### 8.3 Post-emergence evolution of subsurface fields

This is so far a largely unexplored area with mostly speculations and very few quantitative calculations. One outstanding question is whether and to what extent sunspots and active region fields at the surface remain connected to the subsurface flux tubes in the deep solar convection zone. Clearly the answer to this question is intimately related to understanding the process of sunspot formation from rising flux tubes as described in

If the surface fields remain connected to the subsurface flux tubes, then based on the shape of the -loop as shown in Figure 17, it seems difficult to understand why the separation of the two polarities of a solar active region stops at the scales of  100 Mm, without continuously increasing to at least the footpoint separation of the -loop at the base of the convection zone, which is  1000 Mm! Due to the angles between the legs of the -loop and the surface, the magnetic tension of the legs is expected to be pulling the two poles of the bipolar region apart until the legs become vertical. However there is also an attracting force between the two poles due to the emerged magnetic field above the photosphere. If we assume that the field above the photosphere is potential, then the attracting force of the two poles due to the field above can be estimated by considering the attraction between two opposite electric charges of , where and is the total magnetic flux of the active region. This yields for the attracting force, where is the separation between the poles. On the other hand, the lateral force from the subsurface field that acts to pull the two poles apart can be estimated by integrating over the area of each pole the Maxwell stress exerted by the subsurface field from below. This leads to an estimated lateral force of , where and are the photosphere normal field strength and area of each magnetic pole respectively (note that we have ), and is the angle between the slanted subsurface tube and the surface. Balancing the forces from above and below the photosphere (taking a representative value of  = 45°), we find that it is possible for the two poles to reach a lateral force balance if . This suggests that the two poles of the bipolar region need to be very close, with the polarity separation no greater than the radius of each pole, in order for the attracting force from above to balance the force that drives the separation from below. Furthermore it appears that the force balance is an unstable equilibrium because of the dependence of the attracting force which weakens in response to an increase in .

One possible resolution of the above problem is that some other processes that can form -loops of much shorter length scale compared to the undulatory buoyancy instability (Section 4.1) is responsible for the formation of active region emerging tubes, although it is not immediately clear what these processes might be. Another alternative is that the active region magnetic fields on the photosphere become dynamically disconnected from the interior flux tubes. Fan et al. (1994) speculated on the process of “dynamic disconnection” which has the same physical cause as that of the so called flux tube “explosion” described in Section 7. It can be seen from Figure 33, that even for an emerging flux loop with a field strength at the convection zone base that is as high as 105 G, the flux tube is expected to lose pressure confinement and hence “explode” at a height of about 10 Mm below the surface as a result of plasma establishing hydrostatic equilibrium (HE) along the tube. While a flux loop is rising, especially in the top few tens of Mm of the convection zone where the rise speed becomes comparable to the Alfvén speed, the variation with depth of the magnetic field strength can deviate significantly from the conditions of HE. However, after flux emergence at the surface, the plasma inside the submerged flux tube will try to establish HE. Tube plasma near the surface can cool through radiation and probably undergoes “convective collapse”, forming sunspots and pores (see Spruit and Zweibel, 1979Stein and Nordlund, 2000) at the photosphere. However in deeper layers where plasma evolution is still nearly adiabatic, a catastrophic weakening of the magnetic field or flux tube “explosion” (Section 7) may occur, which is caused by an upflow of high entropy plasma as it tries to establish HE along the tube. This weakening of the field leads to the effective dynamic disconnection of the active region fields on the photosphere from the interior flux tubes. Fan et al. (1994) suggested that “dynamic disconnection” can explain

1. why the active region tilt persists once an active region has formed (see more recent work by Longcope and Choudhuri, 2002) and
2. why the 2D surface flux transport models based on diffusion by supergranular motion (see Wang et al., 1989) can explain well the observed dispersal of active regions and the migration of active region flux across latitudes.

A first quantitative calculation of the above process of “dynamic disconnection” has been carried out by Schüssler and Rempel (2005), using a 1-D self-similar vertical flux tube model whose top end has reached the photosphere. The model computes the quasi-static evolution of the flux tube under the effects of radiative cooling, convective energy transport, and a pressure buildup by a prescribed inflow at the bottom of the tube due to the high entropy tube plasma flowing upward to establish hydrostatic equilibrium along the field. The calculation shows that after emergence, the radiative losses near the surface drives an inward propagating cooling front accompanied by a downflow, which leads to a decrease of the gas pressure and an intensification of the magnetic field in the surface layers. In the mean time, the convergence of the radiative cooling driven downflow and the buoyancy driven upflow results in an increase of the gas pressure and a weakening of the magnetic field in the tube at a depth of a few Mms. It is found that for a reasonable range of the upflow speed, the magnetic field weakens to fall below the local equipartition value (i.e. dynamic disconnection takes place) at a depth between 2 and 6 Mm, in a time scale of up to 3 days. This is consistent with the time scale over which the evolution of a newly emerging active region changes from an “active phase” of growth with increasing polarity separation to a “passive phase” which can be well represented by transport by near surface flows and supergranular diffusion.