Online learning of a machine learning cloud fraction closure in a hybrid earth system model

Modeling cloud fraction accurately is essential to closing the radiation budget in climate models where top-of-atmosphere root mean square error biases are still several times larger than the global warming signal. Traditional parameterizations diagnostically compute cloud fraction from simple functions or parametric distributions of grid-scale environmental state variables, such as humidity, temperature, pressure, and cloud water and ice mixing ratios fit offline to data. However, these schemes are often too simple to fully capture the complexity of these interactions at the sub-grid scale. While recent offline machine learning (ML) parameterizations have substantially outperformed traditional models by capturing unparametrized nonlinearity, they need to be embedded stably online and re-tuned to observational data to realize these accuracy benefits in a modeling setting. Here, we present an ML-based cloud fraction parameterization calibrated online to satellite data. First, we train several parametrically dense baseline ML models offline to a library of large eddy simulations (LES), including a neural network and a random forest, to serve as a benchmark. Over the current parameterization, the neural network and random forest yield error reductions of 94 and 95% respectively. We then use symbolic regression to generate a suite of equations and choose a closure that achieves parsimony between model accuracy and parameter number. We embed the new parameterization into the CliMA atmosphere model, CliMAAtmos.jl, a new GPU-compatible hybrid atmosphere model optimized for calibrating parametrically sparse physics- and ML-based closures to observations. Then, using ensemble Kalman inversion, we re-calibrate the parameterization online to MODIS cloud fraction measurements in ClimaAtmos.jl. This work illustrates a framework that leverages the iterability of offline machine learning to design a closure that can be re-tuned for greater accuracy online, providing a template for future hybrid model development.