A&A 488, 1-7 (2008)
DOI: 10.1051/0004-6361:200809435
A. Bonanno1,2 - V. Urpin1,3
1 - INAF, Osservatorio Astrofisico di Catania,
Via S.Sofia 78, 95123 Catania, Italy
2 -
INFN, Sezione di Catania, Via S.Sofia 72,
95123 Catania, Italy
3 -
A.F.Ioffe Institute of Physics and Technology and
Isaac Newton Institute of Chile, Branch in St. Petersburg,
194021 St. Petersburg, Russia
Received 22 January 2008 / Accepted 10 April 2008
Abstract
Context. The MHD instabilities can generate complex field topologies even if the initial field configuration is a very simple one.
Aims. We consider the stability properties of magnetic configurations containing a toroidal and an axial field. In this paper, we concentrate mainly on the behavior of non-axisymmetric perturbations in axisymmetric magnetic configurations.
Methods. The stability is treated by a linear analysis of ideal MHD equations.
Results. In the presence of an axial field, it is shown that the instability can occur for a wide range of the azimuthal wavenumber m, and its growth rate increases with increasing m. At given m, the growth rate is at its maximum for perturbations with the axial wave-vector that makes the Alfvén frequency approximately vanishing. We argue that the instability of magnetic configurations in the ideal MHD can typically be dominated by perturbations with very short azimuthal and axial wavelengths.
Key words: magnetohydrodynamics (MHD) - instabilities - stars: magnetic fields
A wide variety of MHD instabilities can occur in magnetized astrophysical bodies where they play an important role in the evolution and formation of various structures and in enhancing transport processes, among others. The onset of instabilities can be caused both by hydrodynamic motions (for instance, differential rotation) or properties of the magnetic configuration. Even magnetic fields with a relatively simple topology (for example, a purely toroidal field) can be subject to instability. Magnetic fields generated by the dynamo action or stretched by hydrodynamic motions are topologically more complex and can cause this sort of instability as well. Which field strength and topology can sustain a stable magnetic configuration is still rather uncertain despite all the extensive work already done (see Borra et al. 1982; Mestel 1999 for review).
The simplest and best-studied magnetic configuration is most likely a purely
toroidal one. This has been known since the paper by Tayler (1957), where stability
properties of the toroidal field
are determined by the parameter
where s is the cylindrical radius.
The field is unstable to axisymmetric perturbations if
and to
non-axisymmetric perturbations if
.
The growth time of
instability is close to the time taken for an Alfvén wave to travel
around the star on a toroidal field line. Numerical modeling by Braithwaite
(2006) confirms that the toroidal field with
or
s2 is unstable to the m=1 mode (m is the azimuthal wave number)
as predicted by Tayler (1957, 1973). However, even a purely toroidal
field can be stable in the region where it decreases rapidly with s.
A purely toroidal field cannot be stable through the whole star
because the stability condition for axisymmetric modes (
)
is
incompatible with the condition that the electric current in the z
direction has no singularity at
,
which implies
.
The stability of the toroidal field in rotating stars has
been considered by Kitchatinov & Rüdiger (2007), who argue that the
magnetic instability is essentially three-dimensional and that the finite thermal conductivity creates a strong destabilizing effect. Terquem & Papaloizou (1996) and Papaloizou & Terquem (1997) considered the stability of an accretion disk with the toroidal magnetic field and found that the disks containing a purely toroidal field are always unstable and calculated the spectra of unstable modes in the local approximation.
The stability properties of purely poloidal magnetic fields are also well-studied. It has been understood since the papers by Wright (1973) and Markey & Tayler (1973, 1974) that the poloidal field is subject to dynamical instabilities in the neighborhood of points (or lines) where the poloidal field is vanishing (neutral points/lines). These authors recognized first that the magnetic field in the neighborhood of a neutral line resembles that of a toroidal, pinched discharge, which is known to be unstable. The instability of a poloidal field is also rather fast: its growth time can reach a crossing time of few Alfvén times (Van Assche et al. 1982; Braithwaite & Spruit 2006) that is very short, for example, compared to the time-scales of stellar evolution. However, the instability of a poloidal field can be suppressed by the addition of a toroidal field in the neighborhood of neutral points (Markey & Tayler 1973; Wright 1973).
Conversely, the addition of even a relatively weak poloidal field
alters the stability properties of the toroidal field substantially. For
example, if the poloidal field is uniform and relatively weak, the instability
condition of axisymmetric modes reads
,
at variance with the
condition of instability for a purely toroidal field (see, e.g., Knobloch 1992;
Dubrulle & Knobloch 1993), which predicts that an unstable toroidal field
configuration has
.
Therefore, a weak poloidal field has a
destabilizing effect. However, a strong enough poloidal field can
suppress the instability of the toroidal field. It turns out that configurations
containing comparable toroidal and poloidal fields are more
stable than purely toroidal or purely poloidal ones (Prendergast 1956;
Tayler 1980) and, generally, the possibility exists that there are
configurations containing mixed fields, which have no instabilities arising
on a dynamical time-scale.
In his study of
unstable magnetic configurations Tayler (1980) has not found any instability if the axial field
for instance, even though such configurations can be unstable for a wide range of the azimuthal wavenumber m if Bz is weaker. With
numerical simulations Braithwaite & Nordlund (2006) studied the stability of
a random initial field in the stellar radiative zone and found that the stable
magnetic configurations generally have the form of tori with comparable
poloidal and toroidal field strengths.
In this paper, we consider in detail the stability properties of magnetic
configurations containing the toroidal and axial magnetic fields with respect
to non-axisymmetric perturbations. We show that the instability may occur for
a wide range of the azimuthal wavenumber m, and the growth rate is typically
higher for higher m. Unstable modes with large m have a very short vertical
lengthscale, so it can be hard to resolve them in numerical calculations.
Depending on the profile
and the ratio
,
the
instability can occur in two regimes that have substantially different growth
rates.
The remainder of this paper is arranged as follows. In Sect. 2, we derive the equation that governs the eigenfunctions and eigenvalues of the magnetic field. We describe the numerical procedure and present the results of calculations in Sect. 3. A brief discussion of the results is given in Sect. 4.
Let us consider the stability of an axisymmetric cylindrical magnetic
configuration in a high conductivity limit. We work in cylindrical
coordinates (s, ,
z) with the unit vectors (
,
,
). We assume that the azimuthal field
depends on the cylindrical radius alone,
,
but the axial magnetic field Bz is constant.
In the incompressible limit, the MHD equations read
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Our analysis does not include gravity, which is important in stars, so
the stabilizing effect of stratification is neglected. In the case of
magnetic instabilities, this is justified if the work done by a perturbation
against gravity is less than the energy released from the magnetic field.
The corresponding condition for stellar radiative zones has been considered by
Spruit (1999) (see Eq. (44)) and reads in our notations as
,
where N is the buoyancy frequency. Assuming that the characteristic value of
N in stars is
10-3 s-1 and introducing the axial wavelength
,
we obtain that the effect of stratification can be
neglected for perturbations with
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(11) |
Magnetic perturbations can also be affected by the presence of dissipation, which is
not considered in our analysis. Usually, magnetic diffusion is weak in
stellar conditions, and its influence is unimportant if the dissipation rate
that is
is small compared to
when
cm2/s the magnetic diffusivity. This condition yields
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(12) |
We assume that the dependence of the azimuthal magnetic field on s is
given by
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(13) |
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(14) |
where
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(16) |
Equation (15), together with the given boundary conditions at the extrema, is a
two-point boundary value problem that can be solved by using the ``shooting''
method (Press et al. 1992). To solve Eq. (15), we used a
fifth-order Runge-Kutta integrator embedded in a globally convergent
Newton-Rawson iterator. We have checked that the eigenvalue was always the
fundamental one, as the corresponding eigenfunction had no zero
except that at the boundaries.
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Figure 1:
The dependence of ![]() ![]() |
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Figure 2:
The dependence of ![]() ![]() ![]() |
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In Fig. 2, we plot the dependence of
on q for the same m=1
mode but in the presence of a relatively weak axial field,
.
The addition of even very weak Bz changes
the stability properties qualitatively even though the energy contained in the axial field is
very low compared to that of the toroidal field (
1%).
We stress that the presence of an axial field breaks symmetry
.
The physical reason for this is fairly simple. The Lorentz force plays a crucial role in the behavior of
perturbations, and this force contains a component that is proportional to
the cross production of a current flowing in the basic state
and the magnetic field of perturbations (see the last
term on the left hand side of Eq. (6)). If
,
then magnetic perturbations
are determined partly by the axial gradient of velocity perturbations since
d
,
and this contribution has a
different sign for positive and negative kz. Therefore, the stability
properties turn out to be dependent on the direction of an axial wavevector.
However, Eq. (15) still contains some degeneracy because the replacements
or
do not change its shape.
The instability occurs only for a restricted range of negative q,
-(20-30)
< q < -2 depending on the value of
,
and do not appear for
positive q. The growth rate has two clear maxima with the higher maximum
corresponding to
.
By the order of magnitude, the
axial wave-vector of the most rapidly growing perturbation can be estimated
from the condition
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(17) |
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Figure 3:
The dependence of ![]() ![]() ![]() |
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The dependence of
on q in the presence of a weak axial field
(
)
and for different values of m is shown in Fig. 3.
In this figure,
is calculated for
,
such that the
magnetic configuration is unstable even in the absence of the axial field.
It turns out that this profile of
is unstable if
,
as well, but the properties of instability are substantially different. In
contrast to the case of a purely toroidal field with
where
only the m=1 mode can arise (Tayler 1973), the instability occurs for modes
with a wide range of m including m=0.
For
,
the growth
rate reaches its maximum approximately at
,
which
is equivalent to the resonant condition
.
The maximum
growth rate increases with an increase
in m. Therefore, the instability of such magnetic configurations is
probably dominated by the modes with large m and extremely short axial
wavelengths
.
This can cause difficulties in numerical simulations, because very high
resolution in the axial and azimuthal direction would be needed
to resolve the most unstable modes. Moreover a nonlinear interaction of these
modes can produce axial lengthscales that are even shorter than those
predicted by the linear theory.
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Figure 4:
The same as in Fig. 3 but for
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An increase in the axial field makes the magnetic configuration more stable.
In Fig. 4, we plot the growth rate for
and the same values
of other parameter as in Fig. 3. The instability grows more slowly in higher Bz:
an increase in Bz by a factor 5 leads to the decrease in
approximately by a factor 2. Nevertheless, the instability can still occur
for this
,
and it is still efficient because its
growth rate is
.
Like the previous case, the modes
with a wide range of m can be unstable, and the growth rate increases with
m. The axisymmetric mode (m=0) is stable in this case (see Bonanno &
Urpin 2007). The critical value
that
suppresses the instability is clearly dependent on the geometry of the basic state.
For example, by using energy considerations, Tayler (1980)
has not found any instability for
in the configuration
where the magnetic surfaces of the poloidal field are coaxial tori.
On the other hand, for values of
significantly less than 0.3, the author found instability
for a wide range of values of m.
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Figure 5:
The same as in Fig. 3 but for
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To illustrate the dependence of
on the profile of the toroidal field,
we show the growth rate as a function of q for
in Fig. 5. This
profile seems to be particularly interesting for astrophysical applications
since the toroidal field is
s in a neighborhood of the axis of
symmetry. In the absence of the axial field, the profile with
corresponds to the toroidal field that is marginally stable to axisymmetric
perturbations (m=0). If
,
the threshold of instability can
change substantially as argued by Bonanno & Urpin (2007) and,
indeed, perturbations with m=0 turn out to be unstable in this
case. However, the non-axisymmetric perturbations grow faster and are likely to
dominate the instability. The maximum growth rate is very high for them and
comparable to the Alfvén frequency for the toroidal field,
.
Remarkably, the maximum growth rate changes rather slowly with m for
all modes with
.
As in the previous cases, the maximum of
is located at
.
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Figure 6:
The same as in Fig. 3 but for
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In Fig. 6, we show
for the case of a rapid increase in
with s,
or
.
The growth
rate of instability is substantially higher for such
and can reach
the value
2-3 toroidal Alfvén frequencies. A qualitative behavior of
remains same: the maximum growth rate is higher for modes with
higher m, and these maxima correspond to very high values of the
wave-vector
.
The maximum growth rate of the
axisymmetric mode (m=0) is comparable to that of non-axisymmetric ones
(see Bonanno & Urpin 2007). Note that our calculations show some trend in
the growth rate to reach saturation for large m.
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Figure 7:
The same as in Fig. 3 but for
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Figure 7 plots the growth rate for
.
In accordance with Tayler
(1973), such a profile of
should be marginally stable if the axial
field is vanishing. However, the presence of Bz makes the toroidal
field unstable with such
even if the axial field is relatively weak. The growth
rate is not high in this case and is determined by the Alfvén frequency for
the axial field. As usual, the maximum growth rate is higher for higher m,
and these maxima are reached for very large q, which corresponds to a short
axial wavelength. A comparison between Figs. 6 and 7 illustrates the
difference well between two regimes of the instability first noted by Bonanno &
Urpin (2007). If
where
is some
characteristic value that generally depends on Bz, the instability is
relatively weak and grows on the Alfvén timescale characterized by the
axial field. In contrast, if
,
the instability is
much more efficient, and the growth time is of the order of the Alfvén
timescale for the toroidal field.
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Figure 8:
The critical value of ![]() ![]() ![]() |
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In Fig. 8, the critical value of
above which
the system is unstable is plotted as a function of
for
a range of vertical wavevectors relevant to stellar conditions.
In fact, the local approximation in the axial direction applies if
|kz s| > 1 that is equivalent |q| > 1. On the other hand, to neglect gravity can be justified
if
(see Eq. (11)), which generally imposes a stronger
restriction on q since the ratio
is typically >1 in
stars. Therefore, in determining
,
we assume that the allowed |q| should
be large enough and we choose
,
which corresponds to axial
wavelengths shorter than
.
Since the presence
of an axial field breaks symmetry between positive and negative q, critical
curves are different for positive and negative wavevectors. The region of
above the lines corresponds to configurations that are unstable for
a given m. It turns out that true
,
which determines instability,
corresponds to perturbations with positive m and negative q as shown
in Fig. 8 by solid lines. Critical
decreases with decreasing
everywhere except a region
of small
where the dependence
is very
sharp. This particularly concerns the curve m=0, which goes up very sharply
at
and reaches the value 1 at
in
agreement with the result by Tayler (1973). However, the scale of Fig. 8
does not let us see this sharp behaviour. Unfortunately, our code does
not allow to follow the behaviour of critical
when it approaches
the value -1 because of the singular character of the last term on the
left hand side of Eq. (15). The corresponding region is marked by crosses in Fig. 8.
A more refined consideration is needed for this case which should perhaps
include dissipation.
We have considered the linear stability of magnetic configurations containing the toroidal and axial fields, assuming that the behavior of small perturbations is governed by equations of the non-dissipative incompressible magnetohydrodynamics. This approximation is justified if the magnetic field is subthermal and the Alfvén velocity is low compared to the sound speed. The stability of magnetic configurations is a key issue for understanding the properties of various astrophysical bodies such as peculiar A and B stars, magnetic white dwarfs and neutron stars. The magnetic instability can alter qualitatively the properties of configurations generated, for example, by dynamo in stars. Many dynamo models predict that the toroidal field should typically be stronger than the poloidal one, but such configurations can be unstable if the generated toroidal field does not decrease enough rapidly with s. The instability generates large- and small-scale motions that should alter the geometry of a generated magnetic field.
Even though the poloidal field is weaker than the toroidal one in a number of dynamo models, its effect on the stability properties cannot be neglected. This particularly concerns the behavior of the nonaxysimmetric perturbations considered in the present paper. If Bz is relatively weak (
)
then, typically, there exists a wide range of the azimuthal wave-numbers m for which the instability may occur. For any given m, only perturbations within some particular range of the vertical wave-vectors kz can be unstable. The growth rate is maximal for perturbations with kz of about
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(19) |
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(20) |
Depending on the profile of the toroidal field and the strength of the axial field, the instability can arise in two essentially different regimes. In the case of a weak axial field,
,
the value of
that distinguishes between the regimes is
-1/2. If
,
then the instability grows on the Alfvén timescale determined by the toroidal field and is rather fast. In this case, the growth time is
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(21) |
It is fairly difficult to compare our results obtained for a simple model with the available numerical simulations, which usually use completely different basic magnetic configurations. For example, in calculations by Braithwaite (2006, 2007), the basic configuration was assumed to be either purely toroidal or purely poloidal, and the stability properties of such configurations differ qualitatively from those considered in this paper. Recently, Braithwaite & Nordlund (2006) and Braithwaite (2008) have considered stability of the magnetic configuration with random initial fields. A vector potential was set up as a random field containing spatial scales up to a certain value. This random field was then multiplied by some screening function, so that the field strength in the atmosphere was negligible.
This initial configuration contains both the toroidal and poloidal
fields but is very different from our simple model. Nevertheless, some
features seem to be in common even for such different models. Braithwaite
& Nordlund (2006) and Braithwaite (2008) find that the instability can lead
to different equilibrium configurations depending on the screening function.
If the screening function for random fields decreases slowly or does not
decrease at all, then the final equilibrium magnetic configuration is
essentially non-axisymmetric. In contrast, an equilibrium configuration is
closer to axisymmetry (but not axisymmetric) if the screening function
decreases rapidly. This dependence on equilibrium configurations obtained in
numerical calculations can reflect the regimes of ``strong'' and ``weak''
instabilities that correspond to different growth rates depending on the value
of .
In accordance with our analysis, the growth rate at given
is higher for higher
(compare, e.g., Figs. 3 and 7).
Our parameter
mimics to some extent the screening parameter p
introduced by Braithwaite (2008) with decreasing
corresponding to
increasing p. Therefore, we can expect from our analysis that non-aximetric
instabilities should be more efficient for the screening function with p=0
than with p=1 and that the final configuration exhibits stronger departures
from axisymmetry for smaller p. This conclusion seems to be in qualitative
agreement with the results of Braithwaite (2008).
Our simple model does not take into account the stratification that can be
important in many astrophysical applications. Basically, stratification
provides a stabilizing influence if the temperature gradient is sub-adiabatic.
However, this influence is small if perturbations have a relatively short
wavelength inthe axial direction,
,
such that inequality (11) is satisfied. Our results are related to this case. The case when
and stratification is important will be considered elsewhere.
Acknowledgements
This research project was supported by a Marie Curie Transfer of Knowledge Fellowship of the European Community's Sixth Framework Program under contract number MTKD-CT-002995. V.U. also thanks INAF-Ossevatorio Astrofisico di Catania for hospitality.