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I'm trying to understand Box 7.1 "The Energy Budget Framework: Forcing and Response" in the IPCC 2021 document. The link is https://www.ipcc.ch/report/ar6/wg1/chapter/chapter-7/ .

$N(x_{1},x_{2},\ldots,T)$ seems to be the net radiative flux at the top of the atmosphere. So $N$ is inbound $W m^{-2}$ minus outbound $W m^{-2}$. N is a function of the Global Surface Air Temperature (GSAT) T and a bunch of variables $x_{i}$ specifying the state of the atmosphere. So, $x_{1}$ (say) could be the mole fraction of CO2.

The text says Effective Radiative Forcing (ERF) $\Delta F$ (units power per unit area), "quantifies the change in the net TOA energy flux of the Earth system due to an imposed perturbation that are uncoupled to any GSAT change." So, I think this means, \begin{equation} \Delta F=\frac{\partial N}{\partial x_{i}}\Delta x_{i} \end{equation} summed over repeated $i$.

The text also explicitly defines the feedback parameter $\alpha$ (units $W m^{-2} K^{-1}$) as, \begin{equation} \alpha=\frac{\partial N}{\partial x_{i}}\frac{dx_{i}}{dT} \end{equation} Earlier in the text, the following equation was written, \begin{equation} \Delta N=\Delta F +\alpha \Delta T \end{equation} Now mathematically, $\Delta N$ has to be, \begin{equation} \Delta N=\frac{\partial N}{\partial x_{i}}\Delta x_{i}+\frac{\partial N}{\partial T}\Delta T =\frac{\partial N}{\partial x_{i}}\frac{dx_{i}}{dT}\Delta T+\frac{\partial N}{\partial T}\Delta T=\alpha\Delta T+\frac{\partial N}{\partial T}\Delta T \neq\alpha\Delta T +\Delta F \end{equation} The definition in the text for the ERF $\Delta F$ doesn't seem consistent. What am I failing to understand?

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