The Angular Momentum Transport by Unstable Toroidal Magnetic Fields

Home , Strouhal number

arXiv:1404.4288v2 [astro-ph.SR] 30 Oct 2014 unyfricmrsil ud.Ti oaiymti in- nonaxisymmetric This fluids. incompressible for quency u oti nacdvsoiy hc xed h molecu- of the factor exceeds a which by viscosity, value enhanced lar red this momentum massive angular to of more due flux the outward The of 8366239. law KIC giant rotation internal flat pectedly nems oeo h tr elire l 21)rpr the- a report the (2012) profile in al. et (quasi-Keplerian) velocity Ceillier oretical angular star. the the of of lead core profiles innermost transport radial steep momentum without rather giants angular to red magnetic of or formation turbulent the for calculations Model Introduction 1. ta.21) gebre ta.(02 ru htol an only that argue (2012) 3 al. of et than viscosity Eggenberger additional faster Deheuvels 2014). times 2012, al. al. five et et (say) (Deheuvels zone only convection rotate young outer and the to sub-giants seem ob- several giants of the of cores red laws the giants: rotation red flatter much served to lead however, data, h o-asrdgatKC7421 oeta h expo- the that Note 7341231. KIC nent giant red low-mass the with o nfr nua momentum angular uniform for nenlslrrtto,drvdfo eisimlgcmea- 0 helioseismologic to the down from of uniformity surements derived of rotation, degree pro- solar high to internal the value stress viscosity Maxwell by this Rüdiger & duce just core. needed radiative (1996) inner Kitchatinov the of spin-down observed ltd ftetria edi iheog u osnot does but enough high is field toroidal exceed am- the the of if disturbances plitude become nonaxisymmetric they however, against fields, unstable toroidal of presence the Under srnm Astrophysics & Astronomy uut3,2021 31, August h nua oetmtasotb ntbetria magnet toroidal unstable by transport momentum angular The oainlw with laws Rotation icst r osdrd efidta h iesols rat dimensionless ampl the field that find magnetic We and considered. rates are rotation radia viscosity in of observed combinations rotation rigid-body those rotatio almost differential the of explain influence momentto the angular under that fields code toroidal MHD of nonlinear a with demonstrate We accepted Received; Sternwa der An Potsdam, für Astrophysik Leibniz-Institut H oe–teei eakbeiflec ftemgei Pran magnetic stress the Maxwell of the influence and Pm remarkable stress for a Reynolds only is the of there KIC sub-giant contributions w – – red numbers code considered Reynolds the magnetic MHD of for interval multipli unity considered fluid order the rotating For of the is of which number – Reynolds the with grows q nisseradbcmslne n ogrdrn h spin-do the during longer and law. words. longer rotation Key the becomes in and shear shear its the on on depends viscosity effective U φ ol ecieaqaiglci oainprofile rotation quasi-galactic a describe would 1 = Ω A os while const = ≃ Ω with ntblte antcfils–dffso ublne-mag - turbulence - diffusion – fields magnetic - instabilities ∼ > 0 . . .W lofidta h antzdpam eae sanon-New a as behaves plasma magnetized the that find also We 5. Ω 15 A < q ν R × q T = ⊙ 10 /ν ersnstertto law rotation the represents 2 = . B r yrdnmclystable. hydrodynamically are 2 4 ≈ φ aucitn.ruediger˙et˙al no. manuscript cm / 0,sffie opouethe produce to suffices 500, p Ω .Ruie,M elr,F pd,adI Tereshin I. and Spada, F. Gellert, Rüdiger, M. G. 2 Ω smyepanteunex- the explain may /s µ R 0 ∝ 2 ρR ≈ R 2 − os.The const. steAlfvén fre- the as q with q ≃ Kepler 1 . for 6 ABSTRACT fields ,adta h eutn ffcievsoiymyb ihenou high be may viscosity effective resulting the that and n, ieselrcrs nysainr urn-refilsand fields current-free stationary Only cores. stellar tive t 6 -48 osa,Gray mi:[email protected] email: Germany, Potsdam, D-14482 16, rte where 05.W aetu rtt rb h au fP nthe in Pm of value red the considered the probe of to interiors first radiative thus have mag- Subramanian main-sequence We & The inside Brandenburg 2005). (see 0.01. plasma smaller (say) the is however, of exceeding stars, simulations number Pm Prandtl nonlinear for netic numerical possible only far, are So diffusivity. netic m= Pm number Prandtl magnetic the of knowledge the as is proven is interior stellar the in working. not fields magnetic of bility rsn ae r hsmxmmvle.I hyaenot are they If the values. maximum for in enough the thus derived high are by values resulting paper reduced viscosity the present be The that buoyancy’. should so ‘negative damped, transport strongly momentum of that, are angular components show field radial can and the also flow One included, fields. thermodynamics strong with be- for influences negligible these the that come with however, studied know, We be and model. cannot compressibility present 2002) both Spruit linear of (see transport consequences the heat The in (Rüdiger 2014). simulations studied al. nonlinear et been by also has but AMRI approximation, setup, the called cylindrical Within been a (AMRI). has instability’ instability the magnetorotational field, charac- ‘azimuthal magnetic force-free the this of of Because ter domain. current-free considered are which the fields in toroidal for exists even stability h ea ieo h ieeta oaintu depends thus rotation differential the of time decay The oo h ffcieoe oeua viscosity, molecular over effective the of io ihi etitdb ueia osrit ftenonlinea the of constraints numerical by restricted is hich dwt h qaero ftemgei rnt number Prandtl magnetic the of square-root the with ed otettlvsoiy hc smgeial dominated magnetically is which viscosity, total the to tdswihpoiemxmlnmrclvle fthe of values numerical maximal provide which itudes 7341231. no tla core. stellar a of wn nipratbssfrraitcnmrclsimulations numerical realistic for basis important An mcnb rnpre u otemgei instability magnetic the to due transported be can um ν ν η stemlclrvsoiyo h udand fluid the of viscosity molecular the is , t ubro h eaieiprac fthe of importance relative the on number dtl eoyrdnmc (MHD) netohydrodynamics ν T /ν ≃ 0 hntecneto h insta- the of concept the then 500 oinfli,ie h resulting the i.e. fluid, tonian Kepler ν T /ν

linearly , c stars. S 2021 ESO η only t mag- its gh r ic (1) 1 G. R¨udiger et al.: Viscosity by unstable magnetic toroidal ﬁelds

Fig. 2. The diffusion coefficients from the stellar model of KIC 7341231. The vertical dashed line gives the outer boundary of the radiative stellar core. Note that the ge- ometric average of the two diffusivitiesη ¯ = √νη 100 ≈ cm2/s.

The run of the viscosity ν and magnetic diﬀusivity η in the interior of this model, given by

5/2 4 −16 T −25 T ν = νmol + νrad =1.2 10 +2.5 10 , · ρ · κρ η = 1013T −3/2 (2) (all in c.g.s. units), is shown in Fig. 2. The basic result is that a characteristic value for the microscopic magnetic diffusivity is 103 cm2/s, while the magnetic Prandtl number varies between 0.1 and 10. The microscopic magnetic diffusivity is thus of the order of that of the Sun, while the microscopic viscosity is much higher. It makes thus sense to Fig. 1. The stellar model of KIC 7341231 (blue) compared focus the attention to the results as a function of the mag- to a standard solar model (red). The vertical dashed line netic Reynolds number rather than the ordinary Reynolds gives the outer boundary of the radiative stellar core of the number. For diﬀerent stellar models, therefore, the mag- red sub-giant. netic Reynolds number mainly encompasses the diﬀerences of the angular velocity of the inner rotation. The magnetic Reynolds number of the solar core is about 5 1012 and its magnetic Prandtl number is 5 10−3. · ·

2. The stellar model of KIC 7341231 3. Numerical setup and the calculation of the effective viscosity A model for the star KIC 7341231 is calculated using the Yale Rotational stellar Evolution Code (YREC) in its non- After the results of the above stellar structure calculations rotational configuration (Demarque et al. 2008). The code it is thus reasonable to perform simulations with magnetic uses up-to-date input physics, such as OPAL 2005 equations Prandtl numbers between 0.1 and unity. Another approxi- of state (Rogers & Nayfonov 2002) and OPAL opacities mation should also be mentioned, as the calculations focus (Iglesias & Rogers 1996). The treatment of the atmospheric to the angular momentum in the equatorial region. After boundary conditions is based on the Eddington grey T - the Taylor-Proudman theorem for fast rotation the Ω forms τ relation; convection is described according to the mixing cylindrical isolines, Ω = Ω(R). More simplifying, the axial length theory (Böhm-Vitense 1958), with the mixing length structure of the toroidal magnetic field belts is neglected so parameter set to αMLT = 1.82, our solar-calibrated value. that also Bφ = Bφ(R). The resulting viscosity values cer- Our choice of the stellar parameters (e.g. M = 0.84 M⊙, tainly overestimate the more realistic corresponding values Z/X = 1.0, age 12 Gyr, etc.) is based on the best-fitting for a spherical model. model of− Deheuvels≈ et al. (2012, see their table 3). Our We consider the same cylindrical setup used by Rüdiger model of KIC 7341231 has effective temperature Teff = 5380 et al. (2014), to which the reader is referred for further de- K, radius R =2.66R⊙, and a fractional radius at the bot- tails. Cylindrical coordinates (R,z,φ) are adopted, with the tom of the convective envelope of rbce =0.34 at the age of z axis coincident with the axis of the bounding cylinders, of 11.8 Gyr. radii Rin and Rout, respectively. The cylinders are assumed

2 G. Rüdiger et al.: Viscosity by unstable magnetic toroidal fields as rotating with different frequencies so that differential ro- of the instability is the turbulence-induced decay of the rotation can easily be modeled. tation law. To this end, angular momentum must be trans- An important detail of the calculations is the scaling ported into the direction of slow rotation, i.e. of the instability for small magnetic Prandtl numbers. For 1 dΩ rotation laws with q 2, the bifurcation map scales with uRuφ bRbφ = νT R . (7) the Reynolds number≃ Re and the Hartmann number Ha, h i− µ0ρh i − dR 2 Here u and b are the fluctuations of the velocity and mag- Ωin R0 Bin R0 Re = , Ha = , (3) netic field with their average values U and B. ν √µ0ρνη The calculation of the effective viscosity νT is thus pos- while for more flat rotation laws the corresponding pa- sible by the calculation of the cross-correlations uRuφ and b b . Note that for a magnetic-dominated MHDh turbu-i rameters are the magnetic Reynolds number Rm = Pm Re h R φi and Lundquist number S = √PmHa. The radial scale is lence the relation 2 R0 = pRin(Rout Rin). 1 b − νT h i τcorr (8) We imagine the formation of the toroidal field as due ≃ 3 µ ρ to the induction by the differential rotation from a fos- 0 < sil poloidal field Bp; hence Bφ ∼ Rm Bp, or what is the holds (Vainshtein & Kichatinov 1983), which may be used < same, ΩA/Ω ∼ Spol with Spol the Lundquist number of as an estimation by replacing the unknown correlation time the poloidal field: τcorr with the growth time 1/ωgr, where ωgr is the maximal growth rate of the instability (see Spruit 2002). The maxi- Bp R0 Sp = . (4) mum growth rates lead to the shortest time-scale in the sys- √µ0ρη tem. One can indeed show that for the magnetic instability 2 a linear relation of the turnover frequency ωturn = kp u Thus, with Sp of order unity, one finds that always z and the growth rate exists (Fig. 4). In the sense ofh the > Ω ∼ ΩA. (5) mixing length arguments we have only to put equal the turnover time and the correlation time. This is insofar a Note also that Sp 1 is already fulfilled with rather weak ≃ striking result as it allows to express the saturated hydro- poloidal fields, of the order of µG. For the rotation law at dynamic energy by (the square of) the linear phase velocity. the Rayleigh limit (µ =0.25) and the current-free magnetic Rüdiger et al. (2014) have shown that for Pm = 1 the background field Bφ 1/R the flow Uφ and the field Bφ unstable m = 1 modes grow with ωgr/Ωin Re. One also ∝ ∝ have the same dependence on the radius. Chandrasekhar finds for the magnetic energy b2 /B2 Re. Hence, (1956) has demonstrated that all ideal MHD systems with h i in ∝ U B 2 = are stable. In contrast, Fig. 3 demonstrates that for νT Ha nonideal fluids an instability exists for Ω = Ω. Moreover, (9) A ν ∝ Re if the Reynolds number of the fluid exceeds a critical value of (only) 100 then an extended area of instability exists which for AMRI where the maximal growth rates are lo- above this line fulfilling just the condition (5). cated at a line with Ha Re (see Fig. 3) leads to a linear ∝ Tayler (1957, 1973) showed that toroidal fields with relation νT/ν Re. The question is how this relation satu- large enough amplitude become unstable against nonax- rates by the magnetic∝ feedback and whether this maximal isymmetric disturbances, in the absence of rotation. There effective viscosity is large enough to explain the above men- is a local criterion for instability written in cylinder coor- tioned Kepler observations. The results of our numerical 2 dinates, i.e. experiments indeed confirm that νT Ωin R0. The coeffi- cient in this relation must be fixed with∝ numerical calcula- d (RB2 ) > 0, (6) tions. With our numerical tools, however, we could not find dR φ the saturation effect so that the question after the maxi- mally possible viscosities requires better numerical quali- which is fulfilled by the magnetic field Bφ R, due to a ho- mogeneous axial electric current. If an electric∝ current flows ties. What we have nevertheless shown is that indeed i) the unbounded in the axial direction, the critical Hartmann Maxwell stress by the fluctuations of the azimuthal magnetorotational instability mainly forms the effective viscosity number Haout = Bout R0/√µ0ρνη is about 20 which has indeed been realized in an experiment (Seilmayer et al. 2012, and ii) the viscosity for slow rotation runs linearly with Rüdiger et al. 2012). Numerical simulations of this magnetic magnetic Reynolds number. configuration showed, however, that the eddy viscosity re- We shall simulate two different rotation laws, i.e. the two limits Ω 1/R2 and Ω 1/R, bracketing the rotation sulting from this instability remains small compared with ∝ ∝ its microscopic value. law obtained in modeling sub-giant stars without enhanced transport of angular momentum, i.e. Ω R−1.6 (Ceillier If the toroidal field is of the vacuum type, it runs as ∝ B 1/R. Here the symmetry axis, R 0, where the elec- et al. 2012). φ ∼ → We start with the steep rotation law Ω 1/R2, which tric current flows, must be excluded from the domain of cal- ∝ culations. This field, alone, is stable according to the crite- is one of the two basic solutions for the stationary rotation rion (6). The field, however, destabilizes a differential rota- law between two rotating cylinders, i.e. tion which itself is stable in the hydrodynamical regime.The b energy source of this instability is the differential rotation, Ω= a + . (10) R2 which via AMRI drives a nonaxisymmetric magnetic instability pattern; the magnetic energy dominates the energy With the definition µ = Ωout/Ωin and for rin = Rin/Rout = in the kinetic fluctuations. The basic saturation mechanism 0.5 one finds µ = 0.25. In order not to conflict with

3 G. R¨udiger et al.: Viscosity by unstable magnetic toroidal ﬁelds

80 Pm=1, µ=0.255 Pm=1, µ=0.354 µ 60 Pm=1, =0.5 Pm=0.2, µ=0.354

TURNOVER RATE 20

0 0 10 20 30 40 50 60 GROWTH RATE

Fig. 4. The linear relation between the turnover rate and the growth rate for various Pm and µ. The models possess Fig. 3. The curves of marginal instability for Ω 1/R2 if ∝ the maximal growth rates as indicated in the instability rin = 0.5, i.e. µ = 0.25. For small magnetic Prandtl num- domain of Fig. 3. bers, the curves scale with Re and Ha. For Pm = 1, the critical Reynolds number is (only) 100. The dotted lines mark the line of maximum growth rates. The dashed line represents for Pm = 1 the lower limit of the condition (5). nonlinear results for the (vertical) velocity and the cell size π/k. This procedure provides the results plotted in Fig. 4 the Rayleigh criterion for hydrodynamical instability (here which indeed provides the linear relation (11) with α as µ =0.25), the steep rotation in our calculations is defined a function of Pm and the shear. As expected, the maxi- by µ = 0.255. The instability map for the rotational pro- mum α (as a modified Strouhal number) is unity for steep file with µ = 0.255 is given by Fig. 3 for Pm = 1 and, rotation law and large magnetic Prandtl number. It be- for reasons of comparison, for the extremely small value comes smaller for both decreasing Pm and for the flatter Pm = 10−6. We find that the dependence of the bifur- rotation rates. For known α we thus have the puzzling sit- cation map on the magnetic Prandtl number is very weak. uation that the nonlinear turbulence intensity u2 can be h zi The combination of (differential) rotation and toroidal field expressed by the characteristic quantities k and ωgr of the is unstable in the open cone between the lower and the up- linear theory with the simple relation urms αωgr/k. It ≃ per branches of the non-monotonic function Re = Re(Ha), does not automatically mean, however, that the effective where the growth rates vanish by definition. The dashed viscosity can be estimated just with these quantities as the lines for Pm = 1 and Pm = 10−6 mark the locations of the effect of the magnetic fluctuations cannot be neglected. It maximal growth rate, which are strikingly close to the lower might be interesting to compare the traditional estimate 2 2 strong-field line of the instability cone. The eddy viscosities νT α ωgr/k with the nonlinear results. In any case this are numerically computed in this instability cone for fixed relation≃ suggests that the effective viscosity should decline Reynolds number and various Hartmann numbers, with the for smaller magnetic Prandtl numbers and flatter rotation general result that νT peaks at the location of the dashed profiles. We take for the most optimistic case (Pm = 1 and line, i.e. for maximum growth rates. µ =0.25) from Rüdiger et al. (2014) that the maximal drift is 0.4Ωin and the smallest wavenumber is about 3R0 so that the upper limit of the viscosity due to the hydrodynamic 4. Nonlinear simulations transport can be estimated by νT/ν 0.04Re which leads ≃ to νT/ν 4 for Re = 100. We shall see that the nonlinear From Fig. 2 we know that the magnetic Prandtl number values do≃ only exceed this value for much higher Reynolds in the radiative core of the red giants varies between 0.1 numbers. and 10 so that it makes sense to focus the simulations to Now the effective viscosity is nonlinearly calculated by Pm < 1. We first check the validity of the often used relation ∼ computing the RHS of the relation (7) within the instability domain in Fig. 3. For a given Reynolds number, the ωturn αωgr, (11) ≃ Hartmann number is varied until the maximum value of see the discussion below Eq. (8). The factor α represents νT is found. Finally, the maximum viscosity between the a characteristic Strouhal number formed with the growth inner and the outer cylinder is taken (see Fig. 5). The av- rate instead of the characteristic spectrum frequency. The erage procedure in (7) concerns only the azimuth and the growth rates have been calculated from linear models for vertical axis. various Hartmann numbers along the line of maximal For various magnetic Reynolds numbers, this procedure growth in the instability domain as indicated by Fig. 3. yields viscosities which linearly grow for growing Rm. This The applied rotation law varies between the steep rotation is true for all rotation laws between 1/R2 and 1/R includ- law with µ = 0.25 and the flat rotation law with µ = 0.5 ing Kepler rotation (Fig. 6). For the magnetic Reynolds which interval certainly contains the internal stellar rota- numbers of the order of 103 (which our code is able to tion. For the calculation of the turnover rates we need the handle) we do not find any indication of a saturation. One

4 G. R¨udiger et al.: Viscosity by unstable magnetic toroidal ﬁelds

5 6

4 5

4 3 Pm ν / √ T

) 3 ν 2 ν / T ν

( 2 1 Pm=1 1 Pm=0.5 Pm=0.1 0 1 1.2 1.4 1.6 1.8 2 0 R RADIUS R 0 200 400 600 800 1000 1200 1400 in out MAGNETIC REYNOLDS NUMBER

4 Fig. 5. Rotation law Ω 1/R2: The fluctuating radial profiles between the inner and∝ the outer radius of the effective 3.5 viscosity in (7). The thick solid line gives the temporal average. Re = 850. 3 2.5 Pm

also ﬁnds that the resulting viscosity for Pm < 1 scales as √

) 2

νT/ν Rm/√Pm, which can also be written as ν /

∝ T 1.5 2 ν √ ( νT PmΩR (12) Pm=1 ∝ 1 or, which is the same, Pm=0.7 Pm=0.5 ν 0.5 T √Pm Re, (13) Pm=0.1 ν ∝ 0 0 500 1000 1500 with Re the Reynolds number of the fluid. The main result MAGNETIC REYNOLDS NUMBER is the rather weak dependence of the viscosity on the magnetic Prandtl number Pm. Obviously – since Pm runs with Fig. 6. Rotation law Ω 1/R2 (top) and Ω 1/R (bot- the electric conductivity – it is the induction by the radial ∝ ∝ rotational shear that produces the strong correlations of tom): The normalized viscosity grows with the magnetic the magnetic fluctuations given in (7), and this induction Reynolds number Rm without indication of a saturation vanishes for Pm 0. (Pm = 0.1...1, as indicated). From Fig. 6 it→ follows that the missing numerical factor in the relations (12) and (13) is of order 5 10−3. For the · solar core, we thus find the rather high value of νT/ν The role of the Maxwell stress in comparison to the 5 109. One must take into account, however, that i) our≃ Reynolds stress is illustrated in Figs. 7 and 8. We find that calculations· always tend to find the maximal values of the magnetic energy dominates the kinetic energy for large the viscosity and ii) we have only verified that the linear magnetic Prandtl numbers, while the energies are almost relations (12) and (13) hold for Reynolds numbers up to in equilibrium for small magnetic Prandtl numbers. 103, and cannot account for any saturation effect. We can For a fixed Reynolds number, the ratio ǫ only assume that the effective viscosity does not exceed the 2 given value. b ǫ = h i2 (14) The dependence of the resulting eddy viscosity on the µ0ρ u magnetic Prandtl number has consequences for stars with h i small Pm like the solar radiative core. There, the eddy vis- reaches its maximum value for weak magnetic fields and cosity due to the instability of toroidal fields becomes rather approaches unity for strong magnetic fields. The kinetic small. This is not the case, however, for the radiative cores energy never exceeds the magnetic energy (Fig. 7). The of red giants with their magnetic Prandtl numbers of or- dominance of the magnetic energy for Pm 1, however, der of unity. The magnetic-induced outward angular mo- is not too strong. It must remain open whether≃ or not ǫ mentum transport will happen much faster for hot stars in saturates for very large Rm. For numerical reasons, the comparison to solar-type stars. calculations are limited to Reynolds numbers of order 103, Inserting characteristic stellar values for ΩR2 one ob- where no saturation occurs. Within this interval the ratio tains high viscosity values, of the order of 1012cm2/s. It is ǫ slightly grows with growing Rm. important to note, however, that in the present paper only On the other hand, for large magnetic Prandtl numbers upper limits for the viscosity are presented, in order to show the total viscosity is almost entirely due to the Maxwell > that the resulting viscosities are not too small. We did not stress (Fig. 8, bottom). For large Rm and for Pm ∼ 0.5, find the saturation level which is a result of the magnetic no visible differences exist between the total viscosity feedback onto the rotation laws and the viscosity due to the Maxwell stress. The viscosity

5 G. R¨udiger et al.: Viscosity by unstable magnetic toroidal ﬁelds

6000

µ 4 Ω=0.255 5000 2 µ ρ / µ 0 Ω=0.3 4000 3 µ Ω=0.35 ν /

3000 T 2 ν 2 µ Ω=0.5

ENERGIES 2000 1 1000

0 0 100 200 300 400 500 600 700 200 400 600 800 1000 HARTMANN NUMBER REYNOLDS NUMBER

Fig. 7. Magnetic and kinetic energy for Pm = 1 and Re = Fig. 9. The normalized eddy viscosity for Pm = 1 and the 800, normalized to the diﬀusion velocity squared. The rotation laws Ω 1/R2 (µ =0.25), Ω 1/R3/2 (µ =0.35), energies are in equilibrium only for strong magnetic ﬁelds and Ω 1/R (µ∝=0.5). ∝ (µ =0.255 in this plot). ∝

produced by the instability of current-free magnetic back- 3 ground fields is an almost purely magnetic phenomenon. The dominance of the corresponding Maxwell stress over 2.5 Pm=1 the Reynolds stress turns out to be larger than the dominance of the magnetic energy over the kinetic energy. The 2 reason for this is that, due to the induction by the radial differential rotation, an almost perfect correlation appears Pm=0.5 between the radial and the azimuthal magnetic field fluctu- 1.5 ations. Nevertheless, as this induction becomes weaker for decreasing magnetic Prandtl number (Fig. 8, bottom) – and EPSILON 1 also for more shallow rotation profiles – a relation νT Pm Pm=0.1 becomes understandable (see Eq. (13)). For Pm ∝0, the → 0.5 resulting eddy viscosity should approach the small values generated by the Reynolds stress. For the special value of Pm = 1, the influence of the 0 0 200 400 600 800 1000 1200 shear on the eddy viscosity is given in Fig. 9. There is a MAGNETIC REYNOLDS NUMBER nonlinear influence of the shear on the stress tensor which is often ignored in the Boussinesq formulation (7). As Fig. 9 clearly shows, the viscosity is reduced for shallower rotation Pm=1 laws. Obviously, the viscosity itself (and not only the total 1 stress tensor) vanishes for rigid rotation. This leads to the Pm=0.5 important conclusion that the diffusive decay of differen-

0.8 tial rotation is decelerated in time, and the decay time due Pm=0.1 to magnetic instabilities becomes longer and longer dur- tot ν

/ ing the diﬀusion process. The outward transport of angular 0.6 momentum is thus self-regulated: the original steep rota- mag

ν tion law rapidly decays, producing a ﬂatter rotation law 0.4 which then decays slower. If the viscosity itself depends on the shear, the considered ﬂuid is called non-Newtonian. 0.2 This phenomenon, known to exist for current-driven instabilities (e.g. Spruit 2002), also occurs for the current-free

0 azimuthal magnetorotational instability. 500 1000 1500 2000 REYNOLDS NUMBER 5. Discussion Fig. 8. Top: Magnetic to kinetic energy ǫ for various Pm, as In the sense of a proof of existence, nonaxisymmetric mag- indicated. The magnetic energy only dominates the kinetic netic instabilities under the influence of differential rotation > energy for Pm ∼ 1. Bottom: The magnetic-induced part in are shown to transport angular momentum in a direction units of the total viscosity (µ =0.255 in this plot). orthogonal to the rotation axis. The toroidal background field is chosen as current-free (outside the axis), so that it cannot decay. The mass density is assumed as uniform, sup-

6 G. Rüdiger et al.: Viscosity by unstable magnetic toroidal fields pressing the influence of the buoyancy. The model is thus While the instability domain is only slightly modified not applicable to solar type stars on the main sequence, but for Prandtl numbers smaller than unity, this is not true only to the very hot radiative cores of (sub)giants. It is also for the behavior of the second order correlations. The ef- shown, by use of a numerical stellar model, that the high fective viscosity formed by Reynolds stress and Maxwell temperatures in such cores result in microscopic magnetic stress decreases for small Pm. It varies by one order of Prandtl numbers varying between 0.1 and 10. This makes it magnitude when Pm varies by two orders of magnitude. possible to use a nonlinear MHD code which only works for The ratio of magnetic energy to kinetic energy (taken for such Prandtl numbers. The code solves the nonlinear and maximum viscosity) also decreases for small Pm (Fig. 8, nonaxisymmetric MHD differential equations in a cylin- top). Consequently, the Maxwell stress in relation to the dric setup and provides the R-φ-component of the complete Reynolds stress also sinks for small magnetic Prandtl num- stress tensor (Reynolds stress plus Maxwell stress), and av- bers (Fig. 8, bottom). The effectiveness of the magnetic erages the resulting number over the azimuth and the axial perturbations in transporting angular momentum is thus coordinate. reduced for small Pm, i.e. with cooler temperatures. A spectral element code is used based on the hydrody- Our results support the conclusion that magnetic insta- namic code of Fournier et al. (2005). The solutions are ex- bilities of toroidal magnetic fields in the presence of differ- panded into Fourier modes in the azimuthal direction. The ential rotation are a viable mechanism to explain angular remaining meridional problems are solved with a Legendre momentum redistribution in stellar interiors, especially for spectral method. sub-giant and young red giant stars. If a linear code is used to find the classical eigenvalues for the onset of the marginal instability then all solutions References are optimized in the wave number k yielding the smallest Böhm Vitense, E. 1958, Zs. Ap., 46, 108 possible Reynolds numbers. Brandenburg, A., & Subramanian, K. 2005, Phys. Rep., 417, 1 The boundary conditions at the cylinder walls are as- Ceillier, T., Eggenberger, T., et al. 2012, Astron. Nachr., 333, 971 sumed to be no-slip and perfectly conducting. The result- Chandrasekhar, S. 1956, Proc. Natl. Acad. Sci. USA, 42, 273 Deheuvels, S., et al. 2012, ApJ, 756, 19 ing cross correlations thus vanish at the surfaces of the two Deheuvels, S., et al. 2014, preprint cylinders, while they have a maximum near the center of Demarque, P., Guenther, D.B., Li, L.H., Mazumdar, A., & Straka, the gap between the cylinders (Fig. 5). These maximum C.W. 2008, ApSS, 316, 31 values are calculated and transformed into viscosity val- Eggenberger, P., Montalbán J., & Miglio A. 2012, A&A, 544, L4 Fournier, A., et al. 2005, JCoPh, 204, 462 ues by means of Eq. (7). The resulting viscosity values are Iglesias, C.A., & Rogers, F.J. 1996, ApJ, 464, 943 compared for various Hartmann numbers in the instability Rogers, F.J., & Nayfonov, A. 2002, ApJ, 576, 1064 cone but for the same Reynolds number (see Fig. 3). If the Rüdiger, G., & Kitchatinov, L.L. 1996, ApJ, 466, 1078 same procedure is done for various Reynolds numbers and Rüdiger, G., et al. 2012, ApJ, 755, 181 various magnetic Prandtl numbers, one obtains the results Rüdiger, G., et al. 2014, MNRAS, 438, 271 Seilmayer, M., et al. 2012, Phys. Rev. Lett., 108, 244501 presented in Fig. 6 for two different rotation laws. Note Spruit, H. 2002, A&A, 381, 923 that the data for different Reynolds numbers do not be- Tayler, R.J. 1957, Proc. Phys. Soc. B, 70, 31 long to the same Hartmann number. The figures do not, Tayler, R.J. 1973, MNRAS, 161, 365 therefore, contain evolutionary scenarios. As expected, the Vainshtein, S.I., & Kichatinov, L.L. 1983, GAFD, 24, 273 consequence of the results given in Fig. 6 is that the viscosity for given shear linearly grows with the angular velocity. The most interesting result, however, is the dependence of the effective viscosity on the shear: the steeper the rotation law the higher the viscosity value. Hence, the decay of a nonuniform stellar rotation law is a nonlinear process. The decay time-scale does not remain constant in time as it becomes larger and larger. The non-Newtonian behavior of the magnetized conducting fluid is basically connected with the mechanism of the azimuthal magnetorotational instability which exists only for differential rotation. It thus leads automatically to saturation during stellar spin-down process, which is most violent at its beginning and becomes slower at later times. As mentioned, the toroidal magnetic field with the current-free radial profile does not dissipate. As we thus fix the magnetic field for a given magnetic Prandtl number and consider the dependence of the effective viscosity on the angular velocity, the effective viscosity will grow, passes its maximum close to the line of maximum growth rate shown in Fig. 3, and again sinks to zero at the lowest possible Reynolds number. Figure 6 demonstrates that, up to Reynolds numbers of the order of 103, along the line of maximum growth rate there is no saturation of the effective viscosities. The stronger the fields, the higher is the maximum viscosity which can be achieved.