One-Layer Model: Earth's Surface Temperature

Hello! Welcome to the fourth lesson in our module on Foundational Climate Physics.

Introduction

In our previous lessons, we've built our understanding of Earth's energy balance from the ground up.

1. We started with a "no atmosphere" model, treating Earth as a simple black body, and calculated an effective temperature of 255 K (-18°C).
2. We compared this to the actual surface temperature of 288 K (15°C), quantifying the natural greenhouse effect as a 33 K warming.
3. We then delved into the molecular physics, understanding that this warming is caused by trace gases like H₂O and CO₂ absorbing specific frequencies of infrared radiation due to changes in their dipole moments during vibration.

Today, we will bridge the gap between the molecular mechanism and the planetary-scale temperature effect. We will move beyond simply stating there is a 33 K warming and build our first quantitative model that can actually predict it.

Our learning outcome is to: Calculate the Earth's surface temperature using a simple one-layer atmospheric model.

This exercise is a classic application of equilibrium analysis, a concept you are very familiar with from both economics and physics. We will set up a system of energy balance equations and solve for the temperatures of the surface and the atmosphere. This will introduce our first climate model parameter, representing the bulk radiative properties of the atmosphere.

1. Constructing the One-Layer Model

Imagine the Earth's climate system consists of two main components: the surface and a single, uniform atmospheric layer above it. We will analyze the flow of energy between these components and space.

Here is a diagram illustrating the energy flows in this model:

A simplified representation of the one-layer atmosphere model. Incoming solar radiation passes through the atmosphere to the surface. The surface radiates longwave radiation upwards. A fraction of this is absorbed by the atmospheric layer, which then radiates energy both upwards into space and downwards back to the surface.

To make the problem tractable, we'll make a few key simplifying assumptions:

1. The Atmosphere is a Single Layer: We treat the entire atmosphere as one isothermal layer at a single temperature, $T_a$.
2. Transparent to Shortwave: The atmosphere is assumed to be completely transparent to the incoming high-energy solar radiation (shortwave). All absorption of sunlight happens at the surface.
3. Absorptive of Longwave: The atmosphere is not transparent to the infrared (longwave) radiation emitted by the Earth's surface. It absorbs a fraction of this energy, which we will denote by the parameter $ϵ$ (epsilon). $ϵ$ is the emissivity or absorptivity of the atmospheric layer.

By Kirchhoff's law of thermal radiation, a body's absorptivity at a given wavelength equals its emissivity at that same wavelength. Therefore, the atmosphere also radiates longwave energy with the same efficiency $ϵ$. A perfect black body has $ϵ=1$, while a body that doesn't absorb or emit at all has $ϵ=0$.

2. The Energy Balance Equations

For the system to be in equilibrium, the energy flowing into each component must equal the energy flowing out. We can therefore write two separate energy balance equations: one for the atmospheric layer and one for the Earth's surface.

a) Atmospheric Layer Energy Balance

Let's consider the energy budget for our single atmospheric layer.

• Energy In: The only energy input to the atmosphere in this model is the longwave radiation it absorbs from the surface. The surface radiates $\sigma T_s^4$ (where $T_s$ is the surface temperature), and the atmosphere absorbs a fraction $ϵ$ of this. $$\text{Energy In}=ϵ\sigma T_s^4$$
• Energy Out: The atmosphere, at temperature $T_a$, radiates energy away as a grey body. It radiates both upwards into space and downwards back towards the surface. $$\text{Energy Out}=\underset{\text{upwards}}{\underbrace{ϵ\sigma T_a^4}}+\underset{\text{downwards}}{\underbrace{ϵ\sigma T_a^4}}=2ϵ\sigma T_a^4$$

At equilibrium, Energy In = Energy Out:

$$ϵ\sigma T_s^4=2ϵ\sigma T_a^4$$

This simplifies beautifully. The $ϵ$ and $\sigma$ terms cancel, leaving a fixed relationship between the atmospheric and surface temperatures:

$$T_s^4=2T_a^4⟹T_s=2^{1/4}{T}_a\approx 1.189T_a$$

This tells us that in this model, the surface must be warmer than the atmosphere, which makes physical sense.

b) Surface Energy Balance

Now let's do the same for the Earth's surface.

• Energy In: The surface receives energy from two sources: the Sun and the atmosphere.
  • Solar Radiation: From our first lesson, we know the globally averaged solar energy absorbed by the surface is $\frac{S_0(1-\alpha )}{4}$.
  • Atmospheric Radiation: It also receives the downward longwave radiation from the atmosphere, which is $ϵ\sigma T_a^4$.
  $$\text{Energy In}=\frac{S_0(1-\alpha )}{4}+ϵ\sigma T_a^4$$
• Energy Out: The surface loses energy by emitting longwave radiation upwards as a black body. $$\text{Energy Out}=\sigma T_s^4$$

At equilibrium, Energy In = Energy Out:

$$\frac{S_0(1-\alpha )}{4}+ϵ\sigma T_a^4=\sigma T_s^4$$

3. Solving for the Surface Temperature

We now have a system of two equations. Our goal is to find the surface temperature, $T_s$.

1. Atmosphere: $ϵ\sigma T_a^4=\frac{1}{2}ϵ\sigma T_s^4$
2. Surface: $\frac{S_0(1-\alpha )}{4}+ϵ\sigma T_a^4=\sigma T_s^4$

We can substitute the expression for the atmospheric radiation term ($ϵ\sigma T_a^4$) from the first equation into the second equation:

$$\frac{S_0(1-\alpha )}{4}+\left(\frac{1}{2}ϵ\sigma T_s^4\right)=\sigma T_s^4$$

Now, we just need to rearrange the equation to solve for $T_s$. Let's group the $\sigma T_s^4$ terms:

$$\frac{S_0(1-\alpha )}{4}=\sigma T_s^4-\frac{1}{2}ϵ\sigma T_s^4$$ $$\frac{S_0(1-\alpha )}{4}=\sigma T_s^4\left(1-\frac{ϵ}{2}\right)$$

Finally, isolating $T_s$ gives us the solution for the surface temperature in the one-layer model:

$$T_s={\left(\frac{S_0(1-\alpha )}{4\sigma \left(1-\frac{ϵ}{2}\right)}\right)}^{1/4}$$

4. Calculation and Interpretation

Let's see what temperature this model predicts. We will use the same physical constants as in our first lesson:

• Solar constant, $S_0=1361{\text{W/m}}^2$
• Planetary albedo, $\alpha =0.3$
• Stefan-Boltzmann constant, $\sigma \approx 5.67\times 10^{-8}{\text{W m}}^{-2}{\text{K}}^{-4}$

The new parameter is $ϵ$, the atmospheric emissivity. This single number represents the combined effect of all the greenhouse gases and clouds we discussed in the last lesson. A reasonable, empirically-derived value for the modern Earth is $ϵ\approx 0.78$.

Let's plug this into our equation. The numerator term, $\frac{S_0(1-\alpha )}{4}$, is the absorbed solar radiation, which we know is approximately $238{\text{W/m}}^2$.

$$T_s={\left(\frac{238}{5.67\times 10^{-8}\left(1-\frac{0.78}{2}\right)}\right)}^{1/4}$$ $$T_s={\left(\frac{238}{5.67\times 10^{-8}(1-0.39)}\right)}^{1/4}$$ $$T_s={\left(\frac{238}{5.67\times 10^{-8}(0.61)}\right)}^{1/4}$$ $$T_s={\left(\frac{238}{3.4587\times 10^{-8}}\right)}^{1/4}\approx (6.88\times 10^9{)}^{1/4}\approx 288.1\mathbf{K}$$

This result is remarkable. With a single, physically plausible parameter for atmospheric absorptivity, our simple model predicts a surface temperature of 288.1 K (15°C), which is an almost perfect match for the observed global average.

Model Behavior

This model elegantly demonstrates the greenhouse effect. Let's examine its behavior at the limits:

• No Atmosphere ($ϵ=0$): If the atmosphere has zero absorptivity, the equation becomes:
  $T_s={\left(\frac{S_0(1-\alpha )}{4\sigma }\right)}^{1/4}$
  This is exactly the equation for the effective temperature, $T_e$, which we found to be 255 K. The model correctly reduces to the "no atmosphere" case.
• Perfectly Opaque Atmosphere ($ϵ=1$): If the atmosphere absorbs 100% of the infrared radiation from the surface, the equation becomes:
  $T_s={\left(\frac{S_0(1-\alpha )}{4\sigma (1-1/2)}\right)}^{1/4}={\left(\frac{2\cdot S_0(1-\alpha )}{4\sigma }\right)}^{1/4}=2^{1/4}{T}_e$
  This gives $T_s=2^{1/4}\times 255\text{K}\approx 303\text{K}$ (30°C). This provides a theoretical maximum surface temperature for a planet with a perfectly absorbing single-layer atmosphere.

The term $\left(1-\frac{ϵ}{2}\right)$ in the denominator shows how the greenhouse effect works in this model. As $ϵ$ increases from 0 to 1, this term decreases from 1 to 0.5, driving the calculated surface temperature $T_s$ higher.

Conclusion

In this lesson, we have successfully built our first quantitative climate model. By applying the fundamental principle of energy balance, we derived an expression for the Earth's surface temperature that explicitly includes the effect of an energy-absorbing atmosphere.

Key Takeaways:

• A one-layer model treats the atmosphere as a single slab that is transparent to solar radiation but partially opaque to terrestrial radiation, with an absorptivity/emissivity of $ϵ$.
• By setting up and solving energy balance equations for both the surface and the atmosphere, we derived the surface temperature: $T_s={\left(\frac{S_0(1-\alpha )}{4\sigma (1-ϵ/2)}\right)}^{1/4}$.
• Using a physically realistic value for atmospheric emissivity ($ϵ\approx 0.78$), this simple model accurately predicts the Earth's average surface temperature of ~288 K.
• The model provides a clear mathematical relationship between the atmosphere's ability to absorb radiation ($ϵ$) and the magnitude of surface warming.

Preview of the next lesson:
This model provides a static picture of the current climate. But our primary interest is in climate change. The next logical step is to ask: what happens to this energy balance if we change something, like the concentration of CO₂? This brings us to the crucial concept of radiative forcing. In our next lesson, we will define radiative forcing and perform a calculation for the forcing caused by a doubling of atmospheric CO₂, which is a standard benchmark in climate science.