\right] }{\partial r} 
=
- \rho g 
+ \frac{2 p }{r} ,$$ $$\label{eqn:main_energy}
\begin{aligned}
\frac{\partial e }{ \partial t} &
+ \frac{1}{r^2} \frac{\partial \left[ r^2 v \left( e + p \right) \right] }{\partial r} 
= - \rho v g + Q_{\mathrm{h}} - Q_{\mathrm{c}} \\
 & + \frac{1}{r^2} \frac{\partial}{\partial r} \left[ r^2 \kappa_\mathrm{cond} \frac{\partial T}{\partial r} + r^2 \kappa_\mathrm{eddy} \left( \frac{\partial T}{\partial r} + \frac{g}{c_\mathrm{P}} \right) \right] ,
\end{aligned}$$ where $r$ is the radius, $n_j$ is the number density of the $j$th species, $\rho$ is the total mass density, $v$ is the bulk advection speed, is the momentum density, $e$ is the energy density, $T$ and $p$ are the temperature and thermal pressure, $\Phi_{\mathrm{d},j}$ and $S_j$ are the diffusive particle flux and chemical source term of the $j$th species, $g$ is the gravitational acceleration, $Q_{\mathrm{h}}$ and $Q_{\mathrm{c}}$ are the volumetric heating and cooling rates, $\kappa_\mathrm{cond}$ is the thermal conductivity, $\kappa_\mathrm{eddy}$ is the eddy conductivity, and $c_\mathrm{P}$ is the specific heat at constant pressure. Since chemistry and diffusion do not change the total mass density of the gas, Eqn. \[eqn:main\_speciescontinuity\] implies the standard mass continuity equation.

The evolution of the chemical structure of the atmosphere is described by Eqn. \[eqn:main\_speciescontinuity\]. The processes 