The 3D simulation is a time-integrated zero-dimensional model: it steps C·dT/dt = Qin − Qout forward year by year, adds an additive log-CO₂ forcing term, and lets a Budyko–Sellers ice-albedo ramp create hysteresis and Snowball-Earth tipping points. This 2D version is a genuinely different, independently-computed mechanic: a single-layer grey-atmosphere model, solved algebraically for the instantaneous equilibrium every single frame — there is no time integration and no memory of past states, so moving a slider jumps straight to the new equilibrium rather than relaxing toward it.
The atmosphere is treated as one uniform layer with longwave absorptivity/emissivity ε (set by the greenhouse-gas slider). It is transparent to shortwave, so all absorbed sunlight reaches the surface directly. Two coupled balances are solved together:
Surface budget (arrives from Sun): Sabs = S0·pct·(1−α)/4
Atmosphere budget (absorbs from surface, radiates up+down equally):
ε·σ·Ts⁴ = 2·ε·σ·Ta⁴ ⇒ Ta = Ts / 2^(1/4)
TOA budget (energy leaving to space = energy arriving):
Sabs = (1−ε)·σ·Ts⁴ + ε·σ·Ta⁴ = σ·Ts⁴·(1 − ε/2)
Closed-form equilibrium: Ts = [ Sabs / (σ·(1 − ε/2)) ]^(1/4)
This is the standard textbook "single-layer atmosphere" greenhouse derivation (distinct from the 3D model's empirical log-forcing-plus-ODE approach), and it reproduces the same real-world checkpoint: at today's defaults (α=0.30, 280 ppm ⇒ ε≈0.779, S0=1361 W/m²) it returns Ts≈288 K with a surface emission of ≈390 W/m² — the textbook value for Earth's actual longwave surface flux.
- Albedo — directly scales how much of the incoming shortwave the surface absorbs versus reflects straight back to space, drawn as the pale arrow bouncing off the top of the atmosphere.
- Greenhouse gas (CO₂-eq) — sets ε logarithmically around the 280 ppm baseline: more greenhouse gas thickens the atmosphere's longwave opacity, which raises both Ts and the back-radiation arrow.
- Solar constant — scales S0 directly, changing the incoming shortwave arrow before albedo is applied.
Because the solve is algebraic, the "TOA balance check" is always ≈0 by construction — it is here to make visible why it must be exactly zero at equilibrium (absorbed solar = outgoing longwave to space), the defining condition this model enforces every frame instead of approaching over simulated years.