Year 0 Global mean 0 °C Ice line 0 ° Albedo 0 Imbalance 0 W/m² FPS 0
Drag to orbit, wheel or pinch to zoom. Colour on the globe is annual-mean surface temperature (blue cold, red warm); white is ice and snow where the model says the surface is colder than the ice threshold. The planet is a cartoon Earth, not real geography.
Budyko-Sellers Energy Balance: Ice Line, Hysteresis and Snowball Earth
One equation per latitude band: sunlight in, infrared out, heat carried poleward. Because ice reflects, the answer can be two different climates for the same Sun. Change the Sun, CO₂ and heat transport, run the clock, and sweep the Sun up and down to see the hysteresis loop. A teaching model with 36 bands, not a climate forecast.
Pause Spin: on Quality: High
Forcing
Climate system
Clock
Run Sun sweep (hysteresis) Reset to warm start Freeze to snowball
Live readouts
Temperature by latitude Energy by latitude Where absorbed sunlight (orange) exceeds outgoing infrared (blue), heat is exported toward the poles by the diffusion term.
Global mean temperature history Hysteresis: ice line vs Sun strength Press the sweep button. The two curves differ where both a warm and a frozen climate are stable.
Independent numerical checks
Controls Drag / one finger: orbit; wheel / pinch: zoom. Sun strength is a multiple of today's solar constant (1361 W/m²). CO₂ changes outgoing infrared by 5.35 ln(C/280) W/m². Heat transport D sets how efficiently the atmosphere and ocean carry heat to the poles. Quality: High is the default on every device; Low (also ?q=low) uses a smaller pixel ratio and no clouds. The science
The equation. With x = sin(latitude): C ∂T/∂t = (Q/4) s(x) [1 − α(T,x)] − (A + B T) + D ∂/∂x[(1 − x²) ∂T/∂x]. Here (Q/4) = S/4 is the mean sunlight, s(x) = 1 − 0.482 P₂(x) spreads it over latitude (North 1975), A + B T is a linearised outgoing infrared law fitted to satellite data (A = 203.3 W/m², B = 2.09 W/m²/K) and the last term is heat diffusion with D = 0.649 W/m²/K. Bands have equal area in x, so the global mean is a plain average.
Ice-albedo feedback. α = 0.32 + 0.08 P₂(x) for open surface and 0.62 for ice and snow, blended smoothly over about 2 K around the ice threshold (−7 °C here, tuned so today's ice line sits near 70°). Colder → more ice → more reflection → colder. That positive feedback is why two stable climates can coexist and why the path matters (hysteresis).
Greenhouse. Doubling CO₂ lowers outgoing infrared by 5.35 ln 2 ≈ 3.7 W/m² (a standard fit); here it simply reduces A.
Numerics. Backward-Euler in time with a tridiagonal solve (stable at large steps) and the albedo updated by three Picard sweeps per step. The Sun sweep steps Q by 0.02 and relaxes to steady state at each point, carrying the state along, upward from a frozen planet and downward from a warm one.
Simplifications. Annual mean, no seasons, no land-sea contrast, no clouds or water-vapour feedback beyond what is folded into A and B, one heat capacity (C = 3×10⁸ J/m²/K, roughly a 70 m ocean layer), one ice threshold temperature. Parameters are the textbook North (1975) values with the ice threshold and albedo tilt chosen for a sensible present-day state; the transition values (for example the Sun fraction at which the planet freezes) depend strongly on them and should be read as qualitative. Not a prediction of real climate. Globe and stand modelled and baked in Blender, scene by Three.js.