Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity
Syukuro Manabe & Richard T. Wetherald
Observation and Context
Climatological data show that while global absolute humidity changes drastically between seasons, the vertical distribution of relative humidity remains remarkably stable. This suggests that the atmosphere naturally acts to restore a specific distribution of relative humidity as temperatures fluctuate.
In previous atmospheric modeling efforts, such as those by Manabe and Strickler (1964), the absolute moisture content of the air was kept completely fixed. However, if relative humidity is held constant instead, any increase in temperature will naturally expand the atmosphere’s absolute water vapor content. Because water vapor acts as a powerful greenhouse gas, this shifts the effective altitude of outgoing long-wave radiation upward, reducing the planet’s ability to shed heat efficiently according to the Stefan-Boltzmann law.
Hypothesis
The researchers hypothesized that an atmospheric model using a fixed vertical distribution of relative humidity would exhibit a significantly slower thermal relaxation rate and be roughly twice as sensitive to climate forcings — such as changes in the solar constant, cloud cover, or carbon dioxide (CO₂) concentration — than a model utilizing a fixed absolute humidity.
Experiment and Methodology
The scientists designed a one-dimensional radiative-convective equilibrium model. The atmosphere was divided using a normalized pressure coordinate system into 9-level and 18-level vertical resolutions. Solar and long-wave radiative transfer equations were integrated forward in time as an initial value problem until the model reached an asymptotic state of thermal equilibrium.
The core elements of the methodology included:
- Convective Adjustment: If radiative heating caused the vertical lapse rate to exceed a critical threshold of 6.5 °C km⁻¹, a mathematical adjustment simulated convective mixing to reset a stable profile.
- Moisture Coupling: At each numerical time step, the absolute mixing ratio of water vapor was recalculated as a direct function of the changing temperature to keep the baseline relative humidity constant.
- Forcing Experiments: The model was subjected to systematic variations in key climate variables, including doubling CO₂ from 300 ppm to 600 ppm, varying stratospheric water vapor, altering surface albedo, and adjusting percentage cloudiness.
Results and Data
The numerical simulations yielded three primary datasets:
- Equilibrium Time: It took the fixed relative humidity model nearly twice as long to reach a steady thermal state (roughly 400 days) compared to the absolute humidity framework.
- Forcing Sensitivity: The surface equilibrium temperature was twice as responsive to fluctuations in solar radiation, cloud cover, and albedo when relative humidity was fixed.
- Carbon Dioxide Forcing: Doubling the atmospheric CO₂ content from 300 ppm to 600 ppm raised the surface temperature by approximately 2.3 °C under fixed relative humidity conditions, compared to only 1.3 °C under fixed absolute humidity.
Conclusion and Climate Impact
The experiment successfully verified that water vapor acts as a powerful, self-amplifying positive feedback loop within the earth’s climate system. By dynamically adjusting to temperature changes, water vapor effectively doubles the warming or cooling effect induced by external forcing mechanisms. Crucially, the holistic approach of calculating the thermal balance of the entire atmosphere disproved a contemporary theory by Fritz Möller, which had predicted a chaotic, extreme sensitivity to CO₂ variations by only evaluating the heat budget at the earth’s surface.
This breakthrough established the foundational numerical baseline for water vapor feedback. It directly enabled the successful coupling of radiative physics with dynamic hydrological cycles in early global climate frameworks, anchoring decades of modern climate modeling.
Citation
Manabe, Syukuro and Richard T. Wetherald. (1967). “Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity.” Journal of the Atmospheric Sciences, Vol. 24, No. 3, pp. 241–259.