Greenhouse Effect Simulator: Stefan-Boltzmann Radiation Model

simulation intermediate ~12 min
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Formula

T_{\text{eff}} = \left(\frac{S(1-\alpha)}{4\sigma}\right)^{1/4}
T_{\text{surface}} = T_{\text{eff}} \cdot \left(1 + \frac{\tau}{2}\right)^{1/4}
\Delta F = 5.35 \cdot \ln\left(\frac{C}{C_0}\right) \text{ W/m}^2
\text{ECS} = \lambda \cdot \Delta F_{2\times} \approx 3°C
The greenhouse effect is the most fundamental mechanism in climate science: atmospheric gases trap outgoing infrared radiation, warming Earth's surface above the temperature dictated by simple radiative equilibrium with the Sun. This simulator implements the Stefan-Boltzmann framework with greenhouse optical depth. Without greenhouse gases, Earth's effective temperature T_eff is determined solely by the balance between absorbed solar radiation S(1-α)/4 and emitted thermal radiation σT⁴. For Earth's current albedo (α ≈ 0.3) and solar constant (S ≈ 1361 W/m²), this gives T_eff ≈ 255 K = -18°C — far below the observed ~15°C. The difference is the greenhouse effect. Atmospheric gases create an optical depth τ that traps a fraction of outgoing IR, warming the surface to T_surface = T_eff·(1+τ/2)^(1/4). The optical depth increases logarithmically with CO₂ concentration, meaning each doubling of CO₂ produces roughly the same additional warming — about 3°C according to the IPCC's best estimate. Arrhenius was the first to quantify this in 1896, calculating that doubling CO₂ would warm the planet by ~5°C. His physics was sound; the overestimate came partly from neglecting negative feedbacks. Manabe and Wetherald (1967) produced the first rigorous general circulation model confirming the logarithmic relationship. The largest uncertainty in modern climate projections comes from cloud feedback. Low clouds reflect sunlight (cooling), but may decrease in a warmer world, amplifying warming. High clouds trap IR (warming) and may increase. The net effect of clouds on climate sensitivity remains the dominant source of uncertainty in IPCC assessments.

FAQ

How does the greenhouse effect work?

The greenhouse effect occurs when atmospheric gases (CO₂, methane, water vapor) absorb outgoing infrared radiation from Earth's surface and re-emit it in all directions, including back toward the surface. This trapping of energy raises the surface temperature above what it would be without an atmosphere. The effect is quantified by the atmospheric optical depth τ in the Stefan-Boltzmann framework.

Why does CO₂ warming scale logarithmically?

As CO₂ concentration increases, the absorption bands become saturated — most IR radiation at those wavelengths is already being absorbed. Additional CO₂ only broadens the absorption bands at the edges, producing a logarithmic relationship: ΔF = 5.35 × ln(C/C₀) W/m². This means each successive doubling of CO₂ produces the same incremental warming.

What is climate sensitivity?

Equilibrium Climate Sensitivity (ECS) is the long-term temperature increase resulting from a doubling of atmospheric CO₂. The IPCC AR6 (2021) assesses the likely range as 2.5–4.0°C, with a best estimate of 3°C. The uncertainty arises primarily from cloud feedback processes.

What was Arrhenius's contribution to climate science?

Svante Arrhenius published the first quantitative estimate of CO₂-driven warming in 1896, calculating that doubling atmospheric CO₂ would raise global temperatures by about 5°C. He used Stefan-Boltzmann radiation physics and recognized the logarithmic relationship between CO₂ and temperature — principles that remain central to modern climate science.

Sources

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