The Earth Energy Balance Model: One Equation, Two Tipping Points

A planet as a single thermometer

A zero-dimensional energy balance model (EBM) treats the whole Earth as one point with one temperature and asks a single question every instant: is more energy arriving than leaving? Sunlight lands across the planet's cross-section, a disc of area πR², but the whole sphere of area 4πR² radiates it away, which is why the incoming flux is divided by 4. The entire model is one differential equation for the heat content of the ocean-atmosphere system:

C · dT/dt = (1 − α) · S₀/4 − σ·ε·T⁴ + ΔF

C      heat capacity of the ocean-atmosphere system
α      planetary albedo, fraction of sunlight reflected away
S₀     solar constant, ≈ 1361 W/m²
σ      Stefan-Boltzmann constant, 5.67×10⁻⁸ W/m²K⁴
ε      effective longwave emissivity of the atmosphere
ΔF     extra forcing from greenhouse gases, ln(CO₂/CO₂₀) scaled

The left side is how fast the planet's temperature changes; the right side is absorbed sunlight minus the Stefan-Boltzmann outgoing longwave radiation, plus whatever extra forcing greenhouse gases add. Set the right-hand side to zero and you get the equilibrium temperature — the value the system relaxes toward. The simulation on this page integrates this equation forward with a simple explicit step and lets you watch T(t) settle, oscillate, or run away.

Why albedo is the dangerous variable

Ice and snow reflect 60–90% of sunlight; open ocean and bare ground absorb 90% of it. That single fact turns albedo α into a feedback, not a constant: cooling grows ice, growing ice raises α, higher α cools further. In this model α is written as a smooth function of T that shifts from about 0.30 (today's Earth) toward 0.62 as the planet's mean temperature falls through the freezing threshold. Because the feedback reinforces the initial change, it is capable of doing something a simple linear system cannot: producing more than one stable equilibrium for the same solar input.

Two stable worlds and a tipping point in between

Plot absorbed sunlight and outgoing radiation against temperature and, for a band of solar constants, they cross three times: a cold, ice-covered equilibrium near 250 K, a warm, mostly ice-free equilibrium close to today's 288 K, and an unstable equilibrium sitting between them. The middle crossing is not a resting place; it is a ridge. Nudge the temperature very slightly below it and the ice-albedo feedback runs away toward the cold branch — a snowball Earth, a state geologists believe the planet actually reached at least twice, around 720 and 635 million years ago. Nudge it above the ridge and the system slides to the warm branch instead.

At the other extreme, above roughly 305–340 K depending on the atmosphere's optical depth, water vapour itself becomes the runaway ingredient: warmer oceans evaporate more water, water vapour is a potent greenhouse gas, so the planet warms further and evaporates still more, a positive feedback with no stable stopping point until the oceans are gone. This is the textbook runaway greenhouse, thought to be the reason Venus looks the way it does today, and a useful bound on how much forcing Earth's climate could in principle tolerate before crossing a similar threshold.

Where the simplification bites

A single global temperature cannot see the difference between the poles and the tropics, cannot represent ocean heat transport, and updates its albedo instantaneously instead of over the years ice sheets actually take to grow or retreat. Real climate models resolve latitude bands (1-D EBMs), full 3-D circulation, and cloud feedbacks that are still an active area of uncertainty. What the zero-dimensional model keeps, honestly, is the one mechanism that matters most for the bistability: the loop between temperature and albedo, expressed with nothing hidden. That is exactly why it remains the first model taught in every climate course before anyone opens a general circulation model.

Frequently asked questions

Try it live

Everything above runs in your browser — open Earth Energy Balance and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.