Nonlinear regimes in mean-field full-sphere dynamo
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 26 September 2018 (MN L A TEX style file v2.2)
Nonlinear regimes in mean-field full-sphere dynamo
V.V. Pipin (cid:63) Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Irkutsk, 664033, Russia
26 September 2018
ABSTRACT
The mean-field dynamo model is employed to study the non-linear dynamo regimesin a fully convective star of mass 0.3 M (cid:12) rotating with period of 10 days. For theintermediate value of the parameter of the turbulent magnetic Prandl number, P m T =3 we found the oscillating dynamo regimes with period about 40Yr. The higher P m T results to longer dynamo periods. If the large-scale flows is fixed we find that thedynamo transits from axisymmetric to non-axisymmetric regimes for the overcriticalparameter of the α effect. The change of dynamo regime occurs because of the non-axisymmetric non-linear α -effect. The situation persists in the fully non-linear dynamomodels with regards of the magnetic feedback on the angular momentum balanceand the heat transport in the star. It is found that the large-scale magnetic fieldquenches the latitudinal shear in the bulk of the star. However, the strong radial shearoperates in the subsurface layer of the star. In the nonlinear case the profile of theangular velocity inside the star become close to the spherical surfaces. This supportsthe equator-ward migration of the axisymmetric magnetic field dynamo waves. Itwas found that, the magnetic configuration of the star dominates by the regular non-axisymmetric mode m=1, forming Yin Yang magnetic polarity pattern with the strong(>500 G) poloidal magnetic field in polar regions. Key words:
Activity - stars: magnetic field - stars: dynamo
Stars with the extended convective envelopes demonstratethe high level of magnetic activity (Reid & Hawley 2005;Donati & Landstreet 2009; Linsky & Schöller 2015). Itis commonly believed that the magnetic activity of thesestars origins from the hydromagnetic turbulent dynamo ac-tion (Brandenburg & Subramanian 2005; Brun et al. 2014).Extremely high magnetic activity was found on the fully-convective low-mass stars which belong to the M-dwarfsbranch of the low main sequence of the Hertzsprung-Rawsseldiagram. Observations of the M-dwarfs indicated the ratherstrong large-scale magnetic field with strength of several kG(Saar & Linsky 1985; Saar et al. 1986; Johns-Krull & Valenti1996; Linsky & Schöller 2015). The magnetic topology of theM-dwarfs is likely depends on the mass and the rotation pe-riod of a star (Donati & Landstreet 2009; See et al. 2016).Observations indicate that the early type M-dwarfs with themoderate period of rotation about 4-5 days demonstrate thestrong non-axisymmetric magnetic field with the dominanttoroidal component Donati et al. (2008). The extremely fastrotating early type M-dwarfs with period of rotation lessthan 1 day indicate the transition to the axisymmetric dy-namo with the dominant poloidal component of the large- (cid:63) email: [email protected] scale magnetic field. Situation become complicated on themid and late-type M-stars which have the masses less than . M (cid:12) as they could show either the strong axisymmetricdipole-kind large-scale magnetic field, or the low-strengthnon-axisymmetric magnetic field (Morin et al. 2008, 2010).Thus we can conclude about three basic states of the dynamoon the fast rotating M-dwarfs, they are: the strong multipolemagnetic field (hereafter, SM), the strong dipole field (here-after SD) and the weak multipole (hereafter WM) magneticfield. We follow notation suggested by Morin et al. (2011).Interesting that simultaneously with multiply states of thedynamo regimes, the dynamo generated total magnetic fluxdo not show the rotation-activity connection which is knownamong the solar type stars (Mohanty & Basri 2003).Observed magnetic properties of the M-stars initiatedthe number of the theoretical studies employing the mean-field models (see, Chabrier & Küker 2006; Elstner & Rüdi-ger 2007; Kitchatinov et al. 2014; Shulyak et al. 2015) andthe direct numerical simulations (e.g., Dobler et al. 2006;Browning 2008; Dormy et al. 2013; Schrinner et al. 2014).Using the results of the numerical simulations, Morin et al.(2011) suggests that the bi-stability of the magnetic topol-ogy on the late-type M-stars could result from two typesof the convection regimes occurred in the fast rotating con-vective bodies (Roberts 1988). Also, the direct numericalsimulations show the differential rotation is important part c (cid:13) a r X i v : . [ a s t r o - ph . S R ] S e p V.V. Pipin of the dynamo in the fully convective stars. Similar conclu-sions were suggested by Shulyak et al. (2015) after studyingthe linear dynamo regimes.Current interpretation of the dynamo bi-stability givenby Morin et al. (2011) suggests that the strength of thelarge-scale magnetic field is compatible with the nonlinearbalance between the Lorentz and Coriolis force in case ofthe SD-type magnetism and it is established by the Lorentz-inertia force balance in case of the WM magnetism. Later,Schrinner et al. (2014) found that in the anelastic simula-tions the separation between the SD and WM magnetismis less profound than they as well as others (e.g., Simitev& Busse 2009) found with the Boussinescue approximation.The origin of the strong multipolar magnetic field on themoderate rotating early M-stars is barely studied. Results ofthe mean-field models and the numerical simulations suggestthe dynamo on these stars could operate with help of the dif-ferential rotation. The linear analysis of Kitchatinov et al.(2014) show that the axisymmetric magnetic field modeshave the smaller critical threshold of the dynamo instabilitythan the non-axisymmetric ones. Thus, the transition fromaxisymmetric to non-axisymmetric dynamo can occur onlyin the nonlinear regime.The paper we study the nonlinear dynamo models for afully convective star. Here we restrict ourselves to the samecase of the star discussed earlier by Shulyak et al. (2015),i.e., the star of mass 0.3 M (cid:12) of 1Gyr age and rotating withperiod of 10 days. We will address the axisymmetric andnon-axisymmetric dynamo regimes with regards for the non-linear back reaction of the large-scale magnetic field on the α -effect and the large-scale flow. The solution of the dy-namo problem is coupled with the solution of the mean an-gular momentum balance and the mean heat transport inthe convective sphere. The main goal of the paper is to findthe typical topology of the large-scale magnetic field in thenonlinear dynamo for the given rotation period and investi-gate the nonlinear effects on the dynamo. We consider a fully convective star of mass 0.3 M (cid:12) of 1Gyrage and rotating with period of 10 days. The reference in-ternal thermodynamic structure of the star was calculatedusing the MESA stellar evolution code, the version r7503,(Paxton et al. 2011, 2013). It is assumed that compositionof the star is similar to the Sun and the metallicity param-eter is Z = 0 . . In the reference model we neglect effectsof stellar rotation on the hydrostatic equilibrium. The con-vection parameters in the MESA code are determined by α MLT = (cid:96)H p = 1 . , where H p is the pressure stratificationscale. For the given parameters the star has the radius of R (cid:63) ≈ . R (cid:12) , the luminosity of L (cid:63) ≈ . L (cid:12) and thesurface temperature of K. Current understanding of theinternal structure of the fully convective stars is not com-plete. That’s why the theoretical predictions for the stellarradius and the T eff of the M-dwarfs are different from ob-servation(Reid & Hawley 2005). The mean-field heat transport equation takes into accounteffects of the global rotation on the thermal equilibrium. Itis calculated from the mean-field heat transport equation, ρT ∂s∂t + ρT (cid:0) U · ∇ (cid:1) s = − ∇ · (cid:16) F conv + F rad (cid:17) + (cid:15), (1)where (cid:15) is the source function, U is axisymmetric mean flow, ρ and T are the mean density and temperature, and s isthe mean entropy. In what follows, the over-bar denote theaxisymmetric component of the mean field and the anglebrackets are used for the ensemble average of the field whichcould contain the large-scale non-axisymmetric modes con-tributions as well.We employ expression of the anisotropic convective fluxsuggested by Kitchatinov et al. (1994) (hereafter KPR94), F convi = − ρT χ ij ∇ j s, (2)where in the heat eddy-conductivity tensor χ ij we have totake into account both the global rotation and the large-scale magnetic field effects. The expression has the compli-cated form and it is unknown for the general case if boththe Coriolis and the Lorenz forces are not small simulta-neously. This corresponds to conditions in the consideredM-star. The fast rotation regime and Ω ∗ > , holds in wholevolume of the star except the layer above r = 0 . R (cid:63) (see,Fig.1a), where the Coriolis number Ω ∗ = 2 τ c Ω (cid:63) , and τ c isthe turn-over time of convective flow. In the paper we ap-proximate the eddy heat conductivity tensor in following toPipin (2004): χ ij = χ T (cid:18) φ ( I ) χ ( β ) φ (Ω ∗ ) δ ij + φ ( A ) χ ( β ) φ (cid:107) (Ω ∗ ) Ω i Ω j Ω (cid:19) . (3)The effect of the global rotation on the heat transport de-pends on and the functions φ and φ (cid:107) are defined in KPR4.The magnetic feedback on the eddy heat-conductivity de-pends on the functions φ ( I ) χ and φ ( A ) χ ,(see Appendix) andthe parameter β = | B |√ πρu (cid:48) , where | B | is the strength ofthe large-scale magnetic field, and u (cid:48) is the RMS convectivevelocity. Note that for the case β > we have φ ( I ) χ ∼ β − and φ ( A ) χ ∼ β − . Thus the isotropic part of the eddy heatconductivity is quenched stronger than that in direction ofthe rotation axis. However in the case of the weak magneticfield the expression returns to the case discussed in KPR94.The self-consistent model could also include effects theJoule’s heating or sinks of the convective energy into mag-netic activity, see e.g., Brandenburg et al. (1992) and Pipin(2004). We put off discussion of those effects. In limits ofthe slow rotation and the weak magnetic field, i.e., Ω ∗ → ,and β → , the heat conductivity tensor reduces to isotropicform, χ ij = 13 δ ij (cid:96)u (cid:48) , where (cid:96) is the mixing length. The RMSconvective velocity is determined from the mixing-length re-lations u (cid:48) = (cid:96) (cid:115) − gc p ∂s∂r . The integration domain of the mean-field model is from r i = 0 . R (cid:63) to r e = 0 . R (cid:63) , we exclude the central and the c (cid:13) , 000–000 onlinear regimes in mean-field full-sphere dynamo near-surface regions. At the inner boundary the total flux F convr + F radr = L (cid:63) ( r i )4 πr i and for the external boundary, infollowing to Kitchatinov & Olemskoy (2011), we use F r = L (cid:63) πr e (cid:32) (cid:18) sc p (cid:19) (cid:33) . We put other details about the mean-field model of the heattransport in Appendix.The heat transport equation is coupled to equations ofthe angular momentum balance and the mean-field dynamoequations. . In the spherical coordinate system the conserva-tion of the angular momentum (Ruediger 1989) it expressedas follows: ∂∂t ρr sin θ Ω = − ∇· (cid:18) r sin θ (cid:18) ρ ˆ T φ + rρ sin θ Ω U m −(cid:104) B (cid:105) (cid:104) B φ (cid:105) π (cid:19)(cid:19) , (4)where, (cid:104) B (cid:105) is the large-scale dynamo generated magneticfield (see, the Subsection 2.2). The mean flow satisfies thecontinuity equation, ∇ · ρ U = 0 , (5)where U = U m + r sin θ Ω ˆ φ and ˆ φ is the unit vector in theazimuthal direction. The equation for the azimuthal compo-nent of the large-scale vorticity , ω = (cid:0) ∇ × U m (cid:1) φ , is ∂ω∂t = r sin θ ∇ · (cid:32) ˆ φ × ∇ · ρ ˆ T rρ sin θ − U m ωr sin θ (cid:33) + r sin θ ∂ Ω ∂z (6) + 1 ρ [ ∇ ρ × ∇ p ] φ + 1 ρ (cid:20) ∇ ρ × (cid:18) ∇ (cid:104) B (cid:105) π − ( (cid:104) B (cid:105) · ∇ ) (cid:104) B (cid:105) π (cid:19)(cid:21) φ where ˆ φ is a unit vector in azimuthal direction, ˆ T is theturbulent part of the Reynolds and Maxwell stresses and ∂/∂z = cos θ∂/∂r − sin θ/r · ∂/∂θ is the gradient alongthe axis of rotation. The turbulent stresses include the non-disspative part due to the Λ -effect and the anisotropic eddyviscosity. The theory is not complete because it does notcomprise the joint effect of the global rotation and the large-scale magnetic field on the angular momentum transport.We apply the theory developed in Kueker et al. (1996) andPipin (2004) for the case of the arbitrary Ω ∗ and the ar-bitrary strength of the large scale magnetic field. Theirsderivations are valid in the case when the toroidal compo-nent of the large-scale magnetic field dominates the poloidalone. In equation for the toroidal vorticity, Eq.(6) we neglectthe radial derivative of the Lorentz force in compare to thedensity gradient. The details about implementation of theturbulent stress tensor ˆ T are given in Appendix.Figure (1) shows profiles of the internal parameters ofthe mean-field model of the heat transport and the angu-lar momentum balance together with some input parame-ters from the MESA code. It is found that the convectiveturnover time varies from about of 1 day at the near-surfacelayer to about of 200 days near the center of the star. Thisresults to the strong modifications of the turbulent viscos-ity parameters. The resulted differential rotation is ratherweak, it is about of 0.01 Ω (cid:63) , where Ω (cid:63) = 7 . · − rad/sis the stellar rotation rate. The given angular velocity pro-file corresponds qualitatively to results of Kitchatinov et al. (2014). The angular velocity profile shows conical isolinespattern in the bulk of the star. This pattern changes to thecylinder like pattern in the equatorial region. In the portionof the star which occupied by the weakly varying angularvelocity, the given pattern is different to results of the di-rect numerical simulations (cf, Browning 2008; Yadav et al.2015). However, the strong shear in equatorial region is pre-sented in all models. The meridional circulation consists ofthe one cell in each hemisphere with poleward flow in up-per part of the star. The amplitude of the flow is about onemeter per second at the surface. The dynamo model takes into account both the axisymmet-ric and the non-axisymmetric large-scale magnetic field. Itsevolution is described by the mean-field induction equation(Krause & Rädler 1980): ∂ t (cid:104) B (cid:105) = ∇ × ( E + (cid:104) U (cid:105) × (cid:104) B (cid:105) ) (7)where E = (cid:104) u × b (cid:105) is the mean electromotive force; u and b are the turbulent fluctuating velocity and magnetic field re-spectively; and (cid:104) U (cid:105) and (cid:104) B (cid:105) are the mean velocity and mag-netic field. We remind that in the paper, the angle bracketsare used for the ensemble average of the field which couldcontain the large-scale non-axisymmetric modes contribu-tions as well as the axisymmetric modes of the mean fieldwhich is denoted by the over-bar. The mean flow is axisym-metric, i.e., (cid:104) U (cid:105) ≡ U , and it is determined from solution ofthe angular momentum balance. In the fully nonlinear casethe solution of the angular momentum balance is coupledwith the mean-field dynamo equations and the mean-fieldequation for the heat transport.Let ˆ φ = e φ and ˆ r = r e r be vectors in the azimuthal andradial directions respectively, then we represent the meanmagnetic field vectors as follows: (cid:104) B (cid:105) = B + ˜ B (8) B = ˆ φ B + ∇ × (cid:16) A ˆ φ (cid:17) (9) ˜ B = ∇ × (ˆ r T ) + ∇ × ∇ × (ˆ r S ) , (10)where B is the axisymmetric, and ˜ B is non-axisymmetricpart of the large-scale magnetic field, A , B , T and S arescalar functions. Hereafter, the non-axisymmetric part of themean field is denoted by the wave above the symbol.We employ the mean electromotive force in form: E i = ( α ij + γ ij ) (cid:104) B (cid:105) j − η ijk ∇ j (cid:104) B (cid:105) k . (11)where symmetric tensor α ij models the generation of mag-netic field by the α - effect; antisymmetric tensor γ ij controlsthe mean drift of the large-scale magnetic fields in turbulentmedium, including the magnetic buoyancy; tensor η ijk gov-erns the turbulent diffusion. Some details about the E aregiven in appendix, (also, see, Pipin 2008).In our model the α effect takes into account the kineticand magnetic helicities in the following form: α ij = C α ψ α ( β ) α ( H ) ij η T + α ( M ) ij (cid:104) χ (cid:105) τ c πρ(cid:96) (12)where C α is a free parameter which controls the strengthof the α - effect due to turbulent kinetic helicity; α ( H ) ij and c (cid:13)000
Stars with the extended convective envelopes demonstratethe high level of magnetic activity (Reid & Hawley 2005;Donati & Landstreet 2009; Linsky & Schöller 2015). Itis commonly believed that the magnetic activity of thesestars origins from the hydromagnetic turbulent dynamo ac-tion (Brandenburg & Subramanian 2005; Brun et al. 2014).Extremely high magnetic activity was found on the fully-convective low-mass stars which belong to the M-dwarfsbranch of the low main sequence of the Hertzsprung-Rawsseldiagram. Observations of the M-dwarfs indicated the ratherstrong large-scale magnetic field with strength of several kG(Saar & Linsky 1985; Saar et al. 1986; Johns-Krull & Valenti1996; Linsky & Schöller 2015). The magnetic topology of theM-dwarfs is likely depends on the mass and the rotation pe-riod of a star (Donati & Landstreet 2009; See et al. 2016).Observations indicate that the early type M-dwarfs with themoderate period of rotation about 4-5 days demonstrate thestrong non-axisymmetric magnetic field with the dominanttoroidal component Donati et al. (2008). The extremely fastrotating early type M-dwarfs with period of rotation lessthan 1 day indicate the transition to the axisymmetric dy-namo with the dominant poloidal component of the large- (cid:63) email: [email protected] scale magnetic field. Situation become complicated on themid and late-type M-stars which have the masses less than . M (cid:12) as they could show either the strong axisymmetricdipole-kind large-scale magnetic field, or the low-strengthnon-axisymmetric magnetic field (Morin et al. 2008, 2010).Thus we can conclude about three basic states of the dynamoon the fast rotating M-dwarfs, they are: the strong multipolemagnetic field (hereafter, SM), the strong dipole field (here-after SD) and the weak multipole (hereafter WM) magneticfield. We follow notation suggested by Morin et al. (2011).Interesting that simultaneously with multiply states of thedynamo regimes, the dynamo generated total magnetic fluxdo not show the rotation-activity connection which is knownamong the solar type stars (Mohanty & Basri 2003).Observed magnetic properties of the M-stars initiatedthe number of the theoretical studies employing the mean-field models (see, Chabrier & Küker 2006; Elstner & Rüdi-ger 2007; Kitchatinov et al. 2014; Shulyak et al. 2015) andthe direct numerical simulations (e.g., Dobler et al. 2006;Browning 2008; Dormy et al. 2013; Schrinner et al. 2014).Using the results of the numerical simulations, Morin et al.(2011) suggests that the bi-stability of the magnetic topol-ogy on the late-type M-stars could result from two typesof the convection regimes occurred in the fast rotating con-vective bodies (Roberts 1988). Also, the direct numericalsimulations show the differential rotation is important part c (cid:13) a r X i v : . [ a s t r o - ph . S R ] S e p V.V. Pipin of the dynamo in the fully convective stars. Similar conclu-sions were suggested by Shulyak et al. (2015) after studyingthe linear dynamo regimes.Current interpretation of the dynamo bi-stability givenby Morin et al. (2011) suggests that the strength of thelarge-scale magnetic field is compatible with the nonlinearbalance between the Lorentz and Coriolis force in case ofthe SD-type magnetism and it is established by the Lorentz-inertia force balance in case of the WM magnetism. Later,Schrinner et al. (2014) found that in the anelastic simula-tions the separation between the SD and WM magnetismis less profound than they as well as others (e.g., Simitev& Busse 2009) found with the Boussinescue approximation.The origin of the strong multipolar magnetic field on themoderate rotating early M-stars is barely studied. Results ofthe mean-field models and the numerical simulations suggestthe dynamo on these stars could operate with help of the dif-ferential rotation. The linear analysis of Kitchatinov et al.(2014) show that the axisymmetric magnetic field modeshave the smaller critical threshold of the dynamo instabilitythan the non-axisymmetric ones. Thus, the transition fromaxisymmetric to non-axisymmetric dynamo can occur onlyin the nonlinear regime.The paper we study the nonlinear dynamo models for afully convective star. Here we restrict ourselves to the samecase of the star discussed earlier by Shulyak et al. (2015),i.e., the star of mass 0.3 M (cid:12) of 1Gyr age and rotating withperiod of 10 days. We will address the axisymmetric andnon-axisymmetric dynamo regimes with regards for the non-linear back reaction of the large-scale magnetic field on the α -effect and the large-scale flow. The solution of the dy-namo problem is coupled with the solution of the mean an-gular momentum balance and the mean heat transport inthe convective sphere. The main goal of the paper is to findthe typical topology of the large-scale magnetic field in thenonlinear dynamo for the given rotation period and investi-gate the nonlinear effects on the dynamo. We consider a fully convective star of mass 0.3 M (cid:12) of 1Gyrage and rotating with period of 10 days. The reference in-ternal thermodynamic structure of the star was calculatedusing the MESA stellar evolution code, the version r7503,(Paxton et al. 2011, 2013). It is assumed that compositionof the star is similar to the Sun and the metallicity param-eter is Z = 0 . . In the reference model we neglect effectsof stellar rotation on the hydrostatic equilibrium. The con-vection parameters in the MESA code are determined by α MLT = (cid:96)H p = 1 . , where H p is the pressure stratificationscale. For the given parameters the star has the radius of R (cid:63) ≈ . R (cid:12) , the luminosity of L (cid:63) ≈ . L (cid:12) and thesurface temperature of K. Current understanding of theinternal structure of the fully convective stars is not com-plete. That’s why the theoretical predictions for the stellarradius and the T eff of the M-dwarfs are different from ob-servation(Reid & Hawley 2005). The mean-field heat transport equation takes into accounteffects of the global rotation on the thermal equilibrium. Itis calculated from the mean-field heat transport equation, ρT ∂s∂t + ρT (cid:0) U · ∇ (cid:1) s = − ∇ · (cid:16) F conv + F rad (cid:17) + (cid:15), (1)where (cid:15) is the source function, U is axisymmetric mean flow, ρ and T are the mean density and temperature, and s isthe mean entropy. In what follows, the over-bar denote theaxisymmetric component of the mean field and the anglebrackets are used for the ensemble average of the field whichcould contain the large-scale non-axisymmetric modes con-tributions as well.We employ expression of the anisotropic convective fluxsuggested by Kitchatinov et al. (1994) (hereafter KPR94), F convi = − ρT χ ij ∇ j s, (2)where in the heat eddy-conductivity tensor χ ij we have totake into account both the global rotation and the large-scale magnetic field effects. The expression has the compli-cated form and it is unknown for the general case if boththe Coriolis and the Lorenz forces are not small simulta-neously. This corresponds to conditions in the consideredM-star. The fast rotation regime and Ω ∗ > , holds in wholevolume of the star except the layer above r = 0 . R (cid:63) (see,Fig.1a), where the Coriolis number Ω ∗ = 2 τ c Ω (cid:63) , and τ c isthe turn-over time of convective flow. In the paper we ap-proximate the eddy heat conductivity tensor in following toPipin (2004): χ ij = χ T (cid:18) φ ( I ) χ ( β ) φ (Ω ∗ ) δ ij + φ ( A ) χ ( β ) φ (cid:107) (Ω ∗ ) Ω i Ω j Ω (cid:19) . (3)The effect of the global rotation on the heat transport de-pends on and the functions φ and φ (cid:107) are defined in KPR4.The magnetic feedback on the eddy heat-conductivity de-pends on the functions φ ( I ) χ and φ ( A ) χ ,(see Appendix) andthe parameter β = | B |√ πρu (cid:48) , where | B | is the strength ofthe large-scale magnetic field, and u (cid:48) is the RMS convectivevelocity. Note that for the case β > we have φ ( I ) χ ∼ β − and φ ( A ) χ ∼ β − . Thus the isotropic part of the eddy heatconductivity is quenched stronger than that in direction ofthe rotation axis. However in the case of the weak magneticfield the expression returns to the case discussed in KPR94.The self-consistent model could also include effects theJoule’s heating or sinks of the convective energy into mag-netic activity, see e.g., Brandenburg et al. (1992) and Pipin(2004). We put off discussion of those effects. In limits ofthe slow rotation and the weak magnetic field, i.e., Ω ∗ → ,and β → , the heat conductivity tensor reduces to isotropicform, χ ij = 13 δ ij (cid:96)u (cid:48) , where (cid:96) is the mixing length. The RMSconvective velocity is determined from the mixing-length re-lations u (cid:48) = (cid:96) (cid:115) − gc p ∂s∂r . The integration domain of the mean-field model is from r i = 0 . R (cid:63) to r e = 0 . R (cid:63) , we exclude the central and the c (cid:13) , 000–000 onlinear regimes in mean-field full-sphere dynamo near-surface regions. At the inner boundary the total flux F convr + F radr = L (cid:63) ( r i )4 πr i and for the external boundary, infollowing to Kitchatinov & Olemskoy (2011), we use F r = L (cid:63) πr e (cid:32) (cid:18) sc p (cid:19) (cid:33) . We put other details about the mean-field model of the heattransport in Appendix.The heat transport equation is coupled to equations ofthe angular momentum balance and the mean-field dynamoequations. . In the spherical coordinate system the conserva-tion of the angular momentum (Ruediger 1989) it expressedas follows: ∂∂t ρr sin θ Ω = − ∇· (cid:18) r sin θ (cid:18) ρ ˆ T φ + rρ sin θ Ω U m −(cid:104) B (cid:105) (cid:104) B φ (cid:105) π (cid:19)(cid:19) , (4)where, (cid:104) B (cid:105) is the large-scale dynamo generated magneticfield (see, the Subsection 2.2). The mean flow satisfies thecontinuity equation, ∇ · ρ U = 0 , (5)where U = U m + r sin θ Ω ˆ φ and ˆ φ is the unit vector in theazimuthal direction. The equation for the azimuthal compo-nent of the large-scale vorticity , ω = (cid:0) ∇ × U m (cid:1) φ , is ∂ω∂t = r sin θ ∇ · (cid:32) ˆ φ × ∇ · ρ ˆ T rρ sin θ − U m ωr sin θ (cid:33) + r sin θ ∂ Ω ∂z (6) + 1 ρ [ ∇ ρ × ∇ p ] φ + 1 ρ (cid:20) ∇ ρ × (cid:18) ∇ (cid:104) B (cid:105) π − ( (cid:104) B (cid:105) · ∇ ) (cid:104) B (cid:105) π (cid:19)(cid:21) φ where ˆ φ is a unit vector in azimuthal direction, ˆ T is theturbulent part of the Reynolds and Maxwell stresses and ∂/∂z = cos θ∂/∂r − sin θ/r · ∂/∂θ is the gradient alongthe axis of rotation. The turbulent stresses include the non-disspative part due to the Λ -effect and the anisotropic eddyviscosity. The theory is not complete because it does notcomprise the joint effect of the global rotation and the large-scale magnetic field on the angular momentum transport.We apply the theory developed in Kueker et al. (1996) andPipin (2004) for the case of the arbitrary Ω ∗ and the ar-bitrary strength of the large scale magnetic field. Theirsderivations are valid in the case when the toroidal compo-nent of the large-scale magnetic field dominates the poloidalone. In equation for the toroidal vorticity, Eq.(6) we neglectthe radial derivative of the Lorentz force in compare to thedensity gradient. The details about implementation of theturbulent stress tensor ˆ T are given in Appendix.Figure (1) shows profiles of the internal parameters ofthe mean-field model of the heat transport and the angu-lar momentum balance together with some input parame-ters from the MESA code. It is found that the convectiveturnover time varies from about of 1 day at the near-surfacelayer to about of 200 days near the center of the star. Thisresults to the strong modifications of the turbulent viscos-ity parameters. The resulted differential rotation is ratherweak, it is about of 0.01 Ω (cid:63) , where Ω (cid:63) = 7 . · − rad/sis the stellar rotation rate. The given angular velocity pro-file corresponds qualitatively to results of Kitchatinov et al. (2014). The angular velocity profile shows conical isolinespattern in the bulk of the star. This pattern changes to thecylinder like pattern in the equatorial region. In the portionof the star which occupied by the weakly varying angularvelocity, the given pattern is different to results of the di-rect numerical simulations (cf, Browning 2008; Yadav et al.2015). However, the strong shear in equatorial region is pre-sented in all models. The meridional circulation consists ofthe one cell in each hemisphere with poleward flow in up-per part of the star. The amplitude of the flow is about onemeter per second at the surface. The dynamo model takes into account both the axisymmet-ric and the non-axisymmetric large-scale magnetic field. Itsevolution is described by the mean-field induction equation(Krause & Rädler 1980): ∂ t (cid:104) B (cid:105) = ∇ × ( E + (cid:104) U (cid:105) × (cid:104) B (cid:105) ) (7)where E = (cid:104) u × b (cid:105) is the mean electromotive force; u and b are the turbulent fluctuating velocity and magnetic field re-spectively; and (cid:104) U (cid:105) and (cid:104) B (cid:105) are the mean velocity and mag-netic field. We remind that in the paper, the angle bracketsare used for the ensemble average of the field which couldcontain the large-scale non-axisymmetric modes contribu-tions as well as the axisymmetric modes of the mean fieldwhich is denoted by the over-bar. The mean flow is axisym-metric, i.e., (cid:104) U (cid:105) ≡ U , and it is determined from solution ofthe angular momentum balance. In the fully nonlinear casethe solution of the angular momentum balance is coupledwith the mean-field dynamo equations and the mean-fieldequation for the heat transport.Let ˆ φ = e φ and ˆ r = r e r be vectors in the azimuthal andradial directions respectively, then we represent the meanmagnetic field vectors as follows: (cid:104) B (cid:105) = B + ˜ B (8) B = ˆ φ B + ∇ × (cid:16) A ˆ φ (cid:17) (9) ˜ B = ∇ × (ˆ r T ) + ∇ × ∇ × (ˆ r S ) , (10)where B is the axisymmetric, and ˜ B is non-axisymmetricpart of the large-scale magnetic field, A , B , T and S arescalar functions. Hereafter, the non-axisymmetric part of themean field is denoted by the wave above the symbol.We employ the mean electromotive force in form: E i = ( α ij + γ ij ) (cid:104) B (cid:105) j − η ijk ∇ j (cid:104) B (cid:105) k . (11)where symmetric tensor α ij models the generation of mag-netic field by the α - effect; antisymmetric tensor γ ij controlsthe mean drift of the large-scale magnetic fields in turbulentmedium, including the magnetic buoyancy; tensor η ijk gov-erns the turbulent diffusion. Some details about the E aregiven in appendix, (also, see, Pipin 2008).In our model the α effect takes into account the kineticand magnetic helicities in the following form: α ij = C α ψ α ( β ) α ( H ) ij η T + α ( M ) ij (cid:104) χ (cid:105) τ c πρ(cid:96) (12)where C α is a free parameter which controls the strengthof the α - effect due to turbulent kinetic helicity; α ( H ) ij and c (cid:13)000 , 000–000 V.V. Pipin r/R ⊙ [ c m / s ] · u / 3 T 1 ( ∗ ) C O R I O L I S NU M BE R ⋆ a) b Figure 1. a) The Coriolis number Ω ∗ = 2 τ c Ω (cid:63) (black line), where τ c is the turn-over time of convection (from the MESA code), theturbulent diffusivity parameter, red line; the blue line show isotropic eddy viscosity from the heat transport model; b) angular velocityprofiles with contour levels which cover the range of values depicted on the color bar; c) geometry of the meridional circulation, in theNorthern hemisphere. α ( M ) ij express the kinetic and magnetic helicity parts of the α -effect, respectively; η T = ν T /P m T is the magnetic dif-fusion coefficient, P m T is the turbulent magnetic Prandtlnumber and (cid:104) χ (cid:105) = (cid:104) a · b (cid:105) ( a and b are the fluctuating partsof magnetic field vector-potential and magnetic field vector).Both the α ( H ) ij and α ( M ) ij depend on the Coriolis number.Function ψ α ( β ) controls the so-called “algebraic” quenchingof the α - effect where β = (cid:104)| B |(cid:105) / (cid:112) πρu (cid:48) , u (cid:48) is the RMS ofthe convective velocity.The magnetic helicity conservation results to the dy-namical quenching of the dynamo. Contribution of the mag-netic helicity to the α -effect is expressed by the second termin Eq.(12). The magnetic helicity density of turbulent field, (cid:104) χ (cid:105) = (cid:104) a · b (cid:105) , is governed by the conservation law (Pipinet al. 2013): ∂ (cid:104) χ (cid:105) ( tot ) ∂t = − (cid:104) χ (cid:105) R m τ c − η (cid:104) B (cid:105) · (cid:104) J (cid:105) − ∇ · F χ , (13)where (cid:104) χ (cid:105) ( tot ) = (cid:104) χ (cid:105) + (cid:104) A (cid:105)·(cid:104) B (cid:105) is the total magnetic helicitydensity of the mean and turbulent fields, F χ = − η χ ∇ (cid:104) χ (cid:105) is the diffusive flux of the turbulent magnetic helicity, and R m is the magnetic Reynolds number. The coefficient of theturbulent helicity diffusivity, η χ , is chosen ten times smallerthan the isotropic part of the magnetic diffusivity , η χ = η T . The magnetic helicity conservation is determined bythe magnetic Reynolds number R m . In this paper we employ R m = 10 .The numerical scheme employs the spherical harmonicsdecomposition for the non-axisymmetric part of the prob-lem. At the bottom of the domain we put the potentials S and T , as well as the axisymmetric fields, B and A to zero.At the top the poloidal field is smoothly matched to theexternal potential field and the toroidal field goes to zeroThe numerical scheme employs the pseudo-spectral ap-proach for integration along latitude and the finite differ-ences along the radius. Fort the non-axisymmetric part ofthe problem we employ the spherical harmonics decompo-sition, i.e., the scalar functions T and S are represented inthe form: T ( r, µ, φ, t ) = (cid:88) ˆ T l,m ( r, t ) ¯ P | m | l exp ( imφ ) , (14) S ( r, µ, φ, t ) = (cid:88) ˆ S l,m ( r, t ) ¯ P | m | l exp ( imφ ) , (15) Pm T C c r A0S0A1S1 anisotropy
A0, Pm T =20A1A0, Pm T =.85A1 a) b) Figure 2.
The critical threshold parameter C ( cr ) α for isotropicdiffusivity, A = 0 ; b) The dependence of the critical thresholdparameter C ( cr ) α on the anisotropy of the turbulent diffusivity,for P m T = 20 (blue lines), and P m T = 0 . , (red lines). where ¯ P ml is the normalized associated Legendre function ofdegree l ≥ and order m ≥ . The simulations which wewill discuss include 310 spherical harmonics (up to l max =20 ). Note that ˆ S l, − m = ˆ S ∗ l,m and the same for ˆ T . All thenonlinear terms are treated explicitly in the real space. Thenumerical integration is carried out in latitude from the poleto pole and in radius from r b = 0 . R (cid:63) to r e = 0 . R (cid:63) .The thermal equilibrium, the angular momentum bal-ance and evolution of the large-scale magnetic field is con-trolled by the free parameters, which are the angular velocityof the global rotation Ω = 7 . × − rad / s , the turbulentPrandtl number P r T = ν T χ T , the turbulent magnetic Prandtlnumber P m T = ν T η T , the parameter of the magnetic fieldgeneration by the α -effect, C α , and the magnetic Reynoldsnumber R m . We use the mixing-length expression for theeddy heat conductivity, χ T = (cid:96) (cid:115) − gc p ∂s∂r . In the all mod-els we fix
P r T = 34 , P m T = 3 , and R m = 10 . We willdiscuss the possible dependence of results on P m T , as well.We studied the eigenvalue dynamo problem before run-ning the nonlinear models. In the linear model we neglectthe radial dependence of the α -effect and turbulent diffu-sivity. Solutions of the eigenvalue problem showed that thelinear properties of the dynamo model are in agreement c (cid:13) , 000–000 onlinear regimes in mean-field full-sphere dynamo with results reported earlier by Elstner & Rüdiger (2007)and Shulyak et al. (2015). More specifically, the results ofthe linear problem solutions are as follows. Firstly, for thehigh P m T the axisymmetric dynamo has smaller the crit-ical dynamo instability threshold instability than the non-axisymmetric dynamo. The transition from axisymmetric tonon-axisymmetric regimes occurs for P m T ≈ . This is inagreement with findings of Shulyak et al. (2015). Also thesolution shows that the critical threshold for the symmetricand antisymmetric about equator dynamo modes are closeand the symmetric modes have the smaller threshold thanthe antisymmetric ones. Secondly, it was found that for thecase P m T ≈ , when the non-axisymmetric dynamo insta-bility is more powerful than the axisymmetric one, the rota-tionally induced anisotropy of the magnetic diffusivity canpromotes the dynamo instability of the axisymmetric mag-netic field if the amplitude of the eddy diffusivity along therotation axis is twice of that one in the perpendicular di-rection. This result is in agreement with that reported byElstner & Rüdiger (2007). Figures 2(a,b) illustrate our find-ings. In comparing our results with findings from reported inabove cited papers we have to take into account that the pa-rameter of the α -effect in the reduced linear models containthe density stratification factor, ˜Λ ( ρ ) = R (cid:12) ∇ log ρ , and itsmean value in the star is (cid:12)(cid:12)(cid:12) ˜Λ ( ρ ) (cid:12)(cid:12)(cid:12) ≈ . Also models of Shulyaket al. (2015) were normalized for diffusivity cm / s andit is cm / s in our model. Table 1 contains general parameters of our models. Theyare: the B max is the maximum strength of the large-scalemagnetic field in the star, the B (cid:107) and ˜ B (cid:107) are the meanstrength of the axisymmetric and non-axisymmetric large-scale poloidal magnetic field on the surface, the B ⊥ and ˜ B ⊥ is the same for the toroidal magnetic field at the radialdistance R (cid:63) , the M = ˜ E m E m is the ratio of the energy ofthe non-axisymmetric mode of the large-scale magnetic fieldto the total magnetic energy of the large-scale field at theradial distance R (cid:63) , and the parameter ∆ΩΩ is the meanlatitudinal shear on the top of the integration domain, wherethe mean is computed over one dynamo cycle.In the paper we show results for five different runs of thenonlinear dynamo models. In all the runs we put P m T = 3 .In this case the dynamo period is about 40 years. Shulyaket al. (2015) discussed linear dynamo regimes with the longerdynamo period about 100Y. This is because they employedthe higher P m T = 10 in thier models. We have nonlinearruns with P m T = 10 but not for all cases listed in the Ta-ble 1. The higher P m T , the longer dynamo period and ittakes longer evolution time for the model to reach some sta-tionary regime of the dynamo oscillations, especially in thecase of the non-axisymmetric dynamo regimes. Also the ef-fect of the meridional circulation for the case of P m T = 10 requires the better spatial resolution near poles. We madeseparate runs for axisymmetric and non-axisymmetric dy-namo regimes. The latter models take into account both theaxisymmetric and non-axisymmetric magnetic field genera-tion. In this paper we restrict the study of the fully nonlinear
100 110 120 130 140 150 16050050 L A T I T UD E B r ,[ G ]
100 110 120 130 140 150 16010 [ G ] m=0m=1m=2m=3m=4
110 120 130 140 150 16010001500200025003000350040004500 | B | ,[ G ] N O N - AX I SY MM E T R Y a)b)c) [YR] Figure 4.
The model M2, a) the mean strength of the first fivepartial modes of the toroidal magnetic field at the r = R (cid:63) ; b) atthe same r , the index of the non-axisymmetric of the large-scalemagnetic field (dashed line), the mean strength of the axisym-metric toroidal magnetic field (solid line) and the mean strengthof the large-scale toroidal field taking into account both the ax-isymmetric and non-axisymmetric parts of the magnetic field; c)the same as Fig.3b model by the case of the C α when both the axisymmetricand the non-axisymmetric magnetic fields are unstable togeneration in the large-scale dynamo. The initial field in allruns has no preferable parity relative to the equator. α -effect In this subsection we consider the nonlinear models withmagnetic feedback on the generation of the large-scale mag-netic fields by the α -effect. The models remain kinematicrelative to the large-scale flow. The non-linear α -effect takesinto account the dynamical feedback due to magnetic helic-ity conservation (see, the Eq(13)) and the “instantaneous”quenching which is related with magnetic feedback from theLorentz forces on the turbulent convection. This concept wasoriginally formulated by Kleeorin & Ruzmaikin (1982). TheFig.3 show results for the model M1 which illustrates theaxisymmetric dynamo when the parameter of the parame-ter C α is about factor one and half of the critical dynamoinstability threshold. The model show the mixed parity solu-tion with some preference to generation of the antisymmetricabout equator magnetic field. Butterfly diagrams shows thesolar-like equatorial drift of the toroidal magnetic field ofthe strength 4kG at the r = R (cid:63) . The radial magnetic fielddrift to the pole where it reaches strength of the 1kG dur-ing the maximum of the dynamo cycle. The polar drift ofthe poloidal field is supported by the meridional circulation.The migrating dynamo wave of the poloidal magnetic fieldis transformed to the steady one when the meridional circu-lation is neglected. Also, in this case the dominance of theantisymmetric relative to equator magnetic field become lessclear in the nonlinear mixed parity solution. It is found thatthe meridional flow has only a small effect on the amplitudeof the dynamo wave.Results of the linear problem study show that the non-axisymmetric magnetic field is unstable to generation for thesame parameter C α as it is employed in the model M1. The c (cid:13)000
The model M2, a) the mean strength of the first fivepartial modes of the toroidal magnetic field at the r = R (cid:63) ; b) atthe same r , the index of the non-axisymmetric of the large-scalemagnetic field (dashed line), the mean strength of the axisym-metric toroidal magnetic field (solid line) and the mean strengthof the large-scale toroidal field taking into account both the ax-isymmetric and non-axisymmetric parts of the magnetic field; c)the same as Fig.3b model by the case of the C α when both the axisymmetricand the non-axisymmetric magnetic fields are unstable togeneration in the large-scale dynamo. The initial field in allruns has no preferable parity relative to the equator. α -effect In this subsection we consider the nonlinear models withmagnetic feedback on the generation of the large-scale mag-netic fields by the α -effect. The models remain kinematicrelative to the large-scale flow. The non-linear α -effect takesinto account the dynamical feedback due to magnetic helic-ity conservation (see, the Eq(13)) and the “instantaneous”quenching which is related with magnetic feedback from theLorentz forces on the turbulent convection. This concept wasoriginally formulated by Kleeorin & Ruzmaikin (1982). TheFig.3 show results for the model M1 which illustrates theaxisymmetric dynamo when the parameter of the parame-ter C α is about factor one and half of the critical dynamoinstability threshold. The model show the mixed parity solu-tion with some preference to generation of the antisymmetricabout equator magnetic field. Butterfly diagrams shows thesolar-like equatorial drift of the toroidal magnetic field ofthe strength 4kG at the r = R (cid:63) . The radial magnetic fielddrift to the pole where it reaches strength of the 1kG dur-ing the maximum of the dynamo cycle. The polar drift ofthe poloidal field is supported by the meridional circulation.The migrating dynamo wave of the poloidal magnetic fieldis transformed to the steady one when the meridional circu-lation is neglected. Also, in this case the dominance of theantisymmetric relative to equator magnetic field become lessclear in the nonlinear mixed parity solution. It is found thatthe meridional flow has only a small effect on the amplitudeof the dynamo wave.Results of the linear problem study show that the non-axisymmetric magnetic field is unstable to generation for thesame parameter C α as it is employed in the model M1. The c (cid:13)000 , 000–000 V.V. Pipin
Table 1.
Basic parameters, B max is the maximum of the magnetic field strength in the star, B (cid:107) and ˜ B (cid:107) are the mean strength ofthe mean poloidal components of the axisymmetric and non-axisymmetric magnetic field at the surface, B ⊥ and ˜ B ⊥ is the same forthe toroidal magnetic field below surface at the R (cid:63) , M is the ratio of the energy of the non-axisymmetric magnetic field to the totalmagnetic energy at the same radial distance and ∆ΩΩ is the measure of the latitudinal shear at the surface. C α AngularMomentum B max ,[kG] B (cid:107) , ˜ B (cid:107) [kG] B ⊥ , ˜ B ⊥ [kG] M = ˜ E m E m ∆ΩΩ M1 0.04 no 3 0.3, 0 1.5, 0 0 0.014M2 0.04 no 4 0.2, 0.35 1, 2 0.5 0.014M3 0.05 no 8 0.04, 0.8 0.2, 4 0.9 0.014M4 0.04 yes 1.5 0.2 0.35 0 0.012M5 0.05 yes 1.8 0.2,0.3 0.7,1 0.6 0.009 | B | ,[ G ] P O L A R F I E L D ,[ G ] YR L A T I T UD E B r ,[ G ] a)b) B ,[ G ] c) Figure 3.
The model M1, a) evolution of the mean strength toroidal magnetic field at the r = R (cid:63) and the radial magnetic field strengthat the North pole (dashed line); b) the time-latitude diagram for toroidal magnetic field (contours in range ± kG) at the r = R (cid:63) , andthe color image shows the radial magnetic field at the surface; c) snapshot of the large-scale magnetic field distributions at the growingphase of the cycle model M2 takes the non-axisymmetric magnetic field intoaccount. Fig.4 show results for variations of the magneticenergy, index of the non-axisymmetric and the time-latitudediagrams of the axisymmetric magnetic field in the model.We find that the non-axisymmetric dynamo quenches thestrength of the generated axisymmetric magnetic field. Themost important quenching mechanisms are due to effects ofthe magnetic helicity generation from the non-axisymmetricdynamo and another effect is due to the magnetic buoyancywhich is increased when the magnetic energy increases. Inour intepretation, we have to take into account that themagnetic buoyancy can promote the dynamo instability ofthe non-axisymmetric field (Dikpati & Gilman 2001).The non-axisymmetric dynamo wave has a spiral pat-tern which is rigidly rotating (see Fig 5d), which producesYin Yang magnetic polarity pattern on the surface of thestar. The strength of the spiral arms vary in time becauseof interaction with the axisymmetric magnetic field. Thiscauses the long-term variation of the magnetic energy of thenon-axisymmetric field at the given radial distance of thestar.Figure 5 show snapshots of the magnetic field distri-bution inside and outside of the star. Configuration of theaxisymmetric field in the model M2 is similar that in themodel M1. The non-axisymmetric field is distributed along iso-surface of the angular velocity. The polar regions on thesurface of star are occupied by the mixture of the axisym-metric radial magnetic and the non-axisymmetric field mag-netic field of the m=1 mode.In the model M3 the alpha-effect parameter C α is abouttwice of the dynamo instability threshold. The model M3shows the stronger quenching of the axisymmetric dynamothan the model M2. This is illustrated in Figures 6 and 7.In fact, the axisymmetric magnetic field nearly disappearin the stationary phase of the evolution. At the surface thenon-axisymmetric magnetic field shows the large-scale spot-like pattern with angular size about 30 ◦ . Those spots arelocated in the equatorrial and polar regions as well. This subsection contain results about the fully nonlinear dy-namo with regards for the magnetic feedback on the angularmomentum balance and heat transport inside the star. In themodel M4 we neglect effects of the non-axisymmetric fieldon the dynamo.Figure 8 show the time-latitude diagrams for variationsof the magnetic field, angular velocity and the latitudinalcomponent of the meridional flow in the model M4. Thevariations of the angular velocity caused by the dynamo, c (cid:13) , 000–000 onlinear regimes in mean-field full-sphere dynamo AZIMUTH L A T I T UD E B r ,[ G ] B ,[ G ] ˜ B ,[ G ] a) b)c) d) Figure 5.
The model M2, a) snapshot of the axisymmetrictoroidal magnetic field (color image) and the poloidal field atthe end of run ( t = 170 Yr); b) snapshot of distribution the non-axisymmetric magnetic field at the longitude φ = 0 , color imageshows the toroidal magnetic field and contours - the poloidal mag-netic field; c) the non-axisymmetric radial magnetic field at thesurface (color image) and contours show the toroidal field at thesubsurface layer; d) magnetic field lines out of the star and sur-face show the large-scale magnetic field of magnitude 1kG insidethe star. | B | ,[ G ] N O N - AX I SY MM E T R Y [ G ] [YR] L A T I T UD E B r ,[ G ] a)b)c) Figure 6.
The same as the Figure 4 for the model M2. can be observed as the azimuthal flow waves. The modelM4 demonstrate some similarity to the solar case, i.e., thepositive azimuthal flow wave is located on the equatorial sideof the dynamo wave of the large-scale toroidal field. Simulta-neously, the model shows the meridional flows direct to themaximum of the toroidal magnetic field. Zonal variations ofrotation is about 10 percents of the mean latitudinal shear.Variations of the meridional circulation are smaller than onepercent of the mean magnitude. The surface mean latitudi-nal shear in model M4 is about 15 percents smaller than in
AZIMUTH L A T I T UD E B r ,[ G ] ˜ B ,[ G ] B ,[ G ] a) b)c) d) Figure 7.
The same as the Figure 5 for the model M3
40 60 80 100 120 YR L A T I T UD E U ,[ C M / S ]
40 60 80 100 120603003060 L A T I T UD E U ,[ C M / S ]
40 60 80 100 120603003060 L A T I T UD E B r ,[ G ] a)b)c) Figure 8.
The model M4, a) the time-latitude diagram fortoroidal magnetic field (contours in range ± . kG) at the r = R (cid:63) , and the color image shows the radial magnetic field at thesurface; b) color image show the variation of the latitudinal merid-ional flow (positive to the equator) c) the same as b) for variationof the toroidal velocity field at the surface; the kinematic models. Magnetic feedback on the differentialrotations reduces the strength of the toroidal field in themodel by factor 3 in compare to model M1.Figure 9 show variations of the magnetic field and angu-lar velocity on period of half of the magnetic cycle. It is seenthat dynamo wave migrate outward of the rotation axis. Themigration of the dynamo waves induces variations of the an-gular velocity which is separated to zones of the acceleratedand decelerated motions. Those zones are elongated alongthe rotation axis. The dynamo wave inside star distorts dis-tribution of the angular velocity bowing the angular velocityiso-surface to equator. It makes the angular velocity profileinside the star to become close to the spherical surfaces.This supports the equatorial drift of the dynamo wave. In c (cid:13)000
The model M4, a) the time-latitude diagram fortoroidal magnetic field (contours in range ± . kG) at the r = R (cid:63) , and the color image shows the radial magnetic field at thesurface; b) color image show the variation of the latitudinal merid-ional flow (positive to the equator) c) the same as b) for variationof the toroidal velocity field at the surface; the kinematic models. Magnetic feedback on the differentialrotations reduces the strength of the toroidal field in themodel by factor 3 in compare to model M1.Figure 9 show variations of the magnetic field and angu-lar velocity on period of half of the magnetic cycle. It is seenthat dynamo wave migrate outward of the rotation axis. Themigration of the dynamo waves induces variations of the an-gular velocity which is separated to zones of the acceleratedand decelerated motions. Those zones are elongated alongthe rotation axis. The dynamo wave inside star distorts dis-tribution of the angular velocity bowing the angular velocityiso-surface to equator. It makes the angular velocity profileinside the star to become close to the spherical surfaces.This supports the equatorial drift of the dynamo wave. In c (cid:13)000 , 000–000 V.V. Pipin
Figure 9.
Model M4. Snapshots of the magnetic field (left), an-gular velocity, the azimuthal zonal flow distributions (middle) andthe meridional circulation (right, color image is for the latitudinalcomponent and contours are for the radial component) for a halfof the dynamo cycle. the upper part of the star drift of the dynamo waves of thetoroidal field follows the distorted isolines of the angular ve-locity. The drift goes equator-ward up to 30 ◦ latitude. Thedrifting wave of the poloidal field is transformed to nearlysteady on the surface because of effect of the meridionalcirculation. We also see that rather small variations of themeridional circulation are concentrated to the surface.The reduced differential rotation results to a reducedratio between the mean strength of the toroidal and poloidalfield. In the model M4 it is about 2 and in the models M1,M2, M3 it is about 5. The strength of the polar field in themodel M4 is about 500 G which is by factor 2 smaller thanin models M1 and M2.Compare to the previous case, the model M5 takes intoaccount effects of the non-axisymmetric field in the angu-lar momentum balance via the mean Maxwell stresses ofthe non-axisymmetric magnetic field, i.e., terms like − ˜ B i ˜ B j πρ (over-bar means the azimuthal averaging) and contributionsof the non-axisymmetric field to the mean magnetic energy.The latter makes effect to the efficiency of the magneticquenching of the Λ effect and coefficients of the eddy vis-cosity and thermal eddy conductivity. Figure 10 shows evo-lutions of parameters of the large-scale magnetic field andsnapshot for the magnetic field distribution out of the star.It is found that the non-axisymmetric dynamo quenches gen-eration of the axisymmetric magnetic field. However, unliketo the model M3, which has the same parameter C α , the ax-isymmetric dynamo persists in the stationary stage of evolu-tion of the model M5. This is similar to the model M2. Thedynamo wave goes equator-ward in the whole range of lati-tudes. The surface magnetic field is dominated by the m=1mode of the non-axisymmetric magnetic field with the YinYang magnetic polarity pattern as well as the model M2.Figure 11 shows snapshots of the magnetic field and − 00− 400− − − B ,[ G ] U ,[ C M / S ] a) b) AZIMUTH L A T I T UD E B r ,[ G ] c) d) Figure 11.
Except the panel b) the same as Figure 5 for themodel M5. The panel b) shows, the distribution of the mean an-gular velocity (contours), and the background image shows dis-tribution of the zonal variations of rotation. angular velocity distributions in the star for the model M5.The snapshot of the axisymmetric magnetic field is similarto those demonstrated in the other models. The magneticfield is antisymmetric about equator showing three bandsof the toroidal magnetic field propagating outward of therotation axis, along the spherical iso-surfaces of the angu-lar velocity (see, Figure 11b). The zonal variations of theangular velocity are small and their patterns are elongatedalong the axis of rotation. Similar to models M2 and M3, thelarge-scale magnetic field inside the star has a spiral struc-ture because the non-axisymmetric mode m=1 dominatesthe others partial modes.In our simulations, we also have tried the larger valuesof the parameters
P m T and C α . Results for the kinematicmodels with non-linear α effect and P m T = 10 are similarthe model M3. The period of the axisymmetric dynamo incase of P m T = 10 is about 120 years in agreement withexpectations of Shulyak et al. (2015). The model show ratherstrong polar axisymmetric magnetic field with the strengthof 1.5kG. This is because of the meridional circulation effect.Its efficiency ncreases with the increase of the parameter P m T . For the P m T = 10 the axisymmetric regime persistswhen the C α < . . We found the transition to the non-axisymmetric dynamo for the C α = 0 . . The properties ofthe non-axisymmetric mean-field dynamo in the fully non-linear regime for the case of P m T = 10 remain unclear. The previous consideration of the mean-field models of thefully convective stars was restricted to analysis of the eigen-value problems (Elstner & Rüdiger 2007; Shulyak et al.2015) or the kinematic case with uniform density stratifica-tion and the algebraic non-linearity of the α -effect (Chabrier c (cid:13) , 000–000 onlinear regimes in mean-field full-sphere dynamo
80 100 120 140 160 [YR] L A T I T UD E B r ,[ G ]
80 100 120 140 16060080010001200140016001800 | B | ,[ G ] N O N - AX I SY MM E T R Y a)b) c) Figure 10.
The model M5, a) the mean strength of sum the first five partial modes of the toroidal magnetic field at the r = R (cid:63) (redline), the index of the non-axisymmetric of the large-scale magnetic field (dashed line), the mean strength of the axisymmetric toroidalmagnetic field (solid black line); b) the time-latitude diagram of the axisymmetric toroidal magnetic field at the r = R (cid:63) (contours forthe range ± G) and the axisymmetric radial magnetic field at the surface; c) snapshot of the large-scale magnetic field lines out of thestar and the backgorund image shows the radial magnetic field within range of ± G. & Küker 2006). The main progress in theoretical under-standing of the dynamo on the the fully convective starswere made with help of the direct numerical simulations (see,e.g., Dobler et al. 2006; Browning 2008; Gastine et al. 2012;Yadav et al. 2015). The paper for the first time presentsresults of the non-linear mean-field dynamo models of thefully convective star rotating with period 10 days.The key reasons to study the mean-field models is tostudy behavior of the dynamo in varying the governing dy-namo parameters. At the first step, let us discuss the kine-matic dynamos with the nonlinear α -effect. The angularvelocity profile in this case is different to the cylinder-likepattern, which was discussed in the literature (see, e.g.,Moss 2004, 2005; Chabrier & Küker 2006) and which ap-pears in the direct numerical simulations. Our model in-clude effect of the meridional circulation which is importantin the subsurface layer for the case of P m T > . For thecase P m T = 3 , the model M1 shows the strong axisymmet-ric dipole-like magnetic field with magnitude of the polarfield about 1kG. The dominance of the antisymmetric rel-ative to equator magnetic field disappears in the nonlinearmixed parity solution if we neglect the meridional circula-tion. The dynamo waves show the solar-like time-latitudediagrams with toroidal field drifting to the equator and theradial field drifting to the pole.The eigenvalue analysis shows that generation of thenon-axisymmetric magnetic field for the case of P m T > is less efficient than the axisymmetric dynamo because thecritical parameter of the dynamo instability is smaller inthe second case. This is general conclusion of the moststudies of the mean field dynamo starting from the sem-inal paper by Raedler (1986). The conclusion lead to ig-norance of the non-axisymmetric dynamos even for thesuper-critical regimes of the axisymmetric dynamo (cf, Ray-naud & Tobias 2016). However the model M2 show thatin case of the dynamo instability of the non-axisymmetricfield, the non-axisymmetric regime can beat the axisym-metric one. The interaction between axisymmetric and non- a) b) Figure 12.
Snapshots of the model M5, a) the nonlinear α effects (volume contours for ± ± · G /M) axisymmetric magnetic field goes via the nonlinear effects.Those are the conservation of the magnetic helicity and themagnetic buoyancy. Contributions of the magnetic helicityon the α -effect can not be ignored in the mean-field solar dy-namos (Brandenburg & Käpylä 2007). They are importantin the non-axisymmetric dynamo, as well. The change ofthe dynamo regime for the overcritical C α is because of thenon-axisymmetric α -effect, which is produced by the mag-netic helicity conservation in the non-axisymmetric large-scale dynamo. Figures12(a,b) show snapshots of the α φφ (see, Eq.(12) and the mean magnetic helicity density of thesmall-scale field, which is generated because of the magnetichelicity conservation in the model M5. Models M2 and M3show similar distributions. The models produce the non-axisymmetric non-linear α effect and this supports domi-nance the non-axisymmetric magnetic field in the dynamo.In the solar dynamo models the non-axisymmetric α -effectwas employed for explanation of the so-called active longi-tudes of the sunspot formations (Bigazzi & Ruzmaikin 2004;Berdyugina et al. 2006). In our models this effect stems nat-urally from magnetic helicity conservation. c (cid:13)000
Snapshots of the model M5, a) the nonlinear α effects (volume contours for ± ± · G /M) axisymmetric magnetic field goes via the nonlinear effects.Those are the conservation of the magnetic helicity and themagnetic buoyancy. Contributions of the magnetic helicityon the α -effect can not be ignored in the mean-field solar dy-namos (Brandenburg & Käpylä 2007). They are importantin the non-axisymmetric dynamo, as well. The change ofthe dynamo regime for the overcritical C α is because of thenon-axisymmetric α -effect, which is produced by the mag-netic helicity conservation in the non-axisymmetric large-scale dynamo. Figures12(a,b) show snapshots of the α φφ (see, Eq.(12) and the mean magnetic helicity density of thesmall-scale field, which is generated because of the magnetichelicity conservation in the model M5. Models M2 and M3show similar distributions. The models produce the non-axisymmetric non-linear α effect and this supports domi-nance the non-axisymmetric magnetic field in the dynamo.In the solar dynamo models the non-axisymmetric α -effectwas employed for explanation of the so-called active longi-tudes of the sunspot formations (Bigazzi & Ruzmaikin 2004;Berdyugina et al. 2006). In our models this effect stems nat-urally from magnetic helicity conservation. c (cid:13)000 , 000–000 V.V. Pipin
The magnetic feedback on the differential rotation re-duces efficiency of the axisymmetric dynamo. The strengthof the large-scale magnetic field in the model M4 is less thanin the model M1. The cyclic effect of the large and small-scale Lorentz force on the angular momentum fluxes pro-duces phenomena known in the solar magnetic acitvity likethe zonal variations of the angular velocity and variations ofthe meridional flow. Both of them predicted to have muchsmaller amplitude than for the Sun. The rotational velocityat the equator is . km/s, then the predicted magnitudeof the latitudinal shear between equator and pole is onlyabout 14 m/s. Therefore our models demonstrate the dy-namo induced zonal variations are about of 10 percent ofmagnitude of the mean latitudinal shear. The relative vari-ations of the meridional circulation are about 1 percent ofthe mean flow which is much smaller than it is observed onthe Sun. Note, the M-dwarf has much denser plasma thanthe Sun and for the 1kG magnetic field at the top of theintegration domain ( . R (cid:63) ) the Alfven velocity is less than m/s. In the model the toroidal field does not penetrate tothe surface because of the vacuum boundary conditions andthis reduces the magnitude of the large-scale flow variationson the surface. Unlike the Sun (see, eg, Birch 2011; Howeet al. 2011; Kosovichev et al. 2013) the predicted torsionaloscillations have the equal magnitudes in the bulk of the starand at the surface. Variations of the meridional circulationare concentrated to the surface. Note , that the radial pro-file of the meridional circulation is still unclear in the case ofthe Sun, see preliminary results in the papers by Hathaway(2012) and Zhao et al. (2013), who supports concentrationof 11-th year variations of the solar meridional circulationto the surface.It is predicted that magnetic activity produces ratherstrong distortion of the angular velocity profile inside thestar leaving the structure of the meridional flow nearly thesame as it is in the kinematic models. The same results werefound in the direct numerical simulations of Browning (2008)and Yadav et al. (2015). Figure 13a allows comparison totheir results. We find that in the magnetic case (the modelM5) the latitudinal shear persists only in the upper layer ofthe star. Also, there the strong radial shear presents nearthe equator. The same was found in the direct numericalsimulation by Yadav et al. (2015). The model of Browning(2008) showed the uniform angular velocity profile in themagnetic case. We find that in the nonlinear model M5 thepositive radial shear in the equatorial region is stronger thanin the kinematic model M1. Also we see formation of the ra-dial shear at the surface in the polar region in the modelM5. The increase of the magnitude of the subsurface shearas a result of the magnetic field influence on the angularmomentum fluxes is also in agreement with the recent nu-merical simulations on the solar-like stars (Guerrero et al.2013; Käpylä et al. 2014; Guerrero et al. 2016).Our results show that the strength of the surfacepoloidal magnetic field is only factor two or three lesser thanthe strength of the toroidal magnetic field inside the star, seethe Table 1. All the models show rather strong polar mag-netic field, 1kG in the kinematic models and from 100 to 500G in the nonlinear models. Current observations of the stel-lar magnetic activity inform us a lot about the topologicaland spectral properties of the magnetic field distributions atM-dwarfs and cool stars (Morin et al. 2010; See et al. 2016). [ G ] M4, B r M5, B r ,m=0M5, B r ,m=1M5,B ,m=1 r/R ⊙ µ H z M5,0M5,60M1,0M1,60 a) b)
Figure 13. a) The angular velocity radial profile in the kinematic(red lines) and nonlinear models M1 and M5 for the equator ( ◦ )and 60 ◦ latitudes; b)Modes. Figure 13b presents results of the spherical harmonic decom-position for magnetic field predicted by the fully nonlinearmodels M4 (axisymmetric one) and M5. In the axisymmet-ric model M4 we don’t expect any toroidal field out of thesurface because of the boundary conditions. In this case theenergy of the magnetic field outside the star is dominatedby (cid:96) = 3 and (cid:96) = 5 harmonics which is similar to the Sun(Stenflo & Guedel 1988; Stenflo 2013; Vidotto 2016). Thenon-axisymmetric dynamo model M5 show the dominanceof the mode m=1 and (cid:96) = 1 of the large-scale toroidal mag-netic field. The ratio of the energy of the non-axisymmetricand axisymmetric poloidal magnetic field in the model M5is about factor order of the magnitude. The given resultsare in agreement with Morin et al. (2010) for the magneticfield observations for the early types of the M-dwarfs with amoderate rotation rates.Let’s summarize the main findings of the paper. Ourstudy confirm the previous conclusions of Shulyak et al.(2015) that the weak differential rotation of the M-dwarfscan support the axisymmetric dynamo especially for the case
P m T > . For the case
P m T = 3 we find that the gener-ation threshold α -effect parameter C α is lower for axisym-metric magnetic field. However for the overcritical α -effectthe non-axisymmetric dynamo become preferable. The sit-uation is reproduced both in the kinematic and in the fullynonlinear dynamo models. In the non-linear case the differ-ential rotation of the star deviates strongly from the kine-matic case. For the most complete non-linear dynamo modelwe found the non-axisymmetric magnetic field of strengthabout 0.5kG at at the surface mid latitude, it is rigidlyrotating and it is perturbed by the axisymmetric dynamowaves propagating out of the rotational axis. The predicteddynamo period of the axisymmetric dynamo waves in themodel is about 40 Yr for the P m T = 3 and it is longer forthe higher P m T . Acknowledgements
I appreciate Prof D.D. Sokoloff, Prof D. Moss and DrD.Shulyak for discussions and comments. I thank the finan-cial support of the project II.16.3.1 of ISTP SB RAS andthe partial support of the RFBR grants 15-02-01407-a, 16-52-50077-jaf.
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Pipin (2004) found that under the joint action of the Coriolisforce and the large-scale toroidal magnetic field, and whenit holds Ω ∗ > , the eddy heat conductivity tensor could beapproximated as follows χ ij ≈ χ T (cid:18) φ ( I ) χ ( β ) φ (Ω ∗ ) δ ij + φ ( (cid:107) ) χ ( β ) φ (cid:107) (Ω ∗ ) Ω i Ω j Ω (cid:19) , (A1)where φ ( I ) χ = 2 β (cid:32) − (cid:112) β (cid:33) ,φ ( (cid:107) ) χ = 2 β (cid:16)(cid:112) β − (cid:17) . Expression A1 were obtained the standard schemes of themean-field magnetohydrodynamics employing the so-called“second order correlation approximation” and the mixinglength approximations. Also we skip components of the ten-sor along the large-scale magnetic field.The heat transport by radiation reads, F rad = − c p ρχ D ∇ T, where χ D = 16 σT κρ c p , where κ is opacity coefficient. The radial profiles of the grav-ity acceleration, g , the density, ρ , the temperature, T , theheat source, (cid:15) , as well as others thermodynamic parameters,like the c p or the κ are taken form the reference model whichwas calculated with help of the MESA code. Expression of the Reynolds is determined from the mean-field hydrodynamics theory (see, Kitchatinov et al. 1994;Kitchatinov 2004) as follows ˆ T ij = (cid:18) (cid:104) u i u j (cid:105) − πρ (cid:18) (cid:104) b i b j (cid:105) − δ ij (cid:10) b (cid:11)(cid:19)(cid:19) , (A2)where u and b are fluctuating velocity and magnetic fields.The turbulent stresses take into account the turbulent vis-cosity and generation of the large-scale shear due to the Λ -effect (Kitchatinov & Olemskoy 2011): T rφ = ρν T (cid:8) Φ ⊥ + (cid:0) Φ (cid:107) − Φ ⊥ (cid:1) µ (cid:9) r ∂ sin θ Ω ∂r + ρν T sin θ (cid:0) Φ (cid:107) − Φ ⊥ (cid:1) (cid:0) − µ (cid:1) ∂ Ω ∂µ (A3) − ρν T sin θ Ω (cid:18) α MLT γ (cid:19) (cid:16) V (0) + sin θV (1) (cid:17) ,T θφ = ρν T sin θ (cid:8) Φ ⊥ + (cid:0) Φ (cid:107) − Φ ⊥ (cid:1) sin θ (cid:9) ∂ Ω ∂µ + ρν T (cid:0) Φ (cid:107) − Φ ⊥ (cid:1) µ sin θr ∂ Ω ∂r (A4) + ρν T µ Ω sin θ (cid:18) α MLT γ (cid:19) H (1) , where ν T = 45 η T . The viscosity functions - Φ (cid:107) , Φ ⊥ and the Λ - effect - V (0 , and H (1) , are dependent on the Coriolisnumber and the strength of the large-scale magnetic field.They also depends on the anisotropy of the convective flows.Similar to the Subsection5.1 we employ the fast rotatingregime of the magnetic quenching for the eddy viscosity andthe the Λ - effect as it was discussed earlier in (Kueker et al.1996; Pipin 1999; Pipin 2004): Φ ⊥ = ψ ⊥ (Ω (cid:63) ) φ V ⊥ ( β ) , Φ (cid:107) = ψ (cid:107) (Ω (cid:63) ) φ ( I ) χ ( β ) , (A5) V (0) = ( J (Ω (cid:63) )+ J (Ω (cid:63) )+ a ( I (Ω (cid:63) )+ I (Ω (cid:63) ))) φ ( I ) χ ( β ) , (A6) V (1) = ( J (Ω (cid:63) ) + aI (Ω (cid:63) )) φ ( I ) χ ( β ) , (A7) H (0) = J (Ω (cid:63) ) φ H ( β ) , (A8)and H (1) = − V (1) , where the new magnetic quenching func-tions are: φ V ⊥ = 4 β (cid:113) (1 + β ) (cid:16)(cid:0) β + 19 β + 18 (cid:1) (cid:112) (1 + β ) − β − β − (cid:1) , (A9) φ H = 4 β β (cid:113) (1 + β ) − . (A10)We employ the parameter of the turbulence anisotropy a =1 (see discussion, by Kitchatinov 2004). The equation A5)shows that for case of the “fast” rotating fluid the large-scale magnetic field quenches the eddy viscosity anisotropy.This conclusion was obtained for the case when the toroidallarge-scale magnetic field dominates the poloidal component(Pipin 2004). This approximation may be incorrect for thefast rotating fully convective stars.The Lambda effect is modulated by the factor (cid:96) (cid:12)(cid:12)(cid:12) Λ ( ρ ) (cid:12)(cid:12)(cid:12) ≈ α MLT γ , where Λ ( ρ ) = ∇ log ρ . It varies sharplynear the center and the top of the star. To avoid the numer-ical complications we force the Λ -effect to go zero towardthe center of the star, we replaced that factor as follows, α MLT γ = α MLT γ (cid:18) erf (cid:18) (cid:18) rR (cid:63) − . (cid:19)(cid:19)(cid:19) , (A11)where α MLT = 1 . and γ = 53 .The first term in the RHS of the Eq.(6) describes dissi-pation of the mean vorticity, ω . Similarly to Rempel (2005)we approximate it as follows, − (cid:20) ∇ × ρ ∇ · ρ ˆ T (cid:21) φ ≈ ν T φ (Ω ∗ ) ψ ( β ) ∇ ω, (A12)where ν T = 45 χ T , the rotational function φ and the mag-netic quenching function are given in Kitchatinov et al.(1994). We have tried the more general formalism with fullcomponents of the eddy-viscosity tensor for the rotating tur-bulence provided by Kitchatinov et al. (1994). We found re-sults to be similar to the case of the Eq(A12).For the ideal gas the last term in Eq.(6) can be rewrittenin terms of the specific entropy (Kitchatinov & Olemskoy2011), ρ [ ∇ ρ × ∇ p ] φ ≈ − grc p ∂s∂θ . (A13) c (cid:13) , 000–000 onlinear regimes in mean-field full-sphere dynamo The meridional circulation is expressed by a stream function Ψ , U m = 1 ρ ∇ × ˆ φ Ψ . The Ψ and the ω are related via theequation − ρω = (cid:18) ∆ − r sin θ (cid:19) Ψ − rρ ∂ρ∂r ∂r Ψ ∂r . (A14)We employ the stress-free boundary conditions for theEq.(4), the azimuthal component of the mean vorticity, ω ,is put to zero at the boundaries. This section of Appendix describe some parts of the mean-electromotive force. The basic formulation is given in (Pipin,2008) (hereafter, P08). For this paper we reformulate tensor α ( H ) i,j , which represents the hydrodynamical part of the α -effect, by using Eq.(23) from P08 in the following form, α ( H ) ij = 3 (cid:16) Ω · Λ ( ρ ) (cid:17) Ω (cid:26) δ ij f ( a )10 + Ω i Ω j Ω f ( a )5 (cid:27) , (A15)where Λ ( ρ ) = ∇ log ρ . The other parts of the α -effect arerather small because the star is in the regime of the fastrotation, when the Coriolis number Ω ∗ (cid:29) . Moreover, if weneglect terms order of O (cid:18) ∗ (cid:19) in the Taylor expansion ofthe Eq.(A15), we get ((Rüdiger & Kitchatinov 1993)): α ( H ) ij = − π (cid:16) Ω · Λ ( ρ ) (cid:17) Ω (cid:26) δ ij − Ω i Ω j Ω (cid:27) , The functions f ( a )5 , where defined in P08 for the general casewhich includes the effects the hydrodynamic and magneticfluctuations in the background turbulence. In the paper weemploy the case when the background turbulent fluctuationsof the small-scale magnetic field are in the equipartition withthe hydrodynamic fluctuations, i.e., ε = b (cid:48) πρu (cid:48) = 1 , wherethe u (cid:48) and b (cid:48) are intensity of the background turbulent ve-locity and magnetic field. The magnetic quenching functionof the hydrodynamical part of α -effect is defined by ψ α = 5128 β (cid:18) β − − (cid:0) β − (cid:1) arctan (2 β )2 β (cid:19) , (A16)The magnetic helicity part of the α -effect, α ( M ) i,j is expressedby α ( M ) ij = (cid:26) δ ij f ( a )2 (Ω ∗ ) − Ω i Ω j Ω f ( a )1 (Ω ∗ ) (cid:27) . (A17)We employ the anisotropic diffusion tensor which is derivedin P08 and in (Pipin & Kosovichev 2014): η ijk = 3 η T (cid:26)(cid:16) f ( a )1 − f ( d )2 (cid:17) ε ijk + 2 f ( a )1 Ω i Ω n Ω ε jnk (cid:27) (A18) + aη T φ ( g n g j ε ink − ε ijk ) where g is the unit vector in the radial direction, a = 1 is theparameter of the turbulence anisotropy, η T is the magneticdiffusion coefficient. The quenching functions f ( a,d )1 , and φ are given in (Pipin & Kosovichev 2014).The turbulent pumping of the mean-field contains thesum of the contributions due to the mean density gradient (Kitchatinov 1991) and the mean-filed magnetic buoyancy(Kitchatinov & Pipin 1993), γ ( b ) ij ,: γ ij = γ ( ρ ) ij + γ ( b ) ij , (A19)where each contribution is defined as follows: γ ( ρ ) ij = 3 η T (cid:110) f ( a )3 Λ ( ρ ) n + f ( a )1 (cid:16) e · Λ ( ρ ) (cid:17) e n (cid:111) ε inj (A20) − η T f ( a )1 e j ε inm e n Λ ( ρ ) m γ ( b ) ij = α MLT u (cid:48) γ β K ( β ) g n ε inj , (A21)where f ( a )1 , (Ω (cid:63) ) are functions of the Coriolis number, u (cid:48) isthe RMS of the convective velocity, Λ ( ρ ) i = ∇ i log ρ are com-ponents of the gradient of the mean density. The α MLT isthe parameter of the mixing length theory, γ is the adiabaticexponent and the function K ( β ) is defined in (Kitchatinov& Pipin 1993) and g is the unit vector in the radial direction. c (cid:13)000