A dimensionless parameter to predict spontaneous convective aggregation onset in a idealized stochastic-diffusive model of radiative-convective equilibrium

Cloud resolving models run to equilibrium in idealized simulations of radiative-convective equilibrium often show the deep convection spontaneously transitioning from random organization to a state where convection in aggregated into clusters.  This results in a drier mean state and aggregation could be important for climate sensitivity, and is missing in classical parameterization schemes.  However, the onset and nature of the equilibrium, and the sensitivity to lower boundary conditions, differs dramatically between models that use different representations of moist physics and diffusion parameterizations, and varying dynamical cores.  In order to shed light on this, we develop a highly idealized, spatially explicit, stochastic reactive-diffusive model for the column relative humidity in the tropics.

The model is run to equilibrium and it is found that, depending on the model parameter settings and experimental framework, it can produce equilibrium states where the convection remains random, or states where the convection is highly aggregated. Many results of the full-physics models are reproduced, such as their sensitivity to model resolution and domain size, with aggregation more likely using coarse grid sizes and larger domains. The simple model thus allows to explain these sensitivities of the full physics models, with convective nearest-neighbor distances constrained to decrease with smaller domains or higher resolution, which reduces the spatial heterogeneity of column humidity and makes aggregation less likely.  Expanding on these arguments, we use dimensional analysis to combine the model parameters that describe how sensitive convection is to humidity, the subsidence drying rate, and the spreading of humidity by advection and diffusion processes, along with the domain size and resolution.  Using large ensembles of over 1000 simulations, we demonstrate that aggregation occurs at a precise critical value of the resulting dimensionless parameter, which will refer to as the aggregation number.   We suggest that the aggregation number could prove useful to diagnose the differences between full physics models of the atmosphere.