  The time-dependent Ginzburg-Landau equation and the Swift-Hohenberg equation,
both added with a stochastic term, are proposed to describe cloud pattern
formation and cloud regime phase transitions of shallow convective clouds
organized in mesoscale systems. The starting point is the Hottovy-Stechmann
linear spatio-temporal stochastic model for tropical precipitation, used to
describe the dynamics of water vapor and tropical convection. By taking into
account that shallow stratiform clouds are close to a self-organized
criticallity and that water vapor content is the order parameter, it is
observed that sources must have non-linear terms in the equation to include the
dynamical feedback due to precipitation and evaporation. The inclusion of this
non-linearity leads to a kind of time-dependent Ginzburg-Landau stochastic
equation, originally used to describe superconductivity phases. By performing
numerical simulations, pattern formation is observed, specially for cellular
convective phases. These patterns are much better compared with real satellite
observations than the pure linear model. This is done by comparing the spatial
Fourier transform of real and numerical cloud fields. Finally, by considering
fluctuation terms for the turbulent eddy diffusion we arrive to a
Hohenberg-Swift equation. The obtained patterns are much more organized that
the patterns obtained from the Ginzburg-Landau equation in the case of closed
cellular and roll convection.
