7.1 Resolution

In analogy to the shopkeeper’s well-known mantra (location, location, location), one could argue that the three most important factors in improving global-scale solar convection models are resolution, resolution, and resolution.

Global convection simulations are generally well-resolved in the sense that the kinetic, thermal, and magnetic energy spectra peak at relatively large scales ( $l ~ 10 -50$ ) and their amplitude falls off by at least $2 -3$ orders of magnitude before reaching the grid scale. Furthermore, the results converge in the sense that a higher-resolution with the same parameters will give statistically the same results. However, simulations are far from the solar parameter regimes (Section 5.1). Thus, as the resolution is increased, parameters are generally not held constant.

In particular, higher resolution allows for higher Reynolds, magnetic Reynolds, and Peclet numbers, $Re$ , $Rm$ , and $Pe$ , which quantify the relative importance of advection and diffusion. As these ratios are increased, the flow generally becomes more turbulent and the convective patterns and transport properties may change. For example, as the Reynolds number is increased, the downflow lanes and plumes which currently dominate simulations may alter their entrainment properties or even destabilize completely (e.g., Rast, 1998 ). New convective modes may become unstable, characterized by smaller spatial scales and rapid time variability (Zhang and Schubert, 2000 ). Nonlinear processes such as tachocline shear instabilities (Section 8.2) may only occur at sufficiently high Reynolds numbers (Section 8.2). Furthermore, the structure of the overshoot region may be sensitive to the Peclet number (Section 8.1).

The hope and expectation is that these changes only occur up to a point. If enough of the global dynamics is explicitly resolved, smaller-scale dynamics may be reliably treated as an effective diffusion or in terms of a more elaborate sub-grid-scale model (Section 7.2). The question is; how much resolution is enough? At the very minimum, simulations must resolve the energy-containing scales. This has already been accomplished; as $Re$ , $Rm$ , and $Pe$ are further increased, the peaks in the energy spectra will probably not shift significantly. However, spectra only provide part of the story.

Most researchers would agree that the most significant advances in turbulence research over the past few decades have been concerned with coherent structures which arise from self-organization processes such as the selective dissipation of ideal invariants (Cantwell, 1981 ; Hasegawa, 1985 ; Lesieur, 1997 ; Branover et al., 1999 ). Although such structures may occupy a small volume and possess relatively little energy, they often dominate the transport in inhomogeneous and anisotropic turbulent flows. Symmetry breaking induced by rotation, stratification, and magnetic fields can all give rise to self-organization in the solar convection zone.

A goal for solar convection simulations is therefore to resolve all scales which are significantly influenced by rotation and stratification (i.e., buoyancy) in order to capture such self-organization processes¹⁴ . Smaller-scale motions may then behave more like isotropic, homogeneous turbulence which is generally diffusive in nature. This goal may be achievable throughout most of the convection zone. However, magnetic fields will be present everywhere above the magnetic dissipation scale which, at several kilometers, is well beyond the resolution of simulations (Section 5.1). Furthermore, buoyancy effects remain important even at the smallest resolvable scales near the photosphere and overshoot region. Thus, subgrid-scale models must be developed which can reliably take into account the effects of magnetism and buoyancy, which may be non-diffusive (Section 7.2). In any case, it is clear that global simulations are not yet in a regime in which the results are insensitive to viscous, thermal, and magnetic dissipation and consequently, to resolution. Convective patterns and mean flows still depend to some extent to the effective values of $Re$ , $Pe$ , and $Rm$ (Miesch et al., 2000 ; Miesch, 2000 ; Elliott et al., 2000 ; Brun and Toomre, 2002 ; Brun et al., 2004 ).

The transition regions which couple the convection zone to the radiative interior below and the solar atmosphere above are particularly challenging to resolve in global simulations (see Sections 5.1 and 7.3). Granulation in the surface layers will likely remain outside the scope of global models for some time, as will a realistic depiction of penetrative convection and wave dynamics in the overshoot region and tachocline (Section 8). The effect of these transition regions on global-scale dynamics can however be explored in global simulations with the help of appropriate boundary conditions and subgrid-scale models (Sections 7.2 and 7.3).

Improved numerical methods with enhanced resolution near the boundaries and better parallel efficiency may help to mitigate some of the limitations of global simulations in the coming years. Particularly promising in this respect are finite element and finite volume methods which require less inter-processor communication than spectral methods and which, primarily for this reason¹⁵ , are becoming more common in atmospheric and oceanic applications (e.g., Lin and Rood, 1997 ; Marshall et al., 1997 ; Stuhne and Peltier, 1999 ; Fournier et al., 2004 ). Solar convection simulations must always push the limits of available high-performance computing platforms to achieve ever higher spatial resolution. However, the highest-resolution simulations achievable on a given platform are computationally intensive. Not only do they require more calculations per iteration, but they must take smaller time steps to meet CFL stability conditions, implying more iterations for a particular simulation interval. Thus, it is impractical to run the highest-resolution simulations for the long durations necessary to adequately assess sustained dynamo action or to explore dynamics spanning several solar activity cycles. For such investigations, intermediate-resolution simulations will always remain important. Here again we may be guided by geophysical applications where high-resolution development models may be used to verify and calibrate lower-resolution application models (e.g., Williamson, 2002 ).

The continued importance of intermediate-resolution simulations further emphasizes the need for reliable subgrid-scale models to account for motions which are not resolved. These will be discussed further in the next section (Section 7.2).