Calculating Radiative Forcing from CO2 Doubling

Hello! Welcome to the fifth and final lesson of our first module, "Foundational Climate Physics."

Introduction

In our last lesson, we constructed a simple one-layer atmospheric model. By defining an atmospheric emissivity, $ϵ$, we were able to derive an equation that successfully predicted the Earth's average surface temperature of 288 K. This gave us a static, equilibrium view of the climate.

However, our primary goal is to understand climate change. This requires us to analyze how the system responds to perturbations. What happens to the Earth's energy balance when we add more CO₂ to the atmosphere?

Today, we will introduce the central concept used to answer this question: radiative forcing. It's the conceptual bridge between a change in an atmospheric constituent (like CO₂) and the resulting change in global temperature.

Our learning outcome for this lesson is to: Define radiative forcing (ΔF) and calculate the forcing for a doubling of atmospheric CO₂ using the standard logarithmic approximation.

This concept is fundamental to the entire field of climate science and provides a "common currency" for comparing the effects of various drivers of climate change.

1. What is Radiative Forcing?

At its core, radiative forcing is a measure of how much a given change (e.g., an increase in greenhouse gases) throws the Earth's energy budget out of balance. Before the change, we assume the planet is in equilibrium, with incoming solar energy equal to outgoing thermal radiation. A forcing disrupts this balance.

For a good conceptual overview, please watch the first few minutes of this lecture by Professor Joanna Haigh, a distinguished atmospheric physicist.

As Prof. Haigh explains, a positive forcing leads to warming, while a negative forcing leads to cooling. Now, let's refine this with a more formal definition.

For a precise and detailed definition, the Wikipedia article on Radiative Forcing is an excellent resource.

Let's break down the key elements of the IPCC's definition:

• "The change in the net, downward minus upward, radiative flux...": This is the core of the concept. It's the change in the energy balance. The units are Watts per square meter (W/m²).
• "...at the tropopause.": This is a crucial detail. The tropopause is the boundary between the troposphere (the lowest layer of the atmosphere where weather occurs) and the stratosphere. By calculating the imbalance at this altitude, we isolate the purely radiative effect of the forcing agent before the climate system has had time to respond with slower adjustments like changes in convection, clouds, and surface temperature.
• "...due to a change in an external driver...": This distinguishes a forcing from a feedback.
  • Forcing (External Driver): An initial push on the system from outside the "normal" workings of the climate (e.g., a change in the sun's output, a large volcanic eruption, or anthropogenic emissions of CO₂).
  • Feedback (Internal Response): A process within the climate system that amplifies or dampens the initial warming from the forcing (e.g., warmer air holds more water vapor, which is itself a greenhouse gas, amplifying the warming). We will explore feedbacks in detail in Module 3.

In essence, radiative forcing quantifies the initial "punch" that a driver delivers to the climate system's energy budget.

2. The Link Between Forcing and Temperature

The reason radiative forcing is so powerful is that, to a first approximation, the eventual change in the global average equilibrium surface temperature ($\Delta T_s$) is directly proportional to the radiative forcing ($\Delta F$).

This relationship is expressed by a simple equation, which you can find in the "Climate sensitivity" section of the same Wikipedia article.

The equation is:

$$\Delta T_s=\lambda \Delta F$$

Where:

• $\Delta T_s$ is the change in global average surface temperature in Kelvin (K).
• $\Delta F$ is the radiative forcing in W/m².
• $\lambda$ is the climate sensitivity parameter in K/(W/m²).

This linear relationship is a powerful simplification. $\lambda$ represents the overall sensitivity of the climate system, including all the fast feedbacks. Its value is not perfectly known, but a typical estimate is around 0.8 K/(W/m²).

Given your background in econometrics, you can think of this as a simple linear response model. If you were to regress historical temperature changes on various forcings, $\lambda$ would be analogous to the coefficient on the net forcing term, representing the marginal effect of a one-unit change in forcing on temperature.

This framework allows scientists to place very different physical processes—like the absorption of infrared radiation by CO₂ versus the reflection of solar radiation by sulfate aerosols—onto a single, comparable scale: their effect on the planetary energy balance in W/m².

3. Calculating the Radiative Forcing of CO₂

So, how do we calculate $\Delta F$ for a given increase in CO₂? The detailed physics involves complex computer models called "radiative transfer codes," which calculate how radiation is absorbed and re-emitted line-by-line for every absorption frequency, at every level of the atmosphere.

Fortunately, the results of these complex models for CO₂ can be very accurately captured by a much simpler formula.

The standard simplified expression for the radiative forcing from a change in CO₂ concentration is:

$$\Delta F=5.35\ln \left(\frac{C}{C_0}\right)$$

Where:

• $C$ is the new CO₂ concentration.
• $C_0$ is the original (reference) CO₂ concentration.
• The constant 5.35 is an empirical parameter derived from the detailed radiative transfer models. It is sometimes written as $\alpha$.

The Logarithmic Relationship

The key feature of this formula is the natural logarithm. This means that the relationship between CO₂ concentration and its warming effect is not linear. Each additional unit of CO₂ has a progressively smaller warming effect.

Why is this? The primary absorption band for CO₂ in the infrared spectrum (centered at a wavelength of 15 $\mu$m) is already quite opaque. When you add the first 100 ppm of CO₂, it has a huge effect because it's blocking a mostly open window. As you add more CO₂, the center of this absorption band becomes saturated. The additional warming then comes from the absorption band getting wider (a phenomenon known as "pressure broadening") and absorbing radiation in the "wings" of the band. Each doubling of CO₂ concentration produces a similar amount of band-widening, leading to a logarithmic relationship between concentration and forcing.

Calculation for a Doubling of CO₂

A standard benchmark in climate science is to calculate the forcing from a doubling of pre-industrial CO₂ levels. Let's do this calculation.

We set $C=2C_0$. The formula becomes:

$$\Delta F_{2\times }=5.35\ln \left(\frac{2C_0}{C_0}\right)=5.35\ln (2)$$

Using the fact that $\ln (2)\approx 0.693$, we get:

$$\Delta F_{2\times }\approx 5.35\times 0.693\approx 3.71{\text{W/m}}^2$$

This is a cornerstone result in climate science. A doubling of atmospheric CO₂ concentration produces a radiative forcing of approximately 3.7 W/m².

To put this number in perspective, the total absorbed solar radiation is about 240 W/m². A forcing of 3.7 W/m² is therefore an imbalance of about 1.5%—a small but persistent nudge to the entire planet's energy budget that, over time, drives significant warming.

Conclusion

This lesson introduced radiative forcing, the key metric for quantifying how the climate is pushed out of equilibrium. It's the lynchpin connecting the physics of greenhouse gases to the prediction of global temperature change.

Key Takeaways:

• Radiative Forcing ($\Delta F$) is the net change in the energy balance (in W/m²) at the tropopause caused by an external driver, calculated before the climate system has had time to respond.
• It provides a common currency to compare the initial impact of diverse climate drivers (e.g., different greenhouse gases, aerosols, solar variability).
• The resulting equilibrium temperature change is, to a first approximation, linearly proportional to the forcing: $\Delta T_s=\lambda \Delta F$.
• The forcing from CO₂ is logarithmic: $\Delta F=5.35\ln (C/C_0)$, because the central absorption bands are already saturated.
• A doubling of atmospheric CO₂ from pre-industrial levels results in a radiative forcing of approximately 3.7 W/m².

Preview of the next module:
This lesson concludes our module on Foundational Climate Physics. We've established the basic energy balance, explained the physical mechanism of the greenhouse effect, built a simple model to quantify it, and now defined a tool to measure perturbations to it.

Our models so far, however, have treated the atmosphere as a single, uniform slab. This is a major simplification. In our next module, Radiative Transfer and Atmospheric Structure, we will begin to add vertical detail. Our first lesson will derive the hydrostatic equilibrium equation, which explains the fundamental pressure structure of the atmosphere. This will be the first step toward understanding why the atmosphere is layered and why greenhouse gases absorb and emit radiation at specific altitudes, a key feature missing from our current model.