- What is the magnetic power spectrum of the quiet Sun in the six decades spanning the 100 m – 100 Mm range?
- How does the multiscale magnetic field distribution of the quiet Sun originate?
- How does the dynamical magnetic feedback operate, and at which scales?

Some of these questions are already at least partially answerable with nowadays observational and numerical facilities, or they will become so in the near future.

From the observational point of view, we emphasise the need for a statistical description of solar surface MHD turbulence, as opposed to a description in terms of individual “structures” such as flux tubes or magnetic “elements”. The distributions and geometries of magnetic and velocity fields in the quiet photosphere appear to be so different that their large-scale interactions can probably only be understood in statistical terms. A determination of the magnetic energy spectrum of the quiet Sun over a very wide range of scales would notably be extremely useful to understand the physics of MHD turbulence in the quiet Sun and to put constraints on the physical processes at the origin of network and internetwork fields – and consequently on the supergranulation problem.

Our theoretical and numerical understanding of supergranulation-scale MHD convection is scarce. We do have numerical (Ustyugov, 2009; Stein et al., 2009b) hints that advection of weak, small-scale fields and their subsequent clustering can lead to the formation of increasingly energetic magnetic features distributed on larger scales. This phenomenology was already discussed many years ago by Parker (1963) and is included in the n-body model of Crouch et al. 2007. But is it possible to gain a better understanding of this process, starting from the MHD equations?

An essential issue from the point of view of large-scale MHD turbulence is to decipher the preliminary stages of production of quiet Sun magnetic fields up to supergranulation scales. A possible theoretical approach to this problem would be to study the large-scale structure of magnetic eigenmodes in simplified models of turbulent dynamo action, such as the Kraichnan–Kazantsev model (Kazantsev, 1968; Kraichnan, 1968). Such an approach has recently been taken on from a generic perspective by Malyshkin and Boldyrev (2009). Another interesting exercise would be to study the outcome of turbulent induction by a high compressible flow in an extended domain threaded by a weak uniform mean field. This process is distinct from the fluctuation dynamo and could be responsible for the generation of a small-scale magnetic imprint of the global solar dynamo (e.g., Brandenburg and Subramanian, 2005). In the incompressible limit, it produces a simple magnetic energy spectrum (Ruzmaikin and Shukurov, 1982; Schekochihin et al., 2007), not that far from the solar magnetic power spectrum in the 1 – 10 Mm range (Section 4.6.2).

Finally, in order to explain why the supergranulation scale appears to be special in the quiet photosphere, one needs to better understand the statistically steady state of turbulent MHD convection in this region. The main problem is that we do not currently know what simple nonhelical incompressible low MHD turbulence looks like in dynamical regimes – even in small spatial domains – both when the magnetic field is produced by local turbulent dynamo action (e.g., Cattaneo, 1999; Schekochihin et al., 2004, 2007; Vögler and Schüssler, 2007) and in the presence of a net magnetic flux (Cattaneo et al., 2003; Stein et al., 2009b; Ustyugov, 2009). Addressing this question in the context of supergranulation-scale simulations therefore represents a daunting task. On this side, we currently have no choice but to perform mildly nonlinear simulations in regimes with imposed magnetic flux. To gain some insight into the highly nonlinear behaviour of turbulent MHD flows, these efforts should be complemented by dynamical simulations of specific MHD processes, such as the small-scale dynamo, in smaller domains but more extreme parameter regimes (higher and , low ). By combining both approaches, it may eventually be possible to understand nonlinear MHD physics at supergranulation scales.

The general message that we tried to convey in this section is that the supergranulation puzzle may turn out to be a very challenging MHD turbulence problem, the solution to which will certainly require simultaneous progress on MHD theory and observational and numerical solar physics. This dual fundamental physics and astrophysics perspective of the supergranulation problem, we believe, makes it a particularly exciting challenge for the future.

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