5.1 The challenge

The molecular viscosity in the solar interior may be estimated by $-16 5/2 - 1 2 -1 n ~ 1.2 × 10 ~ T r cm s$ , which is valid for a fully ionized hydrogen plasma, neglecting the contribution due to radiation (Parker, 1979

). This yields $n ~ 1 cm2 s-1$ in the upper convection zone, rising to somewhat higher values near the tachocline. If giant cells have an amplitude of $U ~ 100 m s-1$ and scales of $L ~$ 200 Mm, this implies Reynolds numbers of $14 Re = UL/n ~ 10$ . In other words, inertia dominate over viscous dissipation, making solar convection strongly nonlinear and thus highly turbulent.

Although solar convection is certainly not homogeneous and isotropic, a rough estimate of the viscous dissipation scale $d v$ can be obtained by assuming a classical Kolmogorov inertial range (e.g., Lesieur, 1997 ). The result is $-3/4 dv ~ LR e ~ 1 cm$ - more than ten orders of magnitude smaller than the solar radius! As in most other astrophysical and geophysical systems, direct numerical simulations which capture all the dynamical scales of the system are not feasible because computers simply are not efficient enough to perform all the necessary calculations.

The thermal and magnetic dissipation scales are larger than the viscous dissipation scale but are still beyond the resolution of a global numerical model. We can estimate the magnetic diffusivity by again assuming a fully ionized hydrogen plasma where $j = 1013 ~ T -3/2 cm2 s-1$ (Parker, 1979 ). In the solar interior, radiative diffusion dominates over thermal conduction, giving rise to an effective thermal diffusivity of $kr = 16ssbT 3/(3xr2CP )$ , where $ssb$ is the Stefan-Boltzman constant and $x$ is the opacity (Hansen and Kawaler, 1994 ). Entering values from a solar structure model (model S of Christensen-Dalsgaard, 1996 ) yields $kr ~~ j ~ 105 cm2 s-1$ near the surface, with $kr$ increasing to $7 2 -1 ~ 10 cm s$ and $j$ decreasing to $3 2 - 1 ~ 10 cm s$ in the tachocline. These values imply low Prandtl and magnetic Prandtl numbers: $- 3 -6 Pr = n/k ~ 10 - 10$ and $-5 - 6 Pm = n/j ~ 10 -10$ . The corresponding thermal and magnetic dissipation scales are then several meters to several kilometers.

If motions in the Sun were self-similar then the large dynamical range might not be a problem (see Section 7.2). Although this may be a good approximation for the smallest scales, it does not apply throughout because qualitatively different dynamics occur over a wide range of scales in the solar interior. On the largest scales $~ 1000 Mm$ , we have differential rotation and meridional circulation which require the full spherical geometry to be investigated in detail. In the solar surface layers, the strong stratification coupled with ionization and radiation effects drives much smaller-scale motions including granulation ( $~ 1 Mm$ ) and supergranulation ( $~ 30 Mm$ ). Relatively small-scale motions are also driven by the strong rotational shear and the stiff transition from subadiabatic to superadiabatic stratification at the base of the convection zone, where the region of convective overshoot is thought to be less than $10 Mm$ thick (Sections 3.6 and 8). In between, in the bulk of the convection zone, we have so-called giant cells (Section 3.5) which likely occupy a wide dynamic range from hundreds of Mm where most of the buoyancy driving occurs down to, at least, supergranulation scales (Section 7.1). The coupling between the bulk of the convection zone and the distinct dynamics occurring in the upper and lower interface regions and beyond is a challenging problem which remains poorly understood (Section 7.3).

The range of temporal scales which characterize solar interior dynamics is every bit as daunting as the range of spatial scales. Granulation evolves over the course of a few minutes, which is comparable to the oscillation frequency of acoustic waves ( $~ 5 min$ ). Supergranulation timescales in the surface layers and gravity wave periods in the radiative interior are both somewhat longer - about one day and several hours, respectively. Turnover timescales of giant cells are thought to be comparable to the rotation period of about a month, but substantial evolution likely occurs over the course of days and weeks (Section 6.2). These giant cells likely play a crucial role in the 22-year solar activity cycle (Section 3.8), which must be the ultimate target of any comprehensive dynamical model of the solar interior. Variations of this activity cycle such as the well-known Maunder minimum are known to occur on timescales of centuries or millennia (e.g., Usoskin and Mursula, 2003 ; Charbonneau, 2005 ). Meanwhile, thermal relaxation timescales are hundreds of millennia (Section 4.2) and spin-down of the Sun due to magnetic braking and angular momentum loss in the solar wind occurs on still longer timescales - millions to billions of years!

From a modeling perspective, the vast dynamic range of spatial and temporal scales is the most challenging aspect of solar interior dynamics; no single model can hope to capture all the relevant processes. Some approximations must be made.