A&A 443, 1033-1046 (2005)
DOI: 10.1051/0004-6361:20042272
Dissipation of Alfvén waves in complex 3D coronal force-free structures![[*]](/icons/foot_motif.gif)
F. Malara - M. F. De Franceschis - P. Veltri
Dipartimento di Fisica, Università della Calabria,
via P. Bucci, 87036 Rende (CS), Italy
Received 28 October 2004 / Accepted 19 May 2005
Abstract
The propagation and dissipation of Alfvén waves in a three-dimensional inhomogeneous force-free
equilibrium structure is considered. Assuming small wavelengths the wave propagation
is described within the framework of a WKB theory. The equilibrium structure is a complex
magnetic field, resulting from a superposition of several harmonics, which represents a model of the
coronal magnetic field above a quiet-Sun region. The properties of wave dissipation are related to
the topological complexity of the background magnetic field, and the typical scale law of fast dissipation
is recovered. A partial reflection of waves is included in the model by calculating reflection and
transmission coefficients of both Alfvén and magnetosonic waves.
An energy balance is derived between the wave energy dissipated inside the coronal structure and
the energy lost through the lower boundary. We find that even for large values of the Reynolds number,
a non-negligible fraction of the Alfvén wave energy is dissipated inside the corona.
Key words: magnetohydrodynamics (MHD) - waves - Sun: magnetic fields
Among the mechanisms invoked to explain the
nonradiative heating of the solar corona, dissipation of Alfvén waves represents
one of the most studied from the theoretical point of view. The ubiquitous presence of
a strong magnetic field, which represents the backbone of coronal structures,
allows Alfvénic fluctuations to propagate from distant part, carrying both
mechanical and electromagnetic energy. Alfvénic fluctuations have never been
directly observed in the corona. However, they represent the main component of
solar wind fluctuation spectrum (Belcher & Davis 1971) and it is likely
they are also present in the corona, from where the solar wind is emanated.
Due to the very low dissipative coefficients (resistivity and relevant viscosity) of the
coronal plasma, an important point of wave-based theories is how to dissipate
fluctuations before they leave the corona: dissipation can take place only if the
wave energy is efficiently transferred to small scales.
In highly structured magnetic fields as in the corona, interaction between Alfvén
waves and inhomogeneities of the background structure naturally leads to
generation of small scales. In this framework, we are not interested in specific
dissipation mechanisms active at small scales. Rather, we focus
on the dynamical processes that generate small-scale fluctuations.
In a 2D inhomogeneous background, where the Alfvén velocity
varies in a
direction perpendicular to the magnetic field, two mechanisms have been studied
in detail: (1) phase-mixing (Heyvaerts & Priest 1983; Nocera et al. 1986;
Parker 1991; Nakariakov et al. 1997; Moortel et al. 2000;
Botha et al. 2000; Tsiklauri et al. 2001, 2002b,
2003; Tsiklauri & Nakariakov 2002a), in which
differences in phase velocity at different locations progressively bend wavefronts;
and (2) resonant absorption which concentrates the wave energy in a narrow layer
where the wave frequency locally matches a characteristic frequency (Alfvén or
cusp). These processes have been studied both investigating normal modes of
the inhomogeneous structure (Kappraff & Tataronis 1977;
Mok & Einaudi 1985; Steinolfson 1985;
Davila 1987, 1991;
Califano et al. 1990, 1992;
Hollweg 1987a,b),
and by considering the time evolution of an initial disturbance
(Lee & Roberts 1986; Malara et al. 1992, 1996).
Effects of density stratification and magnetic line divergence (Ruderman
et al. 1998), as well as linear (Tsiklauri & Nakariakov 2002a;
Tsiklauri et al. 2003) and nonlinear (Nakariakov et al. 1997,
1998) coupling with compressive modes have also been
considered. Recently, Tsiklauri et al. (2005) have studied the phase-mixing
of Alfvén waves in a kinetic description, showing that the scale law of dissipation derived
within the MHD is also valid in that context.
Similon & Sudan (1989) considered Alfvén wave propagation in
three-dimensional magnetic structures. In this case, there can be regions where
magnetic lines are chaotic: in these regions initially nearby magnetic lines
exponentially move apart with increasing distance along the lines (Zimbardo et al.
1984, 1995). Then, Alfvénic disturbances that
locally follow magnetic lines are stretched exponentially in time, leading to a fast formation of small scales. As a consequence, the typical dissipation time
of fluctuations follows the scaling law:
 |
(1) |
where S is the relevant viscous and/or resistive Reynolds number. Thus, the
dissipation time remains relatively small even for large values of the Reynolds
number, as in the coronal plasma. The mechanism proposed by Similon &
Sudan has been investigated by Petkaki et al. (1998) and by
Malara et al. (2000), who built a model based on a WKB expansion
of incompressible magnetohydrodinamic (MHD) equations. In this model
propagation and dissipation of Alfvénic wavepackets in 3D magnetic fields is
described. From this model they found that the scaling law (1)
corresponding to fast dissipation of wavepackets is recovered in regions of chaotic
magnetic lines. This effect can co-exist with a slower dissipation related to
phase-mixing, in regions where the topology of magnetic lines is quasi-2D.
More recently, Malara et al. (2003) extended the above model to the
case of a cold plasma, which is more suitable than the incompressible
approximation to describe the low-
coronal plasma.
The equations derived by Malara et al. (2003) describing the evolution
of Alfvénic wavepackets are quite general and they can be used independently
of the specific form of the background structure. However, the background
magnetic fields considered by
Petkaki et al. (1998) and by Malara et al. (2000, 2003)
do not represent a realistic
model of coronal magnetic fields. The purpose of the present paper is to apply
the above theory to a more realistic description of coronal magnetic fields.
In the present paper we will consider a force-free
equilibrium magnetic field which can be used as a model of the coronal magnetic
field above a quiet-Sun region. Propagation and dissipation of Alfvénic
wavepackets within such a structure will be described by means of the equations
derived by Malara et al. (2003). This equilibrium magnetic field was
first proposed by Nakagawa & Raadu (1972) within the problem of deriving the
magnetic field in the corona using measurements in the photosphere. The
mathematical expression is quite general and it can describe a complex magnetic
structure with several regions of the two polarities. The complex 3D topology of
magnetic lines results in a fast increase of the wavevector of Alfvénic packets
and fast dissipation. This magnetic field represents a closed structure because
magnetic lines mostly start and end at the coronal base. Alfvénic packets which
propagate along fieldlines are partially trapped inside this closed structure and,
in order to describe their dynamics, it is necessary to represent the partial reflection
process of packets at the coronal base. This will be done with a simple model which
takes into account the geometry of the equilibrium magnetic field. Then, the wave
energy deposition inside the corona will result as a balance between dissipation and
energy losses through the partially reflecting boundary at the coronal base.
The coronal plasma is dominated by the magnetic force and it is then
characterized by a low value of the plasma
(
being
the gas pressure to magnetic pressure ratio). In the corona, typically
.
In this case, a usual representation of large
scale phenomena is that given by the cold plasma MHD equations,
in which pressure and gravity terms are neglected with respect to
the magnetic force in the momentum equation. These equations
are written in the following dimensionless form:
 |
(2) |
 |
(3) |
 |
(4) |
In these equations, lengths have been normalized to a characteristic
length L; the magnetic field is normalized to a characteristic
magnetic field
,
the density to a characteristic density
,
the velocity to the
corresponding Alfvén velocity
,
and the time to the Alfvén time
.
The dimensionless
dissipative coefficients are
,
,
and
,
where
and
are the
viscosity coefficients and
is the magnetic diffusivity, which
are assumed to be constant. Summation over dummy indices is
hereafter assumed.
We consider the propagation and the dissipation of Alfvén waves
in a 3D inhomogeneous magnetic structure, representing
a model of the magnetic field in the solar corona, above a quiet Sun
region. A full description of wave propagation in a complex 3D magnetic field represents a formidable task, since different wave modes
(Alfvén and magnetosonic) are coupled among them and with the
inhomogeneity of the background structure. A possible approach to
such a problem is represented by direct numerical simulations, in which
the full set of MHD equations is solved. These methods have been used
to study the wave evolution in simple 2D equilibria (e.g., Malara et al. 1996).
For a 3D configuration, the limited number of numerical degrees of freedom
makes inaccessible the high Reynolds/Lundquist numbers which are
typical of the coronal plasma. This fact limits the applicability of direct
numerical simulations to model wave dissipation in a complex 3D magnetic structure.
An alternative approach to the problem of Alfvén wave propagation and
dissipation in 3D equilibria was proposed by
Similon & Sudan (1989), and subsequently investigated by
Petkaki et al. (1998) and by Malara et al.
(2000, 2003).
Here we use this same method, which is briefly summarized in the
following. Some assumptions are required: (i) Alfvénic perturbations
are supposed to have a small
amplitude:
,
where
is the typical velocity
fluctuation and
is the background Alfvén velocity; (ii)
perturbations have a small wavelength
,
where
is the wavelength and L is the scale of variation of the
equilibrium magnetic field. Under the above assumptions the MHD equations
are linearized and a WKB expansion is applied. The Alfvénic perturbation
is divided into a large number of packets, each occupying an infinitesimal volume;
thus, each packet is characterized by a position
,
a wavevector
and an energy e=e(t). When the packet propagates inside the
equilibrium structure, these quantities evolve in time. The time evolution of
each packet is calculated by solving the equations:
 |
(5) |
 |
(6) |
 |
(7) |
where
is the unit vector in the direction of
the equilibrium magnetic field
;
is the background density;
is the Alfvén velocity; all these quantities
depends on the position
.
The Reynolds number S is defined by
.
The sign in the upper index identifies the propagation direction of the packet:
+A in the direction of
and -A in the opposite direction.
The same normalization of Eqs. (2)-(4) has been used.
Equations (5)-(7) have been derived by
Petkaki et al. (1998) for an incompressible plasma. Recently,
Malara et al. (2003) starting from Eqs. (2)-(4) showed
that Eqs. (5)-(7) describe the evolution of
Alfvénic packets also in a compressible cold plasma.
Equations (5)-(7) indicate that: (i) each Alfvénic packet propagates
at the Alfvén velocity
along the magnetic field
lines; (ii) wavevector changes are due to spatial variations of
,
which distort the packet; (iii) energy dissipation takes place with a rate
proportional to k2 and to 1/S.
Considering simple 2D configurations of the equilibrium structure, where the
Alfvén velocity
is directed along a given direction (say, z)
and varies along another perpendicular direction (say, x), Eqs. (5)-(7) can be exactly solved (Petkaki et al. 1998):
in this case it is seen that the wavevector k asymptotically increases proportional
to time:
,
while the typical dissipation time
scales according
to the law
.
These features are typical of phase-mixing of the Alfvén waves
(Heyvaerts & Priest 1983).
Since in the corona the dissipative coefficients (relevant viscosity and resistivity) are
very low, the dissipation time due to phase-mixing is exceedingly long; i.e.,
Alfvén waves would be dissipated after traveling over lengths much larger than
the size of coronal structures.
For more complex 3D equilibria, where the Alfvén
velocity
depends on the three spatial coordinates,
regions may exist where magnetic field lines are chaotic, i.e.,
nearby magnetic lines exponentially move apart. Equation (5) indicates
that Alfvénic perturbations propagate locally following magnetic lines; then, in a
chaotic region Alfvénic packets are rapidly stretched, and the wavevector
exponentially increases in time. Equations (5)-(7) describe this
behaviour which leads to the scaling law (1) of dissipation time,
typical of fast dissipation (Petkaki et al. 1998).
Thus, even for very low values of the dissipative coefficients (viscosity and/or
resistivity), relatively short dissipation times are achieved, contrary to the
phase-mixing-driven dissipation. The two phenomena (phase-mixing and 3D
fast dissipation) can co-exist in different regions within the same equilibrium
structure (Malara et al. 2000).
In the above-cited papers the main interest was to investigate the properties of
Alfvén wave propagation and dissipation in complex 3D magnetic field. Thus,
the explicit form used for the background structure was not particularly
representative of the magnetic field in the corona. In the present paper, we
want to study Alfvén wave dissipation in an equilibrium magnetic field which
is a more realistic representation of the coronal magnetic field. In particular, we
will use a force-free magnetic field which was first introduced by
Nakagawa & Raadu (1972) within the problem of extrapolating the magnetic
field in the corona starting from measurements in the photosphere. For completeness,
in the following we briefly describe the derivation of this magnetic
field, which requires some assumptions:
- (i)
- the equilibrium magnetic field
satisfies the
force-free condition
 |
(8) |
with
constant. The force-free condition is a natural requirement, since it
is well satisfied by an equilibrium magnetic field in a low-
plasma, like in
corona. The condition
is a much stronger assumption;
on the other hand, it allows us to simplify the derivation of the equilibrium magnetic
field;
- (ii)
- the curvature due to the spherical geometry of the corona is neglected.
This assumption is valid if relatively small portions of the corona are considered.
This allows us to use a Cartesian reference frame, in which the xy plane represents
the base of the corona, while the z axis is in the upward vertical direction;
- (iii)
- we assume statistical homogeneity along the horizontal directions
x and y. This implies that no dominant structures (e.g., a magnetic
dipole) are present. On the contrary, the magnetic field distribution is characterized
by several regions of different polarities randomly distributed at various
spatial scales. This situation can be found in quiet-Sun regions, and it is illustrated,
for instance, in the magnetogram plotted in Fig. 1 of Schrijver et al. (1997).
In order to describe this kind of configuration, we assume that along the horizontal x and y
directions the equilibrium magnetic field has a characteristic scale of variation
L0 and it is periodic with a periodicity length L. Statistical homogeneity is
reproduced when
(Pommois et al. 1998); in particular, we used
L = 4 L0. Comparing with the magnetogram of Schrijver at al. (1997)
(their Fig. 1), the
characteristic length for the magnetic field variation can be estimated as
km, thus, the unit length will be
km. When using dimensionless variables,
the periodicity length is equal to 1, while the magnetic field characteristic
scale is
l0 = L0/L = 1/4;
- (iv)
- following Nakagawa & Raadu (1972), we assume that
the magnetic field vanishes in the limit
.
This implies that the
magnetic energy in a single periodicity box is finite (Alissandrakis 1981).
Using the above assumptions, a form for the equilibrium magnetic field can be
derived. Using the periodicity condition, the magnetic field can be expanded in
Fourier series in the x and y directions:
![\begin{displaymath}
B_{\rm0i}(x,y,z) = \sum_{\kappa_x,\kappa_y} {\hat B}_{\rm0i}...
...a_y,z)
\exp \left [ {\rm i}(\kappa_x x + \kappa_y y) \right ]
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img67.gif) |
(9) |
where the wavenumbers are
and
,
with nx and ny integers.
The dependence on z is included in the
Fourier coefficients
.
Inserting the expansion (9)
into the force-free condition (8), we derive the equations:
 |
(10) |
 |
(11) |
 |
(12) |
where
,
while
and
are the components in
the horizontal direction, parallel and perpendicular to the wavevector
,
respectively. The only solution of Eq. (10)
which satisfies the above assumption (iv) is in the form:
 |
(13) |
where
.
This condition implies that
is a lower
bound for the horizontal wavevector
.
We will assume that
varies in the range
,
where
corresponds to the typical scale of
,
while
corresponds to the smallest length in the
equilibrium structure. Then, the largest length in the equilibrium magnetic
field structure corresponds to the characteristic horizontal length l0.
In particular,
.
Using the expression (13) the Eq. (11)
is solved, and the solution is used to write the equilibrium magnetic field components
in the form:
The specific form of the magnetic field is determined by the complex Fourier
amplitudes
.
The coefficient
in the expressions (14)
diverges in the limit
.
Though
is strictly
larger than
,
it is useful to eliminate such a divergence by simply
re-defining the amplitude of the Fourier components. In particular, we define:
 |
(15) |
where
is the new complex amplitude.
The reality condition
(where the asterisk indicate complex conjugate)
implies:
 |
(16) |
Using this condition, and the definition (15), after some algebra the
equilibrium magnetic field components can be put in the following form:
where only real quantities appear and the sums are extended only on
half of the
plane.
The expressions (17) indicate that the equilibrium magnetic field is
obtained as a superposition of several harmonics, each characterized by an
horizontal wavevector
which defines a corresponding
scale
.
Each harmonic exponentially decreases with increasing
the altitude z with a rate
;
thus, the
contribution of harmonics with small horizontal wavelengths is limited to
low altitudes, while at higher altitudes only the longest wavelength harmonics
are present (
).
The expression (17) depend on some parameters: namely, the
coefficient
;
the minimum and maximum wavenumber
and
;
the amplitudes
and the phases
of harmonics. In the context of extrapolating the
coronal magnetic field from measures in the photosphere, the above parameters
should be determined using appropriate boundary conditions at the coronal
base z=0. In our approach we are not interested in representing any specific
observed configuration; on the contrary, the equilibrium magnetic field should
represent a generic magnetic structure above a quiet Sun region. In fact, we
are mainly interested in studying the general properties of Alfvén wave
dissipation, which should not depend on the details of the equilibrium structure.
For this reason, we have chosen the above parameters in the following way.
- (i)
- We assume that harmonic amplitudes
depend only
on the modulus
of the horizontal wavevector, according to a power law:
 |
(18) |
where
is the spectral index and c is a normalization constant. In order to choose a
value for
,
we consider the magnetic energy at the coronal base z=0 in a
periodicity box, which can be expressed in terms of the magnetic field Fourier
components by using the Parseval theorem:
 |
(19) |
In the limit of large L, the sum can be approximated by an integral,
according to the prescription
Thus, Eq. (19) gives
 |
(20) |
where we used the assumption that
depends on the
modulus
of the wavevector. If the magnetic field structure is
the results of a turbulent process taking place beneath the coronal base, we
can assume that
 |
(21) |
Comparing Eqs. (20) and (21) we see that
 |
(22) |
Using the Kolmogorov spectral index
of the spectral energy
density
,
we obtain the spectral index
of the amplitude
.
The normalization constant c in Eq. (18) is determined
requiring that
.
Thus, the unit magnetic
field in dimensionless units corresponds to the rms magnetic field at the base.
- (ii)
- Due to the
form (18), the maximum of energy is at the largest wavelengths
.
As discussed above, in order to reproduce statistical
homogeneity we used
l0=L0/L=1/4, corresponding to
.
The maximum wavenuber is
.
With these
values, each component of the equilibrium magnetic field is a superposition
of 133 harmonics, which gives a complex magnetic field but computational times not
exceedingly long.
- (iii)
- The parameter
determines the current density
to the equilibrium magnetic field
ratio. Moreover, in our model the value of
is
bounded by the minimum horizontal wavevector:
.
We considered two different values for the parameter
:
a)
,
corresponding to a potential magnetic field (
);
b)
,
which is close to the upper bound
and
gives a nonvanishing current
.
- (iv)
- Since
does not represent any particular magnetic field, the phases
of harmonics have been randomly
chosen in the interval
.
In Fig. 1 a contour plot of the vertical component
B0z(x,y,0) at the base
z=0 in the periodicity domain is represented, in the case
.
Darker or clearer tones correspond to positive or negative values of B0z,
respectively. It can be seen that B0z is randomly distributed, with structures at
different length scales. This is reminiscent of a portion of the magnetogram in
a quiet Sun region displayed in Schrijver et al. (1997). In Fig. 1 several
magnetic lines are represented; they have
been obtained integrating the Eq. (5). Each magnetic line starts from a
point randomly chosen at the base z=0 and it ends at another location at z=0,
linking two points with a different magnetic polarity (sign of B0z).
Since the magnetic field intensity decreases with the height z, the density of field
lines must decrease, too: thus, most of lines remain at low altitudes and very few
reach higher values of z. Magnetic lines which are limited to low altitudes are
mostly in the form of an arc which crosses a neutral line at the base, defined by
B0z(x,y,0)=0. Magnetic lines which extends to higher altitudes have a
more complex behavior: they are more twisted and do not have a definite shape.
![\begin{figure}
\par\includegraphics[width=8.8cm,clip]{2272fig1.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg126.gif) |
Figure 1:
A contour plot of the vertical component B0z of the equilibrium magnetic
field, along with the trajectories of 70 magnetic lines. The stating point of
each magnetic line is randomly chosen inside the periodicity domain. |
The overall structure of the magnetic field represented in Fig. 1 is quite complex.
Dissipation properties of Alfvén waves are directly related to the topological complexity
of the magnetic field (Petkaki et al. 1998; Malara et al. 2000).
In order to illustrate some features of the topology of the
equilibrium magnetic field we drew some magnetic surfaces in form of flux tubes.
These magnetic surfaces are formed by several magnetic lines. Each line starts from
a point at the coronal base located in a small circle; in other words, the initial section
of the flux tube is circular, and it is located within a region with the same magnetic
polarity. We calculated several of these flux tubes, finding that they can have two
kinds of structures, which we call a "regular'' flux tube and a "singular'' flux tube,
respectively. In Fig. 2 (upper panel) an example of a regular flux tube
is represented: it is a sort
of cylinder linking two regions of different polarities located at the base z=0. The
tube section, initially circular, changes along the tube axis, being larger at higher
altitudes, but it remains a closed line at any location along the tube. Magnetic lines
wrap around the magnetic surface and their twist is related to the presence of a
nonvanishing current.
In Fig. 2 (lower panel) an
example of a singular flux tube is represented. At variance with the previous situation,
in this case the magnetic surface divides into several sheets, each following a different
path: all lines forming this surface start from the same region with a given
polarity, but they end in four different regions with the opposite polarity. The
singularity is in the behaviour of magnetic lines around locations where the
surface separates in different sheets: two initially infinitely close magnetic lines can
either remain close to each other (if they belong to the same sheet) or they can follow
two completely different trajectories (if they belong to two different sheets). This
behaviour is found also into phase space trajectories of chaotic dynamical systems.
Considering the overall 3D structure of the magnetic field, this separation takes
place onto separatrix surfaces: magnetic lines which are on the two sides of the
separatrix will follow diverging paths. Although we did not carry out any
systematic study of these features in the magnetic structure, we expect that it contains
several separatrices. A detailed study of the general properties of separatrices in
3D fields can be found in Priest & Titov (1996).
![\begin{figure}
\par\includegraphics[width=9.4cm,clip]{2272fig2.eps}\par\vspace*{2mm}
\includegraphics[width=8.9cm,clip]{2272fig3.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg127.gif) |
Figure 2:
A regular ( upper panel) and a singular ( lower panel) flux tube. |
Divergence of nearby magnetic lines is related to the increase of the wavevector of
Alfvénic disturbances, which propagate locally following magnetic lines. The
volume of an Alfvénic packet is approximately constant (this is strictly verified for
uniform
,
but it holds also when
varies on a sufficiently large scale).
Thus, when a packet goes through locations where magnetic lines diverge
it is stretched and squeezed. This results in an increase of the wavevector.
Another independent mechanism which determines wavevector increase is related to
the exponential dependence of
on the z coordinate. A simple
configuration where the Alfvén velocity varies exponentially with the altitude was
considered by Ruderman et al. (1998). These authors showed that in
such a case the damping of
an Alfvénic perturbation takes place exponentially with the distance along a
field line. This is due to the fact that two points in the perturbation which
propagate along two nearby magnetic lines move apart exponentially, thus resulting
in an exponential increase of the wavevector. This same effect is present also in the
more complex structure we are considering.
![\begin{figure}
\par\includegraphics[width=6cm,clip]{2272fig4.eps}\par\vspace*{2m...
...ar\vspace*{2mm}
\includegraphics[width=6.5cm,clip]{2272fig6.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg128.gif) |
Figure 3:
The trajectory of a packet ( upper panel); the wavevector k ( middle panel)
and the energy e ( lower panel) of the packet, as functions of time t. |
A description of the dynamics of a single packet can be obtained by integrating the
Eqs. (5)-(7). As an example, the time evolution of the
position
,
the wavevector
and the energy of a packet is
shown in Figs. 3. The initial condition is the
following:
- (i)
- the initial position
is located at the
base;
- (ii)
- the initial wavevector is
,
giving
,
i.e., large enough to
satisfy the WKB approximation;
- (iii)
- the initial energy is e0=1; this value is
not relevant to determine the time evolution: since the Eq. (7) is
linear with respect to e, the physically relevant quantity is the ratio e(t)/e0.
The sign in Eqs. (5) and (6) determining whether
the packet propagates parallel or antiparallel to
is chosen so that
the packet initially propagates in the upward direction. The value of the Reynolds
number is
.
Following a magnetic line, the trajectory of the packet starts and ends at the
base z=0; the time evolution of Fig. 3 corresponds to this path. The wavevector
k increases roughly exponentially in time (Fig. 3, middle panel), in accordance with
the previous considerations; this characterizes the fast dissipation mechanism
(Petkaki et al. 1998). The energy of the packet is fully dissipated
(Fig. 3, lower panel). A time
evolution similar to that shown in Fig. 3 has been found for many other packets
with different initial positions, especially for those whose trajectory reaches large
enough altitudes. In such cases there is enough time for the energy to dissipate
before the packet reaches the base.
The dissipation time
for a packet is defined by
;
increases with increasing the Reynolds number S. In particular,
is proportional to
if the wavevector increases exponentially in
time (Petkaki et al. 1998). The dissipation time
of the packet of Fig. 3 is represented in Fig. 4 for different values
of S:
the rough exponential growth of k (Fig. 3) gives
approximately
proportional to
.
This is typical of fast dissipation.
![\begin{figure}
\par\includegraphics[width=6.5cm,clip]{2272fig7.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg137.gif) |
Figure 4:
The dissipation time
of the same packet as in Figs. 6, as a function
of the Reynolds number S (dots). The full line indicates a scale law
. |
Packets whose trajectory is confined at lower altitudes has a time
evolution different from that displayed in Fig. 3. If the magnetic line where
the packet propagates is too short, only a fraction of the initial energy is dissipated
before the packet reaches the base. Moreover, when the Reynolds number is
increased the dissipation rate becomes smaller; in such a case also packets
reaching higher altitudes do not have enough time to dissipate all their initial energy.
For these reasons it is important to establish what happens when a packet
reaches the coronal base. Since large vertical gradients of the background
quantities are present beneath the base of the corona, we will model the coronal base
as a discontinuity surface. In this case, an Alfvén wave which reaches the
base z=0 will give origin to transmitted and reflected waves, both Alfvénic
and magnetosonic. In particular, the subsequent time evolution of the reflected
Alfvénic packet can be described by using the same Eqs. (5)-(7). Reflection and transmission coefficients determine
how the energy of the incident packet is redistributed into reflected and transmitted
packets. A simple model of the discontinuity at the coronal base, as well as the
detailed calculation of the corresponding transmission and reflection coefficients are given in the Appendix.
The value of reflection
,
and transmission
,
coefficients of Alfvénic and magnetosonic waves calculated in the Appendix
(Eqs. (A.103)-(A.106)) depends on some parameters:
- (i)
- the angle
between the magnetic field
and the horizontal direction. This angle has different values at different
positions at the base z=0;
- (ii)
- the ratio
,
where the upper index + or -
refers to the value just above or below the discontinuity, respectively. The density
ratio between
the corona and the chromosphere has been estimated as
(see Mariska 1992).
More recently, Ofman (2002)
in a 1D model of Alfvén wave energy leakage from the corona used
the value
.
Then, in our model, we considered the
values r=0.01 and r=0.032;
- (iii)
- the wavevector
of the incident Alfvén wave. It can be verified that the reflection and transmission
coefficients do not depend on the modulus but only on the orientation of the
wavevector
.
Moreover, they do not change under the
transformation
.
Sometimes the reflected and/or the transmitted magnetosonic
wave is evanescent (see the Appendix). In such a case, the corresponding
reflection/transmission coefficient vanishes.
As an example, in Fig. 5
we plotted the reflection and transmission coefficients as functions of the
angle
,
for r=0.01 (upper panel) and r=0.032 (lower panel), with
a given value of the wavevector
.
The value of
used in Figs. 5 has been chosen in the
following way. We have verified that during the Alfvénic
packet propagation the wavevector component parallel to
remains roughly constant, while the perpendicular components tend to
increase in time. Then, we considered the parallel component
and the two perpendicular
components
and
.
In Figs. 5 we
chose the wavevector
so that
and
.
At r=0.01 the Alfvén wave reflection coefficient is close
to unity:
,
and it is largely independent of the angle
;
transmission coefficients
and
are much smaller, both being
at most
0.04; the reflected magnetosonic wave is evanescent (
)
for
any angle
.
At r=0.032 the reflection coefficients is slightly
lower:
,
and again mostly independent of
;
both
transmission coefficients are of the order
0.1; the maximum
transmission of Alfvén waves is at
and
(
), while the maximum transmission of magnetosonic waves
is at
(
); the reflected magnetosonic wave is
evanescent for any angle
.
The wavevector
used in Figs. 5 is
only an example (in the model, the actual value of
is determined
by the packet evolution). However, these figures shows that
for sufficiently small values of the parameter r a large fraction of
the energy of the incident Alfvénic packet is transferred to the reflected
Alfvénic packet, while the remaining part is essentially lost in waves
(Alfvénic and magnetosonic) transmitted below the coronal base.
![\begin{figure}
\par\includegraphics[width=6.6cm,clip]{2272fig8.eps}\par\vspace*{4mm}
\includegraphics[width=6.6cm,clip]{2272fig9.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg157.gif) |
Figure 5:
The reflection coefficient
(thick line),
the transmission coefficients
(dashed line) and
(thin line), as functions
of the angle ,
calculated for
and
,
and for r=0.01 ( upper panel) and r=0.032
( lower panel). |
Energy conservation requires that
 |
(23) |
We checked that this condition is verified by our calculation of reflection and
transmission coefficients.
The development described in the Appendix allowed us to determine
all the parameters (wavevector and energy) of the reflected and transmitted
wavepackets, as functions of the parameters of the incident Alfvénic packet.
The procedure can be summarized as follows. We follow the time evolution of
an Alfvénic packet integrating the Eqs. (5)-(7),
- (i)
- when
the packet reaches the level z=0 coming from above, we consider its
wavevector
and its energy
;
- (ii)
- the components of the wavevector are given with respect to the xyz
reference frame. Thus, first we calculate the components of
with respect to the XYZ reference frame (see the Appendix, Fig.
A.2); the direction of the X and Y
axes and the
angle are obtained calculating the equilibrium magnetic
at the packet location onto the plane z=0;
- (iii)
- the vertical (Z) components of the wavevectors of the reflected and
transmitted Alfvénic packets are calculated using the Eqs. (A.57)
and (A.60). The sign of the RHS of Eqs. (A.64) and (A.73)
determines whether the magnetosonic packets are propagating or
evanescent; in the former case, to calculate the components
and
Eqs. (A.67) and (A.75) are used,
while in the latter case we use Eqs. (A.69) and (A.77). The horizontal (x
and y) components of all
reflected and transmitted packets are the same as those of the incident wave;
- (iv)
- the wave amplitude ratios are calculated by numerically solving the system of
complex linear Eqs. (A.79)-(A.82). Such ratios are
used in Eqs. (A.99) and (A.100) to determine
the energy density ratios
,
,
,
and
,
which are used to calculate the
reflection and transmission coefficients (Eqs. A.103)-(A.106));
- (v)
- the energy of reflected Alfvénic
,
reflected magnetosonic
,
transmitted Alfvénic
,
and transmitted magnetosonic
packets are determined by the equations
 |
(24) |
where
is the energy of the incident Alfvénic packet.
3.2 Energy balance of Alfvénic perturbations
We considered an Alfvénic perturbation formed by a large number of packets.
At each time t the energy E(t) of the perturbation is given by the sum of
the energy e(t) of each packet.
The time evolution of each packet inside the equilibrium magnetic field
is determined by integrating the Eqs. (5)-(7),
starting from the initial conditions specified in Sect. 2.3. In particular, the packet
initial position
is located at the base, and it is
randomly chosen within the periodicity domain. If the number
of packets is
large enough, they can represent the evolution of the whole Alfvénic perturbation.
In particular we used
,
and we verified that increasing
the results
do not significantly change. Each time a packet reaches the base z=0 it is
replaced by a reflected Alfvénic packet, which propagates back. The wavevector
and the energy of the reflected packet are calculated as described in Sect. 3.2.
At the same time, the energies of the magnetosonic reflected perturbation and
Alfvénic and magnetosonic transmitted perturbations are stored. The evolution of
each packet is followed until its energy e(t) drops under the value
10-6e0,
where
e0=e(t=0); at that time the initial energy e0 has been completely
dissipated and/or converted in energy of transmitted or magnetosonic reflected
perturbations. The run ends when all the packets have reached this condition.
We performed the above calculation for different values of the parameters of
the problem. In particular, we considered two values for
:
namely,
(potential magnetic field) and
(nonvanishing
equilibrium current). The Reynolds number varies from
up to
.
We considered two values for the ratio r:
r=0.01 and r=0.032. The energy balance is
represented by the dissipated energy
of Alfvénic perturbations, the
energies
and
lost in transmitted Alfvénic and magnetosonic
perturbations, respectively, and the energy
of reflected magnetosonic
perturbations. These energies satisfy the balance equation
 |
(25) |
where E0 is the initial perturbation energy.
In Figs. 6 we plotted the ratios
,
,
and
,
calculated by the above procedure, for different values of the
parameters
,
S and r. The dissipated to initial energy ratio
is derived by the Eq. (25) and it is also plotted.
![\begin{figure}
\par\includegraphics[width=8cm,clip]{2272fi10.eps}\par\vspace*{2mm}
\includegraphics[width=8cm,clip]{2272fi11.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg181.gif) |
Figure 6:
The energy ratios
(full line),
(dashed line),
(dotted line), and
(dash-dotted line) are shown
as functions of the Reynolds number S, for
(no symbols)
and for
(black dots). The upper and lower panels refer to the case
r=0.01 and r=0.032, respectively. |
The upper panel of Fig. 6 shows the results obtained for r=0.01. For low values of the Reynolds
number
the ratio
is very close to
unity. This indicates that the initial perturbation energy E0 is almost completely
dissipated inside the magnetic equilibrium structure. On the contrary, the fraction
of energy lost in waves (both Alfvénic and magnetosonic) transmitted through
the discontinuity at the base is very low (less than 5%), while the fraction of
energy of reflected magnetosonic waves is negligible (
).
With increasing S the dissipated energy decreases, while the energy lost in
transmitted waves increases. This behaviour is the result of the balance between
the dissipation time
of a single packet and the crossing time
,
i.e., the
time the packet takes to propagate all along a magnetic line connecting two points of
the base z=0. The dissipation time
can be defined as the time the
wavevector k of a packet takes to reach a typical value
.
For
the time scale of energy dissipation is
1 (see Eq. (7)); then,
energy dissipation for the packet takes place at
.
The dissipation time
increases roughly proportional to
(see Fig. 4; and Malara et al.
2000).
On the contrary, the crossing time
for a given packet is independent of S. For sufficiently small values of S,
.
In this case, most of the
packet energy is dissipated during the first trip along the magnetic line. Then, only a
small fraction of the initial energy is lost in transmitted waves during the reflection
process. With increasing S, the dissipation time becomes increasingly larger with respect
to
;
then,
the packet undergoes multiple reflections before being dissipated. In this case, the
fraction of initial energy lost in transmitted waves becomes larger. This explains why
decreases while
and
increase with increasing S.
Clearly, both
and
are different for different packets and they depend
on the form of the equilibrium magnetic field; moreover, the wavevector of the
Alfvénic packet is modified by the reflection process. For these reasons, a
quantitative evaluation of
,
,
,
and
has
required the explicit solutions of both Eqs. (5)-(7) for
all the packets, as well as the calculation of quantities involved in each reflection
process. In particular, for
the dissipated energy is
;
then, a non negligible fraction of the initial wave energy
is dissipated inside the equilibrium structure, even for extremely high values of
the Reynolds number. This fact is due to the fast dissipation process (Malara et al.
2000),
which is related to the three-dimensional geometry of the equilibrium magnetic
field
.
In Fig. 6 it is seen that the fraction
of dissipated energy is larger
in the case of a nonvanishing equilibrium current (
)
with respect to
the potential field case (
). This could be due to two different effects:
either the dissipation time
of single packets on average decreases with
increasing
,
which would correspond to a larger wavevector growth rate;
or the crossing time
increases with
,
which could be due to
more twisted and then longer magnetic lines, and/or to a smaller average value
of the Alfvén velocity. This point needs further investigation which is left for
a future work.
In the lower panel of Fig. 6 the quantities
,
,
,
and
are shown for r=0.032. In this case both transmission coefficients
are larger with respect to the case r=0.01 (Figs. 5); then, the energy lost in
transmitted waves is larger, while the dissipated energy is smaller. However, the
dependence of these quantities on the Reynolds number S and on the parameter
is qualitatively similar to that found for r=0.01. In particular, for large
values of S, (
)
the ratio
.
The energy of reflected magnetosonic waves is negligible also for r=0.032.
In this paper we have studied the evolution and dissipation of Alfvénic
perturbations, which propagate inside a 3D inhomogeneous equilibrium magnetic
field. This magnetic field belongs to a wide class of solutions of the force-free
equilibrium equations (Nakagawa & Raadu 1972), obtained under
some assumptions:
namely, constant
;
periodicity in the horizontal directions; vanishing
magnetic field for
.
The magnetic field is a superposition of
contributions (harmonics) with different amplitudes and at different spatial scales.
We used this solution to represent a typical structure of a magnetic field above a
quiet-Sun region, which is characterized by several regions of the two polarities
randomly distributed in different locations and at various spatial scales. To model
this kind of configuration we used a power-law spectrum for the harmonic
amplitudes and random phases, while
,
which determines the current density,
has been used as a free parameter of the model. This equilibrium magnetic field does
not represent any particular structure deduced from observations: it is intended only
as a model for a typical configuration of the coronal magnetic field. The underlying
assumption is that our results on wave dissipation would not be modified if another
equilibrium magnetic field of the same class would be used (for example, using a
different choice for the phases of harmonics).
The propagation and dissipation of Alfvénic perturbations within the equilibrium
structure has been described using an approach based on a WKB expansion of the
cold plasma MHD equations (Malara et al. 2003). Some assumptions
have been used in this approach: first, it has been assumed that
the typical wavelength of fluctuations
is smaller than the spatial scale L0 of the equilibrium structure. Since the
wavevector of perturbations tends to increase in time, if this condition
is verified at the initial time, then it will be satisfied for all
the subsequent times. The initial condition
indicates that the
wavelengths considered in our model do not represent the main contribution in
the spectrum directly produced by photospheric motions. However, in consequence
of the coupling with both transverse and longitudinal inhomogeneities at scale L0, waves at wavelengths smaller than L0 are formed. This has been
suggested, for instance, for waves generated by the chromospheric network activity
(Axford & McKenzie 1992). Moreover, there are indications that in
closed magnetic structures, Alfvénic perturbations at
wavelengths smaller than the loop length are generated by resonance phenomena
(Nigro et al. 2004; Veltri et al. 2004);
such wavelengths would represent the main contribution to the
spectrum of Alfvénic fluctuations inside closed structures. The dissipation
mechanism here studied can be applied to these waves.
The second assumption is that perturbations have a small amplitude
with respect to the equilibrium structure:
.
It is commonly
assumed that in corona the Alfvén velocity is
km s-1 ,
while typical values of velocity fluctuation
km s-1
can be deduced from measures of nonthermal broadening of coronal lines
(Acton et al. 1981; Warren et al. 1997; Chae et al. 1998).
Thus, the small amplitude assumption for fluctuations is verified in the corona.
From a theoretical point of view, this assumption is commonly used to neglect
nonlinear terms in the MHD equations which could give rise
to a transfer of the energy of Alfvénic fluctuations to compressive modes. The
efficiency of such transfer does not depend only on the amplitude but also on the
wavelength of the Alfvén wave. Actually, Nakariakov et al. (1997) have
shown that phase-mixing of Alfvén waves on a purely transverse
inhomogeneity, and the consequent increase of magnetic pressure gradient, gives rise
to magnetosonic waves, even for a small amplitude wave. A similar effect could
be present also in the case considered here, since it is driven by the
increase of the wavevector of Alfvén wave. However, provided that the Alfvén
wave amplitude is not strongly nonlinear (
), the growth
of the driven compressive fluctuation due to phase-mixing saturates at amplitudes
much smaller than that of the Alfvén wave (Botha et al. 2000;
Tsiklauri et al. 2001). This suggests that also in our case
the coupling of Alfvén waves with compressive modes due to nonlinear terms can be
neglected.
Concerning this point, Tsiklauri & Nakariakov (2002a) and
Tsiklauri et al. (2003) have studied the time evolution of a 3D MHD pulse
propagating in a transversely inhomogeneous medium, finding a coupling between
the Alfvénic and the magnetosonic part of the perturbation; such coupling is
present at the linear level. However, their results are not directly comparable with
that of the present paper because their initial condition already contains a mix of a
magnetosonic and an Alfvénic component. On the contrary, here we considered only
purely Alfvénic perturbations, which are polarized perpendicularly both to
and to the local wavevector (Malara et al. 2003). Our configuration corresponds
to the case
of Tsiklauri & Nakariakov (2002a) and Tsiklauri
et al. (2003). In that case, these authors also find no energy transfer from
the Alfvén wave to magnetosonic perturbation.
More recently, Del Zanna et al. (2005) have studied the propagation of
a purely Alfvénic pulse within an arcade. The equilibrium magnetic field they used
can be obtained from our expression (17) for
and retaining the
contribution of a single wavenumber
.
These authors find a coupling
between Alfvén and magnetosonic waves. However, in that case the coupling is
nonlinear, due to the large amplitude of the initial pulse, the velocity perturbation
being comparable with the background Alfvén velocity.
Some general result which had been obtained for simpler background configurations
(Malara et al. 2000) have been recovered also in the more complex
equilibrium magnetic field considered here. In particular, the wavevector of
Alfvénic packets tends to grow exponentially, giving origin to the scaling law (1) for the dissipation time
,
typical of fast dissipation. The dissipation
time determines the typical distance over which a packet is dissipated; then,
in open magnetic configurations the determination of
is crucial in
order to establish whether an Alfvénic disturbance is dissipated before it
leaves the corona. In a closed magnetic structure like that considered here
the dissipation time plays a slightly different role, since
Alfvénic perturbations are confined to propagate
within the structure. However, since the corona is dynamically coupled to
lower layers in the solar atmosphere, this confinement is not perfect and a certain
leakage of fluctuating energy takes place at the coronal base (e.g., Ofman 2002).
Then, the fraction of the initial perturbation energy which is dissipated inside the
corona is determined by a balance between the dissipation time
,
which
depends on the Reynolds number S, and the efficiency of the wave reflection
process, which is quantified by the reflection coefficient
.
In our model we
found that even for very large values of the Reynolds number
a non negligible fraction of the initial perturbation energy is dissipated within
the coronal structure.
The value of the reflection coefficient
depends on how the transition between the corona and lower atmospheric
layers is modelled. In this paper we represented such a transition by a sharp
discontinuity in the background density. Other models have assumed an
exponential dependence of the Alfvén velocity on the vertical coordinate z
(e.g., De Pontieu et al. 2001; Ofman 2002) below the corona.
Although this assumption is more
realistic than a discontinuous density, in the other models both the background
magnetic field and the perturbation wavevector have been assumed to be vertically
directed, so that no magnetosonic perturbations are generated during the reflection
process. On the contrary, in our model we tried to account for the geometry of
the background magnetic field in calculating the wave reflection process; in
particular, we took into account the local orientation of both
and
of the wavevector of the incident wave. Moreover, the generation of
both reflected and transmitted magnetosonic waves has been included in the model.
We point out that in our model the detailed calculation of reflection and transmission
coefficients of Alfvénic and magnetosonic perturbations has been possible as a
consequence of the simplifying assumption of a discontinuous background density.
However, DePontieu et al. (2001) found that, in the geometry they considered, the
reflection coefficient obtained by using an exponential profile for the Alfvén
velocity is lower than in the case of a discontinuous profile. This suggests that
also in the more complex geometry considered here, a continuous profile of
would give a reflection coefficient
lower than what we calculated,
and the fraction of dissipated energy (e.g.,
at
)
would be lowered, as well.
Acknowledgements
This work was partially supported by the MIUR (Ministero dell'Istruzione, dell'Universitá
e della Ricerca) through a National Project Fund (Cofin 2002)
and by the European Community within the Research Training Network "Turbulence in
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5 Online Material
We want to describe how Alfvén waves impinging over a planar density
discontinuity give origin to both reflected and transmitted waves, and to
calculate reflection and transmission coefficients. The discontinuity
represents an extremely simplified model for the sharp density variation
associated
with the solar transition region. The width
km of
the transition region is much smaller than the typical variation scale
km of the background equilibrium structure. We
consider a region
located across the density variation layer; the
typical linear size of such a region is
,
with
.
As a consequence of these inequalities, within the region
we will
represent the background density variation as a real discontinuity, while
on both sides of such a discontinuity the background equilibrium structure
will be considered as uniform. The discontinuity is assumed to be planar
and located at z=0.
Moreover, we assume that the background magnetic field
is continuous through the discontinuity.
In the region
,
on both sides of the discontinuity we
decompose physical quantities as the sum of a uniform background
value plus a small amplitude perturbation:
where
gives the amplitude of perturbations, and the indices
"0'' and "1'' indicate quantities relative to the background structure and
to the perturbation, respectively. The properties of perturbations can be
derived following a standard procedure that we briefly outline. The
expressions (A.1) are inserted into the Eqs. (2)-(4), which are expanded in powers of the parameter
,
retaining only terms at the lowest order
.
A Fourier
transform is carried out, according to
 |
(A.2) |
where
indicates
or a component of
or
of
,
and
is the corresponding
Fourier transform. For any given value
of the wavevector,
nonvanishing solutions are found only when the frequency
satisfies one of the following four dispersion relations:
 |
(A.3) |
 |
(A.4) |
which give the frequency of Alfvén waves and of fast magnetosonic
waves, respectively.
and
determine the propagation direction of Alfvén waves (parallel or
antiparallel to
)
and of magnetosonic waves (parallel or
antiparallel to
), respectively.
The solution of the linearized equations is a
sum of contributions, each due to a single mode:
 |
(A.5) |
where the upper Greek index identifies the contribution of a particular
mode to the total perturbed field. Inserting the form (A.5)
into Eq. (A.2) we obtain the equation
![\begin{displaymath}
f_1(\vec{x},t) = \sum_\alpha \int {{\rm d}\vec{k}\over (2 \p...
...c{k} \cdot \vec{x} - \omega^\alpha (\vec{k})t \right] \right\}
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img221.gif) |
(A.6) |
expressing the field
as a superposition of waves, each
with different wavevector
and belonging to a different mode.
![\begin{figure}
\par\includegraphics[width=6.1cm,clip]{2272fi12.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg228.gif) |
Figure A.1:
The
reference frame is represented, along with the
wavevector
and the magnetic field ,
whose direction is given
by the unit vectors ,
,
respectively. The unit vector
gives the polarizarion of Alfvén waves, while the unit vectors
and
are in the
plane, perpendicular to
and to ,
respectively. The dashed line is parallel to . |
In order to express the solution of
the linearized system, for any given value of
it is useful
to define a Cartesian reference frame
with the
-axis
along the wavevector
,
and the
-plane containing the
direction of the background magnetic field
(see Fig. A.1).
With respect to such reference frame,
for the two magnetosonic modes we find:
where
is the angle between
and
,
and
is the arbitrary (dimensionless) amplitude of the
magnetosonic mode. Note that the velocity perturbation
of magnetosonic waves is directed in the
-plane, perpendicular to
.
In a similar way, for the two Alfvénic modes we find:
|
|
 |
|
|
|
 |
(A.8) |
where
is the amplitude of the Alfvénic mode.
Eqs. (A.7)-(A.8) indicate that the total velocity
perturbation due to both Alfvénic and to magnetosonic perturbations
lies in the plane perpendicular to
.
Actually, at the order
the magnetic force
is
perpendicular to
,
while the pressure gradient
vanishes in a cold plasma. Thus, no velocity perturbations can be
generated parallel to
.
The above expressions of the perturbed fields can be written in a more
general form, independent of the particular reference frame we used.
For this purpose, we introduce the following unit vectors, which are
represented in Fig. A.1:
and
,
directed
along
and
,
respectively;
directed parallel to the Alfvénic
perturbations;
and
in the direction of
magnetosonic magnetic field and velocity perturbation, respectively.
Using such unit vectors, the perturbations relative to magnetosonic and
Alfvénic modes can be written in the form:
 |
(A.9) |
and
 |
(A.10) |
The unit vectors
,
and
are not defined
when
is parallel to
.
In this particular case
the magnetosonic modes have the same properties as the
Alfvén modes: (i)
;
(ii) the velocity and magnetic field perturbations of magnetosonic mode are
parallel and they are both directed perpendicular to
and to Alfvénic perturbations (Eqs. (A.4) and (A.7) for
). Thus, for
parallel to
we can still use the expressions (A.9) and (A.10), where
and
are two arbitrary unit vectors perpendicular to
each other and to
.
![\begin{figure}
\par\includegraphics[width=6.5cm,clip]{2272fi13.eps}
\end{figure}](/articles/aa/full/2005/45/aa2272-04/Timg252.gif) |
Figure A.2:
The
(X, Y, Z ) reference frame is represented. The XY-plane
corresponds to the base of the corona; the background magnetic field is
in the YZ plane, at an angle
with the horizontal direction. |
Within the region
we introduce another Cartesian reference
frame (X,Y,Z), represented in Fig. A.2. This reference frame is obtained by
rotating the reference frame xyz around the z-direction, so that the background
magnetic field
lies in the YZ-plane, forming an angle
with the
Y-axis. Thus, the Z-axis is directed parallel to z, the density discontinuity lies in the XY-plane and the corona lies in the half-space Z>0. In this
reference frame the wavevectors
of the different modes can have an
arbitrary orientation. The XYZ frame is
defined within the small region
,
where the background magnetic field can
be considered uniform; however, in different points of the coronal base z=0 the
orientation of X and Y axes is different, according to local the direction of the
background magnetic field
.
In the (X,Y,Z) reference frame
the above-defined unit vectors have the following expressions:
![\begin{displaymath}
\vec{e}_{\rm A} = {\left( k_Y \sin \phi - k_Z \cos \phi \rig...
..._Y \sin \phi - k_Z \cos \phi \right)^2 + k_X^2
\right]^{1/2}}
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img253.gif) |
(A.11) |
![\begin{displaymath}
\vec{e}_1 = { - k_X \left[ k_Y \cos \phi + k_Z \sin \phi \ri...
...Y \sin \phi - k_Z \cos \phi \right)^2 + k_X^2
\right]^{1/2} }
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img254.gif) |
(A.12) |
![\begin{displaymath}
\vec{e}_2 = { k_X \vec{e}_X +
\left( k_Y \sin^2 \phi - k_Z ...
...k_Y \sin \phi - k_Z \cos \phi \right)^2 + k_X^2 \right]^{1/2}}
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img255.gif) |
(A.13) |
where
,
and
are the unit vectors of
the X, Y and Z axes, respectively. Using these expressions and the
Eqs. (A.7), the perturbations of the magnetosonic mode
in the XYZ reference frame are written as:
![\begin{displaymath}
\rho^{\pm \rm M}_1(\vec{k})={\rho_0 \over k^2}
\left[ \left(...
...Z \cos \phi \right)^2 + k_X^2
\right] a'_{\pm \rm M}(\vec{k})
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img259.gif) |
(A.14) |
![\begin{displaymath}
\vec{v}^{\pm \rm M}_1(\vec{k})= {\sigma_{\rm M} c_{\rm A} \o...
...s \phi k_Y \right) \vec{e}_Z \right ]
a'_{\pm \rm M}(\vec{k})
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img260.gif) |
(A.15) |
![\begin{displaymath}
\vec{B}^{\pm \rm M}_1(\vec{k})= {B_0 \over k^2}
\left \{ - k...
...cos \phi \right ]
\vec{e}_Z \right \}
a'_{\pm \rm M}(\vec{k})
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img261.gif) |
(A.16) |
and the perturbations of the Alfvénic mode are:
 |
(A.17) |
![\begin{displaymath}
\vec{v}^{\pm\rm A}_1(\vec{k})={- \sigma_{\rm A} c_{\rm A} \o...
...e}_Y + k_X \cos \phi \vec{e}_Z \right ]
a'_{\pm\rm A}(\vec{k})
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img263.gif) |
(A.18) |
![\begin{displaymath}
\vec{B}^{\pm\rm A}_1(\vec{k})= {B_0 \over k}
\left [ \left( ...
...}_Y + k_X \cos \phi \vec{e}_Z \right ]
a'_{\pm\rm A}(\vec{k}).
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img264.gif) |
(A.19) |
To simplify the expressions (A.14)-(A.19) we have
re-defined the dimensionless wave amplitude as
![\begin{displaymath}
a'_{\rm\pm M,A}(\vec{k}) =
{k \over \left [ \left( k_Y \sin...
...i \right) ^2 + k_X^2
\right ] ^{1/2}} a_{\rm\pm M,A}(\vec{k}).
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img265.gif) |
(A.20) |
The above expressions cannot be used for
parallel to
.
In fact, in such a case we have
kX=0,
,
and
;
thus, the denominator
of Eq. (A.20) would vanish.
In the particular case when
is parallel to
,
magnetosonic
and Alfvén
waves cannot be distinguished. So, we conventionally choose:
 |
(A.21) |
Thus, for
parallel to
the perturbations of
the magnetosonic mode are given by
 |
(A.22) |
 |
(A.23) |
 |
(A.24) |
while the perturbations of the Alfvénic mode are:
 |
(A.25) |
 |
(A.26) |
 |
(A.27) |
The discontinuity in the equilibrium density is located at the surface Z=0. The
values of the background density for Z>0 and Z<0 are indicated by
and
,
respectively. Some
physical quantity must be continuous across the discontinuity. In order to
find these continuity conditions, it is useful to re-write the cold plasma MHD
Eqs. (2)-(4) in the following conservative form:
 |
(A.28) |
 |
(A.29) |
 |
(A.30) |
Each component of Eqs. (A.28)-(A.30) can be written
in the form:
 |
(A.31) |
where f represents either the mass density or a component of the momentum
density or of the magnetic field, and
is the corresponding flux density.
We consider a cylindrical volume V located across the discontinuity surface
Z=0, with height
.
The bases of the cylinder, which are parallel to the
discontinuity surface, have area S and perimeter p. We integrate the
Eq. (A.31) over the volume V, transforming the volume integral of
the second term into a surface integral, by means of the divergence theorem.
Dividing by S we find:
 |
(A.32) |
where
is the average of f over the volume V;
is the the Z-component of the
flux
averaged over the bases S+ and S-, above and below
the surface Z=0, respectively; and
is average of
the flux over the lateral surface of the cylinder. In the limit of vanishing
,
assuming that all the
above averaged quantities remain finite, we obtain:
Since the basis S of the cylinder is arbitrary, the above equation implies
 |
(A.33) |
indicating that the flux density component
perpendicular to the
discontinuity is continuous across the discontinuity.
We apply the above argument both to the momentum Eq. (A.29) and to the magnetic field Eq. (A.30), and
we will examine the mass density equation later. The continuity of the
momentum flux gives the equations:
![\begin{displaymath}
\left [ \rho v_i v_Z - {B_i B_Z \over 4 \pi} \right ]^+ =
\l...
...[ \rho v_i v_Z - {B_i B_Z \over 4 \pi} \right ]^- , \;\; i=X,Y
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img289.gif) |
(A.34) |
![\begin{displaymath}
\left [ \rho v_Z^2 + {B_X^2 + B_Y^2 - B_Z^2 \over 8\pi}\righ...
...\rho v_Z^2 + {B_X^2 + B_Y^2 - B_Z^2 \over 8\pi}\right ]^-\cdot
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img290.gif) |
(A.35) |
The continuity of the magnetic field flux gives the equations:
![\begin{displaymath}
\left [ B_i v_Z - B_Z v_i \right ]^+ = \left [ B_i v_Z - B_Z v_i \right ]^-
, \;\; i=X,Y
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img291.gif) |
(A.36) |
while the third equation, corresponding to i=Z, is identically satisfied.
We decompose the quantities as a sum of a background equilibrium plus a
small amplitude perturbation, as in Eqs. (A.1). Taking
into account that the background magnetic field has been assumed continuous
across the discontinuity (
), at the first order
in the perturbation amplitude
Eqs. (A.34) and (A.36) give:
 |
(A.38) |
 |
(A.39) |
 |
(A.41) |
Being
,
Eqs. (A.38) and (A.39)
imply
and
This result, along with Eq. (A.37), indicates that the
perturbation magnetic field
is continuous across the
discontinuity.
The Eq. (A.41) can be written in the form
 |
(A.44) |
which is equivalent to
where
is
the unit vector lying in the YZ-plane and perpendicular to
(Fig. A.2). Since the velocity perturbation
is in the plane
perpendicular to
,
the conditions (A.40) and (A.44)
indicate that
is also continuous across the discontinuity.
Concerning the condition (A.43), it expresses
the fact that
is divergenceless across the discontinuity. However,
the Eq. (A.43) does not need to be explicitly used, since
it can be obtained as a direct consequence of the other continuity conditions.
To show this fact, we consider the Z-component of the induction
Eq. (4), which is linearized on both sides of the
discontinuity surface Z=0. Subtracting the two resulting equations we find
 |
(A.45) |
Since all the components of the velocity perturbation
are
continuous across the surface Z=0 for any value of X and Y, this
implies that all the space derivatives in the RHS of Eq. (A.45) are continuous, as well. Then, the RHS of Eq. (A.45) vanishes, giving
which implies
B1Z+(X,Y,t) = B1Z-(X,Y,t) + c(X,Y)
where c(X,Y) depends on the initial condition. If at the initial time
B1Z+ = B1Z-, then c(X,Y)=0, and the Eq. (A.43) follows. Thus, it is not necessary to impose the B1Z continuity condition (A.43) , because it will be
automatically satisfied once the continuity conditions on the perturbation
velocity will be imposed.
Finally, we consider the mass flux
across the discontinuity
surface Z=0. Since the equilibrium density
has different values
on the two sides, while v1Z is continuous, it is clear that the mass flux
is necessarily discontinuous:
.
Differences in the mass flux on the two sides of the discontinuity generate
density fluctuations localized within the width
of the discontinuity.
These density fluctuations are simply due to the undulations of the discontinuity
surface, generated by the component v1Z of the velocity perturbation.
The amplitude of
of such fluctuations can be estimated by
Eq. (A.32), finding
,
where
is
the wave period of the velocity perturbation. However, since in a cold plasma
the gradient pressure term has been dropped, these driven density variations
have no effects on the dynamics of the plasma.
In summary, the continuity conditions which will be imposed at the discontinuity
surface are the following:
B1X+(X,Y,Z=0,t) = B1X-(X,Y,Z=0,t)
|
(A.46) |
B1Y+(X,Y,Z=0,t) = B1Y-(X,Y,Z=0,t)
|
(A.47) |
v1X+(X,Y,Z=0,t) = v1X-(X,Y,Z=0,t)
|
(A.48) |
 |
(A.49) |
Within volumes smaller than the variation scale l0 of the
background structure, the perturbations considered by the WKB
theory can be approximated as a superposition of planar waves, each one
with a given wavevector (e.g., Malara et al. 2003).
Thus, within the region
we consider an Alfvén wave with a given
arbitrary wavevector, incident on the
discontinuity from the coronal side Z>0. At the discontinuity, four waves
are generated: an Alfvén wave and a fast magnetosonic wave, both
reflected back into the corona (Z>0); an Alfvén wave and a fast magnetosonic
wave, both transmitted to Z<0. We will indicate quantities relative to these
waves by the upper indices: "IA'' for the incident Alfvén wave; "RA'' and
"TA'' for the reflected and transmitted Alfvén waves; "RM'' and "TM''
for the reflected and transmitted magnetosonic waves. Each of the five waves involved in this
process have a given wavevector
and a corresponding frequency
determined by the dispersion relations (A.3) and (A.4), where the index
.
The velocity and
magnetic field perturbations associated with such waves have the form given by
Eq. (A.6):
![\begin{displaymath}
\vec{v}_1^\beta (\vec{X},t) = \vec{v}_1^\beta
\exp \left [ ...
...t(\vec{k}^\beta \cdot \vec{X} - \omega^\beta t\right) \right ]
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img312.gif) |
(A.50) |
The total perturbation above
and below the discontinuity is a superposition of the perturbed fields associated with
the single waves. Then, the continuity condition (A.46) takes the form
![$\displaystyle B_{1X}^{\rm IA} \exp \left [ {\rm i} \left(k_X^{\rm IA}X + k_Y^{\...
...rm i} \left(k_X^{\rm RM}X + k_Y^{\rm RM}Y - \omega^{\rm RM} t\right)
\right ] =$](/articles/aa/full/2005/45/aa2272-04/img313.gif) |
|
|
|
![$\displaystyle B_{1X}^{\rm TA} \exp \left [ {\rm i} \left(k_X^{\rm TA}X + k_Y^{\...
...rm i} \left(k_X^{\rm TM}X + k_Y^{\rm TM}Y - \omega^{\rm TM} t\right) \right ] .$](/articles/aa/full/2005/45/aa2272-04/img314.gif) |
|
|
(A.51) |
This equation must hold for any value of X, Y and t, which implies that the X and Y components of all wavevectors, as well as the frequencies, must
be equal:
 |
(A.52) |
 |
(A.53) |
 |
(A.54) |
while the components
of the five wavevectors are not necessarily
equal. The same results (A.52)-(A.54) are obtained
considering the
other continuity conditions (A.47)-(A.49).
The components
of the wavevectors of reflected and transmitted
waves can be calculated in terms of the wavevector
of the
incident wave using the dispersion relations (A.3) and (A.4) and the Eqs. (A.52)-(A.54).
Let us consider the single waves:
The equation
,
along with the dispersion relation (A.3) of Alfvén waves, gives the equation
 |
(A.55) |
where
and
determine the
propagation direction of the incident and of the reflected Alfvén wave,
with respect to the direction of the equilibrium magnetic field. Since the
incident wave propagates downward, while the reflected wave propagates
upward, there
are two possible situations: (i) if B0Z>0 the incident and the reflected
Alfvén waves propagate antiparallel and parallel to
,
respectively;
thus
and
;
(ii) if B0Z<0, then
and
.
The condition B0Z=0 can be
verified only along lines or isolated points in the Z=0 plane, and it will not
be considered. In summary, we have
 |
(A.56) |
where
is the sign of
B0Z. Inserting this relations into the Eq. (A.55) we find the third component of the wavevector of the
reflected Alfvén wave
 |
(A.57) |
In a similar way, the equation
and the dispersion relation (A.3) give the equation
 |
(A.58) |
In this case the incident and the transmitted Alfvén waves both propagate
downward. Then, the relation between the sign of B0Z and the
propagation directions gives
 |
(A.59) |
Inserting this relation into the Eq. (A.58) we find
 |
(A.60) |
where
 |
(A.61) |
is the ratio between the Alfvén velocities on the two sides of the discontinuity.
In the considered configuration, we have
.
The equation
and the dispersion relations
(A.3) and (A.4) give the equation
 |
(A.62) |
Since both B0 and
are positive quantities, from
this equation it follows that
 |
(A.63) |
Moreover, from Eq. (A.62) we derive:
 |
(A.64) |
This equation indicates that
can be either real or imaginary,
according to the sign of the RHS.
In the former case the reflected magnetosonic wave is propagating, while
in the latter case it is evanescent along the Z direction.
Let us suppose first that the reflected magnetosonic wave is propagating
(real
). In this case, the sign of
can be determined in the
following way.
The propagation direction of the reflected magnetosonic wave (parallel or antiparallel
to
)
is determined by
.
Since the reflected
magnetosonic wave propagates upward,
depends on the orientation
of
.
In particular,
 |
(A.65) |
This condition and the Eq. (A.63) determine the sign of
as a function of the incident wave:
 |
(A.66) |
Then, from Eq. (A.64) the following expression for the Z-component
of the wavevector of propagating reflected magnetosonic wave is obtained:
![\begin{displaymath}
k_Z^{\rm RM}=
\sigma_{\rm IA} {\rm sgn} \left(\vec{k}^{\rm ...
...2 k_Y k_Z^{\rm IA} \sin \phi \cos \phi - k_X^2 \right ]^{1/2}.
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img347.gif) |
(A.67) |
If the sign of the RHS of Eq. (A.64) is negative, then the reflected
magnetosonic wave is evanescent (imaginary
). In this case
we write
,
and from Eq. (A.64) we find
 |
(A.68) |
Since the wave amplitude must not diverge for
,
then
must be positive:
![\begin{displaymath}
\gamma^{\rm RM}=
\left [ \left( {k_Y^2 - k_Z^{\rm IA}}^2 \ri...
...2 k_Y k_Z^{\rm IA} \sin \phi \cos \phi + k_X^2 \right ]^{1/2}.
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img352.gif) |
(A.69) |
It is also useful to obtain a simple expression for the modulus of the wavevector
of the reflected magnetosonic wave.
From Eq. (A.62), we get
Taking the absolute value of this equation, and using the fact that
,
we obtain the expression for
:
 |
(A.70) |
Note that the Eq. (A.70) is valid both for non-evanescent
and for evanescent magnetosonic wave, since in both cases
is a
real positive quantity.
The condition
and the dispersion relations (A.3) and (A.4) give the equation
 |
(A.71) |
From this equation, it follows that
 |
(A.72) |
Moreover, from Eq. (A.71) we derive
 |
(A.73) |
Also in this case, the transmitted magnetosonic wave can be either
propagating or evanescent along Z, according to the sign of the RHS of
Eq. (A.73).
Considering first a propagating transmitted wave, the propagation direction
(parallel or antiparallel to
)
is determined
by
.
Since the transmitted magnetosonic wave propagates downward,
the sign
is related to the orientation of
by
This condition and the Eq. (A.72) determine the sign of
:
 |
(A.74) |
Then, the expression for the z component of the wavevector of the transmitted
magnetosonic wave, as a function of the incident wave, follows from
Eq. (A.73):
![\begin{displaymath}
k_Z^{\rm TM} = - \sigma_{\rm IA} {\rm sgn} \left(\vec{k}^{\r...
...\rm IA} \sin \phi \cos \phi \over r^2} - k_X^2 \right ]^{1/2}.
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img366.gif) |
(A.75) |
In the case of evanescent transmitted magnetosonic wave (negative RHS of Eq. (A.73)), we write
,
and from Eq. (A.73) we find
 |
(A.76) |
Since the wave amplitude must not diverge for
,
then
must be negative. Thus
![\begin{displaymath}
\gamma^{\rm TM}=
- \left [ k_Y^2 \left( 1 - {\cos^2 \phi \ov...
...\rm IA} \sin \phi \cos \phi \over r^2} + k_X^2 \right ]^{1/2}.
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img371.gif) |
(A.77) |
In a similar way as for the reflected wave, a simple expression for the modulus
of the wavevector
of the transmitted
magnetosonic wave can be obtained. From Eq. (A.71) we get
Taking the absolute value of the above equation and using the fact that
, we obtain
 |
(A.78) |
This equation is valid both for non-evanescent and for evanescent magnetosonic
wave, since in both cases
is a real positive quantity.
The amplitudes of the waves involved in the reflection process can be determined
by the continuity conditions at the surface Z=0. The continuity of B1X is
expressed by Eq. (A.51); using the conditions (A.52)-(A.54) we obtain the equation
Using the expressions (A.16) and (A.19) for the perturbation magnetic
field, we obtain:
where the apostrophe and the dependence on the wavevector have been dropped
from the wave amplitudes
.
In a similar way, the continuity of
B1Y at the surface Z=0 (Eq. (A.47)), along with the
expressions (A.16) and (A.19) for the perturbation magnetic field,
give the equation
|
|
 |
|
|
|
 |
(A.80) |
Finally, using the continuity of the velocity perturbation
(Eqs. (A.48) and (A.49)), the expressions
(A.15) and (A.18) of the perturbation velocity, and the relations
(A.56), (A.59), (A.63), and (A.72),
we find the equations
|
|
 |
|
|
|
 |
(A.81) |
and
 |
|
|
|
 |
|
|
(A.82) |
The four Eqs. (A.79)-(A.82) allow to determine the
ratios
,
,
,
and
of the reflected and transmitted wave amplitudes to the
incident wave amplitude. These ratios determine the reflection and transmission
coefficients of the various waves. Knowing the wavevector of the incident wave,
these amplitude ratios are calculated by numerically solving
the linear system (A.79)-(A.82). In particular, the
Z-components and the
modulus of wavectors of transmitted and reflected waves which appear
in Eqs. (A.79)-(A.82) are determined by the
relations (A.57), (A.60), (A.67), (A.75)
(or (A.69), (A.77) for evanescent magnetosonic waves),
(A.70), and (A.78) in terms of the wavevector of the incident wave.
The system (A.79)-(A.82) is valid both for propagating and for
evanescent magnetosonic waves. In the latter case, the component
and/or
are imaginary. Thus, in the general case the coefficients of
Eqs. (A.79)-(A.82) are complex, and the ratios
,
,
, and
are complex quantities, as well.
In a cold plasma, where the internal energy is neglected, the energy density
is given by
 |
(A.83) |
We decompose physical quantities as a sum of the background and perturbation
value, according to Eq. (A.1). Thus, the energy density is
divided as a term due to the background structure plus the energy density
due to fluctuations. The latter has the following form:
 |
(A.84) |
where terms of order
have been neglected. According to
Eq. (A.6),
the velocity and magnetic field perturbations are written as a sum of
contributions due to the different propagating modes and expanded in
Fourier integrals:
|
|
![$\displaystyle \vec{v}_1(\vec{x},t)=\sum_\alpha \int {{\rm d}\vec{k} \over (2\pi...
... i} \left [ \vec{k} \cdot \vec{x} -
\omega^\alpha (\vec{k})t \right ] \right \}$](/articles/aa/full/2005/45/aa2272-04/img396.gif) |
|
|
|
![$\displaystyle \vec{B}_1(\vec{x},t)=\sum_\alpha \int {{\rm d}\vec{k} \over (2\pi...
...i} \left [ \vec{k} \cdot \vec{x} - \omega^\alpha (\vec{k})t \right ] \right \}.$](/articles/aa/full/2005/45/aa2272-04/img397.gif) |
(A.85) |
Considering, for instance, the velocity perturbation, we require that it is
a real quantity:
.
Using the expression (A.85), this condition implies
The dispersion relations of Alfvénic and magnetosonic modes give the
following symmetry properties:
 |
(A.87) |
Using these relations, from Eq. (A.86) we derive the following
reality conditions for the Fourier amplitudes of velocity perturbation of the
different modes:
 |
(A.88) |
The same reality conditions hold also for the Fourier amplitudes of magnetic
field perturbation.
Inserting the Fourier expansions (A.85) into Eq. (A.84),
the fluctuation energy density is written in the form
|
|
![$\displaystyle \epsilon_{\rm f}(\vec{x},t)={\varepsilon \over 4\pi} \vec{B}_0 \c...
...t [ {\rm i} (\vec{k}^\alpha \cdot \vec{x} - \omega^\alpha (\vec{k}) t) \right ]$](/articles/aa/full/2005/45/aa2272-04/img404.gif) |
|
|
|
![$\displaystyle \qquad+ \varepsilon^2 \sum_{\alpha ,\beta}
\int {{\rm d}\vec{k} \...
...t(\omega^\alpha (\vec{k}) + \omega^\beta (\vec{k}')\right)t \right ] \right \}.$](/articles/aa/full/2005/45/aa2272-04/img405.gif) |
(A.89) |
Assuming that the perturbation has a duration time T, we calculate
the time average of the total fluctuation energy
 |
(A.90) |
Using the relations
 |
(A.91) |
and the condition that fluctuating fields have a vanishing space average,
we see that the first term of Eq. (A.89) is averaged out. Then,
the average fluctuation energy can be written in the form
![\begin{displaymath}
E_{\rm f} = \varepsilon^2 \sum_{\alpha ,\beta} \int {{\rm d...
... ]
\delta_{\omega^\alpha (\vec{k}), -\omega^\beta (-\vec{k})}.
\end{displaymath}](/articles/aa/full/2005/45/aa2272-04/img408.gif) |
(A.92) |
At any given
,
the frequencies
and
of two modes are equal only when
.
Using
this fact and the relations (A.87) we can express the average fluctuating
energy in the form
|
|
 |
|
|
|
![$\displaystyle \left.\qquad +{1 \over 8\pi} \left(\vec{B}_1^{\rm +A}(\vec{k}) \c...
...vec{B}_1^{\rm +M}(\vec{k}) \cdot \vec{B}_1^{\rm -M}(-\vec{k})
\right) \right ].$](/articles/aa/full/2005/45/aa2272-04/img413.gif) |
(A.93) |
Finally, using the reality conditions (A.88) for the velocity perturbations
and the analogue for the magnetic field perturbations in Eq. (A.93) we find the time-averaged fluctuation energy
 |
(A.94) |
where
 |
(A.95) |
is the perturbation energy density in the
-space of the
-th
mode at wavevector
.
The energy of the modes involved in the reflection process can be calculated
as a function of the amplitude
and of the wavevector. Considering
first the incident Alfvén waves, we insert the expressions (A.18) and (A.19) of the velocity and magnetic field perturbation into
Eq. (A.95), obtaining the energy density of the incident
Alfvén wave:
 |
(A.96) |
In a similar way, the energy density of the reflected and transmitted Alfvén waves
is
 |
(A.97) |
The energy density of the reflected and transmitted magnetosonic
waves are obtained inserting the expressions (A.15) and (A.16) of
the velocity and magnetic field perturbation into Eq. (A.95):
Thus, the ratios between the reflected and transmitted wave energy density and the
incident wave energy density are
 |
(A.99) |
and
Such ratios depend on the amplitude ratios and on the wavevectors of
the involved waves.
The energy flux associated with a wave at a given wavevector
,
belonging to the
-th mode can be defined as
 |
(A.101) |
where
is the group velocity of the
-th mode.
In particular:
 |
(A.102) |
The reflection coefficient for the Alfvén waves is given by the ratio
 |
(A.103) |
The transmission coefficient for the Alfvén waves is given by:
 |
(A.104) |
The reflection and transmission coefficients for the magnetosonic waves are
 |
(A.105) |
and
 |
(A.106) |
respectively. The energy density ratios in the above equations are given by
the relations (A.99) and (A.100).
The Eq. (A.101) holds only in the case of a
non evanescent wave. If a wave is evanescent along a given direction,
then the corresponding component of the energy flux
is
vanishing. Thus, when the reflected and/or transmitted magnetosonic wave
is evanescent (along z), then the reflection coefficient
and/or
the transmission coefficient
will be set equal to zero.
Copyright ESO 2005