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## A Guide to CO2 and Stratospheric Cooling

Temperature trend 1960–2012 versus latitude and pressure. The value for each latitude and pressure is the medians of the trends at individual stations in that (10°) latitude bin. Units are °C per decade. From Sherwood and Nishant (2015).

The figure above, taken from Sherwood and Nishant (2015), is rather reminiscent of the vertical structure of temperature change people talk about in the context of 20th century climate change. The focus of this post is on the cooling stratosphere (and evidence exists for cooling into the mesosphere and thermosphere, as well), the parts of the atmosphere very high up (above ~200 mb or so, depending on latitude). It is well known that we expect such a response to increasing CO2 and other greenhouse gases, but how does this relation come about? It is one aspect of the climate change problem that most people know about but rather few understand.

There is definitely a huge role for ozone depletion/recovery in this story, since O3 is the principle source of very shortwave (UV) absorption in the stratosphere, but that doesn’t explain cooling in the mesosphere. Ozone itself causes the presence of a temperature inversion aloft, whereby temperature increases with height in the stratosphere (some people define the stratosphere on this basis; the tropopause that separates the troposphere and stratosphere is largely determined by this UV absorption…though, note that one could still have a stably stratified layer aloft, that was not convecting, even without a solar absorber. Most people wouldn’t call that a “troposphere”).  But what if Earth had no ozone, would we still expect cooling in the “upper atmosphere” in response to CO2?

To explore this, we can turn to a rather simple model of the climate system, much like one I’ve used in previous posts on the greenhouse effect and feedbacks. We’ll have to amp up the complexity a tiny bit, though. To do this, I’ll call upon climlab, an open-source climate modeling software toolkit being developed by Brian Rose at the University at Albany. This is a nice tool for understanding the physics of climate by playing interactively with a hierarchy of climate models at the “lower end” on the complexity spectrum. I’m going to actually use two models embedded within climab, one of which is a “grey radiation” model, and the other a spectral band model. The choice for showing these two results will become clear.

Both models I’ll show have only a vertical dimension- 31 levels, 30 atmospheric levels plus the surface (which radiates like a blackbody). The grey radiation model has incoming (solar) and outgoing (terrestrial, infrared) energy flows, but otherwise knows nothing about wavelength-dependence in atmospheric radiative transfer. The greenhouse effect is built-in using a longwave emissivity, $\epsilon$ that is constant with height. All wavelength-dependence in the infrared spectrum is thus wrapped up into a single emissivity, and the slab emissivity must be equal to the slab absorptivity (by something known as Kirchoff’s law). So, the radiative balance (not including convection) for any slab in the atmosphere is $\epsilon I = 2 \epsilon \sigma T^{4}$, where $\epsilon I$ is the impinging radiation that originated from any part of the atmosphere (or surface) above or below the slab in question. The factor of two comes from the fact that the atmospheric radiates up and down.

Model Setup

One can also prescribe a shortwave emissivity with some vertical structure to mimic the role of ozone in the climate system. This emissivity can be different than the longwave emissivity. I’ve computed results for the “grey gas” model, both with and without ozone, using an actual ozone concentration profile that was interpolated onto the model’s vertical grid. This just gives some shortwave absorption in the model’s upper atmosphere, and I turn the ozone concentration into a shortwave emissivity simply by multiplying the concentration data by some constant (i.e., absorption is proportional to absorber amount) that gave a reasonable looking stratosphere.

In contrast to the grey atmosphere, the spectral model attempts to build-in some wavelength dependence. The longwave is divided into four “bands”: Band 1 is the window region (between 8.5 and 11 μm), 17% of total flux, band 2 is the CO2 absorption channel (the band of strong absorption by CO2 around 15 μm), 15% of total flux, band 2 is a weak water vapor absorption channel, 35% of total flux, and band 3 is a strong water vapor absorption channel, 33% of total flux. Both models have a convective adjustment built in, so any part of the atmosphere will convect and transport energy upward if the lapse rate exceeds some user-specified critical value. I’ve chose -7 K/km here as that number, so lapse rates cannot be steeper than this (but can be less steep). The spectral model also includes a parameterization that allows water vapor to change by holding the relative humidity constant at every model level, so specific humidity increases with temperature.

In the grey gas model, I’ve tuned the emissivity of the column (and ran to equilibrium) to the value needed to give a surface temperature of 288 K. In the spectral model, there is no single emissivity that is tuned like this, so I tuned the surface albedo to yield the same temperature. These two configurations will be our baseline “control” climate.

Results

The following image shows the vertical temperature structure of the baseline climates.

By construction, all these curves have surface temperatures of 288 K. Given the prescribed convective adjustment, the lapse rate is the same for all curves in the troposphere. As expected, putting ozone in the grey gas model creates a stratospheric temperature inversion. Note that having a solar absorber is not sufficient to do this, it’s necessary to have the right vertical profile, but ozone does and the real concentration data is being input into the model. Interestingly, the stratosphere in the model with different spectral bands is much colder than the stratosphere in the grey gas model (note- I’m not showing the spectral model with ozone here, but if I did it would also exhibit an inversion but with everything shifted to colder temperatures). But, we at least have a clue that the spectral dependence of the absorbing gases is important in the stratosphere.

In the following, I’ll derive an approximate expression for temperature in the stratosphere (with and without ozone) and how it changes with emissivity (with and without ozone). We are restricting ourselves to the grey-gas case for the next few paragraphs. Then, we’ll test the expressions with the model.

Grey Gas

We can approximate the stratospheric (or “upper atmospheric”) temperature in such a model by operating under the assumption that the stratosphere is optically thin (the prescribed emissivity in each slab is ~0.05 in this case since there are so many atmospheric slabs). In such a limit, we can assume that most of the outgoing longwave radiation to space (OLR) originates from below the stratosphere, and so the energy balance of the stratosphere with no solar absorption is $2 \epsilon \sigma T_{t}^{4} \approx \epsilon OLR$, where $T_{t}$ is the stratospheric temperature. Of course, the stratosphere is not just one slab with zero contribution to the OLR, but it gives a reasonable first guess…the OLR in this model is about 240 W/m2, yielding a predicted stratospheric temperature of about 215 K, much like what we see near the top of the “red curve” in figure 1. Note also that if we change emissivity, the stratospheric temperature should remain fixed according to the above equation, provided the OLR does not change (i.e., we are in equilibrium with unchanged solar constant or planetary albedo).

The energy balance with a stratosphere that absorbs UV radiation at a rate $R$ is similarly, $2 \epsilon \sigma T_{tt}^{4} \approx \epsilon OLR + \epsilon_{s} R$, where $T_{tt}$ is now the stratospheric temperature with ozone, and with shortwave emissivity $\epsilon_{s}$. In the following, I solve for the stratospheric temperature and how it varies with longwave emissivity in the grey gas model. Does it exhibit the same insensitivity to emissivity as the no-ozone case?

$2 \epsilon \sigma T_{tt}^{4} \approx \epsilon OLR +\epsilon_{s}R$

or $\displaystyle T_{tt}^{4} \approx T_{t}^{4} + \frac{\epsilon_{s} R T_{t}^{4} }{\epsilon OLR}= T_{t}^{4}[1+\frac{\epsilon_{s}R}{\epsilon OLR}]$

and, now we can solve for the change in stratospheric temperature with epsilon. Since we found that the “no ozone” stratosphere temperature was insensitive to emissivity, I’ll solve for the change in the ratio of the “ozone” stratosphere vs. the “no ozone” stratosphere ($T_{tt}/T_{t}$) and how that varies with emissivity.

$\displaystyle \frac{\partial [T_{tt}/T_{t}]}{\partial \epsilon} = - \frac{\epsilon_{s} R }{4 \epsilon^{2} OLR} [1+\frac{\epsilon_{s}R}{\epsilon OLR}]^{-0.75}$

This is zero if R=0, and is otherwise negative. Since the “no ozone” stratosphere temperature remains invariant with emissivity, it follows that the stratosphere “with ozone” cools as emissivity increases. Let’s test these results by increasing the emissivity in the grey gas model by 4%, both in the ozone and no-ozone cases.

Evidently, the stratosphere does cool a bit in a grey-gas atmosphere when emissivity increases, but only if a solar absorber is present aloft. Otherwise, the upper atmospheric temperature scales with the emission temperature of the planet (defined by the temperature that a blackbody would have if it emitted at a rate equal to the OLR), and this does not change with greenhouse gases. With ozone however, we do see a slight cooling. This is so because the energy balance of the stratosphere is between heating from UV radiation and infrared cooling. Increasing infrared emissivity allows slightly more cooling (at a given temperature) balancing the same solar heating, hence the stratosphere can get cooler. I’ll let the reader convince themselves that the above equation is skillful against the model result, noting that the solar absorption in the topmost levels of the model is ~5 W/m2[note the original emissivity with and without ozone was 0.05411 and 0.053244, respectively, the values I needed to give a baseline climate at 288 K. Both values were perturbed by 4% for the anomaly plot].

Spectral Model

We saw before that the stratosphere in the spectral band model is colder than the grey gas model. We can reconcile this with the equation described earlier if the stratospheric infrared absorptivity is less than its infrared emissivity, which would not violate Kirchoff’s law if the spectrum of upwelling radiation from below were different than the spectrum of stratospheric emission, an issue totally absent in the grey-gas formulation. Indeed, if one uses a grey gas model for research problems, it is likely the stratosphere in that model will be too warm, with possible implications for the tropopause height and convection in that setup.

Let us double CO2 now in the spectral model. This is similar to increasing the emissivity as before, except only in the spectral band sensitive to CO2 (if we actually doubled emissivity before the temperature change would be enormous, which is why I only increased it 4%. Life on a planet where CO2 acted like a grey gas would not be very pleasant). But in a spectral model, only certain wavelength bands care about the increased opacity. The following shows the previous plot, except now with temperature anomaly to 2xCO2 in the spectral model (remember, no ozone).

As before, the stratosphere cools, except the magnitude of the cooling is substantially larger than our grey gas model with ozone. Thus, the stratospheric cooling we talk about in association with increased CO2 must have a lot to do with the split-spectral nature of the terrestrial atmosphere…namely, that there is a window region where the atmosphere can radiate to space rather cleanly, and a opaque band where the atmosphere is highly absorbing. In the following graphic, I’ll show the absorption change (only in the CO2 band) after we’ve increased CO2, both before and after the temperatures have adjusted to equilibrium.

So, what’s the sales pitch on why the stratosphere cools? At the 15micron band, as the opacity increases the stratospheric absorption of upwelling infrared from below will be less, but the upper layers are still good emitters. Having a split-spectral window allows the planet to come into equilibrium by increasing infrared emission to space in relatively transparent bands (while cooling the stratosphere due to decreased upwelling 15 micron flux), an effect we cannot have in a grey atmosphere. This cools the stratosphere. Unlike the grey gas model, the cooling must be in a limited wavelength band if one is to satisfy the constraint of coming back to equilibrium.

More generally, it is not necessary to have ozone in order to cool the “upper atmosphere” of a planet. Indeed, planetary scientists are very interested in dense CO2 atmospheres, not just because it makes the surface very hot, but because it makes the upper atmosphere cold, with implications for “atmospheric escape” physics (the irreversible loss of molecules into space) and the evolution of a planet’s atmosphere.

Summary Remarks

Adding CO2 makes a planet’s atmosphere opaque. In the troposphere, temperature decreases with height- an implication is that absorption of upwelling “warm” photons and emission at a colder temperature means that heat is being effectively trapped in the column (i.e., heat loss is being reduced). Moreover, since the troposphere is stirred by convection, this “trapped heat” will communicated vertically warming the whole troposphere.

However, as one moves higher in the atmosphere, eventually the column becomes optically thin enough such that upwelling radiation goes to space rather than being absorbed by higher layers. Increasing the atmospheric opacity will cause the outer layers of the atmosphere to cool by increasing their emissivity, while reducing the upward infrared flux that warms them from below.

As a side point, if temperature increased with height in the whole atmosphere, adding CO2 would cool the surface by replacing “cool” surface emission with “warm” emission aloft. This occurs to some extent in the stratosphere, since temperature increases with height and the 15micron band is so opaque that photons do not escape to space until they reach the stratosphere. Adding CO2 shifts this “radiating height” upward and increases emission, but that increased emission is more than offset by the decreased emission toward the flanks of the spectral band sensitive to CO2. Thus, CO2 decreases emission to space and warms the planet. This increased 15micron emission can help cool the stratosphere, but the main argument in this post would operate even if CO2 magically increased only in the troposphere (where upwelling infrared flux would be reduced).

### 2 Responses

1. on October 15, 2015 at 6:18 am | Reply Robert Clemenzi

I am having trouble with 2 of the grey atmosphere equations.

I don’t understand why “\epsilon_{s}” ($\epsilon_{s}$) is in the following

\displaystyle T_{tt}^{4} \approx T_{t}^{4} + \frac{\epsilon_{s} R \sigma T_{t}^{4} }{\epsilon OLR}= T_{t}^{4}[1+\frac{\epsilon_{s}R}{\epsilon OLR}]

$\displaystyle T_{tt}^{4} \approx T_{t}^{4} + \frac{\epsilon_{s} R \sigma T_{t}^{4} }{\epsilon OLR}= T_{t}^{4}[1+\frac{\epsilon_{s}R}{\epsilon OLR}]$

In the partial derivative

\displaystyle \frac{\partial [T_{tt}/T_{t}]}{\partial \epsilon} = – \frac{\epsilon_{s} R T_{t}}{4 \epsilon^{2} OLR} [1+\frac{\epsilon_{s}R}{\epsilon OLR}]^{-0.75}

$\displaystyle \frac{\partial [T_{tt}/T_{t}]}{\partial \epsilon} = - \frac{\epsilon_{s} R T_{t}}{4 \epsilon^{2} OLR} [1+\frac{\epsilon_{s}R}{\epsilon OLR}]^{-0.75}$

Why is T_{t} ($T_{t}$) included in

\epsilon_{s} R T_{t}
$\epsilon_{s} R T_{t}$

Perhaps you intended to take the partial of just the new temperature instead of the ratio.

Assuming that, it appears that as epsilon gets smaller the partial derivative gets more negative, which implies that increasing CO2 should make the stratosphere warmer. Needless to say, I am very confused.

Response: Adding CO2 increases absorptivity (and hence emissivity…if I understand your question right). So it looks like the sign is right…but yes, I fixed a couple mistakes in those equations. The dimensions didn’t even make sense before. Thanks. This didn’t affect any of the graphics though – chris

2. on October 15, 2015 at 6:24 am | Reply Robert Clemenzi

Sorry, my problem with the first equation was with $\sigma$ not $\epsilon_{s}$.

(copy and paste error.)