Date of Award

May 2024

Degree Type


Degree Name

Master of Science



First Advisor

Vince Larson

Committee Members

Vince Larson, David Spade, Peter Hinow


Atmospheric modeling, CLUBB, Gaussian mixture, Normal mixture, Relationships of moments


The Cloud Layers Unified By Binormals (CLUBB) model uses the sum of two normal probability density function (pdf) components to represent subgrid variability within a single grid layer of an atmospheric model. This binormal approach, while computationally efficient, restricts the model’s ability to capture the full spectrum of potential shapes encountered inreal-world atmospheric data.

This thesis proposes to introduce a third normal pdf component strategically positioned between the existing two, significantly enhancing the model’s representational flexibility. This trinormal representation allows for a wider range of grid-layer shapes while permitting analytic solutions for certain higher order moments.

The core of this work lies in deriving the necessary mathematical transformations for incorporating the third normal pdf seamlessly into the CLUBB framework. This thesis lists all formulas, inputs, and outputs associated with the extended model as well as gives an outline on how to check those equations. Additionally, it describes certain asymptotic behavior ofthe trinormal pdf under various parameter settings.