7.3 Boundary influences

Most global solar convection simulations assume that the upper and lower boundaries of the computational domain are stress-free and impenetrable. Convection is driven by imposing a heat flux on the lower boundary and either a constant heat flux or a constant entropy on the upper boundary. Magnetic boundary conditions are generally either perfectly conducting or matching to an external potential field. All of these conditions are gross simplifications of the complex dynamics which actually couple the solar convection zone to the extended atmosphere above and the radiative interior below.

Although the uppermost layers of the convection zone account for only a small fraction of its total mass, the precipitous drop in entropy near the surface produces strong buoyancy driving. This, when coupled with radiative transfer and ionization effects, maintains granulation and supergranulation. These motions do not penetrate far below the photosphere but stochastic forcing from ensembles of plumes may have a subtle influence on the deeper convective zone. In particular, coupling between supergranulation, mesogranulation, and deeper convective motions may have some bearing on the near-surface shear layer seen in helioseismic rotational inversions (Section 3.1). Furthermore, magnetic flux dispersion by near-surface convective motions might contribute to global polarity reversals as in Babcock-Leighton dynamo models (e.g., Ossendrijver, 2003 ; Charbonneau, 2005 ).

Coupling between the convective envelope and the corona can occur through magnetic torques and mass exchange via the solar wind. Such processes generally occur on timescales much longer than the solar activity cycle. However, magnetic helicity flux through the solar surface may play an important role both in the operation of the dynamo (Blackman and Brandenburg, 2003 ) and in determining the global configuration of the coronal field (Low, 2001 ; Zhang and Low, 2001 , 2003 ). Understanding the complex process of flux emergence is also essential for interpreting photospheric and coronal observations (e.g., Fan, 2004 ).

Perhaps an even more important factor in improving global simulations is a more realistic treatment of the complex dynamics occurring at the base of the convection zone, where the solar envelope couples to the radiative interior through the overshoot region and tachocline. This transition region is thought to play a critical role in the solar dynamo (Section 4.5) so it must be represented with some fidelity if global simulations are ever to make meaningful contact with observations of magnetic activity.

The primary difficulty in capturing the dynamics of the overshoot region and tachocline in a global simulation lies in their thin extent (Section 3.2, Section 3.6). Like the near-surface layers, relatively small-scale processes occur which are difficult to resolve (Section 8). This small grid spacing sets corresponding limits on the time steps required for numerical stability, further adding to the computational expense. Such restrictions are overcome in current models by either placing the boundary of the computational domain at the base of the convection zone (no penetration) or by artificially decreasing the subadiabatic stratification in the interior, thereby extending the overshoot region.

Nevertheless, global simulations can potentially capture may aspects of the solar dynamo including turbulent pumping of fields into the overshoot region and amplification by differential rotation in the tachocline (cf. Figure 8 ). The subsequent formation and rise of flux tubes by buoyancy instabilities may require much higher resolution to reliably model but it should be present to some degree in global simulations which possess a tachocline.

Establishing and maintaining a tachocline in a global convection simulation is a challenge in itself, since it may involve processes which occur on timescales much longer than the solar activity cycle, (Section 8.5). It also requires minimal vertical diffusion to prevent artificial spreading (high $Re$ , $Pe$ ; cf. Section 7.1). Global simulations have only begun to explore penetrative convection in detail and have not yet achieved a rotational shear layer comparable to the tachocline¹⁶ . The tachocline region is not only important from a dynamo perspective; it also mediates angular momentum transport between the convective envelope and the radiative interior. This may occur through magnetic coupling or by through penetrative convection and gravity waves (Section 8.4). Helioseismic inversions suggest that this transport must be relatively efficient, since the mean rotation rate of the convection zone and radiative interior are comparable (Section 3.1).

Angular momentum exchange between the convection envelope and the deep interior plays an important role in the rotational history of the Sun over evolutionary timescales (Charbonneau and MacGregor, 1993 ). However, it has little bearing on the differential rotation profile of the envelope which is maintained on shorter timescales (Section 4.3). Still, there are several reasons to believe that the differential rotation profile may be sensitive to dynamics near the base of the convection zone.

Turbulent penetrative convection tends to produce poleward angular momentum transport in the overshoot region due to the rotational alignment of downflow plumes (Section 6.3). Since the overshoot region is artificially deep in simulations, we may be overestimating this transport (Miesch, 2005 ). More generally, poleward angular momentum transport in the tachocline by instabilities and turbulence may balance equatorward transport in the convection zone, giving rise to a global angular momentum cycle which would ultimately determine the equilibrium rotation profile (Gilman et al., 1989 ).

Thermal effects may also play an important role. The differential rotation in the lower convection zone is probably in thermal wind balance, maintained by latitudinal entropy gradients (Sections 4.3 and 6.3). The radiative interior provides a large thermal reservoir which can influence this balance depending on how it is tapped by penetrative convection. An example of how this may occur has been described by Rempel (2005 ) in the context of a mean-field model.

In Rempel’s model, differential rotation in the convection zone is maintained by a $/\$ -effect (Section 5.3) and a meridional circulation which is solved for together with the angular velocity and thermal structure by means of the axisymmetric momentum, energy, and continuity equations. Uniform rotation is imposed on the lower boundary and the system is evolved until a steady state is reached. The competition between the $/\$ -effect and the lower boundary condition quickly establishes an artificial ’tachocline’; i.e., a large vertical angular velocity gradient near the base of the convection zone. This, in turn, sets up latitudinal entropy gradients in accord with thermal wind balance. These entropy gradients are then transmitted upward into the envelope by convective motions, here treated as an effective thermal diffusion. The net result is a solar-like differential rotation profile in which departures from cylindrical alignment are maintained by latitudinal entropy gradients originating in the tachocline. The model involves many crude simplifications but it illustrates how thermal coupling between the convection zone and radiative interior may influence the differential rotation profile.

More generally, since the convection zone is nearly adiabatic, even small entropy variations originating in the strongly subadiabatic radiative interior may be significant. The depth to which penetrative convection mixes entropy with the interior and the efficiency by which energy is transported through the surface together determine the entropy content, or in other words, the adiabat of the convection zone. This is one more reason why a realistic modeling of these boundary regions may be important for global simulations.

There are many other phenomena which will require detailed modeling of the upper and lower convection zone to fully account for. An example is light element depletion in the Sun and other late-type stars which may arise from chemical transport by gravity waves (Section 8.4). Momentum transport by gravity waves may account for the tachocline oscillations found in helioseismic inversions (Section 3.3). Incorporating all of these influences into global convection simulations is a tall order but it must eventually be done to some degree if we are ever to have a reasonably complete and integrated model of solar interior dynamics. Meeting these challenges will require a combination of increased resolution near the boundaries, sophisticated SGS models, and carefully chosen boundary conditions.