Radiative forcing and feedbacks strongly influence the Arctic climate. The warming in the Arctic has occurred in a faster pace than the global average, due to greenhouse gas forcing and amplifying feedbacks (Stocker et al., 2013). It requires accurate quantification of the radiative effects of associated feedback variables (surface albedo, atmospheric temperature, water vapor, cloud, etc.,) in order to ascertain their contributions to the climate change of interest. For instance, based on the energy budget balance with regard to the Top-of-Atmosphere (TOA), surface or atmospheric budget and assuming the warming induced thermal radiation (Planck effect) balances the radiation changes caused by feedbacks, one can infer how much global or regional warming, e.g., the Arctic warming amplification, can be attributed to individual feedbacks (Held and Soden 2000; Lu and Cai 2009; Pithan and Mauritsen 2014).

Often assumed in feedback analysis is linear additivity of the radiative effects of different feedback variables. For instance, the widely adopted kernel method (Soden and Held 2006) measures the radiation change caused by a feedback variable ( X ) by multiplying a pre-calculated radiative kernel ( ∂ R ∂ X ) with the climate response ( d X ). Due to its simple concept and computational efficiency, a large number of studies have been conducted using this method (e.g., Soden and Held 2006; Zelinka et al., 2012; Vial et al., 2013; Zhang and Huang, 2014) and pre-computed kernels based on different atmospheric datasets, including climate models, reanalyses and satellite data (e.g., Soden et al., 2008; Yue et al., 2016; Huang et al., 2017).

The nonlinear effects, however, are often too large to ignore. When individual feedback terms are independently measured, such as the non-cloud feedbacks in the clear-sky case in the kernel method, the ignored nonlinear effects may lead to a non-closure of the radiation budget, i.e., the sum of the individual terms cannot reproduce the overall radiation change (e.g., Huang 2013; Vial et al., 2013). In the Arctic, where climate perturbations are of large magnitudes, e.g., in the case of sea ice melt, the non-closure issue is especially noticeable (e.g., Shell et al., 2008; Block and Mauritsen 2013; Zhu et al., 2019). Besides the large perturbations in surface albedo, the Arctic is also noted for its strong and unique lapse rate (e.g., Pithan and Mauritsen 2014) and cloud (e.g., Kato et al., 2006) feedbacks. It should be noted that although some methods exhibit a seemingly good radiation closure, the nonlinear effects are not treated but hidden in the feedback term(s) measured as a residual, e.g., the cloud feedback term in the typical kernel method, including the adjusted cloud radiative forcing (aCRF) technique (Shell et al., 2008; Soden et al., 2008).

Although the existence of the nonlinear effects has been recognized (e.g., Zhang et al., 1994; Colman et al., 1997), their impacts were seldom isolated and quantified. Some recent works have specifically addressed the nonlinearity issue in the radiative feedback analysis. Zhu et al. (2019) for the first time used a neural network model (a nonlinear diagnostic method without linearity assumption) to assess the radiative feedbacks and identified a few strong nonlinear effects, including a strong cloud-water vapor coupling effect in the tropical climate variations and a strong nonlinear dependence of radiation flux on the surface albedo. Using Partial-Radiative-Perturbation (PRP) experiments and brute-force radiation model-based computations, Huang and Huang (2021) verified the cloud-water vapor coupling effect and offered an analytic estimation of this effect on the longwave radiation. Shakirova and Huang (2021) advanced the neural network model of Zhu et al. and demonstrated its advantages particularly for quantifying the albedo feedback.

In this paper, we aim to give an overview of the nonlinear radiative feedback effects in Arctic climate change. Based on a heuristic climate change scenario of broad interest: the abrupt quadrupling of atmospheric CO₂ (4xCO₂), we investigate how the nonlinear radiative effects arise from the univariate and multivariate variations of the feedback variables, such as atmospheric and surface temperature (t), water vapor (q), surface albedo (a) and cloud (c), and measure how the nonlinear effects compare to the linear effects in terms of magnitude and pattern. We note that in this paper we are not concerned with how the changes in these variables are resulted, which if nonlinearly related to the surface warming may also cause nonlinearity in climate feedbacks, but focus on how their changes, as projected by the GCM, lead to nonlinear changes in the TOA longwave (LW) and shortwave (SW) radiation energy fluxes. In the following sections, we will define, demonstrate and discuss the various feedback effects of interest in order.

Here, we define a radiative feedback as the (partial) radiation change, in the units of W m⁻², due to one or multiple feedback variables. This should be distinguished from a feedback parameter, which is normalized by surface temperature change and is in the units of W m⁻² K⁻¹.

Consider the radiation field of interest, e.g., the TOA or surface radiation flux, as a function of the feedback variables: R = R ( x ,   y ,   z ) , where the letters ( x ,   y ,   z ) are generic notations of the feedback variables. The total radiation change in a given climate change scenario can thus be expressed by a Taylor series as Δ R ( x , y , z ) = R ( x 2 ,   y 2 ,   z 2 ) − R ( x 1 ,   y 1 ,   z 1 ) = ∂ R ∂ x Δ x + ∂ R ∂ y Δ y + ∂ R ∂ z Δ z       ( univariate linear effects ) + 1 2 [ ∂ 2 R ∂ x 2 ( Δ x ) 2 + ∂ 2 R ∂ y 2 ( Δ y ) 2 + ∂ 2 R ∂ z 2 ( Δ z ) 2   ( univariate nonlinear effects ) + 2 ∂ 2 R ∂ x ∂ y Δ x Δ y + 2 ∂ 2 R ∂ y ∂ z Δ y Δ z + 2 ∂ 2 R ∂ x ∂ z Δ x Δ z ]           ( multivariate nonlinear effects ) +   O ( Δ 3   ) (1) where the subscripts 1 and 2 denote two different climate states, e.g., those before and after quadrupling CO₂ (noted as 1xCO₂ and 4xCO₂, respectively from now on); such terms as Δ x = x 2 − x 1 denote the climate responses. The terms on the righthand side of Eq. (1) illustrate three types of radiative effects that we aim to elucidate here:

1) The univariate linear effects, such as ∂ R ∂ x Δ x , which we denote as Δ R x ;

To avoid confusion, we denote a univariate feedback, i.e., the overall radiation change due to a single variable, as Δ R ( x ) , which consists of both univariate linear ( Δ R x ) and univariate nonlinear ( Δ R x x ) effects: Δ R ( x ) = R ( x 2 ,   y 1 , ... ) − R ( x 1 ,   y 1 , ... ) = ∂ R ∂ x Δ x + 1 2 ∂ 2 R ∂ x 2 ( Δ x ) 2 + O ( Δ 3 ) = Δ R x + Δ R x x (2) Equation (2) is written following the PRP concept (Wetherald and Manabe 1988) and measures the univariate feedbacks by simply evaluating the radiation flux twice: first with an unperturbed profile, R ( x 1 ,   y 1 , ... ) , and then perturbing x only, R ( x 2 ,   y 1 , ... ) . Note that other unperturbed independent variables than y are omitted in these expressions. If not otherwise stated, unspecified independent variables all take the unperturbed values when the radiation fluxes are evaluated in the following. The evaluation of the radiation fluxes can be done using a physical model, i.e., a radiative transfer model (RTM) (Huang and Huang 2021), or a statistical model, e.g., a neural network model that emulate the radiation fluxes (Zhu et al., 2019).

Similarly, a bivariate feedback can be expressed as: Δ R ( x , y ) = R ( x 2 ,   y 2 ) − R ( x 1 ,   y 1 ) = ∂ R ∂ x Δ x + ∂ R ∂ y Δ y + 1 2 ∂ 2 R ∂ x 2 ( Δ x ) 2 + 1 2 ∂ 2 R ∂ y 2 ( Δ y ) 2 + ∂ 2 R ∂ x ∂ y Δ x Δ y + O ( Δ 3 ) = Δ R x + Δ R y + Δ R x x + Δ R y y + Δ R x y (3) From Eq. 2 and Eq. 3, the bivariate coupling effect Δ R x y can be obtained as Δ R x y = Δ R ( x , y ) − Δ R ( x ) − Δ R ( y ) = R ( x 2 ,   y 2 ) − R ( x 2 ,   y 1 ) − R ( x 1 ,   y 2 ) + R ( x 1 ,   y 1 ) (4)

Based on the above equations and following Huang and Huang (2021), we evaluate the radiation fluxes and isolate the respective feedback effects ( Δ R x ,     Δ R x x ,     Δ R x y ,   e t c . ) , using the Rapid Radiative Transfer Model (RRTM, Mlawer et al., 1997). Using this RTM, the TOA and surface radiation fluxes are computed offline from instantaneous atmospheric profiles generated by the Community Earth System Model, CESM1.2, in a quadrupling CO₂ experiment (Wang and Huang, 2020). More details of the flux computation can be found in Huang and Huang (2021); we note that the radiative transfer computations, including the PRP computations, are based on instantaneous (3-hourly, as opposed to monthly mean) profiles at the original horizontal resolutions (1.9 ° × 2.5 °) of the CESM and then averaged monthly or annually in all the results presented in the following section.

Note that we use a one-sided PRP, starting with the unperturbed climate and then prescribing the change(s) in the variables of interest, to define feedbacks, i.e., how much radiation change is caused by the change of the feedback variable(s) of concern. Some studies opt to use two-sided perturbations (e.g., Colman and McAvaney 1997). In contrast to Eq. (2), one may evaluate the feedback of x as δ R ( x ) = 1 2 R [ ( x 2 ,   y 1 ) − R ( x 1 ,   y 1 ) + R ( x 2 ,   y 2 ) − R ( x 1 ,   y 2 ) ] ≈ 1 2 [ ∂ R ∂ x | y 1 Δ x + ∂ R ∂ x | y 2 Δ x ] ≈ 1 2 [ ∂ R ∂ x | y 1 Δ x + ( ∂ R ∂ x | y 1 + ∂ ∂ y ( ∂ R ∂ x ) | y 1 Δ y ) Δ x = ∂ R ∂ x + 1 2 ∂ 2 R ∂ x ∂ y Δ x Δ y (5) which, as shown in the above expansion, effectively includes nonlinear coupling effects such as 1 2 ∂ 2 R ∂ x ∂ y Δ x Δ y in δ R ( x ) . When individual feedbacks are evaluated this way, their sum can better reproduce the overall radiation change, i.e., achieving a better radiation closure. However, it should be noted these "individual" feedbacks δ R ( x ) differ from the univariate feedbacks Δ R ( x ) as δ R ( x ) contains coupling effects. To disclose these coupling effects, we adopt the one-sided formulations as exemplified by Eq. 2, Eq. 3, and Eq. 4

In this paper, we use the climate change in an abrupt 4xCO₂ experiment of CESM (Wang and Huang, 2020) to provide a context for examining the linear and nonlinear radiative feedbacks. As illustrated in Figure 1 for a few selected variables, this scenario represents strong perturbations in the Arctic climate, including reduction in surface albedo due to sea ice melt, surface warming and atmospheric moistening. The feedback quantifications presented in the following are based on the two months exemplified in Figure 1 if not otherwise noted.

In this paper, we present an overview of the nonlinear effects in both longwave and shortwave radiative feedbacks in the CO₂-driven Arctic warming. Based on brute-force radiation model calculations we disclose the most prominent nonlinear feedback effects and based on simple analytical models we offer explanations of their physcial causes. Although the presentation and discussion are focused on the Arctic feedbacks, the diagnostic framework (Eq. 1) and the theoretical explanations are applicable to global feedback analyses.

We identify these important nonlinear feedback effects:

1) The univariate nonlinear effect in the surface albedo feedback in the shortwave. This nonlinearity can be understood from a simple analytical model [Eq. (11)] that accounts for the coupling, due to multiple-scattering, between the surface and atmosphere (clouds). This coupling makes the radiative sensitivity to surface albedo decrease with the surface albedo value (Figure 6). Because of this effect, it generally leads to an overestimate of the surface albedo feedback when albedo kernels computed from the current climate are used to quantify the albedo feedback in a warming climate (Figure 3; Table 1).

2) The bivariate surface albedo-cloud coupling effect in the shortwave. This effect is attributable to the masking effect of cloud increase that damps the radiative sensitivity to surface albedo. This effect is the most prominent in the summer when solar insolation is strong, as illustrated by Figure 8 for the month of June.

3) The multivariate temperature-cloud feedback in the longwave. This effect is attributable to the fact that the Arctic warming is much stronger at and near the surface than in the upper air, which leads to a damping effect on the temperature feedback. This nonlinear effect should be distinguished from the temperature lapse rate feedback and is found to be the strongest in the winter as illustrated by Figure 7 for the month of January.

4) Although the univariate water vapor feedback largely scales logarithmically with water vapor changes, it is found that the relation deviates from the logarithmic scaling, especially in the window band. This is due to the unsaturated atmospheric absorption in this band and suggests that a hybrid scaling method as proposed by Eq. (10) may improve the accuracy of the kernel-diagnosed water vapor feedback (compare Figure 2C and Figure 5C).

It should be noted that the large nonlinear effects discovered here is not limited to the 4xCO₂ experiment. As shown by Huang and Huang (2021) for the longwave feedbacks and Shakirova and Huang (2021) for the shortwave feedbacks, similar, strong nonlinear effects exist even also interannual climate variations. It is noted that the nonlinear effects may quantitatively differ in different forcing experiments, thus requiring them to be assessed more comprehensively in future work.

The strong nonlinear effects as disclosed here call into question the accuracy of linear methods currently used in the feedback analysis. Nonlinear methods are needed to improve the accuracy of feedback quantificaiton when RTM-based PRP experiments are not feasible due to its forbidding computational demands. Especially in need are replacement of the linear kernels for the surface albedo feedback and cloud feedback quantification. Although a handful of studies have touched this topic, for instance, using quadratic fitting (Colman et al., 1997), histogram (Zelinka et al., 2012) and neural network (Zhu et al., 2019) methods, this challenging problem demands devoted research programs to further develop, test and mature the candidate methods.