A&A 455, 361370 (2006)
DOI: 10.1051/00046361:20064926
Wave propagation in incompressible MHD wave guides: the twisted magnetic Annulus^{}
R. Erdélyi  B. K. Carter
Solar Physics & upperAtmosphere Research Group, Department of Applied Mathematics,
University of Sheffield, The Hicks Building, Hounsfield Road, Sheffield S3 7RH,
UK
Received 27 January 2006 / Accepted 2 May 2006
Abstract
The propagation of MHD waves in a structured
magnetic flux tube embedded within a straight vertical magnetic
environment is studied analytically. The motivation of this
analysis comes from the observations of damped loop oscillations
showing that possibly only part of the loop is homogeneous in the
radial direction. The general dispersion relation of
longitudinal wave propagation is derived for a fully magnetically
twisted configuration consisting of a core, annulus and external
region each with magnetic field of uniform, yet distinct, twist.
Next, a simplified case representing coronal loops is analysed in
detail considering magnetic twist just in the annulus, the
internal and external regions having straight magnetic field.
Modes of oscillations are examined from the general dispersion
relation that is suitable for obtaining information on not just
oscillations but also on some instability properties of this
complex tube structure.
It is shown that there are purely surface
(i.e. evanescent) and hybrid (spatially oscillatory in the twisted
annulus, otherwise evanescent) modes. Except for small
wavenumbers, the surface waves show little dispersion; a property
making them more suitable for observations. The hybrid modes show
a more complex character. Though the frequency range seems to be
rather limited, there is a continuum band of frequencies for a
given wavenumber. Both short and long wavelength approximations
are considered for the symmetrical (sausage) mode and with small
twist in particular.
Key words: magnetohydrodynamics (MHD) 
waves  Sun: oscillations  Sun: magnetic fields
The solar corona is a highly structured and dynamic environment
that has captured major interest for many years. The launch of
such recent missions as the SOlar and Heliospheric Observatory
(SOHO) and Transition Region And Coronal Explorer (TRACE) has
delivered a wealth of high cadence data that has allowed a
multitude of studies to occur (see e.g. Aschwanden 2004, for an extensive
review on observations). A "hot'' topic of study has been
that of coronal loop oscillations; their cause, their damping, and
their role and effect in the age old problem of coronal heating.
For a review of oscillations see e.g. Aschwanden (2003), Roberts (2000), Nakariakov & Roberts (2003)
and Wang (2004). The data obtained has allowed the identification of many
modes of oscillation including transverse (kink) oscillations
(see Nakariakov et al. 1999; Ofman & Aschwanden 2002; Aschwanden et al. 1999). Standing and propagating kink
modes in loops was possibly one of the first successful
observations using these high resolution space instruments and the
detailed measurements of their amplitude, period, damping times
and loop parameters were derived from data. The work and theories
developed in this field have led to the evolution of the subjects
of coronal seismology (see Roberts 2000; Roberts et al. 1984) and of
atmospheric seismology (see Erdélyi 2006; de Pontieu et al. 2005,2004). Within
the solar corona, one specific area of study is that of the
oscillations that are observed to occur in plasma magnetically
confined to specific configurations, namely coronal loops or flux
tubes. Such flux tubes have been studied in great detail by, among
others, Edwin & Roberts (1983). They find that the existence of
inhomogeneities in the form of structuring of the magnetic field
enables loops to act as wave guides for a variety of different
modes. The typical speeds arising are linked closely to the local
Alfvén speed ,
sound speed
and tube speed .
An
additional feature of the flux tube is that of twist. Twist is now
an accepted concept tied to the existence of coronal loops.
Aschwanden (2004) discusses observed noncoplanarity of loops which
implies kinked field lines which, in turn, are intimately related
to twist. Rotational movement along a loop (as observed by
e.g. Chae et al. 2000) also indicates the existence of twist. Besides, it
is unfeasible that every flux tube within the solar atmosphere is
entirely twist free in spite of the random continuous motions
observed at the footpoints. Twisted tubes have been studied in the
past by, among others, Dungey & Loughhead (1954),
(Goedbloed 1971ac, 1983) Parker (1974), Bogdan (1984),
Goossens et al. (1995), Klimchuk et al. (2000) but were mainly concerned with
stability aspects, the general classification of the MHD spectra
or loop geometries. Bennett et al. (1999), however, investigated MHD wave
propagation in the case of a uniformly twisted
incompressible magnetic flux tube and extended the study of
Edwin & Roberts (1983) by analysing the specific modes that occur in such
flux tubes. They found that twist introduces an infinite band of
body modes occurring as torsional, or rotational, modes with phase
speeds
close to the longitudinal Alfvén speed of the loop
.
Another topic of much discussion is that of wave damping. All
oscillations in the corona are subject to some sort of damping.
There are many hypothesised mechanisms by which damping can be
explained; sound wave radiation (Stenuit et al. 1999,1998), energy
absorption due to anomalous viscosity (Nakariakov et al. 1999) and resonant
absorption in a thin boundary layer (Van Doorsselaere et al. 2004; Ruderman & Roberts 2002; Aschwanden et al. 2003). In
particular, Ruderman & Roberts (2002) found that observed damping times would
imply an inhomogeneous layer of between 545% of the tube's
radius. These studies led us (Carter 2005; Carter & Erdélyi 2004) and Mikhalyaev & Solov'ev (2005)
to the concept of MHD wave studies in a flux tube model consisting
of a cord surrounded by a magnetic shell. Carter & Erdélyi (2004) and
Carter (2005) investigated the case of a straight magnetic core
and external field and a twisted magnetic annulus. The general
dispersion relation was found and cases of long and short
wavelength briefly considered. In this paper we give a more
comprehensive analysis and consider some specific applications to
the solar atmosphere. Mikhalyaev & Solov'ev (2005), who considered the case of a
straight magnetic flux tube consisting of a core, shell and an
external magnetic field of different strengths, analysed the
specific modes supported by this configuration and found a variety
of modes occurring. Two slow and two fast modes were found
supported by the cord (oscillating in phase) and the shell
(oscillations with opposite phase). In the present work we extend
the current ideas to a magnetic configuration consisting of a
magnetic core, shell and external ambient field all with an
applied twist. We model the predicted inhomogeneous layer as a
twisted annulus embedded within a straight magnetic field. We
neglect the effect of gravity, the emphasis being on the role of
the magnetic field. We find the general dispersion relation for
the case of twist in each region and for the case of a straight
core and external field for degenerated magnetosonic waves (of
course, in incompressible plasma, the sound speed is infinite and
appropriate limits must be taken as it will be shown in the
paper). The general dispersion relation is solved analytically for
the sausage (m=0) case and in the thin tube (long wavelength) and
short wavelength approximations considering further thin annulus
and small twist approximation. This configuration can support
various modes. For incompressible sausage modes it is found that
only surface waves are supported in the internal and external
regions with both body (i.e. hybrid) and surface modes occurring
in the twisted annulus. It is the absence of twist combined with
the assumption of incompressibility that restricts to solely
surface modes.
Let us restrict the investigation to an incompressible plasma
for which fast waves are removed from the system and Alfvén
and slow waves merge. The modes of oscillation in a twisted
magnetic annulus with straight internal and external magnetic
field will be studied. To do so we first suppose, in cylindrical
polar coordinates, magnetic field of the form

(1) 
embedded in an incompressible plasma. We take the plasma
density
as uniform and the field and plasma pressure p(r)structured in the radial direction so as to satisfy the pressure
balance

(2) 
Here
is the magnetic permeability and
denotes the
equilibrium magnetic field strength.
Let b denote a perturbation in the
magnetic field. The MHD equations, linearised about an equilibrium
given by
for an incompressible plasma,
as found for example in Bennett et al. (1998) can then be
written:
where
is the Lagrangian displacement vector and

(6) 
is the total pressure perturbation for kinetic plasma
pressure p and magnetic field perturbation .
The
magnetic perturbation can then be eliminated from
Eqs. (3), (4) and we can rewrite
Eqs. (3)(6) in component form for the
Lagrangian displacement:
where

(10) 
and the operator
is

(11) 
Let us Fourier analyse the system

(12) 
where m
is the azimuthal
order of the mode, k_{z} is the longitudinal wavenumber, and
is the frequency of the mode. It is now possible to
combine Eqs. (7)(10)
and return two first order coupled differential equations for,
say, the radial component of the Lagrangian displacement and total pressure perturbation :



(13) 



(14) 
where
D 
= 


C_{1} 
= 


C_{2} 
= 


C_{3} 
= 


for which
is the Alfvén frequency defined as
Equations (13) and (14) are a special
(incompressible) case of the compressible equations found by,
amongst others, Appert et al. (1974), Goedbloed (1971a) and Hain & Lüst (1958).
It is now possible to eliminate
from
Eqs. (13) and (14) to leave a second
order differential equation solely in terms of :



(15) 
In general Eq. (15) is a variable coefficient ODE
that can only be solved analytically under special circumstances.
For simplicity we consider the
case for which all three regions have constant longitudinal field
and uniform twist such that
and B_{z}(r) are
constant. We consider such a field and using the notation

(16) 
where
and
are constant. The
densities of the internal, annulus and external regions are
denoted as
and
,
respectively. This
configuration reduces the Alfvén frequency
and the
coefficients D, rC_{1} and C_{3} to constants:
With this simplification one can recover a second order
differential equation in :

(19) 
where

(20) 
in which 0 subscripts (which correspond to the annulus) are
replaced by i, e corresponding to the internal and external
regions, respectively. Equation
is the
modified Bessel's equation with solutions the modified Bessel
functions I_{m}(m_{0}r), K_{m}(m_{0}r), of the first and second kind,
respectively. For m_{0}^{2}<0, Eq. (19) becomes
Bessel's equation with solutions the Bessel functions of the first
and second kind, J_{m}(m_{0}r) and Y_{m}(m_{0}r). Appendix A contains
the full dispersion relation for this general fully twisted
configuration for arbitrary poloidal wavenumber, m. Included is
the detailed analysis of reducing this dispersion relation to the
dispersion relation for the simpler case of a straight magnetic
internal and external region while maintaining the uniform twist
within the annulus.

Figure 1:
The equilibrium configuration for a straight magnetic core and a
twisted magnetic annulus embedded in a straight ambient magnetic field. 
Open with DEXTER 
We consider, specifically, a flux tube consisting of
a twisted magnetic annulus embedded in a straight magnetic field,
as in Fig. 1 given by:

(21) 
in which the constants ,
from configuration (16) are set to 0. For the configuration described by
Eqs. (21), the governing Eq. (19) has solutions like
and
,
the modified Bessel functions of order m. We
require that
is finite at r=0 giving

(22) 
and we require that the solutions are evanescent as
giving

(23) 
for ,
arbitrary constants. Since the
magnetic field has no twist in these two regions we notice that
and
reduce to  k_{z} , meaning
and
are always positive. However, within the annulus m_{0}may be real or purely imaginary. For frequencies that satisfy



(24) 
m_{0}^{2} is indeed negative and so body modes (J_{m}, Y_{m}solutions of the Bessel equation) are found to occur. In this
latter case it is convenient to write
n_{0}^{2} =  m_{0}^{2} > 0,

(25) 
and then the solution to Eq. (19) within the
annulus may be written in general as

Figure 2:
Body and surface waves determined by the value of m_{0}^{2} in each
region. The modified Bessel functions I_{m} and K_{m} for
m_{0}^{2}>0 giving surface waves and Bessel's equation when
m_{0}^{2}<0 with solutions the Bessel functions, J_{m}, Y_{m} of the
first and second kind resulting in body waves. 
Open with DEXTER 
This allows two different mode configurations to occur. The
case of m_{0}^{2}>0 results in the waves everywhere being evanescent
surface waves whereas in the latter case, m_{0}^{2}<0, we find the
solution is in the form of hybrid modes; surface in the internal
and external regions and body in the annulus. These are shown
schematically in
Fig. 2.
For the boundary conditions we require the conservation of
total Lagrangian pressure across the perturbed boundaries and that
the normal perturbation
remains continuous across the
boundaries r=a and r=R:

(26) 

(27) 
These boundary conditions Eqs. (26) and (27), when applied to solutions for
after
some cumbersome algebra and eliminating ,
yield the general
dispersion relation for the more general configuration (i.e.
twisted core, annulus and external field) defined in
Eq. (16). Appendix A gives the full general dispersion
relation for a uniform, yet distinct, twist in all three regions,
namely the core, the annulus and the external magnetic field. It
also outlines the steps involved in reducing this dispersion
relation to a more specific case by setting
and
,
i.e. no twist in the core and ambient external region. It is
possible to express this dispersion relation in terms of the
dispersion relation found by Bennett et al. (1999). Appendix B shows the
dispersion relation Eq. (28) written as the
dispersion relation from Bennett et al. (1999) plus terms due to the
annulus. We obtain the dispersion relation for a uniformly twisted
annulus embedded in a straight magnetic field:



(28a) 
for purely surface waves, and



(28b) 
for hybrid (surfacebody) waves. Here
where X denotes the corresponding Bessel function J, Y or
modified Bessel function I, K, and
is replaced by a,Rfor the corresponding region,
where the dash ' denotes the derivative with respect to the
argument of the Bessel functions. The Alfvén frequencies in
the annulus, internal and external regions are given by
,
and
,
respectively such
that
Equations
b) are the general
dispersion relations for an incompressible twisted magnetic
annulus embedded in an untwisted environment. In the limit of no
core, when
,
we recover the dispersion relation
from Bennett et al. (1999) for the case of a uniformly twisted
incompressible magnetic tube, and further, setting A_{0}=0, we
recover the case for a straight magnetic tube as in Edwin & Roberts (1983).
Let us now consider three specific cases to study the array of
modes given by Eqs. (28) and (28). We examine the effect of varying annulus
width and the effect of applying different degrees of twist on the
propagating sausage (m=0) modes.

Figure 3:
Phase velocity of the m=0 sausage modes determined from
Eqs. (28), (28) for a twisted
annulus embedded in a straight internal,
,
and external,
,
magnetic field with a/R=0.80,
,
.
Shown are plots for increasing twists
,
0.1, 0.2, 0.4 v's normalised phase speed
.
Also shown is the envelope separating the hybrid
(dotted) modes, as an infinite band of eigensolutions in the
neighbourhood of
,
and the surface modes. 
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Figure 4:
Same as Fig. 3 but with fixed twist
.
Shown are plots for annulus width 5%, 10%,
20% and 40% of the tubes radius. 
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Photospheric phenomena such as sunspots, plage regions,
largescale unipolar areas and supergranulation fields at
supergranule boundaries all contain a much higher concentration of
magnetic field within the tubes than in the surrounding atmosphere
(Aschwanden 2004). Figures 3 and 4 show the solutions of the dispersion
relations of the hybrid and surface modes under expected typical
photospheric tube conditions. (Note, however, that stratification
effects have been ignored which could be relevant under
photospheric conditions.) We model numerically the conditions as a
magnetic intensity enhancement within the tube with the
longitudinal magnetic field strength in the internal and annulus
regions being 10 or 20 times that of the external field. Strictly
speaking the ambient field for a photospheric tube should be
considered vanishing, but the current choice of parameters will
give a very good approximation. It also helps to keep the
numerical complications to a minimum. When the magnetic twist is
increased, the hybrid modes cover a wider range of phase speeds,
centered around the annulus' longitudinal Alfvén speed
.
Notice the avoided crossing feature in the two modes
with phase speeds between
and 0.1 in e.g.
Fig. 4. They appear more pronounced as
twist is made large. When the annulus width is increased, the
infinite band of hybrid modes broadens slightly but seems to have
no other significant effect. Also plotted, in
Fig. 5, is the effect of increasing the
twist on the phase speed of the surface mode seen in
Fig. 3 occurring with phase speed
0.1. Parameters are taken the same as those used in
Fig. 3. It is clear from this plot that
increasing the strength of twist in the annulus has a more
significant effect on the phase speed for longer wavelengths. For ka values over around 3, increasing the twist seems to have
little or no effect.
Figure 6 shows the change in period,
as a percentage, from a tube magnetic field enhancement of 10 to
20 times the external field. When the value of ka is larger the
period changes by
independent of the degree of
twist. For longer wavelengths however, increasing the magnetic
field enhancement has more effect on the value of the period of
oscillation. For a magnetic twist of 0.05, reducing ka from 0.4 to 0.2 (long wavelength) gives a rise in percentage form
to .
However, for shorter wavelengths,
reducing ka from e.g. 2.2 to 2.0, there is no change from a increase in period.

Figure 5:
This plot shows the effect of
on the periods of
oscillation for parameters the same as in
Fig. 3 for the surface mode occurring
at a phase speed between 0.09 and 0.11 for a series of
wavelengths (
ka=1.0, 1.5, 2.0, 2.5, 3.0, 3.5). 
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Figure 6:
Percentage difference between a tube magnetic field
enhancement of
compared to
for a twist
of
of the surface wave in
Fig. 3 at a phase speed of
. 
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Figures 7 and
8 illustrate the behaviour of the modes
in parameters approximating a dense tube. We model this as a
density enhancement within the tube taking the external ambient
magnetic field as half that of the longitudinal components of the
internal and annulus fields. We approximate the (internal and
annulus) tube density of twice that in the external region. We
found, for a fixed annulus thickness of 20% of the radius of the
tube and fixed strength of the longitudinal components of the
magnetic field in each region (Fig. 7)
the increase in width of the infinite band of hybrid modes as the
twist is increased. Also observed is the introduction of a mode
when twist is introduced to the annulus and the increase in phase
speed of this mode as the applied twist is made larger. A similar
effect occurs as the annulus thickness, for fixed twist (Fig.
8), is increased. It is, however, less
pronounced than the varying twist case.
Figure 9 shows the change in period of the
surface mode in Fig. 7 as twist is
increased. Note that for lower ka values the period is less for
smaller twist but this changes as twist becomes larger.

Figure 7:
Same as Fig. 3 but with parameters
approximating a dense tube in a magnetic environment.
,
,
a/R=0.80,
,
.
Shown are plots for twist
,
0.1, 0.2, 0.4. 
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Figure 8:
Same as Fig. 7 but with fixed twist
.
Shown are plots for annulus width 5%, 10%,
20% and 40% of the tubes radius. 
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Figure 9:
This plot shows the effect of
on the periods of oscillation
for parameters the same as in Fig. 7.
This plot is for the surface mode with phase speed 0.85
at ka=4. 
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Evacuated tubes (where the density in the tube is less than the
external density) are now modelled and the behaviour of the waves
are shown in Figs. 10 and 11. We model an evacuated tube by
considering an internal and annulus density half that of in the
external region. We have plotted the case of an internal and
annulus longitudinal magnetic field strength double the external
magnetic field. Broadening of the infinite band of hybrid modes is
also seen in this case. Similar to the photospheric case the
avoided crossing is, again, a feature and is clearly visible in
Fig. 10. The relationship between twist
and period is plotted in Fig. 12 for
different values of ka for the surface mode in
Fig. 10. Increasing the twist has a much
more pronounced effect on the period for longer wavelengths. For
ka=1, an increase in twist from 0 to 0.2 results in
increase in period whereas for ka=3 the same increase in
twist sees no change in the period. Another feature of the
pseudomode (one that changes from hybrid to purely surface) in
Fig. 10 is that it has, for twists less
than 0.2, a phase speed minimum occurring at
shown by the crossing of lines in
Fig. 12.

Figure 10:
Same as Fig. 3 but with parameters
approximating coronal conditions.
,
,
a/R=0.80,
,
.
Shown are
plots for twist
= 0.05, 0.1, 0.2, 0.4. 
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Figure 11:
Same as Fig. 10 but with fixed twist
.
Shown are plots for annulus width 5%, 10%, 20% and 40% of the tubes radius. 
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Figure 12:
This
plot shows the effect of
on the periods of oscillation
for parameters the same as in Fig. 10. 
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It is of interest to now consider the cases of large and
small k_{z}a, k_{z}R corresponding to the short and long
wavelength approximations. This allows us to have a first insight
into oscillations and MHD waves in a twisted coreshell problem.
In addition, we investigate what happens to the solutions when the
annulus has small twist and when the annulus is comparatively
thin. To analytically describe the behaviour of the waves for the
long and short wavelength approximations we restrict attention to
the sausage (m=0) modes. The kink (m=1) and flute (m>1)
modes are subject to follow up investigations.
We introduce the following notation:
So that
is the Alfvén speed due to the
longitudinal component of the annulus' magnetic field,
,
are due to the azimuthal component at
each of the boundaries and
is the phase speed of the
modes.
For the long wavelength approximation, since



(30) 



(31) 
we find that the purely surface modes (when m_{0}^{2}>0) occur
when



(32) 
and when



(33) 
Letting
,
and realising the fact that
for this case, the dispersion relation (28)
describing purely surface waves reduces to:



(34) 
which can be rewritten as



(35) 
a transcendental equation determining m_{0}R, and, ultimately
the phase speed,
,
and the eigenfrequency, ,
for a
given wavenumber, k_{z}. It is left to determine
from
rewritten from Eq. (31).
Relation (35), is satisfied, and becomes an
identity, when a=R. However, for the long wavelength
approximation, purely surface modes only occur, from
Eqs. (32), (33), at
phase speeds
.
Considering the hybrid modes we find that m_{0}^{2} <0 when
and we write n_{0}a=x, n_{0}R=y where
n_{0}^{2}=m_{0}^{2}. Letting
,
the dispersion relation (28)
for hybrid modes reduces to:



(36) 
Solutions of Eq. (36) have been plotted for
a/R=0.95, Fig. 13, and for
a/R=0.71, Fig. 14, each for twist,
,
values of 0.01, 0.05, 0.1 and 0.2. Although the
xaxis values are from 0  4, it should be noted that these
plots, as with Figs. 15 and 16, are for the long wavelength approximation
so valid for only smaller values of ka.

Figure 13:
Phase velocity of the m=0 modes determined from
Eq. (36) for a/R=0.95 showing the considerable
widening of the band of infinite hybrid modes as twist is
increased. Also shown is the envelope (dotdash line) separating
hybrid (dashed) modes and surface modes. 
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Figure 14:
Phase velocity of the m=0 modes determined from
Eq. (36) for a/R=0.71 showing the considerable
widening of the band of infinite hybrid modes as twist is
increased. Also shown is the envelope (dotdash line) separating
hybrid (dashed) modes and surface modes. 
Open with DEXTER 
If twist is small then
and it follows from Eq. (36)
that



(37) 
After expansion of the Bessel functions J_{0}, J_{1}, Y_{1}and noting that in the long wavelength limit for small twist
,
Eq. (37) becomes

(38) 
and, since y=Rx/a and
we
obtain

(39) 
a transcendental equation determining x and ultimately the
phase speed
.
We are now, for the solutions to
Eq. (39), left to determine
from
Solutions to Eq. (39) have been plotted for
two different degrees of twist (see Figs. 15,
16). As twist is increased, the infinite band
of hybrid modes broadens.

Figure 15:
Phase velocity of the m=0 modes determined from
Eq. (39) for
.
Also shown is
the envelope (dashed line) separating hybrid (dotted) modes and
surface modes. 
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Figure 16:
Phase velocity of the m=0 modes determined from
Eq. (39) for
.
Also shown is
the envelope (dashed line) separating hybrid (dotted) modes and
surface modes. 
Open with DEXTER 
For the short wavelength
limit,
,
we introduce the following
notation:
We notice, from Eq. (30), that m_{0}^{2}, for
the short wavelength approximation, is greater than 0 unless
so hybrid modes will occur with phase
speeds
only in a close neighbourhood of
.
We now
consider the purely surface modes so that m_{0}^{2}>0. After some
algebra we find from Eq. (28) for short
wavelengths that
This is of too high order in
to be analytically
solvable for the eigenfrequency, .
However, it is possible
to further simplify for the slender annulus
limit.
We are able to rewrite the LHS of Eq. (40) in
the following way:

(41) 
Thus in the slender annulus limit
we find:

(42) 
Using the fact that
(short wavelength) and
(slender annulus), Eq. (40) can be
reduced to give



(43) 
which yields the solutions

(44) 
a similar result to that found by Bennett et al. (1999). These two
modes are the surface wave (as obtained in Cartesian
geometry such as the magnetic surface and slab by
e.g. Roberts 1981a,b) and body waves of the system. The first
solution in Eq. (44) are the trapped
Alfvén waves which, in the limit of thin annulus, all become
the same.
We considered an incompressible medium and, to study the role of
the magnetic field, gravity effects are ignored. For a magnetic
configuration consisting of an internal region, annulus and
external ambient region each with uniform yet distinct twist the
full analytical dispersion relation was obtained. More
specifically, we analysed the case where the internal and external
regions have only magnetic field in the z direction (in a
cylindrical coordinate system), only the annulus having a uniform
twist. Again, the analytical dispersion relation was found and we
investigated numerically for the cylindrically symmetric m=0(sausage) mode the existence and behaviour of the different modes
for a wide range of typical solar atmospheric parameter values. We
also investigated analytically the asymptotic cases of long and
short wavelength approximation. We considered the case of
photospheric tubes, present in, e.g. sunspots, plage regions,
largescale unipolar areas and supergranulation fields at
supergranule boundaries; coronal loops as seen by TRACE, and
finally in evacuated tubes. In each case we solved the appropriate
dispersion relation and plotted the dispersion diagrams for
varying twist and varying annulus thickness in the order to
determine the role of each of these characteristics. It was found,
in each case, that the solutions consisted of an infinite band of
hybrid modes centered around the Alfvén speed due to the
longitudinal component of magnetic field in the annulus,
.
In the incompressible case, when there is no applied twist, there
are no hybrid modes present (see Edwin & Roberts 1983; Bennett et al. 1999), as twist
is introduced the hybrid modes begin to occur. It was found that
as twist is increased, the infinite band of hybrid modes broadens
and as the applied twist becomes large, possible indications of
instabilities (shown as the avoided crossing feature) appear. It
was also apparent that an increase in the annulus thickness (in
comparison to the tube thickness) had a similar effect of
broadening the infinite band of modes and eventual destabalising
effects on slower modes.
The long wavelength approximation analysis confirmed these
findings by also showing a widening of the infinite band of hybrid
modes. The additional assumption of small twist also clearly shows
the widening of the infinite band. In the short wavelength
approximation it was analytically shown that, for hybrid modes, as
then the phase speed
.
For surface modes, including the assumption of slender
annulus, we found solutions were tending to the tube kink speed,
c_{k}, where
.
For photospheric values, taking e.g. ka=1, we found that
increasing the twist from
to
caused an increase in period of
,
and a further increase of twist, to 0.2, resulted in
increase in the period (see
Fig. 5). Also investigated was the
effect that the strength of the magnetic field in the tube has on
the period. Figure 6 shows that for larger
values of ka, the increase in
(the ratio of internal
magnetic field strength to external), has the same effect on the
period (namely an increase of
for
Fig. 6) regardless of the amount of
applied twist. However, for longer wavelengths (when
1), the value of the twist has an effect on how much increasing
the internal magnetic field strength increases the period.
For coronal values, the effect of twist is less pronounced. The
value of ka where the pseudomode (one that changes from hybrid
to purely surface) shown in
Figs. 78
changes from hybrid to purely surface is highly dependant on the
twist. For large values of ka the period of the surface mode is
relatively unaltered by the amount of twist. For smaller kavalues the surface mode only exists for smaller amounts of twist.
Evacuated tube parameters, modelled as an internal density half
that of the external density, were analysed.
Figure 12 shows the importance of twist
on the periods of oscillation (for the surface mode at
). For ka=1, an increase of twist from 0.001
to 0.1 results in an increase in period of
,
and
increasing the twist to 0.2 gives a further increase of .
Further study will hopefully lead to examining the case of a
twisted annulus without the discontinuity in magnetic field at the
inner annulus boundary in order to extend understanding of heating
processes within the solar atmosphere.
Acknowledgements
The authors thank Prof. M. Ruderman and N. Venkov for a number of
useful discussions. R.E. acknowledges M. Kéray for patient
encouragement. The authors are also grateful to NSF, Hungary
(OTKA, Ref. No. TO43741) and to The University of Sheffield (White
Rose Consortium) for the financial support they received.
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Online Material
The general dispersion relation for a magnetic core, annulus and
external field all with uniform, though distinct for each region,
twist given by



(A.1) 
is found:
Here



(A.4) 
where X denotes the corresponding Bessel function and
is replaced by a,R for the corresponding region,
where the dash ' denotes the derivative with respect to the
argument of the Bessel functions. The Alfvén speeds in the
annulus, internal and external regions are given by
,
and
,
respectively such
that
By letting
and
,
so that we now consider the
case of a straight internal and external field with a twisted
annulus, Eqs. (A.6) reduce to:



(A.7) 
so that
and
which are always
positive. We can now rewrite the definitions (A.5) as



(A.8) 
so that we recover the general dispersion relations for a
twisted annulus embedded in a straight internal and external field
given by Eqs. (28), (28).
The general dispersion relation for surface modes,
Eq. (28) can be recast as







(B.1) 
where
which can be compared to the dispersion relation for a
twisted tube in a straight magnetic environment as found by
Bennett et al. (1998) whose result can be seen as the first
two
terms in Eq. (B.1).
Copyright ESO 2006