A linearly increasing radiative forcing produces a fast adjustment over about four years superposed on a slow deep-ocean response over about four hundred years. Once forcing stops increasing at year 70, surface warming continues and approaches the final 3K equilibrium. The right panel of solution gives amodel response normalized to the supplied 2K endpoint and 3K equilibrium.
Write u=t−70 after stabilization and 3−Ts=Afe−u/4+Ase−u/400, with Af+As=1K. For a 70-yearforcing ramp, the normalized step response is 1−bfe−t/4−bse−t/400, bf+bs=1. The ramp endpoint implies
at u=10,15,20years respectively. The usual order estimate after the fast tail has largely disappeared is T˙s∼(3−2)/400≃0.0025Kyr−1, with a still visible fast contribution near year10.
To hold Ts=T∗ fixed, the deep layer obeys Ts−Td∝e−γu/Cd and the surfaceequation requires F=λT∗+γ(T∗−Td). Hence F˙=−γ2(T∗−Td)/Cd. Using bf≃λ/(λ+γ) gives γ/λ≃0.475 and γ/Cd≃1/(400bf)≃0.00369yr−1. With T∗=2K and initial forcing F0=3λ, the initial fractional decline is
The rate declines slowly as the ocean contrast disappears. A coarser estimate treating the post-fast warming as an effective equilibrium change gives 0.0025/3≃0.08% per year; distinguishing the instantaneous surfacefeedback from the equilibrium feedback explains that difference.
