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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, 2000Jump To The Next Citation Point).

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., 1996Jump To The Next Citation Point2002Jump To The Next Citation PointStein and Nordlund, 1998Jump To The Next Citation Point2000Jump To The Next Citation PointHurlburt et al., 2002Vögler et al., 2005Jump To The Next Citation PointRincon et al., 2005Jump To The Next Citation Point), 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., 1991Jump To The Next Citation PointBrummell et al., 1996Jump To The Next Citation Point1998Jump To The Next Citation PointBrandenburg et al., 1996Jump To The Next Citation PointStein and Nordlund, 1998Jump To The Next Citation PointPorter and Woodward, 2000Jump To The Next Citation PointTobias et al., 2001Jump To The Next Citation PointBrummell et al., 2002bJump To The Next Citation PointZiegler and Rüdiger, 2003Jump To The Next Citation Point) or on the Boussinesq approximation where the compressibility of the fluid is neglected outside of the buoyancy driving (Julien et al., 1996aJump To The Next Citation Point,bWeiss et al., 19962002Cattaneo, 1999Cattaneo et al., 2003Jump To The Next Citation Point). 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, 2004Jump To The Next Citation Point).

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., 2003Jump To The Next Citation Point).

Brummell et al. (1996Jump To The Next Citation Point1998) 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., 1996aJump To The Next Citation Point1999Jump To The Next Citation PointBrummell et al., 1996Jump To The Next Citation Point). 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. (1996aJump To The Next Citation Point1999Jump To The Next Citation Point) 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., 1996Jump To The Next Citation PointTobias et al., 2001Jump To The Next Citation PointDorch and Nordlund, 2001Jump To The Next Citation PointZiegler and Rüdiger, 2003Jump To The Next Citation Point).

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.


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