5.3 Reduced models

High-resolution numerical simulations have become invaluable research tools but they do have their limitations. Because of the computational expense, it is difficult to comprehensively explore the sensitivity of the solutions to parameter values, boundary conditions, and the influence of dynamics which are unresolved or otherwise beyond the scope of the model. Furthermore, it is difficult to investigate relatively slow dynamics such as long-term modulations of the solar activity cycle or the spin-down of a star over the course of its main-sequence lifetime. For these and other purposes, a variety of reduced models have been devised, which are based on some approximated form of the full 3D equations of motion.

The most common approach is to average the equations of motion, generally over longitude, and then to introduce parameterizations for the nonlinear advection terms, including the Reynolds stress, the convective energy fluxes [ $F EN$ and $F KE$ ; see Equation (2 )], and the turbulent emf (Section 4.5). These parameterizations may themselves be nonlinear and they may introduce additional variables and additional prognostic equations, but they are designed to be more analytically or computationally tractable than the full, 3D equations of motion. The reduced equations are then solved to obtain the structure and evolution of the mean fields which are the quantities of interest. In a solar physics context, this approach is often referred to as mean-field hydrodynamics or mean-field dynamo modeling but it is closely related to what in the turbulence community is called Reynolds-averaged Navier-Stokes (RANS) modeling.

As a simple example of how a mean-field model may work in practice, we consider Equation (5 ) which is the evolution equation for the differential rotation. In a mean-field model, we may wish to approximate the Reynolds stress in terms of a turbulent viscosity operating on the mean flow (cf. Equation (73 )) and a $/\$ -effect (Rüdiger, 1989 ; Rüdiger and Hollerbach, 2004 ):

$( ) ( ) RS -- 2 @_O_- 2nH-@_O_- ^ F ~ r c nV @r + /\V_O_0 sin h ^r + c r @h + /\H_O_0 cosh h. (23)$

The turbulent viscosity represents diffusive mixing of momentum by turbulent motions and is usually justified using mixing-length arguments. It is in general anisotropic ( $n /= n V H$ ) and inhomogeneous ( $nV = nV(r,h)$ , $nH = nH(r,h)$ ) due to the influence of rotation and stratification. The $/\$ terms are non-diffusive source terms which are intended to represent systematic velocity correlations induced by the Coriolis force. The coefficients $/\V$ and $/\H$ may also depend on the latitude, radius, and rotation rate. Many recent models include quenching mechanisms so the $/\$ coefficients remain bounded as the rotation rate or the magnetic field strength becomes large (e.g., Rüdiger et al., 1998 ).

If one specifies the coefficients $nV$ , $nH$ , $/\V$ , and $/\H$ and also the meridional circulation, then Equation (5 ) may be solved numerically to obtain the equilibrium rotation profile (neglecting the Lorentz force). A more self-consistent approach would be to solve the angular momentum equation together with the longitudinally-averaged meridional momentum and thermal energy equations to obtain the full mean flow and thermodynamic fields. In order to do this, similar parameterizations must be introduced to represent the meridional Reynolds stress and the convective heat flux.

Although an anisotropic viscosity alone can induce mean flows, the differential rotation in many mean-field models is driven mainly by either the $/\$ -effect or by latitudinal variations in the convective heat flux which may drive a thermal wind (Section 4.3.2). The importance of the latter effect in particular has recently been emphasized by Kitchatinov and Rüdiger (1995 ) and Durney (1999 ).

As an example of a multi-equation RANS approach, we consider the $k$ - $e$ model which is commonly used in industrial applications (e.g., Pope, 2000 ; Durbin and Pettersson Reif, 2001 ). Here the Reynolds stress is expressed in terms of an isotropic turbulent viscosity which is proportional to $E2 /e k$ where $E k$ is the kinetic energy of the fluctuating velocity field and $e$ is the energy dissipation rate, which is assumed to be scale-invariant within a self-similar inertial range. This expression may be justified using dimensional arguments for homogeneous, isotropic, incompressible flow at high Reynolds numbers. Diagnostic equations for $Ek$ and $e$ may then be derived from the fluctuating flow equations or from phenomenological arguments. These equations are then solved simultaneously along with the mean-field equations.

Similar multi-equation approaches may be followed in the Sun, but they must be somewhat more sophisticated in order to take into account rotation, stratification, shear, and if they’re ambitious enough, magnetic fields. Canuto et al. (1994 ) have developed a Reynolds stress model based on a hierarchy of equations obtained by taking successive moments of the compressible Navier-Stokes equations and then introducing analytic closures for the highest-order moments. A multi-equation model for the convective energy flux has been developed by Canuto and Dubovikov (1998 ) and has been used by Marik and Petrovay (2002 ) to investigate the structure of the overshoot region.

Mean-field hydrodynamics in a solar context has been thoroughly reviewed by Rüdiger (1989 ), Canuto and Christensen-Dalsgaard (1998 ), and Rüdiger and Hollerbach (2004 ). More general reviews of turbulence modeling are given by Cambon and Scott (1999 ), Pope (2000 ), Durbin and Pettersson Reif (2001 ), and Hanjalić (2002 ).

Mean-field dynamo models are distinct from hydrodynamic models in that many of them are kinematic, based only on the mean induction equation, with a specified mean flow field and parameterizations introduced for the turbulent emf (Section 4.5). Much recent attention has focused on flux-transport dynamo models in which the meridional circulation plays a key role in setting the period of the activity cycle and in establishing emergence patterns of magnetic flux such as the butterfly diagram (Choudhuri et al., 1995 ; Durney, 1995 ; Dikpati and Charbonneau, 1999 ; Dikpati and Gilman, 2001a ; Charbonneau, 2005 ). The literature on mean-field solar dynamo models is vast and we make no attempt to review it here. The reader is referred to Charbonneau (2005 ) and to the other references given in Section 4.5.

Many dynamo models have been developed which do consider the feedback of magnetic fields on the mean flow, often focusing on temporal variations of the differential rotation (Kitchatinov et al., 1999 ; Durney, 2000b ; Covas et al., 2001 , 2004 ). These models have shown that the torsional oscillations in particular (Section 3.3) are likely due to the action of the Lorentz force from the axisymmetric dynamo-generated field in relation to the activity cycle. This was first suggested by Yoshimura (1981 ) and Schüssler (1981 ) soon after the torsional oscillations were discovered.

An alternative to (or in some cases a variation of) mean-field models are phenomenological approaches which are motivated by observations, numerical simulations, or laboratory experiments. Chief among these are the various models which describe solar convection as an ensemble of turbulent plumes (Schmitt et al., 1984 ; Rieutord and Zahn, 1995 ; Rast, 2003 ; Rempel, 2004 ) or eddies (Kumar et al., 1995 ). Another type of phenomenological model has been proposed by Longcope et al. (2003 ) who consider a plasma permeated with thin flux tubes which exert a visco-elastic drag on the mean flow (see also Parker, 1985 ).

Although they can provide valuable insight, the main disadvantage of reduced models of any kind is that it is difficult to verify whether the parameterizations and approximations introduced are reliable representations of the underlying dynamics. The overwhelming majority of reduced models may be classified as mean-field models and of these, nearly all assume scale separation in space and/or time. There is little empirical or numerical evidence that such scale separation is valid for solar convection. Furthermore, some reduced models are not completely self-consistent. For example, the well-known $a$ -effect parameterization commonly used in mean-field dynamo modeling is based in part on the linearity of the induction equation in $B$ (see Section 4.5). This argument is only strictly valid if the velocity field is independent of $B$ which cannot be the case in any real-world, sustained dynamo where the Lorentz force must react back on the flow to curb unlimited field amplification. Even so, mean-field dynamo models are quite successful at reproducing many features of the solar activity cycle, a result which might provide clues into the nature of the dynamo (see Section 6.5).

Mean-field hydrodynamics is built on a more questionable theoretical foundation than mean-field dynamo modeling. The turbulent viscosity formalism in particular is known to be inaccurate even for the simplest turbulent flows where momentum transport is often not directed down large-scale velocity gradients (e.g., Pope, 2000 ). Although experimental verification is difficult in a solar context, some testing and calibration of reduced models can be done by comparing them to solar and stellar observations and numerical simulations (e.g., Kupka, 1999 ).

In principle, there is not a large conceptual gap between mean-field/RANS models and large-eddy simulations (LES); the difference lies mainly in the nature and scale of the averaging. In practice, however, there is usually a substantial gap because mean-field models are generally 2D or much lower resolution. Still, some of the parameterizations and procedures developed for reduced approaches could be incorporated into a large-eddy simulation as a subgrid-scale model (e.g., Canuto, 2000 ). This will be discussed further in Section 7.2.