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The relationship between greenhouse gases and radiative forcing $F(t)$ is given by $F(t)=\eta\frac{\ln\left(M_{AT}(t)\right)-\ln\left(M_{AT}^{1750}\right)}{\ln(2)}+F_{EX}(t)$ where $F_{EX}(t)$ represents forcing from gases other than CO2 in time period $t$ and $\eta$ represents the increase in forcing from the doubling of CO2 in the atmosphere. These other forcings are estimated in a given period by a linear function of the difference between forcings from non-CO2 today and an estimate of those forcings in the year 2100: $F_{EX}(t)=F_{EX}(0)+0.1\left(F_{EX}(2100)-F_{EX}(0)\right)\cdot t.$
Radiative forcing leads the warming in the atmosphere, which then warms the upper ocean, gradually warming the deep ocean. The model is: \begin{align*} T(t) & =T(t-1)+\xi_{1}\left[F(t)-\lambda T(t-1)-\xi_{2}\left(T(t-1)-T_{LO}(t-1)\right)\right]\\ T_{LO}(t) & =T_{LO}(t-1)+\xi_{3}\left(T(t-1)-T_{LO}(t-1)\right) \end{align*} where the $\xi_{i}$ are the transfer coefficients reflecting the rates of flow and thermal capacities of the sinks. In particular, $1/\xi_{1}$ is the thermal capacity of the atmosphere and the upper oceans, $1/\xi_{3}$ is the transfer rate from the upper ocean to the deep ocean, and $\xi_{2}$ is the ratio of the thermal capacity of the deep oceans to the transfer rate from the shallow to deep ocean.
The key parameter is $\lambda$, or climate sensitivity, is a way of representing the equilibrium temperature from doubling the concentration of CO2. If we solve the temperature equation for a constant temperature (i.e. equilibrium), we get $\Delta T/\Delta F=1/\lambda$. If $T_{2xCO_{2}}$ is the equilibrium impact of a doubling of CO2 concentrations, we get $T_{2xCO_{2}}=\Delta F_{2xCO_{2}}/\lambda$ where $\Delta F_{2xCO_{2}}$ is the change in radiative forcing from a doubling of CO2. Therefore, setting $\lambda$ allows us to set the climate sensitivity.