Bayesian estimation of the parameters of the nonlinear cloud and rain equation: finding physically relevant model parameters from large-eddy simulations.

Shallow clouds in the Earth system cover large regions of the oceans and are important for climate predictability. We study emergent behavior of marine stratocumulus cloud desks as they transition between weakly reflective open-cell structures, and highly reflective closed-cell structures. It was shown in previous work that open-cell/closed-cell transitions exhibit predator-prey type oscillations with the rain acting as a predator of clouds. Using this insight, a model of a single delay differential equation has been constructed. The parameters of the model describe the full environmental potential for cloud development, the cloud recovery time, the time delay for rain production, and the droplet concentration.

We are concerned with the question: how should one pick these parameters and how well are these parameters constrained by observations of a marine stratocumulus cloud system? We take a Bayesian approach and construct a prior distribution over the model parameters based on a linear stability analysis. The prior information is sharpened by a likelihood that is constructed by mapping model outputs to features derived from a large-eddy simulation (LES). The feature we use is a "typical" cloud oscillation. The feature is constructed by averaging several oscillations detected in the LES. A Gaussian error model is also derived using the oscillations extracted from the LES. Prior distribution and likelihood jointly define a posterior distribution over the model parameters. The posterior distribution describes our knowledge of the parameters given the model and the LES data.

The result is a simplified representation of the cloud cycles in terms of a scalar delay differential equation with stochastic parameters. The stochasticity is derived from the posterior distribution and describes the uncertainties in the model and LES data. We find a physically relevant distribution of parameters and an overall good fit of the calibrated model to the features derived from the LES data. These results provide new insight into complex cloud and rain interactions and might be useful for representing these systems in climate models.