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Figure 1: Sequence of temperature profiles as a function of depth in the upper part of the model, showing the rapid inward propagation of a cooling front and the establishment of a largely stationary profile after a few hours. Zero depth corresponds to optical depth unity in the external stratification. The uppermost (thick full) curve shows the (time-independent) external temperature. The other curves show the temperature on the sunspot axis after 0.5 h (dotted curve), 1 h (dashed curve), 4 h (dash-dotted curve), 15 h (dash-triple-dotted curve), and 30 h (full curve), respectively, from the onset of radiative cooling (corresponding to the emergence of the flux tube at the surface).

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In the text

Figure 2: Snapshots from the time evolution of the magnetic field strength along the tube axis as a function of depth for the same case as shown in Fig. 1 (note the different depth range). The curves correspond to about 1 h (upper full curve), 10 h (dotted curve), 20 h (short-dashed curve), 30 h (dash-dotted curve), 35 h (dash-triple-dotted curve), and 40 h (long-dashed curve) after the start of radiative cooling (emergence). The lower, thick full curve gives the equipartition field strength with respect to the convective velocities from the mixing length model of the external stratification (Kiefer et al. 2000). After an initial general drop, the field strength increases again down to a depth of about 4 Mm while a local minimum of the field strength develops and moves downward until, after about 40 h, the field strength falls below the local equipartition value at at depth of 4.7 Mm.

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In the text

Figure 3: Profiles of the convective energy flux density (normalized by the undisturbed solar value) along the sunspot axis for the same instants of time as shown in Fig. 2 (note the different depth range). Above the Wilson depression, the energy transport is taken over by the radiative flux (not shown here). The convective transport affects progressively deeper layers in order to supply the (almost constant) surface energy flux density of about 22% of the undisturbed solar value. The corresponding downward extension of the superadiabatic stratification leads to a growth of the region that is stabilized against disconnection through the resulting increase of the magnetic field (cf. Fig. 2.

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In the text

	Figure 4: Disconnection time ( upper panel) and disconnection depth ( lower panel) as functions of the initial upflow velocity at the lower boundary. The mass influx is kept constant in time in the individual simulation runs (indicated by asterisks). The connecting curves represent spline interpolations.
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In the text

Figure 5: Disconnection depth vs. disconnection time for various model runs. Plus signs: runs with constant mass influx (the same runs as in Fig. 4); asterisks: runs with constant inflow velocity in the range 75-400 m s-1; diamonds: runs with different values for the initial pressure at the top, changing the initial mass in the flux tube by about 10%; triangles: runs with different values of the total magnetic flux (1020 Mx and Mx, respectively); crosses: runs for which the magnetic field was determined according to the pressure difference (approximation of thin flux tubes). The curve illustrates the downward progression of the zone with convective energy transport (cf. Fig. 3); it shows, as a function of time, the depth at which the convective energy flux density equals 10% of the undisturbed solar value, about half of the surface flux density of the model sunspot. Except for the largest disconnection depths (which may be affected by the proximity of the lower boundary of the model at 12.5 Mm), the points closely follow the curve, indicating that the disconnection depth is mainly determined by the downward progression of the convective cooling region.

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In the text