Modeling Ice-Albedo Feedback with Temperature-Dependent Albedo

Hello! Welcome back to our course on climate change impacts.

In our last few lessons, we've been building a framework for understanding climate feedbacks. We started by defining the feedback concept mathematically, then applied it to the most powerful feedback in the climate system: water vapor. We performed a back-of-the-envelope calculation showing that the water vapor feedback, with a feedback factor of ~0.5, more than doubles the warming from an initial forcing.

Today, we turn our attention to another critical positive feedback: the ice-albedo feedback. This lesson will address the learning outcome: Incorporate a temperature-dependent albedo into a simple energy balance model to quantify the strength of the ice-albedo feedback.

We will explore the physical mechanism, build it into our energy balance model, and analyze the fascinating, non-linear consequences—including the possibility of climate "tipping points."

1. The Ice-Albedo Feedback Mechanism

The physical principle is straightforward. Ice and snow are highly reflective, while open water and land are much darker.

• Albedo ($\alpha$) is the fraction of incoming solar radiation that is reflected back to space. A perfectly white surface has an albedo of 1, and a perfectly black surface has an albedo of 0.

Here are some typical albedo values:

The feedback loop works as follows:

1. An initial warming (e.g., from increased CO₂) causes snow and ice to melt.
2. This replaces bright, reflective surfaces with darker, absorptive surfaces like open ocean or land.
3. The Earth's overall albedo decreases.
4. A smaller fraction of solar energy is reflected, so more is absorbed by the climate system.
5. This absorption causes additional warming, which melts more ice, reinforcing the cycle.

This is a classic positive feedback. Our goal is to represent this process in a model.

2. Building the Feedback into an Energy Balance Model

Recall the zero-dimensional energy balance model, using the linearized form for outgoing longwave radiation (OLR) that we've used previously:

Where:

• $S_0$ is the solar constant (~1360 W/m²).
• $\alpha$ is the planetary albedo.
• $A+BT$ is the linearized OLR, where $T$ is the global average surface temperature.

Until now, we have treated albedo $\alpha$ as a constant. To model the ice-albedo feedback, we must make it a function of temperature: $\alpha (T)$.

A Step-Function Model for Albedo

The simplest way to model $\alpha (T)$ is with a step function. We can define a critical temperature, $T_c$, below which the planet is largely ice-covered and above which it is largely ice-free.

For example, we could set $T_c=-10^{\circ }\text{C}$, ${\alpha }_{ice}=0.6$, and ${\alpha }_{no-ice}=0.3$.

With this temperature-dependent albedo, our energy balance equation becomes non-linear. The intersections of the left-hand side (Absorbed Solar Radiation) and the right-hand side (Outgoing Longwave Radiation) represent the possible equilibrium climate states.

The following reading from a foundational paper on energy balance models explores exactly this idea.

As you saw in Figure 1 of the North et al. paper, this simple formulation leads to a striking result: three possible equilibrium temperatures for the current solar constant.

• Root I: Represents our current climate.
• Root III: Represents a completely ice-covered "snowball Earth."
• Root II: An intermediate, unstable state. If the climate were perturbed slightly from this point, it would either fall into the snowball state or warm to the present state.

This emergence of multiple equilibria from a simple feedback model is a profound concept, echoing ideas of multiple equilibria you may have encountered in game theory or economic models.

3. Stability, Ramps, and Tipping Points

A step-function is a blunt instrument. In reality, the transition from an ice-covered to an ice-free planet would occur over a range of temperatures, not at a single threshold. A more realistic approach is a linear "ramp" model.

As the Winton paper explains, the stability of the equilibrium depends on the relative slopes of the OLR and absorbed radiation curves. An equilibrium is stable only if, upon a small warming, the increase in outgoing energy (cooling) is greater than the increase in absorbed energy (warming).

• Slope of OLR curve: $\frac{d(A+BT)}{dT}=B$
• Slope of Absorbed Solar Radiation curve: $\frac{d}{dT}\left(\frac{S_0}{4}(1-\alpha (T))\right)=-\frac{S_0}{4}\frac{d\alpha }{dT}$

The stability condition is: $B>-\frac{S_0}{4}\frac{d\alpha }{dT}$.

The term on the right, $g_{albedo}=-\frac{S_0}{4}\frac{d\alpha }{dT}$, is the ice-albedo feedback strength (in W m⁻² K⁻¹). It measures how much additional solar energy is absorbed for each degree of warming.

If the albedo changes too abruptly with temperature (i.e., the "ramp" is too steep), the feedback strength $g_{albedo}$ can exceed the planetary cooling response $B$. This leads to an unstable "tipping point," where a small initial warming can trigger a rapid, self-reinforcing transition to a much warmer, ice-free state.

4. Quantifying the Feedback Strength

We can now use this framework to quantify the ice-albedo feedback. The feedback strength, $g_{albedo}$, has been estimated from climate models and observations.

• A widely accepted value for the ice-albedo feedback strength is approximately $g_{albedo}\approx 0.35$ W m⁻² K⁻¹.

Let's convert this into a feedback factor, $f_{albedo}$, using the same method as in our last lesson. The feedback factor represents the fraction of the final temperature change that is "fed back."

$f_{albedo}={\lambda }_0\cdot g_{albedo}$

Using the no-feedback climate sensitivity parameter ${\lambda }_0\approx 0.3\text{K}/({\text{W m}}^{-2})$:

$f_{albedo}\approx 0.3\frac{\text{K}}{{\text{W m}}^{-2}}\times 0.35\frac{{\text{W m}}^{-2}}{\text{K}}\approx 0.105$

So, the ice-albedo feedback factor is approximately 0.1.

Let's put this in context with the water vapor feedback ($f_{H_{2}O}\approx 0.5$). The total amplification from these two feedbacks can be found using the gain formula:

$G=\frac{1}{1-(f_{H_{2}O}+f_{albedo})}=\frac{1}{1-(0.5+0.1)}=\frac{1}{1-0.6}=\frac{1}{0.4}=2.5$

This means that the water vapor and ice-albedo feedbacks combined cause an initial warming to be amplified by a factor of 2.5. The ice-albedo feedback is responsible for a significant portion of this amplification.

Conclusion

In this lesson, we have moved from a qualitative understanding of the ice-albedo feedback to a quantitative one by embedding it within a simple energy balance model.

Key Takeaways:

• The ice-albedo feedback is a positive feedback where warming reduces ice cover, lowering Earth's albedo and causing more solar absorption and thus more warming.
• Incorporating a temperature-dependent albedo ($\alpha (T)$) into an energy balance model makes the system non-linear and can lead to multiple stable climate states (e.g., our current climate and a "snowball Earth").
• The stability of the climate depends on whether the cooling effect from outgoing radiation can overcome the warming effect from the albedo feedback. If the albedo changes too rapidly with temperature, an unstable tipping point can occur.
• The strength of the ice-albedo feedback is estimated to be $g_{albedo}\approx 0.35$ W m⁻² K⁻¹, which corresponds to a feedback factor of $f_{albedo}\approx 0.1$.

Preview of the Next Lesson:

So far, we have examined feedbacks that amplify warming. But are there any that counteract it? In the next lesson, we will investigate the lapse rate feedback, a negative feedback related to how the vertical temperature structure of the atmosphere changes as the planet warms.