Understanding the coupling between convective clouds and the general circulation, as well as addressing the gray zone problem in convective parameterization, requires insight into the genesis and maintenance of spatial patterns in cumulus cloud populations. In this study, a simple toy model for recreating populations of interacting convective objects as distributed over a two‐dimensional Eulerian grid is formulated to this purpose. Key elements at the foundation of the model include i) a fully discrete formulation for capturing discrete behavior in convective properties at small population sample sizes, ii) object age‐dependence for representing life‐cycle effects, and iii) a prognostic number budget allowing for object interactions and co‐existence of multiple species. A primary goal is to optimize the computational efficiency of this system. To this purpose the object birth rate is represented stochastically through a spatially aware Bernoulli process. The same binomial stochastic operator is applied to horizontal advection of objects, conserving discreteness in object number. The applicability to atmospheric convection as well as behavior implied by the formulation is assessed. Various simple applications of the BiOMi model (Binomial Objects on Microgrids) are explored, suggesting that important convective behavior can be captured at low computational cost. This includes i) subsampling effects and associated powerlaw scaling in the convective gray zone, ii) stochastic predator‐prey behavior, iii) the downscale turbulent energy cascade, and iv) simple forms of spatial organization and convective memory. Consequences and opportunities for convective parameterization in next‐generation weather and climate models are discussed.

中文翻译：

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对流人口离散统计有效建模的二项式随机框架

要了解对流云与总环流之间的耦合，并解决对流参数化中的灰色地带问题，就需要深入了解积云种群的空间格局的形成和维持。在这项研究中，为此目的建立了一个简单的玩具模型，用于重建分布在二维欧拉网格上的相互作用对流物体的种群。该模型基础的关键要素包括：i）完全离散的公式，用于捕获小样本数量的对流属性中的离散行为； ii）对象的年龄依赖性，以代表生命周期的影响； iii）预后数字预算物体相互作用和多种物种的共存。一个主要目标是优化该系统的计算效率。为此目的，通过空间感知的伯努利过程随机地表示物体的出生率。相同的二项式随机算子应用于对象的水平对流，从而保留了对象数量的离散性。评估了大气对流的适用性以及该配方所隐含的行为。探索了BiOMi模型的各种简单应用（微电网上的二项目标），这表明可以以较低的计算成本捕获重要的对流行为。这包括：i）对流灰色带中的二次采样效应和相关的幂律定标，ii）随机捕食者-猎物的行为，iii）湍流能量级联的下级，以及iv）空间组织和对流记忆的简单形式。