QHE-SE Dry Vertical Coordinate

Here we denote $m^{[l]} = \frac{\rho^{[l]}}{\rho^{[d]}}$, with $l \in \mathcal{L}_{\textrm{all}} \equiv \{d, wv, cl, ci, rn, sw\}$.

We let $$\begin{align*} \mathcal{L}_{\textrm{water}} = \{wv, cl, ci, rn, sw\}\\ \mathcal{L}_{\textrm{cond}} = \{cl, ci, rn, sw\} \end{align*}$$

The moist (physical) density is then given by $$\rho = \rho^{[d]} \left(\sum_{l \in \mathcal{L}_{\textrm{all}}} m^{[l]} \right)$$

No changes have been made thus far, as these are physical definitions.

IGL + $T_v$

The IGL governs the gaseous components, not the condensates! The most general governing law for the dry species reads as $p^{[d]} V^{[gas]} = N^{[d]} k_B T$, which gets rewritten in typical atmospheric science fashion as $$p^{[d]} V^{[gas]} - V \rho^{[d]} R^{[d]} T$$ with an analogous equation for $p^{[wv]}$. We make the assumption that $V = V^{[gas]}$, by virtue of assuming that condensate species are incompressible and take up negligible volume. We end up with $$p = \left(\rho^{[d]}R^{[d]} + \rho^{[wv]} R^{[wv]}\right)T$$

This gets rewritten (letting $\varepsilon = \frac{R^{[d]}}{R^{[wv]}}$) as $p = \rho R^{[d]} \left(\frac{1 + \frac{1}{\varepsilon} m^{[wv]}}{\sum_{l \in \mathcal{L}_{\textrm{all}}} m^{[l]}} \right)T \equiv \rho R^{[d]} T_v$

Mass coordinate definition:

The dry-$\eta$ coordinate can be specified in terms of $\eta^{[d]}(s) = h\left(\int_{z' = z(s)}^{z'=\infty} \rho^{[d]}\intd{z'}, \int_{z' = z(1)}^{z'=\infty} \rho^{[d]}\intd{z'}\right)$ with $\eta^{[d]}(0) = 0$ and $\eta^{[d]}(1) = 1$.