Follow stored energy after forcing removal

Inspect the moment an input changes

The imposed forcing is 4 W/m² from year zero up to, but not including, year 25. At year 25 it becomes zero. The ledger shows separate entries just before and just after removal so a discontinuity in the energy-transfer rate is not mistaken for an instantaneous jump in stored energy.

Predict which quantities can change at that instant: forcing, temperature anomaly, feedback flux, net flux and stored energy. Then inspect the two rows. A finite reservoir cannot lose a finite amount of stored energy in zero time under a bounded flux. Its temperature and storage therefore remain continuous.

This experiment removes a synthetic imposed radiative forcing. It is not a scenario in which human CO₂ emissions become zero. The model contains no carbon cycle, atmospheric concentration or relationship between emissions and forcing, so those actions cannot be substituted for each other.

Account for the four-watt jump

Net flux is N = F − λθ, with λ = 3.2 − f. Immediately before removal, Nbefore = 4 − λθ25. Immediately afterward, Nafter = −λθ25. Subtracting gives Nafter − Nbefore = −4 W/m².

Temperature has not jumped, so the baseline and feedback contributions are the same in both rows. Only F changes at that moment. The net therefore drops by exactly the imposed 4 W/m², even if the preceding state was still warming appreciably.

At H = 200 m and f = 1.2, θ25 is approximately 1.7080 K and λ = 2. Just before removal, net is about +0.5839 W/m². Just afterward it is about −3.4161 W/m². One state is still accumulating energy and the other is losing energy, yet their instantaneous temperature and storage agree.

On paper, draw paired ledgers for this example. Keep θ identical and cross out only the forcing term. This action makes the continuity argument concrete rather than relying on the apparent smoothness of a plotted temperature curve.

Follow the remaining stored energy

After removal, the equilibrium target is zero and the exact solution is θ(t) = θ25 exp[−(t − 25)/τ], with τ = C/λ in compatible time units. The positive anomaly decreases toward zero. It does not cross below zero in this stable, unforced one-reservoir calculation.

For the 200 m, f = 1.2 example, the anomaly is still about 0.24935 K at year 50. A positive anomaly during cooling is not a contradiction. Temperature indicates the state relative to the initial reference; net flux indicates whether its energy is increasing or decreasing now.

Stored energy is E = Cθ. Here C is 820 million J m⁻² K⁻¹, so at year 50 storage remains about 204.47 MJ/m² above the initial state. Its decline corresponds to cumulative net energy loss after forcing removal. No energy has disappeared merely because the external input changed.

Separate persistence from ongoing forcing

Compare depths 50 and 200 m at the same feedback. The larger reservoir cools more slowly after removal. Its later positive anomaly is residual stored energy under this model, not evidence that the forcing control secretly stayed on.

The NASA energy-budget explanation gives context for energy accounting, but this transient trajectory is a constructed calculation rather than a measured cooling event. It omits ocean exchange, evolving feedbacks and multiple timescales. The complete real Earth response cannot be inferred from the exponential alone.

Record one sentence about state and one about rate at years 25 after removal and 50. Include signs and units. This prevents “still warm” and “still warming” from being treated as equivalent observations.

Transfer to a different removal state

A paper reservoir has θ = 1.5 K and net damping 2 W m⁻² K⁻¹ immediately before a 4 W/m² forcing is removed. Calculate net flux before and after. Which of temperature, stored energy and warming rate can jump?

Check the forcing-removal transfer

Net is +1 W/m² before and −3 W/m² after removal. Temperature and stored energy remain continuous. Their rates of change can jump because the net flux changes abruptly; the temperature rate is N/C.