{ 2{ the resulting dierential rotation. Our model's adiabatic and density-stratied convection zone with stress-free surfaces and a thickness of 0.3 st

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1 submitted to Astrophysical Journal Dierential Rotation and Meridional Flow for Fast-Rotating Solar-type Stars Gunther Rudiger 1 (gruedigeraip.de) Brigitta v. Rekowski 1 (bvrekowskiaip.de) Robert A. Donahue 2 3 (donahuecfa.harvard.edu) and Sallie L. Baliunas (baliunascfa.harvard.edu) ABSTRACT Observations indicate that normalized surface dierential rotation decreases for fast rotating stars: jj= / ;0:3. An increase of jj= is provided, however, by the present-day Reynolds stress theory of dierential rotation in stellar convection zones without inclusion of meridional ow. We compute both the equator-pole dierence of the surface angular velocity and the meridional drift for various Taylor numbers to demonstrate that the inclusion of meridional ow into the computations for fast rotation yields a systematic reduction of 1 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D Potsdam, Germany. 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA. 3 Mount Wilson Institute, 740 Holladay Road, Pasadena, CA 91106, USA. 4 Center for Automated Space Science and Center of Excellence in Information Systems Tennessee State University, th Ave. N., Nashville, TN 37203, USA.

2 { 2{ the resulting dierential rotation. Our model's adiabatic and density-stratied convection zone with stress-free surfaces and a thickness of 0.3 stellar radii yields the relation jj= / ;(0:15:::0:30) for stars with faster rotation than the Sun in agreement with previous observations (Fig. 5). If the Coriolis number is varied instead of the Taylor number, we nd a maximum dierential rotation of 20%. For stars with fast rotation, exponents up to n 0 ' 0:4 are found. All rotation laws exhibit super-rotating equators. The only exceptions occur at very high rotation and small eddy viscosities (Fig. 10). Subject headings: turbulence { stars { dierential rotation { meridional ow 1. MOTIVATION Recent developments have been made in the observational study of surface dierential rotation. Donahue (1993) used the long-term database of chromospheric activity measurements from Mount Wilson Observatory's HK Project to monitor changes in the seasonal rotation period of lower main-sequence stars. From a sample of 110 stars, 36 had ve or more determinations of rotation period over the twelve years of data available. For some of these stars, there is evidence of trends between rotation period and level of activity, or where measurable, activity cycle phase (Baliunas et al. 1995). While some slowly-rotating stars (P rot > 10d) show an equatorward migration pattern that is similar to the Sun's, the opposite situation in which P rot increases with activity cycle phase is clearly present for several stars (Donahue 1993 Donahue & Baliunas 1994), even for stars with similar spectral types (i.e., mass) and mean rotation period (i.e., age). This `anti-solar' pattern could reect either a poleward migration of active regions against a solar-like surface dierential rotation, or possibly the presence of poleward acceleration. A few stars, for example Comae, possibly have two active latitude bands (Donahue & Baliunas 1992 however, Gray & Baliunas 1997 oer a dierent explanation), although for several of these stars the range of periods observed within each activity band is very small, < 2% (e.g., 1 Ori, Donahue & Baliunas 1994). Young stars (< 1 Gyr) tend to have no discernible pattern of seasonal rotation period with either time or mean level of surface activity. For all the stars studied, the range of rotation period observed, P, increases with the mean rotation period, hp rot i. There isapower-law relationship, jp j/hp rot i 1:3 (1)

3 { 3{ independent of spectral type (Donahue, Saar, & Baliunas 1996). However, for stars with similar Rossby number, K-type stars have larger jp j compared to G stars. It is possible to explain the observed internal solar rotation by means of a Reynolds stress theory in which the inuence of the basic rotation upon the cross correlations of the turbulent convection is determined (Kuker et al. 1993). In a pure Reynolds stress theory, the meridional ow in such calculations is completely neglected. The reason for this ignorance is twofold. First, the observations only provide a slow poleward ow with an amplitude of 10 m/s. There is the assumption that such a slow ow only modies the results of the Reynolds stress theory. On the other hand, inspection of the (observed) rotational isolines does not suggest a strong inuence due to meridional ow with one or two large cells. The meridional circulation tends to align the isolines of angular momentum with the streamlines of the ow (cf. Rudiger 1989 Brandenburg et al. 1990). Thus, the existence of distinct equatorial acceleration indicates that any possible meridional surface drift cannot be too strong. In the other case because of the small distance between the polar regions and the rotational axis, it should be more likely that a polar acceleration (\polar vortex") appears. Moreover, for very large Taylor numbers, the ow system approaches the Taylor- Proudman state where the isolines are two-dimensional with respect to the rotation axis, i.e. there is no variation along the rotational (z) axis. At the same time the meridional circulation breaks, i.e.: z =0 up =0 (2) (cf. Kohler 1969 Tassoul 1978). The rst of these conditions is not seen for the Sun the solar Taylor number, however, is large. From the denition of the Taylor number Ta = 42 R 4 2 T (3) with R as the stellar radius and as the rotation rate, we nd using the following reference value for the eddy viscosity, T = c l 2 corr= corr (4) (where l corr is the correlation length, and corr is the correlation time) the relation Ta = 2 c 2 4 : (5)

4 { 4{ Here, as usual, is the Coriolis number =2 corr (6) and = l corr =R is the normalized correlation length. So with typically assumed values for the Sun of ' 5 c ' 1 and ' 0:1 (`giant cells'),we nd Ta ' 2: : (7) With such a Taylor number the observed dierential rotation should produce a remarkable intensity of the meridional ow. Therefore, we need an additional explanation as to why the solar polar drift is so slow. Kitchatinov & Rudiger (1995) nd the rotational inuence on the eddy heat diusion leads to a warmer pole such that an equatorwards-directed extra drift weakens the meridional circulation. Recent analysis of observational solar data conrms the poleward direction of the ow (between 10 and 60 ). Snodgrass & Dailey (1996) report a value of 13 m/s while Hathaway (1996) derive 20 m/s, and occasionally larger ( < 50 m/s). While Snodgrass & Dailey used magnetogram data to nd the meridional motions by following the motions of small magnetic features in the photosphere, Hathaway worked with Doppler velocity data obtained with the GONG instruments. Any shortcomings concerning the usage of magnetic tracers are thus excluded and the results are of particular importance. Summing up, the following strategy of this paper might be reasonable: the -eect is prescribed in order to reproduce the `solar case' for a certain xed Taylor number and then the Taylor number is varied. The main question concerns the response of the surface rotation law to this procedure. Equation (1) can also be represented as lat = jj / ;n 0 eq (8) with n 0 = 0:3. Recent theoretical papers which did not include meridional ow yield negative n 0 value in contrast to the observations (Kuker et al Rudiger & Kitchatinov 1996). Only by including the meridional ow into the solution of the Reynolds equation has a positive n 0 been found (Kitchatinov &Rudiger 1995). This paper conrms this situation. The inuence of the meridional circulation upon dierential rotation is thus indirectly observed with equation (1) for fast rotating solar-type stars.

5 { 5{ 2. TURBULENCE MODEL A stratied turbulence can never rotate rigidly. The reason is that even in the case of uniform rotation { in contrast to any form of pure viscosity { the turbulence transports angular momentum. The formal description of this phenomenon is given by the non-diusive part Q ij of the correlation tensor, i.e. Q ij = V ipk g j + jpk g i Q ij = hu 0 i(x t) u 0 j(x t)i (9) g p k + H (g )( ipk j + jpk i ) g p k (10) { also called the -eect in Rudiger (1989). In equation (10) is the basic rotation, while g gives the unit vector in radius. as According to equation (10) the turbulent uxes of angular momentum can be written Q r = :::+ T V (0) + V (1) sin 2 sin Q = :::+ T H (1) sin 2 cos (11) where the coecients V (0) V (1) and H (1) are functions of the Coriolis number and the density scale-height with H = jgj ;1 (12) G = r log : (13) The dotted parts in equations (11) are denoting the contributions of the eddy viscosity tensor Q ij which in the simplest (axisymmetric) case yields Q r = ; T r sin + ::: r Q = ; T sin + ::: : (14) Kuker et al. (1993), under application of a model of the solar convection zone by Stix & Skaley (1990) were able to reproduce the main features of the `observed' internal rotation of the Sun's outer envelope. They neglected, however, the angular momentum transport of

6 { 6{ Fig. 1. The solar case: Meridional ow (left) and dierential rotation (right) in outer convective envelopes. -eect is from equation (15), Taylor number is , MLT =5=3. meridional ows and magnetic elds. A simple example of such a theory of the dierential solar rotation is given in Figure 1 produced with the numbers V (0) = ;1 V (1) = H (1) =1:25 (15) (Rudiger & Tuominen 1990 Brandenburg et al. 1990). Again the known main features of the solar rotation are well reproduced. Note the disk-like structure of the -isocontours at the poles and the cylinder-like structure of the -isocontours at the equator. In mid-latitudes the angular velocity of the rotation does not vary with depth. The main nding by Kitchatinov & Rudiger (1993) is the representation for the functions V and H as V (0) = corru 2 2 T [I H 2 0 ( )+I 1 ( )] (16) V (1) = H (1) = ; 2 corru 2 T H 2 I 1 ( ) (17) where u T = q hu 02 i is the rms turbulent velocity, corr is the eddy turnover time, H is the density scale-height. We adopt the approximation 2 corr u 2 T ' l 2 corr ' 2 MLT H 2 (18)

7 { 7{ with MLT from the usual mixing-length relation l corr = MLT H p (19) between the mixing length, l corr, and the pressure scale-height, H p. is the adiabacy index. For MLT = we nd l corr = H. The expressions for I 0 and I 1 read I 0 = ; ; 2 +9 arctan (20) and I 1 = ; ; ; 12 2 ; 45 arctan : (21) These functions describe the rotation-rate dependence of the -eect. In Figure 2 the rotational proles of the functions V (0) and V (1) = H (1) are given. Fig. 2. The inuence of the basic rotation on the -eect coecients V (0) and V (1) = H (1) after Kitchatinov &Rudiger (1993). For stars the Coriolis number exceeds unity. In that case, indeed, the values in equation (15) form a good representation of the -eect. However, one has to incorporate the meridional ow into the theory. To this end, one needs the knowledge of the inuence of rotation onto the turbulent heat transport. If this is known from an established turbulence theory, then one can compute the complete elds of pressure, circulation, and dierential rotation. 3. BASIC EQUATIONS We solve the stationary Reynolds equation for a turbulent ow in the inertial system, (ur) u + 1 div ( Q)+rp = g (22)

8 { 8{ with div(q) as a representation for ( Q j )=x j. g denotes the gravity acceleration which is assumed as uniform. Magnetic elds and deviations from their axisymmetry are not considered. Anelastic uids are only considered, i.e. div(u) =0: (23) After adopting these initial assumptions, the azimuthal component of (22) is x j r sin (u j u + Q j ) =0: (24) Our energy equation might be overly simple. The turbulence is considered to be so intensive that the medium will be isentropic, S = C v log p ; = const: (25) hence there is a direct relation between density and pressure. If this is true the pressure term in equation (22) can be written as a gradient which disappears after application of curling: curl (ur) u + 1 div( Q) =0: (26) It remains to x the ow eld by u r = 1 r 2 sin For normalization, the relations A u = ; 1 r sin A r u = 0 r sin + u : (27) r = Rx = ^ A = R T ^A u = R 0 ^u (28) i.e. u r = T R ^u r u = T R ^u u = R 0 ^u (29) are used. The density stratication equation (13) is then represented by H = ;dr=d log = R ^H: (30) For the conservation law of the angular momentum we get the dimensionless equation 1 x 2 1 x2 ^ ^Q r + x x sin +2^ cos ; 1 ^x sin ^ sin ^ ^Q + ^A +2^ sin x x ^Q ^ cot r + ^Q + x 1 ^A ^x 2 sin =0: (31)

9 { 9{ Here also the turbulent uxes of angular momentum are written in dimensionless form in accord with Equation (31) is linearized by means of the assumption Q i = T 0 ^Q i i = r : (32) ju j 0 R: (33) The meridional equation is formulated under the same condition. Hence the `centrifugal force' is reformulated as The result is 1 x x ; 1 x ; cos Ta 2x u 2 ' r 2 sin r sin 0 u : (34) 1 1 x2 ^ ^Q r + sin ^ ^Q + ^Q r ; cot ^Q ; ^x x ^ sin x2 ^ ^Q rr + sin ^ ^Q r ; ^x 2 x ^x sin x ^Q ; 1 x ^Q ; x (x^u )+ Ta 2x (sin ^u )=0: (35) The Taylor number (equation 3) couples the centrifugal terms in equation (35) to the contributions of the turbulence-originated Reynolds stress. In contrast to equation (32), the latter are normalized here with 2 T=R 2. The normalized components of the (symmetric) correlation tensor are ^Q rr = ;2 x 1 ^x 2 sin ^Q r = ; x ; 1 ^x sin ^A ^A ; 1 1 x x ^x 2 sin ^Q r = ; ^u x + 1 x ^u + 1 V0 + V 1 sin 2 ^u x ^Q = ; 2 ; 1 ^A ; 2 1 x ^x sin x x ^x 2 sin ^Q = cot x ^u + cot x H 1 sin 2 ^u ; 1 ^u x ^Q = ; 2cot x ; 1 ^A ^x sin x ; 2 x 1 ^x 2 sin ^A + 1 x ; 1 ^x sin ^A ^A x ^A : (36)

10 {10{ Fig. 3. A slowly-rotating star: Meridional Fig. 4. A rapidly-rotating star: Meridional ow (left) and dierential rotation (right) in ow (left) and dierential rotation (right) outer convective envelopes. -eect is after in outer convective envelopes. -eect is equation (15), Taylor number is 10 3, MLT = after equation (15), Taylor number is 10 7, 5=3. MLT =5=3. Stress-free boundary conditions are imposed at the top and bottom of the convection zone, i.e. ^A =0 ^Q r =0 ^Q r =0: (37) With the denition in equation (34) and after our normalization procedure, it makes sense to x the azimuthal velocity at the equator to zero: ^u eq =0: (38) 4. RESULTS 4.1. Variation of the Taylor number In all of our calculations, the density stratication in equations (31) and (35) is taken from the model of Stix and Skaley (1990). For the determination of the exponent n 0 in equation (8) it suces to reproduce the solar situation with a well-chosen Taylor number or rotation rate and then to vary the parameter. In principle, one only needs three models describing cases of slow, medium and fast rotation in order to derive the exponent n 0 in equation (8).

11 {11{ We start with a highly simplied model. The -eect as the generator of the nonuniform rotation is xed to the values given in equation (15). Then Figure 1 gives the `solar' case with lat ' 0:3 at the surface. The meridional drift is poleward with an amplitude of about 7 m/s. Figure 3 presents the same model but with a smaller Taylor number (Ta =10 3 ). The normalized dierential rotation is increased and the drift is decreased. The opposite is true for higher Taylor number (Fig. 4). Obviously, the meridional ow tends to smooth the surface rotation prole. Indeed, the exponent n 0 in equation (8) proves to be positive (Fig. 5). If the observations are providing positive n 0, Fig. 5. Solid line: The inuence of the Taylor number on the normalized equator-pole dierence of the surface angular velocity. Dashed: The exponent n' in (8) for the scaling of the normalized dierential surface rotation. -eect by (15), MLT =5=3. then after our calculations it could easily reect the inuence of the meridional ow induced by the -eect. Note that there is even a quantitative agreement with the observations. A value of about 0.15 { 0.30 can indeed be found in the vicinity of the Taylor number used here for the `solar case'. Solar values for the normalization of the meridional ow at the surface adopted ( T =R ' 1 m/s), we nd for the solar case (Fig. 1) about 7 m/s while this value is increased for faster rotation. In Figure 4 using a Taylor number of Ta =10 7 one nds a meridional drift of 15 m/s which is in excess of the observational limit for the Sun.

12 {12{ Fig. 6. The ow-free reference status for models with varying ( MLT = 2:5 = 2 Ta =0). Fig. 7. Meridional ow (left) and dierential rotation (right) in a convection zone of slow rotation ( MLT =2:5 =1) Variation of the Coriolis number One might be unsatised by the simplicity of the above model represented by the Figures 1 5. Of course, the Taylor number and the -eect both depend on the basic rotation rate. Actually, faster rotation also increases the Coriolis number. In consequence, both the -eect as well as the Taylor number is changed by changing. The related expressions are given by equations (16) { (21) and, for the Taylor number, by equation (5). Therefore in the following analysis, only the Coriolis number will be varied. However, this results in the alteration of the eddy viscosity. In Kitchatinov et al. (1994) the structure of the eddy viscosity tensor under the inuence of a global rotation is derived within the framework of a quasilinear theory. The tensor becomes highly anisotropic, making the theory more complicated. In the present paper we apply only the -dependence of the isotropic part of the viscosity tensor, i.e. the coecient of the deformation tensor, replacing (4) by with T = 0 1 ( ) (39) 0 = c l 2 corr= corr (40) and 1 = ;21 ; arctan : (41)

13 {13{ Fig. 8. Meridional ow (left) and dierential Fig. 9. Meridional ow (left) and dierential rotation (right) in a convection zone of moderate rotation (right) in a convection zone of fast rotation ( MLT =2:5 =2). rotation ( MLT =2:5 =5). Again, as in equation (18), we adopt l corr R ' MLT H R ' MLT : (42) For reference, the computation starts with the articial case of Ta = 0(Fig. 6). We have used larger MLT to increase the dierential rotation, MLT =2:5. Again =5=3 =0:1 and c = 1. Except for the equatorial region the isoline contours of the angular velocity agree with results from helioseismology. Of course, the ow pattern is far from that produced in the Taylor-Proudman regime. The surface equator-pole dierence of the angular velocity is about 50%. Let us proceed to the inclusion of the meridional ow. In the Figures 7, 8, and 9 the Coriolis number,, runs from 1 to 5, and the corresponding dierential rotation prole subjects changes drastically. The solar-type rotation is realized for = 2 leading to a Taylor number of ' The surface equator-pole dierence of the angular velocity is reduced by the action of the meridional ow to about 20%. Slower rotation ( = 1 gives Ta ' ) yields spherical isolines of the rotation rate (Fig. 7). Faster rotation ( = 5 gives Ta ' ) basically produces the 2D Taylor-Proudman status (Fig. 9). In the latter case, equator and mid-latitudes are rotating with nearly the same angular velocity. However, we do not see a polar vortex i.e. a situation where the poles rotate faster than the equator. It may be a consequence of the turbulence model used from Kitchatinov & Rudiger (1993), or from a neglect of thermodynamics i.e. the ignorance of any rotationally-reduced anisotropy of the eddy heat

14 {14{ transport (cf. Kitchatinov & Rudiger 1995). For increasing the equator-pole dierence Fig. 10. The normalized dierential surface rotation scaled with the basic rotation rate for various c after (4). Note the existence of maxima of dierential rotation for each c value. For faster rotation the meridional ow strongly smoothes the surface rotation laws ( =0:1 MLT =2:5). of the angular velocity at the surface grows to a maximum of 20% for ' 2:25 and c = 1 (Fig. 10). The maximum magnitude is dened by our choice of the turbulence model parameters MLT and l corr as well as the scaling factor c. On its slow-rotation side it is n 0 ' ;1. Beyond the maximum, on the fast rotation side, the dierential rotation is reduced by the meridional ow. Accordingly, thescaling exponent n 0 changes its sign from negative for slow rotation to positive for fast rotation. In the latter case, the exponent n 0 adopts values of 0.4 for Coriolis numbers up to 4, and might grow further for faster rotation. Hence, a population of young stars should behave quite dierently with respect to surface dierential rotation than sample of very old stars. This is implied observationally by Donahue (1993). The proles in Figure 10 yield the only possibility for tuning the scaling factor c in the denition (4) of the eddy viscosity. c = 1 may be considered as a maximal value. In Figure 10 the results for smaller values are also given. The smaller the value of the proportional constant is, the earlier the maximal (normalized) dierential rotation appears in a run of increasing rotation rates. On the other hand, on the fast-rotation side of the graph the dierential rotation becomes negative so that indeed an equatorial deceleration (`polar vortex') may exist. As the rotational inuence on the eddy heat transport becomes unneglegible in particular for very rapid rotation, the development ofamuch more complex theory proves to be necessary.

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