Cloud Altitude, Optical Depth, and Earth's Energy Balance

Hello! Welcome back to our study of climate feedbacks.

In the previous lesson, we analyzed the lapse rate feedback, a crucial negative feedback that stabilizes the climate by amplifying warming in the upper troposphere. We've now examined feedbacks involving water in its gaseous (water vapor) and solid (ice) forms. Today, we turn to liquid water and ice crystals suspended in the atmosphere: clouds.

Clouds are one of the most familiar features of our climate system, but their role in climate change is profoundly complex. They are responsible for the single largest uncertainty in projections of future warming. This lesson addresses the learning outcome: Perform a simple calculation to illustrate how a cloud's altitude and optical depth determine whether its net effect on the Earth's energy balance is one of warming or cooling.

We will dissect the dual radiative effects of clouds and then, using a simplified physical model, quantify how their properties tip the balance between planetary cooling and warming.

1. The Two Competing Effects of Clouds

Clouds influence the Earth's energy balance in two opposing ways simultaneously. The net effect of any given cloud depends on which of these two effects dominates.

1. The Albedo Effect (Cooling): Clouds are generally white and highly reflective. They reflect a significant portion of incoming shortwave (solar) radiation back to space, preventing it from warming the surface. This is a cooling effect. The strength of this effect depends primarily on the cloud's optical depth—thicker, denser clouds reflect more sunlight.

2. The Greenhouse Effect (Warming): Clouds, being composed of water and ice, are excellent absorbers and emitters of longwave (infrared) radiation. They absorb the heat radiated from the Earth's surface and radiate it back down, trapping heat in the lower atmosphere. This is a warming effect. The strength of this effect depends on the cloud's temperature (and thus its altitude) and its optical depth.

The overall impact of clouds, known as the Cloud Radiative Effect (CRE), is the sum of these two components. Globally, the cooling effect currently outweighs the warming effect, meaning that on average, clouds cool the planet. However, for any specific cloud, the balance can easily shift.

2. Modeling Cloud Radiative Forcing

To move from this conceptual understanding to a quantitative one, we need a model. We will use a set of simplified equations from a scientific paper that parameterizes the longwave and shortwave radiative effects of clouds. These equations will allow us to calculate the net effect based on a cloud's physical properties.

The key resource for our calculation is the paper "A simple model for cloud radiative forcing" by Corti and Peter (2009). This paper boils down complex radiative transfer physics into manageable formulas.

The Longwave (Warming) Effect: CRF_LW

First, let's model the warming effect. A cloud traps heat by absorbing outgoing longwave radiation from the warm surface and emitting it at its own, colder temperature. The higher and colder the cloud, the more effectively it traps heat.

As you've read, the longwave cloud radiative forcing, $CR{F}_{LW}$, can be approximated by:

$CR{F}_{LW}\approx {\sigma }^{\ast }(T_{srf}^{k^{\ast }}-T_{cld}^{k^{\ast }})(1-e^{-{\delta }^{\ast }\tau })$

Let's break this down:

• $(T_{srf}^{k^{\ast }}-T_{cld}^{k^{\ast }})$: This term represents the altitude effect. $T_{srf}$ is the surface temperature and $T_{cld}$ is the cloud-top temperature. High-altitude clouds are very cold (low $T_{cld}$), making this difference large and resulting in a strong warming effect. Low-altitude clouds are warmer (high $T_{cld}$), making this difference small and resulting in a weak warming effect.
• $(1-e^{-{\delta }^{\ast }\tau })$: This is the cloud's effective emissivity, which represents the optical depth effect. For an optically thin cloud (small $\tau$), this term is small, reducing the warming effect. For an optically thick cloud (large $\tau$), this term approaches 1, meaning the cloud acts like a blackbody and its warming effect is maximized for its altitude.

The Shortwave (Cooling) Effect: CRF_SW

Next, let's model the cooling effect. A cloud reflects sunlight, and the amount it reflects depends mainly on its optical depth.

The shortwave cloud radiative forcing, $CR{F}_{SW}$, is given by:

$CR{F}_{SW}\approx -I\cdot t{t}^{\prime \ast }\left[\frac{(1-\alpha )R_c-\alpha R_c^{\prime }}{1-\alpha R_c^{\prime }}\right]$

Here, the key variable controlled by the cloud is its reflectivity ($R_c$ for incoming direct sunlight and $R_c^{\prime }$ for outgoing diffuse light from the surface). These reflectivities increase with optical depth $\tau$. A thicker cloud (larger $\tau$) is more reflective, making $CR{F}_{SW}$ more negative and thus producing a stronger cooling effect. Notice that cloud altitude ($T_{cld}$) does not appear in this equation.

3. Calculation: High vs. Low Clouds

Now we can perform the central task of this lesson: a calculation to see how these properties determine the net effect, $CR{F}_{net}=CR{F}_{LW}+CR{F}_{SW}$.

We will model two idealized cases using the parameters from the paper for tropical conditions:

• Case A: A low, thick cloud (like a stratocumulus cloud)
• Case B: A high, thin cloud (like a cirrus cloud)

Constants and Parameters (from Corti and Peter, 2009):

• Surface Temperature: $T_{srf}=299$ K
• Surface Albedo: $\alpha =0.05$
• Daily Mean Solar Flux: $I=435$ W/m²
• Solar Zenith Angle Cosine: $\zeta =0.636$
• Longwave constants: ${\sigma }^{\ast }=1.607\times 10^{-4}$, $k^{\ast }=2.528$, ${\delta }^{\ast }=0.75$
• Shortwave constants: $t{t}^{\prime \ast }=0.73$, ${\gamma }^{\ast }=7.7$

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Case A: Low, Thick Cloud

Let's assume a cloud at 2 km altitude with a large optical depth.

• Cloud-top Temperature: $T_{cld}=275$ K
• Optical Depth: $\tau =10$

1. Longwave Effect (Warming):
$CR{F}_{LW}\approx 1.607\times 10^{-4}(299^{2.528}-275^{2.528})(1-e^{-0.75\times 10})$
$CR{F}_{LW}\approx 1.607\times 10^{-4}(1.54\times 10^6-1.22\times 10^6)(1-e^{-7.5})$
$CR{F}_{LW}\approx 1.607\times 10^{-4}(3.2\times 10^5)(0.9995)\approx +51.4W/{\mathbf{m}}^2$
The warming effect is modest because the cloud is not much colder than the surface.

2. Shortwave Effect (Cooling):
First, calculate reflectivities $R_c$ and $R_c^{\prime }$:
$R_c\approx \frac{\tau /\zeta }{{\gamma }^{\ast }+\tau /\zeta }=\frac{10/0.636}{7.7+10/0.636}=\frac{15.72}{23.42}\approx 0.67$
$R_c^{\prime }\approx \frac{2\tau }{{\gamma }^{\ast }+2\tau }=\frac{20}{7.7+20}=\frac{20}{27.7}\approx 0.72$
Now, calculate $CR{F}_{SW}$:
$CR{F}_{SW}\approx -435\cdot 0.73\left[\frac{(1-0.05)0.67-0.05\cdot 0.72}{1-0.05\cdot 0.72}\right]$
$CR{F}_{SW}\approx -317.6\left[\frac{0.6365-0.036}{0.964}\right]=-317.6\left[\frac{0.6005}{0.964}\right]\approx -197.8W/{\mathbf{m}}^2$
The cooling effect is very strong because the cloud is optically thick and highly reflective.

3. Net Effect:
$CR{F}_{net}=51.4-197.8=-146.4W/{\mathbf{m}}^2$
The low, thick cloud has a strong net cooling effect.

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Case B: High, Thin Cloud

Now, let's assume a cloud at 12 km altitude with a small optical depth.

• Cloud-top Temperature: $T_{cld}=215$ K
• Optical Depth: $\tau =1$

1. Longwave Effect (Warming):
$CR{F}_{LW}\approx 1.607\times 10^{-4}(299^{2.528}-215^{2.528})(1-e^{-0.75\times 1})$
$CR{F}_{LW}\approx 1.607\times 10^{-4}(1.54\times 10^6-0.58\times 10^6)(1-0.472)$
$CR{F}_{LW}\approx 1.607\times 10^{-4}(9.6\times 10^5)(0.528)\approx +81.4W/{\mathbf{m}}^2$
The warming effect is strong because the cloud is very cold compared to the surface.

2. Shortwave Effect (Cooling):
First, calculate reflectivities $R_c$ and $R_c^{\prime }$:
$R_c\approx \frac{\tau /\zeta }{{\gamma }^{\ast }+\tau /\zeta }=\frac{1/0.636}{7.7+1/0.636}=\frac{1.57}{9.27}\approx 0.17$
$R_c^{\prime }\approx \frac{2\tau }{{\gamma }^{\ast }+2\tau }=\frac{2}{7.7+2}=\frac{2}{9.7}\approx 0.21$
Now, calculate $CR{F}_{SW}$:
$CR{F}_{SW}\approx -317.6\left[\frac{(1-0.05)0.17-0.05\cdot 0.21}{1-0.05\cdot 0.21}\right]$
$CR{F}_{SW}\approx -317.6\left[\frac{0.1615-0.0105}{0.9895}\right]=-317.6\left[\frac{0.151}{0.9895}\right]\approx -48.4W/{\mathbf{m}}^2$
The cooling effect is weak because the cloud is optically thin and not very reflective.

3. Net Effect:
$CR{F}_{net}=81.4-48.4=+33.0W/{\mathbf{m}}^2$
The high, thin cloud has a moderate net warming effect.

4. Visualizing the Result

Our calculation clearly shows the principle: low, thick clouds cool, and high, thin clouds warm. The paper visualizes this for all combinations of altitude and optical depth.

As you can see in Figure 4c of the paper, our results align perfectly with the model's overall behavior:

• Our low, thick cloud (altitude 2 km, $\tau =10$) falls deep in the blue region of negative (cooling) CRF.
• Our high, thin cloud (altitude 12 km, $\tau =1$) falls in the red region of positive (warming) CRF.

The zero-contour line on that plot represents the tipping point where the warming and cooling effects exactly cancel. Clouds above this line warm the climate, and clouds below it cool the climate.

Conclusion

This lesson has demystified the complex role of clouds by breaking it down into two competing radiative effects and quantifying their dependence on key physical properties.

Key Takeaways:

• Clouds have a dual impact: they cool the Earth by reflecting shortwave radiation (albedo effect) and warm it by trapping longwave radiation (greenhouse effect).
• The albedo effect is primarily driven by a cloud's optical depth. Thicker clouds are more reflective and have a stronger cooling effect.
• The greenhouse effect is primarily driven by a cloud's altitude. Higher (colder) clouds have a stronger warming effect.
• Low, optically thick clouds (e.g., stratocumulus) have a strong cooling effect that dominates their weak warming effect, resulting in a net cooling.
• High, optically thin clouds (e.g., cirrus) have a strong warming effect that dominates their weak cooling effect, resulting in a net warming.
• Our simple calculation successfully reproduced this behavior, demonstrating how a cloud's physical characteristics determine its role in the Earth's energy balance.

Preview of the Next Lesson:

We have now examined the major fast feedbacks in the climate system: water vapor, ice-albedo, lapse rate, and clouds. But how do we combine all this information to estimate the total warming from a given forcing? In our next and final lesson for this module, we will define Equilibrium Climate Sensitivity (ECS) and use a simple energy balance model that incorporates a single feedback parameter to calculate this crucial metric.