All Figures

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f01.eps}\end{figure}$ Figure 1: Field lines starting in the outer vortex area for a) the double-dipole field used in the present paper with dipole position y₀=1 and b) the single-dipole field used by Amari et al. (1996b). The plot shows field lines that start on the y axis in the range between the vortex centre and the outer point where the velocity has fallen to 0.1 of the peak value.
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In the text

$\begin{figure} \par\resizebox{18cm}{!} % {\includegraphics[clip]{h4206f02.eps} } \end{figure}$ Figure 2: a) Imposed velocity field for y₀=1. b) Paths of fluid elements in the upper vortex for v₀=10^-2 after $75\tau _{\rm a}$ . The vortex centre is located at $(0,1.03,-\Delta z)$ . The thick arrows indicate the positions where two of the magnetic field line bundles in Fig. 3 start from. c) Profiles of the vortex $u_x(0,y,-\Delta z)$ ( thick line) and of the normal field component B_0z(0,y,0) ( thin line).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f03.eps}\end{figure}$ Figure 3: Different evolution of the three groups of magnetic field lines that are directly influenced by the vortex motions: "central field lines'' (starting in the vicinity of a vortex centre) form the rising and expanding twisted flux tube; "inner field lines'' (initially starting between the vortex centres) form a weakly expanding inverse-S-shaped structure; and "outer field lines'' (initially starting beyond a vortex centre) show the strongest rise and expansion and a forward S shape. The reference run (y₀=1, $\rho _0=B_0^2$ , and v₀=10^-2) is shown at a) t=0 and b) t=75. The start regions of the selected inner and outer field lines are marked by the heavy arrows in Fig. 2b.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f04.eps}\end{figure}$ Figure 4: Evolution of height, length, volume, and apex rise velocity of the central field line of the twisted flux tube for the run shown in Fig. 3.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f05.eps}\end{figure}$ Figure 5: Velocity along the z axis at different times for the run shown in Fig. 3. The times correspond to the datapoints in Figs. 4a-c. Plus signs indicate the position of the respective apex heights of the central field line.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f06.eps}\end{figure}$ Figure 6: Evolution of a) magnetic energy; b) current through the half plane $\{x,y>0,0\}$ ( plus signs: main current, asterisks: return current, diamonds: total current); c) and d) peak current density and peak Lorentz force density, respectively, in the whole box for the run shown in Fig. 3.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f07.eps}\end{figure}$ Figure 7: Isosurfaces $\vert\vec{j}\vert=0.2~j_{\rm max}$ (left) and $\vert\vec{j}\vert=0.5~j_{\rm max}$ (right) for the run shown in Fig. 3 at t=75.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f08.eps}\end{figure}$ Figure 8: $\vert\vec{j}\times\vec{B}\vert/\vert\vec{j}\vert\vert\vec{B}\vert$ along the central field line at t=54 ( solid), t=75 ( dotted), and t=102 ( dashed) for the run shown in Fig. 3.
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In the text

$\begin{figure} {\includegraphics[width=12cm,clip]{h4206f09.eps} } \end{figure}$ Figure 9: Twist of the central flux tube for the run shown in Fig. 3. a) Injected global (end-to-end) twist as a function of footpoint radius $r_{\rm foot}$ at t=75. b) Evolution of the twist at the apex of the flux tube with $r_{\rm foot}=0.01$ ( asterisks) and $r_{\rm foot}=0.1$ ( diamonds). Also shown are the global twist ( dashed), $\alpha L/\Phi _{\rm a}$ at the apex ( triangles), and the (scaled) apex height, 0.1h/h₀ ( solid line).
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In the text

$\begin{figure} \includegraphics[width=12cm,clip]{h4206f10.eps} \end{figure}$ Figure 10: Evolution of apex height and rise velocity of the central field line for $\rho _0=B_0^2$ , v₀=10^-2, and different dipole half distances: y₀=0.5 ( plus signs); y₀=1 ( diamonds); y₀=2 ( asterisks). h₀=0.67, 1.22, 3.24 for y₀=0.5, 1, 2, respectively.
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In the text

$\begin{figure} \includegraphics[width=12cm,clip]{h4206f11.eps} \end{figure}$ Figure 11: Evolution of apex height and rise velocity of the central field line for y₀=1, v₀=10^-3, and different initial density profiles: $\rho_0=\vert\vec{B}_0\vert$ ( plus signs); $\rho _0=B_0^2$ ( asterisks).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f12.eps}\end{figure}$ Figure 12: Velocity along the z axis at different times for the run with y₀=1, v₀=10^-3, and $\rho_0=\vert\vec{B}_0\vert$ . The plus signs indicate the position of the respective apex heights of the central field line.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{h4206f13.eps}\end{figure}$ Figure 13: Apex heights of the continuously twisted or relaxed flux tube during the quasi-static phase of the runs with y₀=1. Plus signs: continuously twisted, $\rho_0=\vert\vec{B}_0\vert$ , v₀=10^-2; diamonds: continuously twisted, $\rho _0=B_0^2$ , v₀=10^-2; asterisks: continuously twisted, $\rho _0=B_0^2$ , v₀=10^-3; squares: relaxed states obtained from the run with $\rho_0=\vert\vec{B}_0\vert$ , v₀=10^-2; and triangles: relaxed states from the run with $\rho _0=B_0^2$ , v₀=10^-3. (The uppermost triangle does not represent a relaxed state, see text for details.)
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f14.eps}\end{figure}$ Figure 14: Force-free parameter $\alpha$ along the central field line in the runs with y₀=1 at a global twist of $\Phi _{\rm g}\approx 2.5\pi$ . Dot-dashed: continuously twisted, $\rho_0=\vert\vec{B}_0\vert$ , v₀=10^-2; solid: continuously twisted, $\rho _0=B_0^2$ , v₀=10^-2; dotted: continuously twisted, $\rho _0=B_0^2$ , v₀=10^-3; dashed: after relaxation from $\rho _0=B_0^2$ , v₀=10^-3.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f15.eps}\end{figure}$ Figure 15: Relaxation runs near the critical twist, started from a system with y₀=1 that was twisted using $\rho _0=B_0^2$ and v₀=10^-3. a) Apex height of twisted flux tube and b) total Lorentz force in the box during relaxation at $\Phi _{\rm g}=2.53\pi$ ( triangles), $2.75\pi$ ( squares), and $2.95\pi$ ( asterisks). The apex height for $\Phi _{\rm g}=2.95\pi$ is scaled to fit into the figure.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{h4206f16.eps}\end{figure}$ Figure 16: Lorentz force density at the z axis for relaxation runs withy₀=1, $\rho _0=B_0^2$ , v₀=10^-3. Thin lines are used for the run at $\Phi _{\rm g}=2.53\pi$ ( t=721, 801, 1041), thick lines are used for the runat $\Phi _{\rm g}=2.95\pi$ ( t=841, 1004, 1935). Solid lines show the system at the start of the relaxation, dotted lines show intermediate states, and dashed lines show the end of the runs. The Lorentz force density changes sign along the z axis during the relaxation at $\Phi _{\rm g}=2.53\pi$ , so |f_z(z)| is plotted in this case for $t\ge 801$ . Crosses mark the respective apex heights of the twisted flux tube.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{h4206f17.eps}\end{figure}$ Figure 17: Apex height as a function of global twist for the runs with y₀=1, $\rho _0=B_0^2$ , and v₀=10^-2 ( dashed) and v₀=10^-3 ( solid). Triangles show apex heights at the end of relaxation runs for states calculated from the run with v₀=10^-3; the final two datapoints show still rising states, which were not relaxed at termination of the runs.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{h4206f18.eps}\end{figure}$ Figure 18: Comparison of the apex heights of the twisted flux tube after (or at termination of) relaxation (symbols as in Figs. 13 and 17) with the rise characteristics $h\propto\exp({\rm const.}\cdot\Phi_{\rm g}^2)$ of axially symmetric configurations from Sturrock et al. (1995). A least-squares fit through the first eight datapoints is shown.
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In the text

$\begin{figure} \par\includegraphics[width=7.2cm,clip]{h4206f19.eps}\end{figure}$ Figure 19: Field lines of the magnetic field ( solid) and current density ( dotted) in the central flux tube projected onto the bottom plane for y₀=1, $\rho _0=B_0^2$ , and v₀=10^-3. a) Driven phase, $\Phi _{\rm g}=2.53\pi$ (t=721). b) Driven phase, $\Phi _{\rm g}=2.95\pi$ (t=841; start of relaxation). c, d) Two states in relaxation run at $\Phi _{\rm g}=2.95\pi$ (t=1065 and t=1785).
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{h4206f20.eps}\end{figure}$ Figure 20: Contours of $\vert\vec{j}(z\!=\!0)\vert$ for the run with y₀=1, $\rho _0=B_0^2$ , v₀=10^-3 after relaxation at $\Phi _{\rm g}=2.06\pi$ . Contours with $\alpha >0$ (j_z>0) are shown solid, contours with $\alpha <0$ (j_z<0) are shown dotted. (The contours of $\alpha$ in the bottom plane are similar to the contours of $\vert\vec{j}\vert$ , see Fig. 3 in Török & Kliem 2002.) B_z>0 in the whole area shown. The vortex centre is marked by a plus sign, the location of peak current density is marked by an asterisk. These are the start points of the two flux tubes shown in Fig. 21.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{h4206f21.eps}\end{figure}$ Figure 21: Projection onto the bottom plane of the flux tubes that extend from the vortex centre and from the area of maximum current density; their apex heights are 2.6 and 2.2, respectively. Field lines of $\vec{B}$ (solid) and $\vec{j}$ (dotted) are shown. Parameters are as in Fig. 20. The contours $\alpha =0$ (dashed) are included.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f01.eps}\end{figure}$	Figure 1: Field lines starting in the outer vortex area for a) the double-dipole field used in the present paper with dipole position y₀=1 and b) the single-dipole field used by Amari et al. (1996b). The plot shows field lines that start on the y axis in the range between the vortex centre and the outer point where the velocity has fallen to 0.1 of the peak value.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f04.eps}\end{figure}$	Figure 4: Evolution of height, length, volume, and apex rise velocity of the central field line of the twisted flux tube for the run shown in Fig. 3.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f05.eps}\end{figure}$	Figure 5: Velocity along the z axis at different times for the run shown in Fig. 3. The times correspond to the datapoints in Figs. 4a-c. Plus signs indicate the position of the respective apex heights of the central field line.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f06.eps}\end{figure}$	Figure 6: Evolution of a) magnetic energy; b) current through the half plane $\{x,y>0,0\}$ ( plus signs: main current, asterisks: return current, diamonds: total current); c) and d) peak current density and peak Lorentz force density, respectively, in the whole box for the run shown in Fig. 3.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f07.eps}\end{figure}$	Figure 7: Isosurfaces $\vert\vec{j}\vert=0.2~j_{\rm max}$ (left) and $\vert\vec{j}\vert=0.5~j_{\rm max}$ (right) for the run shown in Fig. 3 at t=75.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f08.eps}\end{figure}$	Figure 8: $\vert\vec{j}\times\vec{B}\vert/\vert\vec{j}\vert\vert\vec{B}\vert$ along the central field line at t=54 ( solid), t=75 ( dotted), and t=102 ( dashed) for the run shown in Fig. 3.
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$\begin{figure} \includegraphics[width=12cm,clip]{h4206f10.eps} \end{figure}$	Figure 10: Evolution of apex height and rise velocity of the central field line for $\rho _0=B_0^2$ , v₀=10^-2, and different dipole half distances: y₀=0.5 ( plus signs); y₀=1 ( diamonds); y₀=2 ( asterisks). h₀=0.67, 1.22, 3.24 for y₀=0.5, 1, 2, respectively.
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$\begin{figure} \includegraphics[width=12cm,clip]{h4206f11.eps} \end{figure}$	Figure 11: Evolution of apex height and rise velocity of the central field line for y₀=1, v₀=10^-3, and different initial density profiles: $\rho_0=\vert\vec{B}_0\vert$ ( plus signs); $\rho _0=B_0^2$ ( asterisks).
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{h4206f12.eps}\end{figure}$	Figure 12: Velocity along the z axis at different times for the run with y₀=1, v₀=10^-3, and $\rho_0=\vert\vec{B}_0\vert$ . The plus signs indicate the position of the respective apex heights of the central field line.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{h4206f17.eps}\end{figure}$	Figure 17: Apex height as a function of global twist for the runs with y₀=1, $\rho _0=B_0^2$ , and v₀=10^-2 ( dashed) and v₀=10^-3 ( solid). Triangles show apex heights at the end of relaxation runs for states calculated from the run with v₀=10^-3; the final two datapoints show still rising states, which were not relaxed at termination of the runs.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{h4206f18.eps}\end{figure}$	Figure 18: Comparison of the apex heights of the twisted flux tube after (or at termination of) relaxation (symbols as in Figs. 13 and 17) with the rise characteristics $h\propto\exp({\rm const.}\cdot\Phi_{\rm g}^2)$ of axially symmetric configurations from Sturrock et al. (1995). A least-squares fit through the first eight datapoints is shown.
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$\begin{figure} \par\includegraphics[width=8cm,clip]{h4206f21.eps}\end{figure}$	Figure 21: Projection onto the bottom plane of the flux tubes that extend from the vortex centre and from the area of maximum current density; their apex heights are 2.6 and 2.2, respectively. Field lines of $\vec{B}$ (solid) and $\vec{j}$ (dotted) are shown. Parameters are as in Fig. 20. The contours $\alpha =0$ (dashed) are included.
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