🌍 Simple Climate Model

The ice-albedo feedback is a positive feedback loop: as temperature falls, more sea ice and snow accumulate, raising planetary albedo, which reduces absorbed solar energy, further cooling the planet. Conversely, warming melts ice, lowers albedo, absorbs more sunlight, and amplifies warming. This feedback is particularly strong in the Arctic, where sea-ice extent has declined about 13% per decade since satellite records began in 1979. In the model, enabling this feedback creates the possibility of runaway cooling to a Snowball Earth state if temperature drops below ~250 K.

Climate sensitivity is the equilibrium global mean temperature increase caused by a doubling of atmospheric CO₂. The Equilibrium Climate Sensitivity (ECS) estimated from models and palaeoclimate data ranges from 2.5 to 4°C, with a best estimate around 3°C (IPCC AR6, 2021). A higher sensitivity means a given emissions pathway leads to more warming, and hence greater risk. Uncertainties in cloud feedbacks are the dominant source of uncertainty in ECS estimates.

Without any atmosphere — greenhouse effect or clouds — Earth's albedo would be roughly 0.11 (rock and ocean only), and emissivity ε = 1. The energy balance gives T ≈ 270 K (−3°C). With realistic no-GHG albedo of about 0.30 but ε = 1, T ≈ 255 K (−18°C). The greenhouse effect (ε < 1) raises this to the observed ~288 K (15°C), a warming of about 33°C attributable to water vapour, CO₂, methane, and other greenhouse gases.

A 10% increase in solar constant from 1361 to 1497 W/m² would increase T_eq ∝ S₀^(1/4). Multiplying 288 K by (1.1)^(1/4) ≈ 1.024 gives about 295 K (22°C), an increase of roughly 7°C. This is a "forcing" of about 3.4 W/m², comparable to a CO₂ tripling. Over Earth's history, the Sun has brightened by about 30% since the Archean eon (~4 billion years ago), yet the climate remained clement — a puzzle known as the Faint Young Sun paradox, likely resolved by higher CO₂ levels early in Earth's history.

The Stefan-Boltzmann law states that a black body radiates power P = σT⁴ per unit area, where σ = 5.67×10⁻⁸ W/(m²·K⁴). A grey body with emissivity ε < 1 radiates P = εσT⁴. At Earth's current temperature of 288 K, this gives about 390 W/m² for a true black body, but because ε ≈ 0.78 due to greenhouse gases, only about 240 W/m² escapes to space — which must balance the absorbed solar flux of also ~240 W/m² at equilibrium.

This zero-dimensional EBM treats Earth as a single point with a uniform temperature — it has no geography, no ocean circulation, no seasonal cycle, and no atmospheric dynamics. General circulation models (GCMs) used by the IPCC divide Earth into a three-dimensional grid of millions of cells, solving fluid dynamics equations for atmosphere and ocean at resolutions of 10–100 km. GCMs capture regional patterns, precipitation changes, and cloud feedbacks that the EBM cannot, but the EBM correctly identifies the fundamental forcing-feedback framework that governs global mean temperature.

Snowball Earth refers to episodes in Earth's history (~720 Ma and ~635 Ma) when the planet may have been almost entirely glaciated, with sea-ice extending to the tropics. In the energy balance model with ice-albedo feedback, if temperature drops below about 250 K the high-albedo ice state becomes self-sustaining — even at current solar forcing, the ice-covered planet reflects so much sunlight that it cannot warm. This hysteresis means escape from Snowball Earth requires a massive CO₂ build-up from volcanic outgassing over millions of years.