Magnetic pinch-type instability in stellar radiative zones
aa r X i v : . [ a s t r o - ph . S R ] J a n Cosmic Magnetic Fields: from Planets, to Stars and GalaxiesProceedings IAU Symposium No. 259, 2008K.G. Strassmeier, A.G. Kosovichev & J. Beckmann, eds. c (cid:13) Magnetic pinch-type instability in stellarradiative zones
G¨unther R¨udiger Leonid L. Kitchatinov , and Marcus Gellert Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germanyemail: [email protected] Institute for Solar-Terrestrial Physics, P.O. Box 291, Irkutsk, 664033, Russia
Abstract.
The solar tachocline is shown as hydrodynamically stable against nonaxisymmetricdisturbances if it is true that no cos θ term exists in its rotation law. We also show that thetoroidal field of 200 Gauss amplitude which produces the tachocline in the magnetic theoryof R¨udiger & Kitchatinov (1997) is stable against nonaxisymmetric MHD disturbances – butit becomes unstable for rotation periods slightly slower than 25 days. The instability of suchweak fields lives from the high thermal diffusivity of stellar radiation zones compared with themagnetic diffusivity. The growth times, however, result as very long (of order of 10 rotationtimes). With estimations of the chemical mixing we find the maximal possible field amplitude tobe ∼
500 Gauss in order to explain the observed lithium abundance of the Sun. Dynamos withsuch low field amplitudes should not be relevant for the solar activity cycle.With nonlinear simulations of MHD Taylor-Couette flows it is shown that for the rotation-dominated magnetic instability the resulting eddy viscosity is only of the order of the molecularviscosity. The Schmidt number as the ratio of viscosity and chemical diffusion grows to valuesof ∼
20. For the majority of the stellar physics applications, the magnetic-dominated Taylerinstability will be quenched by the stellar rotation.
Keywords.
Sun: rotation, stars: interiors, instabilities, turbulence
1. Introduction
We ask for the stability of differential rotation in radiative stellar zones under thepresence of magnetic fields. If the magnetic field is aligned with the rotation axis thenthe answer is simply ‘magnetorotational instability’ (MRI). If the field is mainly toroidal(the rule rather than the exception) the answer is more complicated. Then the Rayleighcriterion for stability against axisymmetric perturbations of Taylor-Couette (TC) flowwith the rotation profile Ω = Ω ( R ) reads1 R dd R ( R Ω ) − Rµ ρ dd R (cid:18) B φ R (cid:19) > . (1.1)Hence, almost uniform fields or fields with B φ ∝ /R (a current-free field) are stabilizing the TC flow and no new instability appears.More interesting is the question after the stability against nonaxisymmetric perturba-tions. Tayler (1973) found the necessary and sufficient conditiondd R ( RB φ ) < B φ ∝ /R are stable.We have probed the interaction of such stable toroidal fields with stable flat rotation1 G. R¨udiger, L. L. Kitchatinov & M. Gellertlaws and found, surprisingly, the Azimuthal Magnetorotational Instability (AMRI) whichfor small magnetic Prandtl number scales with the magnetic Reynolds number Rm ofthe global rotation similar to the standard MRI (R¨udiger et al. 2007).In the following, as an astrophysical application of these nonaxisymmetric instabilitiesthe magnetic theory of R¨udiger & Kitchatinov (1997, 2007) of the solar tachocline ispresented. In the last Section we return to a TC flow under the presence of strong enoughtoroidal fields presenting first results of the eddy viscosity and the turbulent diffusion ofchemicals for the Tayler instability (TI).
2. Solar tachocline
The tachocline is the thin shell between the solar convection zone and the radiativeinterior of the Sun where the rotation pattern dramatically changes.
Figure 1.
The tachocline formation on the basis of a fossil poloidal field of 10 − Gauss confinedin the radiative solar interior (R¨udiger & Kitchatinov 1997). Left: the stationary rotation profile;right: the isolines of the resulting toroidal field with its amplitude of 200 Gauss.
The nonuniform rotation of the solar convection zone – which is due to the interactionof the convection with the global rotation – has no counterpart in the solar core butthe convection zone rotates in the average with the same angular velocity as the interiordoes. The radial coupling is thus large. This phenomenon cannot be explained by viscouscoupling (the viscosity below the convection zone is by more than 10 orders of magnitudesmaller) but it can be explained with a weak fossil poloidal field which is confined in thesolar radiative interior. For the amplitude of this field only values of order mGauss arenecessary resulting in a tachocline thickness of about 5% of the solar radius (Fig. 1). Theresulting toroidal field amplitude inside the tachocline of about 200 Gauss mainly dependson the magnetic Prandtl number (and the rotation velocity) for which Pm = 5 · − has been used in the model (R¨udiger & Kitchatinov 1997, 2007). One can estimate theresulting toroidal field in terms of the Alfv´en velocity V A = B φ / √ µ ρ simply as V A = √ Pm U (2.1)with U the linear velocity of rotation, U = R Ω . For Pm ≃ ≃ − thefield strength is reduced to only 1 kGauss so that we have carefully to check its stability.The magnetic theory holds for two main conditions: i) the field must completely be agnetic instability in radiative zones (h) Figure 2.
Neutral hydrodynamic instability lines for the rotation law without cos θ -term in3D. Only A1 and S2 modes are unstable. The A1 mode for wavenumber k → a = δ Ω / Ω ≃ . confined in the radiation zone and ii) the magnetic Prandtl number must be small enough.There are several possibilities to fulfill the first condition (cf. Garaud 2007; R¨udiger &Kitchatinov 2007) which, however, shall not be discussed in the present paper.2.1. Hydrodynamic stability
To fulfill the second condition the radiative tachocline must hydrodynamically be stable.On the first view this should not be a problem. As the differential rotation in latitudeforms a shear flow its amplitude, δ Ω = Ω eq − Ω pole , decides the stability properties. Byuse of a 2D approximation which ignores the radial coordinate Watson (1981) derived forideal fluids the condition δ Ω / Ω < .
286 for stability. This rather large value would leadto a stable tachocline. The radial velocity components, however, are small but not zero.Also the latitudinal profile of the angular velocity is more complicate than the simplecos θ -law used by Watson. We have thus to rediscuss the stability of shear flows with Ω ( θ ) = Ω (cid:18) − a ((1 − f ) cos θ + f cos θ ) (cid:19) (2.2)where δ Ω / Ω = a ; f is the contribution of the cos θ term which describes the shape ofthe rotation law in midlatitudes. At the solar surface we have a ≃ .
286 and f = 0 . N = (cid:18) gC p ∂S∂r (cid:19) . (2.3) S is the entropy. The frequency N in the upper solar core is by more than a factor of100 larger than the rotation frequency Ω . So the equation system ∂ u ∂t + ( U ∇ ) u + ( u ∇ ) U = − (cid:18) ρ ∇ p (cid:19) ′ + ν ∆ u T (cid:18) ∂s∂t + ( U ∇ ) s + ( u ∇ ) S (cid:19) = C p χ ∆ T ′ (2.4)must be solved for the flow perturbation u and the entropy fluctuation s . It is div u = 0; s = − C p ρ ′ /ρ . T ′ and ρ ′ are the fluctuations of the temperature and the density, resp.The mean flow U is given by (2.2). The equations are solved with a Fourier expansion G. R¨udiger, L. L. Kitchatinov & M. Gellertexp(i( kr + mφ − ωt )) in the short-wave approximation kr ≫ m as the azimuthalwave number. As usual, in latitude a series expansion after Legendre polynomials is used.In both latitude and longitude the modes are global. The parameter including thedensity stratification is ˆ λ = Nkr Ω , (2.5)so that ˆ λ → ∞ reproduces the 2D approximation by Watson (1981) and Cally (2001).They showed that only nonaxisymmetric modes with m = 1 can be unstable and thesame is true in the present 3D approximation. The modes with u θ antisymmetric withrespect to the equator are marked with Am and the modes with u θ symmetric withrespect to the equator are marked with Sm .Let us start with the rotation law ( f = 0). The Prandtl number is fixed asPr = νχ = 2 · − , (2.6)where ν and χ are viscosity and thermal conductivity.The main result is given in Fig. 2 which shows the neutral-stability lines for various a = δ Ω / Ω . Only A1 and S2 are obtained as unstable (S1 is stable!). For k → a = 0 .
286 is reproduced. With radial stratification, however, this critical value isreduced to a = 0 .
21. For ideal fluids Cally (2003) found instability for a = 0 .
24 whichalso fits our result. Hence, for a < .
21 the solar tachocline remains hydrodynamicallystable. There is thus no shear-induced turbulence. Note also that for N → Figure 3.
Hydrodynamic-stability map for various values f for the power of the cos θ -termin the rotation law (2.2). This term destabilizes the S1 mode. For high f already rather smallshear values become unstable. The calculations have been repeated with the cos θ -term included in the rotation law.Then the S1 mode becomes dominant and reduces the critical shear. For f = 0 . a = 0 .
16 (Fig. 3).This is a rather small value. Both our modifications of the Watson approach are desta-bilizing the differential rotation. If the shape of the surface rotation with f = 0 . and the tachocline then only 50% of the surfaceshear can indeed produce hydrodynamic turbulence in the tachocline. If the shear is not agnetic instability in radiative zones f = 0) then the minimum shear for turbulence is 0.21, large enough to ensurestability.A dramatic stabilization , however, of the latitudinal shear results from the inclusion ofthe tachocline rotation law as observed with its large radial gradients into the calcula-tions. With a 3D code without stratification one can show that in this case all rotationlaws with a < . a .
15 and f ≃
0. After our analysis (and also after theirs)such a rotation law is hydrodynamically stable . Obviously, we have to know in detail thespace dependence (and time dependence) of the internal solar rotation.2.2.
MHD stability
We consider the tachocline as hydrodynamically stable. The magnetic Prandtl numberis thus the microscopic one for which we use Pm = 0 .
005 as a characteristic value. Forthe instability of toroidal fields the ratio of Pm and Pr, the Roberts numberq = χη (2.7)plays a basic role. For the Sun the typical value 2500 is used. The interaction of rotationand toroidal magnetic fields (of a simplified structure) will be demonstrated with theseparameters. The general result is that for small q the field must be strong to becomeunstable while for q ≫ Ω A is defined by B φ = r sin θ √ µ ρ Ω A , (2.8)and is considered as constant (see Cally 2003). The global rotation is assumed as rigid.For this case Fig. 4 (left) reveals the fields with Ω A > Ω as always unstable with growthrates > ∼ Ω A . Toroidal fields with amplitudes V A > ∼ U eq , i.e. ∼ Gauss (for the Sun)cannot stably exist in the radiative solar core. Even weaker fields with Ω A > ∼ . Ω canbe unstable but only for very high heat-conductivity. For q = 0 the weak fields are stable.For q ≫ − . The growthtime τ growth in units of the rotation period is τ growth τ rot = 12 πγ (2.9)with the normalized growth rate γ = ℑ ( ω ) / Ω . The maximum growth time is then > Ω A ≃ . Ω . For the Sun, therefore, the maximumstable field in the model is ∼
600 Gauss.The model is insofar correct as the microscopic diffusivity values (viscosity, magneticdiffusivity, heat-conductivity) have their real amplitudes (for ideal fluids, see Cally 2003).The model, however, is insofar not correct as the radial profiles of the fields are assumed asnearly uniform. Arlt et al. (2007) work with a 3D code without buoyancy (q → ∞ , Pm =0 .
01) for toroidal field belts with strong radial gradients and find instability for weakfields with amplitudes of order 10 Gauss. G. R¨udiger, L. L. Kitchatinov & M. Gellert
Figure 4.
Normalized growth rates for the magnetic instability in rigidly rotating stars. Thecurves are marked with their values of the Roberts number (2.7). Left: one magnetic belt peakingat the equator; right: two magnetic belts with zero-field at the equator. There are two instabilitydomains: for strong magnetic fields ( Ω A > Ω ) the growth rates are high ( ≃ Ω A / Ω ) and for weakmagnetic fields ( Ω A < Ω ) they are very small ( ≃ ( Ω A / Ω ) ). The latter domain only exists forq ≫ Figure 5.
A star spinning down moves from the top (stable area) to the bottom (unstable area).It is shown that a toroidal magnetic field of 200 Gauss is stable for the solar rotation period butthe stability is lost for slower rotation.
Not surprisingly, for two belts with equatorial antisymmetry (the field vanishes at theequator) there are some differences to the one-belt model, but the maximal stable fieldamplitudes are always of the same order (Fig. 4, right).2.3.
Effect of stellar spin-down
The question arises whether a solar-type star is always able to form a tachocline. The olderthe star the slower its rotation. Hence the rotational quenching of the Tayler instabilitybecomes weaker and weaker for older stars so that the instability becomes more efficient.By its spin-down the star moves to the right along the abscissa of both the Figs. 4. The(slow) magnetic decay goes in the opposite direction; this effect is still neglected. Weassume that the total amount of the latitudinal differential rotation remains constantduring the star’s spin-down. This is a well-established assumption (see Kitchatinov &R¨udiger 1999; K¨uker & Stix 2001). agnetic instability in radiative zones δ Ω ≃ .
06 day − (the solar value). The rotation period is normalized with 25days in Fig. 5 so that the horizontal yellow line represents the Sun. In the upper part ofthe plot the 200 Gauss are stable while in the lower part of the plot they are unstable.Obviously, the Sun lies in the stable area but very close to the instability limit. We arethus tempted to predict that G2 stars older than the Sun (or better: of slower rotation)should not have a tachocline. When the toroidal field becomes unstable then the resultingturbulence is able to destroy the tachocline rather fast.2.4. Chemical mixing
The flow pattern of the magnetic instability also mixes passive scalars like temperatureand chemical concentrations. The instability, therefore, could be relevant for the so-called lithium problem. In order to explain the observed lithium concentration at thesolar surface one needs a turbulent mixing beneath the convection zone which enhancesthe microscopic value of the diffusion coefficient of 30 cm /s by (say) two orders ofmagnitude. Note the smallness of this quantity; only a very mild turbulence can providesuch a small value of the diffusion coefficient D T ≃ h u ′ i τ corr . (2.10)This relation is used here as a rough estimate, a quasilinear theory of turbulent mixinghas been established by R¨udiger & Pipin (2001) also for rotating turbulences. For acorrelation time of the order of the rotation period the desired mixing velocity is only 1cm/s.One can estimate the characteristic time by τ corr ≃ l /D T with l as the radial scaleof the instability and D T ≃ cm /s. For the radial scale the value 1000 km has beenfound by Kitchatinov & R¨udiger (2008). With this value it follows τ corr ≃ s whichcorresponds to a very small normalized growth rate of τ rot /τ ≃ − . The resultingtoroidal field which fulfills this condition is smaller than 600 Gauss (Fig. 6). Strongerfields would produce a too strong mixing which would lead to much smaller values forthe lithium abundance in the solar convection zone than observed.Our result in connection with the observed lithium values also excludes the possi-bility that some dynamo works in the upper part of the solar radiative core. If sucha (‘Tayler-Spruit’) dynamo existed then the resulting toroidal fields with less than 600Gauss are much too weak to influence the magnetic activity of the Sun with magneticfields exceeding 10 kGauss.
3. Tayler instability in Taylor-Couette systems
To simplify matters we consider the pinch-type instability in a Taylor-Couette systemfilled with a conducting fluid which is nonstratified in axial direction. The stationaryrotation law between the cylinders is Ω = a + b/R where a and b are given by the fixedrotation rates of the cylinders. In a similar way the stationary toroidal field results as B φ = AR + B/R . In the following we have fixed the values at the cylinders to Ω out =0 . Ω in and mostly µ B = B out /B in = 1 is used. The outer cylinder radius is fixed to 2 R in .Reynolds number Rm and Hartmann number Ha are defined asRm = Ω in R η , Ha = B in R in √ µ ρνη , (3.1)the magnetic Prandtl number here is always put to unity. G. R¨udiger, L. L. Kitchatinov & M. Gellert Figure 6.
Growth rates in units of Ω for the two-belts model and for a fixed radial scale of1000 km. The right-hand scale gives the estimated values for the diffusion coefficient for passivescalars and the uppermost scale gives the magnetic field amplitude in kGauss. A detailed description of the used nonlinear MHD code for incompressible fluids isgiven by Gellert et al. (2007). In the vertical direction periodic boundary conditions areused to avoid endplate problems. In this approximation the endplates rotate with thesame rotation law as the fluid does. The height of the virtual container is assumed as6 D with D the gap width between the cylinders. The cylinders are considered as perfectconductors. The code was first tested for the nonaxisymmetric AMRI which appearsif stable rotation laws and stable toroidal fields (current-free, µ B = 0 .
5) are combined(R¨udiger et al. 2007). Figure 7 (left) shows the instability domain (solid) which for given(supercritical) magnetic field always lies between a lower Reynolds number and an upperReynolds number. For too slow rotation the nonaxisymmetric modes are not yet excited,but for too fast rotation the nonaxisymmetric instability modes are destroyed so thatthe field becomes stable again.
60 80 100 120 140 160 180 200150200250300350400 Ha R e AMRI (instability of m=1) for Pm=1, µ Ω =0.5 linearsim: no inst.sim: AMRI Figure 7.
Pm = 1 , µ B = 0 . m -spectrumfor Re = 250, Ha = 110. The m = 1 mode contains 69% of the total magnetic energy. Between the limiting Reynolds numbers the instability is no longer monochrome butalso other modes than m = 1 are nonlinearly excited. The m = 1 mode often containsthe majority of the total magnetic energy (Fig. 7, right). There are also cases, however,where m = 0 and m = 1 contain nearly the same amount of magnetic energy. Note that agnetic instability in radiative zones Figure 8.
Left: The instability map for µ Ω = 0 . µ B = 1, Pm = 1. The hatched area isstable. Right: The growth rate (normalized with the rotation rate of the inner cylinder) alongthe horizontal line at Re = 500. The instability in the upper domain (AMRI) grows slowly whilein the lower domain (TI) it grows much faster. Now with µ B = 1 positive axial currents between the cylinders are allowed. Again thereare two different instability domains for the nonaxisymmetric modes with m = 1. Theyare separated by a stable domain (Fig. 8, left). The upper one is for fast rotation and weakfields (Re > Ha) and the lower one is for strong fields and slow rotation (Ha > Re). Thegrowth rates in both domains are very different: The weak-field instability is slow andthe strong-field instability is very fast (Fig. 8, right). The rotation-dominated instabilitydisappears for rigid rotation while the magnetic-dominated instability even exists withoutrotation (Re = 0). The latter one is the Tayler instability (TI) under the stabilizinginfluence of the basic stellar rotation (Pitts & Tayler 1985). The rotation-dominated(‘upper’) instability appears to be the AMRI which also exists under the modifyinginfluence of weak axial currents in the fluid. Note again that i) too fast rotation finallystops both the instabilities, and ii) its growth rate is very small. One must also stressthat the presented results only concern the most simple case of Pm = 1. For very strongfields the stable domain between AMRI and TI disappears.The condition for the existence of TI as given in Fig. 8 (left) is B φ > √ µ ρR Ω whilethe condition for AMRI is R Ω > B φ / √ µ ρ . Both the instabilities, however, only exist ifthe rotation is not too fast. Nonaxisymmetric modes are always stabilized by sufficientlyfast rotation.Ap stars (with 10 kGauss and a rotation period of several days) and neutron stars (with10 Gauss and a rotation period of 10 ms) are rotation-dominated. Their instabilitiesare not of the Tayler-type. In the following we have thus considered the AMRI in moredetail. For given Ha (= 500) the eddy viscosity, the diffusion coefficient D T and theSchmidt number Sc = ν T D T (3.2)are computed. The eddy viscosity ν T is the ratio of the angular momentum transportby Reynolds stress and Maxwell stress and the differential rotation. We find a ν T of theorder of the microscopic value (Fig. 9, left). The maximum exists as the instability –as mentioned – disappears for too fast rotation. The averaging procedure concerns thewhole container so that the values in Fig. 9 are lower limits.0 G. R¨udiger, L. L. Kitchatinov & M. Gellert
700 900 1100 130001234 MAGNETIC REYNOLDS NUMBER E DD Y V I S C O S I T Y
700 900 1100 13000102030 MAGNETIC REYNOLDS NUMBER S CH M I D T NU M BE R Figure 9.
Left: the eddy viscosity in units of the microscopic viscosity for µ Ω = 0 . µ B = 1,Pm = 1 and Ha = 500. It grows for faster rotation. Right: the Schmidt number (3.2). So far the diffusion coefficient for chemicals could only be estimated by D T ≃ h u ′ R i / Ω .It proves to be much smaller than the viscosity. Brott et al. (2008) have shown that fortoo strong mixing the stellar evolution is massively affected. A better theory must solvethe diffusion equation.Accepting this approximation the resulting Schmidt number (3.2) reaches values of20 . . .
30 (Fig. 9, right). Obviously, the angular momentum is mainly transported by theMaxwell stress while the diffusion of passive scalars is due to only the Reynolds stresswhich is much smaller. Quite similar results have been obtained by Carballido et al.(2005) and Johansen et al. (2006) for the Schmidt number of the standard MRI. Maeder& Meynet (2005) for a hot star with 15 solar masses and for 20 kGauss find muchhigher values of the Schmidt number (10 ). Also Heger et al. (2005) work with magneticamplitudes of 10 kGauss for which Ω A < Ω hence the growth rates are small. References
Arlt, R., Sule, A. & R¨udiger, G. 2005,
A&A
A&A
FIRST STARSIII , AIPC 990, p. 273Cally, P.S. 2001,
SoPh
MNRAS
MNRAS
ApJ
The Solar Tachocline , p. 147Gellert, M., R¨udiger, G. & Fournier, A. 2007, AN ApJ
MNRAS
A&A
A&A
A&A
A&A
MNRAS AN A&A
New J. Phys.
9, 302R¨udiger, G., Hollerbach, R., Schultz, M., Elstner, D. 2007,
MNRAS
MNRAS