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 q_vand q_c 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 C_cterm. This term controls the exchange between water vapour and condensed water ensuring the sum is conserved. The saturation density is included in the q_s 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)