Modeling Climate Feedback Loops

Hello! Welcome to the first lesson in our module on Climate Feedbacks and Sensitivity.

In the previous modules, we established the foundational physics of Earth's energy balance. We saw how to calculate Earth's temperature with and without a simple atmosphere and quantified the initial energy imbalance, or radiative forcing (ΔF), caused by greenhouse gases.

Today, we will build on that foundation to explore one of the most critical concepts in climate science: feedbacks. The climate system isn't a passive recipient of forcing; it responds in complex ways that can either amplify or diminish the initial effect. Our goal in this lesson is to develop a simple but powerful mathematical framework to understand and quantify these feedback loops.

Specifically, by the end of this lesson, you will be able to formulate the mathematical expression for a climate feedback loop and use it to calculate the final temperature change for a given forcing and feedback strength.

1. The Baseline: A World Without Feedbacks

Before we can understand feedbacks, we need a clear baseline: how much would the Earth warm from a given forcing if the only response was an increase in heat radiation back to space? This fundamental response is governed by the Stefan-Boltzmann law ($OLR=\sigma T^4$) and is often called the Planck feedback. It is a stabilizing, negative feedback that is always present.

Let's formalize this. An initial radiative forcing, $\Delta F$, creates an energy imbalance. The Earth's surface temperature, $T$, will rise until the outgoing longwave radiation (OLR) increases by an equal amount, restoring equilibrium.

As the reading explains, for small changes in temperature, we can linearize the Stefan-Boltzmann law around an initial equilibrium temperature $T_0$:

$$\Delta OLR\approx \frac{d(\sigma T^4)}{dT}{|}_{T_0}\Delta T=4\sigma T_0^3\Delta T$$

At the new equilibrium, the change in outgoing radiation must balance the initial forcing: $\Delta F=\Delta OLR$. Substituting the expression above and rearranging for the temperature change, $\Delta T$, gives us the temperature response in a world with only the Planck feedback:

$$\Delta T_0=\left(\frac{1}{4\sigma T_0^3}\right)\Delta F$$

We call $\Delta T_0$ the no-feedback temperature response. The term in the parentheses is a constant that relates forcing to temperature change. We'll define it as ${\Lambda }_0$, the no-feedback climate sensitivity parameter:

$${\Lambda }_0=\frac{1}{4\sigma T_0^3}[\text{in units of K}/(\text{W}/{\text{m}}^2)]$$

So, our simple relationship is:

$$\Delta T_0={\Lambda }_0\Delta F$$

For Earth, with an effective emission temperature of 255 K, a more detailed calculation (as noted in the resource) gives ${\Lambda }_0\approx 0.3\text{K}/(\text{W}/{\text{m}}^2)$. This means that for every 1 W/m² of forcing, the Earth's temperature would increase by about 0.3 K, if no other feedbacks were involved.

2. Modeling a Feedback Loop

Now, let's introduce other feedbacks. A climate feedback is a process that is initiated by a change in temperature and, in turn, causes an additional radiative forcing, further influencing the temperature.

• Positive feedback: Amplifies the initial temperature change.
• Negative feedback: Dampens the initial temperature change.

We can model the energy contribution from a feedback as being proportional to the temperature change itself. Let's call this feedback flux $\Delta F_{feedback}$:

$$\Delta F_{feedback}=\lambda \Delta T$$

Here, $\lambda$ is the feedback parameter in units of W/m²/K.

• If $\lambda >0$, the feedback is positive (e.g., warming causes more water vapor, which traps more heat).
• If $\lambda <0$, the feedback is negative (e.g., warming causes changes in clouds that reflect more sunlight).

Our planet's energy budget at the new equilibrium must now account for both the initial forcing and this new feedback-induced forcing. The total forcing must be balanced by the Planck response:

$$\Delta F+\lambda \Delta T=\Delta OL{R}_{planck}$$

Since we know $\Delta OL{R}_{planck}=\Delta T/{\Lambda }_0$, we can substitute this in:

$$\Delta F+\lambda \Delta T=\frac{\Delta T}{{\Lambda }_0}$$

This is a simple algebraic equation for $\Delta T$. Let's solve for it:

$$\Delta F=\frac{\Delta T}{{\Lambda }_0}-\lambda \Delta T$$ $$\Delta F=\Delta T\left(\frac{1}{{\Lambda }_0}-\lambda \right)$$ $$\Delta T=\frac{\Delta F}{\frac{1}{{\Lambda }_0}-\lambda }$$

To make this equation more intuitive, we can multiply the numerator and denominator by ${\Lambda }_0$:

$$\Delta T=\frac{{\Lambda }_0\Delta F}{1-\lambda {\Lambda }_0}$$

Notice that the numerator, ${\Lambda }_0\Delta F$, is just our no-feedback temperature change, $\Delta T_0$. This leads to a very elegant formulation.

3. The Gain Factor

The equation we just derived reveals a powerful concept used widely in engineering and systems analysis: gain.

Let's define two terms from the reading:

1. Feedback Factor ($f$): A dimensionless quantity defined as $f=\lambda {\Lambda }_0$. It represents the strength of the feedback relative to the stabilizing Planck response.
2. Gain ($G$): The ratio of the final temperature change (with feedbacks) to the no-feedback temperature change: $G=\Delta T/\Delta T_0$.

Substituting these into our equation for $\Delta T$:

$$\Delta T=\frac{\Delta T_0}{1-f}$$

This gives us the final, simple expression for the gain:

$$G=\frac{1}{1-f}$$

And the total temperature change is simply the no-feedback change multiplied by the gain:

$$\Delta T=G\cdot \Delta T_0$$

This framework is powerful because it separates the initial forcing from the system's internal amplification. The climate system has many feedbacks, and their effects are additive. If we have multiple feedback processes with parameters ${\lambda }_1,{\lambda }_2,...,{\lambda }_n$, the total feedback parameter is ${\lambda }_{total}=\sum {\lambda }_i$, and the total feedback factor is $f_{total}=\sum f_i=\sum ({\lambda }_i{\Lambda }_0)$.

Let's analyze the behavior of the gain, $G$:

• No feedback ($f=0$): $G=1/(1-0)=1$. The final warming is just the no-feedback warming.
• Negative feedback ($f<0$): Let $f=-0.5$. Then $G=1/(1-(-0.5))=1/1.5\approx 0.67$. The feedback dampens the initial warming.
• Positive feedback ($0<f<1$): Let $f=0.6$. Then $G=1/(1-0.6)=1/0.4=2.5$. The feedback amplifies the initial warming by a factor of 2.5.
• Runaway feedback ($f\to 1$): As $f$ approaches 1, $G$ approaches infinity. This would imply a "runaway" warming. In reality, the linear approximations would break down, but it highlights the potent nature of strong positive feedbacks.

4. Putting It All Together: A Worked Example

Let's use this framework to perform a calculation. A doubling of atmospheric CO₂ from pre-industrial levels corresponds to a radiative forcing of approximately $\Delta F_{2x}\approx 3.7\text{W}/{\text{m}}^2$ (often rounded to 4 for simplicity). We'll use our value of ${\Lambda }_0\approx 0.3\text{K}/(\text{W}/{\text{m}}^2)$.

Step 1: Calculate the no-feedback temperature response ($\Delta T_0$)

$$\Delta T_0={\Lambda }_0\Delta F_{2x}=(0.3\text{K}/(\text{W}/{\text{m}}^2))\times (3.7\text{W}/{\text{m}}^2)\approx 1.1\text{K}$$

This is the warming we'd expect from doubling CO₂ if only the Planck response acted to restore equilibrium.

Step 2: Incorporate the feedbacks
Climate models and observations suggest that the sum of all major fast feedbacks (water vapor, lapse rate, albedo, clouds) is positive. A central estimate for the total feedback factor is $f_{total}\approx 0.6$.

Step 3: Calculate the gain ($G$)

$$G=\frac{1}{1-f_{total}}=\frac{1}{1-0.6}=\frac{1}{0.4}=2.5$$

This means the climate system's internal feedbacks amplify the initial warming by a factor of 2.5.

Step 4: Calculate the final equilibrium temperature change ($\Delta T$)

$$\Delta T=G\cdot \Delta T_0=2.5\times 1.1\text{K}=2.75\text{K}$$

This value, the equilibrium temperature response to a doubling of CO₂, is a crucial metric known as the Equilibrium Climate Sensitivity (ECS). Our simple model yields an ECS of ~2.8 K, which falls squarely within the range estimated by comprehensive climate models.

---

An Alternative Formulation

It's useful to know that the feedback framework can be expressed in a slightly different but equivalent way. The Wikipedia resource on climate feedbacks presents it this way, which you may also encounter in scientific literature.

In this formulation, the energy balance equation is written as:

$$\Delta F+{\lambda }_{total}\Delta T=0(\text{at equilibrium})$$

Here, ${\lambda }_{total}$ is the sum of all feedback parameters, including the Planck response. Since the Planck response is $\Delta OLR=(\Delta T/{\Lambda }_0)$, its feedback parameter is ${\lambda }_{planck}=-1/{\Lambda }_0$.

So, ${\lambda }_{total}={\lambda }_{other\_feedbacks}+{\lambda }_{planck}$.
Solving for $\Delta T$ gives:

$$\Delta T=-\frac{\Delta F}{{\lambda }_{total}}$$

This looks different, but it is mathematically identical to our gain-based formula. Since ${\lambda }_{total}$ is dominated by the large negative Planck feedback, its value is negative, so $\Delta T$ has the same sign as $\Delta F$. This formulation is common in research for diagnosing the strength of individual feedbacks from model outputs.

Conclusion

In this lesson, we have constructed a mathematical model for climate feedbacks. You should now be comfortable with the key concepts and their mathematical relationships.

Key Takeaways:

• The climate system's response to a forcing begins with the no-feedback response ($\Delta T_0={\Lambda }_0\Delta F$), which is determined by the fundamental Planck feedback.
• Other processes, called feedbacks, are triggered by temperature changes and add their own forcing ($\lambda \Delta T$), either amplifying (positive) or dampening (negative) the initial change.
• The total effect of these feedbacks can be elegantly described by a gain factor, $G=1/(1-f)$, where $f$ is the dimensionless sum of all feedback factors.
• The final equilibrium temperature change is the no-feedback response multiplied by the gain: $\Delta T=G\cdot \Delta T_0$.
• In Earth's climate, the sum of feedbacks is significantly positive ($f\approx 0.6$), leading to a substantial amplification ($G\approx 2.5$) of the initial warming caused by greenhouse gases.

Preview of the Next Lesson:

We've established the "what" and "how much" of feedbacks in a general sense. In our next lesson, we will begin to explore the "why" by examining the physical mechanisms of the most important individual feedbacks. We will start with the single largest positive feedback: the water vapor feedback, and derive its behavior from the fundamental Clausius-Clapeyron relation.