# 15 Feedback

*“CO _{2}‘s a gas they say can warm the planet, but it doesn’t act alone” — sung to the tune of The Beatles “Get Back”*

Our investigations into the hydrologic cycle have shown the importance of heating on atmospheric circulation. In response to any isolated heating, atmospheric circulations develop to redistribute this energy, affecting rainfall patterns and weather systems along the way. But heatings that develop in response to global temperature changes also can end up affecting the global temperature itself. This is the definition of a **feedback**: a heating anomaly that develops in response to global temperature changes, and affects global temperature itself.

Feedbacks can be contrasted with forcings, which affect global temperatures directly, and do not develop in response to temperature changes. Forcings can be caused by humans (e.g., increasing heat-trapping gas concentrations or air pollution) or natural (e.g., volcanoes or sunspots), while feedbacks are only due to natural responses of the climate system.

Positive feedbacks amplify an initial warming, while negative feedbacks attenuate an initial warming. Feedbacks work in both directions, so in response to an initial cooling, a positive feedback amplifies the cooling, while a negative feedback attenuates an initial cooling.

The most prominent climate feedbacks are:

**water vapor**, which increases in response to temperature, because warmer air can hold more moisture. Because it is such a strong heat-trapping gas, it traps more longwave radiation at higher temperature. Thus water vapor is a**positive feedback**.**lapse rate**, which decreases in response to temperature, meaning the levels that radiate to space warm more quickly than the surface. Lapse rate feedback is thus a**negative feedback**.**ice-albedo feedback**, in which higher albedo frozen surfaces disappear at higher temperature, absorbing more solar radiation. Ice-albedo feedback is a**positive feedback**.**Cloud feedbacks**, which can be usefully separated into several categories. Longwave cloud feedbacks are confidently positive, while shortwave cloud feedbacks are quite uncertain. Altogether, cloud feedbacks lead to the most uncertainty in feedbacks among climate models.

Let’s perform some mathematical analysis on the globally averaged energy balance equation to gain some quantitative backing for our understanding of feedbacks.

Consider a radiative forcing *ΔF*, applied to a planet in equilibrium. Let’s initially calculate the response of a climate with no feedbacks, i.e., in which only the Planck response of outgoing longwave radiation (OLR), governed by the Stefan-Boltzmann equation, balances the forcing.

[latex]\Delta F = \Delta OLR \\ = \Delta ( \sigma T^4)\\ = 4 \sigma T_0^3 \Delta T[/latex]

where we’ve linearized about the average emission temperature *T _{0}*. Solving for the temperature change,

[latex]\Delta T = {1 \over 4 \sigma T_0^3} \Delta F\\ = \Lambda_0 \Delta F[/latex]

defining Λ_{0} = (4 σ T_{0}^{3})^{-1}. For Earth emission temperatures of 255 K, our simple calculation gives Λ_{0} = 0.26 K/(W/m^{2}). For more comprehensive climate models, the strength of the Planck response, calculated by suppressing all other feedbacks, is calculated as a slightly larger value, Λ_{0} = 0.31 K/(W/m^{2}).

We can now estimate the equilibrium climate response to a radiative forcing with no radiative feedbacks. In 2022, industrial-caused radiative forcing was estimated in this paper to be 2.91 W/m^{2} (2.19-3.63 W/m^{2} 5-95% confidence interval). Although the climate is likely far from equilibrium with the current forcing, this would suggest an equilibrium warming with no feedbacks of 0.9 K (0.7-1.1 K). The Earth has already warmed more than this amount (with 95% confidence), meaning the effect of feedbacks must indeed be substantial.

A doubling of carbon dioxide induces a radiative forcing of around 4 W/m^{2}, and the equilibrium climate sensitivity is defined as the steady-state response to this forcing. For our no-feedback climate, the temperature response is 1.2 K, again much smaller than climate models predict.

*“Water vapor, clouds, and ice-albedo feedback, sensitivity’s unknown”*

Let’s add a feedback to the equation, i.e., an additional radiative nudge proportional to the temperature change, λ_{1} ΔT. Now our energy budget equation is

[latex]\Delta F + \lambda_1 \Delta T = \Lambda_0^{-1} \Delta T[/latex].

Solving for ΔT,

[latex]\Delta T = {\Lambda_0 \Delta F \over (1 - \lambda_1 \Lambda_0)}[/latex]

Let’s define the feedback factor f_{1} = λ_{1} Λ_{0} and the gain G = ΔT/ΔT_{0}, where ΔT_{0} = Λ_{0 }ΔF is the climate response with no feedbacks. Then our equation is

[latex]G = {1 \over 1 - f_1}[/latex]

For larger and larger negative feedbacks, the temperature changes becomes closer and closer to zero, but always remains positive. For positive feedbacks, however, the temperature change becomes very large as the feedback factor approaches 1. For f ≥ 1, the system “runs away,” that is, it leads to infinite temperature change (or more likely, a violation of the linearity assumption).

We leave it as an exercise for the reader to show that with the incorporation of additional feedbacks with strengths λ_{2}, λ_{3}, etc.,

[latex]G = {1 \over 1 - f}[/latex]

where f = λ_{1} Λ_{0} + λ_{2} Λ_{0} + λ_{3} Λ_{0} + …

*“Feedbacks, feedbacks set how the Earth will warm… Feedbacks… will they amplify or reduce the harm?”*

The expression for gain, which takes the form 1/(1-f), explains so much of the response of the Earth to industrial emissions, so deserves a detailed analysis.

The gain vs feedback factor relation is non-dimensional, meaning it is valid not just for the climate system, but for any linear feedback problem. To translate it back into Earth units, just recall that the gain is defined as the amplification of the no-feedback Planck response (the radiative forcing times 0.31 K per W/m^{2}), and that the feedback factors are scaled by the Planck feedback (the energy looped back into the system in units of W/m^{2}/K times 0.31 K/W*m^{2}).

Consider a situation with no feedbacks, or equivalently, one in which there are positive and negative feedbacks which have feedback factors that add to zero. In this situation, the gain is 1, and the equilibrium temperature change is the same as the no feedback situation. If instead, the sum of feedbacks were negative, the gain is less than one, and the equilibrium temperature response is less than the no-feedback response.

Positive feedbacks, on the other hand, are anything that loop back more energy through the system with increased temperatures. Positive feedbacks add in strange ways. While one feedback with feedback factor f = 0.4 causes 67% more warming than the no-feedback warming, two feedbacks with f = 0.4 cause 5 times the no-feedback warming. The addition of a third feedback with even half the amplitude is enough to cause the system to blow up, and not reach an equilibrium.

This tendency for positive feedbacks to combine to create a much larger temperature response is an unfortunate fact about global warming, with far-reaching implications. Have you ever notice that the ranges given for future warming always have a larger uncertainty on the higher end than the lower? E.g., a quote from Chapter 7 of the IPCC AR6 WG1 report states “Based on multiple lines of evidence the best estimate of equilibrium climate sensitivity (ECS) is 3°C, the *likely *range is 2.5°C to 4°C, and the *very likely *range is 2°C to 5°C.” The reason the uncertainty range is twice as high on the upper end than on the lower end of the 3°C best estimate is quite simply the gain versus feedback factor equation and plot above.

Let’s examine the strength of individual climate feedbacks, which will help to identify what feedback factor range the Earth is likely in right now.

Water vapor and lapse rate feedbacks

*“Tropical air can hold a lot of moisture, and winter air is oh so dry. *

*So when the planet warms, things will be getting sticky, and our greenhouse will amplify.”*

Water vapor (WV) is the strongest positive feedback in the climate system. Why is it usually added to the lapse rate (LR) feedback in studies of feedbacks? The reason is that they tend to be negatively correlated across models. Those with a slightly more positive water vapor feedback also have a more negative lapse rate feedback. These tend to be models that increase their humidity more rapidly than the average. Those that humidify slightly less rapidly, on the other hand, tend to have a weaker positive water vapor feedback and a less negative lapse rate feedback. Adding the two together allows for a smaller uncertainty range because of this.

Water vapor wins out in magnitude over lapse rate, so the water vapor + lapse rate (WV + LR) feedback is quite positive. The WV + LR feedback is estimated to be 1.3 W/m^{2}/K (likely range: 1.2-1.4 W/m^{2}/K), equivalent to a feedback factor of f = 0.4. Individually, the central estimate for WV feedback alone is 1.8 W/m^{2}/K, and LR feedback is -0.5 W/m^{2}/K.

Ice-albedo feedback

*“Arctic lands are cold and often snow covered, and snow reflects a lot of light. *

*So when the warming comes, there is a lot of melting, and heating there is out of sight.”*

The ice-albedo feedback is crucial to climate change in the far north, where temperatures already have warmed over twice the global average rate. But its global significance is surprisingly small. This feedback is estimated to be 0.35 W/m2/K (likely range 0.25-0.45 W/m2/K), equivalent to f = 0.1.

Cloud feedbacks

*“Puffy clouds are white and you might think they’d cool us, but they have a darker side. *

*In the longwave, they’re keeping all the heat in, but the model spread is wide.”*

Clouds cool the planet by reflecting sunlight, but also have a heat-trapping effect, preventing longwave radiation from escaping to space. Different types of clouds have a different balance of the heating and cooling effects. Thick, puffy stratocumulus clouds reflect a lot of shortwave radiation, but have a small heat-trapping effect, so cool the planet. Wispy cirrus clouds, on the other hand, allow most shortwave radiation to pass right through, but have a large heat-trapping effect, meaning these delicate tails of ice crystals warm our Earth.

What matters for feedbacks, though, is how clouds change with warming. We’ll discuss mechanisms for cloud changes in the next chapter, but report the total cloud feedback numbers: 0.42 W/m2/K (likely 0.12-0.72 W/m2/K), giving a best-estimate feedback factor of f = 0.13. The likely range is 3 times as large as the ice-albedo or WV+LR feedbacks, but actually is quite a bit smaller than previous estimates. Understanding of cloud feedbacks has improved substantially, with much more confidence in mechanisms by which clouds amplify warming.

*“With our finest models and piles and piles of data, it’s still hard to comprehend. *

*While clouds may react to make it a bit better, we can’t rule out the upper end… of climate sensitivity, that is.”*

This final lyric of Feedback was written with Dr. Mark Zelinka and Prof. Dennis Hartmann several years back. Since then, much has been learned about cloud feedbacks, and much has been bad news. Clouds remain the largest uncertainty, but likely are not a negative feedback.

All in all, the feedback factor is likely 0.6, meaning a gain of 2.5, or 150% larger eventual temperature change to a forcing than the no-feedback response. For doubled CO_{2}, the expected equilibrium response is a 3 K warming. This equilibrium response to doubling carbon dioxide is defined as the **equilibrium climate sensitivity**, or ECS. If the current radiative forcing were to persist, we’d expect an eventual temperature change of 2.3 K. Adding the variances of the three feedbacks results in likely total feedback factor range of 0.5-0.7, gains of between 2-3.33, and ECS of 2.4-4 C.

If instead the “very likely” confidence is used (meaning 5-95% confidence intervals instead of 10-90% for “likely”), the feedback factor range is instead f = 0.41-0.79, meaning gains from 1.69-4.76, and an ECS range of 2-5.7 K. Notice the big expansion of uncertainty on the upper end, a full 1.7 K larger, even though the low end decreased only 0.4 K. The way positive feedbacks combine means that it is difficult to rule out high climate sensitivities with absolute certainty. Low climate sensitivity, e.g., of 1 or 1.5 K, on the other hand, is virtually impossible.