Blog - doubling CO2 part two: optical depth and atmospheric profiles

**Last time I gave some basics about …**

Last time, worked up till infrared radiation interaction with greenhouse gases. Before we consider this quantitatively, introduce structure of atmosphere, and give . Because of its importance in the climate sensitivity, derive profiles of in lower atmosphere.

Our aim is to deduce the relation between temperature and atmospheric carbon dioxide concentrations, $T = f ([CO_2])$. More precisely, since we know the current temperature and concentration, we would like to find an expression that involves the future temperature change corresponding to a change in concentration: $\delta T / \delta c$. Perhaps some of you have heard talking about a logarithmic relation, so in fact we will be looking for something of the form $\delta T \propto \delta log (c/c_0)$. Our first aim would then be to derive the functional relationship (log), the second aim to estimate the constant of proportionality. To do so we have to investigate the earth’s energy balance and the greenhouse effect in detail.

Now there are at least two hidden assumptions in a formula as $\delta T = \hat{T} \delta log (c/c_0)$. The first concerns the “temperature”, the second concerns the “concentration”. What temperature and concentration are we actually talking about? Probably the temperature at the earth’s surface, but the earth’s surface extends over longitudes and latitudes, and earth surface temperatures do depend on those coordinates. So we will have to argue how we can still obtain a meaningful temperature after averaging over longitude and latitude. (Of course we’re also averaging over an ensemble of earths to eliminate weather – if you’re uncomfortable with this, think that we take a time average long enough to eliminate weather, but short enough compared to the timescales involving the change in carbon dioxide concentration. If you still feel uncomfortable because we’re adding CO2 at a rather fast pace, leaving not too much room for an averaging window, reassure yourself that what the (dead) earth system response concerns, the temporal rate of CO2 increase does not matter. if we would include biology etc, it may be a different matter, but we’re already deviating too far from the main topic). So we’ll derive an average temperature for the earth and use this for T. Then, what concentration are we talking about? CO2 well mixed within the atmosphere, derive expression for this, and can be parameterized by say, CO2 levels at sea level. c_0 and c_0(0) with the underscore the time parameter and the functional dependence. Also longitude and latitude averaged out (here).

Studying the earth’s energy balance and the greenhouse effect primarily involves a study of radiation balance in the atmosphere. The laws for radiation transfer are well known, and so is the radiative spectrum of carbon dioxide. Because radiation depends on temperature, we will also have to study the temperature profile within the atmosphere. To derive the functional relationship and the constant of proportionality, we can break up the problem in the following steps: -we investigate the radiation transfer for one specific wavelength -we investigate radiation balance for all wavelengths -we take earth feedback processes into account (this last option will lead us to far, and we won’t include it in this series)

Now there are actually only three mechanisms/causes **(???)** one has to understand in order to understand the logarithmic relation: -why some wavelengths matter more than others -why temperature drops linearly with height in the troposphere -why carbon dioxide drops exponentially with height

These three causes will be explained in this blog post. To understand why some wavelengths matter more than others, we need to look closer at a quantity called optical depth.

Length: glasses of beer of different size

Particles: milk and water in glass

add atmosphere 10 meter thick if same density of water

Tropopause, troposphere, stratosphere (other layers neglected)

Side note: half of matter below tropopause, half above

Presence, heating from earth’s surface, heating at top of stratosphere, somewhere minimal. Minimum important, because acts as barrier (explain convection and stratification). Reason heating earth’s surface can be additionally motivated by lower tropopause near the tropics.

Where does heating come from? Earth’s surface higher absorption of solar radiation than air, heating practically by convection. In stratosphere heating by absorption by ozone, more ozone higher up (ozone layer) because there created by UV radiation on oxygen.

Derive profiles in troposphere. Mention lower stratosphere rather constant temperature, before rise higher up.

I’ve mentioned the Earth’s temperature. Isn’t the Earth a 3D object, and if we talk about surface temperature, shouldn’t we talk about a 2D field? So how can we talk about just one number?

To start with the principle of energy conservation is often a successful approach, and we will also need it here.**Mention T-fourth** Basically, the sun sends radiation to the earth, and the earth is heated until it reaches a certain temperature when the radiation emitted by the earth balances the incoming radiation by the sun. Well, that sounds fine for a 0-dimensional earth. We’re adding an atmosphere to the problem so we need at least a 1-dimensional approach (I advise you to think of a radial dimension, not an ordinary spatial dimension with an axis running to infinity at both sides). That complicates matters a bit… But, in steady state (or better quasi-steady state since the earth will be warming up, but slowly with respect to the timescale needed to reach energy balance in the form we’ll write it up) there should still be an energy balance for each point on our radial atmosphere. Conceptually it will be the easiest to consider an energy balance at the top of the atmosphere (usually denoted TOA) where the incoming radiation of the sun is balanced by that part of the outgoing radiation of the earth that manages to make its way through the atmosphere…

How, does not all radiation make it through? Indeed not, and another important piece in our story is the presence of carbon dioxide in the atmosphere, because carbon dioxide absorbs and re-emits part of the longwave radiation emitted by the earth. Where is the $CO_2$ exactly in the armosphere? Well, luckily, carbon dioxide is well-mixed, so we can consider the *standard* expressions for air density to find an answer to that. Carbon dioxide has absorption bands **add major bands**, and some wavelengths are more affected than others. Please keep this in the back of your mind.

We’ve been talking about radiation, so we will certainly need some law to describe how the radiation is scattered, absorbed and reflected. In fact, for our current purposes, we don’t really need to consider the incoming radiation, which mainly passes through the atmosphere at wavelengths outside the major absorption bands of carbon dioxide, and I’ve mentioned we’ll consider the balance at the TOA. What’s more important is the difference between scattering and absorption. (remember that the sky looks blue due to scattering http://www.xkcd.com/1145/) Scattering can be taken into account by adding some factors, we’ll absorb these in a redefinition and then we can use the formulas for direct radiation.

The outgoing radiation is governed by Schwarzschild’s equation, which describes the vertical changes in radiation due to local emission and absorption. Later in this little blog series we’ll discuss the Schwarzschild equation in more detail, but if you want to know more about it right now, (Professor, John?) Baez has already written some material about it in his blog post Mathematics of the environment (part 3) . For our purpose what’s important about this equation is that, at each point in our 1D atmosphere, its absorption term is proportional to the radiation, and that **the emission term depends on the local temperature**. Both emission and absorption are influenced a factor that depends on the concentration of carbon dioxide, and on the pressure. It’s the same factor due to the ‘reciprocity’ nicely explained in the linked post.

In our forthcoming analysis of the Schwarzschild equation we will have to introduce the concept of optical depth because it will *simplify* the discussion (no irony here!). Optical depth sounds a little mysterious, but it roughly means how rapidly the outgoing radiation for a given wavelength gets attenuated when it’s passing through the atmosphere. It’s monotonically decreasing when we move upwards into the atmosphere, and what’s very important, **optical depth scales with the carbon dioxide concentration**, due to the ‘factor’ mentioned in the previous paragraph.

Related to the optical depth, we’ll need to introduce another concept, the effective emission height. For a certain wavelength, the photons that manage to leave the atmosphere do not emanate from the surface, but from this effective height. How we obtain this height will have to wait, but it turns out that **the effective emission height is logarithmically dependent on the optical depth** because pressure and density have an approximately exponential profile. Since the emission depends on the local temperature, it is the temperature at the effective emission height that matters the most for our problem. It turns out that **the wavelengths for which the outgoing radiation is most sensitive to changes in optical depth** (induced by changes in carbon dioxide concentration), **have an effective emission height somewhere in the troposphere. In the troposphere, the temperature drops linearly with height**, relative to the temperature at the ground. Why does the temperature drop linearly there? Well, that has to do with some basic and approximate principles of mechanics and thermodynamics. We’ll explain that in the next post in our series.

So I’ve already mentioned the major mechanisms in boldface, each of which has either a logarithmic or linear dependency, but before we put them all together, let’s contemplate how strong our argument actually is. From a conceptual point of view, the most important approximation is that we will discuss the mechanism for one wavelengths only, whereas temperature is actually related to all wavelengths together. But that’s the price we have to pay for simplicity. Another shortcut we should keep in mind is that we’ll directly consider the temperature response to changes in carbon dioxide concentration. The more correct way would first lead us to investiate the response of radiative forcing first, which is:

$\Delta F \propto \Delta \log(C/C_0).$

Welcome back to our little series to explain the log forcing relationship between the earth’s temperature and carbon dioxide concentration in the atmosphere. Last time, we saw a summary of the mechanisms behind the log forcing, such as:

- Optical depth linear dependence on local carbon dioxide concentration
- Carbon dioxide is well-mixed and follows air density profile
- Pressure and density exponential profile, therefore relation between effective emission height and optical depth is logarithmic
- Emission at effective emission height depends on temperature there
- Temperature in troposphere linear drop

What remains is to discuss these mechanisms in much more detail. What will we discuss today? We’ll restrict ourselves to the profiles in the lower atmosphere for temperature, pressure and density. I saw lower atmosphere, because the atmosphere is huge,

Add picture of 1D atmosphere (from John’s post)

and we shall only consider the troposphere and the stratosphere (there is hardly air above). Compared to the third part of the series this second part is shorter, and the physics is more widely known. So, to save our seriousness for next time when we’ll certainly need it, the tone of today will be more lighthearted.

Let us do some guesswork first: we want to relate the temperature at height $z$ – $T(z)$ – to the height and some parameters. What could matter? For sure the temperature at the ground, and the height $z$ above the ground, but with those two we can’t build a dimensionless relation. If we consider that our gas is ideal, we may expect that temperature has some relation with pressure and density. Therefore it’s likely that the gravitational attraction will matter, so $g$ enters, and since we’re talking about gases, $R$ is usually present too. But could the type of gas matter? Let us assume it doesn’t, although the weight of the gas molecules might play a role in the density profile and thus the temperature profile, otherwise there would still be helium gas in the atmosphere, or not? Anyway, let’s follow a minimalistic approach for now. We notice that $R$ has the dimensions $m^2/(s^2 K)$ in fundamental SI-units, so we are happy with our four parameters, because we can expect a simple relation of the form

(1)$T(z)/T_0 = f(\frac{g z}{R T_0})$

Oh, by the way: anyone who is willing to suggest that the rotation of the earth may influence the temperature profile is (a) kindly requested to estimate the magnitude of this effect and (b) ponder how it would matter when we’re considering a static environment. Adding the temperature of the sun and some absorption cross section to our dimensional analysis could be argued, but more about that later. Right now we’re staying close to the earth, so we would rather expect that the temperature is dictated by $T(0)$ (which will be influenced by the sun, but that’s indirectly). For example, let us invoke a simple analogy, if you keep a thermometer on your skin while you’re standing next to a radiator, it will measure a temperature that’s closer to your body temperature, not closer to a radiator’s temperature. Is this true? I haven’t tried it.

After the guesswork, let us think a bit deeper and invoke some equations to determine the vertical ($z$) profiles in the lower atmosphere. But let’s first restate clearly that we don’t move too far out of the earth, so the earth still looks flat, and we consider it to be horizontally homogeneous, so we can look at the $z$ dependence only. And there’s no wind either, we’re considering a static environment, to neglect all time variation and velocities.

The linear drop of temperature in the troposphere can be understood from very basic principles:

- The atmosphere is in mechanical equilibrium.
- The ideal gas law is valid.
- The resulting atmospheric profiles are isentropic (if an air parcel would move reversibly, it wouldn’t add to the entropy). This is an approximation.

If desired, please should work out the derivation by yourself, it’s a nice and short exercise. Two hints: (a) divide the first equation by the second, and plug in into the third; and (b) look up the second law of thermodynamics. We then obtain the linear relationship:

(2)$T(z)/T_0 = 1 - \frac{g z}{c_p T_0}$

With $c_p$ the specific heat constant at constant pressure for air, that’s roughly ` 1000\,\mathrm{J/(K kg)}`

. If we compare that with $1000\,\mathrm{J/g$, it turns out that about every 100 m the temperature drops by one degree (this is called the lapse rate). We see that our original guess was not correct: it matters that we consider air ($c_p$ is gas specific). Do you have any idea why the type of gas matters?

Well, if you can determine the temperature profile, you can also find the density and pressure profiles. Let us simply note that they are both approximately exponential, with

From these we can determine a scale height for pressure and density.

Plugging in some numbers, we find that they’re both around 10 km. That’s really close to the height of the tropopause, and we should naturally wonder if that’s pure coincidence. But let us not distract from this blog post, whether coincidence or not, that’s another question than the log forcing that we’re trying to understand now!

It’s about time to start talking about carbon dioxide. Carbon dioxide is **well-mixed**, but it’s a **trace gas**. This former means that its concentration is proportional to the air density, and the latter means that it’s concentration is low enough not to influence the air density and pressure in a significant manner.

So, what’s now the summary for the logarithmic forcing, how do all the above mechanisms interact? When carbon dioxide levels double, optical depth doubles, and the effective emission height shifts proportional to the logarithm of two. Higher up in the troposphere the temperature is lower, and the outgoing radiation would drop. Simplistically, to retain the same amount of outgoing radiation we need the temperature at the higher altitude to become equal to the temperature at the original effective emission height before doubling. Temperatures in the troposphere are connected to the temperature to the ground, so the outgoing radiation remains equal under a doubling of carbon dioxide concentration if the temperature at the ground responds logarithmically.

Next time, we’ll discuss the underlying mechanisms more closely, starting with atmospheric profiles in part two, and discussing the Schwarzschild equation in part three.