Atmos Model

\[\frac{\partial \rho}{\partial t} + \nabla\cdot (\rho\vec{u}) = \rho \mathcal{\hat S}_{q_t}.\]

\[\frac{(\partial \rho\vec{u})}{\partial t} + \nabla\cdot \left[ \rho\vec{u} \otimes \vec{u} + (p - p_r) \vec{I}_3\right] = - (\rho - \rho_r) \nabla\Phi - 2\vec{\Omega} \times \rho\vec{u} \\ - \nabla\cdot (\rho \vec{\tau}) - \nabla\cdot\left( \vec{d}_{q_t} \otimes \rho\vec{u} \right) + \nabla\cdot \left( q_c w_c \vec{\hat k} \otimes \rho \vec{u} \right) + \rho \vec{F}_{\vec{u}}\]

\[ \frac{\partial(\rho e^\mathrm{tot})}{\partial t} + \nabla\cdot \left( (\rho e^\mathrm{tot} + p)\vec{u} \right) = -\nabla\cdot (\rho \vec{F}_R) - \nabla\cdot \bigl[\rho (\vec{J} + \vec{D})\bigr] + \rho Q \\ +\nabla\cdot \left(\rho W_c \vec{\hat k} \right) - \nabla\cdot (\vec{u} \cdot \rho\vec{\tau)} %+ \rho \vec{u} \cdot \vec{F}_{\vec{u}} \\ - \sum_{j\in\{v,l,i\}}(I_j + \Phi) \rho C(q_j \rightarrow q_p) - M\]

\[\frac{\partial (\rho q_t)}{\partial t} + \nabla\cdot (\rho q_t \vec{u}) = \rho \mathcal{S}_{q_t} - \nabla\cdot (\rho \vec{d}_{q_t}) + \nabla\cdot \bigl(\rho q_c w_c \vec{\hat k} \bigr) \equiv \rho \mathcal{\hat S}_{q_t}\]

\[\frac{\partial (\rho q_{p,i})}{\partial t} + \nabla\cdot \left[\rho q_{p,i} (\vec{u} - w_{p,i} \vec{\hat k}) \right] =\\ \rho \left[C(q_t \rightarrow q_{p,i}) + C(q_{p,k} \rightarrow q_{p,i}) \right] -\nabla\cdot (\rho \vec{d}_{q_{p, i}})\]

\[\frac{(\partial \rho \chi_i)}{\partial t} + \nabla\cdot \left(\rho \chi_i \vec{u} \right) = \rho \mathcal{S}_{\chi_i} - \nabla\cdot (\rho \vec{d}_{\chi_i}) + \nabla\cdot (\rho \chi_{i} w_{\chi, i} \vec{\hat k})\]

\[\frac{\partial \vec{Y}}{\partial t} = - \nabla \cdot (\vec{F}_{nondiff} + \vec{F}_{diff} + \vec{F}_{rad} + \vec{F}_{precip}) + \vec{S}\]

\[\vec{Y}=\left( \begin{array}{c} \rho \\ \rho\vec{u} \\ \rho e^{\mathrm{tot}}\\ \rho q_k\\ \rho q_{p,i}\\ \rho \chi_j \end{array} \right).\]

\[ \mathrm{F}_{nondiff}=\left( \begin{array}{c} \rho \vec{u} \\ \rho \vec{u} \otimes \vec{u} + (p - p_r) \vec{I}_3 \\ \rho e^{\mathrm{tot}} \vec{u} + p \vec{u}\\ \rho q_k \vec{u}\\ \rho q_{p,i} \vec{u} \\ \rho \chi_j \vec{u} \end{array} \right).\]

\[\mathrm{F}_{diff}=\left( \begin{array}{c} \rho\vec{d}_{q_t} \\ \rho\vec{\tau} + \rho\vec{d}_{q_t} \otimes \vec{u}\\ \vec{u} \cdot \rho\vec{\tau} + \rho (\vec{J} + \vec{D}) \\ \rho\vec{d}_{q_k}\\ \rho \vec{d}_{q_{p, i}}\\ \rho \vec{d}_{\chi_j} \end{array} \right).\]

\[\mathrm{F}_{rad} = \left( \begin{array}{c} \vec{0} \\ \vec{0} \\ \rho \vec{F}_R \\ \vec{0} \\ \vec{0} \\ \vec{0} \end{array} \right)\]

\[\mathrm{F}_{fall} = - \left( \begin{array}{c} \rho q_{c} w_{c} \vec{\hat k} \\ q_c w_c \vec{\hat k} \otimes \rho \vec{u} \\ \rho W_c \vec{\hat k} \\ \rho q_{k} w_{k} \vec{\hat k} \\ \rho q_{p,i} w_{p, i} \vec{\hat k} \\ \rho \chi_{i} w_{\chi, i} \vec{\hat k} \end{array} \right)\]

\[\mathrm{S}(\vec{Y}, \nabla\vec{Y})= \left( \begin{array}{c} -\rho C(q_t \rightarrow q_p) \\ -(\rho - \rho_r) \nabla\Phi - 2 \vec{\Omega} \times \rho\vec{u} + \rho \vec{F}_{\vec{u}} \\ \rho Q - \sum_{j\in\{v,l,i\}} (I_j + \Phi) \rho C(q_j \rightarrow q_p) - M \\% + \rho \vec{u} \cdot \vec{F}_{\vec{u}} \\ \rho C(q_p \rightarrow q_k) + \rho \sum_j \rho C(q_j \rightarrow q_k) \\ \rho \sum_k C(q_k \rightarrow q_{p, i}) - \rho \sum_j C(q_{p, i} \rightarrow q_{p, j})\\ \rho \mathcal{S}_{\chi_i} \end{array} \right)\]