Lapse Rates and Atmospheric Feedback

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

In our last few lessons, we've explored the powerful positive feedbacks that amplify global warming—specifically, the water vapor and ice-albedo feedbacks. We saw how these mechanisms can more than double the initial warming from a forcing like increased CO₂.

Today, we shift our focus to a crucial counteracting force: the lapse rate feedback. This is the most important negative feedback in the climate system. This lesson will address the learning outcome: Using the concept of the lapse rate, perform a back-of-the-envelope calculation to demonstrate how a warmer, moister atmosphere creates a negative feedback.

We will dissect the physics of how the atmosphere's vertical temperature profile responds to warming and then quantify how this response acts to stabilize the climate.

1. The Lapse Rate: Dry vs. Moist

First, let's revisit a concept from Module 2: the lapse rate, denoted by $\Gamma$, which is the rate at which temperature decreases with altitude ($\Gamma =-dT/dz$).

• Dry Adiabatic Lapse Rate (${\Gamma }_d$): As a parcel of dry air rises, it expands and cools. This cooling happens at a nearly constant rate of ${\Gamma }_d\approx 9.8$ K/km.
• Moist Adiabatic Lapse Rate (${\Gamma }_m$): In a moist atmosphere, a rising air parcel also cools, but once it becomes saturated, water vapor condenses into liquid, releasing latent heat. This heat release partially offsets the cooling.

The result is that the moist adiabatic lapse rate is always less than the dry one (${\Gamma }_m<{\Gamma }_d$). The actual observed average lapse rate in the troposphere is around 6.5 K/km, indicating that moisture plays a dominant role.

To understand the feedback, the crucial point is that the amount of latent heat released depends on how much water vapor the air holds.

As you just read, the logic is direct:

1. A warmer atmosphere holds more water vapor (due to the Clausius-Clapeyron relation we studied earlier).
2. When this warmer, moister air rises, more water vapor is available to condense.
3. More condensation means more latent heat is released.
4. This enhanced heat release further offsets the cooling from expansion.

Therefore, as the climate warms, the moist adiabatic lapse rate ${\Gamma }_m$ decreases. The temperature drop with height becomes less steep.

2. The Feedback Mechanism: Amplified Upper Tropospheric Warming

How does a change in the lapse rate affect Earth's energy balance?

Recall that Earth radiates energy to space not from the surface, but from an effective emission height in the upper troposphere (around 5-6 km). The temperature at this height, $T_e$, determines the rate of outgoing longwave radiation (OLR), approximately given by the Stefan-Boltzmann law, $OLR=\sigma T_e^4$.

If the lapse rate decreases as the surface warms, the upper troposphere must warm more than the surface. Let's see why:

• Initial State: $T_e=T_s-\Gamma \cdot z_e$
• Warmed State: $T_e^{\prime }=T_s^{\prime }-{\Gamma }^{\prime }\cdot z_e$

Here, $T_s$ is the surface temperature, $z_e$ is the emission height, and ${\Gamma }^{\prime }<\Gamma$. The change in emission temperature is:
$\Delta T_e=T_e^{\prime }-T_e=(T_s^{\prime }-T_s)-({\Gamma }^{\prime }-\Gamma )z_e=\Delta T_s-\Delta \Gamma \cdot z_e$
Since warming causes the lapse rate to decrease, $\Delta \Gamma$ is negative. This means the second term is positive:
$\Delta T_e=\Delta T_s+|\Delta \Gamma |\cdot z_e$
This equation shows that the temperature change at the emission height ($\Delta T_e$) is greater than the temperature change at the surface ($\Delta T_s$). This phenomenon is known as amplified upper tropospheric warming.

This amplified warming at the emission height allows the planet to shed heat more effectively. For a given surface warming $\Delta T_s$, the OLR increases more than it would if the atmosphere warmed uniformly. This enhanced cooling is a negative feedback—it counteracts the initial warming.

The following resources explain and visualize this process.

3. Back-of-the-Envelope Calculation

Let's quantify this effect with a simplified calculation. We'll compare the radiative response to a 1 K surface warming in two scenarios: one without the lapse rate feedback and one with it.

Assumptions:

• Initial surface temperature: $T_s=288$ K
• Initial lapse rate: $\Gamma =6.5$ K/km
• Effective emission height: $z_e=5$ km
• Stefan-Boltzmann constant: $\sigma =5.67\times 10^{-8}$ W m⁻² K⁻⁴

Initial State:

1. Emission Temperature: $T_e=T_s-\Gamma \cdot z_e=288-(6.5\times 5)=255.5$ K
2. Outgoing Longwave Radiation (OLR): $OLR=\sigma T_e^4=5.67\times 10^{-8}\times (255.5{)}^4\approx 240.4$ W/m²

Now, let's impose a surface warming of $\Delta T_s=1$ K.

Scenario A: No Lapse Rate Feedback (Uniform Warming)

Here, the lapse rate remains constant (${\Gamma }^{\prime }=\Gamma$). The entire atmospheric column warms by 1 K.

1. New Surface Temperature: $T_s^{\prime }=289$ K
2. New Emission Temperature: $T_e^{\prime }=T_e+\Delta T_s=255.5+1=256.5$ K
3. New OLR: $OL{R}^{\prime }=\sigma (256.5{)}^4\approx 244.2$ W/m²
4. Change in OLR: $\Delta OL{R}_A=244.2-240.4=3.8$ W/m²

This 3.8 W/m² increase in outgoing radiation for a 1 K warming is the Planck feedback, the fundamental cooling response of the planet.

Scenario B: With Lapse Rate Feedback

Here, the warming causes the lapse rate to decrease. Let's assume the 1 K surface warming reduces the lapse rate by 0.3 K/km (a value consistent with climate models).

1. New Surface Temperature: $T_s^{\prime }=289$ K
2. New Lapse Rate: ${\Gamma }^{\prime }=6.5-0.3=6.2$ K/km
3. New Emission Temperature: $T_e^{\prime }=T_s^{\prime }-{\Gamma }^{\prime }\cdot z_e=289-(6.2\times 5)=258.0$ K
4. New OLR: $OL{R}^{\prime }=\sigma (258.0{)}^4\approx 251.6$ W/m²
5. Change in OLR: $\Delta OL{R}_B=251.6-240.4=11.2$ W/m²

Analysis:

By including the lapse rate change, the increase in outgoing radiation for the same 1 K surface warming was 11.2 W/m², compared to just 3.8 W/m² from the Planck feedback alone. The lapse rate feedback provided an additional cooling effect.

We can now calculate the feedback strength, $g_{LR}$. Following the standard convention where a negative feedback has a negative $g$ value:
$\Delta OL{R}_{total}=(B-g_{LR})\Delta T_s$
Where $B$ is the Planck feedback strength ($3.8$ W m⁻² K⁻¹ from Scenario A).

$11.2{\text{W/m}}^2=(3.8-g_{LR})\times 1\text{K}$
$g_{LR}=3.8-11.2=-7.4{\text{W m}}^{-2}{\text{K}}^{-1}$

This is a large negative value, demonstrating a strong damping effect. While our simple model overestimates the magnitude (the accepted value is closer to -0.8 W m⁻² K⁻¹), it successfully demonstrates the physical mechanism and sign of the feedback. The discrepancy arises because this calculation implicitly includes a part of the positive water vapor feedback, a subtlety we won't detail here but which is discussed in the optional reading below.

4. The Interplay of Feedbacks

It's impossible to discuss the lapse rate feedback in isolation. It is intrinsically linked with the water vapor feedback.

• Lapse Rate Feedback (Negative): Upper troposphere warms more than the surface, increasing OLR.
• Water Vapor Feedback (Positive): This same upper tropospheric warming allows for a large increase in water vapor (a potent greenhouse gas) at high altitudes, which traps OLR.

These two effects, both driven by the moist adiabatic process, partially offset each other. In the tropics, the negative lapse rate feedback is stronger than the associated water vapor feedback. Globally, however, the total water vapor feedback (which also includes the effect of uniform warming) is much larger than the lapse rate feedback.

Conclusion

This lesson has demonstrated how the fundamental physics of a moist, convecting atmosphere gives rise to a powerful negative feedback.

Key Takeaways:

• The lapse rate ($\Gamma$) is the rate of temperature decrease with height. In a moist atmosphere, latent heat release during condensation makes the moist lapse rate (${\Gamma }_m$) smaller than the dry one (${\Gamma }_d$).
• As the climate warms, the atmosphere holds more moisture. This leads to more latent heat release in rising air, causing the lapse rate to decrease.
• A decreasing lapse rate results in amplified warming in the upper troposphere relative to the surface.
• Because Earth radiates to space from the upper troposphere, this amplified warming allows for more efficient cooling, creating a negative feedback that dampens surface warming.
• Our back-of-the-envelope calculation confirmed that a changing lapse rate significantly increases the outgoing radiation response to surface warming.
• The lapse rate feedback is physically intertwined with the positive water vapor feedback, and the two effects partially cancel each other out.

Preview of the Next Lesson:

We've now covered feedbacks related to water in its vapor and solid (ice) forms. But what about liquid water in clouds? In the next lesson, we will tackle the cloud feedback, the single largest source of uncertainty in climate projections, and perform a calculation to see how a cloud's properties determine whether it warms or cools the planet.