Zero-dimensional energy balance
The simplest EBM treats Earth as a single uniform body with heat capacity $C$ (J m⁻² K⁻¹). The energy budget equates absorbed shortwave radiation to outgoing longwave:
\[C\frac{dT}{dt} = \frac{S_0(1-\alpha)}{4} - \epsilon\sigma T^4\]| Symbol | Meaning | Typical value |
|---|---|---|
| $S_0$ | Solar constant | 1361 W m⁻² |
| $\alpha$ | Planetary albedo | 0.30 |
| $\epsilon$ | Effective emissivity | 0.78 |
| $\sigma$ | Stefan-Boltzmann constant | 5.67 × 10⁻⁸ W m⁻² K⁻⁴ |
| $C$ | Heat capacity (mixed layer) | ~2 × 10⁸ J m⁻² K⁻¹ |
At radiative equilibrium ($dT/dt = 0$):
\[T_{eq} = \left(\frac{S_0(1-\alpha)}{4\epsilon\sigma}\right)^{1/4}\]With the values above, $T_{eq} \approx 255$ K. The observed surface temperature of ~288 K is warmer by 33 K — the greenhouse effect.
Greenhouse effect parameterization
A simple way to include the greenhouse effect is to modify the outgoing longwave radiation (OLR) with an emissivity $\epsilon < 1$ or equivalently write:
\[\text{OLR} = A + BT\]where $A$ and $B$ are empirically fitted constants (Budyko linearization, $A \approx -326$ W m⁻², $B \approx 1.9$ W m⁻² K⁻¹). The equilibrium temperature becomes:
\[T^* = \frac{S_0(1-\alpha)/4 - A}{B}\]A CO₂ doubling shifts $A$ by $\Delta A \approx -3.7$ W m⁻², giving equilibrium warming:
\[\Delta T = -\frac{\Delta A}{B} \approx \frac{3.7}{1.9} \approx 1.95\text{ K}\]This is the no-feedback climate sensitivity (Planck response alone).
Stefan-Boltzmann law and linearization
The Stefan-Boltzmann law $F = \sigma T^4$ is highly nonlinear. Near a reference temperature $T_0$, expand to first order:
\[\sigma T^4 \approx \sigma T_0^4 + 4\sigma T_0^3(T - T_0)\]The Planck feedback parameter is:
\[\lambda_0 = -4\epsilon\sigma T_0^3 \approx -3.2 \text{ W m}^{-2}\text{ K}^{-1}\]This negative value ensures stability: warming increases OLR, restoring equilibrium. The e-folding timescale for relaxation is:
\[\tau = \frac{C}{|\lambda_0|} \approx \frac{2\times10^8}{3.2} \approx 2 \text{ years}\]Albedo feedback and ice-albedo instability
Albedo $\alpha$ depends on surface type. Snow and ice reflect ~0.6–0.85 of incoming solar, while open ocean reflects only ~0.06. Write:
\[\alpha(T) = \begin{cases} \alpha_i & T < T_1 \text{ (ice-covered)} \\ \alpha_i + (\alpha_w - \alpha_i)\frac{T - T_1}{T_2 - T_1} & T_1 \le T \le T_2 \\ \alpha_w & T > T_2 \end{cases}\]with $T_1 = -10°C$, $T_2 = 0°C$, $\alpha_i \approx 0.62$, $\alpha_w \approx 0.30$. The ice-albedo feedback parameter is:
\[\lambda_{ice} = \frac{S_0}{4}\frac{d\alpha}{dT} > 0\]This positive feedback amplifies warming (or cooling). The total feedback is $\lambda_{tot} = \lambda_0 + \lambda_{ice}$; stability requires $\lambda_{tot} < 0$.
Budyko-Sellers one-dimensional latitudinal model
The 1D EBM distributes temperature $T(\phi)$ with latitude $\phi$ and includes meridional heat transport:
\[C\frac{\partial T}{\partial t} = \frac{S_0}{4}s(\phi)(1-\alpha(T)) - (A + BT) + D\nabla^2 T\]where $s(\phi)$ is the meridional solar distribution ($\int s\,d\mu = 1$), and $D \approx 0.55$ W m⁻² K⁻¹ is the diffusion coefficient for dry static energy transport. In terms of $\mu = \sin\phi$:
\[C\frac{\partial T}{\partial t} = Qs(\mu)(1-\alpha) - (A + BT) + \frac{\partial}{\partial\mu}\left[D(1-\mu^2)\frac{\partial T}{\partial\mu}\right]\]Expanding in Legendre polynomials $P_n(\mu)$, the $n=0$ mode gives the global mean and the $n=2$ mode gives the equator-to-pole gradient. The equilibrium solution has an ice line at latitude $\phi_s$ where $T(\phi_s) = T_{freeze}$.
Snowball Earth bifurcation
Reducing insolation (or albedo feedback with growing ice) can push the system through a saddle-node bifurcation. Plotting equilibrium $T$ vs. solar constant $Q = S_0/4$:
- For large $Q$: warm, partially ice-covered equilibrium
- Decreasing $Q$: ice line advances equatorward
- At critical $Q_c$: ice line reaches tropics — no stable warm equilibrium; system collapses to fully glaciated Snowball state ($T \approx -50°C$)
The Snowball-to-warm transition requires very high CO₂ (volcanic outgassing over millions of years) to overcome the high albedo, creating hysteresis:
\[\Delta Q_{hysteresis} = Q_{melt} - Q_{freeze} \approx 0.1 Q_0\]This bistability was recognized in Budyko (1969) and Sellers (1969) and provides a model for Neoproterozoic glaciation events (~635–720 Ma). The bifurcation structure can be analyzed via the potential:
\[V(T) = -\int \left[\frac{S_0(1-\alpha(T))}{4} - (A + BT)\right]dT\]Stable equilibria correspond to minima of $V(T)$; the unstable equilibrium between warm and Snowball states is a local maximum (separatrix).