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:

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 in order to have a temperature profile like that described above with an OLR of about 240 W/m^{2}, 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 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 for anomalies, so at the surface, we have . Likewise, and .

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,

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/m^{2} less than before. Thus, we say that the radiative forcing is +2.3 W/m^{2}.

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/m^{2} forcing, or ECS=1.3 K/(W/m^{2}). 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/m^{2} 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 to denote feedback, it’s just:

, 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/m^{2} 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:

As you can see, we’re adding the surface temperature to each layer and quantifying the OLR change. In this case, the new OLR is ~251.4 W/m^{2}, so the Planck feedback is:

In this case, the system isn’t very sensitive. Just one degree of warming lets the system shed a whopping 3.8 W/m^{2} 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,

or,

In this case, the new OLR is ~245.1 W/m^{2}, so the lapse rate feedback is:

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

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 . 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/m^{2} 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 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 where 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?

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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 can be assigned an infrared emissivity of such that it emits radiation at a rate according to the Stefan-Boltzmann law. The limit shown in the above figure is where , 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 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 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 (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 , where is the direct emission from the slab and is the transmission of radiation through the slab that originated from lower levels. It follows that and the contribution of radiation, , from any level is . The surface is a blackbody, so and .

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

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/m^{2} 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., increases as CO2 concentration goes up. We have:

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 . In this case,

In other words, there could be no greenhouse effect in an isothermal atmosphere! Conversely, if temperature *increases* with height, i.e, , 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,, to this OLR and their change in contributions? We have:

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/m^{2} 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 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.

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(comments enabled there)

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A common reason that water vapor is treated differently from CO2 in the present Earth-based climate is that water exists in multiple phases (i.e., between liquid, solid, and vapor) and the partitioning between water in those various forms on Earth is a function of temperature. For our purposes, we are interested in the atmospheric component, which is predominately vapor. We define two quantities that will be useful for this post: the first, is the saturation vapor pressure, which is the maximum partial pressure of water vapor that can build up in an air mass. This quantity is a rapidly increasing function of temperature. For an Earth-like background pressure of ~1000 mb, then above about 350 K, water vapor would change from a trace constituent to one of the dominant atmospheric molecules by volume (like diatomic nitrogen or oxygen). We also define relative humidity, as the ratio of the actual vapor pressure to its saturation value. A relative humidity of one (or commonly “100%”) means the pressure of water vapor can be fully described by its saturation value, ), which is given by the Clausius-Clapeyron law (the below is an integrated form assuming constant latent heat of vaporization):

where A and T* are constants (with dimensions of pressure and temperature, respectively) that depend on the gas in consideration. T* is 5419 K for condensation into liquid and 6148 K for condensation into ice, while A is a large constant for water vapor (~2.56 x 10^{11} Pa). Note that 1 Pa = 0.01 mb. This formula thus predicts that is a rapidly increasing function of temperature.

It is easy to visualize the distinction between water vapor and the more “permanent” constituents of the atmosphere. Shown in the figure below, is a model result showing the evolution of water vapor concentration following an instantaenous removal or doubling of water vapor concentration. Convergence to the original equilibrium is very fast for doubled water vapor since relative humidity above 100% leads to rapid condensation and rainout. Evaporation is also fast-acting when starting with zero relative humidity, but the vertical and horizontal transport of moisture requires about a week to approach its equilibrium amount. Thus, on timescales of days to weeks, the “virtual forcing” of changing water vapor is no longer felt by the climate system. Thus, it is appropriate to think of water vapor as a feedback response to a climate change that must be initiated externally, rather than an independent forcing. This would be true for any condensing substance in a fluid dynamical setting, such as with methane on Saturn’s moon Titan.

Because water vapor is a greenhouse gas, its concentration increasing with temperature means that atmospheric opacity increases with temperature, thus providing a positive feedback to climate change. The physics of feedbacks were discussed in the last post (if the reader has not been initiated into the language of feedbacks, it may be worth visiting that post). The rapid adjustment of water vapor to the climate state also means one must be cautious with popular statements like “water vapor is the most important greenhouse gas.” This is true in the sense it makes up the bulk of infrared opacity, but its concentration is a slave to the current climatic boundary conditions. Aside from the solar energy received by the planet, CO2 concentration is the next most fundamental long-term determinant of Earth’s climate, and the most readily available “knob” that can be turned to shift Earth’s climate on decadal to geologic timescales.

A common remark one sometimes hears is that the water vapor feedback can be traced back to increased surface evaporation as the climate warms. This is a misguided line of thought, which mistakes the flux of water into the atmosphere with a reservoir. In fact, in a warming climate, water vapor could accumulate to its new (higher) saturation value even if evaporation remained constant. Evaporation (and precipitation) are largely controlled by the energy constraints at the surface and troposphere, and the changes in those variables in a warming climate *do not* scale with the amount of water vapor in the atmosphere (e.g., Held and Soden, 2006; O’Gorman et al (2012)).

Another common misconception is that Clausius-Clapeyron alone inherently means that the water vapor concentration of the atmosphere must increase with temperature. In fact, Clausius-Clapeyron is only an upper bound, and one needs to consider the large-scale dynamics of the atmosphere in order to determine the degree to which an air column will remain sub-saturated, and how that might change in a new climate (the Sahara desert is a clear example of where hot temperatures will not necessarily lead to abundant water vapor amounts). The distribution of relative humidity and the factors governing subtropical dryness is therefore a first-order determinant to Earth’s climate, as the behavior of outgoing radiation (OLR) is intimately connected with the humidity field.

The underlying physics has been explored at the interface of theory, e.g., the “advection‐condensation” paradigm based on trajectory calculations. Results such as this indicate that the observed humidity distribution in the troposphere can, to first order, be reconstructed by considering the large-scale circulation and temperature structure. Basically, this circulation advects a passive tracer that is allowed to precipitate its condensate when it exceeds 100% relative humidity returning the parcel to saturation. Within this framework, saturation occurs as air ascends and cools; upon descent, the parcel of air will conserve its water vapor mixing ratio, and so the parcel’s humidity will equal the lowest saturation value it has experienced since leaving the boundary layer. The humidity can be understood in terms of the point at which air parcels last experienced saturation (the saturation vapor pressure at the temperature of the last saturation point). This is for a point. For a volume, one can treat water vapor in a volume as the linear combination of all locations of last saturation. The relative humidity in a location is thus a numerator which accounts for the circulation and non-local temperature influences, divided by a saturation humidity corresponding to a local temperature. This simplistic view, despite limitations such as ignoring the evaporation of falling condensate (as occurs in areas of deep convection), can boast immense explanatory ability in reconstructing the observed humidity field. It has been applied in applications ranging from idealized models of the PDF of relative humidity to Lagrangian and Eulerian-based computations. This type of thinking has also impacted isotopic-based studies examining how the heavy and light isotopes of water are transported through the atmosphere, which has implications for the interpretation of natural archives used in paleoclimate reconstructions.

In a last‐saturation framework, a global warming situation translates to an increase in the temperature of last saturation and an associated increase in saturation specific humidity at the target point. So Clausius-Clapeyron does in fact enter into the heart of things, but one must consider its usage in proper context (i.e., one cannot just say that water vapor increases because the surface temperature goes up). Instead, it is linked to C-C through the statistics of last saturations. That water vapor feedback is substantially positive, and that global relative humidity changes are quite small when compared to the Clausius-Clapeyron dependence, is now a remarkably robust and uncontroversial result that has been established in many observational studies, and in models of varying complexity.

As it turns out, a great starting point for thinking about the water vapor feedback in climate changes is to consider a relative humidity which is invariant with temperature, such that the actual vapor pressure scales with Clausius-Clapeyron. This does not, however, mean that the distribution of relative humidity is constant. Consider the following figure, which shows the relative humidity difference on a pressure-latitude grid (averaged zonally) between 2xCO2 and control runs in several models. This emergent structure is robust amongst GCMs and to simpler models and within the simple advection-condensation philosophy.

_{Figure 1: Zonally (east-west) averaged relative humidity change as a function of latitude and height. See Wright et al (2010) }

Note that relative humidity increases at the tropopause outside the tropics, in the tropical deep convective zone, in the stratosphere, and decreases in the subtropics (near the positions of the mean-climate subtropical relative humidity minima). One typically sees an upward shift in the relative humidity field and poleward expansion of the descending (drying) cells of the Hadley circulation in global warming experiments. The circulation physics that leads to the low humidity conditions in the subtropics has been explored in a great many studies, including those elaborating on the simplistic view above and in more comprehensive models (see e.g., Galewsky et al., 2005 ; Pierrehumbert et al. 2007; Sherwood et al., 2010a ; Sherwood et al., 2010b). Paul O’Gorman and Tapio Schneider also have several good papers on this matter.

The fundamental determinant to climate sensitivity is to ask how the surface temperature is connected to the slope of the net radiation flux as a function of temperature, as explored in the last post. If, for example, OLR increases only weakly with temperature, than sensitivity will be higher than if the OLR increases very efficiently with temperature (and thus is quicker at restoring a new equilibrium). A question to ask concerning the water vapor feedback is what locations in the atmosphere contribute what to the perturbations in the radiative balance. Summing over all *k* locations,

where for constant RH,

The spatial contribution to water vapor feedback can be understood from this diagram in Soden et al (2008), which shows max sensitivity to water vapor in the mid to upper troposphere in the intertropical regions. The dry free troposphere is important, and clouds can mask the the effects of lower level moistening even in the tropics, while shifting peak contributions to the tropical upper atmosphere. Most of the water vapor feedback occurs above 800 millibars, since the reduction in OLR depends on the temperature gradient between the surface and layers aloft. See “ScienceofDoom” for a qualitative introduction into this figure and a broader discussion of the “radiative kernel” approach in climate sensitivity studies, which decomposes feedbacks into components determined by how a climate variable changes with temperature, and how that climate variable then modifies the radiative budget of the planet.

**The Runaway Greenhouse**

In the last post on feedbacks, I defined the feedback factor as:

with the terms defined in the previous post, with being the net radiation budget (OLR-absorbed shortwave) and in this case being the water vapor concentration. It was also demonstrated that climate sensitivity was proportional to 1/(1-*f*). Water vapor is a positive feedback since the above expression for *f* is positive. As the planet gets warmer, we can expect that the outgoing radiation to space should increase in order to restore the climate to a new equilibrium point. This equilibrium will be established as the absorbed shortwave radiation equals the outgoing thermal radiation, which in the optically thick limit, is “originating” from the upper troposphere, since all the exiting radiation from the surface will be absorbed before it reaches the optically thin regions where the optical depth becomes less than unity.

It is reasonable to ask what the temperature structure in a high-temperature, and water vapor enriched atmosphere would look like. Shown below, I have plotted a family of curves for the temperature as a function of pressure (decreasing with height). Curves are shown for surface temperatures of 250, 300, 350, 425, and 475 K. The temperature structure typically behaves like a moist adiabat, which is determined by the cooling of air parcels as they rise and expand, and the warming as the parcels release their latent heat. The net effect is for cooling with height.

However, once water vapor becomes a dominant constituent of the atmosphere, the moist adiabatic temperature profile simplifies to the saturation vapor pressure curve. In essence, on can re-arrange the first equation in this post, in order to solve for the temperature structure of a saturated atmosphere in which water vapor is the prevailing component.

In this limit, we see that the temperature structure of the upper atmosphere becomes fixed toward the saturation vapor pressure curve. Increasing the surface temperature by whatever means (and thus adding more and more water vapor via feedback) can then be viewed as increasing the total mass of the atmosphere and extending the warmer temperatures down to new (higher) pressures that did not previously exist. For example, at close to 500 K the total atmospheric pressure has now increased to over 10 times its modern value of 1000 mb. However, the temperature profile has become fixed aloft. Because all the radiation to space is emanating from the high, optically thin regions of the atmosphere, and because the temperature structure of this region is now fixed, we can intuit that the outgoing radiation must also be capped at a limiting value. We call this the Nakajima-Simpson limit (see Nakajima et al (1992) .

We can now see how the runaway greenhouse effect operates. In essence, there is competition between the OLR increasing with temperature, and OLR decreasing with water vapor content, as in the preceding equation for *f*. Eventually the latter term wins, and OLR no longer goes up (or even decreases) with temperature. It is possible to overshoot the limiting OLR values by different processes, such as relative humidity decreasing, or mixing in other gases into the atmosphere. The following figure is from Sugiyama et al (2005) (also see a more descriptive review of various radiation limits that have been found in the review by Goldblatt and Watson, 2012). The physics here has a long history extending at least back to the 1969 paper by Ingersoll, and evidently even in a 1920’s paper by Simpson (which I haven’t read). Kasting (1988) studied the problem with a non-grey (i.e., where absorption is spectrally resolved) model in connection with the evolution of Venus, where a runaway may have occurred.

The criteria for a runaway greenhouse to sustain itself is that the absorbed solar radiation exceeds the limiting OLR. In this case, there is no equilibrium solution, and no radiative balance. The planet becomes locked into a state in which it continues to absorb more solar radiation than can be emitted to space, and temperatures can runaway into excess of 1000 K. This state of affairs can eventually be escaped, either by terminating the water vapor feedback (e.g., when all the oceans are in the atmosphere, and the atmospheric mass no longer increases with temperature) or when the planet becomes hot enough to emit light near the visible or near-infrared, where atmospheric opacity is weaker.

Some claims have surfaced (e.g., by NASA’s Jim Hansen) that a runaway greenhouse is possible if we burned all the CO2. Unfortunately, there is no evidence in the planetary science literature to support the claim, and it can be dismissed based on the fairly trivial fact that the amount of sunlight that Earth absorbs does not even come close to the limiting OLR values typically found in the literature (usually > 300 W/m^{2}). Thus, even if the temperature is shifted to higher values, there will always be a radiative equilibrium. Carbon dioxide can cause a runaway in some of the “extreme” situations where you have overshot the OLR limit, as in the lower relative humidity cases above. CO2 will remove this hump, and can push you into a runaway, but you still need enough solar radiation to sustain it. That said, in principle, CO2 could still raise surface temperatures many hundreds of Kelvin if you continued to increase its concentration (a runaway wouldn’t occur at the modern boiling point of 373 K, since atmospheric pressure has increased quite a bit…one would need to wait until the critical point, 647 K, was reached for an Earth-sized ocean to be lost). As the Goldblatt and Watson review paper pointed out, clouds are a typical ? factor that could change the “no runaway possible” argument, but it is quite farfetched to think that the net effect of clouds could come close to making up the difference in energy needed to trigger a runaway.

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I write the energy balance equation as before, which is typically a close balance between the incoming solar radiation and the outgoing longwave radiation (OLR). The latter depends on the atmospheric temperature profile and opacity, as discussed in previous posts.

where *R* is the net top-of-atmosphere radiation budget, which is zero in steady state, defined so that R > 0 means the planet cools down, and R < 0, it warms up. We will assume the planet radiates according to where is the surface temperature, and is now introduced to represent the bulk planetary emissivity (dominated by the atmosphere's ability to trap exiting surface radiation…the surface itself has an emissivity very close to one). If the radiation to space emanated exclusively from the ground, then . In reality, the ground radiates about 390 W/m^{2}, and because of the greenhouse effect, about 240 W/m^{2} exits out of the top of the atmosphere. This establishes a temperature gradient between the surface (~288 K) and the effective emission temperature of the planet many kilometers above the surface (~255 K) corresponding to a bulk emissivity of . We can write,

Suppose we are interested in the case in which we alter the atmospheric greenhouse concentration (say, doubling CO2), by changing , but leave the incoming solar radiation unchanged. Then taking the derivative,

For the present Earth case, we have noted that . The radiative forcing for doubling CO2 is (i.e., every doubling of CO2 reduces the outgoing radiation (or equivalently, the planetary emissivity) by roughly 4 W/m^{2} at fixed temperature). Solving for the temperature change above, we have:

which reduces to dT_{s} = -(288/4)(-4/240) = 1.2 K, so we expect every doubling of CO2 to change the global temperature by just over a degree.

The analysis above, in the language of climate scientists, is one that ignores feedback effects. In other words, only the temperature adjusts to the change in CO2 concentration (and correspondingly, the outgoing radiation of the planet, which is a pre-requisite to reach a new steady state). The increased or decreased radiation loss to space, as the planet warms or cools (respectively) is an extremely strong stabilizer of climate. We call this the *Planck feedback*. This is the default reference system used by the climate community by which we can evaluate other feedback effects (and as a reference system, is not often called a “feedback”). Feedbacks do not exert a direct forcing themselves (like external knobs such as CO2) but modify the net TOA energy balance indirectly because of their response to changes in temperature. This can further amplify or dampen the final temperature response. One such feedback already encountered was the ice-albedo feedback. Another is the water vapor feedback, which is likely the most important feedback for modern global warming.

**Brief Water Vapor Feedback detour**

On Earth, there is a substantial reservoir of water at the surface in the form of ocean or ice. If left undisturbed for a long enough time, water vapor would continue to enter the atmosphere until it reached a point in which its partial pressure equaled the saturation vapor pressure, which is an exponentially increasing function of temperature. Therefore, the content of water vapor in our atmosphere will increase or decrease as temperature goes up or down, respectively; since water vapor is a greenhouse gas, the additional water vapor will amplify whatever caused the initial warming (and same for cooling influences). This idea can be generalized to any greenhouse substance that exists in a temperature regime allowing it to be condense on a planet or moon in consideration (such as methane on Titan). As it happens, the atmosphere does not usually reach saturation, but it turns out that we can think of the fractional change in water vapor (i.e., the slope) as a great approximation to the slope of the saturation vapor curve (i.e., the fixed relative humidity assumption, which is borne out by observation and comprehensive models). This means the actual water vapor content increases quasi-exponentially with temperature, producing a positive (or destabilizing) feedback. A negative (or stabilizing) feedback suppresses the impact of the initial perturbation. I will explore more of the nuances of the water vapor feedback in a later post.

Feedback Physics

Let *⌀* be a control parameter (say CO2 concentration), and *r* a feedback factor that depends on temperature, i.e., *r = r(T)*, both of which modify the net TOA energy budget, *R*. Then,

The new temperature response (now including feedbacks) is therefore:

where,

These expressions may look complicated, but are rather simple when you examine the terms individually. The numerator in equation A is a measure of the radiative forcing (i.e., how the net TOA energy budget is altered with changes in the control parameter, like CO2 concentration). The bracketed term in the denominator is a measure of how the energy budget is altered with changes in temperature, since increase in temperature in almost all circumstances lead to an increase in the outgoing radiation. With no other feedbacks, this slope is .

Climate sensitivity is proportional to the *1/1-f* term. As can see in equation B, this is a function of how a feedback variable responds to temperature, and how that alteration actually influences the planetary energy budget. For example, we can replace *r* with the water vapor concentration of the atmosphere. Water vapor increases with temperature (i.e., is positive) while the outgoing radiation is reduced as water vapor increases ( is negative). This makes the entire expression for *f* positive, implying a positive feedback.

For *f* greater than zero, but less than one, the feedback is positive but still smaller than the stabilizing influence of the denominator (the increased radiation loss to space as the planet warms). This is to say that OLR still increases with temperature, even if the atmospheric opacity is increasing with temperature too. This increases climate sensitivity but does not result in a climate system that “runs away.” See the figure below. This diagram shows the outgoing radiation as a function of surface temperature, and two different values for the absorbed incoming radiation. CO2 is fixed in this diagram. You can think of the applied forcing as the change in sunlight between the red and green horizontal lines. The black curve corresponds to a planet that radiates according to (i.e., with only the Planck radiation “feedback” operating). The blue curve corresponds to the same case except with a water vapor feedback, which allows the atmospheric opacity to increase with temperature. This reduces some of the curvature you’d expect with a pure T^{4} dependence. In steady state, the outgoing radiation curves intersect at the incoming solar radiation lines. Therefore, the climate sensitivity for the black OLR curve case is the difference in the distance between the two blue squares as the solar radiation is increased. The climate sensitivity for the blue OLR curve is the difference in the distance between the red circles as the solar radiation is increased. One can see that you need to increase the temperature more in the latter case to reach the same radiative equilibrium. Put another way, the more sensitive system is less efficient at shedding its infrared emission to space, and so the temperature must increase by that much more in order to come back to balance. One can include shortwave feedbacks into this picture by plotting *R* on the vertical axis instead of OLR, or adjusting the position of the horizontal lines (as in the ice-albedo post).

Emergent Properties of the Feedback Theory

There are important consequences to the dependence of climate sensitivity on *1/1-f*. One is that the effects of feedbacks on climate sensitivity are not simply additive. The second is that how uncertainties in the feedback strengths themselves propagate onto the system response, depends itself on the “true” value of the feedback strength. Some illustrative examples will clear this up.

We can re-write the above expressions as:

where is the “no-feedback” temperature response (i.e., the response to a given forcing with only the reference system operating, such as the Planck response, and no other feedbacks). is called the system multiplier, which amplifies or dampens the sensitivity. If its value is greater than one, feedbacks are net positive; if its value is less than one, feedbacks are net negative. We can now shed light on the two properties outlined above.

**1) **Suppose two feedbacks are operating, both of which on their own would amplify sensitivity by 20% () relative to the case where only the Planck radiative response was working. This corresponds to both feedbacks factors, f_{1} and f_{2}, both of which have values of 0.17. The total system response with *both* feedbacks operating is:

If the feedbacks impacted the system response in an additive fashion, you’d expect a 40% amplification of sensitivity (), but in this case the response is actually 1.52 times the no-feedback reference system. This is because the feedbacks must also act upon each other, and not just on the reference system. The deviation from the additive property would be even more striking if the individual feedback multipliers were larger than 20%.

**2)** Suppose we don’t know the true value of *f* for a feedback of interest, but want to know how that uncertainty might correspond to uncertainty in the system response. Then we can differentiate the temperature with respect to a feedback factor,

The dependence on means that for some uncertainty in the feedback, , the uncertainty in the actual temperature response is larger for higher climate sensitivities. This property has been used by Roe and Baker, 2007 (for example) to suggest that narrowing the range of climate sensitivity may be quite difficult (however, see Hannart et al., 2009). A symmetric distribution of the uncertainty in the strength of the feedbacks results in a skewed distribution in the climate sensitivity itself, with a high probability of large values, though it should be noted that there’s no good *a priori* reason for uncertainties in feedbacks to be symmetric given information from observational and paleoclimate data. As a more practical matter, when using observations, sensitivity distributions depend on the prior assumptions made for the feedbacks. It took the community a while to figure out why some studies showed so dramatically different results from others. It depends largely on whether symmetric uncertainties were assumed for the feedback or the inverse of it. A linear relationship between observational data and the radiative restoring efficiency of the planet, implies that the derivative of the sensitivity with respect to the data becomes zero in the limit of high sensitivity (i.e., it would be harder to distinguish the difference between a 5 and 10 C sensitivity based on looking at the transient response to a volcanic eruption, than it would the difference between a 1.5 and 2 C sensitivity; see e.g., Frame et al., 2005 ).

**The f=1 regime**

One might expect based on the above equations that as *f* becomes close to one, the climate sensitivity blows up to infinity and a runaway effect develops (physically, the total feedback effect begins to overwhelm the stabilizing influence of the Planck radiation response). We encountered this with the Snowball Earth discussion, whereby a runaway ice-albedo feedback became sufficiently strong at low enough CO2 concentrations (or sunlight) that it forced the entire planet into an ice-bound state. This runaway feedback then ceases once the planet is covered in ice.

Of course, one cannot use this linear analysis anymore as *f* becomes too close to one. Feedbacks can be thought of in terms of a Taylor expansion series, and linearization corresponds to just the first term in such an expansion. The next term involve those that depend on df/dT (i.e. the dependence of the feedbacks on the climate state), then d^{2}f/dT^{2}, etc. One can imagine that which feedbacks operate, or the relative importance of those feedbacks, depends on the climate state, and so the linear analysis is no longer valid for large enough perturbations. In the next figure, I plot the net TOA energy budget (this time, *R*) on the vertical axis vs temperature on the horizontal axis. We apply a radiative forcing, , that brings the climate state from the blue curve to the red curve (for instance, by adding CO2 and lowering the OLR). If there are a set of processes (like an ice-albedo feedback) that happens to give that climate the feedback structure shown in the diagram, then f = 1 actually doesn’t mean a runaway, it means a bifurcation from the first blue circle to the next. f=1 could mean a runaway if there were no equilibrium point on the other side of the bifurcation, but no information can be obtained about where such an equilibrium point might be (or if it exists) without knowledge of the non-linear Taylor expansion terms.

_{See discussion in the Pierrehumbert lecture linked below}

That is a bit of introduction into thinking about feedbacks from a theoretical perspective. Later posts will be sure to elaborate on this, and aim at more practical diagnostics of sensitivity for the modern case, in additions to discussions of other extreme bifurcations aside from Snowball Earth (e.g., the runaway greenhouse).

Some recommended reading/viewing on this is Hansen et al., 1984, Roe, 2009, and this lecture by Ray Pierrehumbert, the last one focusing on bifurcations specifically.

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In Part 1 a first, simple model of planetary temperature was discussed, all based on knowledge of how much starlight a planet receives () and how “reflective” that planet is (i.e, its albedo, , an effect elaborated on in this post).

In the first post, an “effective temperature” was solved for of the form:

where is a geometrical redistribution term that accounts for how well the input of stellar energy is evened out across the planet by rotation/thermal inertia and planetary motions. For a sufficiently rotating planet with an advecting atmosphere, it takes the value of ~4, but it would be more appropriate to take on a different *local* value on an airless body incapable of transporting much heat around (concerning the question of habitability, this could set up a regime in which water could exist in liquid form over parts of a planet but not others).

We now introduce the effect of an atmosphere that can interact with the radiation entering or exiting the planet. On Earth, that interaction is predominately in the infrared (the outgoing energy) via absorption/emission processes and is accounted for by trace gases in the atmosphere, which we refer to as *greenhouse gases* (water vapor, carbon dioxide, methane, nitrous oxide, etc); the interaction can also occur with aerosol particles in the atmosphere and with clouds, though these latter two also tend to scatter shortwave solar energy and cause a net cooling effect on Earth (there are exceptions to this, such as black carbon). Ozone interacts in the infrared as a greenhouse gas, but also absorbs incoming UV radiation high in the stratosphere to set up a persistent thermal inversion where temperatures increase with height.

On Venus, the predominant atmospheric constituent is CO_{2} which acts as a strong greenhouse agent, although contributions exist from sulfur dioxide, water vapor, and sulfuric acid based clouds. On Titan, one of Saturn’s moons, a greenhouse effect is arises due to N_{2}, CH_{4}, and H_{2}. It is a widespread notion that symmetric, diatomic molecules cannot behave as greenhouse gases (as is the case with both main constituents of the Earth’s atmosphere- N_{2} and O_{2}). In order to be a good infrared absorber, the molecule must **1)** have quantum energy transitions whose energy corresponds to the infrared spectrum **2)** a dipole moment, or a charge distribution such that a disproportionate amount of the electron clouds “negative charge” is clumped up to one side and the “positive charge” to another side. Symmetric, diatomic molecules do not exhibit a static electric dipole moment (as with water vapor) nor is there the possibility to vibrationally induce a temporary dipole moment, as in the case of CO_{2}, where the “bending” mode allows for interaction of thermal radiation at ~15 micron wavelength.

However, in sufficiently dense atmospheres (as on Titan and on gas giant planets) collision-induced absorption leads to absorption features in diatomic gases. The collision-induced dipole that forms from colliding molecules forms transitorily, but can lead to broad absorption bands. Collision-induced absorption dominate the far-infrared spectra of Jovian planets. This is a direct radiative effect, but the diatomic molecules also have an influence in broadening the absorption lines of the typical greenhouse substances in our atmosphere. This pressure broadening effect becomes more important as the gas pressure increases, since collisions will be more abundant. The lack of any substantial atmosphere on Mars (1/100th the pressure of Earth’s atmosphere), despite consisting mostly of CO_{2}, cannot generate a strong greenhouse effect because of this.

Figure 1 shows a typical thermal radiation spectrum emitted by the Earth (as seen from a viewer in outer space).

The colored curve corresponds to the emission of Earth; the solid, light curves in the background correspond to the emission that would emanate from a “blackbody” (essentially a perfect radiator). Radiation is emitted across a spectrum of wavelengths, and the intensity at all wavelengths increases as the temperature is made higher (the family of curves shown on the diagram is the wavelength distribution of radiant intensity at various temperatures). The relevant equation to describe this relationship is the Planck law which can be integrated over all wavelengths and directions to yield .

In regions where the atmosphere is transparent to thermal radiation (e.g., between 8-12 microns, with the exception of ozone) the radiation is seen (from space) to emerge from the warm surface. In contrast, at opaque wavelengths (e.g, 15 microns) the radiation from the surface is absorbed by the atmosphere and will be shielded by the viewer in space. Radiation that interacts strongly with the atmosphere can only exit to space from the upper, thin layers of the atmosphere where it is quite cold. This effect is manifested as a “dip” in spectrum at those opaque wavelengths, which of course must physically correspond to a reduction in OLR at that temperature. In fact, regions of low OLR in the tropics for instance are often used as a proxy for areas of deep convection in satellite analyses, since the low thermal radiation comes from the cold, clouds tops. Figure 1b contains the same information, except showing the transmissivity () as a function of wavelength, which approaches zero as the opacity becomes large. can be related to a quantity called optical depth, , which is a measure of the opacity. It depends on the density of the absorber and the pressure interval upon which a beam of light is traveling. At normal angle incidence, where we remember that both quantities are wavelength-dependent.

Basic Radiative Transfer

Let us begin with the Earth-like case in which we add a substance, such as CO_{2}, that is opaque in the infrared but transparent to incoming solar radiation. This addition of greenhouse gas will reduce the planetary outgoing longwave radiation (OLR) as seen from an observer in space. We define a greenhouse parameter, :

and now, the surface temperature becomes:

Figure 2 below shows the OLR of a planet that radiates as a blackbody (in the black line) according to and the red curve shows the same situation except with 400 parts per million (ppm) of carbon dioxide added to the atmosphere. The horizontal curve is a constant Earth-like value for the absorbed solar radiation, . The equilibrium temperature must correspond to the intersection of these curves. The addition of the greenhouse substance reduces the OLR at any given temperature, since some portion of the energy is now being blocked; alternatively, the intersection point must occur at a higher temperature.

We can formulate the vertical temperature profile as a function of optical depth, , employing useful approximations. We will use the so-called Eddington approximation for a “grey” atmosphere (grey meaning that the absorption is wavelength-independent, which is clearly unrealistic, but a useful starting point for conceptualizing the problem). We have,

and, we can write the ground temperature, , as function of the planetary emission temperature ( as defined in the first link),

where is the column infrared grey optical depth. is the air temperature at , or the skin temperature of the planet.

When greenhouse gases are added to our atmosphere, the column optical-depth increases and the profile moves upward. Suppose, for instance, that the atmosphere was optically thick throughout the infrared and eventually became optically thin enough at some high altitude, such that the observer in space could only see to the level. The observer would be blind to all events happening below , much in the same way as we cannot “look into” the sun’s outer photosphere very far (as all radiation has been absorbed before exiting into space). As greenhouse gases are added, the height of the surface will move up, such that radiation does not escape to space until it reaches a higher altitude than before. Of course, where a surface is located is wavelength dependent. On the gas planets, it is not often desirable to know about the entire atmosphere right down to the interior of the planet, so for many purposes it is sufficient to consider radiation down to the point where the fluid becomes dense enough that it radiates like a blackbody. This acts like a “surface” (just in the same way that you don’t need to know the temperature profile right down to the core of the Earth in order to do atmospheric radiation).

It is typical that the radiative equilibrium profile described above introduces a strong temperature discontinuity between the surface and overlying air column. This results in convection that transports heat from the surface to overlying atmosphere. The concept of radiative-equilibrium is explored, for example, in Manabe and Wetherald, 1967 (see also Isaac Held’s useful summary). Typically, if the vertical temperature discontinuity becomes too large, we think of the atmospheric temperature profile being relaxed to some critical value called the *lapse* *rate* via convection, which has a value of on Earth (where c_{p} is a specific heat value). The actual lapse rate on Earth tends to be closer to a moist adiabat, which is less steep (-6.5 K/km or so) due to the latent heat of condensate being released which partially offsets the cooling induced by a rising parcel of air as it expands under decreased pressure.

The following diagram, from Manabe and Strickler, 1964 (Figure 3 here, Figure 4 from the paper) shows a radiative-equilibrium profile along with a a dry and moist adiabatic lapse rate typical of the convecting part of Earth’s atmosphere. Lapse rates of this sort emerge in other planetary atmospheres as well.

**Anti-Greenhouse Effects**

It is possible to modify the picture somewhat by including the effects of solar absorption in the atmosphere. Earth’s atmosphere is ~80% transparent to incoming sunlight, though even the energy deposited throughout the troposphere doesn’t substantially impact the above argument, as the whole column is yoked together by convection, requiring us to think about the energy budget of the whole surface+troposphere column. However, it is possible to substantially decouple the surface from the atmosphere with enough atmospheric solar absorption, or to absorb the sunlight high enough (without impacting the reflection) such that there is no communication with the surface. On Titan, this “anti-greenhouse effect” partially compensates for the traditional greenhouse influence. It arises from the absorption of sunlight by haze and CH_{4} in the upper atmosphere (e.g., McKay et al (1991)). This haze is opaque at visible wavelengths but is virtually transparent to thermal wavelengths, in contrast to the greenhouse case of familiarity. This is also similar to the “nuclear winter” problem, and has also been proposed to be important on early, Archean Earth.

Following McKay et al (1999), with this type of solar absorption acting, the surface temperature can be expressed as:

where is a measure of the anti-greenhouse effect strength (i.e., the portion of solar energy blocked by the anti-greenhouse layer but not reflected back to space). In the extreme case where there is only a strong anti-greenhouse effect acting () with no infrared opacity (), then the surface actually becomes colder than the emission temperature of the planet by a factor of (1/2)^{0.25}. This can be imagined by envisioning a layer between the surface and space; the high-altitude layer absorbs all the solar energy, upon which it then emits half back to space and half to the surface as thermal radiation. The surface then radiates all of the emission to space, thus receiving an amount of radiation half of the incident value but returning it all to space.

One could also make the atmosphere isothermal via solar absorption, in which case the greenhouse effect diminishes due to the lack of temperature differential between the surface and top of the atmosphere (recalling the definition of ). Physically, one can see this based on the infrared spectrum plotted in Figure 1. If the upper atmosphere radiated to space at the same temperature as the surface, there would be no “dips” in the spectrum and no need for the atmosphere to heat up. When a “dip” occurs, it represents a portion of energy that would have otherwise escaped to space but no longer is. This means the planet is now taking in more energy than it is losing. That lost energy “removed” from the total OLR must be accompanied by an increase in column temperature. This increases the emission at other wavelengths by an amount equal to the area of the “bite.”

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There has been a large interest during the last couple of days in the Venus transit, where the second planet in our solar system passed directly between Earth and the sun, which was seen by many people and shown in the video above. For us, this phenomenon will not happen again until 2117. However, a viewer on a planet orbiting a distant star might see a Venus transit every Venusian year (~225 Earth days) if they resided in the same plane of orbit. Likewise, it would be possible for that observer (or for example, an observer on Mars) to see an Earth transit.

This leads to the subject of the post, which may be a bit more descriptive than others. Unknown to many yesterday, the world had a front row seat to a common method that is used to detect exoplanets (distant planets that orbit around stars other than our own, often tens of light-years away). The Venus transit will be used a test of the quality of the technique. The detection of such exoplanets is the goal of the Kepler satellite, one of the greatest scientific missions of our time. Kepler is NASA’s first mission capable of detecting Earth-size planets in orbit around other solar-like stars. So far, well over 1200 candidate planets have been discovered since 2009, with sizes ranging from less than Earth to twice as large as Jupiter (and with orbital periods shorter than a day to more than a year). It is a photometric space-based mission with the intent of finding bodies that orbit in the so-called habitable zone of their host star. This is the region where liquid water water is capable of being sustained on the surface of a planet (the factors that govern these limits will be discussed in later posts). From an observational and climate perspective, it is also possible to retrieve information about the atmospheres of such planets based on spectroscopy techniques of selected transiting planets.

As we’ll see, transiting planets offer the possibility to detect and characterize atmospheres of giant planets like Jupiter, and should also provide an access to detecting biomarkers (atmospheric constituents indicating the presence of life) in the atmospheres of Earth-like exoplanets. This science is in its infancy but is growing rapidly, and offers the most promising way to detect life outside our solar system. But how does it all work?

When a planet passes in front of its host star, it blocks a very small amount of starlight and there is a slight reduction in brightness when viewed from Earth. The transit reduces the stellar brightness in an amount equal to the ratio of planet-to-star area. Assuming the flux from the planet is negligible compared to that of the star, then:

The left hand side is the fractional brightness change, and *r* represents the radius of the planet or star.

For those interested, when the radiation passes through the atmosphere and absorbed, the emerging radiation has of course been attenuated by a certain factor. The intensity of the resulting radiation at wavelength is:

where is the slant optical depth integrated through the planet’s atmosphere along the observer’s line of sight (a measure of the atmospheric absorption efficiency, neglecting additional scattering), is the cosine of the viewing angle, and is the initial intensity.

The reduction in brightness is on the order of ~1% for a Jupiter-sized planet orbiting our sun and ~0.01% for an Earth-sized planet. This makes it a challenging problem but still accessible with current technology. The timing between subsequent “dips” in the brightness curve is related to the orbital period, and in principle the size and duration of those brightness “dips” will be constant.

Thus, detections of repeated reductions in stellar brightness (of similar magnitude) indicate the presence of a transiting body around the star. It also turns out that the probability of observing a transit from Earth is related to the distance of the planet to its host star. That probability increases for planets closer in. Fortunately, it is now known that there is a large population of planets orbiting very close to their host stars. These are so-called hot-Jupiters, hot-Neptunes, or even hot super-Earths, and they reside in the neighborhood of just 5% the distance between Earth and Sun! Temperatures of these bodies can exceed 1000-2000 K, and the close proximity of the planet to the star forces such planets into a tidally-locked state in which one side is in perpetual daytime and one side in darkness. The distribution of stellar radiation and resulting atmospheric circulation is therefore unlike any in our solar system, but may be a common occurrence in general.

Absorption of starlight passing through the planet’s atmosphere during transit can give information concerning the composition and scale height of the exoplanet atmosphere. The passage of the starlight that has passed through the atmosphere carries with it a signature of the atmospheric composition, as the planet’s absorption features become superimposed on the observed stellar flux. As seen in Figure 2 (directly above), when the planet occults a portion of the stellar disk, a fraction of light from the star is seen that has traversed a part of the atmosphere around the planet’s limb (the smaller white area surrounding the black circle that represents the planet). Figure 3 shows the same thing in a different way. Light that penetrates to the lower atmosphere will not likely emerge “from the other side” and will be invisible to an Earth-based observer, though light that “skims” the atmosphere will make it through and be viewable from an Earth observer. This depends on the nature of the atmosphere itself (e.g., its height) and how well it interacts with radiation. The spectral signature of the starlight that passes through the upper, optically thin region can be compared with observations of the planet-star system when not in transit, and the planet’s transmission spectrum measured from the difference.

Of course, planets in circular orbits that pass in front of the star must also disappear behind the star (Figure 2). When this happens, only starlight can be observed from Earth, and not the light from the planet+star combined (planets emit energy in the infrared wavelengths, and these “hot” planets emit generous amounts, within detection limits). On Earth, the received light from the planet+star can be contrasted with observations of the star only after the planet disappears in the secondary eclipse. The difference is an estimate of the planetary emission alone. Of course, the secondary eclipse will correspond to a smaller reduction in brightness than in the primary eclipse (since the star is being blocked in the primary case, and the planet being blocked in the secondary case) (see figure below).

It actually turns out the size of the smaller dip (at second eclipse) can be related to the brightness temperature of the planet, although this is not a full estimate of surface temperature, since other things like the presence of a greenhouse effect may come into play.

As some examples, Charbonneau et al., detected sodium absorption in the atmosphere of planet HD 209458b (Charbonneau et al. 2002), and methane in the atmosphere of planet HD189733b (Swain et al., 2008, see figure below). This was the first detection of any carbon-bearing molecule on a planet outside our Solar System, all possible because that planet can be seen to transit its star from Earth!

Measurements of the difference in spectra during the secondary eclipse was done, for example, in 2005 by Deming et al. on the extra-solar planet HD 209458b (Deming et al. 2005). Water vapor, carbon dioxide, and methane have all been detected in exoplanetary atmospheres (e.g., Swain et al., 2009; Tinetti et al., 2010). Moreover, thermal inversions have been detected on a number of exoplanets indicating a solar absorber in the upper atmosphere (see Seager (2010) for a review).

CO_{2} features have been found in the HD 189733 thermal emission spectrum, which is somewhat surprising since at very high temperatures atmospheres which are dominated by molecular hydrogen are expected to have carbon primarily in the form of CO or CH_{4}.

These examples are all “large” planets; characterizing the atmospheres of Earth-mass planets with transmission spectroscopy is extremely challenging because of the small extent of their gaseous envelopes. One can imagine from the second and third figure that obtaining good enough light signatures (that have passed through an exoplanet atmosphere) is very difficult if those atmospheres are more compact.

**Venus as an Exoplanet**

Testing the above methods will be useful for the planetary science community since we know the atmosphere of Venus rather well through independent methods, such as visits from monitors like the Venus express mission. It is a hellish place. The temperature of the Venusian surface is roughly 735 K (860 Fahrenheit), hot enough to melt lead. Its primary atmospheric constituent is carbon dioxide, though with trace amounts of water vapor and sulfur dioxide. The greenhouse effect generated by CO_{2} is largely responsible for this high temperature, and is a far-away extreme case of anything humans are capable of doing on Earth, though the physics is precisely the same. Venus has multiple cloud decks composed of sulfuric acid particles, and exerts an atmospheric pressure of roughly 90 Earth atmospheres (equivalent to about half a mile of ocean water above you).

A theoretical transmission spectrum of the atmosphere of Venus that will be tested with spectroscopic observations during yesterday’s transit was provided by Ehrenreich et al (2012) (a solar transit of Venus in the 1700’s was actually the first time it was discovered that Venus had an atmosphere). Shown below are typical temperature profile and profiles of constituents in the atmosphere, provided by the Venus Express mission.

The transmission spectrum covers a range of 0.1–5 μm and probes the limb between 70 and 150 km in altitude. This is caused by droplets of sulfuric acid composing an upper haze layer above the main deck of clouds, so someone on Earth cannot “see” radiation passing through Venus at altitudes lower than this. The lowest altitude reached by transmission spectroscopy is determined by the dominant scattering regime (Rayleigh scattering on Earth caused by the dominant atmospheric constituent N_{2}, which preferentially scatters blue light and explains the color of the sky; in contrast, Venus is primarily in a Mie scattering regime, caused by cloud cover, and less so by the main constituent, CO2). The difference is caused by the relative size of the particles doing the scattering compared to the wavelength of light.

**Earth as an Exoplanet**

A holy grail of exoplanet detection, and arguably of science itself, is the detection of a life-bearing world. Transmission spectroscopy has also been applied to Earth as well from satellites looking back (e.g., Kaltenegger and Traub, 2009). Ozone and diatomic oxygen are present in these measurements as expected, even though the lower part of the troposphere is not accessible for detection using this method for reasons mentioned earlier.

On Earth, oxygen is very reactive and could not exist in substantial amounts unless it was constantly produced. That source is photosynthesis. Oxygen and ozone together are Earth’s two most robust biosignatures. There are other ways to have high oxygen loadings in the atmosphere, however, which would not indicate life. Such a condition could arise if the planet’s ocean were vaporized into the atmosphere. This is the classical “runaway greenhouse” phenomenon thought to have occurred on ancient Venus. A byproduct of this is large quantities of water at high altitudes in the atmosphere, where high-energy radiation can break apart the water molecule and the light hydrogen species can escape into outer space while the heavier oxygen molecule is left behind (and O_{2} builds up in the atmosphere for a short period of time before reacting with the crust). The relative ratios of oxygen and methane in our atmosphere are not compatible with an abiotic world either, although it may be difficult to establish both spectral features for a far away exoplanet. It may therefore be challenging to find robust biosignatures.

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One of the fundamental characteristics of a planetary atmosphere is its wind distribution with height. Shown in the opening figure is the east-west component (or the *u* component) of wind speed on a latitude-height grid. The vertical axis is pressure, which decreases upward with height, since there is less air above you as one progresses higher in the air column.

One of the most noticeable features of the figure is that winds become more westerly (i.e., stronger towards the east) with height. The question motivating this post is what drives this observed phenomenon, and can we come up with a relationship between vertical wind shear (or the change in wind speed/direction with height) and horizontal temperature gradients? It is certainly not intuitive that such a relationship would exist. By the end of this post, we will conclude that the presence of westerly vertical shear is a direct consequence of the uneven solar heating of the Earth (more heating at the equator) coupled with the dominance of something known as the *thermal wind balance*, particularly in mid-latitudes. Broader application of this thermal wind balance will arise in future posts.

I must back up for a moment and lay out a few fundamental physical relationships that will set the stage for our exploration of thermal wind. The physical steps I will outline here may seem random, but hopefully it will all come together in the end.

Consider first the question of how “thick” a layer of atmosphere is between two pressure surfaces. That is, pick your two favorite two pressure levels above the surface (say 800 millibars as the level closer to the surface, and 400 millibars further up in the atmosphere). Then, what is the physical distance between those two locations in the atmosphere? A starting point to explore this question is the *hydrostatic equation*, which can be derived from a simple force balance: Gravity is always trying to pull air molecules close to the surface. In contrast, there exists a vertical pressure gradient which compels air molecules at higher pressure (close to the surface) upward to regions of lower pressure (further above the surface). The competing influence of these two effects is why our atmosphere does not fly away into space or get dragged down to a thin layer near the surface. Hydrostatic equilibrium is obeyed to very good accuracy in planetary atmospheres, and even in the sun for example. The mathematical expression for this balance is:

the left hand side of the equation is the differential of pressure with respect to height, is density, and is the acceleration due to gravity. We also know of another, more familiar fundamental law that emerges from thermodynamics: the equation of state for an ideal gas, which is , where R is a gas constant appropriate to the atmosphere in consideration and *T* is temperature. We can combine these two equations to yield,

We can integrate this relationship between two pressure levels, which gives a relation:

where p_{1} > p_{2} and the height z_{2} > z_{1}. Since *R* and *g* are constants, and we have fixed what pressure levels we are interested in, we end up with the conclusion that the thickness between the two pressure levels is directly proportional to the temperature of the layer.

As a next step in this analysis, we have learned in high school that horizontal (east- west, or north-south) wind is caused by pressure gradients. Wind is directed toward regions of lower pressure, but is deflected by the coriolis effect that occurs in a rotating frame of reference. In particular, we call the north-south direction the *y* direction and the east-west direction the *x* direction; the velocity in those two directions (the wind) is the *v* and * u* component, respectively. Ignoring a few small terms, the accelerations are given by:

and,

The term *f* is called the Coriolis parameter, which is proportional to the sine of the latitude, and also increases for higher planetary rotation rates.

In the mid and high latitudes, the influence of the coriolis effect and the pressure gradients are typically an order of magnitude larger than horizontal accelerations. In particular, when one moves away from the surface (where friction is important, one of those “small terms” mentioned before), then we can define a force balance between the horizontal pressure gradient and Coriolis effect. This gives rise to the *geostrophic * component of the total wind, which in many regions of the planet constitutes a substantial fraction (or nearly all) of the total wind field. The deviation from geostrophy is usually very small about ~1 km. It is also convenient to eliminate density from these equations and define a term called geopotential, . The total geostrophic wind component can be expressed as blowing parallel to lines of constant geopotential heights (or heights), (the vector is the total geostrophic wind and is the vertical direction that one applies the cross-product to, which defines the wind direction. For those uncomfortable with such vector notation, the graphical expression of the above equation is shown below (see description in caption):

The vertical gradient of the geostrophic wind is:

**The Conclusion**

Finally, because the layer-averaged temperature is directly related to the layer-averaged thickness between layers, we can write an expression for the vertical shear in the geostrophic wind. For both the east-west geostrophic wind () variation with pressure, and the north-south geostrophic wind component (), we write:

and,

This confirms the rather remarkable conclusion that the vertical gradient of the horizontal winds is proportional to the horizontal temperature gradient! The physics is straightforward from the second figure: Increasing slopes of constant pressure surfaces, which is related to gradients in geopotential height, which in turn is determined by gradients in temperature. Because the geostrophic wind increases in magnitude as one tightens pressure gradients, the increased slopes in Figure 2 imply increasing geostrophic wind. See Holton’s “Introduction to Dynamic Meteorology” or Jonathan Martin’s “Mid-Latitude Atmospheric Dynamics” for similar presentations and a more complete discussion/application of thermal wind balance.

From the first equation for example, one “broad” application of the first of these two equations is that temperature decreases poleward. That is dT/dy is negative. Therefore du_{g}/dp is negative (alternatively, du_{g}/dz is positive if you intuit things better in height coordinates). This confirms our model and observational result that winds become more westerly in the mid latitudes. In the region between the equator and the Sahel (near 30 N) in Africa, temperatures actually increase poleward and an easterly jet is found.

Note that in the oceans, an equation of state can be used to give density gradients in terms of temperature/salinity variations. Neglecting salinity gradients, if ocean temperatures decrease toward the poles, currents will become increasingly westward as a function of depth.

Application of the thermal wind has extremely important use for meteorologists and atmospheric dynamicists. It has been used extensively by atmospheric scientists (and at least one study used its application in order to use wind measurements as a proxy for temperature trends in the upper, tropical atmosphere, given the large uncertainties that have plagued the robust detection of this phenomenon- the relative warming of the upper atmosphere relative to the surface in a warming world has strong implications for tropical dynamics and stability). I will return to application of the thermal wind equation in later posts on the general circulation of our atmosphere.

**A couple of notes:**

The thermal wind equation can be applied to other atmospheres as well, although this post is a bit Earth-centric in that geostrophic balance is not always a good approximation. In particular, the Coriolis component of that balance becomes much less important at lower rotation rates (as on Venus, or tidally-locked exoplanets) and other terms become more important. Geostrophy relates the 3-dimensinal structure of the winds and temperatures at a given time but says nothing about the flow’s time evolution (note in this post that we removed the acceleration term from the momentum equations in order to come up the geostrophic balance, which lost any time dependent terms). In a rapidly rotating atmosphere, both the temperatures and winds will generally evolve together, and maintain approximate geostrophic balance as they do so. On Venus, in contrast, the curvature term (latitude and planet radius ) is the dominant force term along with the pressure gradient force. This yields a “cyclostrophic balance” that neglects the coriolis term (this is also important in tornadoes, for example). One can come up with a thermal wind version that is applicable to a flow predominately in cyclostrophic balance, and in fact the temperature gradient becomes directed toward the equator regardless of whether the wind shear is westerly or easterly with height, in this regime. It is also worth adding that it is often preferable to use the so-called virtual temperature in determining layer thickness as opposed to real temperature. This is a minor correction for the presence of water vapor in the atmosphere and its impact on air density (this way one could retain the dry gas constant *R* in the thermodynamic relation). It’s a very small effect however.

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where T_{s} is the surface temperature, and the other terms are defined as before. I will focus this discussion on the ice-albedo feedback, since the extent to which a planet is covered in ice will be intimately connected to temperature. One can intuit that changing the ratio of ice surface to land/ocean surface, in response to climate change, will modify a planets reflectivity to sunlight and amplify the initial cause of the change. One can also speak of albedo changes due to desertification or re-forestation, for example. However, the ice-albedo feedback is a common example of thinking about surface albedo changes, and one that also enters prominently into the “snowball Earth” issue that I want to shed light on (and has broader connections to planetary habitability as one moves farther away from a star). To move forward with the discussion, I work under two assumptions:

**1) **We are interested in a water version of the ice-albedo feedback on an ocean-covered world. This is because water is unique amongst expected planetary substances in that its ice form is less dense than its liquid form, so the ice can float on an ocean. Ice composed of carbon dioxide or methane floating on an ocean of CO2/methane would sink, preventing surface albedo fluctuations. It is also possible that one can develop glaciers on a land surface, although then the problem becomes more complicated than simply knowing the freezing point. Rather the details of where precipitation falls become vital. Glaciers will not form without precipitation, even if the local temperature resides below the freezing point of the substance in consideration. Mars, for example, does not have extensive glacial deposits on the surface even though it is always below the freezing point of water.

**2)** I assume there is a sufficient difference in reflectivity between liquid water and ice/snow. This difference in reflectivity generates the feedback. This assumption is true for Earth, where reflectivity of ice is high in the solar spectrum (liquid water is dark at those same wavelengths). One thus expects a colder planet with more ice in the ocean to raise the planetary albedo, a destabilizing tendency. However, if Earth were placed around another type of star (for example M-type stars, which are colder and have a spectrum shifted to infrared wavelengths relative to our sun) then the feedback would be suppressed. This is because ice becomes “dark” at wavelengths longer than solar, typically past ~1 micron wavelength, and colder stars emit a greater fraction of energy at those wavelengths. It is a consequence of something known as Wien’s displacement law that the emitted spectrum of colder bodes are typically shifted to longer wavelengths. See e.g., Joshi and Haberle (2012), Astrobiology for a description of this effect around redder, dwarf stars that compromise most of the stars that we know of (and have thus become an interesting target for planet hunters looking for prospective habitable worlds where liquid water might exist). Shown in the below figure is a typical reflectivity spectrum for various surfaces, including lake water and snow (fresh snow is a bit more reflective than ice at solar wavelengths, but in both cases much more so than standard ocean water conditions).

**Ice-Albedo Feedback **

We can think about the ice-albedo feedback from a global or local perspective. We can assume temperature increases equatorward and thus the ice line migrates in that direction as the planet becomes colder. In the local perspective, we aim our focus on the latitude at which the ice-line resides, which occurs when that latitude reaches a critical temperature (for example, -10 C). The ice-line is thought of as being zonally (east-west) uniform. Let be the latitude of the ice-line, and let the planetary albedo vary smoothly between a low-albedo state where ice is located only near the poles, and a high-albedo state where ice advances closer to the equator. One expects the albedo dependence (i.e., the slope of the albedo vs. latitude of ice-line) to vary more strongly as ice moves equatorward.

The local energy balance at the ice line is not just radiative, but depends also on heat convergence/divergence from the ocean and atmosphere. We can assume the heat convergence/divergence is proportional to the temperature difference at the ice line, and the global mean temperature, , so where *c* is a positive constant. When the local temperature is higher than global-mean, the divergence term is positive indicating heat export from the source, and vice versa. Heat transport efficiency increases with higher values of *c*. Thus, the energy balance at the location of the ice-line is:

where , , and denote the local insolation, local albedo, and local outgoing (longwave) radiation at the ice-line, respectively. The outgoing radiation here is a constant due to the imposed condition of the temperature being at a certain value at the ice-line.

It follows that as the ice-line advances, and because the local solar insolation increases equatorward, the heat flux divergence must also increase locally. Ice will advance if this cooling term exceeds the warming caused by increased sunlight at lower latitudes. From these arguments, it becomes possible to construct a plot of how the ice-line latitude varies as a function of the solar constant, *Q* (or some other parameter such as carbon dioxide concentration in the atmosphere). The argument here has been developed in a large number of paper (see e.g., Budyko, 1969 , Held and Suarez, 1974 , Ghil, 1976 , Lindzen and Farrell, 1977 , North, 1990 , a more recent discussion connecting to ancient Neoproterozoic climates is in Pierrehumbert et al, 2011 , see also more advanced modeling treatments, e.g., Yang et al., 2012). An example of this is in the following figure:

Note that the y-axis is the sine of the latitude, moving poleward as you go up the axis. It is a consequence of something known as the “slope-stability theorem” that regions with positive slope on this diagram are stable equilibria, and negative slopes are unstable equilibria. The albedo feedback thus gives rise to the possibility of a bifurcation in the system (loosely, a tipping point) in which a small change in a parameter such as the solar constant can produce a radical transition in climate state. Suppose, for instance, one was in a climate at 1.2 times the modern solar constant, residing in the ice-free branch at the top. Suppose further that the solar constant was gradually lowered (moving left along this horizontal branch); eventually, it will get cool enough that some ice will exist in the Polar Regions.

However, at some point, a further gradual reduction in sunlight will result in enough albedo feedback (as the ice-line advances equatorward) that one abruptly moves downward to the ice-covered branch. Because of the high-albedo, one can then increase the solar constant again to its original value (moving right along the bottom horizontal line now) yet still remain in the ice-covered state. This introduces dependence on the history that the climate took to get to a point.

**From a global perspective**

Let us focus on the case where we gradually lower the solar constant. From a global perspective, recall from the previous post that the outgoing radiation of a planet depends on temperature (to the fourth power). It follows that the cooling induced by decreasing the solar constant will be stabilized by less planetary emission to space. This “Stefan-Boltzmann response” is not often described as a feedback in the climate science community, but it is the classical way planets reach an equilibrium point when they are perturbed. All other feedbacks are thought of as modifying the strength of that restoring response. When ignoring any of those other feedbacks, such as albedo variation, the sensitivity is . is the forcing resulting from an incremental change in sunlight, greenhouse gases, or whatever. In our case it is equivalent to *dQ*.

Sensitivity expressed in this way is equivalent to the radiative restoring tendency by a planet that radiates according to Stefan-Boltzmann. It is the inverse of the no-feedback “climate sensitivity” metric, that is commonly used to help diagnose the magnitude of future global warming. A more sensitive system will respond to greater temperature fluctuations for a given solar forcing; equivalently, the radiative restoring tendency will be less for a given temperature change than for a less sensitive system.

When we include the albedo change, the temperature sensitivity to a change in solar constant becomes:

where,

*R * is the net radiation flux change of the planet due to the change in the albedo. The feedback strength is thus a function of how sensitive albedo changes are to temperature, and how strongly that albedo change impacts the planets radiation balance. In this linear analysis, it is seen that when f = 1 the destabilizing albedo feedback becomes sufficient to outweigh the restoring response caused by decreased planetary emission to space, and a runaway snowball ensues. (I will have more to say about feedbacks, the origin of the above equation, and this interpretation of f=1 in a later post). It can be shown (e.g., in the Roe and Baker, 2010 paper linked in the first figure caption) that the global and local interpretations are equivalent- that is, f=1 occurs when the ice line is at some latitude, the same latitude as when the slope of y_{i} vs. Q becomes negative. Equivalently, if one envisions a curve of radiative flux vs. surface temperature, the equilibrium is stable if the slope of the solar curve is less than the OLR curve at the intersection point.

In thinking about contemporary global warming, the albedo feedback enters into the picture in a similar way. Less ice cover means more solar energy absorbed at the surface and amplified warming. However, the role that the surface albedo has on the total planetary albedo becomes somewhat muddied when you bring clouds and atmospheric scattering into the picture. Moreover, ice does not exist in substantial amounts on the present Earth except very near the Poles where there is already weak sunlight to begin with. So the modern albedo feedback should be thought of as very important locally but small globally, and is a decidedly secondary effect relative to some other feedbacks we will encounter in later posts. It is used as a illustration of feedbacks quite often however, since it behaves very intuitively. The bifurcation arguments play out prominently in studies of the “Snowball Earth” hypothesis, proposed to explain the Neoproterozoic glacial episodes that occurred twice in the period 750–550 million years ago, and suggesting that the Earth was nearly or globally covered by ice/snow during these events. I will return to this time period in later posts.

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Keeping this goal in mind, one could argue that a reasonable starting point for understanding a planet is to simply know how much energy it receives and how that might translate into a temperature if you stood somewhere on the planet. By “energy it receives,” we are accustomed to thinking about the energy from the host star like our sun. Earth receives virtually 100% of its energy from the sun, although it is quite possible to receive significant amounts of energy from sources other than a star (this is the case for several of the gaseous planets in our own solar system). I will ignore those terms in this post, and will return to them in a later article.

As a starting point, I like to envision a spherical rock in space- essentially a “planet” that has been stripped of any atmosphere and which gains energy by absorbing a certain fraction of stellar energy. Let be the power output of the star, called the stellar luminosity (measured in Watts for example, with dimensions of energy per unit time). The luminosity for our own sun is approximately 3.85 x 10^{26} W. The instantaneous stellar flux, reaching a planet that orbits its star at a distance is,

(more generally, a factor of appears in the denominator for planets that orbit their star in a non-circular orbit, where the factor is called the *eccentricity*. Annual-mean insolation (INcoming SOLar radiATION) thus increases for higher eccentricity. is close to zero (circular orbit) for most solar system planets, though indications from extrasolar planetary systems suggests this is not a general rule).

Imagine now a solid, square surface that stood in space perpendicular to the incoming star light. On Earth, we call the solar constant the amount of energy that impinges on such a surface at the mean Earth-sun distance. Measured at Earth, . The units “Watts per square meter” are helpful because in calculating temperature it turns out to be more useful than just knowing the total power the entire planet receives.

When starlight impinges upon the planet, not all of it is absorbed at the surface. Surfaces typically reflect some fraction of that energy away, depending on the nature of the surface and the wavelengths of the electromagnetic spectrum that strike the surface. Let us define a parameter to be called the planetary albedo, the bulk fraction of reflected stellar energy (across all wavelengths). It follows that is the fraction absorbed. Earth reflects about 30% of all the energy it receives from the sun right back to space, though most of this is actually due to clouds, but in this simple no-atmosphere model the planetary albedo is the same as the surface albedo. Although we typically think of albedo as a property of the planet, it depends also on the spectrum of starlight received, since atmospheric scattering processes and surface reflectivities are a function of wavelength on the electromagentic spectrum. If Earth were placed around a different type of star, with other climate conditions held constant, one would expect it to have a different planetary albedo. Moreover, albedo can change as climate does, possibly amplifying or dampening the magnitude of change that occurs. Such a feedback process will be discussed in following articles.

We next need to consider the spherical geometry of the planet, and how the starlight is distributed over the sphere. In this simple model, we do something that might appear silly- take the 1370 W/m^{2} solar energy impinging on a perfectly perpendicular plate facing the sun and distribute it over the whole planet evenly, such that every square meter of the planet is receiving equal amounts of sunlight. What value then should we use in this average for the whole planet?

We can think of a shadow that is cast on a plane sitting “behind” the Earth as the cross-section of a sphere (a circle). Imagine holding a basketball between a projector and the white screen on a board. The image displayed on the screen will be the shadow of a circle, and the area of that circle is the appropriate term to multiply by in order to obtain the total power received by Earth (or whatever value of is appropriate for a planet in consideration). For a planet with radius , the total incoming stellar energy (in Watts) is thus,

**Outgoing Energy**

Obviously our planet does not continue to just soak up sunlight forever without doing anything else. All objects with a temperature also emit radiation, and it turns out that the amount of radiation being emitted is a function of temperature. The emission of such radiation is how the planet sheds the heat it receives from the sun back into space. I will have more to say about the nature of this emitted planetary radiation in many future posts, but for now suffice to say that

– The amount of energy radiated by a body is proportional to the fourth power of its absolute temperature. The relevant constant of proportionality is called the Stefan-Boltzmann constant, often represented by , which is a constant that emerges from quantum physics and reflects the influence of the microscopic world on the planetary scale. Its value is 5.67 x 10^{-8} W/(m^{2} K^{4}).

– On Earth, this outgoing radiation is emitted from a planet much colder than the sun, and is thus in the infrared portion of the electromagnetic spectrum (which is why we don’t “see” energy emitted from objects other than very hot surfaces such as a stovetop). I will return to the nature of radiation in a later post to elaborate on this.

-In order to characterize the temperature of the planet, we can assume the planet to be in steady state, such that it receives an equal amount of starlight as it emits energy back into space. This near-perfect balancing act is assured to us by the temperature dependence of the outgoing radiation. If a planet were to suddenly receive more stellar energy, its temperature would rise, and by the Stefan-Boltzmann law, emit more energy until the radiative fluxes balanced again.

– We can assume in this simple model a spherically uniform temperature, such that all points on the planet are characterized by an equilibrium temperature and the energy is emitted over the whole sphere (of area ).

Energy balance then assures that:

The planetary radius cancels and becomes irrelevant. Re-arranging terms,

or,

Note that temperature goes as the fourth root of the radiation flux, but from equation 1 the radiation flux goes down as the squared distance from the star, so that we expect planetary temperature to drop fairly slowly (as the square root) as you move away from the star.

**Limitations/Things to Come**

Finally, some notes on the limitations of this energy-conserving zero-dimensional model, much of which will provide fodder for later discussions:

**1) **The “uniform planet” model is not valid for a bare rock, since one would expect rapid changes in the temperature between day and night. However, it works by coincidence for Earth, Venus, and other bodies that in reality have an atmosphere (or oceans) to transport heat around. Since we are measuring temperature in Kelvins, the absolute value fluctuates by only a few percent between day and night, or equator to pole. However, the approximation is not useful for Mercury, the moon, or other airless bodies. What is relevant then may be the hemispheric-averaged temperature, or the instantaneous temperature at a point assuming radiative equilibrium, which affects the term one ends up using in the geometrical considerations (e.g., for the local, instantaneous temperature at the equator, at noon, on a bare rock one would ignore the division by 4).

**2)** We have implicitly ignored any temperature dependence on albedo. This, in general, is not a good assumption since we expect the characteristics of a planet (like cloud or ice cover) to change with temperature. In considering a broad range of climate-space, this temperature dependence of albedo gives rise to a family of solutions for what equilibrium climate one ends up with for a given stellar luminosity, and can introduce dependence on the history of what path a planet took to get to a given state. This gives rise to bifurcation structures and hysteresis, which will be explored in the next post. This is perhaps an unsatisfactory limitation of the simple model, in that the albedo is largely determined by the intrinsic properties of a planet, so one cannot obtain a unique solution for temperature on some distant exoplanet by only knowing the distance to the star.

**3)** The calculated equilibrium temperature depends only on the absorbed starlight. However, a fundamental characteristic of many planets is its atmosphere, and all known bodies with a significant atmosphere in some way alter the flow of energy into and/or out of the planet. On Earth, , below freezing, yet the true global mean temperature is closer to 290 K. This is due to a greenhouse effect generated by the atmosphere which inhibits radiative heat loss to space. Venus’ true temperature is a hellish 735 K due to its atmosphere, even though its calculated is just over 230 K (colder than Earth because its high albedo over-compensates for the closer distance to the sun). Moreover, there are no feedbacks in this model: in reality, if one made the Earth that cold, high-albedo ice would begin to cover the oceans and reflect more sunlight forcing to be much less than 255 K. Similarly, Venus would not retain its high albedo (~75%) if it actually had no atmosphere, yet it would still be much colder than currently observed. It should be kept in mind however that the parameters one uses in the energy balance calculation are not always self-consistent with the assumptions used. In a future post, we will see that will no longer correspond to the surface temperature when we include an atmosphere, but rather a temperature encountered at a certain height above the surface. This, and building on other assumptions, will also be developed in a later post.

**4)** Once we cover the basics in detail, I will discuss in a later post that for a planet with liquid oceans, must reside below a critical value, typically 275 K or so, in order for that ocean to exist without vaporizing completely into the atmosphere (this value increases for higher values of the gravitational acceleration of a planet). This fact is crucial to the question of habitability on other worlds. 255 K is well below that limit, and in fact Venus resides even below this (~235 K). It is a necessary but not sufficient condition, however, since Venus and Mars both meet that criteria yet neither have liquid water.

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