# A Single-Layer Atmosphere Model

## ACS Climate Science Toolkit | How Atmospheric Warming Works

Arrhenius and his 19th century predecessors proposed that __absorption of radiant energy__ by Earth’s atmospheric gases is responsible for warming its surface. This page of the ACS Climate Science Toolkit introduces a model that illustrates the fundamental mechanism for atmospheric warming. A problem involved in introducing this model is that most of us have little knowledge or intuition about the behavior of low-temperature black bodies. The radiation they emit is invisible and we have essentially no direct experience of phenomena related to their properties. Thus, it sometimes requires extra effort to develop a sense of how the model works.

This model is based on the concept of emissivity of thermal radiation from a body (solid, liquid, or gas) at any temperature above zero Kelvin. The emissivity at a specified wavelength is the ratio of the amount of energy emitted by the body to the amount of energy emitted by a __black body__ at the same temperature. Emissivity is often symbolized by ε (Greek lowercase epsilon). Emissivities range from 0 (no emission) to 1 (for a black body).

This figure compares the emission from a black body (full area under the top black curve) at 280 K to the emission from a grey body with an emissivity of 0.80 at the same temperature (the area under the lower grey curve) over the same frequency range.

A body with an emissivity less than unity and the same at all wavelengths is called a grey body, by comparison with a black body that has unit emissivity at all wavelengths. The emission from a grey body with ε = 0.80 is illustrated by the lower grey emission curve in the figure. The total energy emitted by a grey body is equal to the energy emitted by a black body at the same temperature multiplied by the emissivity, that is, εσT^{4}, where σ is the __Stefan-Boltzmann constant__, 5.67·10^{–8} W·m^{–2}·K^{–4}. __Aerosol particulate matter__ suspended in the atmosphere, is a pretty good approximation of a grey body.

Grey bodies, like black bodies, absorb electromagnetic radiation. The absorptivity of a grey body is the ratio of the amount of energy absorbed by the body to the amount of energy absorbed by a black body at the same temperature. Note that the definition of the absorptivity is parallel to the definition of emissivity. If the components of a grey body, the particles and molecules in a sample of an atmospheric aerosol, for example, are in thermal equilibrium, then the emissivity and absorptivity for thermal radiation must be equal. If the emissivity and absorptivity were not the same, the sample could spontaneously develop cooler and warmer regions, which violates the second law of thermodynamics. Since the emissivity and absorptivity are the same for a grey body in thermal equilibrium, we will use the same symbol, ε, for both emissivity and absorptivity in the single-layer model for the atmospheric warming mechanism illustrated in this figure.

The __energy from the sun__ that the surface absorbs is given by the expression (1 – α)S_{ave}. S_{ave} is the average incoming solar energy per unit area of the Earth, 342 W·m^{–2}, and α is the Earth’s average albedo, 0.30, which accounts for the fraction of the total incoming radiation that is reflected away, as shown in the figure. The warmed surface emits radiation as a black body at a temperature of T_{p}. The total energy emitted is σT_{p}^{4}.

In this model, the atmosphere is represented by a single homogeneous layer of gases in thermal equilibrium at temperature T_{a} acting as a grey body with an emissivity and an absorptivity given by ε. The figure shows that the atmosphere absorbs part of the energy emitted by the warmed surface. The amount of energy absorbed, εσT_{p}^{4}, is governed by the atmospheric absorptivity. The remainder of the radiation from the surface, (1 – ε)σT_{p}^{4}, passes unabsorbed through the atmosphere and into space outside the atmosphere. A grey body, like a black body, emits in all directions. This is represented in the figure by atmospheric emission both in toward the surface and out into space. The amount emitted in any direction is given by the expression for emission from a grey body, εσT_{a}^{4}.

At the steady state when the temperatures are constant, incoming solar radiation energy absorbed by the Earth must be balanced by outgoing radiation from the surface and atmosphere. The equation representing the planetary energy balance is:

(incoming) (1 – α)S_{ave} = (1 – ε)σT_{p}^{4} + εσT_{a}^{4} (outgoing) ^{. . . . . . . . . . . . . . }(1)

Similarly, the energy absorbed by the atmosphere must equal the energy emitted by the atmosphere, in order for its temperature to be constant. The equation representing this equivalence is:

(absorbed) εσT_{p}^{4} = 2 εσT_{a}^{4} (emitted) ^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .} (2)

Solving this equation for T_{a}^{4} gives:

T_{a}^{4} = (1/2) T_{p}^{4} ^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .} (3)

The term “grey body” has nothing to do with the visible color of a body, but is used only by comparison to a “black body” when considering emission and absorption of thermal IR radiation. A grey body may appear to be any color. For example, for the purposes of our model here, we take the varicolored surface of the Earth (on average) as a black body, but it is probably better described as a grey body with a very high emissivity.

Note that the temperature of the atmosphere will always be lower than the temperature of the surface. We can substitute for T_{a}^{4} in the planetary balance equation (1) and solve for T_{p}, as shown in the sidebar. A plot of T_{p} as a function of emissivity, ε, gives this graph for the single-layer atmosphere model.

(1 – α)S_{ave} = (1 – ε)σT_{p}^{4} + εσ(1/2) T_{p}^{4}

(1 – α)S_{ave} = T_{p}^{4} [(1 – ε)σ + εσ(1/2)]

T_{p}^{4} = [2 (1 – α)S_{ave}]/ [σ(2 – ε)]

T_{p} = {[2 (1 – α)S_{ave}]/ [σ(2 – ε)]}^{1/4}

If the emissivity and absorptivity are zero, no radiation from the surface will be absorbed. This is identical to the energy balance for Earth acting as a black body in the absence of an atmosphere, for which the planetary temperature is __calculated to be 255 K__, just as in this graph. If the emissivity and absorptivity are unity, the atmosphere is a black body and all radiation from the surface is absorbed. For this model, the graph shows that the surface temperature for ε = 1 is 303 K. In this case, the atmospheric temperature is:

T_{a} = (1/2)^{1/4} T_{p} = 0.841 × (303 K) = 255 K

These conclusions should come as no surprise, since the limiting conditions are built into the model.

That is, the atmosphere emits outward into space as a black body at 255 K, just as it must to balance the incoming solar energy absorbed by the planet.

The average temperature of the Earth’s surface is about 288 K. For the single-layer atmosphere model, the graph shows that this temperature would correspond to an atmospheric emissivity of about 0.8. Thus, the model succeeds in its main purpose, demonstrating how an atmosphere that absorbs and re-emits some of the radiation from a planet’s surface results in a surface that is warmer than if there were no atmosphere.

The single-layer atmosphere model captures the essence of the physical processes governing atmospheric warming, but numerical values related to the model must be taken with a grain of salt.

Note that the energy emitted into space comes from two sources. If the emissivity is 0.8, part of the energy, (1 – ε)σT_{p}^{4}, comes from the planetary surface at about 288 K. The rest of the emission, εσT_{a}^{4}, comes from the top of the atmosphere, which is at a temperature T_{a} = 0.841 × (288 K) = 242 K. An observer (or satellite) outside the atmosphere will see an emission spectrum that is the sum of the emissions from these two sources, as represented in this figure.

To satisfy the energy balance equation, the rounded values in this paragraph have to be replaced by these more precise values: ε = 0.78; T_{p} = 288.2 K; T_{a} = 242.4 K.

Assuming that emission from each source has the emission profile of a black body (true for this model), the observed spectrum can be deconvoluted (broken into its components) to give the temperatures and relative contributions of the two sources. The model demonstrates that we can infer properties of both the surface and the atmosphere from observations of their emissions from outside the atmosphere.

It is obvious, however, that the model needs modifications. In particular, the atmosphere is not isothermal and the __experimental data__ below show that the observed emission from Earth into space does not fit a simple grey body model. See __A Multilayer Atmosphere Model__ for the next steps in modifying the model to provide an interpretation of these data in the __Application to Earth’s Atmosphere__ page.

The solid black curve in this figure represents the emission that would be observed outside the imaginary single-layer atmosphere and shows how its composition is the sum of the surface (blue) and top-of-the-atmosphere (red) emissions.