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The linear spectrum of twisted magnetic flux tubes
in viscous MHD
B. Pint'er, R. Erd'elyi and M. Goossesns
Centre for Plasma Astrophysics, K.U.Leuven,
Celestijnenlaan 200B, B­3001 Heverlee, Belgium
(Received )
In the present paper we investigate the influence of tensorial viscosity on the linear MHD
spectrum for isolated twisted magnetic flux tubes surrounded by vacuum. The magnetic flux tubes
are modelled as 1D cylindrical plasma columns. Viscosity is described by its classical tensorial form
with compressive and shear viscosity. The equilibrium parameters of the flux tube are choosen to
model coronal loops.
Numerical simulations show that in ideal MHD a poloidal magnetic field can destabilize the mag­
netic structure. They also show that viscosity can strongly destabilize the magnetic structure.
This destabilization depends on the poloidal component of the magnetic field.
I. INTRODUCTION
In the solar atmosphere we can observe very stable magnetic configurations, like
coronal loops, arcades, sunspots. Although magnetic confinement is possible in na­
ture, it still remains impossible to achieve a stable plasma on a long timescale in
tokamaks. Recently a lot of effort has been invested in the study of the stability of
plasmas (see reviews e.g. Lifschitz 1988 , Freidberg 1987). Kerner (1989) reviewed
numerical aspects of spectral problems.
Ideal magnetohydrodynamics (MHD) is often a very good approximation of plasma
phenomena, like magnetic loops in the solar corona. The equations of ideal MHD
for a 1D equilibrium state possess two regular singularities associated with spatial
non­square­integrable solutions. These singularities give rise to the Alfv'en and slow
continua.
Investigations of the stability of dissipative MHD may open a new branch of physics.
Computations of the linear spectrum in resistive MHD were carried out very ex­
tensively (i.e. Goedbloed 1975, Kerner et al. 1985). The ideal Alfv'en and slow
continua disappear when the plasma has finite electric conductivity as the singu­
larity is removed from the equations. The resistive MHD normal modes -- with a

finite number of exceptions -- do not converge to the ideal continuum modes (see
e.g. Riedel 1986, and Kerner et al. 1986).
In the present paper we examine the viscous MHD spectrum of cylindrical flux
tubes under solar coronal conditions (a TCP --Tokamak­like Current Profile-- ap­
proximation is used). This is done in a stepwise manner. First, the linear spectrum
of the magnetic loops is studied in ideal MHD. The possible role of an azimuthal
magnetic field for introducing exponential instability is investigated. Second, the
linear spectrum is studied for a non­magnetic cylinder in the presence of viscosity.
Finally, the linear spectrum is studied for magnetic loops in viscous MHD. Special
attention is given to the effects of viscosity and magnetic twist.
Compressive and shear viscous effects are taken into account. A computational
code (called LEDA) based on a numerical technique of finite element discretization
combined with the Galerkin method was developed by Kerner et al. (1985) to in­
vestigate the resistive instability in a 1D MHD model. This code was extended by
including the classical viscous stress tensor in the equations.
II.VISCOUS MHD EQUATIONS AND BOUNDARY CONDITIONS
The equilibrium quantities are assumed to be continuous functions of the radial
coordinate, r, only and gravitational effects are ignored. The momentum equation
for the linear MHD perturbations around a static equilibrium in viscous MHD is
ae 0
@v
@t
= \Gammarp 1 + (r \Theta B 0 ) \Theta B 1 + (r \Theta B 1 ) \Theta B 0 \Gamma r \Delta ú:
The subscript '1' denotes an Eulerian perturbation. In the momentum equation ú
denotes the classical stress tensor , viz.,
ú fffi = \Gammaj 1
`
@v ff
@x fi
+ @v fi
@x ff
\Gamma 2
3 ffi fffi r \Delta v
'
\Gamma j 0 ffi fffi r \Delta v;
where j 0
is the compressive viscous coefficient and j 1
is the shear viscous coefficient
(see also Braginskii 1965). The two viscosity coefficients are assumed to be constant
in the present paper.
The non­linear partial differential MHD equations are linearized. The perturbed
quantities are Fourier analyzed and the temporal behaviour of every perturbed
quantity f 1 is given by f 1 (r; t) = f 1 (r) exp(–t), where – is the eigenvalue of the
perturbation.
The plasma that surrounds the coronal loop has a very low density and is approx­
imated by a vacuum. At the plasma­vacuum boundary we have continuity of the
normal component of the Lagrangian displacement
¸ n (r = r p ) = ¸ n;vac (r = r p );
and continuity of the Lagrangian perturbation of the total pressure
–
`
p 1 + B 0 \Delta B 1 \Gamma B 0;vac \Delta B 1;vac
¯
'
\Gamma v r
/
B 2
0'
\Gamma B 2
0';vac
¯r
!
= 0;

where v = @¸=@t. The Lagrangian perturbations for all the three components of
the magnetic field also satisfy at the boundary
B 1 = B 1;vac :
These boundary conditions together with two regularity conditions at the axis guar­
antee a uniquely determined solution. The solution in the external vacuum can be
obtained in analytical form.
III. NUMERICAL RESULTS
The details of the numerical technique for obtaining solutions of the linear system
of ODE's are given very extensively in e.g. Van der Linden (1991). The unknown
functions are approximated by finite, linear combinations of local expansion func­
tions. Finite elements are used for the spatial discretization. Because of the finite
element discretization we will find only discrete eigenvalues. By increasing the num­
ber of gridpoints we obtain a strongly convergent and highly accurate spectrum.
The present paper studies the eigensolutions and stability of the equilibrium model
of Poedts, Goossens, & Kerner (1990). Of course, solar coronal loops are not sur­
rounded by a perfectly conducting wall. Therefore the wall has been removed to
infinity (r w !1).
III.A. Computations in ideal MHD
In order to understand the effect of magnetic twist and of classical tensorial viscosity
on the linear spectrum we have splitted our numerical investigation into three parts.
The first part is concerned with the linear spectrum in ideal MHD.
If the magnetic field is purely parallel to the axis of the flux tube there is no
physical reason to cause instability. Perturbations of the coronal loop can oscillate
harmonically at discrete frequencies and also on frequencies in the slow and Alfv'en
continuum. The entire spectrum should be exactly on the imaginary axis. However,
the eigenmodes found by the numerical code actually have a real part. This real
part is always very small in absolute value and unphysical but due to numerical
inaccuracies in the order of 10 \Gamma15 \Gamma 10 \Gamma13 . Actually these real parts specify the
numerical accuracy.
Let us now see how the spectrum is changed with an azimuthal magnetic field. If
the direction of the magnetic field is only slightly different from the direction of the
axis of the flux tube there is no recognizable difference from the spectrum of the
purely axial case. However, when the poloidal magnetic field is further increased it
will attain a critical value, B 0' crit at which the flux tube becomes unstable (Fig.
1).
This critical value is the threshold for instability. At this value of the poloidal com­
ponent of magnetic field appears the first and at the same time the most unstable
discrete eigenvalue of the system (Fig. 1a). By increasing further the poloidal mag­
netic field more and more (and smaller and smaller) eigenvalues appear (Fig. 1b
and Fig 2a).

Fig. 1. Above B 0' crit = 0:1525 appear some real discrete eigenvalues (in the order of 10 \Gamma4 \Gamma
10 \Gamma1 ) in the spectrum. These are unstable ideal eigenmodes of the system.
Fig. 2 Ideal instabilities caused by poloidal magnetic field. Fig. 2b shows that in the intervallum
close to B 0' crit the eigenvalues of the most unstable eigenmodes fit a parabola.
These instabilities are well­known ideal instabilities. Here the main qualitative re­
sult is the introduction of exponentially unstable modes by the azimuthal magnetic
field, B 0' . Fig. 2a also shows that above a given poloidal magnetic field the value
of the eigenvalues is becoming decreasing.
Numerical results indicate also that close to the threshold value of the poloidal
magnetic field, B 0' crit , the eigenvalues of the most unstable eigenmodes are pro­
portional to the square of the difference between the actual poloidal magnetic field
and the threshold for instability:
real(–) ¸
p
B 0' \Gamma B 0' crit
The parabolic interpolation to the points in Fig. 2b differs only by a few hundredth
of a percent.
III.B. Viscous computations
Let us now see how viscosity affects the linear spectrum. In case of zero poloidal
magnetic field and small non­zero viscosity a parabola­like continuum appears at
the positive real part in the spectrum. This means that viscosity can destabilize
the magnetic flux tube. However, even if the poloidal magnetic field is switched on
and it is kept small enough the spectrum will show practically the same unstable
eigenvalues (Fig. 3).

Fig. 3. The spectrum at small poloidal magnetic field (B 0'
= 2:5 \Theta 10 \Gamma3 ). Fig. 3b shows the
unstable parabolic continuum.
In what follows we have increased further the poloidal magnetic field. When the
poloidal magnetic field is bigger than a threshold value, B 0' crit , then again discrete
(Fig. 4a,b) and parabola­like continuous (Fig. 4c,d) unstable eigenvalues appear in
the spectrum. The critical value of the poloidal magnetic field is independent of the
value of the viscosity and is the same as found in ideal MHD. The discrete eigenval­
ues are several order of magnitude larger than the eigenvalues of the parabola­like
unstable continuum.
Fig. 4. The whole spectrum at and above the critical poloidal magnetic field, B 0' (Fig. 4a,b).
The lower figures (Fig. 4c,d) show the continuous unstable part of the spectrum.
Again, close to the B 0' crit , even the eigenvalues of less unstable modes depend on
the poloidal magnetic field as we found for the most unstable modes (see Fig. 5).
The real part of the discrete unstable eigenvalues decreases linearly with increasing
of viscosity at a constant poloidal magnetic field,
(real(–(j 0
0
) \Gamma real(–(j 00
0
))) ¸ (j 0
0
\Gamma j 00
0
);

Fig. 5. The proportionality real(–) ¸
p
B 0' \Gamma B 0' crit for the real parts of the eigenvalues is
valid again independently from the different viscosity.
Fig. 6. The real part of the eigenvalue of the most unstable eigenmodes is proportional to the
value of viscosity at a given poloidal magnetic field, B 0' .
as it is given in Fig. 6.
ACKNOWLEDGEMENT
The authors thank to Dr. M.S. Ruderman for his usefull comments to their research.
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