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## A better use of layer models in teaching the greenhouse effect

I have been absent from writing this blog for nearly two years, which sometimes happens when one has an interest in starting a blog. This is a post I felt was worth writing down though. Some of the context might not be entirely clear to those who haven’t gone through an atmospheric science program or picked up some textbooks dealing with climate physics, so I will try to clear that up.

I will try to outline what I feel can be improved in practical instruction of the greenhouse effect, just based on some modification to a toy model that has already been employed in many classrooms and textbooks (some examples are Atmosphere, Ocean and, Climate Dynamics; Global Physical Climatology; Global Warming: Understanding the forecast, all great reads by the way). The framing here can be presented to an upper-undergraduate or graduate classroom, but the calculus can be stripped away for courses with minimal math requirements. Here, my focus is in making better contact with how the greenhouse effect actually works rather than going through algebraic exercises that yield numbers which don’t have any particular special meaning.

I’m pretty picky about how the greenhouse effect is talked about- you could say I like to provide and search for good explanations like an espesso lover wants to find a good cup. I’ve usually been critical of this “layer model” that I’ll describe below, but I started thinking more about this after a graduate class I take in climate modeling, and seeing it applied in a better way than I’ve seen outlined in the above examples, I thought I’d type away at the keyboard and add my own few wrinkles. Fundamentally, there’s nothing different than what I wrote nearly three years ago here but maybe this is more of a useful template for “teaching.”

Background

Whenever students, either undergraduate or at the graduate level, learn about the greenhouse effect in a classroom setting they will almost always at some point go through a “layer model” exercise on the blackboard and in homework assignments. I’ve seen this worked out in many forms- a casual google search of this model will reveal many class notes and book chapters setting up the model, which is shown above in schematic form.

The essence of the formulation is to represent an atmosphere only in its vertical structure, with a distinct surface and one or more atmospheric “slabs” that interact with radiation, thus providing the toy model with a “greenhouse effect.” A convenient starting point is to treat these slabs as being transparent to solar radiation and opaque to terrestrial (infrared) radiation. A complete distinction between these two streams of radiation (a two-stream approximation) is one of the few convenient simplifications nature has provided for us, entirely due to the very different temperatures of the Sun and planets encountered in our solar system (in the exoplanet realm, one could imagine a situation in which roaster planets at several thousand Kelvin orbit stars not too much hotter than this, resulting in a loss of this simplification, but we’ll ignore this here).

Once you move beyond this starting point one can easily modify the properties of these slabs to allow for some absorption of solar radiation or “leakiness” in the outgoing infrared radiation (OLR), with only slight increases in the complexity of what you’re solving. Aside from the distinction between star and planet radiation, the toy model typically knows nothing about any other wavelength dependence, and a slab at temperature $T$ can be assigned an infrared emissivity of $\epsilon$ such that it emits radiation at a rate $\epsilon \sigma T^{4}$ according to the Stefan-Boltzmann law. The limit shown in the above figure is where $\epsilon = 1$, meaning each layer acts like a blackbody in the infrared spectrum (absorbs and emits perfectly). Representing the interaction with entire infrared spectrum with a wavelength-independent emissivity is called a grey-gas approximation.

Usually, the prescribed input into the system is the absorbed solar radiation (ASR) and we ask questions about the resulting temperature structure of the model, operating under the constraint that the planet (and each slab for that matter) is in radiative equilibrium. That ASR = OLR in equilibrium is the fundamental boundary condition that constrains the global climate of all terrestrial planets, and the temperature dependence of the right term provides the fundamental stabilizing feedback that allows equilibrium to even be possible over a range of climate change scenarios. Therefore, in the context of the layer model, it is of interest to know how OLR varies with temperature and emissivity (e.g., for fixed solar absorption, the slope of OLR vs. surface temperature is inversely related to the magnitude of climate sensitivity).

For the most part (in my experience at least), working out problems in the context of the layer model becomes mostly an algebraic exercise in solving a system of equations, i.e., in the case above one would have three equations and three unknowns (the temperature of each slab). The resulting revelation is typically that surface temperature increases as $\epsilon$ of each layer, or the number of layers, increases. With any absorption, this temperature exceeds the temperature of the top slab where emission is also escaping. There are additional things we can learn- such as the fact that the surface becomes colder than the emission temperature to space when you have strong incoming solar absorption in the high atmosphere (the “nuclear winter” problem) such that the solar absorption is no longer occurring at the surface boundary.

Although we have called upon this toy model as the vehicle used to examine the mechanistic basis for the greenhouse effect, we are still left with a fuzzy intuition on how exactly it works. Sure, we have made a more opaque “infrared blanket” so it seems like the surface should be warmer…after all, it does have another term of energy coming from an atmospheric slab above it. In general, once we’ve asked a question and exhausted what we can learn from a model we need to either build the complexity of the model or ask a different question. But can we interrogate this model a little further, in order to think about the physics of the problem? I think so, but it calls for a slight re-framing of the question we’re answering.

In the following, I’ll think about it from the perspective of how $\epsilon$ perturbs the energy budget of the planet, and the individual contributions of our slabs to that perturbation, rather than trying to solve for the temperature structure. My claim is that this will reveal for us more explicitly how the “heat-trapping” is working. In fact, the analysis above does not tell us why the observed temperature structure is what it is. In reality, the troposphere is not in radiative equilibrium. The absorption of sunlight at the surface and emission of infrared aloft (on their own) would create an atmospheric temperature gradient that is way too steep. This column becomes dynamically unstable and must convect (moving energy upward) until a temperature profile is established that satisfies buoyancy constraints. This is the reason that we have a troposphere.

Emissivity and the OLR

In the following analysis, rather than trying to model convection we are simply going to prescribe the temperature structure of the atmosphere (instead of solving for it). We’ll also fix the surface temperature to the observed value of 288 K. Instead, the target question is how changes in emissivity, which we can think of as being modified by CO2 concentration for example, alter the outgoing radiation of the planet given a fixed temperature profile.

The above schematic shows a two-layer model with some emissivity in the atmospheric slabs (assume it is the same in both slabs), with arrows on the left showing the direct emission by the slab and arrows on the right showing the transmitted component of radiation from layers below.

Note that for slab $n$ (where n=0 at the surface, n=1 in the first atmospheric slab, and n=N at the topmost level), the radiation seen coming from the level at which the slab resides (looking down) is $F_{n} = E_{n} + t_{n} = E_{n} + (1-\epsilon)F_{n-1}$, where $E_{n}$ is the direct emission from the slab and $t_{n}$ is the transmission of radiation through the slab that originated from lower levels. It follows that $OLR=F_{N}$ and the contribution of radiation, $C_{n}$, from any level is $C_{n} = E_{n}(1-\epsilon)^{N-n}$. The surface is a blackbody, so $E_{n=1,N} = \epsilon \sigma T_{n}^{4}$ and $E_{n=0} = \sigma T_{0}^{4}$.

In the two-layer model above, the OLR it is the emitted plus transmitted component from the top slab:

$\displaystyle OLR = \epsilon \sigma T_{2}^{4} + (1 - \epsilon) \epsilon \sigma T_{1}^{4} + (1 - \epsilon)^{2} \sigma T_{0}^{4}$

Let’s pick some temperatures now. We’ll use 288 K for the surface (layer 0), 270 K for layer 1 (the first atmospheric slab), and 240 K for the top slab, giving us a temperature profile that gets colder with height, as is the case in the troposphere. Given these inputs, the required emissivity using the above equation to yield an Earth-like OLR of ~240 W/m2 is about 0.64. Obviously, if we had more layers we’d need to choose a smaller emissivity. These are entirely tuned quantities and their numeric detail aren’t interesting for our purposes.

Now suppose we are interested in some perturbation in emissivity, e.g., $\epsilon$ increases as CO2 concentration goes up. We have:

$\displaystyle \frac{\partial OLR }{\partial \epsilon} = \sigma T_{2}^{4} - \sigma T_{1}^{4}(2\epsilon - 1)-2(1-\epsilon)\sigma T_{0}^{4}$

Given our temperature profile, I’ll leave it as an exercise to the reader to convince yourself that the right-hand side is always negative, for any given emissivity (by definition, between zero and one). That is, if temperature is fixed, OLR decreases with increased absorption. This is what we call radiative forcing.

An important note is that OLR only decreases in the case of a temperature profile declining with height, as occurs in Earth’s troposphere. Let’s consider a perfectly isothermal case where $T_{0}=T_{1}=T_{2}=T$. In this case,

$\displaystyle \frac{\partial OLR }{\partial \epsilon} = \sigma T^{4} [1 - (2\epsilon - 1) -2 (1-\epsilon)] = 0$

In other words, there could be no greenhouse effect in an isothermal atmosphere! Conversely, if temperature increases with height, i.e, $T_{0} < T_{1} < T_{2}$, then OLR increases with increased emissivity, such that the planet is now emitting more energy to space and more energy than it is receiving. This reveals for us that if for some reason the temperature increased with height in Earth’s troposphere, adding CO2 would cool the surface. What about the contribution of each slab,$C$, to this OLR and their change in contributions? We have:

$C_{0}=\sigma T_{s}^{4}(1-\epsilon)^{2}$

$C_{1}=\epsilon \sigma T_{1}^{4}(1-\epsilon)$

$C_{2}=\epsilon \sigma T_{2}^{4}$

$\frac{\partial C_{0} }{\partial \epsilon} = -\sigma T_{0}^{4}(1-\epsilon)$

$\frac{\partial C_{1} }{\partial \epsilon} = \sigma T_{1}^{4}(1-2\epsilon)$

$\frac{\partial C_{2} }{\partial \epsilon} = \sigma T_{2}^{4}$

The math shows that regardless of emissivity, the surface layer contributes less to the outgoing radiation than before, and the topmost layer contributes more. The intermediate layer can give either more contribution or less contribution, depending on the initial value of emissivity.  The physical interpretation is that adding absorbers in our atmosphere shifts the height at which the bulk of emission takes place to higher levels. This makes sense- if you are standing over a clean pond, you can see further down to the bottom than if the pond were murky. Absorbers make the atmosphere murkier in an infrared radiation sense, and so a satellite looking down at the Earth would see emission coming from closer and closer to the sensor as the opacity went up.

I’ve built a computer program using a temperature profile from the NCEP reanalysis product (here). I interpolated the global, annual-mean vertical temperature profile out to 200 hPa (before we encounter the stratosphere) to 27 atmospheric slabs, plus a surface, giving 28 locations contributing to the OLR. I then used this temperature profile as the “inputs” into the layer model that I’ve outlined here. When you do this, I find that tuning the emissivity of each slab to a value of ~0.083 reproduces the modern OLR. Below, I plot the change in contribution of my 27 layers (layer zero at the surface, 27 at the top) to the OLR after perturbing the emissivity of the slabs by 1 % and 5%.In this experiment, the OLR was reduced by 0.79 and 3.9 W/m2 in the two cases.

As in the more simple two-layer case, we recover the phenomena of interest- that when we increase absorption, more emission is emanating from higher levels aloft, and less emission from near the surface. This is true regardless of the temperature profile. But we are now armed with the tools and intuition to wrap this together, and actually understand the physics of the greenhouse effect, something I’m not quite sure the formulation in many textbooks hit the mark on.

1) Adding greenhouse gases shifts the height at which radiation escapes to space. It is the greenhouse effect that determines the fate of longwave energy from its initial emission at the surface to its final destination out to space from the top of the atmosphere. Indeed, in the modern atmosphere, very little surface emission escapes directly to space (see the small “atmospheric window” contribution in this figure).

2) Because of what we call Kirchoff’s law, the emissivity of a substance must be equal to its absorptivity (at a given wavelength, but is true in the whole infrared band in this grey-gas formulation). The total emission is the emissivity multiplied by $\sigma T^{4}$ appropriate to the emitting substance. However, absorptivity of most substances is not very temperature-dependent, and in our model we have specified the absorption fraction (relative to a blackbody) which is equal to its emissivity. Yet, the total emission is very temperature dependent. As a result, warm surface emission gets absorbed by a layer and is re-emitted at a much colder temperature, and of lesser intensity. This reduces the OLR and provides a critical heat-trapping mechanism.

3) If temperature did not decrease with height, any addition of opacity would not change the OLR. Any sensor in space looking down could not distinguish the height of two surfaces of identical temperature, since both are radiating at the same intensity. In this case, any absorption would result in re-emission of the same intensity and no “heat trapping.”  Thus, it is the temperature gradient that sets the potential for infrared absorbers to create a greenhouse effect. Radiation, however, will typically demand a temperature profile that decreases with height, at least in a setting where the column is mostly transparent to sunlight and opaque to thermal energy.

4) We have not explored this component here, but any reduction in OLR implies a radiative imbalance. Eventually, our layers must increase in temperature in order to emit the extra energy that we absorbed, in order to satisfy the planetary energy budget. This is the greenhouse effect.

5) Instruction of the mechanics behind the greenhouse effect should stress the way in which absorbers alter the OLR given a fixed temperature structure, and how that temperature profile determines the strength of the greenhouse effect. This perspective centered on the top-of-atmosphere radiative budget (which the surface temperature is slaved to) will lead to better understanding of atmospheric radiative transfer.