Analysis of the flux growth rate in emerging active regions on the Sun
  We studied the emergence process of 42 active region (ARs) by analyzing the
time derivative, R(t), of the total unsigned flux. Line-of-sight magnetograms
acquired by the Helioseismic and Magnetic Imager (HMI) onboard the Solar
Dynamics Observatory (SDO) were used. A continuous piecewise linear fitting to
the R(t)-profile was applied to detect an interval, dt_2, of nearly-constant
R(t) covering one or several local maxima. The averaged over dt_2 magnitude of
R(t) was accepted as an estimate of the maximal value of the flux growth rate,
R_MAX, which varies in a range of (0.5-5)x10^20 Mx hour^-1 for active regions
with the maximal total unsigned flux of (0.5-3)x10^22 Mx. The normalized flux
growth rate, R_N, was defined under an assumption that the saturated total
unsigned flux, F_MAX, equals unity. Out of 42 ARs in our initial list, 36 event
were successfully fitted and they form two subsets (with a small overlap of 8
events): the ARs with a short (<13 hours) interval dt_2 and a high (>0.024
hour^-1) normalized flux emergence rate, R_N, form the "rapid" emergence event
subset. The second subset consists of "gradual" emergence events and it is
characterized by a long (>13 hours) interval dt_2 and a low R_N (<0.024
hour^-1). In diagrams of R_MAX plotted versus F_MAX, the events from different
subsets are not overlapped and each subset displays an individual power law.
The power law index derived from the entire ensemble of 36 events is
0.69+-0.10. The "rapid" emergence is consistent with a "two-step" emergence
process of a single twisted flux tube. The "gradual" emergence is possibly
related to a consecutive rising of several flux tubes emerging at nearly the
same location in the photosphere.
