A&A 485, 351-361 (2008)
Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany
Received 12 February 2008 / Accepted 11 April 2008
Context. The stability properties of toroidal magnetic flux tubes are relevant for the storage and emergence of magnetic fields in the convective envelope of cool stars. In addition to buoyancy- and magnetic tension-driven instabilities, flux tubes are also susceptible to an instability induced by the hydrodynamic drag force.
Aims. Following our investigation of the basic instability mechanism in the case of straight flux tubes, we now investigate the stability properties of magnetic flux rings. The focus lies on the influence of the specific shape and equilibrium condition on the thresholds of the friction-induced instability and on their relevance for emerging magnetic flux in solar-like stars.
Methods. We substitute the hydrodynamic drag force with Stokes law of friction to investigate the linear stability properties of toroidal flux tubes in mechanical equilibrium. Analytical instability criteria are derived for axial symmetric perturbations and for flux rings in the equatorial plane by analysing the sequence of principal minors of the coefficient matrices of dispersion polynomials. The general case of non-equatorial flux rings is investigated numerically by considering flux tubes in the solar overshoot region.
Results. The friction-induced instability occurs when an eigenmode reverses its direction of propagation due to advection, typically from the retrograde to the prograde direction. This reversal requires a certain relative velocity difference between plasma inside the flux tube and the environment. Since for flux tubes in mechanical equilibrium the relative velocity difference is determined by the equilibrium condition, the instability criterion depends on the location and field strength of the flux ring. The friction-induced instability sets in at lower field strengths than buoyancy- and tension-driven instabilities. Its threshold is independent of the strength of friction, but the growth rates depend on the strength of the frictional coupling between flux tube and environment.
Conclusions. The friction-induced instability lowers the critical magnetic field strength beyond which flux tubes are subject to growing perturbations. Since its threshold does not depend explicitely on the friction parameter, this mechanism also applies in case of the quadratic velocity dependence of the hydrodynamic drag force. Whereas buoyancy- and tension-driven instabilities depend on the magnetic field strength alone, the dependence of hydrodynamic drag on the tube diameter gives rise to an additional dependence of growth times on the magnetic flux.
Key words: magnetic fields - magnetohydrodynamics (MHD) - instabilities - Sun: magnetic fields - stars: magnetic fields
The magnetic field observed on the surfaces of cool, solar-like stars is assumed to originate in the bottom of the stellar convection zone. It is amplified by the shear flow in the tachocline and stored in the stably stratified overshoot region at the interface to the radiative core. The toroidal magnetic fields are subject to perturbations through overshooting gas plumes penetrating from the convection zone above. Beyond a critical field strength, unstable perturbations lead to growing magnetic flux loops, which rise through the convection zone and eventually emerge at the stellar surface (e.g. Spruit & van Ballegooijen 1982; Schüssler et al. 1996). In the solar overshoot region, critical field strengths are of the order of (e.g. Ferriz-Mas & Schüssler 1995,1993). Simulations of rising flux tubes with such initial field strengths yield emergence properties which are consistent with observed properties of sunspots such as their eruption latitudes, tilt angles, proper motions, and asymmetries between the leading and the following component of bipolar spot groups (Moreno-Insertis et al. 1994; Fan et al. 1994; D'Silva & Choudhuri 1993; Caligari et al. 1995).
The stability properties of magnetic flux tubes are relevant for the amplification, storage, and emergence of magnetic fields. An efficient generation of magnetic flux requires the field to remain within the convection zone for time periods which are comparable with or longer than the amplification timescale of the (here, unspecified) dynamo mechanism. Instability mechanisms limit the storage time and lead to the leakage of magnetic fields from the overshoot region with field-dependent growth times. The main driving mechanism of unstable, non-axisymmetric perturbations is magnetic buoyancy, which gives rise to undulatory, Parker-type instabilities (Parker 1955). If the flux tube dynamics is dominated by the magnetic tension force, an axial-symmetric poleward slip instability can occur (Spruit & van Ballegooijen 1982).
The hydrodynamic drag force reduces the relative motion of a flux tube perpendicular to its environment, similar to the case of a solid cylinder immersed in an external flow. The intrinsically dissipative interaction with the environment can cause under certain conditions the onset of flux tube instabilities (e.g. Joarder et al. 1997; Ryutova 1988). A friction-induced instability occurs if the relative flow velocity along the magnetic flux tube is higher than a critical value. In Paper II of this series (Holzwarth et al. 2007), we have investigated friction-induced instabilities of straight, horizontal flux tubes with parallel flows to elucidate the basic instability mechanism. In that case, the velocity difference between the internal and external plasma was a free parameter. Here we consider magnetic flux rings in a stratified environment to investigate the influence of the toroidal geometry and of the equilibrium condition on the stability properties. Both aspects affect the velocity difference between the internal and external plasma.
In Sect. 2 we describe the linearisation of the equations of motion in the framework of the thin flux tube-model, including a frictional interaction with the environment. In the linear stability analysis in Sect. 3, analytical instability criteria are derived for some special cases. The general case is analysed numerically on the basis of flux rings located in the solar overshoot region. Section 4 contains the discussion of the results and Sect. 5 the conclusions.
The stability analysis is carried out in the framework of the thin flux tube approximation (Spruit 1981). The radius of the flux tube is small compared with the radius of curvature, the wavelength of a perturbation, and the pressure scale height. Cross-sectional variations are taken to be negligible. All quantities are given by their values on the tube axis and are functions of time, t, and arc length, s, only. The flux tube is assumed to always retain a circular cross section.
The dynamics of the magnetic flux tube is determined through the
equation of motion
In the limit of infinite conductivity, plasma cannot cross the magnetic
surface of a flux tube. Consequently, a motion of the flux tube perpendicular to its tube axis
implies a flow of the external plasma around the tube.
Analogous to a cylinder immersed in a non-ideal streaming fluid, this
flow gives rise to a hydrodynamic drag force
Disregarding any differential rotation or meridional circulation, the
stellar stratification in the co-rotating reference frame is determined
We consider magnetic flux rings in mechanical equilibrium located parallel to the equatorial plane. A detailed description of this equilibrium and how it is obtained is given by Moreno-Insertis et al. (1992) and Caligari et al. (1995).
In the following, the index ``0'' indicates equilibrium values.
Since the equilibrium flux ring is axially symmetric, the tangential
component of Eq. (7) vanishes.
From the binormal component follows that the density contrast is nil,
because buoyancy provides the only force component parallel to the
rotation axis. With
the normal component of Eq. (7) yields
The linear stability properties are determined through the first-order
perturbation terms of Eq. (7). In the co-moving Frenet basis, the tangential, normal, and binormal components are
Except for the drag force, which comes from the perturbation of the
exterior, the velocity of the external plasma is taken to be unchanged
by the perturbation of the flux tube, that is,
The perturbations of the internal flow velocity and of the tube's axis
given by Eqs. (A.5) and (A.1), respectively, yield
the perturbation of the relative perpendicular flow velocity:
The linearised equations of motion are transferred to non-dimensional
form by multiplying Eqs. (11)-(13) with the
For the numerical evaluation of linear stability properties, we shall refer to a magnetic flux ring located in the middle of the solar overshoot region. At radius r0= 5.07 , the local density is , the gravitation is g0= 5.06 , the pressure scale heigh is , and the superadiabaticity is . The solar rotation rate is taken to be (i.e. ) and the ratio of specific heats . The local Brunt-Väisälä frequency is N0= 3 and .
We consider perturbations in the form of discrete harmonic functions.
The exponential ansatz
An explicit calculation of eigenfrequencies is not essential for the
determination of the stability criteria of a dynamical system, which
can be inferred from the coefficients of the dispersion polynomial by
using theorems from stability and control theory.
In our case, an instability implies a root of the dispersion equation
in the upper half of the complex plane.
We therefore use a generalised version of the Routh-Hurwitz theorem
(Marden 1966, Sect. 39): assume the monic complex polynomial
for z real, and define the quantities
The method is used in the following sections, which focus on special cases of flux tube perturbations.
The flux tube equilibrium is indifferent with respect to axially
symmetric perturbations in the azimuthal direction.
Such perturbations are marginally stable and the associated
The remaining eigenfrequencies are determined by the quartic dispersion
The -criterion implies the condition
In our analysis of the -criterion, we follow the approach of
Ferriz-Mas & Schüssler (1995) and combine the coefficients c0 and c2to eliminate
The resulting expression is used to eliminate c0 from the
-criterion, which yields a quadratic expression in c2.
If the discriminant,
The -criterion applies to flux tubes outside the equatorial
plane only and implies
The poleward slip instability is the governing mechanism regarding
axially symmetric perturbations of flux rings outside the equatorial
plane. This is shown by verifying that the sign change in the
-sequence first occurs between its last two elements.
At the -threshold, defined by equality of both sides of
relation (39), that is c0= 0, we have
In the equatorial plane, buoyancy-driven instabilities occur if c2>
0. Since the discriminant
is positive, the
-criterion covers the range
In the determination of the critical equilibrium parameter, beyond
which magnetic flux rings are subject to unstable axially symmetric
perturbations, the dependence of the magnetic Brunt-Väisälä
frequency on the field strength must be taken into account.
To this end, we use Eq. (24) to substitute
in Eqs. (39) and (42).
The equation for the threshold of the poleward slip instability is
|Figure 1: Thresholds of the poleward slip instability and axially symmetric equatorial instability for different length scale ratios . Flux tube equilibria with parameters located above (below) a curve are insusceptible (susceptible) to the respective instability.|
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This instability occurs in a stably stratified environment (i.e.
if the equilibrium parameter is sufficiently large (Fig. 1).
In contrast, a flux ring located in a convectively unstable environment
is insusceptible to this instability mechanism if the local
Brunt-Väisälä frequency is higher than
Both poleward slip and equatorial instabilities are not caused by friction in the first place and occur in the frictionless case as well. There are no specific friction-induced instabilities regarding axially symmetric perturbations of magnetic flux rings, though friction may affect the occurrence of overstabilities in the presence of differential rotation. The absence of friction-induced instabilities is consistent with our finding in 2007AetA...469...11H that their occurrence depends on the relation between flow and phase velocity. Since axially symmetric perturbations do not yield propagating wave signals the method of principal minors recovers the (modified) instability mechanisms which also occur in the frictionless case.
We now consider flux rings in the equatorial plane and derive analytical criteria for friction-induced instabilities with azimuthal wave numbers . In the equatorial plane binormal perturbations are decoupled from perturbations within the osculating plane, so that the dispersion equation factorises into a quadratic and a quartic polynomial.
The eigenfrequencies of binormal perturbations,
The eigenfrequencies of perturbations in the osculating plane are
determined by a monic quartic polynomial, whose coefficients are given
by Eqs. (B.10)-(B.13) in Appendix B.3.
It is sufficient to analyse the lowest-order eigenmode, because
increase with the azimuthal wave
An instability is indicated by
Taking the field dependence of the magnetic Brunt-Väisälä
frequency into account, the threshold of the -instability is
|Figure 2: Thresholds of the friction-induced instability (solid line) and of buoyancy-driven instabilities (broken lines) for non-axial symmetric perturbations (for f0= 0.1). The dotted, horizontal line indicates the stratification parameter in the middle of the solar overshoot region.|
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Since there is only one sign change in the -sequence, only one eigenmode is unstable yielding growing perturbations. A comparison with Eq. (24) shows that Eq. (62) is independent of the pressure scale height, so that the underlying instability is not caused by buoyancy. It does not exist in the frictionless case and is therefore specifically induced by the frictional interaction between the magnetic flux tube and its environment. The driving mechanism of the friction-induced instability has been described in detail in 2007AetA...469...11H.
The basic condition which discriminates between stable and unstable flux tube equilibria is the -criterion with azimuthal wave number m= 1. Beyond that threshold, higher-order eigenmodes can be unstable with growth rates higher than those of the m=1 mode. The determination of the fastest growing perturbations requires the explicit calculation of all eigenfrequencies. Keeping the equilibrium parameter constant, growth rates decrease if friction is strong (Fig. 3). This is caused by the dissipative nature of friction, which opposes relative plasma motions perpendicular to the tube axis. If friction is weak, the dissipative effect is outbalanced by the energy transfer from the relative motion of internal and external plasma through frictional coupling. Further to friction, the growth times of unstable perturbations still depend on magnetic buoyancy and tension. Growing perturbations of flux tubes which in the frictionless case are subject to Parker-type instabilities experience in the frictional case still a strong magnetic buoyancy, which gives rise to relative high growth rates (cf. Fig. 3, lower-right corner).
|Figure 3: Growth rates, , of magnetic flux tubes in the equatorial plane. The threshold to the left is determined by the friction-induced instability with azimuthal wave number m= 1. In the region enclosed by dotted lines, m= 2 modes have the highest growth rates. The dimensionless Brunt-Väisälä frequency is (cf. Fig. 2) and f0= 0.109.|
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An important aspect of our approach using the principal minors of the coefficient matrix is that, for a polynomial of degree n with , the last in the sequence depends linearly on the absolute term of the dispersion polynomial: . This can be seen by comparing Eqs. (53) with (B.13) and also in Eq. (34). A sign change of the absolute term of the dispersion polynomial thus indicates a friction-induced instability with, at least, one eigenfrequency being zero at the threshold c0= 0. This result agrees with our finding in 2007AetA...469...11H, that the onset of a friction-induced instability is characterised by the reversal of the propagation direction of a wave mode from retrograde, with , to prograde, with .
In case of magnetic flux rings parallel to but outside the equatorial plane, gravitation causes a coupling of perturbations parallel and perpendicular to the rotation axis. The eigenfrequencies are determined by a 6th-degree dispersion polynomial, whose coefficients are given in Appendix B.1. The method based on the principal minors is still applicable, but the analytical expressions for are cumbersome and not of much practical use. Guided by our previous results, we take the reversal of the propagation direction of an eigenmode as the criterion for the friction-induced instability, implying . The resulting instability criterion is formally identical with condition (59).
|Figure 4: Latitude dependence of the friction-induced instability threshold for different Burger numbers; in the solar reference case, it is 10-2. Note that the equatorial value increases with the Brunt-Väisälä frequency of the stratification (i.e. for high Bu).|
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|Figure 5: Growth rates of Parker-type and friction-induced instabilities of magnetic flux rings in the middle of the solar overshoot region. Shaded areas indicate unstable equilibria, with dotted lines marking boundaries of Parker-type instabilities in the frictionless case (panel a). Growth rates first increase with friction (panel b), then reach a maximum (panel c), and decrease for very strong friction (panel d). Arrows in panel b) indicate the latitudes shown in Fig. 7.|
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The latitude-dependence of the instability threshold, Eq. (62), is mediated through the dependence of Rossby number and equilibrium parameter on the curvature of the equilibrium flux
tube. Consider, for example, similar magnetic flux rings at a given depth of
a spherically symmetric star. Since the Brunt-Väisälä frequency is the same for each flux tube
at different latitudes, the critical flow velocity decreases with
Owing to the mechanical equilibrium condition (9), this
implies that at higher latitudes the friction-induced instability sets
in at lower field strengths (Fig. 4).
If the latitude dependence of all parameters is taken into account, the
threshold is determined through the bi-cubic polynomial
|Figure 6: Sign changes of , , and in the solar reference case, corresponding to the instability diagrams in Fig. 5. Tickmarks along each line indicate the downhill direction of the -functions. Within the shown range of field strengths and latitudes, .|
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We numerically investigate the friction-induced instability properties of flux rings outside the equatorial plane on the basis of the solar reference case (Fig. 5). The calculation of the -sequence confirms that the -criterion determines the onset of friction-induced instabilities (Fig. 6). For field strengths beyond this threshold, the sign change in the -sequence can occur at or , but in the parameter range considered here there is always only one sign change, indicating a single unstable eigenmode. Given that the friction-induced instability sets in once an eigenmode reverses its propagation direction, the retrograde and prograde slow wave modes are liable to become unstable since the moduli of their phase velocities are smallest. Owing to the prograde plasma flow inside the flux tube, the retrograde wave is advected against its propagation direction and (the modulus of) its phase velocity closest to zero. In most cases it is the retrograde eigenmode which becomes frictionally unstable once its propagation reverses into prograde direction. An exception to this rule occurs if the threshold of the friction-induced instability coincides with strong driving caused by magnetic buoyancy. In the frictionless case, Parker-type instabilities are characterised by a merging of two eigenmodes: the phase velocities of both modes are identical and the growth rates of opposite sign (cf. Fig. 7, panel a). If the merging coincides with the threshold of the friction-induced instability, the phase velocities of both eigenmodes are zero (panel c). In that situation, the prograde eigenmode can reverse its propagation to the retrograde direction and become frictionally unstable (panel d).
|Figure 7: Oscillation frequencies of slow eigenmodes near to the threshold of friction-induced instabilities in the solar reference case. Each panel corresponds to a latitudinal cut in the stability diagram Fig. 5, panel b); the friction parameter is . Thick lines indicate an unstable eigenmode. In the gray shaded regions the instability is predominantly driven by magnetic buoyancy, outside by frictional coupling. Since the azimuthal wave number is m= 1, we have .|
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In contrast to the frictionless case, where magnetic flux rings at high latitudes may be stable up to high field strengths, flux rings subject to friction are generally unstable if their field strength is beyond the threshold given by Eq. (62). The growth rates depend on the dominating driving mechanisms, which in the parameter range considered here are frictional coupling and magnetic buoyancy; tension-driven poleward slip instabilities typically occur at higher field strengths. If friction is weak, growth rates are significant only if the flux tube equilibrium is in a parameter domain of Parker-type instabilities (Fig. 5, panel b). If friction is strong, a differentiation between Parker-stable and Parker-unstable domains becomes irrelevant, since the growth times are comparable. In case of very strong friction, growth rates decrease as flux tube motions are more and more harnessed by the environment.
|Figure 8: Like Fig. 5, but for a rapidly rotating star with a rotation period of . The Burger number is Bu= 8.04 10-5. In accordance with the results shown in Fig. 4, the latitude dependence of the threshold is weaker than in the solar reference case and only effective at very high latitudes.|
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The criteria for friction-induced instability of straight and toroidal
magnetic flux tubes are identical.
From the equilibrium condition, Eq. (9) follows
In contrast to the case of straight flux tubes, in which the flow velocity is a free parameter, the relative flow velocity inside toroidal flux tubes is fixed by the equilibrium condition. This implies an explicit dependence of the friction-induced instability on the curvature and field strength of the flux ring. The instability criterion, which relates the flow velocity with the phase velocity of a perturbation, thus determines a critical magnetic field strength (or, more general, Alfvén velocity) beyond which the friction-induced instability sets in. This critical field strength is lower than the threshold of the buoyancy-driven instability. A preliminary numerical parameter study indicates that this property applies to a broad range of stellar rotation rates and equilibrium depths in the overshoot region. From this, we conjecture that it is the friction-induced instability which determines whether flux ring equilibria in the overshoot region are stable or not.
Virtually all flux tube equilibria beyond the instability threshold are unstable. This is important, for example, in the case of fast stellar rotation. Stability analyses of flux rings in rapidly rotating stars (e.g. Holzwarth & Schüssler 2003; Ferriz-Mas & Schüssler 1995) show that, in the frictionless case, for high magnetic field strengths isolated regions of stable equilibria exist, in which flux tubes are not subject to the Parker-type instability (see Fig. 8, panel a). These field strengths are well beyond the threshold of the friction-induced instability. Therefore, if friction is taken into account, these equilibria are unstable with high growth rates (Fig. 8, panels c and d).
The possibility to store magnetic flux in the overshoot region depends
on the growth rate of the instabilities.
An evaluation of the growth rates of friction-induced instabilities
requires an estimate for the friction parameter .
A comparison of drag force, Eq. (2), and friction force,
Eq. (4), yields
Thin magnetic flux tubes are weeded out of the overshoot region on shorter time scales than thick flux tubes, and should therefore occur more often. This agrees with the trend indicated by umbral areas observed on the Sun, which follow a log-normal distribution with small surface features being more numerous than large features (e.g. Baumann & Solanki 2005). If the higher number of small surface features is a consequence of higher friction-induced emergence rates, the dynamics and stability properties of thin magnetic flux tubes are dominated by friction. In that case, the latitudinal distribution of surface features may also depend on the size of the magnetic feature, since thinner flux tubes experience a stronger deflection to higher latitudes by meridional circulation (e.g. Holzwarth et al. 2006). The influence of meridional circulation would possibly be diluted by the action of convective motions on rising flux tubes. But since the former effect is systematic and the latter random, a discernable signature may persist in the latitudinal distribution. Even if thin flux tubes do not make it to the surface due to convective interaction, their susceptibility to friction-induced instabilities implies a decrease of magnetic flux in the overshoot region, which affects the budget and amplification of magnetic fields in the tachocline region.
The numerical results shown in Sect. 3.4 are based on magnetic flux rings located in the middle of the solar overshoot layer. Owing to the strong dependence of stability properties on the superadiabaticity, flux tubes located deeper in the overshoot region require significantly higher field strength to become buoyancy unstable. The friction-induced instability eases the need for very high field strengths, since its threshold is below the critical field strength of the buoyancy-driven instability. For example, a flux ring located in the equatorial plane close the radiative core, in an environment with (N= 3 ), becomes frictionally unstable for field strengths above 3.5 , which is well below the critical field strength of the Parker-type instability at 6.2 .
The friction-induced instability of toroidal flux tubes is also relevant in other astrophysical contexts, such as in circumstellar accretion discs. Accretion of mass implies the transport of angular momentum to larger radii, which requires an efficient coupling of adjacent disc annuli. The coupling of Keplerian shear flow through weak magnetic fields gives rise to the magneto-rotational instability (MRI), which generates magnetic turbulence and efficiently increases the viscous coupling and angular momentum transport (Balbus & Hawley 1991). The MRI requires a poloidal magnetic field and is independent of the toroidal field component. This axi-symmetric shear instability occurs for vertical wavelengths larger than a critical, field-dependent value, though growth rates and threshold are independent of field strength. It is an interesting possibility that the friction-induced instability of toroidal fields complements the MRI by supporting the radial magnetic coupling of a disc. Both instability mechanisms produce their highest growth rates on large length scales, yet the friction-induced instability does intrinsically not saturate at high field strengths as the MRI. However, it is unclear what mechanisms aside of Keplerian shear can produce the required velocity difference between the plasma inside and outside the flux tube.
Schramkowski (1996) investigated the dynamics and evolution of small flux rings in an accretion disc, including the drag force caused by shear flows. If a flux ring is deformed by a strong shear flow, variations of the tension force and density contrast occur along the flux tube, which produce two loops rising vertically through the disc until they emerge into the disc corona. These investigations confirm the strong influence of the drag force on the dynamics of flux tubes with small minor tube radii, but they do not consider friction-induced instabilities. A stability analysis including the drag force would be possible, because there is a flow component perpendicular to the axis of the equilibrium flux tube. Since differential rotation has a significant influence on the stability properties of magnetic flux rings (e.g. Ferriz-Mas & Schüssler 1995), it will be necessary to include the effect of shear flows and more general equilibrium conditions in the stability analysis of the friction-induced instability in the framework of accretion discs.
It is not clear to which extend the results based on a linear velocity dependence of friction carry over to the non-linear case. Given the -independence of the instability thresholds, the actual functional form of friction, that is, linear or quadratic, seems to be of secondary importance, provided that it is anti-parallel to the flow velocity perpendicular to the tube axis. Numerical simulations based on the drag force confirm the properties of friction-induced instability derived here. A detailed analysis of the non-linear case and parameter study is the topic of a forthcoming paper in this series.
The friction-induced instability determines the stability properties of flux rings at the bottom of the convection zone, as its threshold is at lower field strengths than those of buoyancy- and tension-driven instabilities. The stability properties depend on the magnetic field strength and on the magnetic flux. The threshold also depends on the location of the flux ring, since the toroidal geometry causes a connection between the instability criterion and the equilibrium condition. The analytical results confirm that a magnetic flux tube subject to friction becomes unstable when a backward wave occurs, that is, when an eigenmode reverses its direction of propagation. We expect that friction-induced instabilities are also relevant in other astrophysical contexts, such as the angular momentum transport in accretion discs.
The author thanks Manfred Schüssler, Robert Cameron, and Dieter Schmitt for fruitful discussions and helpful comments.
We summarise the expressions describing adiabatic perturbations of toroidal magnetic flux tubes in mechanical equilibrium. Their detailed derivation can be found, for example, in Schmitt (1998) and Ferriz-Mas & Schüssler (1995).
of the flux tube from its equilibrium position change the
tangential, normal, and binormal vectors,
respectively, of the Frenet basis, the curvature, ,
velocity, u, and the acceleration of the internal plasma:
The flux tube is in pressure equilibrium with its environment. Assuming adiabatic perturbations, the field strength, density, and density contrast, , change according to the following equation:
Based on the linearised equations of motion in the form of Eq. (27), the eigenfrequencies of magnetic flux rings in mechanical equilibrium parallel to the equatorial plane are determined
by the monic dispersion polynomial
In the special case of axial symmetric perturbations the dispersion
equation reduces to the monic quartic polynomial in Eq. (30).
The friction-independent part of the coefficients are:
For flux rings in the equatorial plane the dispersion equation
factorises in a quadratic and quartic polynomial, which describe the
perturbation perpendicular and parallel to the equatorial plane,
respectively. The coefficients of the latter are: