arXiv:1601.04659v1 [astro-ph.SR] 18 Jan 2016 ertn&Mee 18)cniee o h iigi h sol the in chemic mixing the passive for of considered zones transport (1987) Maeder radiation extra & mild Lebreton stellar such stratified of stably generator the the magni- instabi in of of appearing sort orders some – two suggested or 1977) deca (1969, one Schatzman lithium by tude. the coefficien diffusion molecular explain effective the the under to exceed Gyr km order 1....10 about 40.000 In of region destroyed. time betw burning is process the bottom diffusion and its z a zone after convection the which the but lithium rigidly contains rotates still core solar present-day 2008). in- mass al. seems in et which abundances (Brott mixing, surface stars intensive of too observations a but with to compatible momentum lead angular not does transports which which far- stars the in stab main-sequence progen exist always must the are instability dwarfs of alternative White an cores and stratified the stars the neutron with As newborn of viscosity. explained tor the be ang of cannot of values which transport molecular km/s outward happened 10 an momentum than Obviously, lar 2005). lower t al. velocities higher et show (Berger velocities observations but equatorial have km/s, Whit 15 to the predicted like objects are compact p Less Dwarfs provide longer. observations much are the which but ods stars of neutron periods newborn rotation the yield for explosions Numerica supernova re periods. are of rotational physics ulations of stellar findings modern empirical the to in lated problems open the of Some Introduction 1. srnm Astrophysics & Astronomy uy1,2021 11, July iigo asv clrb h ntblt fadifferenti a of instability the by scalar passive a of Mixing vnslrosrain edt iia ocuin.The conclusions. similar to lead observations solar Even mligta nytria antcfilso re Gcnex can words. kG Key order of fields magnetic toroidal only that implying ol togyece h au faot10nee odescri to suppres strongly needed be 100 will about coefficient of diffusion value the however, the exceed strongly would caza 17) h ueia au ftecefiin nt in coefficient the Mm of at value numerical The (1977). Schatzman iers htas ierrlto ewe h dydiffus eddy the between normalize relation the linear a between also relation that the so linear va number diffusivity Reynolds eddy the fixed state, For latter this for Just theorem. ceceig01d aial rvd nt dydiffusivit eddy finite peak Mm provide concentration basically number global do a 0.1 of exceeding process Sc decay the by measured 0 oain h iiu antcPadlnme sdis T used a number within Prandtl out magnetic carried minimum is tem study The or Our rotation. beryllium considered. lithium, is like pinch scalar axial passive a of mixing The accepted Received; Sternwa [email protected] der An [email protected], Potsdam, Astrophysik f¨ur Leibniz-Institut . 1 and ≃ 2 2 hr h atrvlecaatrzsteslrtachocline. solar the characterizes value latter the where n ilrpdydces o ag m eas ftehg va high the of Because Mm. large for decrease rapidly will and ntblte antcfils-Dfuin-Truec Ma - Turbulence - Diffusion - fields Magnetic - Instabilities ≃ 2 h o rnisfo h lwrtto eiet h fast-r the to regime slow-rotation the from transits flow the aucitn.paredes no. manuscript .Prds .Glet n .R¨udiger G. and Gellert, M. Paredes, A. oaigailpinch axial rotating developed uspa nterdpnec nteRyod ubro h gl the of number Reynolds the on dependence their in peak lues must t sim- l 0 vt n h oeua icst eut xcl ntesen the in exactly results viscosity molecular the and ivity e yteslrrtto fismgei ahnme fulfills number Mach magnetic its if rotation solar the by sed ms 1 aus eas n htfrqaiKpe oaina magn a at rotation quasi-Kepler for that find also We values. y ABSTRACT . as – t 6 -48 osa,Gray e-mail: Germany, Potsdam, D-14482 16, rte 05 han one een eri- als. lity eaueflcutosdet h antcTye instabilit Tayler magnetic the to due fluctuations perature yo-oet eu o w oainlw:qaiKpe an quasi-Kepler laws: rotation two for setup aylor-Couette ive i eain eecle h caza ubr ilraha reach will number, Schatzman the called here relation, his s nteslrtachocline. solar the in ist u- ar ly i- y e - hl h oeua cmd ubrS ftefli aisbetw varies fluid the of Sc number Schmidt molecular the while oae ewe ohclne al.W n htol models only that find We walls. cylinder both between located dydfuiiyadteShitnme ftefli salway is fluid the of number Schmidt the and diffusivity eddy d etemxn eo h oa ovcinzn.Atrorresu our After zone. convection solar the below mixing the be nefciedfuiiycefiin o h iigi numer is mixing the for coefficient diffusivity effective An h oa lsasotyblwteslrtcoln.Bu et Brun tachocline. solar with the work below (1998) shortly plasma solar the oa aitv oesat rmtemlclrvalue molecular the from starts cm zone radiative solar ilns ffs oaindet h o nfriyo h fie the of than uniformity more non of the viscosity to the A due by lines. formed rotation is fast problem rotation of A The lines. ‘islands’ field core. the the along within uniform comes field magnetic interac poloidal rotation Kitchatinov differential fossil R¨udiger & internal magnitude. v that of molecular assume orders its (1996) two exceed than must more viscosity by the interior solar the al et (Brun instability hydrodynamical postulated tically etv ifsvt ugssamgei akrudo h p v the momentum angular of transport background fluctuations Magnetic magnetic nomena. a suggests diffusivity fective th to (2012). core al. the et from Ceillier transport envelope momentum cod evolution angular stellar fast extra the times than without five less is least early This at surface. spinning the the core than of a profile with 7341231 rotation KIC internal giant an derive 2014) gi (2012, red obtaine the results K of asteroseismic law the with rotation consistent internal 8366239 an produce KIC to (2012) al. et u uhatfiilpasi h eutn oainlaws. rotation resulting the in peaks artificial such out hsi ahrsalvlewihlasto leads which value small rather a is This h oaino caza,seZh 90 with 1990) Zahn see Schatzman, of notation the oe relation a model 2 nteohrhn,i re orpouetergdrtto of rotation rigid the reproduce to order in hand, other the On htteefcievsoiyi tograpie hntee the than amplified stronger is viscosity effective the That h au of value The sadreaches and /s nthdoyais(MHD) gnetohydrodynamics uso h oa atannme,teShtmnnumber Schatzman the number, Hartmann solar the of lues EPLER tto eiedmntdb h Taylor-Proudman the by dominated regime otation iso eke l 21) lo eevl tal. et Deheuvels Also, (2012). al. et Beck mission D 3 Re D ∗ · 10 ∗ = ∗ ≃ 4 ≃ Re cm 10 20 ∗ 2 2 ν .... si lorpre yEggenberger by reported also is /s hl oerfie oe fthe of model refined more a while o h ifso ofcet(after coefficient diffusion the for 4 10 ne h nuneo heuris- a of influence the under 4 [email protected], cm 2 si eddt smooth to needed is /s D epooe by proposed se farotating a of y ∗ blrotation. obal solid-body d ∼ < Mm maximum tcMach etic

10 c Re 3 S 2021 ESO ically ally ≫ ∗ with cm D een swt a with ts lts, 1999). . ≃ syield es 1 s 2 ≃ sfor /s by d 100 alue be- red he- ant 10 al. er ld ia f- 1 e . A. Paredes, M. Gellert, G. R¨udiger: Mixing of a passive scalar ∂B the Maxwell stress but they do not transport chemicals. Most Rm curl(U B) = ∆B (7) sorts of MHD turbulence should thus provide higher eddy vis- ∂t − ×   cosity values rather than eddy diffusivity values, i.e. both Prandtl and div U = div B =0. These equations are numerically solved numbers as well as the Schmidt number must not exceed unity. for no-slip boundary conditions and for perfect-conducting In the present paper the Tayler instability (Tayler 1957, 1973) cylinders which are unbounded in axial direction. Those bound- under the presence of (differential) rotation is probed to pro- ary conditions are applied at both Rin and Rout. The dimension- duce diffusion coefficients for passive scalars. By linear theory less free parameters in Eqs. (7) are the Hartmann number (Ha) the instability map is obtained for the unstable nonaxisymmetric and the (Re), mode with m = 1. The eigenvalue problem is formulated for a cylindrical Taylor-Couette container where the gap between B R Ω R2 both rotating cylinders is filled with a conducting fluid of given Ha = in 0 , Re = in 0 , (8) magnetic . Inside the cylinders homogeneous ax- √µ0ρνη ν ial electric-currents exist which produce an azimuthal magnetic field with the fixed radial profile Bφ R which – if strong where R0 = Rin(Rout Rin) is the unitoflengthand Bin the enough – is unstable even without rotation.∝ It is known that for azimuthal magnetic field at− the inner cylinder. With the magnetic of order unity a rigid rotation strongly Prandtl numberp ν suppresses the instability but – as we shall show – a differen- Pm = (9) tial rotation with negative shear re-destabilizes the flow so that η a wide domain exists in the instability map wherein the nonlin- the of the rotation is Rm = Pm Re. ear code provides the spectra of the flow and field fluctuations. For the magnetic Prandtl number of the solar tachocline Gough Between the rotating cylinders a steep radial profile for the con- (2007) gives the rather large value Pm 0.05. There are even centration of a passive scalar is initially established which de- smaller numbers down to Pm = 10−4≃under discussion (see cays in time by the action of the flow fluctuations. The decay Brandenburg & Subramanian 2005). However, for the aforemen- time is then determined in order to find the diffusion coefficient. tioned red giants one finds Pm of the order of unity (R¨udiger et al. 2014). The code which solves the equation system (7) is 2. The rotating pinch described in Fournier et al. (2005) where also the detailed for- mulation of the possible boundary conditions can be found. For In a Taylor-Couette setup, a fluid with microscopic viscosity ν the present study only perfect-conducting boundaries have been and magnetic diffusivity η =1/µ0σ (σ the electric conductivity) considered. and a homogeneous axial current J = curl B are considered. The equations of the system are ∂U 1 1 + (U )U = P + ν∆U + curl B B, (1) ∂t · ∇ ρ∇ µ0ρ × ∂B = curl (U B)+ η∆B (2) ∂t × with div U = div B = 0 where U is the actual velocity, B the magnetic field and P the pressure. Their actual values may be split by U = U¯ +u, and accordinglyfor B and the pressure. The general solution of the stationary and axisymmetric equations is b U¯ = RΩ = aR + , U¯ = U¯ =0, (3) φ R r z

B¯φ = AR, B¯r = B¯z =0 (4) with µ r2 1 µ a = − in Ω , b = − Ω R2 (5) 1 r2 in 1 r2 in in − in − in and with A = Bin/Rin. Here a and b are constants and A rep- resents the applied electric-current. The rotating pinch is formed by a uniform and axial mean-field electric-current. The solutions U¯φ and B¯φ are governed by the ratios

Rin Ωout rin = , µ = , (6) Rout Ωin Fig. 1. Stability map for m = 1 modes for the pinch with rigid where Rin and Rout are the radii of the inner and the outer cylin- Ω Ω rotation (top) and with quasi-Kepler rotation law (µ = 0.35, der, in and out are their rotation rates. bottom). The critical Hartmann number for resting cylinders is Equations (1)–(2) in its dimensionless form become Ha = 35.3 for all Pm. The curves are marked with their values U of Pm, the curve for Pm = 0.001 also represents all curves for ∂ 2 Re + (U )U = P + ∆U + Ha curl B B, smaller Pm. rin =0.5, perfect-conducting boundaries. ∂t · ∇ −∇ ×   2 A. Paredes, M. Gellert, G. R¨udiger: Mixing of a passive scalar

Figure 1 (top) shows the map of marginal instability for the rigidly rotating pinch with rin = 0.5 and for various Pm. The rotating fluid is unstable under the presence of a magnetic field with parameters on the right-hand side of the lines. It also pro- vides the influence of the magnetic Prandtl number on the rota- tional suppression. The Pm-influence completely disappears for the resting pinch with Re = 0. Note that the rotating pinch is massively stabilized for magnetic Prandtl numbers Pm 0.1. For very small magnetic Prandtl number the curves become≥ in- distinguishable, i.e. the marginal instability values under the in- fluence of rigid rotation scale with Re and Ha for Pm 0. This is a standard result for all linear MHD equationsin the induc→ tion- less approximation for Pm = 0 (if such solutions exist). On the other hand, the rigidly rotating pinch belongs to the configura- tions with the same radial profiles for velocity (here U¯φ R) ∝ (a) Re = 500 (b) Re = 600 and magnetic field (here B¯φ R) defined by Chandrasekhar (1956). One can even show that∝ all solutions fulfilling this con- dition scale with Re and Ha for Pm 0 (R¨udiger et al. 2015a). These facts imply that for a fixed magnetic→ resistivity smaller molecular destabilize the rotating pinch. The situation changes if the two cylinders are no longer ro- tating with the same angular velocity because also the shear en- ergy is now able to excite nonaxisymmetric magnetic instability patterns by interaction with toroidal fields which are current-free within the fluid. In this paper we shall present the results for the interaction of shear with the azimuthal magnetic field which is due to the axial electric-current which defines the pinch. The bottom panel of Fig. 1 gives the instability map for the magnetic instability of a fluid in quasi-Kepler rotation (µ = 0.35) for various Pm. It shows the influence of the differential rotation on the instability map of the rotating pinch. Again, of (c) Re = 700 (d) Re = 800 course, the critical Hartmann number for resting cylinders does not depend on the magnetic Prandtl number but in addition, the Fig. 2. Isolines for the radial component of the velocity in units borderlines of the unstable region for all Pm 1 do hardly dif- of ν/D. After the Taylor-Proudman theorem for faster rotation fer. For the given Reynolds number ranges, the≤ rotational sup- the axial wavelength becomes longer and longer and the radial pression almost disappears for Pm < 1. For Pm = 1 and for rms value of the velocity sinks. Ha = 80, µ =0.35, Pm =0.1. Re < 400 the instability becomes even subcritical and the ro- tational stabilization changes to a rotational destabilization. For too fast rotation, however, the subcritical excitation disappears Fig. 1, bottom). This is an important point in the following dis- but the rotational suppression is weaker than it is for rigid rota- cussion. tion. According to Fig. 1 the value Pm = 0.1 which is mainly The plots of Fig. 2 represent the radial velocity which basi- used in the calculations below already belongs to the small-Pm cally produces the radial mixing. The instability is nonaxisym- system. metric, the velocity amplitude does hardly vary for fast rotation but the rms velocity of u decreases by a factor of 1.6 between The flow pattern of the instability is shown in Fig. 2 R Re = 500 and Re = 700 reaching a saturation value while the for quasi-Kepler rotation of growing Reynolds numbers. The axial flow perturbation starts to rise (Fig. 3). Under the influence Hartmannnumber is fixed at Ha = 80. The magnetic Mach num- of fast rotation a turbulence field which is isotropic in the lab- ber of rotation oratory system becomes more and more anisotropic reaching a 2 2 2 Ωin √PmRe relation u u + u for the volume-averaged velocity. Mm = = (10) h zi ≃ h Ri h φi ΩA Ha One finds from Figs. 2 and 3 that the anisotropy – or, with other words, the transition from slow rotation to fast rotation – starts reflects the rotation rate in units of the Alfv´en frequency at Re 600 or Mm 2, respectively. Ω 2 ≈ ≃ A = Bin/ µ0ρR0. Almost all cosmical objects possess large This statement is supported by the behavior of the axial magnetic Mach numbers. E.g., the White Dwarfs rotate with wavelength. Within the same interval also the axial wavelength– about 2 km/sp while the observed magnetic field with (say) 1 MG which after the Taylor-Proudman theorem should grow for faster leads to an Alfv´en-velocity of about 3 m/s so that Mm 700. rotation – seems to jump by the same factor. The question will This value even exceeds unity if the largest ever observed≃ mag- be whether for Reynolds numbers of about 600 also the diffusion netic fields of 100 MG are applied. Inserting the characteristic coefficient jumps. 10 values for the solar tachocline (R0 = 1.5 10 cm, ρ = 0.2 It is interesting that the estimation uR,rmsL/3 for any kind 3 · g/cm ) one finds Mm = 30/Bφ with Bφ in kG so that with of turbulent diffusivity leads to the maximum value 10ν for < > ≈ Bφ ∼ 1 kG also the tachocline with Mm ∼ 30 belongs to the the diffusion coefficient. This rather small value does not fulfill class of rapid rotators. The upper panel of Fig. 1 demonstrates the constraints by Schatzman and Lebreton & Maeder described that pinch models with Mm > 1 and rigid rotation are stable above. The nonlinear simulations will show whether this prelim- but they easily become unstable if they rotate differentially (see inary result is confirmed or not.

3 A. Paredes, M. Gellert, G. R¨udiger: Mixing of a passive scalar

30 saturation is achieved when the magnetic and kinetic energy of U R each mode is saturated. Second, the transport equation (11) is U switched on and several simulations with different Sc numbers z 25 are performed. This two steps are repeated for several Re while all other parameters remain fixed. Since the diffusion leads to a homogenization of the pas-

rms 20 sive scalar profile, the quantity C¯ will exponentially decay in U a characteristic time τ which is directly related to the effective Schmidt number. This process will occur whether the magnetic- 15 induced instability is present or not. It will be considered two decay times, i.e. τ ∗ and τ. The first is computed from a simula- tion where the instability is present and the second is computed 10 ¯ 500 700 900 1100 1300 1500 from a simulation where C evolves alone. Both decay times can REYNOLDS NUMBER inversely be related to their diffusivity, i.e. ∗ Fig. 3. The radial and the axial rms velocity components uR and D τ < = ∗ 1. (15) uz for Mm ∼ 6. Ha = 80, µ =0.35, Pm =0.1. D τ − To compute the decay time, the maximum of the radial profile 3. The diffusion equation C¯ is plotted at fixed time steps. The characteristic time τ is the e-folding of the resulting profile. For the Tayler unstable system the eddy diffusion of a passive scalar in radial direction is computed. To this end, the additional dimensionless transport equation 4. Results ∂C 1 Numerical simulations are carried out in a Taylor-Couette con- + (UC)= ∆C (11) ∂t ∇ · Sc tainer with periodic boundary conditions in the axial direction. Additionally to the boundary conditions for the velocity and the for a passive scalar, C, is added to the equation system (7). This magnetic field, Neumann boundary conditions at both Rin and passive scalar can be the temperature or a concentration func- Rout are applied for the passive scalar C. In order to focus only tion of chemicals like lithium or beryllium. Here the microscopic on the radial transport, the initial condition for the passive scalar Schmidt number is chosen to be axisymmetric and constant in the -direction. ν C0 z Sc = (12) These characteristics on C0 will allow the evaluation of the in- D crement on the diffusivity in the radial direction. Thus C0 is is used in Eq. (11), where D is the molecular diffusivity of the taken as a radius-dependent Gaussian centered at the middle of fluid. The Schmidt number for gases is of order unity while it the gap, i.e. is O(100) for fluids. Gough (2007) gives Sc 3 with a (radia- 2 2 r r0 tive) viscosity of 27 cm /s for the plasma of the≃ solar tachocline. C0 = exp − (16) − 0.1 In the present paper the molecular Schmidt number is varied   ! from Sc = 0.1 to Sc = 2. In their simulations with a driven with r0 = 0.5(Rin + Rout). Since the boundary conditions are turbulence probing the Boussinesq type of the diffusion process periodic in the z-direction and Neumann at cylinder walls, the Brandenburg et al. (2004) are also using Sc =1 for the molecu- initial condition will evolve towards a constant profile in the en- lar Schmidt number. tire cylinder. The maximum of C¯ at each time step is plotted and When the instability is completely developed, it will influ- the characteristic decay time can be computed. ence the transport properties of the fluid. This influence might We mainly work with a quasi-Kepler rotation profile which be isotropic or anisotropic, thus different in radial and axial di- is unstable for the given value of Re = 500 (Fig. 4). The thresh- rection. For the latter Nemri et al. (2012)and Akonur& Lueptow old value for the onset of the instability is Ha 35 from which (2002) find a linear dependence between D and Re for the hy- value on the effective diffusivity grows. At Ha≃= 158 the mag- drodynamic system with resting outer cylinder. We focus on the netic Mm becomes unity defining the regimes of radial direction. If D is considered as the molecular diffusivity, slow rotation (Mm < 1) and fast rotation (Mm > 1). In the slow its modification can be modeled by an effective diffusivity rotation regime, the normalized diffusivities grow with growing ∗ Ha while they sink with decreasing Ha in the fast rotation regime D = D + D , (13) eff where the rotation is fast compared to the magnetic field. Figure ∗ where D∗ is only due to the magnetic-induced instability. The 4 also demonstrates that D /D linearly increases for increasing ∗ Sc so that simply D∗ ν results for Sc > 0.1. It is the molec- goal is to compute the ratio D /D as a function of Re and Ha. ∝ When averaging Eq. (11) along the toroidal and azimuthal direc- ular viscosity alone which determines the diffusion effect of the tions, the mean-field value C¯ follows magnetic-induced instability – just in the sense of Schatzman. The resulting ratio D∗/D for quasi-Kepler rotation and for ¯ a fixed Hartmann number is shown in Fig. 5. Now the magnetic ∂C U¯ ¯ 1 ¯ + C = ∆C, (14) Mach number becomes unity at Re . In the slow rota- ∂t ∇ · Sceff = 253 tion regime, the effective diffusivity hardly changes. The fact  ∗ with the effective Schmidt number Sceff = ν/Deff . that even without rotation the ratio D /D is different from zero To measure the ratio D∗/D, two steps are followed. First, is because the flow is unstable even for Re =0. In the fast rota- a numerical simulation of Eqs. (7) is performed until the insta- tion regime, the ratio D∗/D increases in a monotonic way until bility is fully developed and energy saturation is reached. The it peaks at about Mm 2. Finally, for faster rotation (Mm > 2) ≃

4 A. Paredes, M. Gellert, G. R¨udiger: Mixing of a passive scalar

Mm by the instability which modifies the radial velocity magnitude 8 4 2 1 0.8 and the wave number. As shown in Fig. 2 the magnitude of the 12 radial velocity component hardly changes from Re = 400 to Re = 800 while the wavelength increases between Re = 400 ∗ 10 and Re = 700. Thus the decreasing of the ratio D /D is due to 2 the decreasing of the wave number. 8 16 /D

* 6 D 1 14 4 12 0.5 2 10 Ha=200, Re=1200 /D

* 8 0 D Ha=120, Re=800 0 50 100 150 200 6 Ha 4 Ha=80, Re=500 Fig. 4. The normalized diffusivity D∗/D vs. the Hartmann num- 2 ber (bottom horizontal axis) and the magnetic Mach number (top 0 horizontal axis). The curves are marked with the values of the 0 0.5 1 1.5 2 Schmidt number Sc. The value Mm = 1 separates the regimes Sc of slow and fast rotation. Re = 500, µ =0.35, Pm =0.1. Fig. 6. D∗/D as a function of Sc for those Reynolds numbers yielding the maximal instability-induced diffusivities. The mag- Mm netic Mach number slightly exceeds 2 in all cases. µ =0.35, Pm =0.1. 0 1 2 3 4 5 6 8 Figure 6 shows the normalized diffusivity for µ = 0.35 7 for fixed Ha as a function of Sc. For all Reynolds numbers the 2.0 induced diffusivities D∗ are different from zero and the ratio 6 D∗/D scales linearly with Sc. The figure only shows the rela- 5 tion for those Reynolds numbers for which the maximum dif- fusivities for magnetic Mach numbers about 2 are reached. For

/D ∗

* 4 Sc 0 the diffusivity D appears to vanish. Hence, for molecu-

D lar→ Schmidt numbers Sc > 0.1 the essence of Fig. 6 is the linear 3 1.0 relation between D∗/D and Sc. In the notation of Schatzman (1977) it means that 2 ∗ ∗ D = Re ν (17) 1 0.5 ∗ with the scaling factor Re (which indeed forms some kind of a 0 Reynolds number). Also the Figs. 4 and 5 demonstrate that lin- 0 400 800 1200 1600 ear relations hold for all considered Re and Ha. The Schatzman ∗ Re factor Re after Fig. 6 grows with growing Ha while for all three models the magnetic Mach number is nearly the same. We find ∗ ∗ Fig. 5. The same as in Fig. 4 but in relation to the Reynoldsnum- Re ∼< 4 for Ha = 80 growing to Re ∼< 8 for Ha = 200; a sat- ber (bottom horizontal axis) and the magnetic Mach number (top uration for larger Ha is indicated by the results presented in Fig. horizontal axis) for Ha = 80. The value Mm = 1 separates the 6. It is not yet clear whether for large Ha an upper limit exists ∗ regimes of slow and fast rotation. Note the reduction of D /D for Re∗ due to numerical limitations. for Mm > 2. µ =0.35, Pm =0.1.

5. Conclusions the effective diffusivity decays due to the rotational quenching. In all cases, however, the effective normalized diffusivity grows The influence of the current-induced instability on the effective with growing Schmidt number so that also here simply D∗ ν diffusivity in radial direction of a rotating pinch has been stud- without any influence of the microscopic diffusivity. The mi∝ss- ied for different rotation laws. The diffusion equation is numer- ing factor in this relation is simply given by the curve for Sc =1 ically solved in a cylindric setup under the influence of stochas- in Fig. 5. tic fluctuations which are due to the magnetic Tayler instability. For all Schmidt numbers, the ratio D∗/D increases mono- The conducting fluid between two rotating cylinders becomes tonically until a certain Re is reached, beyond it decreases. This unstable if an axial uniform electric-current is strong enough. behavior can be understood by the fact that the ratio D∗/D The rotation law between the two cylinders is fixed by their ro- must be a direct function of the radial velocity magnitude and tation; our main application is a quasi-Kepler rotation which re- the wave number of the solution, since D∗ is produced solely sults when the cylinders are rotating like planets. The magnetic

5 A. Paredes, M. Gellert, G. R¨udiger: Mixing of a passive scalar

Prandtl number of the fluid is fixed to the value of 0.1 while its 9 Schmidt number (12) is a free parameter of the model. Pm=0.1 The main result is a strictly linear relation between the result- 8 Ha=200 ∗ ing normalized eddy diffusivity D /D and the given Schmidt 7 number. The Schatzman relation (17) – which also describes our result that for small Schmidt number the eddy diffusivity is neg- 6 ligibly small – has thus been confirmed in a self-consistent way. 5 The model also provides numerical values for the scaling Ha=120 ∗ 4 factor Re which increases for increasing Hartmann number of (Schatzman) * the toroidal field. For Schmidt numbers ∼< 0.1 the diffusivity due 3 ∗ Re to the magnetic instability is negligibly small, however Re is 2 Ha=80 different from zero and even exceeds unity in our computations. ∗ Already for the value Ha = 80 one finds Re ∼< 4 for quasi- 1 Ha=60 Kepler rotation and this value increases for increasing Hartmann 0 number (Fig. 6). 0 400 800 1200 1600 The second result concerns the role of the magnetic Mach Re number Mm which represents the global rotation in relation to the magnetic field strength. For slow rotation the eddy diffusiv- 8 ity runs linear with Mm but in all cases a maximum of D∗/D Ha=120 exists at Mm 2. For faster rotation the induced diffusion is 7 suppressed and≃ finally seems to remain constant (Fig. 5). This 6 Pm=0.05 fast-rotation phenomenon may be a consequence of the Taylor- Proudman theorem after which the axial fluctuations are favored 5 in expense of the radial ones. Also the correlation lengths in ax- 4 ial direction grow for growing Reynolds numbers. Both conse- (Schatzman) quences of the Taylor-Proudman theorem appear to retard the * 3 Pm=0.1 growth of the radial diffusion in stars. Re 2 Figure 7 summarizes the results of this study by presenting numerical values of Re∗ for Kepler rotation laws. In the upper 1 ∗ Pm=0.5 panel Re is given for four values of the Hartmann numbers as a 0 function of the Reynolds number. Without rotation one finds that 0 200 400 600 800 1000 ∗ Re Ha is realized which for solar/stellar Ha-values would Re produce∝ even higher Re∗-values that the ones found in our com- putations. With rotation, a maximum of Re∗ exists for approx- imately one and the same magnetic Mach number (Mm 2) 9 ∗ ≃ where the value of Re strongly increases with the Hartmann 8 number. The higher the Hartmann number the larger Re∗, but this relation is not strictly linear as a mild saturation may exist. 7 ∗ For Mm 2 the rotational quenching leads to Re even smaller 6 than the values≫ for Re =0. 5 The numerical restrictions of our code prevent calculations for higher Reynolds numbers but we are able to vary the mag- 4 (Schatzman) * netic Prandtl number. This might be necessary as Pm in the solar 3 tachocline is certainly smaller than 0.1. Figure 7 (middle panel) Re shows the clear result that Re∗ is anticorrelated with the mag- 2 ∗ netic Prandtl number. The smaller Pm the larger Re , without 1 rotation there is even Re∗ 1/Pm. For fast rotation the results do not exclude the possibility∝ that a saturation may occur for 0 Pm 0.1 so that the influence of Pm becomes weaker as the 0 2 4 6 8 10 close≪ lines of marginal instability in Fig. 1 for small Pm suggest. Mm ∗ The bottom panel completes the picture showing Re as a ∗ Fig. 7. The Schatzman number vs. Re as functions of Ha function of Mm for all simulations used in this study. It shows Re ∗ (top panel), of Pm (middle panel) and of Mm (bottom panel). that Re has a maximum at Mm 2 and rapidly decreases for ∗ In the bottom panel all the simulations of this study are shown. large Mm for which it seems to saturate≃ around Re =1. ∗ µ =0.35. A final answer of how large the Schatzman number Re may become by the presented mechanism of nonaxisymmetric insta- bilities of azimuthal fields can not yet be given, mainly due to numerical limitations. All the used dimensionless numbers such and by rescaling to the real solar/stellar Hartmann numbers the as the Reynolds number and the magnetic Mach number are dif- resulting Re∗-values would become much too high. The curves ferent from the real (solar) numbers by orders of magnitudes. in Fig. 7, however, demonstrate a dramatic rotational suppres- That the model, proposed in this paper to estimate the magnetic sion of the diffusion process for higher Mm so that the small induced extra diffusivity, directly leads to the original formula- values Re∗ 100 which are necessary to explain the obser- tion of Schatzman seems to be a highly motivating result. Using vations may≃ easily result from the intensive quenching by the the maximal Re∗ values of our curves (which hold for Mm 2) global rotation belonging to magnetic Mach numbers exceeding ≃

6 A. Paredes, M. Gellert, G. R¨udiger: Mixing of a passive scalar

(say) 30. A theoretical explanation of the slow diffusion effects Brandenburg, A., & Subramanian, K. 2005, Phys. Rep., 417, 1 with magnetic instabilities, therefore, requires differential rota- Brott, I., Hunter, I., Anders, P., et al. 2008, AIP Conf. Proc., 990, 273 tion and weak fields (of order kG) as otherwise the mixing would Brun, A. S., Turck-Chi`eze, S., & Morel, P. 1998, ApJ, 506, 913 Brun, A. S., Turck-Chi`eze, S., & Zahn, J. P. 1999, ApJ, 525, 1032 be too effective. Chandrasekhar, S. 1956, Proc. Natl. Acad. Sci. USA, 42, 273 Ceillier, T., Eggenberger, T., et al. 2012, Astron. Nachr., 333, 971 Deheuvels, S., Garcia, R., Chaplin, W.J., et al. 2012, ApJ, 756, 19 10 Deheuvels, S., Dogan, G., Goupil, M.-J., et al. 2014, A&A 2014, 564, 27 Eggenberger, P., Montalb´an J., & Miglio A. 2012, A&A, 544, L4 Fournier, A., Bunge, H.-P., Hollerbach, R., & Vilotte, J.P. 2005, J. Comput. Phys., 204, 462 Gough, D. 2007, in The Solar Tachocline, ed. D. Hughes, R. Rosner & N. Weiss

η (Cambridge University Press) / * 1 Lebreton, Y., & Maeder, A. 1987, A&A 175, 99 η Nemri, M., et al. 2012, ChERD, 91, 2346 R¨udiger, G., & Kitchatinov, L.L. 1996, ApJ, 466, 1078 R¨udiger, G., Gellert, M., Schultz, M., et al. 2014, MNRAS, 438, 271 Ha=200,Re=1200 R¨udiger, G., Schultz, M., Stefani, F., & Mond, M. 2015a, ApJ, 811, 84 0.1 R¨udiger, G., Gellert, M., Spada, F., & Tereshin, I. 2015b, A&A, 573, 80 10-3 10-2 10-1 R¨udiger, G., Paredes, & Gellert, M. 2016, in prep. Schatzman, E. 1969, Astrophys. Lett., 3, 139 Pm Schatzman, E. 1977, A&A, 56, 211 ∗ Tayler, R.J. 1957, Proc. Phys. Soc. B, 70, 31 Fig. 8. The normalized instability-induced diffusivity η /η for Tayler, R.J. 1973, MNRAS, 161, 365 the differentially rotating pinch as a function of Pm for the Zahn, J. P. 1990, in Rotation and Mixing in Stellar Interiors, ed. M.-J. Goupil, & ∗ Reynolds and Hartman number yielding the maximal Re . The J.-P. Zahn, Lecture Note of Physics 336 (Springer Verlag) p. 141 magnetic Mach number is about 2 (see Fig. 7). µ =0.35.

The presented instability model bases on the simultaneous existence of differential rotation and toroidal magnetic field. It will thus finish after the decay of one of the two ingredients. The question is which of them decays faster by the instability- induced diffusion. Provided the characteristic scales of the dif- ferential rotation and the magnetic field are of the same order (as it is the case for the magnetized Taylor-Couette flows) then the ratio of the decay times of the magnetic field and the differential rotation is ∗ τmag ν + ν = ∗ . (18) τrot η + η As the angular momentum transport is also due to the Maxwell stress of the fluctuations the turbulent viscosity always consid- erably exceeds the molecular viscosity which – for small Pm – is not the case for the magnetic resistivities. We always find for the rotating pinch the η∗ to be of order of η (Fig. 8). The instability-induced η∗ result from the defining relation η∗ = uφbR uRbφ /2A of the instability-originated axial compo- nenth of− the electromotivei force u b (R¨udiger et al. 2016). At least for small magnetich Prandtl× i number, therefore, the instability does not basically accelerate the decay of the fossil ∗ magnetic field. Hence, τmag/τrot (ν /ν) Pm. This expression decreases with decreasing magnetic∝ Prandtl number as also the normalized turbulent viscosity ν∗/ν sinks for decreasing Pm for fixed Reynolds number (R¨udiger et al. 2015b). For small mag- netic Prandtl number, therefore, the differential rotation never decays faster than the magnetic field which by itself decays at the long microscopic diffusive timescale. Only for large mag- netic Prandtl number the magnetic angular momentum transport might stop the instability prior to the decay of the fossil field.

Acknowledgements. This work was supported by the Deutsche Forschungsgemeinschaft within the SPP Planetary Magnetism.

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