tokens: et al. (2011) . These swirls are obviously caused by the turbulent plasma motions in the surface layers of the convection zone. However, the Hydro simulation shows a higher density of swirls than the MHD simulations. This is because the magnetic fields hamper the formation of vortices in the convective layers of the simulation. The photosphere of the MHD 200 G time instant harbors an almost homogeneous structure of vertically oriented swirls and only a few swirl arches are present near the τ 500 = 1 surface. In the MHD 50 G case, the vertical swirls are clustered in funnel-shaped structures and the arches are more prominent than in MHD 200 G. Different from these two cases, vertical swirls are essentially absent in the Hydro simulation, while swirl arches are omnipresent in the photosphere. Again, these results well agree with and confirm earlier results of Moll et al. (2011 , 2012) . A similar behavior is observed in the chromosphere: the stronger the initial magnetic field, the more abundant the vertical swirls are. However, we observe that the strongest swirls appear at the top of the MHD 50 G simulation. This suggests that too strong magnetic fields can possibly inhibit the formation of strong vertical vortices in the chromosphere. In the Hydro simulation, we see a quasi-isotropic distribution of swirls at chromospheric levels, which are most probably related to hydrodynamic shocks taking place in this region of the field-free solar atmosphere (see, e.g., Moll et al. 2012 ). Therefore, not all swirls found at chromospheric levels are necessarily related to the magnetic field, even though the magnetism clearly plays a cardinal role in the generation of vertical vortices. The above more qualitative assessment is confirmed by the quantitative analysis of Fig. A.2 , which shows for each simulation a bi-dimensional histogram of the inclination angle cos( θ ) of the swirling strength vector at each height z . The angle θ is defined through being the angle between the swirling strength vector λ and the unit vector in the vertical direction . Therefore, horizontally oriented swirls are binned close to cos( θ ) = 0, while vertically oriented swirls are close to cos( θ ) = 1. We choose to plot the histogram of cos( θ ) instead of θ in order to easily recognize isotropy. In this way, an isotropic angle distribution at a certain height z is characterized by a flat distribution in the histograms of Fig. A.2 . In the convection zone, that is, for z ≲ 0 km, all three simulations show a flat distribution, which confirms the nearly isotropic distribution of swirls in that region. Also, as estimated before, there are on average more swirls in the Hydro simulation (panel c) than in the MHD 50 G (panel b) and the MHD 200 G (panel a) simulations. Near the optical surface, that is, around z = 0 km, there is a sudden break of isotropy. Two families of vortices can be spotted. First, a cluster of horizontally oriented swirls (cos( θ )≈0), visible in all three
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