.) b) The dependence of
the absorption coefficient A on the wave frequency ! at three
indicated angles of incidence `.
is nicely seen in Fig.3b showing the vertical view at the
surface from Fig.3a.
The reason why such a cut appears can be deduced
from Fig.4 where a 3D plot shows the location of the cusp
resonance z c (`; OE) for the same wave with ! = 0:8. The
domain of angles ` and OE where z c ! 1, clearly coincides
with the domain in which the absorption coefficient could
not be calculated.
At these angles, namely, the incoming slow magneto
acoustic mode propagates through the region 2 with a
frequency that is practically equal to the cusp resonant
frequency which is the very limit of the propagation do­
main for this mode. From the physical point of view, then
it becomes meaningless to talk about a propagating wave
and the computational difficulties begin to occur.
Some cross sections showing the absorption variation
with one of the angles taking the other angle constant
are given in Fig.5. The curves end at angles when the
cusp frequency in the region 2 gets close to the slow mode
frequency (or when z c ! 1) as noted above.
How the absorption of the slow mode depends on the
wave frequency is presented in Fig.6: a) for OE = 0 ffi and
several values of ` and b) for ` = 50 ffi and taking OE as a
parameter. These two fixed values where chosen as they
yield the maximalvalue for A in Fig.3. The plots now show
that the absorption rate increases with the frequency to
high values close to 90%.
Next we investigated the effects of the cusp resonance
on the fast magneto acoustic mode incident from the re­
gion 2. Since these waves have higher frequencies that are
above the upper cutoff frequency ! II (1) at z = 1, they
can reach first the Alfv'en resonance z A and then also the
cusp resonance z c that are located within the layer so that
1 – z A ? z c . Thus one can have both resonances con­
tributing to the absorption of the fast mode provided the
wave frequency ! does not exceed the value of the cusp
frequency at z = 0, i.e. for ! c (0) – ! – ! II (1). At higher
frequencies when ! – ! c (0) the absorption occurs from
the Alfv'en resonance solely.
The only possibility of avoiding the Alfv'en resonance
for the fast mode is to consider waves propagating with
OE = 0 ffi . The resulted absorption coefficient plots are shown
in Fig.7. A general conclusion is that the absorption of
these waves is less efficient if compared to the slow mode
and it does not exceed 60%. The curves end again when
the wave propagation limits are closely approached.
Fig. 8. The dependence of the absorption coefficient on the
incident angle ` and the azimuth angle OE for the fast MHD
mode.

V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ... 9
Fig. 9. The fast MHD mode with the frequency ! = 0:8. a)
The dependence of the absorption coefficient A on the incident
angle ` at three indicated azimuthal angles OE. b) The depen­
dence of the absorption coefficient A on the azimuthal angle OE
at three indicated incident angles `.
Finally, if OE 6= 0 ffi the Alfv'en resonance sets in too
and how the the total absorption coefficient depends on
angles OE and ` is shown in Fig.8. The cross sectional plots
in Fig.9 clearly indicate that the absorption is again the
most efficient at incident angles ` = 50 ffi \Gamma 60 ffi and that it
decreases with OE.
As to the frequency dependence, we see from Fig.10
that lower frequencies are absorbed more efficiently.
All the results obtained so far are related to fi = 0:6
and to the ratio of the Alfv'en speeds v A (1)=v A (0) = 0:2
and they show that the highest values for the absorption
coefficient occur when ` ú 50 ffi and ! ú 6. How the ab­
sorption coefficient now changes if fi is varying is shown
in Fig.11.
Finally, the dependence of the absorption coefficient
on the ratio vA (1)=v A (0) is given in Fig.12 for three wave
Fig. 10. The dependence of the absorption coefficient A on
the wave frequency ! for the fast MHD mode for: a) OE = 10 ffi
and b) ` = 70 ffi and other parameters as indicated.
frequencies for the slow mode only. Other parameters are
OE = 0 ffi , ` = 50 ffi and fi = 0:6.
So far, we have been studying general properties of
MHD wave absorption due to the cusp resonance in a
nonuniform plasma layer that separates two homogeneous
regions.
To see now the significance of the obtained results in
the context of solar conditions, we can, for example, con­
sider the region 1 as the corona with the basic state pa­
rameters evaluated at z = 0, the transitional layer is the
chromosphere while the region 2 is the photosphere where
the incoming magneto acoustic modes are generated and
the basic state parameters are evaluated at z = 1.
Taking v s (0) = 100 km=s and v s (1) = 8 km=s for
the sound speeds corresponding to temperatures T 0 (0) ú
10 6 K and T 0 (1) ú 6; 000 K for the corona and the pho­
tosphere respectively and taking fi = 0:6, one obtains
for the related Alfv'en speeds v A (0) ú 140 km=s and

10 V. M. Ÿ
CadeŸz, '
A. Cs'ik, R. Erd'elyi and M. Goossens: The effect of the cusp resonance ...
Fig. 11. The dependence of the absorption coefficient A on
the parameter fi for the fast and the slow mode when: ! = 6
and ` = 50 ffi
and OE as indicated.
Fig. 12. The dependence of the absorption coefficient A on
the ratio of the Alfv'en speeds vA(1)=vA(0) for the slow mag­
neto­acoustic mode when: fi = 0:6, ` = 50 ffi and OE = 0 ffi .
vA (1) ú 10 km=s or v A (1)=v A (0) ú 0:1 for their ratio.
Fig.12 shows that a wave with the dimensionless frequency
! ú 1 with an angle of incidence ` = 50 ffi , can be absorbed
close to 50% if the Alfv'en speed ratio is ú 0:1. The value of
! ú 1 corresponds to the time period of t o ú L=23 in sec­
onds if the scaling thickness of the layer L is given in kilo­
meters. Thus, if L = 2; 300 km which is a typical value for
the thickness of the chromosphere, we obtain t o ú 2 min
for the wave period which does not differ significantly from
the period of 5 min of the observed oscillations related to
the solar p­mode. In other words, the solar p­mode oscil­
lations can play a role in the process of coronal heating
through the mechanism of enhanced absorption of MHD
modes at the cusp resonance.
Another example shows that the cusp resonance causes
a significant absorption of MHD modes also in the case
when the waves are generated locally in the corona itself.
To examine this possibility, we take that both uniform
domains are located in the corona and separated by a
nonuniform layer of thickness L. Let the waves propagate
through the region with the lower temperature of T 0 (1) ú
10 6 K and be evanescent in the other domain with the
higher temperature of T 0 (0) ú 2 \Theta 10 6 K. Taking fi = 0:6,
the related values for the sound and the Alfv'en speeds
are then: v s (1) = 100 km=s, v s (0) = 140 km=s, vA (1) ú
140 km=s and vA (0) ú 200 km=s respectively. The time
period t o of the waves that are being absorbed in the layer,
is then
t o = 2ú
!
L
vA (0) ú 0:03 L[km]
!
[s]
where ! is the dimensionless frequency. We can take now
L ú 60; 000 km and ! = 6 which gives the time period
of t o = 5 min for the considered wave. Its absorption
rate finally follows from the Fig.12 to be around 90% at
vA (1)=v A (0) ú 0:7.
To conclude, the resonant absorption at the cusp res­
onance under the considered conditions can be an effi­
cient mechanism for the coronal heating, mostly by slow
magneto­acoustic waves. The fast magneto­acoustic mode
is less absorbed at this resonance.
Acknowledgements. V. Ÿ
CadeŸz acknowledges the financial sup­
port by the 'Onderzoeksfonds K.U.Leuven' (the senior research
fellowship F/95/62) while '
A. Cs'ik expresses his acknowledge­
ment to the Hungarian Soros Foundation. This work was partly
carried out while R. Erd'elyi was a Research Fellow of the Re­
search Council of the Katholieke Universiteit Leuven, Belgium.
The support by the K.U.L. Research Council is greatly ac­
knowledged. R. Erd'elyi was also supported in this work by the
grant GR/K43315 from the PPARC of the U.K.
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