# Mathematical Models for Cloud Formation

#### Authors: Benjamin Lees, Jorge Lindley, Faizan Nazar, Huan Wu

Supervisors: Kody Law, James Robinson

Cloud simulation is an important aspect in weather prediction, and is an ongoing and complex field. In this project we adapt a model from [Yamamoto (2008)]. In the original paper, the model is used to simulate clouds for computer graphics in which they would use appropriate forcing to shape the clouds into a desired shape. We seek existence and uniqueness results for the model and we apply two different numerical techniques to generate some simulations.

Clouds form when the density of the water vapour in the air reaches a critical point and has to condense out, often onto airborne particles such as dust, soot or sea salt called cloud condensation nuclei. This saturation density can be described using the Clausius-Clapeyron equation [Rogers & Yau (1989)], one approximated form can be found in the *q _{s}* term of our model.

As a preliminary investigation we considered the scenario described by the simple diagram of cloud formation taken from [Yamamoto (2008)]. In Figure (a) a moist air parcel is heated up by the ground, Sun or some other heat source. In Figure (b) the air begins to rise above the cooler surrounding air and expands to match the ambient pressure. In Figure (c) the air parcel reaches the saturation height and water vapour begins to condense out into water droplets to form a cloud. Using the Clausius-Clapeyron equation with a standard formula for how air pressure varies in the atmosphere we can compute the height at which a cloud forms from a rising parcel of warm, moist air.

#### Acknowledgements

We acknowledge and thank the help of our supervisors Dr Kody Law and Prof James Robinson.

We also acknowledge the funding body EPSRC and the support from MASDOC CDT.

Contact: b.lees at warwick.ac.uk, j.v.m.lindley at warwick.ac.uk, f.q.nazar at warwick.ac.uk, huan.wu at warwick.ac.uk