Optical Depth, Atmospheric Structure, and Greenhouse Gas Layers

Hello! Welcome back to our course on the physics of climate change.

Introduction

In our last lesson, we established the fundamental principle of radiative attenuation: the Beer-Lambert law. We defined optical depth ($\tau$) as the key dimensionless parameter that quantifies the opacity of an atmospheric layer, with the fraction of transmitted radiation given by $T=e^{-\tau }$.

So far, we have only considered absorption. However, the atmosphere doesn't just absorb radiation; it also emits it. Today, we will combine the concepts of absorption and emission with our knowledge of atmospheric structure (how temperature and density change with height).

This brings us to today's learning outcome: to synthesize the concepts of optical depth and atmospheric structure to explain why greenhouse gas absorption and emission occur in distinct layers. We will discover that for wavelengths where the atmosphere is opaque, the radiation that escapes to space originates not from the ground, but from a specific "emission altitude" high in the atmosphere. Understanding this mechanism is central to understanding the greenhouse effect.

1. From Absorption to Radiative Transfer

The Beer-Lambert law, $dI=-I\mu dz$, only tells half the story—the absorption part. To get a complete picture, we must also account for emission.

According to Kirchhoff's Law of thermal radiation, any object that absorbs radiation at a certain wavelength also emits it at that same wavelength. An ideal blackbody absorbs and emits with 100% efficiency. A real object, like a slab of atmosphere, with absorptivity $A_{\lambda }$, will emit a fraction $A_{\lambda }$ of the blackbody radiation corresponding to its temperature, $B_{\lambda }(T)$.

In the previous lesson, we saw that for a small path $dz$, the absorptivity is $d\tau =\mu dz$. Therefore, the change in intensity $dI$ as radiation passes through an infinitesimal layer of optical thickness $d\tau$ is the sum of what's lost to absorption ($-Id\tau$) and what's gained from emission ($+B(T)d\tau$).

This gives us the fundamental Radiative Transfer Equation (RTE) in its simplest form:
$\frac{dI}{d\tau }=-I+B(T)$
Here, $I$ is the intensity (or radiance) of the radiation, $B(T)$ is the Planck blackbody function for the local temperature $T$ of the atmospheric layer, and $\tau$ is the optical depth. This first-order linear ODE governs the flow of thermal radiation through the atmosphere.

The following resource provides a more formal derivation of this equation.

2. The Emission Altitude

Now, let's use the RTE to answer a crucial question: from where does the infrared radiation that escapes to space actually originate?

• If the atmosphere were perfectly transparent at a given wavelength ($\tau =0$), the answer would be simple: the Earth's surface. A satellite would measure the surface temperature.
• However, if the atmosphere is optically thick ($\tau \gg 1$) at that wavelength, any radiation from the surface is absorbed long before it reaches space. The radiation that does escape must be emitted from higher up in the atmosphere.

We can define an emission altitude as the characteristic level from which radiation is emitted to space. By convention, this is the altitude where the remaining optical depth to the top of the atmosphere is approximately one (${\tau }_{upward}\approx 1$). From this height, a photon has a good chance ($e^{-1}\approx 37\%$) of escaping directly to space without being re-absorbed.

The resource from the previous section also formalizes this idea.

3. Synthesis: Why Emission Occurs in Distinct Layers

We can now synthesize these ideas to achieve our learning outcome. The altitude of the $\tau =1$ emission layer is not fixed; it depends critically on the wavelength of the radiation, because the absorption coefficient of greenhouse gases is highly variable with wavelength.

This next reading provides an excellent conceptual explanation, tying together the ideas of wavelength-dependent absorption, atmospheric temperature structure, and emission altitude.

Let's break down the synthesis using the examples from the reading:

1. Atmospheric Window (e.g., 8-12 µm):

   • Optical Depth: Here, absorption by greenhouse gases is very weak. The atmosphere is optically thin ($\tau \ll 1$).
   • Emission Layer: The $\tau =1$ level is never reached in the atmosphere. Radiation from the surface travels almost unimpeded to space.
   • Result: A satellite looking at Earth at 11 µm sees the thermal emission from the warm surface.

2. CO₂ Absorption Band (e.g., 15 µm):

   • Optical Depth: CO₂ absorbs very strongly here. The atmosphere is optically thick ($\tau \gg 1$).
   • Emission Layer: The $\tau =1$ level occurs very high in the atmosphere, near the tropopause (~10-15 km), where the air (and thus the CO₂) is thin enough to become transparent.
   • Result: A satellite looking at 15 µm sees thermal emission from the cold tropopause (around 215-220 K).

3. H₂O Absorption Band (e.g., 20 µm):

   • Optical Depth: Water vapor absorbs moderately here. However, water vapor concentration drops off very rapidly with altitude.
   • Emission Layer: The atmosphere becomes transparent to these wavelengths in the mid-troposphere (~5 km), where the air becomes sufficiently dry.
   • Result: A satellite looking at 20 µm sees thermal emission from the cool mid-troposphere (e.g., 260 K).

This is the essence of the "distinct layers": the effective emission layer is not a fixed physical boundary but an altitude that is a function of wavelength, determined by the interplay between gas concentrations and their specific absorption cross-sections.

4. The Greenhouse Mechanism Revisited

This framework provides a more precise physical explanation for the greenhouse effect. What happens when we add more CO₂ to the atmosphere?

1. Optical Depth Increases: The concentration of CO₂ increases everywhere, so the optical depth of the atmosphere in the 15 µm band increases.
2. Emission Altitude Rises: To find the new, higher $\tau =1$ level, we must go to a higher altitude where the air is thinner to compensate for the increased concentration. The emission altitude for the 15 µm band rises.
3. Emission Temperature Drops: In the troposphere, temperature decreases with altitude. This new, higher emission altitude is colder than the old one.
4. Outgoing Radiation Decreases: The radiation emitted to space is proportional to $T^4$. Since the emission temperature $T$ has dropped, the outgoing radiation at this wavelength decreases.
5. Planetary Energy Imbalance: The Earth is now absorbing the same amount of solar energy but radiating less thermal energy to space. This imbalance, called radiative forcing, causes the entire system (surface and troposphere) to warm up until a new equilibrium is reached.

The diagram below, from the lecture notes we viewed earlier, perfectly illustrates this process.

Conclusion

In this lesson, we have moved beyond a simple picture of absorption to a more complete model of radiative transfer, allowing us to pinpoint where in the atmosphere greenhouse gases have their effect.

Key Takeaways:

• The Radiative Transfer Equation ($dI/d\tau =-I+B(T)$) models how radiation intensity changes due to both absorption and emission within the atmosphere.
• For optically thick wavelengths, radiation escapes to space from an emission altitude, conventionally defined as the layer where the optical depth to space is approximately one ($\tau \approx 1$).
• The height of this emission layer is wavelength-dependent. It is low (at the surface) in the transparent "atmospheric window" and high (in the upper troposphere) in strong absorption bands like that of CO₂.
• This explains why absorption and emission occur in "distinct layers": the effective source of outgoing radiation is a layer whose altitude and temperature depend on the atmospheric opacity at that specific wavelength.
• Adding more greenhouse gases makes the atmosphere more opaque, which raises the emission altitude to a colder level, reducing outgoing radiation and causing the planet to warm.

Preview of the Next Lesson:

We have now developed the theoretical tools to interpret Earth's energy budget. In the next lesson, we will apply them by analyzing a real plot of Earth's outgoing longwave radiation (OLR) spectrum. We will identify the features caused by different greenhouse gases and estimate the crucial role of the "atmospheric window" in allowing the planet to cool.