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## Diagnosing feedbacks in simple layer models

In this post, I’ll revisit the simple two-layer column model described in the last post. This time we’ll use the model as a way to start thinking about the way we define “climate feedback.”

The top-of-the-atmosphere energy budget has been the vehicle we’ve employed in order to explore how planets come into equilibrium with radiation from their stellar host, and here we wish to further investigate how that equilibrium is obtained. It follows that we ought to be interested in how the outgoing longwave radiation (OLR) varies with surface temperature. By now, you are hopefully familiar with the fact that increasing emissivity (i.e., the greenhouse effect) in the atmospheric column will reduce OLR by shifting the mean emission height to higher, colder levels (keeping temperatures held fixed). We call this radiative forcing, or the instantaneous effect of the increase in column absorption on emission to space. Furthermore, we also know that the temperatures must then increase in order for the OLR to return to its original value. We’ll assume that shortwave absorption by the planet is unchanged, so the OLR (at equilibrium) in the “perturbed emissivity” climate is exactly identical to the value in the unperturbed climate. That zero anomaly in OLR can thus be decomposed into two parts- the radiative forcing that decreased OLR, and an equal but opposite flux that arises from all changes that occurred in the column to get back to equilibrium. It is the latter we’d like to further diagnose, and hopefully make contact with how scientists actually use models operationally in order to quantify feedbacks.

As before, the structure of the atmosphere is as follows:

And we write an expression for OLR:

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

We’ll prescribe the temperatures like in the first post on this model. The surface temperature will be 288 K, layer 1 will be 270 K, and layer 2 will be 240 K. Again, right now we’re not interested in the convective processes, etc., that cause those temperatures to actually be what they are- we’ve already seen that a shortcoming of just including radiation is that the lapse rate is too steep and a temperature discontinuity exists between the surface and air immediately above it. In other words, the opacity is sufficiently large that the temperature gradient would need to be unrealistically large if radiation alone were to move energy upward effectively in the troposphere. The troposphere is actually in radiative-convective equilibrium, with vertical motions being the major form of vertical energy transport. So, like before, we’re just going to fix the temperatures of our surface and atmospheric slabs to sidestep all this.

Using the first equation, with a handy little root solver I find that $\epsilon = 0.642$ in order to have a temperature profile like that described above with an OLR of about 240 W/m2, similar to the rate of energy Earth must emit to space in order to balance the incoming, absorbed solar radiation. This is just a tuned quantity in this simple model that has no special meaning- it would be a smaller number if we changed the number of atmospheric slabs from two to eight, for example, and shouldn’t be connected to the behavior of real greenhouse gases (remember, this is a “grey gas” model in which infrared absorption is independent of wavelength. In fact, actually “doubling CO2” in this setup would cause temperature changes far outside an Earth-like experience).

So we will not double CO2, but what if $\epsilon$ were increased to 0.655? That’s about a 2% increase. One could write a system of equations and solve for a new set of layer temperatures, but instead I’m going to take a different approach and just declare what the new temperatures will be. Any difference between what you’d calculate and my declaration could be attributed to some missing feedback or process. Again, these numbers don’t come from anywhere, and this is intentional…I’m just assuming there’s a set of processes which create the following temperature anomalies: the surface is warmed by 3 K, the first atmospheric slab by 4 K, and the second atmospheric slab by 5 K. I’ll use a prime (‘) symbol to indicate the new temperatures and a $\delta$ for anomalies, so at the surface, we have $T_{0}' = T_{0}+\delta T_{0} = 291 K$. Likewise, $T_{1}' = T_{1}+\delta T_{1} = 274 K$ and $T_{2}' = T_{2}+\delta T_{2} = 245 K$.

Because of my imposed declaration, we see that the temperature anomalies actually increase with height, so the upper levels warm more than the surface. This could mimic the effect of changes in latent heat release from an ascending air parcel, for instance…but again, we’re not actually modeling such a process so you can use your imagination in what sets this anomaly pattern.

Before we proceed, we should first ask what the radiative forcing is? Remember, we hold temperatures fixed and calculate the net flux change at the top-of-atmosphere (which is entirely longwave radiation here). Thus,

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

This is the change in OLR keeping everything else except emissivity fixed (including T but also any of the “missing processes” that may be relevant in determining the eventual climate). The new OLR is about 2.3 W/m2 less than before. Thus, we say that the radiative forcing is +2.3 W/m2.

What is the equilibrium climate sensitivity of this model? Sometimes people report sensitivity in temperature dimensions (K), e.g., the surface temperature response to 2xCO2, just because that’s a conventional forcing benchmark that people like to talk about. Other times we talk in dimensions of surface temperature change per unit forcing, so in this case the sensitivity is ECS=3 degrees per 2.3 W/m2 forcing, or ECS=1.3 K/(W/m2). I prefer the latter convention- although we ultimately care about temperature, to me it makes more sense to talk about a “more sensitive” system as being more sensitive because of the feedbacks inherent in that system. If we start talking about a system being more sensitive only because you’ve pushed it harder, then separating forcing and feedback in order to learn something doesn’t serve much purpose anymore. As it happens, CO2 forcing follows an approximate logarithmic function for most situations we encounter, so every doubling produces something like 4 W/m2 forcing (in reality, not in this simple model) so it’s easy to relate the two conventions by multiplying or dividing by four. But not every doubling needs to generate the same forcing, and in fact this approximation breaks down a bit when you start talking about deep-time hothouse climates or post-Snowball Earth greenhouses.

Back to the point, what is the net feedback in this model? Using $\lambda$ to denote feedback, it’s just:

$\displaystyle \lambda_{net} = -ECS^{-1}= -0.77 (W/m^{2})/K$

, or the inverse of climate sensitivity. The sign is negative just to indicate that the feedback is net negative, allowing stability and the presence of a new equilibrium. In other words, for every K surface temperature increase, the system must radiate another 0.77 W/m2 to space. So when rising temperature plays tug of war against increasing opacity, it takes 2.3 K of warming to increase emission by the same amount that we lost from the radiative forcing. This is actually the longwave component of the feedback, but there’s no shortwave feedback being considered here.

Planck feedback

We can further interrogate the effect of different feedbacks. Scientists do this all the time by artificially constructing different scenarios in which we ask what a particular effect would have in the absence of some other effect. Remember that I prescribed the temperature changes in the model above, and we can’t relate them to any physical process like water vapor, since our model knows nothing about them. But I will introduce two conventional feedbacks that we can at least quantify in this simple model. One of which is the so-called Planck feedback. This just says that emission to space increases with temperature, as it does in this model. More specifically, we calculate the Planck feedback by assuming that the temperature change at the surface is the same temperature change we’d encounter everywhere in the column. This is just a definition that reality doesn’t need to adhere to (and it doesn’t, nor does this model) but it’s useful to partition the vertical structure of warming into various components, the simplest is just uniform vertical warming. So, what would the new OLR be if just the Planck feedback operated? We’ll return to our unperturbed emissivity state, and increase the temperature to the Planck feedback profile. Then:

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

As you can see, we’re adding the surface temperature $T_{0}$ to each layer and quantifying the OLR change. In this case, the new OLR is ~251.4 W/m2, so the Planck feedback is:

$\displaystyle \lambda_{planck} = -(251.4 - 240)/3 = -3.8 (W/m^{2})/K$

In this case, the system isn’t very sensitive. Just one degree of warming lets the system shed a whopping 3.8 W/m2 to space, so you’d only need less than a degree of warming to counteract the radiative forcing we found before. But we found before that our system was less efficient than this at emitting energy to space, thus demanding a larger temperature increase to restore balance.

Lapse Rate feedback

In reality, the warming is not vertically uniform. Based on my imposed temperature change, we see the warming increases with height. Physically, we’d expect this to decrease climate sensitivity, since the upper layers are warming more (than in the scenario where we had uniform vertical warming). This increases outgoing longwave radiation more than it otherwise would, and takes some slack of how much the surface would need to warm in order to restore equilibrium. Conversely, if the upper layers didn’t warm very much, the surface would then have to pull the weight and warm up even more in order to increase OLR. We’ll verify this expectation below. Typically, in more realistic models we see a top-heavy warming structure in the troposphere, so that’s what I chose for the anomaly profile. Note we haven’t actually attached a physical process behind this structure right now.

So what is this lapse rate feedback, then? It is based on the departure from vertical uniform warming. More precisely, it is defined as the change in TOA longwave flux per unit surface temperature change that would arise because of deviations from uniform vertical warming. So, we have,

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

or,

$\displaystyle OLR_{LR} = \epsilon \sigma (T_{2}'- \delta T_{0})^{4} + (1 - \epsilon) \epsilon \sigma (T_{1}'- \delta T_{0})^{4} + (1 - \epsilon)^{2} \sigma T_{0}^{4}$

In this case, the new OLR is ~245.1 W/m2, so the lapse rate feedback is:

$\displaystyle \lambda_{LR} = -(245.1 - 240)/3 = -1.7 (W/m^{2})/K$

Like the Planck feedback, the effect of the lapse rate change is to increase OLR and thus reduce climate sensitivity. Note that,

$\displaystyle \lambda_{LR} + \lambda_{planck} = -5.5 (W/m^{2})/K$

Because of these two negative feedback, we have a very high radiative restoring efficiency and an insensitive system. We actually found before that the net feedback was $-0.77 (W/m^{2})/K$. This implies the presence of some positive feedback that would be required for this model to make sense given my imposed temperature changes.

In practice, this is how individual feedbacks must be diagnosed. Individual feedbacks must be calculated using a radiative transfer model based on the actual changes (for example, in water vapor concentration or lapse rate change, often from an actual GCM simulation) but then seeing how the TOA net radiative flux responds perturbing only that variable/process. Therefore, the decomposition of how OLR increases on the path to equilibrium, based on individual components leading to that increase, is a model-based exercise, and a good example of how models aid us in understanding the system.

Exercises to help with understanding

1) Under what temperature-profile conditions would increasing emissivity increase the OLR (negative radiative forcing)?

2) Verify that the sum of the two feedbacks we encountered here (Planck + Lapse Rate) is equal to what you’d get if you did the actual calculation, i.e., adding the true temperature anomaly profile to each layer with the unperturbed emissivity and calculating the OLR.

3) Do this whole exercise using your favorite computer programming language, this time with 30 layers of equal mass. Give this model a reasonable temperature profile (such as a dry adiabatic lapse rate) with a surface at 288 K. All atmospheric layers can all have the same emissivity, but the surface radiates like a blackbody. You will need to code an expression for the OLR, given the contribution from all slabs (and the surface) below. Tune the model’s emissivity until you get an OLR of 240 W/m2 given this temperature profile. Perturb the emissivity you found by increasing it 2% and calculate the radiative forcing. Add a temperature anomaly profile of the form $\delta T(n)=\delta T_{0}+0.15n$ where n is the slab counting from one above the surface (n=1 is the first atmospheric slab, etc). Calculate the Planck and lapse rate feedback. Note: the dry adiabat using pressure as a vertical coordinate is $T(n)=T_{0}(p_{n}/p_{0})^{2/7}$ where $p_{n}$ is the pressure level of the nth atmospheric slab. Surface pressure is 1000 hPa.

4) Using the model in exercise (3), verify that changing emissivity does not generate a radiative forcing in an isothermal column. Why not?