Today, I’d like to look at two things:

- Energy recirculation/recycling as a way to conceptualize how energy fluxes between the Earth and its atmosphere can be larger than the incoming and outgoing energy fluxes.
- A simplified thermal model in which we look at the effects of an “atmosphere” consisting of one or more distinct layers which can transfer heat only by radiative heat transfer.

The thermal model I’ll be using is an example of what physicists sometimes call a “toy” model. It’s a simple model that can be easily analyzed, and which is in a very limited way like a real system of interest—but is so over-simplified that it leaves out many of the important features of that real system. Yet, despite their limitations, the behavior of “toy” models can nonetheless offer useful insights into aspects of how real systems function.

How longwave thermal radiation fluxes between the Earth’s surface and its atmosphere can be as large as they are is a topic that leads to seemingly endless confusion among people who doubt that longwave-radiation-absorbing gases can warm planets. So, this topic is likely to be of interest.

One way of making sense of this is to think in terms of energy being recirculated or recycled. It’s a framework that can be its own source of confusion for some people, but for others it can be helpful. So, I’ll share this way of thinking about things, and then we’ll move on to how these issues are more conventionally thought about.

[I’ve also written a newer essay that explores this topic of energy recirculation.]

## Table of Contents

- Radiative model of an atmospheric layer
- Energy recycling and single-layer radiative atmosphere
- Multi-layer radiative atmosphere
- Does the model violate the Second Law of Thermodynamics?
- Saturation
- How many “layers” does Earth’s atmosphere have?
- Conclusions
- Addendum: Similar models

## Radiative model of an atmospheric layer

Let’s start by considering Figure 1 below.

Solar short-wave radiation, with a mean radiant flux, Fₛ(1-A)/4, is absorbed by the Earth’s surface. The Earth’s surface, at an effective radiative temperature, T₀, emits long-wave thermal radiation with a flux, σT₀⁴, in accordance with the Stephan-Boltzmann law. A fraction (1-f) of this surface-emitted long-wave radiation passes through the atmosphere and reaches space, while a fraction f is absorbed by the atmosphere.

A particular layer of the atmosphere is assumed to be at a temperature, T₁. This temperature is the temperature that equalizes the flows of energy entering and leaving that layer. According to the diagram, the layer will emit long-wave radiant energy equally in all directions, with a flux fσT₁⁴ being sent upward and an equal flux being sent downward.

It should be noted that this diagram is intended for general education, and oversimplifies some details that a serious climate modeler would take into account. In particular, I see the following simplifications:

- The diagram depicts the total absorbed mean solar irradiance, Fₛ(1-A)/4, being absorbed by the Earth’s surface. However, something like 27% of that is actually absorbed into the atmosphere (via clouds, water vapor, dust, and ozone).
- The long-wave flux emitted by the Earth’s surface actually has the form 𝜀₀σT₀⁴, where 𝜀₀ is the mean emissivity of the surface, which has been measured to be 0.94 (at least according to one source).
- How much long-wave radiation is emitted by a layer of the atmosphere depends on the thickness of that layer. Saying the radiant flux is fσT₁⁴ reflects a few implicit assumptions, namely that (a) the layer has sufficient optical depth that it absorbs most of the incident radiative at the wavelengths of interest and (b) the temperature doesn’t vary much across the layer. Serious modeling would involve formulas for the radiative properties of a thin layer of atmosphere, as well as accounting for convection, etc.
- The radiant flux emitted by an atmospheric layer is given as fσT₁⁴, but would more accurately be given by 𝜀σT₁⁴ where 𝜀 is the emissivity of the gas. It’s likely to be approximately true that 𝜀 ≈ f, but this may not be precisely the case. Additionally, the overall emissivity of a gas depends somewhat on temperature, so the radiated flux may not scale precisely as T₁⁴.

Yet, the purpose of the diagram is public education, not rigorous modeling. For that purpose, the diagram has its uses.

## Energy recycling and single-layer radiative atmosphere

One person referencing this diagram noted that “half of all captured flux is returned to the surface as back radiation and recycled.” That’s right, or would be, if the atmosphere only had one layer. Sometimes people forget there are many layers in the atmosphere, and mistakenly think a diagram like this means the atmosphere should direct equal amounts of radiation to the surface and out to space. That doesn’t happen when there are multiple layers involved, as we’ll see a bit later.

The person interpreting the diagram also noted that “Because the intercepted energy flux is being recycled this feed-back loop is… endless … It has the mathematical form of a geometric series, and is a sum of the descending fractions…”

Let’s look at a diagram that illustrates the energy recycling process they were talking about.

In Figure 2, sunlight with power S is absorbed by the surface of the Earth. (For simplicity in sorting out the concepts, we’ll ignore the solar irradiation that is directly absorbed by the atmosphere.)

Because the surface of the Earth is assumed to be neither gaining or losing net energy (when averaged over a day or a year), the amount of power absorbed by the surface must lead to an equal amount of energy leaving the surface. The power leaves the surface via a combination of thermal radiation and convective transport of latent heat (water vapor) and sensible heat (hot air).

Suppose we assume that, for every unit of energy flux that leaves the surface, a fraction (1-β) is radiated into space, and the remaining fraction, β, is returned to the surface via long-wave back-radiation. (In steady-state, on average, the energy flux leaving the atmosphere must equal the energy flux entering the atmosphere. Hence, any energy flux that doesn’t reach space must be returned to the surface, for energy flux balance to hold.)

For each energy flux that reaches the surface, an equal energy flux leaves the surface and enters the atmosphere. A fraction (1-β) reaches space, and a fraction β is returned to the surface. The energy flux returned to the surface must lead to an equal flux leaving. This results in another cycle of some energy reaching space, and some being returned to the surface. In principle, this recycling continues forever, with ever smaller fluxes. Because each round of the cycle reduces the flux by a fixed proportion, the fluxes form a geometric series, making it easy to sum the infinite series. Computing these sums, one finds that the total power radiated into space is S, the same as the energy flux absorbed by the Earth. That’s as one would expect.

One finds that the total back-radiated energy flux, B, is given by B = β⋅S/(1-β).

Climate models don’t usually include a figure like Figure 2 above, in which each iteration of the energy recycling process is shown. Diagrams like Figure 2 are useful for instructional purposes, but aren’t as practical as other ways of depicting things.

Instead, climate models often offer a diagram of total energy fluxes, like Figure 3 below. This diagram shows the net result, after all the recycled energy flows have been added together.

This diagram shows solar flux, S, being absorbed by the surface. There is also an energy flux S/(1-β) leaving the surface (via thermal radiation and other heat transfer), and a back-radiation energy flux B = β⋅S/(1-β) from the atmosphere to the surface. The radiant energy flux leaving the top of the atmosphere is S, equaling the amount of solar irradiance that was absorbed by the Earth. The thickness of the lines qualitatively suggests the differing magnitudes of these energy fluxes.

For Earth, the data in Kiehl and Trenberth (1997), indicate a ratio of back-radiation to absorbed insolation, B/S = 1.38. This corresponds to a recycling fraction β = 0.58. (These calculations pretend all absorbed solar irradiance is absorbed by the surface.)

Sometimes people are incredulous at the idea that the back-radiation flux, B, is greater than the absorbed insolation, S. Yet, this is what is measured to be true.

The energy recirculation diagram, Figure 2, should make clear how this can and does happen, without requiring that anything “fishy” be going on.

It might be reassuring to look at heat flow, instead of the usual energy balance diagram (like Figure 3 above) which mixes heat flows with radiant energy flows. Recall that heat flow is the net energy flow, so that a heat flow (unlike an energy flow) is only in one direction. Translating Figure 3 into an equivalent heat flow diagram yields the diagram below.

If one takes the combined energy flux away from the surface, S/(1-β), and subtracts the back-radiation flux, β⋅S/(1-β), one finds that the heat flux from the surface to the atmosphere is S, exactly the same as the heat flux absorbed from the Sun, and the heat flux radiated into space.

There is nothing contrary to energy conservation happening here. It all adds up.

To some people, it seems counter-intuitive to some that energy recirculation can result in recirculating energy fluxes higher than the initiating absorbed energy flux. But, this result, while perhaps surprising, is not wrong. The math is quite straightforward, as I think I’ve shown.

* * *

One thing to be clear about in all this is that the atmospheric layer doesn’t literally re-emit a fixed fraction of any absorbed energy flux that it receives. Nor is this narrative of energy recirculating back and forth how physicists typically think about the exchange of thermal radiation.

The narrative I’ve offered about thermal energy recirculating and building up does happen, when temperatures are in transition. But, there’s nothing like a fixed fraction of the energy that gets returned during each round trip.

The fixed-fraction recirculates is a mathematical fiction. It’s not really how things work, but:

- The fixed-fraction recirculation idea can be helpful in understanding how a low power level entering from outside could build up to a higher power level within the region where thermal radiation is recirculating.
- When temperatures finally get to their equilibrium state, one can always, after the fact, come up with a recycling fraction, β, that corresponds to the ratio of radiation fluxes in each direction that one is seeing. (But, other than producing the “right” answer, the calculated value of β doesn’t say much about the actual physics.)
- Energy recirculating between two different things that absorb and emit thermal radiation really can cumulatively build up in a way that increases longwave power fluxes. So, in a sense, it’s true that power recirculating back and forth allows the total power to build up to a higher level. Yet, relative to the speed at which radiation travels back at forth (around 1 foot or 30 cm per nanosecond), it’s a long, slow process. Matter needs to accumulate heat, and increase in temperature, before it can emit more thermal radiation. And if something else is draining heat away, that affects whether or not temperature will end up increasing.

The way that scientists and engineers more typically talk about of talking about this sort of situation involves look at the net heat flows between various bits of matter, tracking how a net heat flow into a some matter leads to its the thermal energy inside increasing, thereby increasing its temperature, while a net heat flow out decreases its temperature. Then, what energy flux a bit of matter emits is purely determined by the temperature of that bit of matter, as well as its emissivity,

Things get a little more complicated than that when the matter is partially transparent, but I won’t try to address that in any general way in this essay.

Temperature gradients also affect what happens. Putting a material like carbon dioxide around something doesn’t automatically make it warmer. It all depends on the heat flows and the relative temperatures. Again, I won’t go into all that any further in this essay―except by looking at a multi-layer model, which does offer one example of a type of heat gradient. (It’s the sort of heat gradient that arises when radiative heat transfer is the only heat transfer mechanism involved.)

## Multi-layer radiative atmosphere

The atmosphere has more than one layer. To consider the behavior of the atmosphere as a whole, one needs to consider the aggregate effect of many layers interacting with one another.

To illustrate this, let’s look at a “toy model” of the atmosphere, consisting of N layers, each of which behaves like the atmospheric layer in Figure 1.

Sunlight with an average flux, S, is absorbed by the planetary surface.

The surface emits a total flux of long-wave radiation, σT₀⁴. For simplicity, we assume a fraction (1-f) of this thermal radiation has wavelengths that pass through the atmosphere unhindered, while a fraction f is at wavelengths which are totally absorbed by each layer of the atmosphere.

Each layer of the atmosphere has a distinct temperature, and radiates equally in both directions, with a radiant flux fσT⁴.

For simplicity, I assume that only radiative heat transfer is relevant. I assume that the radiant long-wave flux from space is negligible.

This model is not a realistic representation of Earth’s atmosphere. But, solving this problem is likely to be informative, nonetheless.

Using energy balance, we can solve for all the temperatures. This is done easily using a corresponding heat flow diagram.

Here’s how the calculation works. Feel free to skip these details. I denote the heat flux from the surface to the atmosphere, Q. Because the heat flowing to and away from the surface must balance, we know Q=S-(1-f)σT₀⁴. Energy balance also tells us that the heat flowing to and from each atmospheric layer matches, so that Q flows between each layer, and out of the final layer. Comparing the amount flowing out of the final layer in Figure 5 and 6 allows us to solve for the temperature of the last layer, Tₙ, in terms of Q. That allows one to solve for the temperature of each layer in turn, finally yielding a formula for T₀ in terms of Q. Combining this with the previous formula for Q allows us to eliminate Q and solve for T₀ in terms of S and N.

The layers of the atmosphere have T⁴ values which are spaced linearly between the value for the surface, T₀⁴, and zero (the assumed value for space). In this model, the atmosphere gets monotonically colder as one moves to higher layers.

Key results include:

Q = S⋅f/[1+N(1-f)]

T₀⁴ = (S/σ)⋅(N+1)/[1+N(1-f)]

Tᵢ⁴ = (S/σ)⋅(N+1-i)/[1+N(1-f)] = [(N+1-i)/(N+1)]⋅T₀⁴

B/S = N⋅f/[1+N(1-f)]

Relating this model to the prior energy-recycling model, one finds that the energy recycling fraction associated with N atmospheric layers is:

β = N⋅f/(N+1)

For an atmosphere that is opaque to long-wave radiation (i.e., f=1), then a single layer (N=1) yields β=½ and B = S, as assumed by W&M. However, in general, for an atmosphere opaque to long-wave radiation, the energy recycling fraction is β = N/(N+1) and back-radiation flux is B = N⋅S.

**As long as an atmosphere has more than one layer, it is entirely possible for the recycling fraction to be greater than ½, and for the back-radiation flux to be arbitrarily large, compared to the absorbed insolation.**

How can we make intuitive sense of this result?

An atmosphere as a whole is not at a single temperature.

In our “toy model”, the top layer is much colder than the bottom layer. The radiant flux downward to the surface (the “back radiation”) is determined by the temperature of the bottom layer. The radiant flux upward to space is determined by the temperature of the top layer. **Because of the temperature difference between the top and bottom layers, it is entirely natural that the atmosphere as a whole directs more radiation downward to the surface than it does upwards to space.**

* * *

I mentioned before that some people appear to believe something like “for all energy fluxes intercepted by the atmosphere, half of the flux is directed upwards and lost to space, and half of all captured flux is returned to the surface as back radiation and recycled”―and that this belief is false. While it would be true if the the atmosphere consisted only of one thin layer of air, it’s a *false* assumption for the atmosphere as a whole, as demonstrated by our model of a multi-layer atmosphere.

The hypothesis that the atmosphere as a whole behaves sends equal radiative fluxes upwards and downwards is also contradicted by measurements. Those measurements show significantly more flux being directed downwards to the surface than is directed upwards and lost to space (by a factor of around 1.38).

* * *

Some people have a belief that something which involves internal power fluxes larger than the level of power entering the system must be a violation or energy conservation, or of the Second Law of Thermodynamics. We’ve already addressed energy conservation, by noticing that at the level of heat flows, no heat is being gained or lost. But what about the other concern?

**Does the model violate the Second Law of Thermodynamics?**

For an atmosphere opaque to long-wave radiation (f=1), my toy model of a multi-layer atmosphere predicts a surface temperature given by T₀⁴ = (S/σ)⋅(N+1). This has no upper limit, as the number of layers in the atmosphere increases.

It appears that the model is predicting that, with a sufficiently thick long-wave-absorbing atmosphere, a planet could achieve a surface temperature hotter than the Sun. That would be a violation of the Second Law of Thermodynamics. That can’t happen in reality. So, what is going on here?

The solution is very simple. If a planetary surface gets sufficiently hot, the surface will start to emit more and more of its thermal radiation as short-wave radiation. That short-wave radiation will pass through the atmosphere unhindered, just like the incoming solar radiation did. So, once a planet becomes hot enough to emit short-wave radiation, it can efficiently cool its surface.

As a result, a very thick long-wave absorbing atmosphere can never warm a planetary surface to be as warm as the Sun.

* * *

The other imagined violation of the Second Law that some people worry about relates to energy flowing from a cooler heat reservoir (the atmosphere) to a warmer heat reservoir (the surface of the planet).

But, the Second Law doesn’t say no energy can flow from cooler to warmer. It simply requires that the heat flow (i.e., the *net* energy flow), must be from warmer to cooler. As illustrated in the heat flow illustrations (Figure 4 and Figure 6), even with energy recirculation, heat always flows from warmer to cooler.

There is no violation of the Second Law.

**Saturation**

One of the naïve arguments against the possibility of increasing CO₂ having an effect on climate involves arguing that “CO₂ fully absorbs radiation after it travels a relatively short distance through the atmosphere, so how could adding more CO₂ make any difference?”

My toy model offers some insights regarding this issue.

In the model, each layer is assumed to absorb 100% of the longwave radiation within the fraction f of wavelengths that are absorbed. Yet, despite this, each added layer of atmosphere increases the surface temperature.

*Whether or not the atmosphere absorbs long-wave radiation many times over is irrelevant to the potential of increasing greenhouse gases to lead to more warming.*

However, there is a different type of “saturation” that does have an element of reality.

The energy recycling fraction in my multi-layer model of the atmosphere is given by β = N⋅f/(N+1). For large N, this become approximately β ≈ f⋅(1 − 1/N). So, as the total number of long-wave-opaque layers (or equivalently, the concentration of greenhouse gases) increases, additional layers do have smaller and smaller impacts on the energy recycling fraction.

This is vaguely like the assertion that the impact of increasing CO₂ levels is logarithmic, so that you need to keep doubling CO₂ levels to get comparable changes.

But, the mathematical form for this “saturation” effect isn’t quite the same. Where does our toy model go wrong?

The toy multi-layer model assumes that various wavelengths of long-wave radiation are either 100% transmitted or 100% absorbed. Yet, for real gases, there is a continuum in to the degree to which various wavelengths are absorbed.

One way of thinking about it is that the number of “opaque layers” in the atmosphere varies with wavelength. So, even if increasing gas concentrations has little impact on one wavelength, it might have a significant impact at another wavelength.

Another way of thinking about it is that, as you increase the concentration of long-wave absorbing gases, you effectively increase f, the fraction of wavelengths for which outbound long-wave radiation will be absorbed.

So, it makes sense that as you increase the concentration of long-wave-absorbing gases, the impact of additional increases declines, but in principle, there will still be an impact.

**How many “layers” does Earth’s atmosphere have?**

I calculated how an atmosphere with only radiative heat-transfer might affect surface temperature, as a function of how many “layers” the atmosphere has.

But, how do we decide how many layers there are in an atmosphere, for purposes of applying this model?

Recall that, in discussing Figure 1, I said that the formulas involved require one to assume that (a) the layer has sufficient optical depth that it absorbs most of the incident radiative at the wavelengths of interest and (b) the temperature doesn’t vary much across the layer.

This is a rough model, and there is no hard number one what constitutes absorbing “most” of incident radiation. But, an optical depth 2 would absorb 86% of incident radiation, so maybe that would be the minimum optical depth we’d want to associate with a layer?

For radiation with a wavelength of 15 microns, where CO₂ absorption peaks, the optical depth of Earth’s atmosphere may be around 100. So, that might suggest the use of as many as 50 layers in our toy model. But, at a wavelength of 14 or 16 microns, the optical depth is around 10, corresponding to no more than 5 layers. (However, if you divided the atmosphere into only 5 layers, the assumption that temperature doesn’t vary much across a layer would be unlikely to be valid.)

In general, optical depth varies strongly with wavelength. And, for many wavelengths, atmospheric temperature can be expected to vary over the distances needed for full absorption of those wavelengths.

The bottom line is that the toy model cannot be expected to model the behavior of the real atmosphere. (Let’s not forget that the toy model omits convection, which makes it even more unlikely that it could quantitatively describe the real atmosphere.)

Real climate models make use some of the ideas I’ve included in the toy model, but they fill in an enormous number of details that I’ve left out.

**Conclusions**

How things play out in a real atmosphere is, of course, vastly more complicated than can described quantitatively by models as simple as a what I’ve offered here.

Yet, a simple model like what I’ve shared here can help illustrate general mechanisms, and clarify some otherwise mystifying phenomenon. These simple models explain things like how back-radiation fluxes can be larger than the absorbed solar flux, and how more atmospheric radiation can reach the surface than reaches space.

I hope this has been helpful.

## Addendum: Similar models

The model I’ve offered above is not the only radiative layered model of the atmosphere:

- The ACS website discusses a similar atmospheric model, including a single-layer model, a multilayer model, application of the 3-layer model to Earth.

- The Denning Research Group at Colorado State University offers an interactive webpage which solves a version of the 2-layer model.

There are both similarities and noteworthy differences between the various layered radiative atmosphere models.

A summary is:

- Both the above model and the ACS model rely on the spectrum of thermal radiation not changing between layers. However, the two models make different assumptions about the spectral dependence of absorption in order to achieve that unchanging radiation spectrum.
- The Denning model appears to be a special case of the ACS model, specialized to 2 layers. Their diagrams suggest they might be ignoring transmission (which must happen when 𝜀 < 1), which would be non-physical. But the radiation balance equations include transmission through layers. So, likely their model is correction and similar to the ACS model, but simply uses poor graphics to illustrate the model.

In more detail:

I’ll call the model analyzed above the Fixed-Transmittance Model (FT). The FT model assumes:

- Atmospheric longwave absorption is zero for some portion of the longwave spectrum, allowing a fixed fraction (1-f) of surface emissions to reach space.
- Atmospheric longwave absorption in the remainder of the spectrum is high enough that total absorptance is near 1 within the thickness of a single atmospheric layer.
- Because layers must each be optically “thick” (with total absorptance near 1), layers are assumed to a similar mass of atmosphere, not to have a similar thickness measured in kilometers.
- The parameter f is the spectrally-averaged emissivity 𝜀 of the layer.

I’ll call the ACS model the Spectrally-Uniform-Absorption model. The SUA model implicitly assumes:

- Atmospheric longwave absorption is spectrally uniform, so that for a given atmospheric layer i, absorptance and emissivity are 𝜀ᵢ across the spectrum and transmittance is (1-𝜀ᵢ) across the spectrum.
- Because atmospheric layers need not be optically “thick,” the layer thickness can be chosen in any way one likes. ACS seems to suggest choosing layers to be similar in thickness measured in kilometers.

The FT and SUA models are both valid, within the limitations of their assumptions. They will predict slightly different results, because of those differing assumptions.

The FT and SUA models make different assumptions about the spectral characteristics of absorption. The absorption certainly isn’t spectrally uniform as is implicitly assumed by the SUA model. This is nonuniformity is readily seen in the spectrum of emissions to space. That spectrum include emissions as well as non-absorbed surface radiation. For each portion of the spectrum where emissions are less than the 320 K blackbody spectrum, the atmosphere typically absorbs surface emissions significantly fully, and the emissions are from the atmosphere. That makes the FT model seem perhaps more reflective of the atmosphere than is the SUA model. But, the FT model doesn’t take into account the different optical thicknesses in different parts of the spectrum, so it’s not entirely right either.

So, both models capture some aspect of radiative heat transfer in the atmosphere. But, neither get the spectral dependencies entirely right. (And of course, both models ignore non-radiative heat transfer.)

I think both the FT and SUA models have merit for purposes of trying to understand the radiative behavior of the atmosphere.