  This work is intended to be a contribution to the study of the morphology of
the rising convective columns, for a better representation of the processes of
entrainment and detrainment. We examine technical methods for the description
of the interface of expanding clouds and reveal the role of \emph{fingering}
instability which increases the effective length of the periphery of the cloud.
Assuming Laplacian growth we give a detailed derivation of the time-dependent
conformal transformation that solves the equation of the \emph{fingering}
instability. For the phase of slower expansion, the evolution of complex poles
with a dynamics largely controlled by the Hilbert operator (acting on the
function that represents the interface position) leads to \emph{cusp}
singularities but smooths out the smaller scale perturbations.
  We review the arguments that the rising column cannot preserve its integrity
(seen as compacity in any horizontal section), because of the penetrative
downdrafts or the incomplete repulsion of the static environmental air through
momentum transfer. Then we propose an analytical framework which is adequate
for competition of two distinct phases of the same system.
  The methods exmined here are formulated in a general framework and can be
easily adapted to particular cases of atmospheric convection.
