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The Model

We use the following model adapted from [Yamamoto (2008)]. Here, we have u is the velocity of the air, T is the temperature and qv and qc are the densities of water vapour and condensed water, respectively.

\begin{align*} \frac{\partial \mathbf{u}}{\partial t} \,\,\,\,\, &= \quad \, - (\mathbf{u} \cdot \nabla) \mathbf{u} + \nu \Delta \mathbf{u} - \frac{1}{\rho} \nabla p + \mathbf{B} \\ \nabla \cdot \mathbf{u} \,\,\,\, &= \quad \quad \, 0 \\ \mathbf{B} \,\,\,\,\,\,\,\, &= \quad \,\left( \frac{ \, T}{T_{amb}} - \Big( 1 + \frac{q_{c}}{\rho} \Big) \right) g \mathbf{z} \\ \frac{\partial q_{c}}{\partial t} \,\,\,\,\, &= \quad \, - (\mathbf{u} \cdot \nabla) q_{c} + C_{c} \\ \frac{\partial q_{v}}{\partial t} \,\,\,\,\, &= \quad \, - (\mathbf{u} \cdot \nabla) q_{v} - C_{c} \\ C_{c} \,\,\,\,\,\, &= \quad \, \alpha ( q_{v} - q_{s} ) \\ q_{s} \,\,\,\,\,\,\, &= \quad \, \text{min} \left( \, \frac{A}{R_{v}T} \, \text{exp}\Big( - \frac{B}{T + C} \Big), q_{v} + q_{c} \right) \\ \frac{\partial T}{\partial t} \,\,\,\,\, &= \quad \, - (\mathbf{u} \cdot \nabla) T + \mu \Delta T+ QC_{c} \end{align*}

In this model we have the Navier-Stokes equation governing the air flow which is forced by a buoyancy force B generated by the cloud formations. We then have equations governing the advection of the vapour and cloud densities which are sourced by the Cc term. This term controls the exchange between water vapour and condensed water ensuring the sum is conserved. The saturation density is included in the qs term to facilitate the condensation of water vapour.

[Yamamoto (2008)] Y. Dobashi, K. Kusumoto, T. Nishita, T. Yamamoto, Feedback Control of Cumuliform Cloud Formation based on Computational Fluid Dynamics, ACM Transactions on Graphics (2008)