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Homework question:

Show that the absorption optical thickness of the atmosphere at pressure $p$ (i.e., between $p$ and the top of the atmosphere) as a function of the mixing ratio of the absorbing gas $r$ and the absorption coefficient per unit mass ${k}_{ν}$ can be expressed as:

$τ = \frac{k_νrp}{g}$.

Assume $r ≪ 1$

(So we can approximate $r ≃ \frac{ρ_a}{ρ}$, where $ρ_a$ is the density of the absorbing gas), and that $r$ and $k_ν$ are vertically uniform, and assume that there is no scattering.

Has the anthropogenic increase in $CO_2$ shifted the emission level to higher or lower altitude? Prove your answer.

Attempt:

Given the equation $τ = \frac{k_νrp}{g}$

I took the $ln(\tau) = ln(\frac{k_νrp}{g})$

$ln(\tau) = ln(k_ν) + ln(r) + ln(p) - ln(g)$

$dln(\tau) = dln(k_ν) + dln(r) + dln(p) - dln(g)$

$\frac{dln\tau}{\tau} = \frac{dlnk_ν}{k_v} + \frac{dlnr}{r} + \frac{dlnp}{p} - \frac{dlng}{g}$

We will assume that $g$ and $k_v$ are constant, so disregard for a moment here. It is a battle between the mixing ratio $r$ and $p$.

$\frac{dln\tau}{\tau} = \frac{dlnr}{r} + \frac{dlnp}{p}$

If we want to know if peak $\tau$ moves up or down, I say that $\tau$ is constant. If I increase the mixing ratio $r$ (increase $CO_2$ concentration in atmosphere), pressure must decrease in order to keep $\tau$ constant, and decreased pressure means moving up in altitude. I conclude that increasing $CO_2$ concentrations shifted the emission level to a higher altitude.

Help

I am unsure how to arrive at the initial given equation. My understanding of $\tau$ is minimal, and I just see too many variations and integrals concerning $\tau$. I see how the Ideal Gas Law and the Hypsometric Relation will be used to replace variables

$ρ = ρ_o*e^\frac{-z}{H}$ where $H = \frac{RT}{g}$

$P = ρRT$

(just trying to work backwards and piece the puzzle together, but the vision is unclear), but what was the original equation that was used, and ultimately--why?

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    $\begingroup$ Hi PattyWatty, and welcome to SE. This is not primarily a homework help site, and in general if you just copy/paste a question in here people will not answer it. However, if you need help with a specific concept, that's a different matter. Please have a read of earthscience.meta.stackexchange.com/a/317/39 . $\endgroup$ – Semidiurnal Simon Sep 29 '18 at 20:17
  • $\begingroup$ I'm voting to put this question on hold as off-topic because it is a homework question with no attempt at an answer. The asker can edit it to be more acceptable if they wish. $\endgroup$ – Semidiurnal Simon Sep 29 '18 at 20:18
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    $\begingroup$ @SemidiurnalSimon Thank you for the advice, I do need help with specific content. I will edit accordingly. Please, do not delete/down-vote, the atmosphere is my passion, this is the only community I know of that I can create a dialogue with. $\endgroup$ – PattyWatty27 Sep 29 '18 at 20:34
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    $\begingroup$ Not my down-votes :-) But don't worry - standard procedure here is to "vote to close", which means (once a few more reviewers see it) it gets put "on hold" to give you time to edit the question. If you edit a question that's on hold, it automatically gets sent to reviewers to see if it should be re-opened. $\endgroup$ – Semidiurnal Simon Sep 29 '18 at 20:39

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