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Core Equations in Climate Science Simulation

1. Equilibrium Temperature of Earth

$$ T = \left( \frac{S(1 - \alpha)}{4\sigma} \right)^{1/4} $$

This formula gives the Earth's effective temperature without greenhouse gases, where:

• S = Solar constant (W/m²)
• \(\alpha\) = Earth's albedo (reflectivity)
• \(\sigma\) = Stefan-Boltzmann constant

2. Radiative Forcing from CO₂

$$ \Delta F = 5.35 \cdot \ln\left(\frac{C}{C_0}\right) $$

This represents the change in radiative forcing (in W/m²) from increased CO₂ concentrations:

• \(C\) = current CO₂ (ppm)
• \(C_0\) = reference (pre-industrial) CO₂ = 280 ppm

3. Total Temperature Increase with Feedback

$$ \Delta T = \frac{\lambda \cdot (\Delta F_{CO_2} + \Delta F_{CH_4} + \Delta F_{N_2O})}{1 - f} $$

This formula shows the temperature response adjusted by feedback:

• \(\lambda\) = climate sensitivity parameter (°C/W/m²)
• \(f\) = net feedback factor (dimensionless)

Where individual contributions are:

$$ \Delta F_{CH_4} = 0.036 \cdot (\sqrt{CH_4} - \sqrt{CH_{4,0}}) $$ $$ \Delta F_{N_2O} = 0.12 \cdot (\sqrt{N_2O} - \sqrt{N_{2}O_{0}}) $$

These formulas encapsulate the delicate energy balance of our climate system.

1. Earth's Equilibrium Temperature (Without Greenhouse Effect)

The Earth receives solar radiation:

$$ \text{Incoming power} = S \cdot \pi R^2 \cdot (1 - \alpha) $$

The Earth emits blackbody radiation:

$$ \text{Outgoing power} = \sigma T^4 \cdot 4\pi R^2 $$

At equilibrium:

$$ S(1 - \alpha) \cdot \pi R^2 = 4\pi R^2 \sigma T^4 \Rightarrow \frac{S(1 - \alpha)}{4\sigma} = T^4 \Rightarrow T = \left( \frac{S(1 - \alpha)}{4\sigma} \right)^{1/4} $$

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2. Radiative Forcing from CO₂

Empirically derived from spectral radiative transfer models (Myhre et al., 1998):

$$ \Delta F = 5.35 \cdot \ln \left( \frac{C}{C_0} \right) $$

Where:

• \( C \): current CO₂ concentration (ppm)
• \( C_0 \): reference (pre-industrial) CO₂ concentration

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3. Total Temperature Increase with Feedback

Let:

• \( \lambda \): climate sensitivity parameter (°C/W/m²)
• \( f \): net feedback factor (dimensionless)

The feedback-amplified temperature response:

$$ \Delta T = \lambda \cdot \Delta F \cdot (1 + f + f^2 + \dots) $$

This is a geometric series:

$$ \sum_{n=0}^{\infty} f^n = \frac{1}{1 - f}, \quad \text{for } |f| < 1 $$

So:

$$ \Delta T = \frac{\lambda \cdot \Delta F}{1 - f} $$

With multi-gas forcing:

$$ \Delta T = \frac{\lambda \cdot (\Delta F_{\text{CO}_2} + \Delta F_{\text{CH}_4} + \Delta F_{\text{N}_2\text{O}})}{1 - f} $$