Based on the energy balance equation, CFRAM considers the entire atmospheric column and the surface as a coupled surface-atmosphere system. CFRAM is different from traditional TOA-based feedback diagnostic methods, such as the Partial Radiative Perturbation (PRP) method (Wetherald and Manabe, 1988) and the Cloud Radiative Forcing (CRF) analysis method (Cess et al., 1990), in which only the radiative feedback processes can be explicitly considered while the non-radiative feedback processes are all hidden in changes of the lapse rate, because it directly measures the addible contributions of individual feedback processes to the total temperature changes and explicitly distinguishes the radiative and non-radiative feedback processes (Cai and Lu, 2009; Lu and Cai, 2009).
For a surface-atmosphere column at a given horizontal grid point, CFRAM describes the difference (∆) of the total energy balance between two climate states as
\begin{equation} \Delta S-\Delta R+\Delta Q_{\rm NRad}=0 . (1)\end{equation}
Note that here we consider the difference between the model climatology and the climatology from ERA-Interim. ∆ R and ∆ S are respectively the difference in divergence of the longwave radiation flux and the convergence of shortwave radiation flux between FGOALS-s2 and ERA-Interim. ∆ Q NRad represents the difference between FGOALS-s2 and ERA-Interim in convergence of the total energy flux mainly due to the surface sensible heat flux (∆ Q SH), surface latent heat flux (∆ Q LH), dynamic processes (∆ Q dyn_sfc) involving net energy convergence at the surface (i.e., land and oceanic surface energy transport and energy storage), and dynamic processes in the atmosphere at all scales (∆ Q dyn_atm) including turbulence, convection, and large-scale atmospheric motions:
\begin{equation} \Delta Q_{\rm NRad}=\Delta Q_{\rm SH}+\Delta Q_{\rm LH}+\Delta Q_{\rm dyn_sfc}+\Delta Q_{\rm dyn_atm} . (2)\end{equation}
According to the linear approximation in CFRAM, the nonlinear interactions among various radiative feedback processes are assumed negligible. Thus, ∆ S and ∆ R can be linearly decomposed into the sum of partial energy perturbations due to individual radiative processes such as incoming solar energy flux at TOA (o), ozone (O3), water vapor (w), cloud (c), surface albedo (α), and that due to the temperature differences (∆ T) between FGOALS-s2 and ERA-Interim:
\begin{eqnarray} \label{eq2} \Delta S&\approx&\Delta F_o+\Delta S_{{\rm O}_3}+\Delta S_w+\Delta S_c+\Delta S_\alpha ,\\ \label{eq3} \Delta R&\approx&\Delta R_{{\rm O}_3}+\Delta R_w+\Delta R_c+\left(\dfrac{\partial R}{\partial T}\right)\Delta T , (4)\end{eqnarray}
where (∂ R/∂ T) is the Planck feedback matrix whose jth column corresponds to the vertical profile of the radiative energy perturbation due to 1-K warming at the jth layer from the ERA-Interim state to the FGOALS-s2 state; (∂ R/∂ T)∆ T represents the difference in divergence of the longwave radiation energy flux due to the temperature difference ∆ T. By substituting Eqs. (2-4) into Eq. (1), rearranging the terms and multiplying both sides of the resultant equation by (∂ R/∂ T)-1, we obtain
\begin{eqnarray} \label{eq4} \Delta T&\approx&\left(\dfrac{\partial R}{\partial T}\right)^{-1}[\Delta F_o\!+\!\Delta (S\!-\!R)_{{\rm O}_3}+\Delta (S\!-\!R)_w\!+\!\Delta (S\!-\!R)_c+\nonumber\\ &&\Delta S_\alpha+\Delta Q_{\rm SH}+\Delta Q_{\rm LH}+\Delta Q_{\rm dyn_sfc}+\Delta Q_{\rm dyn_atm}]. (5)\end{eqnarray}
It follows that the local T s difference at each grid point between FGOALS-s2 and ERA-Interim (∆ T) can be directly attributed to the model biases in representing the radiative processes including [left to right in Eq. (5)] incident solar radiation at the TOA, ozone, water vapor, cloud, and surface albedo; and non-radiative processes including surface sensible heat flux, surface latent heat flux, surface dynamics, and atmospheric dynamics. Among the radiative processes, the difference in solar forcing and surface albedo only contribute to the shortwave radiative energy perturbation, while ozone, cloud, and water vapor can contribute to both the shortwave and the longwave radiative energy perturbation.
To calculate the radiative energy perturbation terms in Eq. (5), the Fu-Liou radiative transfer model (Fu and Liou, 1992, 1993) is adopted in CFRAM. Specifically, the radiative forcing ∆ Fo is given by
\begin{equation} \Delta F_o=[F(o_{\rm M},{\rm O}_{3{\rm E}},w_{\rm E},c_{\rm E},\alpha_{\rm E})-F(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm E}, c_{\rm E},\alpha_{\rm E})] , (6)\end{equation}
where the subscripts "M" and "E" indicate that the variable is for the model or for the ERA-Interim. Similarly, the radiative energy perturbation terms ∆ (S-R) O3,∆ (S-R)w,∆ (S-R)c, and ∆ Sα are given by
\begin{eqnarray} \Delta(S-R)_{{\rm O}_3}&=&[\Delta(S-R)(o_{\rm E},{\rm O}_{3{\rm M}},w_{\rm E},c_{\rm E},\alpha_{\rm E})-\qquad\nonumber\\ &&\Delta(S-R)(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm E},c_{\rm E},\alpha_{\rm E})] ,\\ \Delta(S-R)_w &=&[\Delta(S-R)(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm M},c_{\rm E},\alpha_{\rm E})-\nonumber\\ &&\Delta(S-R)(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm E},c_{\rm E},\alpha_{\rm E})] ,\\ \Delta(S-R)_c&=&[\Delta(S-R)(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm E},c_{\rm M},\alpha_{\rm E})-\nonumber\\ &&\Delta(S-R)(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm E},c_{\rm E},\alpha_{\rm E})] ,\\ \Delta S_\alpha&=&[S(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm E},c_{\rm E},\alpha_{\rm M})-\nonumber\\ &&S(o_{\rm E},{\rm O}_{3{\rm E}},w_{\rm E},c_{\rm E},\alpha_{\rm E})] . \end{eqnarray}
For the non-radiative processes, the energy perturbation by surface sensible heat and latent heat fluxes can be directly obtained from the difference between the model and ERA-Interim. However, because layer-by-layer data fields of energy fluxes related to dynamic processes are generally unavailable in both model and reanalysis datasets, ∆ Q dyn_sfc and ∆ Q dyn_atm have to be inferred from Eqs. (1) and (2) as the residual of energy fluxes between radiative processes and the surface sensible and latent heat processes,
\begin{equation} \label{eq5} \Delta Q_{\rm dyn_sfc}+\Delta Q_{\rm dyn_atm}=-(\Delta S-\Delta R)-\Delta Q_{\rm SH}-\Delta Q_{\rm LH} . (11)\end{equation}
Since by definition ∆ Q dyn_sfc has zero values in the atmospheric layers, while ∆ Q dyn_atm has zero values at the surface layer in CFRAM, we can further distinguish ∆ Q dyn_sfc and ∆ Q dyn_atm from each other within the residual.
Before examining the partial contributions from each of the individual physical processes listed in Eq. (5), it should be pointed out that the differences in incident solar radiation at TOA between FGOALS-s2 and ERA-Interim are not model biases. This difference exists because the solar constant applied to define the solar radiation in FGOALS-s2 is 1365-1366 W m-2, which is recommended by CMIP5, while that applied in ERA-Interim is 1370 W m-2, which is estimated from multiple sources of data but has been shown to, on average, overestimate global solar radiation by about 2 W m-2 (Dee et al., 2011). The difference in the annual mean T s between HadCRU4 (Hadley Climate Research Unit version 4) and ERA-Interim reported in (Park et al., 2013) confirms the possible effect of the overestimated solar coefficient.