  Clouds play a central role in climate physics by interacting with
precipitation, radiation, and circulation. Although the self-aggregation of
clouds is a fundamental problem in convective organization, a theoretical
explanation of how it occurs has not been established owing to its complexity.
Here, we introduce an idealized mathematical model of the phenomenon in which
the state of the system is represented solely by the atmospheric columns'
vertically integrated water vapor content. By analyzing the nonlinear dynamics
of this simplified system, we mathematically elucidated the mechanisms that
determine the onset of self-aggregation and the spatial scale of the
self-aggregated state. Nonlocal coupling between atmospheric columns makes the
system bistable with dry and moist equilibria, reflecting the effect of
circulation driven by horizontal differential heating due to convection and
radiation. The bistable self-aggregated state is realized when destabilization
by nonlocal coupling triggered by finite-amplitude disturbances in the uniform
state overwhelms the stabilization by diffusion. For globally coupled systems
in which all the columns are equally coupled, the perturbation of the maximum
wavelength has the maximum growth rate. A solution with an infinitely long
wavelength exists, which can be understood as the dynamical system's
heteroclinic trajectories describing the steady state's spatial evolution. In
contrast, for nonlocally coupled systems with finite filter lengths,
perturbation of the wavelength close to the characteristic length of the
coupling is preferred. The results revealed that the balance between nonlocal
coupling and diffusion is essential for understanding convective
self-aggregation.
