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6.6 Comparisons with mean-field theory

Global convection simulations have provided much insight into solar interior dynamics in general and the maintenance of mean flows and fields in particular. However, in light of their limitations (Section 7), it is prudent to also consider alternative modeling approaches. Mean-field models seek to reproduce the structure and evolution of large-scale flows and fields in the Sun using turbulence models or other physical parameterizations for the Reynold stress, convective heat flux, Maxwell stress, and turbulent emf. The motivation and methodology behind mean-field models was discussed briefly in Section 5.3. Here we review some of the results and insights gained from mean-field modeling and compare them with global convection simulations.

This section is not intended as a comprehensive review. For much more discussion of mean-field hydrodynamics see Rüdiger (1989Jump To The Next Citation Point), Canuto and Christensen-Dalsgaard (1998), and Rüdiger and Hollerbach (2004Jump To The Next Citation Point). For a review of mean-field dynamo models see Ossendrijver (2003Jump To The Next Citation Point), Rüdiger and Hollerbach (2004Jump To The Next Citation Point), and Charbonneau (2005Jump To The Next Citation Point).

6.6.1 Mean-field hydrodynamics

In mean-field hydrodynamics, the components of the Reynolds stress which transfer angular momentum are typically represented in terms of a diffusive contribution represented by an anisotropic turbulent viscosity n t and a non-diffusive contribution known as the /\-effect [Equation (23View Equation)]. These two contributions are generally comparable in amplitude so the relative strength of the Coriolis force and the Reynolds stress may be quantified by the Taylor number based on the turbulent viscosity: Ta = (2_O_0R2 o. /nt)2. This is in effect the inverse square of the Rossby number defined in Equation (9View Equation).

Many early mean-field models of the solar internal rotation relied only on Reynolds stress parameterizations, with the meridional circulation specified or neglected altogether. An example is the model of Küker et al. (1993) who obtained solar-like rotation profiles using the /\-effect theory of Kitchatinov and Rüdiger (1993Jump To The Next Citation Point). However, their solutions were inconsistent with the thermal wind balance Equation (11View Equation) because they did not take into account the Coriolis-induced circulations which would be driven by the rotation profiles they achieved. At the large Taylor numbers required by their model, Equation (11View Equation) implies cylindrical rotation profiles in the absence of baroclinic effects. Other estimates for the amplitude of the turbulent viscosity have similar implications (Rüdiger, 1989Durney, 1999Jump To The Next Citation PointRüdiger and Hollerbach, 2004Jump To The Next Citation Point). This is the “Taylor number puzzle” discussed by Kitchatinov and Rüdiger (1995Jump To The Next Citation Point) and Rüdiger and Hollerbach (2004Jump To The Next Citation Point).

As in global convection simulations, cylindrical rotation profiles can be avoided in two ways, either the Reynolds stress must be substantial (implying smaller Taylor numbers) or latitudinal entropy gradients must be established which maintain a thermal wind differential rotation. Most mean-field models now rely on the latter to achieve solar-like rotation profiles (Kitchatinov and Rüdiger, 1995Durney, 1999Küker and Stix, 2001Jump To The Next Citation PointRüdiger and Hollerbach, 2004Jump To The Next Citation PointRempel, 2005Jump To The Next Citation Point). These models are typically based on anisotropic parameterizations for the convective heat flux obtained from mixing-length theory, modified to account for the influence of rotation. An exception is the mean-field model developed by Rempel (2005Jump To The Next Citation Point) in which the required entropy perturbations originate from thermal wind balance in the tachocline and spread upward into the convection zone without the need for an anisotropic parameterization of the thermal diffusivity (this model is discussed further in Section 7.3). In mean-field models which incorporate convective heat transport, the meridional circulation is usually solved for together with the angular velocity and entropy profiles.

The meridional circulation is generally more sensitive to the parameterizations than the differential rotation (Küker and Stix, 2001Jump To The Next Citation Point). This is again consistent with global simulations where the meridional circulation is maintained by a delicate balance of forces (Section 6.4). For moderate values of the mixing length parameter and for solar-like rotation rates, Küker and Stix (2001Jump To The Next Citation Point) find that the circulation has two cells in radius per hemisphere, with equatorward circulation at the top and bottom of the convection zone and poleward circulation in between. The equatorward surface flow is inconsistent with photospheric measurements and helioseismic inversions (Section 3.4), but multiple-cell structure in depth is also exhibited by global convection simulations (Section 6.4).

The equatorward surface circulation in the Küker and Stix (2001) model can be attributed to the Reynolds stress parameterization. The Kitchatinov and Rüdiger (1993) theory of the /\-effect yields an outward angular momentum flux near the surface. If the meridional circulation is to balance this outward Reynolds stress as expressed by Equation (8View Equation) and if the angular velocity profile is to be solar-like, then the circulation must be equatorward. Conversely, an inward angular momentum flux by the Reynolds stress near the surface implies a poleward circulation. If the Reynolds stress parameterization exhibits inward angular momentum transport near the surface, not only can it produce a poleward circulation as suggested by observations, but it may also establish a subsurface increase in angular velocity analogous to the near-surface shear layer found in helioseismic inversions (Section 3.1). This is indeed the case for the mean-field model developed by (Rempel, 2005Jump To The Next Citation Point). Rempel’s model provides important insight into how the solar differential rotation may be maintained but there is currently little physical justification for the Reynolds stress parameterizations which best match observational data.

In global convection simulations, the angular momentum transport by Reynolds stresses is typically outward as shown in Figure 14View Image, although it is nearly balanced by inward viscous diffusion. As higher Reynolds numbers are achieved and the viscous diffusion is reduced, the Reynolds stress and meridional circulation must adjust accordingly if they are to maintain a solar-like differential rotation as well as a poleward surface circulation.

6.6.2 Solar dynamo theory

No review of solar interior dynamics would be complete without some mention of the thriving field of mean-field solar dynamo modeling. These models seek to reproduce observational manifestations of the solar activity cycle, including the butterfly diagram (Section 3.8), the propagation and phase relationship between axisymmetric poloidal and toroidal fields, and long-term or sporadic cycle variations such as the Maunder minimum. Solar dynamo models are generally quite successful in this regard and have provided much insight into the origin of cyclic magnetic activity on the Sun.

Despite this success, there is still much uncertainty with regard to the primary physical mechanisms responsible for regenerating poloidal field from toroidal field (the a-effect) and with regard to the role (or lack thereof) of the meridional circulation (Mestel, 1999Jump To The Next Citation PointOssendrijver, 2003Jump To The Next Citation PointCharbonneau, 2005Jump To The Next Citation Point). Furthermore, the theoretical foundation of many solar dynamo models remains questionable. Global simulations can be used to help validate mean-field theory although they do not yet possess the resolution or physical conditions to explicitly capture many of the processes which are currently parameterized in dynamo models. Examples include flux-tube instabilities in the tachocline and magnetic diffusion in the solar surface layers due to the decay of active regions. The latter is a fundamental component of Babcock-Leighton dynamo models Mestel (1999); Charbonneau (2005Jump To The Next Citation Point). Global MHD simulations have not yet achieved a shear layer at the bottom of the convection zone comparable to the solar tachocline so they cannot currently be used to validate or motivate interface dynamo models in which the toroidal and poloidal field generation occurs in spatially separated regions. However, this will soon change as global convection simulations incorporate a tachocline either self-consistently or by imposed forcing (Section 7.3).

Many of the approximations commonly used in mean-field dynamo theory are not justified by global convection simulations. In particular, there is no clear scale separation in space or time13 so there is no guarantee that series expansions such as that in Equation (22View Equation) will converge Ossendrijver (2003Jump To The Next Citation Point). Furthermore, the amplitude of the fluctuating magnetic fields exceeds that of the mean fields and the non-axisymmetric analogue of the turbulent emf v× B - <v× B > is not small relative to the other terms in the fluctuating induction equation, calling into question the first-order smoothing approximation which is implicit in most mean-field models (Moffatt, 1978Ossendrijver, 2003Jump To The Next Citation Point). Although their justification formally breaks down, mean-field models may be still used to interpret some aspects of global MHD convection simulations. The highest resolution achieved to date in such simulations is represented by Case M3, discussed in Section 6.5. Here the toroidal field regeneration due to differential rotation is comparable to that due to the turbulent emf. Thus, Case M3 might be classified as an a2-_O_ dynamo in the terminology of mean-field theory. This might help to explain its non-cyclic behavior. In mean-field theory, a-_O_ dynamos are generally more likely to yield cyclic, dipolar solutions than a2 or a2-_O_ dynamos (Charbonneau and MacGregor, 2001Rüdiger et al., 2003Rüdiger and Hollerbach, 2004Jump To The Next Citation Point) In the more laminar MHD convection simulations by Gilman (1983) and Glatzmaier (1985a) differential rotation plays a bigger role, the dynamo was more akin to a-_O_ type, and cyclic, dipolar solutions were found. Moreover, the poleward propagation of magnetic flux in these simulations over the course of a cycle is consistent with the Parker-Yoshimura sign rule of mean-field theory (Charbonneau, 2005Jump To The Next Citation Point).

Numerical simulations of MHD convection can be used not only to evaluate mean-field models but also to calibrate them by providing estimates for model parameters such as a and jt. Furthermore, simulations can provide important insight into nonlinear saturation mechanisms which are often parameterized in mean-field models as quenching of a, jt, and /\. Such efforts have proliferated in recent years (reviewed by Ossendrijver, 2003Jump To The Next Citation PointBrandenburg and Subramanian, 2004Rüdiger and Hollerbach, 2004), although most of this work has focused on Cartesian geometries. Further progress in this area promises to improve our understanding of dynamo processes and to improve the reliability of solar and stellar dynamo models.

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