tokens: shows the maximum value of the swirling strength, λ max , as a function of z for the three simulations. From it, we notice that in the convection zone the stronger the initial magnetic field, the weaker the swirls. However, the overall behavior is similar and the local peak is reached in all three cases around z ≈ −500 km. Thus, relatively strong vortices can develop in the convection zone, independently of the initial magnetic configuration of the simulation. In the photosphere, a local minimum in swirling strength is reached. It is a result of the slowdown of vortical motions in the rapidly expanding convective overshoots as a consequence of angular momentum conservation ( Nordlund et al. 1997 ). For the two MHD simulations, the minimum is reached in the lower half of the photosphere, while the Hydro simulation reaches its minimum at z ≈ 500 km. This difference can be attributed to strong arches of swirling strength in the hydrodynamical simulation, which permeate the photosphere and extend precisely up to z ≈ 500 km, as it can be seen in Fig. A.1 . These arches are less developed in the MHD simulations, probably as a consequence of the magnetic canopy. Fig. A.1. Volume rendering of the swirling strength seen from the side for three different simulations at arbitrary time instants: ( top ) MHD simulation with an initial vertical, homogeneous magnetic field of 200 G; ( middle ) MHD simulation with an initial field of 50 G; ( bottom ) purely hydrodynamical simulation. The red surface indicates the τ 500 = 1 surface. Swirls with a period larger than 10 min ( λ 2.09 × 10 −2 Hz) have been neglected and values above 1 Hz are saturated. Finally, looking at the photosphere and chromosphere, we observe that λ max increases exponentially for the MHD 50 G simulation, while this growth is only linear for the MHD 200 G simulation. In the Hydro case instead, there is a growth only in the low chromosphere up to approximately z ≈ 1200 km. Above this height, the swirls become weaker again. Clearly, magnetic fields favor the generation of swirls. However, it seems that too strong magnetic fields can also suppress the formation of strong vortices. We surmise that there exists an optimal value of the initial magnetic field for which the strength of the vortices is maximal. Fig. A.2. Distribution of the inclination cos( θ ) of the swirling strength vector as a function of altitude z for three different simulations: ( a ) MHD simulation with an initial vertical, homogeneous magnetic field of 200 G, ( b ) MHD simulation with an initial field of 50 G, and ( c ) purely hydrodynamical simulation. The inclination θ refers to the angle between the swirl axis and the vertical direction. Horizontally oriented swirls are binned close to cos( θ ) = 0, while vertically oriented swirls are close to cos( θ ) = 1. The maximum value of swirling strength λ max at each height z , for each simulation, and with a minimum of 10 occurrences per bin, is shown in panel d . The
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