Atmos Model

This page provides a summary of a specific type of balance law within the ClimateMachine source code, the AtmosModel. This documentation aims to introduce a user to the properties of the AtmosModel, including the balance law equations and default model configurations. Both LES and GCM configurations are included.

Conservation Equations

The conservation equations specific to this implementation of AtmosModel are included below.


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


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


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


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

Precipitating Species

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

Tracer Species

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

Equation Abstractions

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

State Variables

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


Nondiffusive Fluxes

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

Diffusive Fluxes

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

Radiation Fluxes

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

Fluxes of precipitating species

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


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