A&A 47, 0 022 (2007) DOI: 0.05/0004-636:2007748 c ESO 2007 Astroomy & Astrophysics Agular mometum coservatio ad torsioal oscillatios i the Su ad solar-lie stars A. F. Laza INAF Osservatorio Astrofisico di Cataia, via S. Sofia 78, 9523 Cataia, Italy e-mail: uccio.laza@oact.iaf.it Received 6 March 2007 / Accepted 7 Jue 2007 ABSTRACT Cotext. The solar torsioal oscillatios, i.e., the perturbatios of the agular velocity of rotatio associated with the eleve-year activity cycle, are a maifestatio of the iteractio amog the iterior magetic fields, amplified ad modulated by the solar dyamo, ad rotatio, meridioal flow ad turbulet thermal trasport. Therefore, they ca be used, at least i priciple, to put costraits o this iteractio. Similar pheomea are expected to be observed i solar-lie stars ad ca be modelled to shed light o aalogous iteractios i differet eviromets. Aims. The source of torsioal oscillatios is ivestigated by meas of a model for the agular mometum trasport withi the covectio zoe. Methods. A descriptio of the torsioal oscillatios is itroduced, based o a aalytical solutio of the agular mometum equatio i the mea-field approach. It provides iformatio o the itesity ad locatio of the torques producig the redistributio of the agular mometum withi the covectio zoe of the Su alog the activity cycle. The method ca be exteded to solar-lie stars for which some iformatio o the time-depedece of the differetial rotatio is becomig available. Results. Illustrative applicatios to the Su ad solar-lie stars are preseted. Uder the hypothesis that the solar torsioal oscillatios are due to the mea-field Loretz force, a amplitude of the Maxwell stresses B r B φ > 8 0 3 G 2 at a depth of 0.85 R at low latitude is estimated. Moreover, the phase relatioship betwee B r ad B φ ca be estimated, suggestig that B r B φ > 0below 0.85 R ad B r B φ < 0 above. Coclusios. Such prelimiary results show the capability of the proposed approach to costrai the amplitude, phase ad locatio of the perturbatios leadig to the observed torsioal oscillatios. Key words. Su: rotatio Su: activity Su: magetic fields Su: iterior stars: rotatio stars: activity. Itroductio Doppler measuremets of the surface rotatio of the Su show bads of faster ad slower zoal flows that appear at midlatitudes ad migrate toward the equator with the period of the eleve-year cycle, accompayig the bads of suspot activity. The amplitude of such velocity perturbatios, called torsioal oscillatios, is 5 ms ad faster rotatio is observed o the side equatorward of the suspot belt (Howard & LaBote 980). Helioseismology has revealed that the torsioal oscillatios are ot at all a superficial pheomeo but ivolve much of the covectio zoe, as show, for example, by Howe et al. (2000), orotsov et al. (2002), Basu & Atia (2003) ad more recetly by Howe et al. (2005, 2006). The amplitude of the agular velocity variatio is δω/2π 0.5 Hz at least dow to 0% 5% of the solar radius, although the precise depth of peetratio of the oscillatios is difficult to establish give the preset ucertaities of the iversio methods i the lower half of the solar covectio zoe (e.g., Howe et al. 2006). I additio to such a low-latitude brach of the torsioal oscillatios, helioseismic studies have detected the presece of a high-latitude brach (above 60 latitude) that propagates poleward ad the amplitude of which is about δω/2π 2 Hz (e.g., Toomre et al. 2000; Basu & Atia 200). Such a brach seems to propagate almost all the way dow to the base of the covectio zoe. A geeral descriptio of the perturbatio of the agular velocity of the torsioal oscillatios, give the preset accuracy of the observatios, is provided by the simple formula (e.g., orotsov et al. 2002; Howe et al. 2005): ω(r,θ) = A (c) (r,θ)cos(σt) + A (s) (r,θ)si(σt) = A(r,θ)si[σt + φ(r,θ)], () where r is the distace from the cetre of the Su, θ the colatitude measured from the North pole, σ the frequecy of the eleve-year cycle, t the time, ad the amplitude fuctios A (c) A si φ, A (s) A cos φ deped o the amplitude A ad the iitial phase φ. Moreover, the velocity perturbatio is symmetric with respect to the equator: ω(r,θ) = ω(r,π θ). (2) Several models have bee proposed to iterpret the torsioal oscillatios begiig with the pioeerig wor by Schüssler (98) ad Yoshimura (98) who cosidered the Loretz force associated with the magetic fields i the activity belts as the cause of the velocity perturbatios observed i the solar photosphere. Later models, based o the effects of the Loretz force o the turbulet Reyolds stresses, were proposed, by e.g., Küer et al. (996) ad Kichatiov et al. (999), followig a origial suggestio by Rüdiger & Kichatiov (990). Spruit (2003) proposed that the low-latitude brach of the torsioal oscillatios is a geostrophic flow drive by temperature variatios due to the ehaced radiative losses i the active regio belts. Article published by EDP Scieces ad available at http://www.aada.org or http://dx.doi.org/0.05/0004-636:2007748
02 A. F. Laza: Torsioal oscillatios i the Su ad stars More recet studies by Covas et al. (2004, 2005) preset models based o the simultaeous solutio of o-liear meafield dyamo equatios ad the azimuthal compoet of the Navier-Stoes equatio with a uiform turbulet viscosity. They reproduce the gross features of the torsioal oscillatios ad of the solar activity cycle with a appropriate tuig of the free parameters. Rempel (2006, 2007) cosiders the role of the meridioal compoet of the Navier-Stoes equatio i mea-field models ad fids that the perturbatio of the meridioal flow caot be eglected i the iterpretatio of the torsioal oscillatios. His models suggest that the low-latitude brach of the torsioal oscillatios caot be explaied solely by the effect of the mea-field Loretz force, but that thermal perturbatios i the active regio belt ad i the bul of the covectio zoe do play a active role, as proposed by Spruit (2003). I the preset study, the agular mometum coservatio is cosidered ad the relevat equatio i the mea-field approximatio is solved aalytically for the case of a turbulet viscosity that depeds o the radial co-ordiate. A geeral solutio is derived idepedetly of ay specific dyamo model, allowig us to put costraits o the localizatio of the torques producig the torsioal oscillatios. A illustrative applicatio of the proposed methods is preseted usig the available data. The observatios of youg solar-lie stars by meas of tomographic techiques based o high-resolutio spectroscopy have provided recet evidece for time variatio of their surface differetial rotatio (e.g., Doati et al. 2003; Jeffers et al. 2007). Laza (2006a) has recetly show how such variatios ca be related to the itesity of the magetic torque produced by a o-liear dyamo i their covective evelopes, i the case of rapidly rotatig stars for which the Taylor-Proudma theorem applies. I the ear future, the possibility of measurig the time variatio of the rotatioal splittigs of p-mode oscillatios i solar-lie stars may provide us with iformatio o the chages of their iteral rotatio, although with limited spatial resolutio. I the preset study, we exted the cosideratios of Laza (2006a) to the case of a geeric iteral rotatio profile, ot ecessarily verifyig the Taylor-Proudma theorem, to obtai hits o the amplitude of the torque leadig to the rotatio chage. 2. The model 2.. Hypotheses ad basic equatios We cosider a iertial referece frame with the origi i the barycetre of the Su ad the z-axis i the directio of the rotatio axis. A spherical polar co-ordiate system (r,θ,ϕ)is adopted, where r is the distace from the origi, θ the co-latitude measured from the North pole ad ϕ the azimuthal agle. We assume that all variables are idepedet of ϕ ad that the solar desity stratificatio is spherically symmetric. The equatio for the agular mometum coservatio i the mea-field approach reads (e.g., Rüdiger 989; Rüdiger & Hollerbach 2004): t (ρr2 si 2 θ Ω) + Θ = 0, (3) where ρ(r) is the desity, Ω(r,θ,t) the agular velocity ad Θ the agular mometum flux vector give by: Θ = (ρr 2 si 2 θ Ω)u (m) +r si θ ρu u ϕ r si θ µ (BB ϕ + B B ϕ ), (4) where u (m) is the meridioal circulatio, u the fluctuatig velocity field, µ the magetic permeability, B the mea magetic field ad B the fluctuatig magetic field; agularacets idicate the Reyolds average defiig the mea-field quatities. The Reyolds stresses ca be writte as: ( ρu i u j = η ui t + u ) j +Λ ij, (5) x j x i where u is the mea flow field, η t (r) is the turbulet viscosity, assumed to be a scalar fuctio of r oly, ad Λ ij idicates the odiffusive part of the Reyolds stresses due to the velocity correlatios i a rotatig star (see Rüdiger 989; Rüdiger & Hollerbach 2004, for details). The coservatio of the total agular mometum of the covectio zoe implies: Θ r = 0forr =, R, (6) where is the radius at the loweoudary of the covectio zoe ad R is the radius of the Su. The equatio for the coservatio of the agular mometum ca be recast i the form: Ω t ( ) r 4 Ω η ρr 4 t η t r r ρr 2 ( µ 2 ) µ where µ cos θ ad the source term S is give by: [ ( µ 2 ) 2 Ω ] =S, (7) µ τ S = ρr 2 ( µ 2 ), (8) ad τ is a vector whose compoets are: [ τ i = r si θ Λ iϕ µ ( Bi B ϕ + B i B ϕ )] + ρr 2 si 2 θωu (m)i. (9) The boudary coditios give by Eq. (6) ca be writte as: Ω r = 0forr =, R, (0) whe we assume τ r = 0 at the surface. Note that helioseismic measuremets idicate the presece of a subsurface shear layer with Ω r < 0 at low latitudes (Corbard & Thompso 2005), but we prefer to adopt the stress-free boudary coditio (0) at the surface to esure the coservatio of the total agular mometum of the covectio zoe i our model. The solar agular velocity ca be split ito a timeidepedet compoet Ω 0 ad a time-depedet compoet ω, i.e., the torsioal oscillatios: Ω(r,µ,t) =Ω 0 (r,µ) + ω(r,µ,t). () The equatio for the torsioal oscillatios thus becomes: ω t ( ) r 4 ω η ρr 4 t η [ t ( µ 2 ) 2 ω ] =S r r ρr 2 ( µ 2, (2) ) µ µ where the perturbatio of the source term is: S = τ ρr 2 ( µ 2 ), (3) with τ i =r si θ [ Λ (p) iϕ µ ( Bi B ϕ + B i B ϕ )] +ρr 2 si 2 θ Ω 0 u (p) (m)i, (4)
A. F. Laza: Torsioal oscillatios i the Su ad stars 03 where Λ (p) iϕ ad u(p) (m)i are the time-depedet perturbatios of the o-diffusive Reyolds stresses ad of the meridioal circulatio, respectively, ad Ω 0 is the average of the solar agular velocity over the covectio zoe. Note that the Maxwell stresses appear i Eq. (4), but ot i the correspodig equatio for Ω 0 because the solar magetic field has o time-idepedet compoet. Moreover, i derivig Eq. (4), the variatio of the agular velocity over the covectio zoe has bee eglected i the term cotaiig the perturbatio of the meridioal circulatio sice ω Ω 0 (e.g., Rüdiger 989; Rempel 2007). Equatio (2) must be solved together with the boudary coditios: ω r = 0forr =, R. (5) 2.2. Solutio of the agular mometum equatio The geeral solutio of Eq. (2) with the boudary coditios (5) ca be obtaied by the method of separatio of the variables ad expressed as a series of the form (e.g., Laza 2006b, ad refereces therei): ω(r,µ,t) = =0,2,4,... =0 α (t)ζ (r)p (,) (µ), (6) where α (t)adζ (r) are fuctios that will be specified below ad P (,) (µ) are Jacobia polyomials, i.e., the fiite solutios of the equatio: d ( µ 2 ) 2 dp(,) + ( + 3)( µ 2 )P (,) = 0, (7) dµ dµ i the iterval µ icludig its eds (e.g., Smirov 964a). The Jacobia polyomials form a complete ad orthogoal set i the iterval [, ] with respect to the weight fuctio ( µ 2 ). Oly the polyomials of eve degree appear i Eq. (6) because the agular velocity perturbatio is symmetric with respect to the equator (see Eq. (2)). For the asymptotic expressio of the Jacobia polyomials is (e.g., Gradshtey & Ryzhi 994): cos [ ( + 3 P (,) 2 (cos θ) = )θ ] 3π 4 [ ( ) ( π si θ 2 cos θ 3/2 + O( 2)] 3 2 ). (8) The fuctios ζ are the solutios of the Sturm-Liouville problem defied i the iterval r R by the equatio: ( ) d r 4 dζ η t ( + 3)r 2 η t ζ + λ ρr 4 ζ = 0 (9) dr dr with the boudary coditios (followig from Eq. (5)): dζ = 0atr =, R. (20) dr We shall cosider ormalized eigefuctios, i.e.: R ρr 4 ζ 2 dr =. The eigefuctios ζ for a fixed, form a complete ad orthoormal set i the iterval [, R ] with respect to the weight fuctio ρr 4 that does ot deped o.we recall from the theory of the Sturm-Liouville problem that the eigevalues verify the iequality: λ 0 <λ <...λ <λ + <... ad that the eigefuctio ζ has odes i the iterval [, R ] for each. For = 0, the first eigevalue correspodig to the eigefuctio ζ 00 is zero ad the eigefuctio vaishes at all poits i [, R ], as it is evidet by itegratig both sides of Eq. (9) i the same iterval, applyig the boudary coditios (20) ad cosiderig that ζ 00 has o odes. For > 0, all the eigevalues λ 0 are positive, as ca be derived by itegratig both sides of Eq. (9) i the iterval [, R ], applyig the boudary coditios (20) ad cosiderig that ζ 0 has o odes. I view of the iequality give above, all the eigevalues λ are the positive for 0. Moreover, it is possible to prove that λ λ if > ad that (see Smirov 964b): ( ) ( 2 π 2 l p + ( + 3)q 2 π 2 2 λ M ) l P + ( + 3)Q 2, (2) m where P, Q ad M are the maximum values of the fuctios r 4 η t, r 2 η t ad ρr 4 i the iterval [, R ], respectively, ad p, q ad m their miimum values i the same iterval, respectively; ad l R ρ/ηt dr. For the asymptotic expressio for the eigefuctio ζ is (e.g., Morse & Feshbach 953): ( λ r ) ζ (r) (ρη t ) 4 r 2 cos ρ/ηt dr. (22) The time depedece of the solutio i Eq. (6) is specified by the fuctios α that are give by: dα + λ α (t) = β (t), (23) dt where the fuctios β appear i the developmet of the perturbatio term S : S (r,µ,t) = ad are give by (Laza 2006b): β = (2 + 3)( + 2) 8( + ) R The solutio of Eq. (23) is: α (t) = α (0) + exp( λ t) β (t)ζ (r)p (,) (µ), (24) ρr 4 ( µ 2 )S (r,µ,t)ζ (r)p (,) (µ)drdµ. (25) t 0 β (t )exp(λ t )dt, (26) which allows us to specify the geeral solutio of Eq. (2) with the boudary coditios (5) whe the perturbatio term S (r,µ,t) ad the iitial coditios are give. 2.3. Solutio for the solar torsioal oscillatios To fid the solutio appropriate to the solar torsioal oscillatios as specified by Eqs. () ad (2), it is useful to derive a alterative expressio for the fuctios β as follows. Substitutig Eq. (3) ito Eq. (25) ad taig ito accout that the elemet of volume is d = 2πr 2 drdµ, the right had side of Eq. (25) ca be recast i the form of a volume itegral exteded to the solar covectio zoe: β = F ( τ )ζ P (,) d, (27) where: F (2 + 3)( + 2) 6π( + ) (28)
04 A. F. Laza: Torsioal oscillatios i the Su ad stars is a factor comig from the ormalizatio of the Jacobia polyomials. It is possible to further simplify Eq. (27) by cosiderig the idetity: [ζ P (,) τ ] = ζ P (,) ( τ ) + τ [ζ P (,) ]. (29) Itegratig both sides of Eq. (29) over the volume of the covectio zoe ad cosiderig that the itegral of the left had side vaishes thas to the Gauss s theorem ad the coditio that the radial compoet of the stresses τ r is zero o the boudaries, we fid: β = F τ [ζ P (,) ]d. (30) The time depedece of the solar torsioal oscillatios specified i Eq. () suggests that a similar depedece for the perturbatio term should be cosidered: τ (r,µ,t) = τ (c) (r,µ)cos(σt) + τ(s) (r,µ)si(σt), (3) from which a similar expressio for the β follows by substitutio ito Eq. (27). If we put such a expressio for β ito Eq. (26) ad perform the itegratios with respect to the time, we fid the statioary solutio: α (t) = F λ 2 {[ +σ2 [ + ζ P (,) ζ P (,) ( λ τ (c) ) ] σ τ(s) d cos(σt) } ( ) ] λ τ (s) +σ τ(c) d si(σt). (32) This expressio ca be substituted ito Eq. (6) to give the agular velocity perturbatio. It ca be writte i the form of Eq. () with: [ A (c) (r,µ)= G (r,µ,r,µ ) τ (c) G ] 2(r,µ,r,µ ) τ (s) d, [ A (s) (r,µ)= G (r,µ,r,µ ) τ (s) +G ] 2(r,µ,r,µ ) τ (c) d, (33) where the symbol d meas that the volume itegratio is to be performed with respect to the variables r ad µ,adthe fuctios G ad G 2 are Gree fuctios defied as: G (r,µ,r,µ ) = F λ λ 2 + σ2 ζ (r)p (,) (µ)ζ (r )P (,) (µ ) G 2 (r,µ,r,µ σ ) = F λ 2 + σ2 ζ (r)p (,) (µ)ζ (r )P (,) (µ ). (34) The Gree fuctios are cotiuous with respect to the argumets r, µ, r, µ, but their partial derivatives with respect to r, µ have discotiuities of the first id i the poits where r = r or µ = µ. The covergece of the series i Eqs. (34), here used to represet the Gree fuctios, is assured by the geeral theory of the Gree fuctio (e.g., Smirov 964b) ad is also prove i Appedix A. Equatios (33) ca be used to compute the agular velocity perturbatio whe τ is ow. Note that i the case i which the Loretz force due to the mea field ad the meridioal flow are the oly sources of agular mometum redistributio, Eq. (4) gives: τ = µ B p (r si θb φ ) + 2ρ Ω 0 (r si θ)u (m)s, (35) where B p is the mea poloidal magetic field ad u (m)s the compoet of the meridioal flow i the directio orthogoal to the rotatio axis. To obtai Eq. (35) we made use of the soleoidal ature of the mea poloidal field ad of the cotiuity equatio for the meridioal flow. 2.4. Localizatio of the source of the torsioal oscillatios i the solar covectio zoe The results derived above allow us to itroduce methods to localize the torques producig the torsioal oscillatios i the covectio zoe. Suppose that the observatios provide us with the fuctios A (c) (r,µ)ada (s) (r,µ) appearig i Eq. (). The fuctios α (t) ca be writte as: α = a (c) cos(σt) + a(s) si(σt), (36) where the costats a (c,s) a (c,s) = 2πF R Similarly, we ca write: β (t) = b (c) with the relatioships: are give by: A (c,s) (r,µ)ρr 4 ( µ 2 )ζ P (,) drdµ. (37) cos(σt) + b(s) si(σt), (38) b (c) = λ a (c) + σa(s), b (s) = λ a (s) σa(c), (39) that follow from Eq. (23). The divergece of τ (c,s) ca be obtaied from Eqs. (3) ad (24) as: τ (c,s) = ρr 2 ( µ 2 ) b (c,s) ζ (r)p (,) (µ). (40) Moreover, it is possible to costruct a localized estimate of the perturbatio term τ by cosiderig a fuctio f (r,µ) the gradiet of which is differet from zero oly withi a give volume f. It ca be developed i series of the eigefuctios i the form: f (r,µ) = c ζ (r)p (,) (µ), (4) where the coefficiet c are give by: c = 2πF R f f (r,µ)ρr 4 ( µ 2 )ζ P (,) drdµ. (42) Let us cosider the equatio: ( τ (c,s) f ) d = f τ(c,s) [c ζ (r)p (,) (µ)] d, (43)
A. F. Laza: Torsioal oscillatios i the Su ad stars 05 obtaied by meas of Eq. (4). Cosiderig Eqs. (30) ad (38), Eq. (43) ca be recast as: ( τ (c,s) f ) d = c b (c,s) f F. (44) Moreover, if we itroduce the volume average of the modulus of the perturbatio τ with respect to the weight fuctio f, i.e.: τ (c,s) τ (c,s) f f f d, (45) f d f the Eq. (44) gives a lower limit for it i the form: τ (c,s) F c b (c,s) f (46) f d f The miimum dimesios of the volume f are set by the spatial resolutio of the measuremets of the agular velocity variatios. They deped o the accuracy of the rotatioal splittig coefficiets, the iversio techique ad the positio withi the covectio zoe (e.g., Schou et al. 998; Howe et al. 2005). The miimum order of the Jacobia polyomials N m eeded to reproduce a agular velocity variatio with a latitudial resolutio θ is N m 2π θ. Similarly, the miimum order of the radial eigefuctios K m is set by the radial resolutio r as K m 2(R ) r. Therefore, it is possible to trucate the series i Eqs. (44) ad (46) to those upper limits for ad because the coefficiets c will decrease rapidly for > N m ad > K m givig a egligible cotributio to the sum. The statistical errors i the measuremets of the agular velocity variatios ca be easily propagated through the liear Eqs. (37), (39), (40) ad (44) to fid the errors o the estimates of τ (c,s) or the average of τ. For istace, if we cosider the stadard deviatios σ i of the data d i, i.e., the rotatioal splittigs or the splittig coefficiets from which the iteral rotatio is derived, the stadard deviatio σ I of the itegral i Eq. (44) is: σ 2 I e 2 i σ2 i, (47) where F 2 e i 2πF R c 2 λ2 i ( µ 2 )ρr 4 c i (r,µ)ζ P (,) drdµ, (48) ad the fuctios c i (r,µ) are the rotatioal iversio coefficiets defied i Eq. (8) of Schou et al. (998). Note that a costat relative error ɛ = ω/ω i the measuremets of A (c,s) leads to the same relative error i Eq. (40) ad i Eqs. (44) ad (46), give the liear equatios that relate the correspodig quatities. Ideed, there is also a systematic error i our iversio method related to the poor owledge of the turbulet viscosity η t (r) that determies the form of the radial eigefuctios ζ. 2.5. Kietic eergy variatio ad dissipatio The variatio of the ietic eergy of rotatio associated with the torsioal oscillatios, averaged over the eleve-year cycle, ca be computed accordig to Laza (2006b) ad is: T c = T c, (49) where T c = { [a (c) 4F ]2 + [a (s) ]2}. (50) The average dissipatio rate of the ietic eergy of the torsioal oscillatios due to the turbulet viscosity is: dt = 2 λ T c. (5) dt c 2.6. Torsioal oscillatios due to mea-field Loretz force Most models of the torsioal oscillatios assume that they are due to the Loretz force produced by the mea field as derived from dyamo models. Therefore, let us cosider the case i which oly the mea-field Maxwell stresses cotribute to the perturbatio, i.e.: τ = µ (r si θ)b φ B p. (52) If the mea radial B r ad toroidal fields B φ are give by: ( ) B r = B 0r cos 2 σt, ( ) B φ = B 0φ cos 2 σt +Π r, (53) where Π r is the phase lag betwee the two field compoets, the compoets of τ r i Eq. (3) are: τ (c) r = 2 cos Π r si θ r µ τ (s) r = 2 si Π r r si θ µ B 0r B 0φ, B 0r B 0φ. (54) A estimate of τ (c) r ad τ (s) r ca be obtaied from the method outlied i Sect. 2.4, cosiderig a localizatio fuctio f (r) that depeds oly o the radial co-ordiate. Specifically, Eq. (44) ca be used to compute a volume average of τ (c) r ad τ(s) r from which the average stress amplitude B r B φ ad phase lag Π r ca be determied. Aalogous cosideratios are valid for the meridioal compoet of the mea field B θ ad B φ. Adoptig a localizatio fuctio f (θ) depedigoly oθ, it is possibleto estimate B θ B φ ad the phase lag Π θ betwee B θ ad B φ. Such results are importat to costrai mea-field dyamo models of the solar cycle, as discussed by, e.g., Schlichemaier & Stix (995) (see also Sect. 3.3). 2.7. Applicatio to solar-lie stars Sequeces of Doppler images ca be used to measure the surface differetial rotatio of solar-lie stars ad its time variability, as doe by, e.g., Doati et al. (2003) ad Jeffers et al. (2007) i the cases of AB Dor ad LQ Hya. Laza (2006a) discussed the implicatios of the observed chages of the surface differetial rotatio o the iteral dyamics of their covectio zoes, assumig that the agular velocity is costat over cylidrical surfaces co-axial with the rotatio axis. The preset model allows us to relax the Taylor-Proudma costrait o the iteral agular velocity, but some differetassumptiosmust be itroducedto obtai the iteral torques i the active stars. Here we assume that the variatio of the surface differetial rotatio is etirely due to
06 A. F. Laza: Torsioal oscillatios i the Su ad stars the Maxwell stresses of the iteral magetic fields, localized i the overshoot layeelow the covectio zoe, as i the iterface dyamo model by, e.g., Parer (993). The iterior model of the Su ca also be applied to AB Dor ad LQ Hya because they have a similar relative depth of the covectio zoe. Therefore, the basic quatities ca be scaled accordig to the stellar parameters, as explaied i Laza (2006a). Followig Doati et al. (2003), we cosider a surface differetial rotatio of the form: Ω(µ, t) =Ω eq (t) dω(t)µ 2, (55) where Ω eq is the equatorial agular velocity ad dω the latitudial shear, both regarded as time-depedet. Sice P (,) 0 = ad µ 2 = 5 + 4 5 P(,) 2, it ca be recast i the form of Eq. (6) leadig to: Ω eq 5 dω= α 0 (t)ζ 0 (R), 4 5 dω= α 2 (t)ζ 2 (R), (56) where R is the radius of the star. The fuctios β (t) ca be computed by assumig that the timescale of variatio of the differetial rotatio t DR is sigificatly shorter tha the timescales for agular mometum trasport, as give by λ. This assumptio is justified i the case of rapidly rotatig stars by the observed variatio timescales of the order of a few years together with the expected rotatioal quechig of the viscosity (e.g., Kichatiov et al. 994). Therefore, Eq. (23) leads to α (t) t DR β (t). The agular velocity ca be writte i a form similar to Eq. (33) by itroducig a appropriate Gree fuctio. For istace, cosiderig the first of Eqs. (56), we fid: Ω eq 5 dω= 3 ( τ ) G s (R, r )d, (57) 8π where: G s (r, r ) = ζ 0 (r)ζ 0 (r ), (58) ad the itegratio is exteded over the stellar covectio zoe. Assumig that τ is differet from zero oly i a overshoot layer of volume o ad applyig cosideratios similar to those of Sects. 2.3 ad 2.4, we fid a lower limit for the variatio of the averaged perturbatio term: ( ) 8π Ω eq ( τ ) o 5 (dω) o M, (59) 3t DR where Ω eq ad (dω) are the amplitudes of variatio of the differetial rotatio parameters, ad M is the maximum of G s (R, r ) i the overshoot layer. This result made use of the limited iformatio we ca get from surface differetial rotatio. However, i the ear future, the observatios of the rotatioal splittigs of stellar oscillatios promise to give iformatio o the iteral rotatio ad its possible time variatios. Sice oly the modes of low degrees (l 3) are detectable i dis-itegrated measuremets, the spatial resolutio of the derived iteral agular velocity profile is very low. Lochard et al. (2005) cosidered the case i which oly the mea radial profile of Ω is measurable. I view of such a iformatio accessible through asteroseismology, let us cosider a more geeral case i which some average of the iteral agular velocity of the star, say ω m (t), ca be measured as a fuctio of the time: ω m (t) = R ω(r,µ,t)w(r,µ)drdµ, (60) where w is a appropriate weight fuctio that taes ito accout the averagig effects of the limited spatial resolutio, ad is the radius at the base of the stellar covective evelope. We itroduce a auxiliary weight fuctio: w(r,µ) w (r,µ) ( µ 2 )ρr, (6) 4 which will ot diverge toward the poles (µ = ±) ad the surface (ρ = 0) because the weight fuctio w is localized i the stellar iterior ad goes to zero rapidly eough toward the rotatio axis ad the stellar surface. We ca develop the fuctio w as: N w K w w (r,µ) = w ζ (r)p (,) (µ), (62) where the summatio ca be trucated at some low orders, say, N w ad K w, because the fuctio w is ot sharply localized i the stellar iterior. Cosiderig Eq. (60), we fid: N w K w ω m (t) = w α. (63) 2πF For the sae of simplicity, we assume ow that the time scale of variatio of the fuctios α (t) is log with respect to λ so that Eq. (23) gives: β λ α. Cosiderig Eq. (30), we fid: N w K w w ω m (t) = τ [ ] ζ P (,) 2πλ. (64) Equatio (64) ca be used to fid a lower limit to the average of τ over the covectio zoe volume. If we idicate by M w the maximum of the fuctio: N w K w w [ ] ζ P (,) (65) 2πλ over the volume of the covectio zoe, we fid: τ (t) τ d ω m(t) M w (66) 3. Applicatio to the Su 3.. Iterior model, eigevalues ad eigefuctios A model of the solar iterior ca be used to specify the fuctios ρ(r) adη t (r) that appear i our equatios. While the desity stratificatio ca be determied with a accuracy better tha 0.5%, the turbulet dyamical viscosity is ucertai by at least oe order of magitude ad it is estimated from the mixiglegth theory accordig to the formula: η t = 3 α MLρu c H p, (67) where α ML is the ratio of the mixig-legth to the pressure scale height H p ad u c is the covective velocity give by: u c = ( ) αml 3 L, (68) 40πr 2 ρ
A. F. Laza: Torsioal oscillatios i the Su ad stars 07 Fig.. The ratio of the desity to the desity at the base of the covectio zoe (ρ/ρ 0 solid lie) ad the ratio of the turbulet dyamical viscosity to the turbulet viscosity at the base of the covectio zoe (η t /η 0 dotted) versus the fractioary radius r/r i our solar iterior model. where L is the lumiosity of the Su. I our computatios we adopt α ML =.5 ad assume the solar model S for the iterior quatities (Christese-Dalsgaard et al. 996). I that model, the base of the covectio zoe is at r = 0.73 R. We cosider also the effect of a overshoot layer extedig betwee r = 0.673 R ad the base of the covectio zoe withi which the turbulet dyamical viscosity is assumed to icrease liearly from zero up to the value at the base of the covectio zoe. The desity ad the turbulet viscosity are plotted i Fig. where their values have bee ormalized to the values at the base of the covectio zoe (i.e., ρ 0 = 0.875 g cm 3 ad η 0 = 2.56 0 2 gcm s, respectively). The basic equatios of our model (i.e., 2, 3 ad 4) ca be made odimesioal by adoptig as the uit of legth the solar radius R, as the uit of desity ρ 0, i.e., the desity at the base of the solar covectio zoe, ad as a uit of time t 0 = ρ 0 R 2 /η 0, where η 0 is the turbulet viscosity at the base of the covectio zoe. Ideed, the value of the diffusio coefficiets estimated from the mixig-legth theory leads to a too short period for the solar cycle i mea-field dyamo models. Covas et al. (2004) adopted a turbulet magetic diffusivity ν t = 3 0 cm 2 s to get a suspot cycle of yr. This implies η 0 ρν t = 5.62 0 0 gcm s i our model. The radial eigefuctios ζ ad the Jacobia polyomials P (,) have bee computed from the respective Sturm-Liouville problem equatios by meas of the Fortra 77 subroutie sleig2.f 2 (Bailey et al. 200). For the radial eigefuctios ζ, the Sturm-Lioville problem has bee solved with Neuma boudary coditios at both eds, set at 0.675 R ad 0.99 R to avoid divergece at the surface. For the Jacobia polyomials, limit poit boudary coditios have bee adopted at µ = ±. Note that, to be rigorous, it would be better to use the helioseismic estimate of Ω r at r = 0.99 where we fixed our outer boudary for the computatio of the radial eigefuctios istead of the stress-free boudary coditio that is valid oly at the surface. However, the differeces are cofied to the outermost layer of the solar covectio zoe, the momet of iertia of which is so small that there are o pratical cosequeces. The eigevalues λ ad the eigefuctios ζ computed by sleig2.f have bee compared with those computed by meas of the code itroduced by Laza (2006b). The relative differeces See http://bigcat.ifa.au.d/ jcd/solar_models/ 2 http://www.math.iu.edu/sl2/ i the eigevalues ad i the eigefuctios are lower tha.5% for 4, 0. However, some problems of covergece of the umerical algorithm used by sleig2.f have bee foud for 20, particularly for 30, so we decided to limit its applicatio up to = 9. The Jacobia polyomials ad the eigevalues computed by sleig2.f are very good up to = 30, as it has bee foud by compariso with their aalytic expressios up to = 0 ad their asymptotic expressios for 2. We coclude that for 30 ad 20 it is better to use the asymptotic formulae (22) ad (8) istead of the umerically computed ζ ad P (,). The eigevalue λ gives the iverse of the characteristic timescale of agular mometum trasfer of the mode correspodig to ζ uder the actio of the turbulet viscosity. The logest timescale correspods to the lowest eigevalue, i.e., λ 20 = 0.078 i odimesioal uits. It correspods to a timescale of 0.86 yr with the turbulet viscosity give by the mixig-legth theory ad to 39.5 yr with η 0 = 5.62 0 0 gcm s. 3.2. Localizatio fuctios for the source of the torsioal oscillatios The available data o the torsioal oscillatios are displayed with a typical radial resolutio of 0.05 R ad a latitudial resolutio of 5 i Howe et al. (2005, 2006). The rotatioal iversio erels of Schou et al. (998) show a higher radial resolutio close to the surface, but, give the small amplitude of the torsioal oscillatios, the choice of a uiform resolutio of 0.05 R seems to be better. We have foud that the best results of the localizatio of the source term with the method outlied i Sect. 2.4 are obtaied with localizatio fuctios that deped o r or µ oly ad that have a smooth derivative. As a typical fuctio to probe the radial localizatio, we adopt: 0 forr r, f (r) = + si [ π ( ) ] r r r 2 r π 2 for r r r 2, (69) 2 forr r 2. This fuctio has zero derivative, except i the iterval ]r, r 2 [ where its derivative varies smoothly reachig a maximum i the mid of the iterval. I Fig. 2 we plot the modulus of the derivative d f dr, as give by the sum of the series of the radial eigefuctios trucated at = 9, for three differet itervals cetered at 0.725, 0.85 ad 0.965 R, all with a amplitude of r 2 r = 0.05 R. The derivative is well approximated by the trucated series, with small sidelobes the amplitude of which icreases toward the surface of the Su because the eigefuctios scale as (ρη t ) 4, accordig to the asymptotic expressio (22). The performace of the localizatio method itroduced i Sect. 2.4 has bee tested with simulated data i the absece of oise. The results of two such tests are plotted i Fig. 3 where we cosider the case of a purely radial perturbatio τ = τ ˆr localized withi a iterval of amplitude 0.05 R. Its time depedece is assumed to be purely cosiusoidal ad the correspodig coefficiets β are computed by meas of Eq. (30) up to the orders = 38 ad = 9. From the β,thecoefficiets α are computed by meas of Eq. (26) ad the simulated agular velocity perturbatio follows from Eq. (6). Whe the iterval i which τ is localized coicides with oe of the iversio itervals [r, r 2 ], the lower limit for τ turs out to be 80% 90% of the value assumed i the simulatio
08 A. F. Laza: Torsioal oscillatios i the Su ad stars Fig. 2. The modulus of the radial localizatio erel versus the relative radius for the fuctio (69), as obtaied by trucatig the series of the radial eigefuctios at K m = 9, for three differet itervals cetered at r = 0.725 (upper pael), 0.85 (middle pael) ad0.965 R (lower pael), all with a amplitude of r 2 r = 0.05 R. The modulus of the derivative has bee ormalized to its maximum value. Fig. 4. The modulus of the latitudial localizatio erel versus µ for the fuctio (70), obtaied by trucatig the series of the Jacobia polyomials at N m = 30, for three differet itervals of µ i the Norther hemisphere, i.e., [0, 0.26] (upper pael), [0.5, 0.7] (middle pael) ad [0.87, 0.99] (lower pael). Note the symmetry of the modulus of the erel with respect to the equator. The value of d f dµ is ormalized at its maximum at µ = ±. Fig. 3. Upper pael: test case of the applicatio of the iversio method from Sect. 2.4. A iput profile with τ =.0ˆr i odimesioal uits betwee 0.80 ad 0.85 R (solid lie) is used to simulate a oiseless profile of agular velocity perturbatio. The recostructed lower-limit profile accordig to Eq. (46) is plotted as a dotted lie. Lower pael: same as i the upper pael, but with a iput profile localized betwee 0.775 ad 0.825 R. (Fig. 3, upper pael). The agreemet icreases up to 90% 95% if we cosider a case i which τ r has a costat sig ad put the value of the derivative d f dr i the deomiator of Eq. (46) istead of its modulus. This happes because the modulus of the derivative icreases the effects of the sidelobes by icreasig the value of the deomiator i Eq. (46). Whe the iterval i which the assumed τ is localized does ot coicide with a iterval of the iversio grid, the iversio method still performs well distributig the cotributios amog eighbour itervals early i the correct proportio (Fig. 3, lower pael). The case of simulatios icludig a oise compoet is straightforward to treat thas to the liear character of our iversio method. The iverted value turs out to be the sum of the iverted oiseless value ad of the cotributio comig from the iversio of the oise. Their amplitude ratio is equal to the ratio of the amplitudes of the iput oise to the iput sigal (for costat relative sigal errors), as discussed i Sect. 2.4. The localizatio i latitude ca be sampled by meas of a localizatio fuctio of the id: 0 for <µ µ 2, + si [ π ( ) µ+µ 2 µ 2 µ π 2] for µ2 µ µ, f (µ) = 2 for µ µ µ, + si [ π 2 π ( )] (70) µ µ µ 2 µ for µ µ µ 2, 0 forµ 2 µ<. Such a fuctio is symmetric with respect to the equator so that its derivative d f dµ is atisymmetric, as it is the source term τ leadig to a symmetric perturbatio of the agular velocity. The localizatio fuctio has a derivative always equal to zero except i two itervals ] µ 2, µ [ad]µ,µ 2 [, symmetric with respect to the solar equator. Its represetatio by meas of the Jacobia polyomials with degree up to N = 30 is give i Fig. 4. Note that the maximum of d f dµ is reached i two sharp peas at µ = ±. However, they are so arrow that their cotributio to the itegral i Eq. (46) is modest i compariso to those of the broader peas i the itervals [ µ 2, µ ]ad[µ,µ 2 ]. Several tests have bee performed, as i the case of the radial localizatio, to assess the performace of the proposed method. The results are similar to those obtaied for the radial case ad are ot discussed here. We coclude that our choice of N m = 30 ad K m = 9 is perfectly adequate to ivert the available data o the solar torsioal oscillatios to derive iformatio o the locatio of the perturbatio term withi the covectio zoe. 3.3. Results o the sources of the torsioal oscillatios The data plotted i Fig. 4 of Howe et al. (2006) ca be used for a illustrative applicatio of the iversio methods itroduced i Sect. 2.4. They are give with a samplig of 5 betwee the equator ad 60 of latitude, i.e., i the rage i which the rotatioal iversio techiques perform better (Schou et al. 998). The features distiguishable o the plots idicate a actual radial resolutio of 0.05 R, i agreemet with the samplig adopted i Fig. 3 of Howe et al. (2005). A average statistical error of about 30% ca be assumed for the amplitude, whereas the phase errors become very large below 0.80 R, especially at low latitudes, because of the ucertaity i the recostructio of the
A. F. Laza: Torsioal oscillatios i the Su ad stars 09 sigal i the deep layers. Note that the error itervals reported i Fig. 4 of Howe et al. (2006) idicate oly how the particular method solutio (here a OLA iversio of SoHO/MDI data) would vary with a differet realizatio of the iput data affected by a rado Gaussia oise. Ufortuately, they do ot give the statistical rages i which the true values of the amplitude ad phase are liely to lie. This is ot a major limitatio i the cotext of the preset study because we aim at illustratig the capabilities of the proposed approach rather tha derive defiitive coclusios. To perform our iversio, we iterpolate liearly the values of the amplitude ad phase over the grid used to compute the radial eigefuctios ad the Jacobia polyomials. We assume that the amplitude is zero at the poles ad icreases liearly toward 60 of latitude whereas the phase is costat poleward of 60 of latitude. The divergece of the agular mometum flux perturbatio τ ca be obtaied from Eq. (40). We defie its amplitude ad phase as: A τ [ τ (c) ]2 + [ τ (s) ]2, (7) Φ τ ta τ (c) τ (s) (72) They are plotted i Figs. 5 ad 6, respectively, i the case of η 0 = 2.56 0 2 gcm s. A suitable smoothig has bee applied to have a resolutio of 0.05 R i the radial coordiate ad 5 i latitude. I order to show the effect of a smaller value of the turbulet viscosity, we plot i Figs. 7 ad 8 the isocotours of A τ ad Φ τ for η t = 5.62 0 0 gcm s.the amplitude of the perturbatio term is higher close to the equator because the data i Fig. 4 of Howe et al. (2006) maily sample the low-latitude brach of the torsioal oscillatios. The relative maxima of A τ are reached close to the base of the covectio zoe ad at a radius of 0.85 R ad their locatios show oly a mior depedece o the value of η 0. Coversely, the value of η 0 sigificatly affects Φ τ, as it follows from Eqs. (39). Whe η 0 is large, the timescales for agular mometum exchage, as give by the iverse of the lowest eigevalues, are sigificatly shorter tha the eleve-year cycle ad the perturbatios are almost i phase with the agular velocity variatios. Whe η 0 is sufficietly low, the timescales for agular mometum exchage become comparable or loger tha the eleve-year cycle so the torsioal oscillatios lag behid the perturbatios. Cosiderig Fig. 6, we see that the phase is i agreemet with that i Fig. 4 of Howe et al. (2006), except for r > 0.93 R. The disagreemet i those layers is due to the small values of τ close to the surface that maes the correspodig phases ucertai (cf. Fig. 5). The phase lag is apparet by comparig Fig. 8 with Fig. 6, especially close to the equator for 0.72 r/r 0.92. It is less evidet ear the base of the covectio zoe ad i the upper layers due to the smaller values of τ i those regios. It is iterestig to ote that the depedece of the amplitude ad phase of the torsioal oscillatios o the turbulet viscosity ca lead to its estimate i the framewor of mea-field models (e.g., Rüdiger et al. 986; Rüdiger 989). Specifically, Rempel (2007) fids that a mea turbulet iematic viscosity about oe order of magitude smaller tha the mixig-legth estimate is eeded to reproduce the polarach of the torsioal oscillatios. Lower limits for the modulus of the agular mometum flux vector τ ca be derived from Eq. (46). From the lower limits Fig. 5. The isocotours of the fuctio A τ (r,µ) as defied by Eq. (7) i the case of η 0 = 2.56 0 2 gcm s. The scale o the left idicates the rages correspodig to the differet colors ad is i uits of 0 5 gcm s 2. The relative statistical ucertaity of A τ is 30%, as it follows from the relative ucertaity of the data. Fig. 6. The isocotours of Φ τ (r,µ) as defied by Eq. (72) i the case of η 0 = 2.56 0 2 gcm s. The phase rages from π 2 to π 2.The scale o the left idicates the phase rages correspodig to the differet colors ad is i radias. Fig. 7. As for Fig. 5 i the case of η 0 = 5.62 0 0 gcm s.the scale o the left is i uits of 0 3 gcm s 2. The relative statistical ucertaity of A τ is about 30%. o τ (c) ad τ(s), a lower limit o τ = [τ (c) ]2 + [τ (s) ]2 ca be derived ad it is plotted i Figs. 9 ad 0 for the radial ad latitudial localizatios described i Sect. 3.2, respectively. Give the ucertaity i the peetratio depth of the torsioal oscillatios, i additio to the results for a peetratio dow to the base of the covectio zoe, we also plot those for peetratio depths of 0.80 ad 0.90 R, respectively. They are obtaied simply by
020 A. F. Laza: Torsioal oscillatios i the Su ad stars Fig. 8. As for Fig. 6 i the case of η 0 = 5.62 0 0 gcm s. assumig that the oscillatio amplitude has the value i Fig. 4 of Howe et al. (2006) above the peetratio depth ad drops to zero immediately below it. Whe the oscillatios peetrate dow to the base of the covectio zoe, the maximum of the perturbatio term is reached betwee 0.80 ad 0.85 R. It is maily localized ito two latitude zoes, i.e., withi ±5 from the equator ad betwee 30 ad 60. Note that our data refer maily to the low-latitude brach of the oscillatios, so the possible source at latitude >60 caot be detected. Whe the turbulet dyamical viscosity is reduced with respect to the mixig-legth value, the amplitude of the perturbatio term drops, but its radial ad latitudial localizatios are ot greatly affected, except for a shift of the early equatorial bad towards higher latitudes. Accordig to the hypothesis that the Loretz force is the oly source of the torsioal oscillatios, the amplitude of the Maxwell stress B r B φ ca be estimated from the lower limit of τ r ad is 2.7 0 5 G 2 i the case of η 0 = 2.56 0 2 gcm s. Cosiderig that the poloidal field B r is of the order of 0 G, this leads to very high toroidal fields i the bul of the covectio zoe, which would be highly ustable because of magetic buoyacy. O the other had, with η 0 = 5.62 0 0 gcm s, the Maxwell stress is B r B φ 8. 0 3 G 2 which leads to a toroidal field itesity of the order of 0 3 G. It may be stably stored for timescales comparable to the solar cycle thas to the effects of the dowward turbulet pumpig i the covectio zoe (Bradeburg 2005, ad refereces therei). Note that the maximum radial stress B r B φ is reduced by a factor of 30 whe η t is decreased from 2.56 0 2 to 5.62 0 0 gcm s, whereas the maximum of B θ B φ is reduced oly by a factor of 4 (Figs. 9 ad 0). This maily reflects the predomiace of the radial gradiet over the latitudial gradiet of the agular velocity perturbatio i the deeper layers of the solar covectio zoe. The average phase lags Π r ad Π θ betwee B r ad B θ ad the azimuthal field B φ, as derived by the method i Sect. 2.6, are plotted i Figs. ad 2, respectively. It is iterestig to ote that B r B φ > 0below 0.85 R whe η t = 5.62 0 0 gcm s, i agreemet with the fidig of most dyamo models i which the azimuthal field is produced by the stretchig of the radial field i the low-latitude regio, where Ω r > 0for r < 0.95 R (e.g., Rüdiger & Hollerbach 2004; Schlichemaier & Stix 995). Above 0.85 R, the phase relatioship betwee the radial ad the azimuthal fields leads to B r B φ < 0 with a phase lag of π, i agreemet with the early fidig that the photospheric zoe equatorward of the activity belt is rotatig faster tha that at higher latitudes (Rüdiger 989). Note that Ω r Fig. 9. Upper pael: the lower limit of the amplitude of the perturbatio term τ r averaged over spherical shells of thicess 0.05 R versus the relative radius; differet liestyles ad colors refer to the depth at which the torsioal oscillatios are assumed to vaish: blac solid lie oscillatios extedig dow to the base of the covectio zoe; gree dotted lie oscillatios extedig dow to 0.80 R ; red dashed lie oscillatios extedig dow to 0.90 R. Results are obtaied assumig a turbulet dyamical viscosity at the base of the covectio zoe η 0 = 2.56 0 2 gcm s. Lower pael: as i the upper pael, but for η 0 = 5.62 0 0 gcm s. The relative statistical ucertaities are i all cases about 30%, as follows from the ucertaities of the data. Fig. 0. Upper pael: the lower limit of the amplitude of the perturbatio term τ θ averaged over differet latitude zoes versus their average value of µ.differet liestyles ad colors are used to idicate the results for differet peetratio depths of the torsioal oscillatios, as i Fig. 9. Results are obtaied assumig a turbulet dyamical viscosity at the base of the covectio zoe η 0 = 2.56 0 2 gcm s. Lower pael: as i the upper pael, but for η 0 = 5.62 0 0 gcm s. The relative statistical ucertaities are i all cases about 30%, as follows from the ucertaities of the data. becomes egative for r > 0.95 R at low latitudes, leadig to a reversal of the phase relatioship betwee the two field compoets i the outer layers of the solar covectio zoe. Our limited spatial resolutio ad the ucertaity of the measuremets of the torsioal oscillatios iside the Su might explai why we fid the trasitio from a mostly positive to a egative B r B φ at 0.85 R. The phase lag betwee B θ ad B φ depeds remarably o the colatitude for η t = 2.56 0 2 gcm s, whereas it is almost costat for η t = 5.62 0 0 gcm s, leadig i the latter case mostly to B θ B φ < 0. This, together with Ω θ > 0, suggests that the toroidal field is maily produced by the stretchig of the radial field. O the other had, if the agular mometum trasport leadig to the torsioal oscillatios is produced oly by a
A. F. Laza: Torsioal oscillatios i the Su ad stars 02 shows a remarable decrease because of the smaller momet of iertia of the surface layers. Fig.. Upper pael: the phase lag Π etwee B r ad B φ as derived by Eqs. (54) averaged over spherical shells of thicess 0.05 R versus the relative radius. Differet liestyles are used to idicate the results for differet peetratio depths of the torsioal oscillatios, as i Fig. 9. Results are obtaied assumig a turbulet dyamical viscosity at the base of the covectio zoe η 0 = 2.56 0 2 gcm s. Lower pael: as i the upper pael, but for η 0 = 5.62 0 0 gcm s. Fig. 2. Upper pael: the phase lag Π θ betwee B θ ad B φ, as derived by equatios aalogous to (54), averaged over differet latitude zoes versus their average value of µ. Differet liestyles are used to idicate the results for differet peetratio depths of the torsioal oscillatios, as i Fig. 9. Results are obtaied assumig a turbulet dyamical viscosity at the base of the covectio zoe η 0 = 2.56 0 2 gcm s. Lower pael: as i the upper pael, but for η 0 = 5.62 0 0 gcm s. perturbatio of the meridioal flow, as i the thermal wid model, the a very small perturbatio follows from the lower limit of τ.forη 0 = 2.56 0 2 gcm s,wefidamiimum amplitude of the meridioal flow compoet oscillatig with the eleve-year cycle of 3cms at a depth of 0.85 R ad a latitude of 45. Note, however, that if we estimate τ from its divergece ad the typical legthscale of its variatios i Fig. 5, we fid a value about oe order of magitude larger, that is i agreemet with the estimate by Rempel (2007). A similar argumet applies also to the estimate of the Maxwell stresses made above. The average ietic eergy variatio associated with the torsioal oscillatios, as computed from Eq. (49), is T c = 4.88 0 28 erg, whereas the average dissipated power is 6.7 0 26 erg s,wheη 0 = 2.56 0 2 gcm s, ad oly.5 0 25 erg s,wheη 0 = 5.6 0 0 gcm s. Whe the torsioal oscillatios are assumed to be cofied to shallower ad shallower layers, the locatio of the perturbatio term is shifted closer ad closer to the surface ad it becomes more ad more uiformly distributed i latitude. Its amplitude 4. Applicatio to solar-lie stars The results of Sect. 2.7 ca be applied to the variatio of the surface differetial rotatio observed i, e.g., LQ Hya betwee 996.99 ad 2000.96 (see Table 2 of Doati et al. 2003). Assumig a iteral structure aalogous to the solar oe ad that the differetial rotatio variatios observed at the surface are represetative of those at a depth of 0.99 R, we ca estimate a lower limit for τ i the overshoot layer from Eq. (59). This is used to estimate the Maxwell stresses by assumig that: B r p B φ o µ δr τ,wherer o = 0.67 R is the radius of the overshoot layer ad δr = 0.04 R its thicess. As such, the miimum magetic field stregth is: B mi = B p B φ 2500 G. However, if we assume a poloidal field stregth of 00 G, as idicated by the Zeema Doppler imagig, we fid a azimuthal field of B φ 6.2 0 4 G, which is i the rage of the values estimated by Laza (2006a) i the framewor of the Taylor-Proudma hypothesis. Note that the result is idepedet of η 0 i the limit t DR λ. Although this applicatio is purely illustrative, it suggests that our method ca be applied to derive estimates of the iteral magetic torques (as well as other sources of agular mometum trasport) whe asteroseismic results are available, i combiatio with surface rotatio measuremets, to further costrai rotatio variatios i solar-lie stars. 5. Discussio ad coclusios We itroduced a geeral solutio of the agular mometum trasport equatio that taes ito accout the desity stratificatio ad the radial depedece of the turbulet viscosity η t.its mai limitatio is due to the ucertaity of the turbulet viscosity i stellar covectio zoes. It is iterestig to ote that the method of the separatio of variables to solve Eq. (3) ca be applied also whe η t is the product of a fuctio of the radius by oe of the latitude. However, whe η t depeds o the latitude, the agular eigefuctios are o loger Jacobia polyomials. Our formalism ca be applied to compute the respose of a turbulet covectio zoe to prescribed time-depedet Loretz force ad meridioal circulatio. From the mathematical poit of view, it is a geeralizatio of that of Rüdiger et al. (986) ad ca be easily compared with it by cosiderig that: P (,) (µ) = 2 dp + +2 dµ = 2siθ +2 P (θ), where P is the Legedre polyomial of degree ad P = dp dθ. Moreover, our method ca be used to estimate the torques leadig to the agular mometum redistributio withi the solar (or stellar) covectio zoe, thus geeralizig the approach suggested by Komm et al. (2003). The mai limitatio, i additio to the ucertaity of the turbulet viscosity, comes from the low resolutio ad the limited accuracy of the preset data o solar torsioal oscillatios, particularly i the deeper layers of the covectio zoe. I fact, these layers are the most importat because torsioal oscillatios with a amplitude of 0.5% % of the solar agular velocity extedig dow to the base of the covectio zoe lead to Maxwell stresses with a itesity of at least 8 0 3 G 2 aroud 0.85 R or a perturbatio of the order of several percets of the meridioal flow speed at the same depth (e.g., Rempel 2007). If the torsioal oscillatios are due solely to the Maxwell stresses associated with the mea field of the solar dyamo, we ca also estimate the phase relatioships betwee B r, B θ ad B φ. Our prelimiary results idicate that B r B φ > 0 i the layers below 0.85 R ad B r B φ < 0