A&A 478, 1-8 (2008)
DOI: 10.1051/0004-6361:20077172
L. L. Kitchatinov^{1,2} - G. Rüdiger^{1}
1 - Astrophysikalisches Institut Potsdam, An der Sternwarte 16,
14482, Potsdam, Germany
2 -
Institute for Solar-Terrestrial Physics, PO Box
291, Irkutsk 664033, Russia
Received 26 January 2007 / Accepted 14 October 2007
Abstract
Aims. Two questions are addressed as to how strong magnetic fields can be stored in rotating stellar radiation zones without being subjected to pinch-type instabilities and how much radial mixing is produced if the fields are unstable.
Methods. Linear equations are derived for weak disturbances of magnetic and velocity fields, which are global in horizontal dimensions but short-scaled in radius. The linear formulation includes the 2D theory of stability to strictly horizontal disturbances as a special limit. The eigenvalue problem for the derived equations is solved numerically to evaluate the stability of toroidal field patterns with one or two latitudinal belts under the influence of rigid rotation.
Results. Radial displacements are essential for magnetic instability. It does not exist in the 2D case of strictly horizontal disturbances. Only stable (magnetically modified) r-modes are found in this case. The instability recovers in 3D. The minimum field strength
for onset of the instability and radial scales of the most rapidly growing modes are strongly influenced by finite diffusion, the scales are indefinitely short if diffusion is neglected. The most rapidly growing modes for the Sun have radial scales of about 1 Mm. In the upper part of the solar radiation zone,
G. The toroidal field can exceed this value only marginally, for otherwise the radial mixing produced by the instability would be too strong to be compatible with the observed lithium abundance.
Key words: instabilities - magnetohydrodynamics (MHD) - stars: interiors - stars: magnetic fields - Sun: magnetic fields
Which fields the stellar radiative cores can possess is rather uncertain. The resistive decay in radiation zones is so slow that primordial fields can be stored there (Cowling 1945). Whether the fields of 10^{5} G that can influence g-modes of solar oscillations (Rashba et al. 2006) or whether still stronger fields that can cause neutrino oscillations (Burgess et al. 2003) can indeed survive inside the Sun mainly depends on their stability.
Among several instabilities to which the fields can be subjected (Acheson 1978), the current-driven (pinch-type) instability of toroidal fields (Tayler 1973) is probably the most relevant one because it proceeds via almost horizontal displacements. The radial motions in radiative cores are strongly suppressed by buoyancy. Watson (1981) estimated the ratio of radial (u_{r}) to horizontal () velocities of slow (subsonic) motions in rotating stars as where is the basic angular velocity and N the buoyancy (Brunt-Väisälä) frequency. This ratio is small in radiative cores (Fig. 1). If the radial velocities are completely neglected, the stability analysis can be done in a 2D approximation with purely horizontal displacements (Watson 1981; Gilman & Fox 1997). Some radial motion still is, however, excited. How much radial mixing the Tayler (1973) instability produces is not well-known so far. As the radial mixing is relevant to the transport of light elements, a theory of the mixing compared with the observed abundances can help to restrict the amplitudes of the internal magnetic fields (Barnes et al. 1999). In the present paper the vertical mixing produced by the Tayler instability is estimated and then used to evaluate the upper limit on the magnetic field amplitude.
In this paper, the equations governing the linear stability of toroidal
magnetic fields in a differentially rotating radiation zone are
derived. They are solved for two latitudinal profiles of the
toroidal field with one and with two belts in latitude but only for
rigid rotation. The unstable modes are expected to have the longest
possible horizontal scales (Spruit 1999). Accordingly, our
equations are global in horizontal dimensions. They are, however,
local in radius, i.e. the radial scales are assumed to be short,
(k is radial wave number). The computations confirm that the
most rapidly growing modes have
,
but they are global
in latitude. The derived equations reproduce the 2D approximation as
a special limit. An exactly solvable case shows that the
Tayler instability is missing in the 2D approximation. It recovers,
however, in the 3D case. Finite diffusion is found important for the
instability. It prevents the preference for the indefinitely short
radial scales found for ideal MHD. The disturbances that first
become unstable as the field strength grows have small but finite
radial scales. For those scales, the minimum field producing the
instability is strongly reduced by allowing for finite thermal
conductivity. This field amplitude is about 600 G for the upper part
of the solar radiation zone. Considering the transport of chemical
species by the Tayler instability, we find that the field strength
can only be slightly above this marginal value. Otherwise, the
intensity of radial mixing would not be compatible with the observed
lithium abundance.
Figure 1: The buoyancy frequency (2) in the upper part of the radiative core of the Sun after the solar structure model by Stix & Skaley (1990). The convection zone of the model includes the overshoot layer so that is large immediately beneath the convection zone. | |
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in terms of the Alfvén angular frequency . In this equation, is the density, are the usual spherical coordinates, and is the azimuthal unit vector. Equation (1) automatically ensures that the toroidal field component vanishes - as it must - at the rotation axis. Centrifugal and magnetic forces are assumed weak compared to gravity, and . Deviation of the fluid stratification from spherical symmetry can thus be neglected.
The stabilizing effect of a subadiabatic stratification of the radiative core is characterized by the buoyancy frequency,
where is the specific entropy of ideal gas. The ratio is high in the radiative cores of not too rapidly rotating stars like the Sun (see Fig. 1).
The larger N, the more the radial displacements are opposed by the buoyancy force. Radial velocities should therefore be low. They are often neglected in stability analysis, which might be dangerous as certain instabilities may even be suppressed by the neglect (we shall see later how this indeed happens for the Tayler instability).
The consequences of neglecting the radial velocity
perturbations, u'_{r}, can be seen from the expression for
(divergence-free) velocity, ,
in spherical geometry in
terms of the two scalar potentials for the poloidal,
,
and toroidal, ,
flows
It can be seen from Eq. (3) that the horizontal part of the poloidal flow can remain unchanged when the radial velocity is reduced if the radial scale of the flow is reduced proportionally. The disturbances can thus avoid the stabilizing effect of the stratification by decreasing their radial scale instead of losing their poloidal parts.
Our stability analysis is local in the radial dimension; i.e. we use Fourier modes with . It will be confirmed a posteriori that the most unstable modes do indeed prefer short radial scales. The analysis remains, however, global in horizontal dimensions. Our formulation is very close to the thin layer approximation by Cally (2003) and Miesch & Gilman (2004), though we do not impose sharp boundaries and use other dependent variables. We also include finite diffusion and focus on marginal stability instead of growth rates for highly supercritical fields.
The instabilities of toroidal fields or differential rotation proceed via non compressive disturbances. The characteristic growth rates of the instabilities are low compared to p-modes frequencies. In the short-wave approximation, the velocity field can be assumed divergence-free, . Note that even a slow motion in a stratified fluid can be divergent if its spatial scale in the radial direction is not small compared to scale height. In the short-wave approximation (in radius), the divergency can, however, be neglected.
= | |||
= |
The perturbations are considered as Fourier modes in time, azimuth, and radius in the form of . For an instability, the eigenvalue should possess a positive imaginary part. Only the highest order terms in kr for the same variable were kept in the same equation.
When deriving the poloidal flow equation, the pressure term was
transformed as
In order to use normalized variables, the time is measured in units
of
(
is a characteristic angular
velocity), the velocities are scaled in units of ,
the
normalized eigenvalue (
)
is measured in units of ,
and the other normalized variables are
The equation for the poloidal flow then reads
For the toroidal field, two simple geometries are considered. First,
the quantity
is taken as a constant so that the
toroidal field has only one belt symmetric with respect to the
equator. Second, two magnetic belts are considered with equatorial
antisymmetry, i.e. with a node of
at the equator,
Two types of equatorial symmetry are allowed by the equations for the
linear disturbances in Sect. 2.2. Different dependent
variables of the same symmetry type can, however, have different
equatorial symmetries. As the toroidal flow disturbances are present
in all instabilities allowed by our equations, the symmetry
notations are related to the potential W of that flow. Standard
notations, Sm and Am, are used for the modes with symmetric and
antisymmetric W about the equator, respectively, and m is the
azimuthal wave number. The symmetry of magnetic disturbances with the same
symmetry notation differ, however, between the cases of one and two
belts of background toroidal field. For the case of one belt,
,
it is
The 2D approximation, however, misses the instability. Though the result
is obtained for the particular case of constant
,
it
is most probably valid in general.
Indeed, the toroidal field can be understood as consisting of
closed flux tubes. Noncompressive disturbances conserve the volume and
magnetic flux of the tubes so that the magnetic energy of a tube is
proportional to the square of its length (Zeldovich et al.
1983). The 2D disturbances also conserve the area of a segment on a
spherical surface encircled by a magnetic field line. The circular
lines of background field have minimum length for any given encircled
area. Any 2D disturbance increases the length, thereby increasing
magnetic energy. Therefore, there is no possibility of feeding a 2D
instability by magnetic energy release. A more formal proof of this
statement can use the expression for magnetic disturbances
The non-existence of Tayler instability in 2D clearly shows that radial displacements are necessary for the instability. Therefore, radial mixing is unavoidable. Transport of chemical species by the Tayler instability will be estimated in Sect. 3.3.2.
The conclusion about the absence of 2D instabilities is of course restricted to the case of rigid rotation. Rotational shear can excite its own 2D instability (Watson 1981; Dziembowski & Kosovichev 1987; Garaud 2001) fed by rotational energy release, and magnetic field can catalyze the process (Gilman & Fox 1997).
Gilman et al. (2007) find that 3D instabilities in the solar tachocline with strong toroidal fields differ only slightly from their 2D counterparts. This is probably because of large differential rotation in the tachocline. We shell see that the results for rigid rotation differ strongly between the 2D and 3D cases. The amount of differential rotation required for the difference to fade is still unknown.
From Sect. 3.1 we know that the instability can appear only in 3D
formulation when radial displacements are allowed. Figure 2
gives the resulting stability map. Two features are remarkable: (i)
rotation suppresses the instability so that subeqiupartition fields,
,
are stable, and (ii) the unstable
disturbances for the smallest field producing the instability have
indefinitely short radial scales,
.
Buoyancy frequency, N, enters the equations for linear
disturbances of Sect. 2.2 in combination (10)
with radial wave number, k. Therefore, the disturbances can avoid
the stabilizing effect of stratification by reducing their radial scale.
This is the strategy the Tayler instability follows. Another
possibility would be zero entropy disturbances (the
-parameter (10) enters the equations as a factor
of entropy perturbation). Entropy perturbations can be avoided with
2D disturbances only. Tayler instability cannot use this opportunity
because it has no 2D counterpart.
Figure 2: The stability map for constant and zero diffusivities. There is no instability for weak fields with . The threshold field strength for the instability increases with increasing vertical wavelength . | |
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Rotational quenching of Tayler instability is a controversial issue. Spruit (1999) finds that rotation modifies the instability but does not switch it off. Cally (2003), however, concluded that the polar kink instability (m=1 mode of Tayler instability) is totally suppressed if is smaller than near the pole. Simulations of Braithwait (2006) also show that the instability does not develop when the toroidal field is below a critical value roughly equal to the equipartition level of .
Figure 3: The small part of the stability map for two belts of toroidal magnetic field (19). Rotation does not suppress instability in this case. Parameter b is the polar value of the ratio . | |
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The controversy can be resolved by observing that the rotational quenching is sensitive to details of the toroidal field profile. Already, Pitts & Tayler (1985) suggested that suppression of the instability by rotation may be exceptionally strong when the ratio of Alfvén velocity to rotation velocity is uniform, , but rotation is unlikely to lead to complete stability for the general configuration of the toroidal field. Computations for the case of two belts of the toroidal field of Eq. (19) confirm this expectation. The stability map of Fig. 3 shows that the disturbances with sufficiently small radial scales remain unstable when the (normalized) toroidal field b is small^{}.
The high sensitivity of the Tayler instability in ideal fluids to details of the profile is mainly of academic interest. Figures 2 and 3 show that the smallest b producing the instability corresponds to indefinitely small radial scales. Finite diffusion, therefore, must be included. We shall see that the strong difference between the cases of one and two belts of toroidal field fades when diffusion is allowed for.
For ideal fluids the Tayler instability operates with extremely
small radial scales. When there is no stabilizing stratification,
however, finite vertical scales are preferred (Arlt et al.
2007). The thermal conductivity decreases the stabilizing
effect of the stratification and reduces the critical field
strengths for the instability if the (normalized) radial scale
is not too small.
Figure 4: The same as in Fig. 2 but with the finite diffusivities (22). The critical field strengths for the onset of the instability are strongly reduced compared to ideal MHD. | |
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Figure 5: The normalized growth rate as function of the magnetic field amplitude for the S1 mode and for (where the line of Fig. 4 has a minimum). The dotted lines for strong, , and weak, , fields give approximations by the linear, , and parabolic, , laws, respectively. | |
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For strong fields ( ) the basic rotation and diffusion are not important and there only characteristic frequency to scale the growth rates is . The dependence of Fig. 5 does indeed approach the relation ( is the growth rate) in the strong-field limit. The growth rate drops by almost four orders of magnitude when is reduced below 1. In the weak-field regime it is (Spruit 1999). The growth rates for weak fields ( ), where the instability exists due to finite diffusion, are rather small. Nevertheless, the growth rate of, e.g., means the e-folding time of about 1000 yr for the Sun, which is very short compared to evolutionary time scales. We shall see that the instability of weak fields is actually too vigorous to be compatible with observed lithium abundance.
The structures of the unstable modes differ between strong and weak
field regimes. For the weak fields, the growing disturbance pattern
is distributed over the entire sphere. For strong fields, it is much
more concentrated at the poles (Fig. 6) but still remains
global in latitude.
Figure 6: Toroidal field lines of the most rapidly growing eigenmodes of Fig. 5 for weak background toroidal field, ( upper panel), and strong field, ( lower panel). | |
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Figure 7: Stability map for the field model (19) with two belts and equatorial antisymmetry. | |
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In physical values, one finds for the upper part of solar radiation
zone (with
0.2 g/cm^{3})
Figure 8: Growth rates in units of for b=0.01 ( left) and b=0.1 ( right). The highest rates exist for independent of the magnetic field amplitude, all for most rapidly growing S1 modes. | |
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The flow field of the instability also mixes chemical species in
radial direction. Such an instability can thus be relevant to the
radial transport of the light elements (Barnes et al. 1999).
The effective diffusivity,
(u' and
are rms velocity and correlation length in radial direction)
can be roughly estimated from our linear computations assuming that
the instability saturates when mixing frequency approaches the growth rate
and
.
With Eq. (24), this yields
Figure 9: The growth rate as a function of the toroidal field amplitude for . The dotted line shows the parabolic approximation . The scale on the right gives the radial diffusivity of chemical species estimated after (25). | |
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We have shown that the pinch-type instability of toroidal fields require nonvanishing radial displacements. The instability does not exist in a 2D approximation with zero radial velocities. The maximum amplitude of stable toroidal magnetic fields for the Sun we found is about 600 G. This value results only for rigid rotation. It will most probably increase if the stabilizing influence of the positive radial gradient of in the equatorial region of the tachocline is included in the model.
The field strength in the upper part of the solar radiative interior can only marginally exceed the resulting critical values. Otherwise the instability would produce too strong a radial mixing of light elements. After our results, all the axisymmetric hydromagnetic models of the solar tachocline (Rüdiger & Kitchatinov 1997; MacGregor & Charbonneau 1999; Garaud 2002; Sule et al. 2005; Kitchatinov & Rüdiger 2006) have stable toroidal fields. On the contrary, the strong fields 10^{5} G, which are able to modify g-modes, or even stronger fields that may be relevant to neutrino oscillations (Burgess et al. 2003) are strongly unstable with e-folding times shorter than one rotation period.
The 3D computations of joint instabilities of toroidal fields and differential rotation (Gilman & Fox 1997; Cally 2003; Rüdiger et al. 2007) will be discussed in a separate paper. Another tempting extension is to include of the poloidal field. The field can be important in view of the very short vertical scales of the unstable modes.
Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft and by the Russian Foundation for Basic Research (project 05-02-04015).