Geometrical aspects of the interaction between expanding clouds and environment

TL;DR

This paper investigates the geometrical behavior of expanding convective clouds, focusing on interface instabilities and their impact on cloud morphology, using conformal mapping and complex analysis to model fingering and cusp formation.

Contribution

It introduces a detailed analytical framework for modeling cloud interface instabilities, especially fingering, using Laplacian growth and conformal transformations, advancing understanding of cloud-environment interactions.

Findings

Fingering instability increases the cloud's effective perimeter.

Complex poles dynamics lead to cusp singularities.

The rising column cannot maintain integrity due to environmental interactions.

Abstract

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.