5.1 Introduction

Until recently, only surface measurements of the solar rotation rate were available. Since the mid-1980s, with the advent of helioseismology, much has been learned about the internal rotation of the Sun through the inversion of ^-mode frequency splittings. As was noted in Section 1.2.2, it now appears that the observed surface pattern of differential rotation with latitude prevails throughout most of the solar convection zone, with equatorial regions moving faster than higher latitudes. In contrast, the underlying radiative core appears to rotate nearly uniformly down to r ^ 0.1-0.2R0, at a rate that is intermediate between the polar and equatorial rates of the photosphere. Within the central region r < 0.2R0, some measurements suggest that the angular velocity increases with depth, implying rotation at a rate between 2 and 4 times that of the surface; other measurements strongly suggest, however, that the solar inner core rotates rigidly down to the center.

The problem presented by the observed solar differential rotation is one of long standing and many efforts have been made to formulate a plausible flow pattern that reproduces the large-scale motions in the solar atmosphere. Following Lebedinski's (1941) pioneering work, many theories have been proposed to explain how the equatorial acceleration originated and is maintained in the solar convection zone. Broadly speaking, they can be divided into two classes, depending on the mechanism proposed to produce and maintain the equatorial acceleration: (i) the interaction of rotation with local turbulent convection and (ii) the interaction of rotation with global turbulent convection in a rotating spherical shell. Till the late 1980s, however, the most detailed models invariably predicted rotation profiles that were constant on cylinders concentric to the rotation axis. Obviously, these solutions are at variance with the current observations, which suggest an angular velocity that is constant on radii in the convection zone, at least at mid-latitudes. In Section 5.2 we shall explain how the disparities between the rotation profiles deduced from the helioseismological data and what has been predicted by these early models can be resolved.

Now, a number of recent observations has shown that solar-type stars undergo rotational deceleration as they slowly evolve on the main sequence (see Eq. [1.7]). As we shall see in Section 7.2, this spin-down is presumably the consequence of angular momentum loss via magnetically channeled stellar winds and/or sporadic mass ejections emanating from the surface layers. The central question is how this inexorable braking of the outer convection zone will affect the rotational state of the radiative interior. In the case of the Sun, the absence of marked differential rotation in the outer layers of its radiative core implies that angular momentum redistribution within that region must be very efficient indeed. Within the framework of the eddy-mean flow interaction presented in Section 3.6, three distinct mechanisms for angular momentum redistribution might be operative: (i) large-scale meridional currents, (ii) turbulent friction acting on the differential rotation, and (iii) large-scale magnetic fields. In Section 5.3 we discuss the time-dependent meridional flow in the Sun's radiative interior, taking into account the development with age of a gradient of mean molecular weight in the hydrogen-burning core. Section 5.4 presents quantitative studies of the rotational evolution of the Sun's radiative interior, with angular momentum being removed from the convective envelope to simulate the effects of the solar wind and/or episodic mass ejections.

The interaction of rotation with convection appears to be the most likely mechanism for the generation of the observed solar differential rotation. Two different approaches have been proposed to explain the maintenance of differential rotation and concomitant meridional circulation. One class of models is based on the appealing assumption that the variations in angular velocity arise mainly from the nonlinear interaction of rotation with the largest scales of convection, when a radial superadiabatic gradient of temperature prevails. These global-convection models resolve numerically as many of the large scales as possible in a rotating spherical shell and parameterize, via eddy diffusivities, the transport of momentum and heat by all the smaller unresolved scales. In the other class of models, the role of global convection is assumed to be unimportant. What is essential in these mean-field models is the effect of the large-scale azimuthal flow on the local convective motions that are not greatly influenced by the Sun's spherical shape. As usual, following closely the method presented in Section 3.6, the role of this turbulent convection is parameterized by the use of eddy viscosities, which are specified functions of rotation. Unavoidably, this mathematical description of the interaction between turbulent convection and rotation depends on adjustable parameters.

In this approach the large-scale motions in the solar convection zone are described by means of stationary, axially symmetric flow patterns, with turbulent convection giving rise to Reynolds stresses and eddy viscosity coefficients. In spherical polar coordinates (r, 9, y), the mean velocity v is of the form v = u + Qr sin9ly, (5.1)

where u is the two-dimensional meridional velocity. Because we have assumed axial symmetry for the mean flow, mass conservation implies that div(p u) = 0, (5.2)

where p is the mean density.

For mean steady motions, the y component of Eq. (3.123) becomes pu ■ grad(Q^2) = ^ d(r3ary) + d(sin2 9 09y), (5.3)

r2 dr sin 9 d9

where m = r sin 6. This equation merely expresses the fact that turbulent friction acting on the differential rotation can be made to balance the transport of specific angular momentum, Qm2, by the meridional flow. If the influence of rotation and gravity was negligible, the turbulent transport of momentum would occur downward along the gradient of angular velocity, so that the Reynolds stresses orv and a6v would be proportional to dQ/dr and 9Q/96, respectively. However, because anisotropy prevails in a rotating fluid embedded in a gravitational field, the stresses arv and a6v contain both diffusive and nondiffusive parts. Following Section 3.6, one has dQ . .

d r and dQ

where s — ¡¡h/¡v is the anisotropy parameter. It is immediately apparent that the nondiffusive parts, which are proportional to the mean rotation rate, maintain rather than smooth out differential rotation in the solar convection zone. They depend on two independent parameters, XV and XH, which define the anisotropies in the vertical (i.e., along the effective gravity) and horizontal directions. Appropriate expansions for these parameters are

Note that the parameter XH, which is related to the anisotropy of turbulence in planes perpendicular to the effective gravity, vanishes at the poles. In principle, the radial functions XV0, XV1, and XH1 may be derived from the equations governing the fluctuating part of the instantaneous velocity field (e.g., Rudiger 1989).

Neglecting the inertial terms u-gradu, one can also rewrite Eqs. (3.125) and (3.126) for mean steady motions in the compact form

where Fp (u) is the poloidal part of the turbulent viscous force per unit volume acting on the meridional flow (see Eq. [3.123]) and lm is the radial unit vector in cylindrical polar coordinates (m, z). Taking the curl of Eq. (5.8), one obtains

p2 d z where, for shortness, R(u) is the curl of the viscous force. If R(u) makes a negligible contribution to this equation, one readily sees that any barotropic model for the solar convection zone has the angular velocity constant on cylinders aligned with the rotation axis; that is, p = p(p) implies that Q = Q(m), and conversely. This result is a mere consequence of the Poincare-Wavre theorem (see Section 3.2.1). It is an important result, however, because we know that the angular velocity is not constant on cylinders within the solar convection zone. Since detailed models for the Sun indicate that R(u) is indeed negligible in the bulk of that zone, it follows that strict barotropy is most certainly an inadequate approximation for the solar rotation problem.

In Section 3.3.2 we have shown that the anisotropy of turbulent convection due to the preferred direction of gravity can produce differential rotation and meridional circulation in the solar convection zone. Since the early 1970s, a variety of models have been calculated, taking into account in an approximate manner the convective energy transport. To complete Eqs. (5.2), (5.3), and (5.8) we thus let pTu • grad S + div(F + Fc) = 0. (5.10)

The specific entropy is given by

p5/3

where cV is the specific heat at constant volume. The radiative flux is given by the standard expression

4ac T3

3 Kp

(see Eqs. [3.5] and [3.6]), and the convective flux is taken to be of the form

where Kc(r) is the turbulent heat transport coefficient. One also has

ff where f is the mean molecular weight. As usual, this set of equations must be solved with appropriate boundary conditions at the base and at the top of the rotating spherical shell.

Baroclinic models based on the concept of anisotropic eddy viscosity exhibit angular velocity profiles that are not constant on cylinders. They also produce a slow meridional flow, with typical surface velocities of the order of 1 m s-1. Moreover, all these baroclinic models have very small (^ 1 K) pole-equator temperature differences. Unfortunately, in order to reproduce the observed equatorial acceleration, the anisotropy parameter s (= fH /fxV) must be larger than one. This is a most surprising result since one expects turbulent convection to provide more transport in the radial than in the horizontal directions. This inadequacy of these solutions strongly suggests that Lebedinski's (1941) anisotropic eddy viscosity might not be the ultimate cause of the Sun's differential rotation.

As was originally pointed out by Weiss (1965), the solar differential rotation could be generated by meridional currents driven by a pole-equator temperature difference. This approach is based on the fact that rotation has a small but significant influence upon turbulent convection, thus resulting in a convective heat transport that depends on heliocentric latitude. This gives rise to an inexorable meridional flow that transports angular momentum toward the equator and thus sustains the differential rotation. Following this idea, several authors have developed models of differentially rotating spherical shells - assuming a latitude-dependent heat transport coefficient kc(r, d) and an isotropic eddy viscosity. Many of these models succeed in maintaining angular velocity profiles that are not constant on cylinders. However, some of them have meridional velocities at the surface that are too large, while others have pole-equator temperature differences that are too large.

More recently, Kitchatinov and Rudiger (1995) have pointed out that the conflict between mean-field models and solar observations can be resolved by taking into account an anisotropic eddy viscosity as well as an anisotropic turbulent heat transport. Thus, instead of letting Kc — Kc (r) or Kc — Kc(r, d) in Eq. (5.13), they prescribe that the convective heat flux has the components, in Cartesian coordinates, where (d T/dXj )ad is the adiabatic gradient of mean temperature and xij is a tensor describing the turbulent heat transport (i — 1, 2, 3). As was done for the eddy viscosities and related coefficients, the components of this tensor can be obtained from the equations governing the fluctuating quantities (e.g., Rudiger 1989). Fortunately, these models involve only one adjustable parameter, which is the ratio of the mixing length to the pressure-scale height. Figure 5.1 illustrates one particular solution. Note that the angular velocity distribution closely fits the helioseismological data reported in Section 1.2.2, with the rotation becoming virtually rigid below the convection zone. This model has a small (^ 5 K) pole-equator temperature difference, which is consistent with the observations. However, it also predicts a slow equatorward meridional motion on the free surface, which is not observed in the Doppler measurements (see Section 1.2.1). Nonetheless, this is the first mean-field model that satisfies almost all the observational constraints. Given this result, it thus seems highly probable that anisotropy plays a key role in the solar rotation problem, since calculations involving isotropic transport coefficients always yield angular velocities that are constant on cylinders in the models. This effect is illustrated in the bottom part of Figure 5.2, which depicts a model corresponding to an isotropic thermal conductivity.

In the global-convection theories of the Sun's differential rotation the largest convective cells are influenced by rotation, leading to a continuous redistribution of angular momentum, which we observe as a differential rotation. Actually, it is the combined effect of the spherical geometry and the Coriolis force acting on these large-scale convective motions that generates variations with latitude and radius of the angular velocity. Extensive numerical calculations have been made, independently, by Gilman and Glatzmaier in the early 1980s. Their models solve the nonlinear, three-dimensional, time-dependent equations for thermal convection in a rotating spherical shell of compressible fluid. Both sets of models are based on the assumption that the convective velocities are small compared to the local sound speed, thus filtering out the pressure waves. Moreover, because it is not possible to resolve all scales of motion, from the largest to the smallest, it is also assumed that the small unresolved scales give rise to viscous and thermal diffusivities, which are specified functions of the coordinates.

Fig. 5.1. Theoretical results for the Sun. (a) The rotation profiles for the equator (solid line), for a 45° latitude (dashed line), and for the poles (dashed-dotted line); (b) the surface rotation rate derived from the model (solid line) and from Doppler measurements (dashed line); (c) deviations of temperature from its latitude-averaged value; (d) the surface meridional velocity, with negative values meaning an equatorward flow. Bottom: The isolines of angular velocity and temperature along with the streamlines of meridional circulation, with solid lines meaning a counterclockwise motion. The dotted line indicates the basis of the convection zone. Source: Kitchatinov, L. L., and Rudiger, G., Astronomy Letters, 21, 191, 1995.

Fig. 5.1. Theoretical results for the Sun. (a) The rotation profiles for the equator (solid line), for a 45° latitude (dashed line), and for the poles (dashed-dotted line); (b) the surface rotation rate derived from the model (solid line) and from Doppler measurements (dashed line); (c) deviations of temperature from its latitude-averaged value; (d) the surface meridional velocity, with negative values meaning an equatorward flow. Bottom: The isolines of angular velocity and temperature along with the streamlines of meridional circulation, with solid lines meaning a counterclockwise motion. The dotted line indicates the basis of the convection zone. Source: Kitchatinov, L. L., and Rudiger, G., Astronomy Letters, 21, 191, 1995.

Although the numerical techniques employed in these models are quite different, the results obtained by Gilman and Glatzmaier are qualitatively the same. In particular, it is found that their simulated global convection in a rotating spherical shell tends to take the form of north-south (banana) rolls, the tilting of which yields Reynolds stresses to drive the zonal flows that maintain differential rotation. Unfortunately, in these early models the simulated angular velocity in the convection zone is constant on cylinders coaxial with the rotation axis, which is not in agreement with the helioseismological data reported in Section 1.2.2.

As was pointed out by Glatzmaier and Toomre (1995), however, these pioneering studies of global convection in a rotating spherical shell have been restricted by computational resources to deal with nearly laminar regimes for the largest scales of convection. One plausible explanation for the disparities between theory and observation is that the numerical resolution of these global-convection models is insufficient to attain the fully turbulent regimes that are observed in the solar convection zone. Indeed, various studies have shown that the transport properties of turbulent convection can be very different from those of laminar convection (e.g., Brummell, Hurlburt, and Toomre 1998). Accordingly, extension of the models into fully turbulent regimes might provide angular velocity profiles that are in agreement with the observational data. Three-dimensional numerical simulations of fully turbulent convection in a rotating spherical shell have been produced. Advances in computation permit these simulations to have a spatial resolution about tenfold greater in each dimension than those of the earlier studies. In particular, it is found that the north-south roll-like convective cells have broken up with the increased nonlinearity; this orderly convection is replaced by convection dominated by intermittent plumes of matter, with the downflow motions stronger in amplitude than the upflow motions. Although these extensions to fully turbulent regimes are quite promising, it is not yet clear at this writing to what extent the new global-convection models adequately describe the observed rotation profile in the solar convective zone.

Because the conditions in the radiative zone of a rotating star are not spherically symmetric, the transport of radiation will in general tend to heat the polar and equatorial regions unequally, thus causing a large-scale flow of matter in planes passing through the rotation axis. This problem was already discussed in Sections 4.3 and 4.4, where we obtained consistent solutions for the meridional flow and concomitant differential rotation in the radiative envelope of a nonmagnetic, early-type star. This section is concerned with the large-scale circulation generated by the small departures from spherical symmetry in the Sun's radiative core. Not unexpectedly, the development with age of a gradient of mean molecular weight /i in the hydrogen-burning core makes this problem much more intricate since, then, any model that possesses full internal consistency necessarily becomes time dependent in its mean properties. To be specific, starting from an initially homogeneous radiative core in an unevolved solar model, we shall discuss the effects of a growing /i-gradient on the meridional flow - making allowance for hydrogen-core burning as the model leisurely evolves away from the zero-age main sequence. However, since we are chiefly interested in the interaction between the /i-distribution and the rotationally driven currents, we shall "turn off' the solar-wind torque that slows down the outer convective envelope.

In an inertial frame of reference, the large-scale velocity field v is the combination of a rotation and a meridional flow, as defined in Eq. (5.1). To complete the continuity equation (Eq. [5.2]) and the momentum equation (Eq. [3.123]) we must add the energy equation, pTu ■ grad S = peNuc + div(x grad T), (5.16)

and Poisson's equation,

where eNuc is the rate of energy released by the thermonuclear reactions, x is the coefficient of radiative conductivity, and G is the constant of gravitation. Let us rewrite the equation of state in the form pT

p /i where ft is the universal gas constant. Neglecting diffusion altogether, we must also prescribe that

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