<p><strong>Homework question</strong>:</p>

<blockquote>
  <p>Show that the absorption optical thickness of the atmosphere
  at pressure <span class="math-container">$p$</span> (i.e., between <span class="math-container">$p$</span> and the top of the atmosphere) as a function of the mixing ratio
  of the absorbing gas <span class="math-container">$r$</span> and the absorption coefficient per unit mass <span class="math-container">${k}_{ν}$</span> can be expressed as:</p>
  
  <p><span class="math-container">$τ = \frac{k_νrp}{g}$</span>. </p>
  
  <p>Assume <span class="math-container">$r ≪ 1$</span> </p>
  
  <p>(So we can approximate <span class="math-container">$r ≃ \frac{ρ_a}{ρ}$</span>, where <span class="math-container">$ρ_a$</span> is the density of the
  absorbing gas), and that <span class="math-container">$r$</span> and <span class="math-container">$k_ν$</span> are vertically uniform, and assume that there is no scattering. </p>
  
  <p>Has the anthropogenic increase in <span class="math-container">$CO_2$</span> shifted the emission level to higher or lower altitude? Prove your answer.</p>
</blockquote>

<p><strong>Attempt</strong>:</p>

<p>Given the equation <span class="math-container">$τ = \frac{k_νrp}{g}$</span></p>

<p>I took the <span class="math-container">$ln(\tau) = ln(\frac{k_νrp}{g})$</span> </p>

<p><span class="math-container">$ln(\tau) = ln(k_ν) + ln(r) + ln(p) - ln(g)$</span></p>

<p><span class="math-container">$dln(\tau) = dln(k_ν) + dln(r) + dln(p) - dln(g)$</span></p>

<p><span class="math-container">$\frac{dln\tau}{\tau} = \frac{dlnk_ν}{k_v} + \frac{dlnr}{r} + \frac{dlnp}{p} - \frac{dlng}{g}$</span></p>

<p>We will assume that <span class="math-container">$g$</span> and <span class="math-container">$k_v$</span> are constant, so disregard for a moment here.  It is a battle between the mixing ratio <span class="math-container">$r$</span> and <span class="math-container">$p$</span>.  </p>

<p><span class="math-container">$\frac{dln\tau}{\tau} = \frac{dlnr}{r} + \frac{dlnp}{p}$</span></p>

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

<p><strong>Help</strong></p>

<p>I am unsure how to arrive at the initial given equation.  My understanding of <span class="math-container">$\tau$</span> is minimal, and I just see too many variations and integrals concerning <span class="math-container">$\tau$</span>.  I see how the Ideal Gas Law and the Hypsometric Relation will be used to replace variables </p>

<p><span class="math-container">$ρ = ρ_o*e^\frac{-z}{H}$</span> where <span class="math-container">$H = \frac{RT}{g}$</span></p>

<p><span class="math-container">$P = ρRT$</span></p>

<p>(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?</p>

CC BY-SA 4.0
2018-09-29T20:17:48.847Z
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 https://earthscience.meta.stackexchange.com/a/317/39 .
Semidiurnal Simon
CC BY-SA 4.0
2018-09-29T20:18:59.097Z
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.
Semidiurnal Simon
CC BY-SA 4.0
2018-09-29T20:34:57.840Z
@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.
PattyWatty27
CC BY-SA 4.0
2018-09-29T20:39:10.970Z
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.
Semidiurnal Simon
<atmosphere><climate-change><homework><mathematics>
Optical Thickness of the Atmosphere
PattyWatty27
# Optical Thickness of the Atmosphere

(Asked by: username_0 on 2018-09-29)

**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?

**Tags:** atmosphere, climate-change, homework, mathematics

> **Comments:**
> **username_1**: 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 https://earthscience.meta.stackexchange.com/a/317/39 .  
> **username_1**: 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.  
> **username_0**: @username_1 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.  
> **username_1**: 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.  

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## 0 answers

