List of Footnotes

1 In solar physics, an appropriate length unit is the megametre (Mm), also 1000 km.
2 Hinode is led by the Japanese Aerospace Exploration Agency (JAXA) in collaboration with NASA, the Science and Technology Facilities Council (STFC), and the European Space Agency (ESA). Hinode is a Japanese mission developed, launched, and operated by ISAS/JAXA, in partnership with NAOJ, NASA, and STFC (UK). Additional operational support is provided by ESA and NSC (Norway). The project website can be found at External Link
3 ℓν can be understood physically as the scale at which viscous effects start to dominate over inertial effects, so that the Reynolds number at this scale Re ℓν = ℓνV ℓν∕ν = 1, where Vℓν ∼ Re −1∕4V is the typical velocity at scale ℓν in the framework of the Kolmogorov theory.
4 ℓη is the scale at which resistive effects take over magnetic field stretching, corresponding to a scale-defined magnetic Reynolds number Rm ℓη = ℓηVℓη∕η = 1.
5 Similarly to ℓν and ℓη, this scale corresponds to the scale-defined Péclet number Peℓκ = Vℓκℓκ∕κ = 1.
6 At granulation scales, the spectral power density is less than 300 km3 s–2. We recall that granules have a much larger typical velocity than supergranules though, of the order 1 – 2 km s–1 (Section 3.1). The difference comes from the definition of the spectral power density at wavenumber k, E(k) ∼ k−1Vk2, which introduces an extra k factor.
7 The Roberts number q = 1∕ζ is also used in the context of convection in planetary cores. In the Earth’s core, q ≪ 1, see, e.g., Zhang and Jones (1996).
8 Note that a previous work of Simon et al. (1991) already used a purely kinematic model to model mesogranular flows and exploding granules.
9 Stratified simulations with a bottom wall tend to exhibit more small-scale turbulent activity in deep layers that their open-wall counterparts, though (see Figure 12View Image). This behaviour may be related to the enhanced shear and recirculations generated at the bottom wall, or with the fact that most of these simulations use a constant dynamical viscosity which, combined with density stratification, enhances the local Reynolds number as one moves deeper down.
10 Idealised simulations evolve nondimensional equations such as (2View Equation), so their results are not given in solar units. However, even in this kind of idealised simulations, granulation-like cells clearly appear in a thin thermal boundary layer at the upper boundary (Figure 12View Image provides horizontal temperature maps extracted from a similar simulation exhibiting this phenomenon). This simple observation usually serves to “calibrate” the size of the dynamical structures present in the simulations with respect to the size of granules.
11 Amongst the significant differences between idealised simulations and the realistic ones, Nordlund et al. (1994Jump To The Next Citation Point) pointed out that using “wall-type” boundary conditions, as is standard in idealised simulations, can alter significantly the shape of the convective pattern. Indeed, this type of boundary conditions allows for a return flow after plumes smash down onto the bottom wall, which of course does not occur in the Sun until the very deep layers of the solar convection zone are reached by descending plumes. Nordlund et al. (1994) suggested to use stronger stratifications in simulations in closed domains to attenuate this effect. Finally, upflows seem to play a much more important role in the process of vertical heat transport in idealised simulations than in realistic ones (Stein and Nordlund, 1994). This may have some important consequences regarding the most energetic scales of the flows, which correspond to the mesoscales in idealised simulations.
12 This is not actually specific of thermal convection. Decaying homogeneous turbulence experiments, for instance, exhibit decreasing large-scale kinetic energy spectra – Saffman–Birkhoff or Batchelor spectra (see, e.g., Davidson, 2004, Chap. 6).
13 Available magnetic power spectra are derived from polarimetric measurements which mostly track the vertical component of the magnetic field at disc centre.
14 We are grateful to one of the referees for pointing this out to us.
15 All such simulations (Cattaneo, 1999Jump To The Next Citation Point; Vögler and Schüssler, 2007Jump To The Next Citation Point) are for P m ∼ 1 or larger and asymptotically not large Re. How saturation takes place and whether equipartition should be expected in both (P m ≫ 1, ≫ Re ≫ 1) and (Pm ≪ 1, Re ≫ ≫ 1) limits is unknown (Schekochihin et al., 2004Jump To The Next Citation Point; Yousef et al., 2007Jump To The Next Citation Point; Schekochihin et al., 2007Jump To The Next Citation Point; Tilgner and Brandenburg, 2008; Cattaneo and Tobias, 2009), so it is currently very difficult and potentially risky to predict how and at which level of magnetic energy the putative solar surface dynamo saturates.