6.5 Conclusions

The numerical study of supergranulation has become a very active topic in the last decade, as simulations presented in Section 6.4 attest. Table 1 provides a list of the most important numerical efforts dedicated to this problem so far.

Table 1: Numerical simulations dedicated to the study of mesogranulation to supergranulation scale dynamics. The “Type” entry specifies whether the simulations are idealised (I) Boussinesq (Bouss.), anelastic (Anel.) or polytropic (Poly.) simulations or realistic (R) simulations (Section 6.2). The box size and duration of simulations in Mm and solar hours (sh) or days are given for qualitative comparative purposes only, as these quantities are actual input parameters of the realistic simulations (and of the global simulations by DeRosa et al., 2002Jump To The Next Citation Point) but not of idealised simulations (only the aspect ratio and density stratification are). For local idealised simulations, the size and duration of the simulation in solar units is estimated empirically by setting the horizontal size of the “granules” visible in the upper 10% surface layers of the simulations to 1 Mm, their vertical extent to 150 km (the width of the entropy jump in realistic simulations; see, e.g., Stein and Nordlund, 1998) and the typical turnover time of the mesoscale cells described in Section 6.4.4 to 1 h (see corresponding discussions in Section 6.4.3).
Reference Type MHD Resolution Box size (Mm3) Duration
Cattaneo et al. (2001Jump To The Next Citation Point) I-Bouss. Yes 10242 × 96 ∼ 30 × 30 × 1.5 ∼ 35 sh
DeRosa et al. (2002Jump To The Next Citation Point) I-Anel. No 1024 × 512 × 128 4400 × 4 400 × 56 80 d
Miesch et al. (2008) I-Anel. No 2048 × 1024 × 257 4400 × 4400 × 190 560 d
Rieutord et al. (2002) R No 3152 × 82 30 × 30 × 3.2 7 sh
Rincon et al. (2005Jump To The Next Citation Point) I-Poly. No 10242 × 82 ∼ 64 × 64 × 1.5 ∼ 15 sh
Ustyugov (2008) R No 6002 × 168 60 × 60 × 20 24 sh
Ustyugov (2009Jump To The Next Citation Point) R Yes 6002 × 204 60 × 60 × 20 24 sh
Stein et al. (2009aJump To The Next Citation Point) R No 10002 × 500 96 × 96 × 20 64 sh
Stein et al. (2009bJump To The Next Citation Point) R Yes 5002 × 500 48 × 48 × 20 48 sh

The most recent large-scale three-dimensional models represent numerical tours de force and are increasingly successful at reproducing observational features of solar surface magnetoconvection in the granulation to supergranulation range. One of their main achievements has been the test of standard helioseismic diagnostic tools with numerical data sets: the results compare reasonably well with those extracted from real data (e.g., Georgobiani et al., 2007Couvidat and Birch, 2009). Another important point on the topic of “virtual observations” is that large-scale simulations have helped validate granule-tracking techniques to reconstruct velocity fields at the solar surface (Rieutord et al., 2001).

On the specific problem of the origin of supergranulation, an important result is that the supergranulation scale does not seem to emerge as a particular scale in purely hydrodynamic simulations incorporating the ionisation of Hydrogen and Helium, which tends to disprove the “classical” Simon and Leighton (1964Jump To The Next Citation Point) supergranulation theory. Finally, it is encouraging for the future that the most recent MHD simulations to date (Ustyugov, 2009Jump To The Next Citation PointStein et al., 2009bJump To The Next Citation Point) start to be large enough for meaningful numerical studies of the process of network formation and supergranulation-scale MHD to be possible.

It is fair to say, however, that even the most advanced and impressive numerical efforts to date can only be considered as preliminary with respect to the supergranulation puzzle. As shown in Section 4.2, recent observations indicate that the supergranulation scale is an undeniable feature of the horizontal velocity power spectrum of solar surface convection, whatever method is used to compute the spectrum. This observation has not been reproduced by any numerical simulation so far. The superrotation rate of supergranules is another open question that no simulation can address at the moment. Finally, the interactions between supergranulation, the magnetic network, and internetwork fields are still poorly understood.

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