Inclusion of Moisture

Defining $\rho_{w,j}$ as the mass of water species $j$, then the total air mass or density of a volume of moist air is defined as

$$\rho = \rho_d + \sum_{j=1}^{N_w} \rho_{w,j}$$ (22)

where $\rho_d$ represents the dry air density. The total air inverse density, $\alpha$, (i.e. including moisture) is related to the dry air density through the ratio

$$\varphi_q = \frac{\alpha}{\alpha_d} = \frac{\rho_d}{\rho} = \frac... ...{j=1}^{N_w}r_j} = \frac{1}{1 + r} \hskip.5in \left(0 < \varphi_q \leq 1 \right)$$ (23)

where the mixing ratio for water species $j$ is $r_j=\rho_j/\rho_d$ (for $N_w$ total water species) and $r$ is the total water mass (sum of all water component mixing ratios).

The thermodynamic equation (Eq. 13) is now modified by simply replacing $\theta$ by $\theta_\rho$ (which is a thermodynamic variable used in some models often referred to as the density potential temperature). This thermodynamic variable is defined as

$$\theta_\rho = \varphi_q\,\left[1 + r_v\,\left(\frac{R_v}{R_d}\rig... ...ta = \varphi_q\, \varphi_m\,\theta = \varphi_q\,\theta_m = \varphi_\rho\,\theta$$ (24)

where $R_d$ and $R_v$ represent the gas constants for dry air and water vapor, respectively. $\theta_m$ is a moist potential temperature which depends on water vapor alone (it is used in WRFv4 as the internal energy variable for example). Here we call $\varphi_\rho$ the density temperature factor. Note that $\theta_\rho$ includes all suspended water species (in addition to water vapor: this is in contrast to the so-called virtual temperature which only includes water vapor effects).

The pressure (Eq. 7) including the effect of moisture is now given by

$$p ={\frac{R_d\,T_\rho}{\alpha}} ={\frac{R_d\,\theta_\rho \,\Pi}{\... ...t( {\frac{R_d\,\theta_\rho}{p_0\,\alpha_d\,\varphi_q}} \right)}^{C_{pd}/C_{vd}}$$ (25)

thus increases in water lead to corresponding decreases in the inverse density. Note that pressure can also be expressed in terms of the other aforementioned thermodynamic variables as

$$p = p_0 {\left( {\frac{R_d\,\theta_m}{p_0\,\alpha_d}} \right)}^{C_{pd}/C_{vd}}$$ (26)

$$= p_0 {\left( {\frac{R_d\,\varphi_m\,\theta}{p_0\,\alpha_d}} \right)}^{C_{pd}/C_{vd}}$$ (27)