Figure 1: Map of G_{A} (Eq. (13)) for a single magnetic region executing a simple translational motion towards the right (arrow). The normal magnetic field component B_{n} is uniform. The grey levels shows the strength of G_{A} with middle grey being 0, lighter grey positive, and darker grey negative (a color version is available in the electronic version at http://www.epsciences.org with red/blue coding the positive/negative values). The computation has been done with U_{0} R B_{0}^{2}=2 (Eq. (30)). | |
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Figure 2: Maps of G_{A} and (Eqs. (13) and (19)), top and bottom panel respectively, for two magnetic regions of opposite polarity executing a simple translational motion away from each other (as indicated by arrows). B_{n} is uniform in both magnetic polarities. Both G_{A} and have two polarities in each magnetic region but with opposite sign and different magnitude ( is lower by about a factor 10, see Fig. 4). The shading convention is the same than in Fig. 1 (a color version is available at http://www.edpsciences.org). The values used are: U_{0} R B_{0}^{2}=3 and D/R=10/3. | |
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Figure 3: Sketch of the general connectivity of two field lines, a and c. The field line a stretches from , where the magnetic field flux is , to with flux . Similarly, the footpoint positions of the c field line, and , are respectively associated with a positive flux and a negative one . Conservation of the magnetic flux along the flux tubes gives: and . | |
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Figure 4: Total positive fluxes for G_{A} and for two separating magnetic regions as in Fig. 2. The total fluxes (with G_{A}>0, Eq. (39)), (with , Eq. (40)) and their ratio are given in function of the separation distance D normalized to the radius R of the magnetic regions (for D/R=2 the magnetic regions are in tangential contact). As D/R increases, decreases rapidly towards zero, while saturates (giving the case of Fig. 1 for each magnetic region). The fluxes are drawn with U_{0} B_{0}^{2} R^{3} = 27. | |
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Figure 5: Top panel: different profiles of as described by Eq. (41). Bottom panel: ratio of (Eq. (39)) with (Eq. (40)) for the different B_{n} profile in function of the separation distance D normalized to the radius R of the magnetic regions. The ratio is larger as the field is more concentrated to the centre of the magnetic region. | |
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Figure 6: Maps of G_{A} (Eq. (13), top panel) and (Eq. (19), bottom panel) in the case of one magnetic polarity (P_{-}) having a solid rotation around another one (P_{+}) and injecting positive magnetic helicity (motions are indicated with arrows). B_{n} is uniform in each magnetic region. G_{A} is strongly bipolar while has no negative value. The shading convention is the same as in Fig. 1 (a color version is available at http://www.edpsciences.org). | |
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Figure 7: Total positive and negative helicity flux for G_{A} and total positive flux for (no negative values) in the case of one magnetic region having a solid rotation around another one (Fig. 6). The abscissa is the relative distance D/R as in Fig. 5. The fake flux, both positive and negative, given by G_{A} is of comparable magnitude to the real flux ( ). | |
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Figure 8: Rows from top to bottom: maps of G_{A}, , (Eqs. (13), (19) and (28) with f=1/2) in the case of the emergence of a twisted flux tube (as defined in Sect. 4.4). The helicity flux densities are presented at four different times with the flux tube rising from left to right. The number of turns in half the torus is N=0.1, and the aspect ratio . The shading convention is the same as in Fig. 1 with a range which depends on the panel (a color version is available at http://www.edpsciences.org). The continuous (dashed) lines are positive (negative) isocontours of B_{n}, respectively. | |
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Figure 9: Total positive and negative helicity flux for G_{A}, , and in the case of the emergence of a twisted flux tube with N=0.1 (Fig. 8). The abscissa is the relative height, Z, of the central part of the torus, above the "photosphere''. The curves start (on the left) when the top of the torus crosses the "photosphere'' ( ) and end (right) when the torus is half emerged (Z=1). The thin vertical dotted lines correspond to the height when the top of the tube (its central cross section) is half emerged (Z=0) and completely emerged ( ). | |
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Figure 10: Dependence on R of the Fake Relative Fluxes, and (ratio between the fake and the real helicity flux for , Eq. (47)). The curves show two values of the number of turns N: 0.03 and 0.1. | |
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Figure 11: Dependence on the number of turns N of the Fake Relative Fluxes ( and , Eq. (47)) for two values of the small radius R: 1 and 4 (the main radius is ). | |
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Figure 12: Maps of (Eq. (19)) in the case of an emerging twisted flux tube for different number of turns N of the field lines. From left to right, is drawn respectively with N=0.2, N=1 and N=6. Themaps are for with . The arrows represent the horizontal motions of the footpoints of the field lines (Eq. (7)).The shading convention is the same as in Fig. 1 with a range which depend on the panel (a color version is available at http://www.edpsciences.org). The continuous (dashed) line represents one isocontour of B_{n}>0 (B_{n}<0), respectively. | |
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