5.2 3D numerical simulations

High-resolution numerical simulations provide a powerful means by which to investigate the diverse and complex dynamics occurring in the solar interior. They have as their basis the fundamental equations of mass, energy, and momentum conservation in a magnetized or neutral fluid and explicitly resolve nonlinear interactions over a wide range of spatial scales. As such, they can capture dynamical processes which lie outside the scope of other modeling approaches and they have therefore become an essential tool in solar physics and throughout turbulence research (e.g., Pope, 2000

Global-scale phenomena such as differential rotation and the solar activity cycle must ultimately be studied using global models. However, in light of the formidable computational challenges highlighted in Section 5.1, much progress can still be made by considering local Cartesian domains intended to represent a small subvolume of the solar envelope. The results, limitations, and promise of global convection simulations will be reviewed at length in Sections 6 and 7.

Although many high-resolution local simulations of solar convection focus on dynamics in the surface layers such as granulation and its interaction with magnetic fields (Weiss et al., 1996 , 2002 ; Stein and Nordlund, 1998 , 2000 ; Hurlburt et al., 2002 ; Vögler et al., 2005 ; Rincon et al., 2005 ), others are concerned with more fundamental fluid dynamical processes which occur throughout the convection zone. These models have been based either on the fully compressible fluid equations (Cattaneo et al., 1991 ; Brummell et al., 1996 , 1998 ; Brandenburg et al., 1996 ; Stein and Nordlund, 1998 ; Porter and Woodward, 2000 ; Tobias et al., 2001 ; Brummell et al., 2002b ; Ziegler and Rüdiger, 2003 ) or on the Boussinesq approximation where the compressibility of the fluid is neglected outside of the buoyancy driving (Julien et al., 1996a ,b ; Weiss et al., 1996 , 2002 ; Cattaneo, 1999 ; Cattaneo et al., 2003 ). This is in contrast to recent global models which are based on the anelastic equations described in Appendix A.2. Anelastic models have been developed in local domains but these have thus far focused mainly on the dynamics of magnetic flux structures rather than convection (reviewed by Fan, 2004 ).

With a few exceptions (e.g., Porter and Woodward, 2000 ), most local models employ spectral methods for the horizontal dimensions which are treated as periodic. The Cartesian geometry permits the use of fast Fourier transforms (FFTs) which are more computationally efficient than the Legendre transforms necessary for the spherical harmonic algorithm currently used in global simulations. Because of this greater efficiency and the simplified geometry, local models can generally achieve somewhat higher resolution and more turbulent parameter regimes than global simulations and are therefore well equipped to study the fundamental coupling between turbulent convection, rotation, and magnetic fields.

Local simulations were the first to demonstrate the granulation-like character of turbulent compressible convection; broad, relatively weak, relatively laminar upflows surrounded by a network of strong turbulent downflow lanes and plumes where vorticity and magnetic fields are highly concentrated. These strong vortical downflow plumes were identified as the dominant structures of the flow which could remain coherent over multiple density scale heights. Although Boussinesq simulations are symmetric about the mid-plane, they also exhibit an interconnected network of lanes and plumes flowing away from the boundaries which resembles granulation near the top of the layer (e.g., Cattaneo et al., 2003 ).

Brummell et al. (1996 , 1998 ) found that in the presence of rotation, turbulent plumes tend to align with the rotation axis, altering the Reynolds stress relative to more laminar flows. Quasi-2D vortex interactions among plumes, enhanced by rotation, alter their entrainment and transport properties (Julien et al., 1996a , 1999 ; Brummell et al., 1996 ). In particular, vortex interactions can lead to enhanced horizontal mixing and a decorrelation of the temperature and vertical velocity in a plume, reducing the buoyancy driving. The resulting decrease in the convective enthalpy and kinetic energy flux must be compensated by thermal diffusion, leading to a larger super-adiabatic entropy gradient in the convection zone relative to comparable non-rotating flows. The Boussinesq simulations by Julien et al. (1996a , 1999 ) possess both upward and downward plumes which dominate the convective heat flux even though they have a small filling factor. However, downward plumes dominate in compressible flows with a substantial density stratification and the resulting downward kinetic energy flux can nearly balance the upward enthalpy flux such that the plumes contribute little to the net vertical energy transport (Cattaneo et al., 1991 ). This asymmetry between upflows and downflows also leads to a net downward pumping of magnetic fields from the convection zone to the stably-stratified radiative interior, a process which has also been investigated in detail with local simulations (Brandenburg et al., 1996 ; Tobias et al., 2001 ; Dorch and Nordlund, 2001 ; Ziegler and Rüdiger, 2003 ).

The highest-resolution simulations of solar convection to date have achieved roughly $10003$ spatial grid points. Thus, even the most ambitious models can only capture a fraction of the vast dynamic range which characterizes solar interior dynamics (Section 5.1). For this reason, all simulations of solar convection should be viewed as large-eddy simulations (LES) in which unresolved subgrid-scale (SGS) processes must be parameterized or otherwise modeled (Section 7.2). Most current models simply treat unresolved motions as an effective turbulent diffusion of momentum, heat, and magnetic fields which is many orders of magnitude larger than the molecular diffusion. Thus, such simulations may also be regarded as direct numerical simulations (DNS) of a hypothetical physical system which is not the Sun.