4.1 Flux emergence in lower convection zone

The solar convection zone encompasses a density contrast of about 106, leading to vast range of length and time scales as well as convection regimes. While the bottom of the convection zone hosts strongly subsonic flows (Ma ∼ 10 −4), convective motions in the solar photosphere turn supersonic. The pressure scale height ranges from about 50 Mm at the base of the convection zone to about 100 km in the photosphere, convective overturning time scales range from weeks to minutes. Modeling the deeper layers of the convection requires filtering out sound waves (to avoid overly severe time step constraints) while fully accounting for stratification, which is achieved through the anelastic approximation (see Glatzmaier, 1984, for full 3D approaches) or the thin flux tube approximation in simplified models. The assumptions underlying the anelastic as well as thin flux tube approximation break down in the upper most 10 – 20 Mm of the convection zone, which requires taking compressibility fully into account.

Early models of the flux emergence process (Choudhuri and Gilman, 1987; Fan et al., 1993Jump To The Next Citation Point, 1994Jump To The Next Citation Point; Moreno-Insertis et al., 1994; Schüssler et al., 1994; Caligari et al., 1995Jump To The Next Citation Point) were based on the thin flux tube approximation. Studying the time evolution of a closed 1-dimensional flux loop in a background stratification taken from a solar convection zone model, these studies were able to explain large scale properties of active regions, such as the low latitude of emergence, latitudinal trend in tilt angles as well as asymmetries between leading and following spots. Necessary condition for this agreement was an initial field strength at the base of the convection zone in the 100 – 150 kG range. Very similar values for the field strength were also found in independent studies of flux storage in the subadiabatic overshoot region (Ferriz-Mas and Schüssler, 1993, 1995).

Based on two-dimensional MHD simulations it was early realized by Schüssler (1979) that untwisted magnetic flux tubes cannot rise coherently and fragment. It was shown later by Moreno-Insertis and Emonet (1996) and Emonet and Moreno-Insertis (1998) that this fragmentation can be alleviated provided that flux tubes have enough initial twist.

More recently also 3D MHD simulations of rising flux tubes based on the anelastic approximation have become possible (Fan, 2008Jump To The Next Citation Point) and give support for results from earlier simulations based on the thin flux tube approximation. It was, however, found by Fan (2008Jump To The Next Citation Point) that there is a very delicate balance between the amount of twist required for a coherent rise and the amount of twist allowed to be in agreement with observations of sunspot tilt angles (twist with the observed sign produces a tilt opposite to the effect of Coriolis forces on rising tubes).

The simulations presented above consider the flux emergence process decoupled from convection. First attempts to address flux emergence in global simulations of the convection zone were made recently by Jouve and Brun (2007, 2009). Understanding the interaction of emerging flux with the ambient convective motions in the convective envelope is a crucial step toward more realism; however, currently the focus on the global scale limits the resolution required to resolve this interaction in detail. A complementary approach that circumvents the resolution problem was recently taken by Weber et al. (2011). They simulated the rise of a thin flux tube through a convection zone taken from a 3D global convection model, where the coupling between flux tube and surrounding convective flow field is accomplished through the drag force. Overall, the interaction between rising flux and convective flows reduces the sensitivity of results with respect to the initial field strength, although the best agreement with observed active region properties is found for about 50 kG initial field strength. The latter is a factor of about 2 – 3 lower than values previously inferred from simulations not considering the interaction with convective motions.

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