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The virtual temperature is a pseudo-temperature taking into account water vapour in the atmosphere in such a way, that the resulting quantity can be used with the ideal gas law and the molar mass for dry air. Although the most significant in the troposphere, water vapour is not the only trace gas for which the concentration varies through the atmosphere. Are there any similarly defined pseudo-temperatures, taking into account varying volume mixing ratios for other trace gases?

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  • $\begingroup$ I liked your comment about methane atmospheres on titan. Without knowing much about the topic, it looks like those equations are suitable for any vapour. Is it the case that for calculations in the earth's atmosphere they're just used for water, because that's the only really significant atmospheric vapour for the purposes of CAPE? $\endgroup$ – naught101 Apr 16 '14 at 1:50
  • $\begingroup$ @naught101 Yes, other gases have a negligible impact on the mass of an air parcel, I think. $\endgroup$ – gerrit Apr 16 '14 at 4:11
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It is true that water is not the only variable constituent of the atmosphere, but it is capable of large variations (0-5%+) that I'm not sure other trace constituents are capable of outside of plumes from smokestacks / cars / factories / etc. A virtual temperature for other gases would only be meaningful if their variability imparts a significant change in mass in a parcel such that it becomes important for buoyancy.

That being said, you could easily derive such a virtual temperature, as the virtual temperature is just a proxy for density changes and we can easily calculate this.

For each constituent gas, you can use the ideal gas law as

$$p_i = \rho_i R_i T$$

such that each density of gas $i$ is

$$\rho_i = \dfrac{p_i}{R_i T}$$

and then sum these with dry air ($p_d$)

$$\rho = \dfrac{p-\sum p_i}{R_d T} + {\large\sum} \dfrac{p_i}{R_i T} $$

and then solving for $p$ in terms of $\rho$, $R_d$ and collecting the rest of the terms into a variable and calling it $T_v$.

This process could be used to generically define a virtual temperature for any number of trace gases (single or multiple) as long as you know their mixing ratios with dry air. I'm not aware of anyone using (on Earth) a virtual temp for anything other than water vapor and suspended hydrometeors (e.g. density potential temperature, $\theta_\rho$) but that certainly doesn't mean no one is.

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