D-X3 · The ocean slows the warming

Climate Energy Balance

Why does Earth keep warming after the heating stops increasing? A steady extra heat input keeps adding energy until warming increases heat loss enough to balance it. The deep ocean absorbs some of that energy, so the surface warms quickly at first and then approaches its final temperature much more slowly.

Add a steady heat input and watch the surface warm ahead of the deep ocean. Change the heat exchange to discover why the final temperature takes much longer to arrive.

Surface temperature change $1.91\,\mathrm K$

Deep-ocean temperature change $0.131\,\mathrm K$

Net energy input $1.41\,\mathrm W\,\mathrm m^{-2}$

Final surface temperature change $3.08\,\mathrm K$

Follow the heat

Warm the surface, then the ocean

Extra heating

3.7

F/(\mathrm W\,\mathrm m^{-2})

Cooling response to warming

1.2

\lambda/(\mathrm W\,\mathrm m^{-2}\,\mathrm K^{-1})

Surface–deep ocean exchange

0.7

\gamma/(\mathrm W\,\mathrm m^{-2}\,\mathrm K^{-1})

Drag the gold source up or down to change the extra heating. Sliders work with the arrow keys.

Model, scale and sources

This synthetic, global two-box model follows temperature changes from an initially balanced state after a constant step in radiative forcing. A positive restoring coefficient means warming increases energy loss to space; the coefficient includes all feedbacks as one constant. Surface and deep-ocean heat capacities are eight and one hundred watt-years per square metre per kelvin. Ocean exchange transfers heat between the boxes without creating energy. The drawing shows changes in energy flows, with enlarged arrows and temperature colours; it is not a map of regional temperatures. The model omits weather, spatial circulation, changing clouds and ice, carbon-cycle feedback and time-dependent emissions. Its adjustable parameters and trajectories are teaching examples, not fitted observations or climate projections.

IPCC AR6 · Energy balance and two-layer emulators NOAA GFDL · Transient and equilibrium warming

Simulated data

Two layers, two response times

A constant heating step begins at year zero. The moving marker follows the replay.

Surface temperature change

Deep-ocean temperature change

Start with the essentials

Focus question: What sets the final warming, and what sets how quickly it arrives?
One-sentence intuition: Radiative restoring sets the final warming; heat capacity and ocean exchange set the delay.

Core mathematical model

Heat entering the surface layer

C_s\frac{dT_s}{dt}=F-\lambda T_s-\gamma(T_s-T_d)

The surface receives the extra heating, loses more radiation as it warms, and transfers heat to the colder deep ocean. Time is in years; heat capacities use watt-years per square metre per kelvin.

Heat entering the deep ocean

C_d\frac{dT_d}{dt}=\gamma(T_s-T_d)

The same exchange leaves one layer and enters the other. It disappears when their temperature changes become equal.

Whole-system energy balance

N=F-\lambda T_s=\frac{d}{dt}(C_s T_s+C_d T_d),\qquad T_{s,\infty}=F/\lambda

A positive restoring coefficient means warming increases energy loss to space. The final surface temperature follows from zero net heat input. It is an equilibrium limit, not a forecast for a particular year.

Common difficulties

Steady heating can still change temperature

Typical misconceptionOnce extra heating stops increasing, the temperature should stop rising.

Better mental modelTemperature stops changing when heat entering and leaving balance. A fixed extra input keeps adding energy until warming has raised the outgoing flow enough.

Run the experiment

01
Let the ocean take some heat
Compare The ocean delays warming with Disconnect the deep ocean.
What to observe: Exchange with the deep ocean reduces early surface warming. Both runs have the same eventual surface temperature.
02
Change the restoring response
Choose More warming from the same heat, then move Cooling response to warming.
What to observe: Weaker radiative restoring allows more warming before the added heat can escape.
03
Reverse the energy flow
Choose A cooling experiment or drag the gold heating source down.
What to observe: The surface cools first. The deep ocean follows gradually, transferring heat in the opposite direction.