INTERACTIVE COMPANION · EARTH'S ENERGY BUDGET · CLIMATE PHYSICS

The Thermal Inertia Bench

A live model accompanying “Why the Planet Has Not Finished Warming Yet”

Two boxes. One on top, holding the atmosphere, the land and the ocean mixed layer. One underneath, holding the rest of the ocean. A pipe between them. That is the entire apparatus, and it is enough to explain why the surface temperature you measure today is not the temperature that today's greenhouse gases have already paid for.

Both models below run the same four equations the club's analysis script runs, in your browser, with no library and nothing fetched. On their default settings they reproduce the article's headline numbers exactly: a fast timescale of 3.95 years, a slow one of 247.7 years, an equilibrium sensitivity of 3.05 K against a transient response of 1.90 K, and 73.61% of the eventual warming delivered a century after the forcing changes.

Model 1. Watch the two clocks run

Set a forcing and press run. The upper box warms fast, because it is thin. The lower box warms slowly, because it is not. While the lower box is still cold it drains heat out of the upper one, and that drain holds the surface below the temperature the forcing would otherwise produce. The shaded region on the chart is the difference: warming that has been bought and not yet delivered.

C · dT/dt = F − λT − γ(T − T₀)
C₀ · dT₀/dt = γ(T − T₀)

With C = 7.3 and C₀ = 106 W yr m⁻² K⁻¹, the CMIP5 multimodel means. Integration is fourth-order Runge-Kutta at a step of 0.05 years, the same scheme the analysis script uses at a finer step, and the curve is drawn against the closed-form solution so you can see them lie on top of each other.

Radiative forcing F 3.45 W m⁻² 3.45 is a doubling of CO₂, half the published 4×CO₂ forcing of 6.9. Try 2.72, the total anthropogenic forcing in 2019 assessed by IPCC AR6, and watch where the equilibrium line lands relative to the 1.1 K the world has actually warmed. The shape of the curve never changes when you move this slider. Only its height does, because the equations are linear. Heat uptake efficiency γ 0.70 W m⁻² K⁻¹ Drag this to zero and the deep box disconnects entirely. The surface then reaches its full equilibrium inside forty years and the shaded pipeline vanishes. Every bit of the delay in this model comes from the ocean underneath. Put it back at 0.70, the CMIP5 ensemble mean. Time axis ends at 1000 years Pull it out to 3000 and the surface and deep curves finally meet. Pull it in to 100 and the run looks almost finished, which is the illusion this whole article is about. Animation Run sweeps the clock from year zero. If your system asks for reduced motion the page skips the sweep and draws the finished curve straight away. The animation also pauses whenever this tab is hidden, so it will not spin in the background.

year: – surface: – deep: – realized: – still committed: – imbalance: –

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Fast timescale: –
Slow timescale: –
Equilibrium: –
RK4 minus closed form: –

That last line is the model grading its own homework. The chart draws the Runge-Kutta integration; the numbers come from the two-exponential closed form; and the difference between them is printed so you can watch it stay at the level of floating-point noise while you move the sliders.

Model 2. Where the gap between transient and equilibrium comes from

Equilibrium climate sensitivity is what the planet does eventually. Transient climate response is what it has done by the time carbon dioxide doubles, seventy years into a 1%-per-year ramp. The two are different numbers and the distance between them is not a measurement error. It is the ocean.

Two sliders. λ is the climate feedback, which sets where the system ends up. γ is the heat uptake efficiency, which sets how long it takes to get there. Watch what each one does to the two bars, and to the two timescales underneath them.

Climate feedback λ 1.13 W m⁻² K⁻¹ The CMIP5 multimodel mean is 1.13 with an intermodel standard deviation of 0.31. Push it to 1.70 and you get the effective feedback the observed historical record implies, which the article argues is biased low by the pattern effect. Notice that a less sensitive climate is also a faster one: the slow timescale falls as you raise λ. Heat uptake efficiency γ 0.70 W m⁻² K⁻¹ The 16 CMIP5 models calibrated by Geoffroy and colleagues span 0.5 to 1.2, shaded on the chart. Move γ from one end of that band to the other and the equilibrium bar does not budge by a thousandth of a kelvin while the transient bar moves by a fifth of one. Two models can agree perfectly on the destination and disagree badly on the journey. Upper box heat capacity C 7.3 W yr m⁻² K⁻¹ This is the one parameter that barely matters here. Double it and the fast timescale doubles while the slow one hardly moves, because after the first few decades the pipeline is the deep ocean and nothing else. That is why the article's Monte Carlo holds C fixed and samples the other three. Compare against The dotted reference line on the chart. Switch to the energy-budget case and you see the claim the article disputes: a model tuned to the historical record says less warming is in the pipeline, because it has absorbed the pattern effect into λ.

ECS: – TCR: – TCR/ECS: – τ_fast: – τ_slow: – realized at 100 yr: –

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Three things worth trying

Set γ to zero in either model. The two-box system becomes a one-box system, the slow mode disappears, and the fast timescale becomes exactly C/λ = 6.46 years. Committed warming goes to essentially nothing within a human lifetime. This is the sanity check the analysis script runs as validation V2, and it is also the clearest demonstration that the ocean is doing all of the delaying.

Then set λ to 1.70 in Model 2. Equilibrium sensitivity drops from 3.05 K to 2.03 K, the realized fraction at 100 years climbs from 73.6% to 81.2%, and the pipeline shrinks from 0.81 K to 0.38 K. That is what happens when you read the feedback off a century of observations instead of a long model run, and the article's section 10 explains why that reading is probably too optimistic.

Last, put γ at 1.2 and λ at 0.82, the two published extremes that both slow the system down. TCR/ECS falls to 0.454 and the realized fraction at 100 years to 60.8%, the corner of the article's parameter grid where the pipeline is deepest: 1.65 K of an eventual 4.21 K still missing a century after the step. Nothing about the eventual warming was settled by moving those two sliders. Only the wait.