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Astron. Astrophys. 354, 334-348 (2000)
3. Diverging magnetic field and no gravitational stratification: no dissipation
To understand each physical effect clearly, we first examine the
effect of a diverging field on the phase mixing of Alfvén
waves. Hence, we eliminate gravitational effects by setting the
pressure scale height to infinity, i.e.
. In the rest of this paper, we
assume that the plasma is structured in the
-direction. When referring to
stratification, we mean radial stratification due to gravity while a
diverging atmosphere refers to the area change due to spherical
geometry.
If we neglect dissipation, i.e. ,
the solution for the magnetic field and velocity perturbations are
given by
![[EQUATION]](img60.gif)
and
![[EQUATION]](img61.gif)
where and J is a Bessel
function of order either or
.
The ideal MHD solutions in spherical coordinates may be
approximated by a simple WKB solution (see Appendix) of the form
![[EQUATION]](img65.gif)
where . Although the exact
analytical solutions for the perturbed magnetic field and velocity
differ, the approximate WKB solutions are both the same since the
leading asymptotic terms of the Bessel functions agree on using the
large argument approximation and
(Abramowitz & Stegun). The
solutions (15) and (16) show that area divergence decreases the
wavelength in agreement with the numerical solutions shown in
Figs. 1 and 2. Figs. 2 (a) and 2(b) confirm that the
solutions for the perturbed magnetic field and the perturbed velocity
are the same as predicted by Eq. (17).
When , Eq. (14) becomes the
standard wave equation for phase mixing in a Cartesian,
non-dissipative system, i.e.
![[EQUATION]](img70.gif)
![[FIGURE]](img77.gif) |
Fig. 1. (left) A contour plot of the perturbed magnetic field in a diverging atmosphere, with . (right) The behaviour of the wavelength with height (at ) with . The solid line is the solution for a diverging atmosphere, the dashed line corresponds to the Cartesian case.
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![[FIGURE]](img97.gif) |
Fig. 2. A cross-section of the perturbed a magnetic field and b velocity for a radially diverging background magnetic field at with , and . c A cross-section of the perturbed magnetic field at with , and .The dotted lines represent the corresponding solutions in Cartesian coordinates at .
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Therefore, the solutions for the perturbed velocity and magnetic
field in Cartesian coordinates are given by
![[EQUATION]](img99.gif)
where .
From Fig. 2 (c) we see that low down, i.e. near
, and for small initial wavelengths
(see Eq. (6)), the spherical and the
Cartesian case indeed agree extremely well. The results in Fig. 1
show clearly that, unlike gravitational stratification which lengthens
the wavelengths, area divergence shortens the wavelengths while the
wavelengths remain constant in the Cartesian case. Indeed, in this
case the Alfvén speed and the wavelength
behave like
. The amplitude of both the perturbed
magnetic field and the perturbed velocity are constant in height as we
expected. Wright & Garman (1998) and Torkelsson & Boynton
(1998) showed that in the large wavenumber limit the amplitudes of the
Alfvén waves behave as and
and as
is constant with height in this
case, the result follows. This suggests that phase mixing will be more
efficient in a diverging medium than in a non diverging medium as the
short length scales, necessary for efficient dissipation, will be
created much faster. Therefore, heat could now be deposited at lower
heights. Similar results were obtained by Ruderman et al. (1998).
These results also show that, unlike the results due to
stratification, the effect of the divergence of the background field
is the same whether resistive or viscous heating is considered.
To obtain an estimate of where this heat would be deposited if
dissipation were included, we now consider the current density,
. In spherical coordinates, only
including the dominant terms, is
given by
![[EQUATION]](img107.gif)
while in Cartesian coordinates,
is given by
![[EQUATION]](img108.gif)
where and
. The numerical results obtained for
both the magnetic field and the velocity indicate that the behaviour
of the current density and the vorticity will be similar. Therefore we
concentrate on the current density alone.
From Fig. 3 (a) we see that even when there is no phase
mixing, i.e. , the current density
builds up very rapidly in a diverging magnetic field. In the
non-diverging atmosphere, i.e. the Cartesian case, this current
density remains constant as gradients in the vertical or horizontal
direction do not build up. As there is no phase mixing, the growth in
in the diverging atmosphere is
entirely due to the flaring out of the background field lines, i.e.
the radial derivatives are building up. When we include phase mixing,
i.e. , we see from Fig. 3 (b)
that the build up of the current density,
, increases when
increases. However, when the
magnetic field is diverging, the effect of increasing
is larger than in the Cartesian
case. Indeed, from comparing Eqs. (17) and (19), we see that a change
in causes a bigger change in the
spherical -derivatives than in the
Cartesian x-derivatives. At
both the spherical r-derivatives and the Cartesian
z-derivatives stay the same when the phase mixing parameter
changes.
![[FIGURE]](img130.gif) |
Fig. 3. A cross-section of the current density for a diverging background magnetic field with and at for a b different values of (solid line: , dashed line: ) The thin lines represent the corresponding solutions in Cartesian coordinates at .
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Fig. 4 (a) shows the change in the current density
when we change the initial
wavelength , through changes to
either the frequency or the
background Alfvén velocity .
Since (Eq. (6)), doubling
has the same effect as doubling
or halving
and so on. Therefore we concentrate
on the effect of varying just the one parameter
. Changing
also causes
to change as
. We notice that
starts off with a higher initial
value when the initial wavelength is
smaller, which is clear from Eqs. (17) and (19). We still see that the
current density builds up faster in
the spherical case and even that the difference between the spherical
and the Cartesian case is more pronounced as the initial wavelength
gets smaller. From Eqs. (17) and
(19) we see that all derivatives, apart from the Cartesian
z-derivative, contain a
factor. From Eq. (21), we see that, in the Cartesian case, the
dimesionless variables introduce a factor
in front of the square of the
z-derivative. These means that in both the spherical and
Cartesian case, which implies that
the difference between the spherical and Cartesian results gets 4
times bigger when we halve the initial wavelength
. From Fig. 4 (b) we see that
the current density builds up higher and that the difference between
the spherical and the Cartesian case gets larger as we decrease the
value of the parameter . However,
from Eqs. (17) and (19) it is clear that all the derivatives remain
unchanged when we vary . The
parameter only appears in
expression (20) and (21) for the current density
. We see that the spherical
-derivatives and Cartesian
x-derivatives are multiplied with a factor
or that these derivatives will
become 4 times larger when we halve
. Again this implies that the
difference between the spherical and Cartesian
will get larger as
gets smaller.
![[FIGURE]](img170.gif) |
Fig. 4. A cross-section of the current density for a diverging background magnetic field with at for a and different values of (solid line: , dot-dashed line: , dashed line: ) and b and different values of (solid line: , dot-dashed line: , dashed line: ). The thin lines represent the corresponding solutions in Cartesian coordinates at .
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From studying the non-dissipative case, we expect phase mixing to
be enhanced when we increase the phase mixing parameter
or decrease either the intial
wavelength or the parameter
. We also expect more heat to be
deposited into the plasma when we consider a diverging magnetic field
compared to the Heyvaerts and Priest model in Cartesian coordinates.
We now want to examine if these effects remain the same when we
include dissipation in the system.
© European Southern Observatory (ESO) 2000
Online publication: January 31, 2000
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