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Consider an air parcel with relative humidity $H$ and vapor pressure $e$ that experiments an adiabatic lifting process. Obviously the saturating vapor pressure is going to change since said process is going to lower the temperature of the system, and assuming it's politropic (heat capacities are conserved), we can determine a saturated adiabatic lapse rate:

$$\Gamma_s=-\frac{dT}{dz}=\frac{g}{c_p}\frac{1+\frac{l_vr_s}{RT_0}}{1+\frac{l_v^2r_s}{c_pR_vT_0^2}}$$

Where $c_p$ is the specific heat capacity at constant pressure and not the molar heat capacity, and $R=R^*/M_d$; $R_v=R^*/M_v$.

I'm asking this because if vapor pressure is conserved, we can use the Clausius-Clapeyron equation for two points such that:

$$\ln\left( \frac{e_{s,1}}{e_{s,0}} \right)=-\frac{l_v}{R_v}\left(\frac{1}{T_1}-\frac{1}{T_0} \right) \rightarrow e_{s,1}=e_{s,0}\cdot\mathrm{exp}\left(\frac{l_v}{R_v}\left(\frac{1}{T_1}-\frac{1}{T_0} \right) \right)$$

Where $e_{s,0}$ is some known saturating vapor pressure. After that, since $H_1=e_1/e_{s,1}$, then if $e_1=e$ the calculation is trivial knowing the initial vapor pressure.

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For the adiabatic ascent of a moist parcel of air there are two scenarios:

  1. if the parcel is not saturated it will just get colder and expand as it goes upwards. Parcel vapor pressure will remain constant as there are no condensation processes.

  2. once the parcel gets saturated (when it reaches its lifting condensation level), any further ascent (cooling) will result in condensation of some of its vapor into water droplets, so the vapor pressure of the parcel will start to decrease, and will continue decreasing during the rest of its ascent.

Summarizing, the vapor pressure of the parcel is conserved for non saturated adiabatic ascent processes, and is not conserved for saturated adiabatic ascent processes.

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  • $\begingroup$ I think the parcel could become also supersaturated... although it is perhaps beyond the thermodynamic model implied here. $\endgroup$
    – Roger V.
    Commented Jul 5, 2022 at 10:15
  • $\begingroup$ I do not know if supersaturations take place for a while during adiabatic ascent processes, but is not so relevant, the key think is that once condensation starts, and that is sure that happens because there exist clouds produced by adiabatic ascent in the atmosphere, the parcel starts loosing water vapor $\endgroup$
    – ramon vazquez
    Commented Jul 5, 2022 at 11:37
  • $\begingroup$ Do I understand correctly that it is not only the vapor pressure that is changing, but the overall atmospheric pressure as well? (And hence the total pressure in the parcel). $\endgroup$
    – Roger V.
    Commented Jul 5, 2022 at 11:54
  • $\begingroup$ yes, the pressure of the parcel is decreasing, because while ascending the parcel is expanding (so increasing its volume) as it encounters lower pressure air on its way up $\endgroup$
    – ramon vazquez
    Commented Jul 5, 2022 at 12:07
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Vapor pressure (e) is NOT conserved during unsaturated adiabatic ascent. However, mixing ratio (q) is conserved during unsaturated ascent. I see this error in textbooks as well. Vapor pressure is a function of dewpoint, which in turn depends on pressure, which is decreasing during ascent, and dewpoint decreases as an air parcel ascends. This can be seen on thermodynamic charts as well.

Once saturation occurs, e=e_s, and q=q_s, and both are decreasing as an air parcel ascends as a cloud. The subscripts denote saturation.

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