Abstract
We model the appearance of solar active regions using percolation theory. Our motivation: magnetic fields of active regions presumably are released and rise from some deep site of the solar dynamo. We attempt to bundle all the very complicated magnetic phenomena into two dimensionless parameters. The main parameter is the probability, Pst, that the release and rise of one flux tube stimulates the subsequent release and rise of a neighboring flux tube. A second parameter measures the lifetime of flux once it has arrived at the surface. With this hypothesis we can reproduce several properties of the distribution of active regions on the Sun: (1) The active regions persist for a long time. Magnetic flux emerges mostly where there is flux already. (2) There are persistent empty regions, reminiscent of coronal holes. (3) The dependence on Pst is that of a phase transition. For example, for Pst near a critical value, a change in Pst by merely 1% can change the area covered by active regions by a factor of 10, comparable to the change observed between cycle minima and maxima. Surface activity provides a highly amplified signal of small changes at the site of the dynamo during the solar cycle. (4) The size distribution of the active regions is close to exponential, as observed.