 Solution

$F$ is the imposed radiative forcing; $-\lambda T_s$ is the restoring change in outgoing radiation; and $-\gamma (T_s-T_d)$ transfers heat from the surface layer to the deep ocean, whose equal and opposite gain appears in its own energy balance. The additional $-{\lambda }^{\prime }(T_s-T_d)$ is a radiative adjustment correlated with the surface–deep-ocean temperature contrast, representing a pattern-dependent effective feedback rather than another conserved interlayer heat flux.

For constant forcing, final equilibrium has $T_s=T_d=F/\lambda$. During transient warming $T_d<T_s$, so the radiative response inferred from a recent global energy budget is ${\lambda }_{\mathrm{eff}}=\lambda +{\lambda }^{\prime }(1-T_d/T_s)>\lambda$. If that response were mistaken for the final feedback, the inferred equilibrium climate sensitivity would be too small. Summing the two equations makes the distinction explicit: $d(C_s{T}_s+C_d{T}_d)/dt=F-\lambda T_s-{\lambda }^{\prime }(T_s-T_d)$.