CNOP-P-Based Parameter Sensitivity Analysis for North Atlantic Oscillation in Community Earth System Model Using Intelligence Algorithms

Abstract

Model error, which results from model parameters, can cause the nonnegligible uncertainty in the North Atlantic Oscillation (NAO) simulation. Conditional nonlinear optimal perturbation related to parameter (CNOP-P) is a powerful approach to investigate the range of uncertainty caused by model parameters under a specific constraint. In this paper, we adopt intelligence algorithms to implement the CNOP-P method and conduct the sensitivity analysis of parameter combinations for NAO events in the Community Earth System Model (CESM). Among 28 model parameters of the atmospheric component, the most sensitive parameter combination for the ${\text{NAO}}^{+}$ consists of parameter for deep convection (cldfrc_dp1), minimum relative humidity for low stable clouds (cldfrc_rhminl), and the total solar irradiance (solar_const). As for the ${\text{NAO}}^{-}$, the parameter set that can trigger the largest variation of the NAO index (NAOI) is comprised of the constant for evaporation of precip (cldwat_conke), characteristic adjustment time scale (hkconv_cmftau), and the total solar irradiance (solar_const). The most prominent uncertainties of the NAOI ($\mathrm{\Delta }\text{NAOI}$) caused by these two combinations achieve 2.12 for ${\text{NAO}}^{+}$ and −2.72 for ${\text{NAO}}^{-}$, respectively. In comparison, the maximum level of the NAOI variation resulting from single parameters reaches 1.45 for ${\text{NAO}}^{+}$ and −1.70 for ${\text{NAO}}^{-}$. It is indicated that the nonlinear impact of multiple parameters would be more intense than the single parameter. These results present factors that are closely related to NAO events and also provide the direction of optimizing model parameters. Moreover, the intelligence algorithms adopted in this work are proved to be adequate to explore the nonlinear interaction of parameters on the model simulation.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

North Atlantic Oscillation (NAO) has the most prominent seesaw-type oscillation in atmospheric circulation between subtropical high and subpolar low [1]. In recent decades, NAO presents the characteristic of abnormal and frequently changed, especially for winter NAO [2]. Recent research indicates that the ${\text{NAO}}^{+}$ (${\text{NAO}}^{-}$) leads to increased (reduced) temperature [3], and extremely cold weather along with snowstorms in Europe is closely associated with ${\text{NAO}}^{-}$ in a large-scale NAO circulation [4, 5]. The low-frequency change and seasonal features of NAO have a significant impact on the global climate [6, 7], thus estimating the uncertainty of NAO simulation and prediction has practical implications [8].

NAO is a complicated nonlinear system of atmosphere, and it is widely considered to be mainly triggered by Rossby wave breaking [9]. Beyond that, the origins of the NAO fluctuation are complex and varied, including sea surface temperature (SST) anomalies in both tropics and extratropics, extreme stratospheric events [10], tropical atmospheric heat anomalies [11], and intensity of geomagnetic activity [12]. Several models have been developed to describe the nonlinear scale interaction and the life cycle of NAO [13, 14], such as the CERFACS forecast system [15], CMCC-INGV [16], and Community Earth System Model (CESM) [17]. However, the climate simulation using dynamic models would be influenced by both initial condition errors and model errors [18], which have become two major predictability problems [19]. Thereinto, model error is considered as one of the main reasons that cause inaccurate modeling, and it would make the uncertainty exist in the NAO simulation and prediction [20, 21]. Previous researches have pointed out that the prediction skills can be improved by reducing errors of sensitive parameters [22, 23].

The first step in optimizing the model via parameter correction is to determine the sensitive parameters and evaluate corresponding sensitivities. Sensitivity analysis is a method to quantify the sensitivity according to the uncertainty of objective function, and it can help us to identify critical relations between variables and prediction. In previous studies, a few approaches have been adopted to explore the sensitivity of physical parameters in numerical models [24]. The hybrid approach, which is coupled with Morris screening method, response surface method, and RSMSobol method, has been successfully applied to evaluate the impact of global parameters in the distributed time-variant gain model (DTVGM), thereby providing a scheme to improve parameter estimation of the model [25]. Griensven et al. have adopted a novel sampling strategy combined with Latin-hypercube and one-factor-at-a-time sampling in a hydrological model to conduct global sensitivity analysis for parameters with a limited number [26]. In the research on soil moisture, the global sensitivity analysis based on the advanced integral equation model highlights the quantitative and systematic evaluation of parameter sensitivities [27]. Although the uncertainties in these climate simulations have been recognized and the corresponding sensitivity analysis schemes with single parameters have been performed, current studies may overlook the dependencies and interactions among these parameters and have not simultaneously considered multiple parameters. Besides, sensitive parameters in their researches are selected based on experience, and potential sensitive parameters may be missed out.

As mentioned above, previous studies have investigated the uncertainty owing to model parameters in simulating several climate events. However, the research on the estimation of model error for the NAO is almost in blank. In this paper, we aim to explore the maximal extent of uncertainty caused by sensitive parameters in the NAO simulation using CESM. CESM is one of the leading earth system frameworks [28] to simulate the evolution of the NAO fluctuation, and a flood of achievements have obtained in numerical modeling of NAO using CESM [29, 30]. The conditional nonlinear optimal perturbation related to parameter (CNOP-P) method, which was introduced by Mu et al. in 2010 [31], is utilized in this work. Using the CNOP-P method, the range of model uncertainty and parameter errors are solved under a reasonable constraint at prediction time. The CNOP-P method has been applied in multiple fields in geography, such as El Niño-Southern Oscillation (ENSO) [32], grassland ecosystems [33], terrestrial ecosystems [34], solid moisture [35], double-gyre variation [36], and net primary production (NPP) [37]. In the predictability study of ENSO, it is indicated that the bias of model parameters obtained by the CNOP-P method yields a large prediction error [32]. In addition, the extent of grassland ecosystem variation triggered by the CNOP-P-type perturbation is much higher than that caused by linear-type vectors [33]. For double-gyre variation in the Regional Ocean Modeling System (ROMS), the effect of sensitive parameters (combinations) on model simulation is verified using an improved simulated annealing algorithm [36]. These research results proved that the CNOP-P method is appropriate for the investigation of the parameter sensitivity in numerical models, especially for multiparameter combinations.

According to the definition, the parameter perturbations obtained by CNOP-P have the most significant effect on prediction results, and it can be tackled as the constrained extreme value problem. To implement the CNOP-P approach, the adjoint-based algorithm is generally used; for instance, the Spectral Projected Gradient (SPG) method [38]. This kind of algorithm has its limitations because it relies on the gradient information provided by the adjoint model, and it is not applicable for models that do not have an adjoint model, like CESM. Thus, we adopt adjoint-free algorithms to implement CNOP-P and select the variation of NAO index $\mathrm{\Delta }\text{NAOI}$ as the objective function to evaluate impacts. After control experiments, 28 potential sensitive parameters are chosen from all the parameters of atmospheric component and ice component. These potential sensitive parameters are grouped into combinations with two or three parameters. Then two intelligence algorithms are applied to further analyze the impact of model parameters. Particle swarm optimization (PSO) is adopted for single parameters and double-parameter combinations. PSO is a kind of swarm intelligence algorithm to solve the nondeterministic polynomial hard (NP-hard) problem and has been widely adopted in solving CNOP-I-type problems [39–41]. As for the combinations that contain more parameters, we utilize the method called the covariance matrix adaptation evolution strategy (CMA-ES) to achieve fast convergence. CMA-ES is commonly used for global optimization problems, such as the parameter adjustment in the neural network [42], large-scale overlapping problems [43], and feature weighting problems [44]. The objective of this work is (1) find out which single parameter or parameter combination can cause the maximal extent of uncertainty for two types of NAO events, and record their values, (2) investigate how much of uncertainties that the single parameter or parameter combination in (1) can achieve, and (3) explore the magnitude of parameter perturbations that can cause the extreme NAO events. The results demonstrate that even a tiny parameter error may cause a large margin of model error. Besides, the comprehensive influence of parameter combinations is greater than that of the single parameter. The most sensitive parameter combinations are parameter for deep convection cloud fraction (cldfrc_dp1), minimum relative humidity for low stable clouds (cldfrc_rhminl), the total solar irradiance (solar_const), and the constant for evaporation of precip (cldwat_conke), characteristic adjustment time scale (hkconv_cmftau), the total solar irradiance (solar_const) for ${\text{NAO}}^{+}$ and ${\text{NAO}}^{-}$, respectively. Our study is also conducive to parameter optimization to improve forecasting skills in future work.

The structure of this paper is organized as follows: Section 2 describes the basic information of CESM, the principle of PSO algorithm, and the strategies in CMA-ES. Experiments and results are displayed in Section 3. This paper ends with discussions and conclusions in Section 4.

2. Materials and Methods

2.1. CESM

As a new generation of the large-scale unity-coupled climate model, CESM provides efficient simulation for global climate and natural variability, and it has been widely used in the research of the midlatitude ocean variability [45], interannual variability of summer surface air temperature [46], and air quality [47], etc. Meanwhile, several scholars also utilized CESM to simulate the evolution of NAO with related factors, such as temperature [30]. The CESM assembles atmosphere, ocean, land, and sea-ice component models coupled through a coupler. The corresponding atmospheric component is derived from the Community Atmosphere Model (CAM), and the land component is the Community Land Model (CLM).

2.2. CNOP-P

CNOP is a practical approach to study the initial error (CNOP-I) and the model parameter error (CNOP-P), which have the largest effect on prediction results. The research of CNOP-I is on the basis of the condition that prediction errors are entirely from the initial state. However, the model parameters, which are determined by observation experiments, can also cause the prediction error. Therefore, such kind of predictability problem was raised to explore the impact due to model parameters.

We can assume that the basic state $\mathbf{U}$ satisfies the following partial differential equation:$$\begin{array}{}\text{(1)}& \begin{array}{c}\frac{\partial U}{\partial T}=F\mathbf{U},\mathbf{P},\\ {\mathbf{U}}_{t=0}={\mathbf{U}}_0,\end{array}\end{array}$$where $F$ denotes the nonlinear propagator, and ${\mathbf{U}}_0$ is the initial state. $\mathbf{P}$ is a collection of parameters with default value in this model. Let $M_t$ be the nonlinear propagator related to equation (1) from time $0$ to $t$, then the solution at time $t$ can be written as:$$\begin{array}{}\text{(2)}& \mathbf{U}t=M_t{\mathbf{U}}_0,\mathbf{P}.\end{array}$$

The solution without perturbation (initial perturbation ${\mathbf{u}}_0$ or parameter perturbation ${\mathbf{p}}_0$) can be viewed as the reference state at time $t$. If we superimpose a parameter perturbation ${\mathbf{p}}_0$ on the initial parameter(s), the solution at time t becomes $M_t{\mathbf{U}}_0,\mathbf{P}+{\mathbf{p}}_0$ and may have difference with the reference state. The difference is described as$$\begin{array}{}\text{(3)}& dt=M_t{\mathbf{U}}_0,\mathbf{P}+{\mathbf{p}}_0-M_t{\mathbf{U}}_0,\mathbf{P}.\end{array}$$

Since our aim is to find the parameter perturbation that has the largest gap with the reference state, the objective function of CNOP-P method represents as follows:$$\begin{array}{}\text{(4)}& J{\mathbf{p}}_0^{\ast }={\max }_{{\mathbf{p}}_0\in \Omega }J{\mathbf{p}}_0.\text{(5)}& J{\mathbf{p}}_0=\text{d}t=M_t{\mathbf{U}}_0,\mathbf{P}+{\mathbf{p}}_0-M_t{\mathbf{U}}_0,\mathbf{P},\end{array}$$where ${\mathbf{p}}_0^{\ast }$ is the optimal perturbation, and $J$ is the objective function of this extreme value problem. $\mathrm{\Omega }$ is the feasible domain of the parameter perturbation and satisfies ${\mathbf{p}}_0\le \beta \mathbf{P},{\mathbf{p}}_0\epsilon \mathrm{\Omega }$. $\beta$ denotes the proportion of the default value and is set to $20\%$ to ensure the rationality. In this work, the integral operator $M_t$ can be viewed as the quantization process of NAOI. To portray the NAO definitely, we choose a blocking indicator proposed by Liu [48] as the NAOI, which is defined as SLP anomaly projection on the NAO pattern:$$\begin{array}{}\text{(6)}& \text{NAOI}=\frac{{\text{SLP}}_{\text{NAO}},{\text{SLP}}_a}{{\text{SLP}}_{\text{NAO}},{\text{SLP}}_{\text{NAO}}},\end{array}$$where $<>$ denotes the Euclidean inner product; ${\text{SLP}}_{\text{NAO}}$ is the NAO anomaly pattern extracted by empirical orthogonal function (EOF), and the ${\text{SLP}}_a$ is obtained by subtracting the climatological mean from the SLP output. Combining the equations (5) and (6), the objective function is described as$$\begin{array}{}\text{(7)}& J{\mathbf{p}}_0=\Delta \text{NAOI}={\text{NAOI}}_{\text{pert}}-{\text{NAOI}}_{\text{refer}}=\frac{M_t{\mathbf{U}}_0,\mathbf{P}+{\mathbf{p}}_0-M_t{\mathbf{U}}_0,\mathbf{P},{\text{SLP}}_{\text{NAO}}}{{\text{SLP}}_{\text{NAO}},{\text{SLP}}_{\text{NAO}}}.\end{array}$$

Since the NAO has two types of phases, the positive ${\text{NAO}}^{+}$ and the negative ${\text{NAO}}^{-}$, we focus on the maximum of the absolute value of the NAOI difference. To simplify the problem, the NAO events are considered to happen when $\mathrm{\Delta }\text{NAOI}>1$.

For searching CNOP-P, we apply two novel intelligence algorithms to deal with different scales of parameters: PSO and CMA-ES. These two algorithms break the limitation of traditional methods for solving CNOP and have a better performance. The architecture of the solving system is displayed in Figure 1.

[figure omitted; refer to PDF]

In order to obtain a more comprehensive view of the model parameters, we perform the preliminary screening from all 237 parameters in these components. 48 available parameters with numeric type are sifted through their types and definitions. These parameters distribute in three components, atmosphere, ice, and river, and most of them are contained in the atmosphere component. In order to select sensitive parameters among them, we superpose the linear perturbations on the basic state for each parameter. The values of the perturbations are set from $1\%$ to $20\%$ of the default values, and the step size is $1\%$. The parameters that would cause the variation of the NAOI can be regarded as the sensitive parameters. After the above steps, we obtain 28 sensitive parameters, which are displayed in Table 1. Thereinto, p1–p25 are parameters in the atmosphere component, and p26–p28 are in the ice component. Then these sensitive parameters are analyzed with further selection and combination in subsequent sections. First, we calculate $\mathrm{\Delta }\text{NAOI}$ by varying selected parameters on a one-by-one basis, then each parameter is searched by PSO with 30 particles and 50 iterations. After that, the top 10 parameters for ${\text{NAO}}^{+}$ and ${\text{NAO}}^{-}$ are, respectively, combined into pairs, and $2\ast C_{10}^2=90$ groups of parameter combinations are built. In this step, the CNOP-Ps are also solved by PSO, and the length of the parameter vector is 2. Finally, we select 5 most sensitive parameters (combinations) from 90 groups, and parameters are combined into triple-parameter subsets with $2\ast C_5^3=20$ groups. The sensitivities of triple-parameter groups are explored by the CMA-ES method. A total of 166 CNOP-Ps with different scales are optimized in light of their fitness values.

Table 1

The 28 sensitive parameters screened by linear perturbations.

Number	Parameter	Description	Default value
p1	ch4vmr	${\text{CH}}_4$ volume Mixing ratio	$1.76\ast {10}^{-6}$
p2	co2vmr	${\text{CO}}_2$ volume Mixing ratio	$3.67\ast {10}^{-4}$
p3	f11vmr	${\text{CCl}}_3\text{F}$ volume Mixing ratio	$6.53\ast {10}^{-10}$
p4	f12vmr	${\text{CCl}}_2{\text{F}}_2$ volume Mixing ratio	$5.35\ast {10}^{-10}$
p5	n2ovmr	${\text{N}}_2\text{O}$ volume Mixing ratio	$3.16\ast {10}^{-7}$
p6	cldfrc_dp1	Parameter for deep convection cloud fraction	$0.1$
p7	cldfrc_dp2	Parameter for deep convection cloud fraction	$500.0$
p8	cldfrc_premib	Bottom height for midlevel liquid stratus fraction	$7.5\ast {10}^4$
p9	cldfrc_rhminh	Minimum relative humidity for high stable clouds	$0.77$
p10	cldfrc_rhminl	Minimum relative humidity for low stable clouds	$0.92$
p11	cldfrc_rhminl_adj_land	Adjustment to rhminl for land without snow cover	$0.1$
p12	cldfrc_sh1	Parameter for shallow convection cloud fraction	$0.04$
p13	cldfrc_sh2	Parameter for shallow convection cloud fraction	$500.0$
p14	cldsed_ice_stokes_fac	Factor applied to the ice fall velocity	$1.0$
p15	cldwat_conke	Tunable constant for evaporation of precip	$5.0\ast {10}^{-6}$
p16	cldwat_icritc	Threshold for autoconversion of cold ice	$1.8\ast {10}^{-5}$
p17	cldwat_icritw	Threshold for autoconversion of warm ice	$2.0\ast {10}^{-4}$
p18	cldwat_r3lcrit	Critical radius at which autoconversion become efficient	$1.0\ast {10}^{-5}$
p19	fcrit2	Critical froude number squared	$1.0$
p20	hkconv_c0	Rain water autoconversion coefficient	$1.0\ast {10}^{-4}$
p21	hkconv_cmftau	Characteristic adjustment time scale	$1.8\ast {10}^3$
p22	solar_const	Total solar irradiance ($\text{W}/{\text{m}}^2$)	$1.36\ast {10}^3$
p23	zmconv_c0_lnd	Autoconversion coefficient over land	$3.5\ast {10}^{-3}$
p24	zmconv_c0_ocn	Autoconversion coefficient over ocean	$3.5\ast {10}^{-3}$
p25	zmconv_ke	Tunable evaporation efficiency	$1.0\ast {10}^{-6}$
p26	dt_mlt_in	Melt onset temperature tunable parameter	$1.5$
p27	r_snw	Snow grain radius tunable parameter	$1.5$
p28	rsnw_melt_in	Maximum melting snow grain radius tunable parameter	$1.5\ast {10}^3$

Thereinto, p1–p5, p9–p10, and p22 are parameters directly related to physical variables and have specific ranges. Through consulting the historical data, the scopes of these parameters are listed in Table 2.

Table 2

The ranges of some parameters according to historical data.

Number	Parameter	Parameter range	Unit
p1	ch4vmr	$1.70-1.90$	ppm
p2	co2vmr	$370-430$	ppm
p3	f11vmr	$260-500$	ppt
p4	f12vmr	$530-830$	ppt
p5	n2ovmr	$311-328$	ppb
p9	cldfrc_rhminh	$33\%-61\%$	—
p10	cldfrc_rhminl	$33\%-61\%$	—
p22	solar_const	$1.356\ast {10}^3-1.364\ast {10}^3$	W/m2

The experiments are conducted on the Tianhe-2 supercomputer, with two Xeon E5 12-core CPUs and three Xeon Phi 57-core coprocessors for each node.

4. Results and Analysis

4.1. The Maximum Uncertainties due to Parameter Perturbations for Two Phases of NAO

The maximal extent of uncertainty is quantified via $\mathrm{\Delta }\text{NAOI}$ in equation (7). $\mathrm{\Delta }\text{NAOI}$ describes the NAOI difference compared with the reference state. The reference state can be obtained by running CESM with default parameter values. The greatest variation of positive $\mathrm{\Delta }\text{NAOI}$ and negative $\mathrm{\Delta }\text{NAOI}$ triggered by single parameters is illustrated in Figures 3 and 4. The range of $\mathrm{\Delta }{\text{NAOI}}^{+}$ is from 0.22 to 1.45, and p10 (Minimum relative humidity for low stable clouds) achieves the largest difference, while p1 (${\text{CH}}_4$ volume mixing ratio) has the least influence on the ${\text{NAO}}^{+}$. In Figure 4, $\mathrm{\Delta }{\text{NAOI}}^{-}$ ranges from −0.54 to −1.70. The sensitivity extent of each parameter is approximately similar to the ${\text{NAO}}^{+}$. p10 (minimum relative humidity for low stable clouds) is still the leading parameter in the aspect of sensitivity, and p22 (total solar irradiance) becomes the most relevant factor. According to our definition in Section 2, NAO events are considered to occur when $\mathrm{\Delta }\text{NAOI}>1$. As shown in Figure 4, some of the parameters can lead the ${\text{NAO}}^{-}$ events to happen, and the maximum value of $\mathrm{\Delta }\text{NAOI}$ is even greater than 1.5 (see p22 in Figure 4). Although not all the cases develop into NAO events, the uncertainties between prediction and observation owing to parameter perturbations are nonnegligible.

[figure omitted; refer to PDF]

For further experiments, the top 10 parameters that achieve higher positive $\mathrm{\Delta }\text{NAOI}$ and lower negative $\mathrm{\Delta }\text{NAOI}$ are listed in Tables 3 and 4. In these tables, parameter values at the maximum of the objective function are also reported.

Table 3

The top 10 parameters that cause the largest uncertainty for ${\text{NAO}}^{+}$.

Number	$\mathrm{\Delta }\text{NAOI}{\text{NAO}}^{+}$	Parameter value
p10	$1.45$	$0.61$
p8	$1.06$	$8.81\ast {10}^4$
p15	$0.86$	$4.0\ast {10}^{-6}$
p9	$0.85$	$0.55$
p6	$0.82$	0.12
p22	$0.81$	$1.36194\ast {10}^3$
p7	$0.79$	$591.53$
p24	$0.78$	$2.84\ast {10}^{-3}$
p20	$0.76$	$8.27\ast {10}^{-5}$
p21	$0.74$	$1.85\ast {10}^3$

Table 4

Similar to Table 2, but for ${\text{NAO}}^{-}$.

Number	$\mathrm{\Delta }\text{NAOI}{\text{NAO}}^{-}$	Parameter value
p21	$-1.70$	$1.99\ast {10}^3$
p22	$-1.49$	$1.36230\ast {10}^3$
p20	$-1.37$	$1.20\ast {10}^{-4}$
p24	$-1.35$	$3.86\ast {10}^{-3}$
p15	$-1.32$	$5.54\ast {10}^{-6}$
p10	$-1.30$	$0.58$
p14	$-1.27$	$1.20$
p16	$-1.25$	$1.72\ast {10}^{-5}$
p28	$-1.17$	$1.64\ast {10}^3$
p7	$-1.15$	$574.87$

To explore which parameter(s) have the highest sensitivities to NAO events more comprehensively, in addition to single parameters, parameter combinations need to be considered. The top 10 sensitive parameters (Tables 3 and 4) of ${\text{NAO}}^{+}$ and ${\text{NAO}}^{-}$ are combined in pairs, and totaling 90 groups of two-parameter combinations are constructed. We implement the sensitivity experiments on these combinations and compare them with single parameters. PSO is still employed to find the extremum of the objective function, with 30 particles and 50 iteration steps for each combination. The parameter perturbation is organized into the vector with a size of 2 and is assigned as the position of particles. We plot heatmaps to visually display distributions of the largest (lowest) $\mathrm{\Delta }\text{NAOI}$ due to these combinations. Figure 5 shows the results of 45 parameter sets related to the ${\text{NAO}}^{+}$, and the diagonal refers to the largest uncertainties corresponding to single parameters. For instance, the first block at the upper left denotes the $\mathrm{\Delta }\text{NAOI}$ triggered by p6 only. Values on the diagonal are smaller than that of the same row or the same column, indicating a more significant effect of parameter combinations. In Figure 5, we can see some grids with darker colors obviously, such as p10 (Minimum relative humidity for low stable clouds) and p20 (Rain water autoconversion coefficient), p10 and p15 (Constant for evaporation of precip), and p10 and p22 (Total solar irradiance). The areas with darker colors are also concentrated in the combination of p9 and p10 with other parameters. Conversely, combinations with p7 (parameter for deep convection cloud fraction 2) or p21 (characteristic adjustment time scale) are in lighter colors with lower sensitivities.

[figure omitted; refer to PDF]

Figure 6 depicts the results of two-parameter combinations for ${\text{NAO}}^{-}$. Similarly, the ranks of sensitivity for single parameters is relatively lower. However, there are exceptions, such as the effect of p7 (parameter for deep convection cloud fraction 2) and p20 (rain water autoconversion coefficient) combination is inferior to that of p7 or p20, and the same goes for p6 (parameter for deep convection cloud fraction 1) and p7. A cross-dark ribbon appears in Figure 6, and it demonstrates that the parameter combinations containing p15 (Constant for evaporation of precip) or p22 (Total solar irradiance) are highly sensitive to the ${\text{NAO}}^{-}$. Compared to the other nine parameters, p28 (Maximum melting snow grain radius tunable parameter) has less contribution to the form of the ${\text{NAO}}^{-}$ events.

[figure omitted; refer to PDF]

In the next group of experiments, 10 parameters (5 for ${\text{NAO}}^{+}$, and the other 5 for ${\text{NAO}}^{-}$) would be determined according to the results of two-parameter combinations, and relative insensitive parameters are eliminated. Then they would be grouped into combinations with three parameters, for a total of 20 sets. At first, we list the top 10 parameter combinations with the largest $\mathrm{\Delta }\text{NAOI}$ value in Figure 5, sequentially recorded as c1 to c10. It can be seen that p10 (minimum relative humidity for low stable clouds) occurs most often in Table 5. What’s more, p10 is also the single parameter most closely related to the ${\text{NAOI}}^{+}$. p9 (minimum relative humidity for high stable clouds) is another critical parameter that can promote the occurrence of the ${\text{NAO}}^{+}$. The complex association has been detected between relative humidity and the NAO in winter (DJF) [50], and the relative humidity can also be affected by the NAO forcing [51]. p9 and p10 are selected for the following experiments, and other parameters (p6, p15, and p22) are picked by balancing the fitness value and the average ranking of $\mathrm{\Delta }\text{NAOI}$. In Table 5, the $\mathrm{\Delta }\text{NAOI}$ are all greater than 1 and are significantly higher than the single-parameter results in Table 3. The top 10 parameter combinations for ${\text{NAO}}^{-}$ named c11 to c20 are presented in Table 6, and parameters (p15, p20, p21, p22, and p24) are chosen in the same way. The high sensitivities of p22 (Total solar irradiance) are demonstrated for most sets. With decreasing solar activity, the increasing pressure difference between Azores low and Iceland high is observed. The values of $\mathrm{\Delta }\text{NAOI}$ are further reduced compared to Table 4, and most of them are less than −2. We can see that a large margin of error can be caused due to deviations of several parameters. It is entirely possible that the qualitative analysis of the prediction would make a huge difference because of parameter errors.

Table 5

The top 10 two-parameter combinations that cause the largest uncertainty for ${\text{NAO}}^{+}$.

Number	Parameter	$\mathrm{\Delta }\text{NAOI}{\text{NAO}}^{+}$	Parameter value
c1	p10, p20	$1.78$	$0.38$, $1.03\ast {10}^{-4}$
c2	p10, p15	$1.66$	$0.51$, $4.78\ast {10}^{-6}$
c3	p10, p22	$1.64$	$0.43$, $1.36222\ast {10}^3$
c4	p9, p24	$1.62$	$0.53$, $3.67\ast {10}^{-3}$
c5	p7, p10	$1.56$	$0.77$, $0.61$
c6	p9, p22	$1.55$	0.35, $1.35875\ast {10}^3$
c7	p6, p10	$1.52$	$0.097$, $0.56$
c8	p8, p10	$1.51$	$7.25\ast {10}^4$, $0.55$
c9	p9, p10	$1.50$	$0.54$, $0.41$
c10	p6, p15	$1.48$	$0.103$, $4.85\ast {10}^{-6}$

Table 6

Similar to Table 5, but for ${\text{NAO}}^{-}$.

Number	Parameter	$\mathrm{\Delta }\text{NAOI}{\text{NAO}}^{-}$	Parameter value
c11	p14, p22	$-2.65$	$0.95$,$1.35650\ast {10}^3$
c12	p10, p22	$-2.50$	$0.48$,$1.36269\ast {10}^3$
c13	p15, p21	$-2.47$	$4.98\ast {10}^{-6}$,$1.53\ast {10}^3$
c14	p7, p22	$-2.44$	$504.14$,$1.35886\ast {10}^3$
c15	p15, p28	$-2.34$	$4.90\ast {10}^{-6}$,$1.53\ast {10}^3$
c16	p21, p24	$-2.18$	1.99 ∗ ${10}^3$, 4.20 ∗ ${10}^{-3}$
c17	p15, p20	$-2.00$	$9.78\ast {10}^{-5}$,$4.77\ast {10}^{-6}$
c18	p20, p24	$-1.94$	$1.05\ast {10}^{-4}$,$3.67\ast {10}^{-3}$
c19	p16, p22	$-1.92$	$1.72\ast {10}^{-5}$,$1.36207\ast {10}^3$
c20	p21, p22	$-1.83$	$1.83\ast {10}^3$,$1.35805\ast {10}^3$

The experiments with three-parameter combinations are performed with parameters that appear more frequently in Tables 5 and 6. In settings of CMA-ES, the initial solution is set to the vector containing default values of the three parameters, and the initial standard deviation is set to 0.5. These combinations are sorted in descending order (ascending order for ${\text{NAO}}^{-}$) by fitness values $\mathrm{\Delta }\text{NAOI}$, marking them as c21 to c40. The results of the two phases are illustrated in Figures 7 and 8 with parallel coordinate systems. The red dot stands for the default value, and the fitness value displays in the last coordinate axis. Taking c30 in Figure 7 as an example, the c30 represented by the red line is composed of p6, p15, and p22 (p9 and p10 are set as the default). When the fitness value achieves a maximum of 1.43, the values of these three parameters are 0.102, $4\text{.}98\ast {10}^{-6}$ , and $1.36206\ast {10}^3$, respectively. Other combinations are processed in the same manner, and the details are reported in Table 7. The multiple factors mutually promote the process of the NAO, and the final state is not equivalent to a simple superposition of the effects of each parameter acting alone. $\mathrm{\Delta }\text{NAOI}$ values are generally larger (smaller) than 1.5 (−1.5). In addition, a large gap is between these fitness values, especially for the ${\text{NAO}}^{-}$, which is distributed between −1.5 and −3 (Table 8).

[figure omitted; refer to PDF]

Similar work was carried out for two-parameter combinations (see Figures 14 and 15), and SLP difference patterns for these combinations are plotted to investigate whether anomalous NAOI and typical meridional dipole occur simultaneously. The SLP difference is generally intensive than that of single parameters, and it ranges from −5000 (−4000) to 4000 (5000) Pa in ${\text{NAO}}^{+}{\text{NAO}}^{-}$. What’s more, pressure cores are more prominent and concentrated. In Figure 14, the pattern of c7 (combined with p6 and p10) has close similarity with that of p10. The SLP difference triggered by c7 or c10 show a negative-over-positive tilted dipole with larger NAOI, deriving from the mutual promotion between two parameters. As for the ${\text{NAO}}^{-}$, the SLP patterns of the 10 two-parameter combinations display an analogous structure in Figure 15. The positive pressure stretches over southern Greenland and western Europe, and the positive pressure center and the negative pressure center are symmetric about the 50 N, exhibiting a typical dipole with no tilt. Compared with single parameters, two-parameter combinations have more potential to develop into NAO events.

Figures 16 and 17 depict the SLP difference for three-parameter combinations, with a range of −5000 (−4500) to 4500 (5500) Pa. All combinations form dipole pattern in the North Atlantic sector at a high intensity. The typical NAO mode proves that the parameter perturbation solved by CNOP-P approach can make the initial state evolve into NAO events, and the combination with three sensitive parameters can also cause the larger uncertainty than that of single parameters.

5. Discussions and Conclusions

Model parameter error is a critical factor that can trigger the uncertainty of the prediction result. Thus, the correction of parameters is beneficial to improve the prediction skill of numerical models. To provide direction on parameter correction, the sensitivity of the model parameter is investigated in this work. We adopt CESM to simulate the NAO and explore the impact of parameters that are sensitive to the NAO events. The research aims to find out the factors that are closely related to the atmospheric circulation of NAO. Our conclusions confirm the results of previous studies and also have some new discoveries.

From the result of the single parameter and parameter combinations, the parameter p22, which is related to solar irradiance, has a very significant impact on the simulation of both ${\text{NAO}}^{+}$ and ${\text{NAO}}^{-}$ in winter. It has been found that downward surface short-wave radiation has a strong relationship with the winter NAO index, even in the low-frequency variability [52]. A recent article that examined the effects of solar activity cycles on winter NAO indicated that the mechanism of solar influence relies on the meridional temperature gradient, zonal winds, and the impact of ultraviolet radiation variability on heating rates in the tropical upper stratosphere [53]. The study compared constant solar radiation with time-varying solar irradiance, confirming the effect of sun on atmospheric heating and dynamic response. In addition, the research of Chiacchio et al. also found that cloud cover is a potential factor that can easily enhance the variability of the winter NAO [54]. As the parameter of deep convective cloud volume, p6 and p7 also display high NAO sensitivity in the cases of the single-parameter and parameter combinations. The above conclusions are consistent with the results of our work. If the model error, which caused by solar irradiance and volume of deep convective cloud, can be corrected, it would have a certain significance for improving the NAO prediction skills.

The precipitation is also a common predictor of NAO, with relative humidity as its direct relevant factor. In previous studies, the relationship between precipitation on the Iberian Peninsula and the NAO has been explored with simulating the NAOI change in DJFM (December, January, February, and March) [55]. In the single-parameter sensitivity analysis, p9 and p10, which denote the relative humidity of high stable cloud and low stable cloud, respectively, become the parameters that cause the maximum uncertainty of the prediction of ${\text{NAO}}^{+}$. Moreover, p10 is also an important factor in triggering the maximum uncertainty of ${\text{NAO}}^{-}$ in a single parameter. The effects of greenhouse gas, sea surface temperature, and stratification variability on NAO variation have also been identified. In this paper, the parameters p1–p5 associated with greenhouse gases are covered in the group of potentially sensitive parameters. Some early studies have shown that increased greenhouse gas concentrations affect the stratospheric cycle and thus have an impact on climate change [56–58]. The concentration of greenhouse gases has increased with years in recent years. The main reason for the increase of ${\text{CO}}_2$ concentration is the large-scale exploitation of fossil fuels, while the bio-fermentation process increases ${\text{CH}}_4$ [59]. Although ${\text{N}}_2\text{O}$ and CFC-11 ${\text{CCl}}_3\text{F}$ and CFC-12 ${\text{CCl}}_2\text{F}$ are chemically inert in the troposphere, they can be photolyzed by solar radiation in the stratosphere, and the net increase rate is slower [60]. Our result is consistent with the work of the predecessors.

In our work, the sensitivity of the single parameter and parameter combinations are analyzed and compared. We find that the combined influence of multiple parameters is more significant than the single parameters. The parameter perturbation causes a large difference between perturbation state and reference state, and it makes the basic state develop into a typical NAO mode at prediction time. It is also proved that CNOP-P is an effective method to conduct sensitivity analysis. Besides, the intelligence algorithms (PSO and CMA-ES) overcome the limitations of the adjoint model and result in a satisfying performance. Our work validates the related factors of the NAO and provides the correction direction of model parameters. In future work, we will optimize the sensitive parameters that would cause model errors and improve the prediction skill for the NAO events.

Acknowledgments

The calculation of this work was performed on Tianhe-2. Thanks for the support of National Supercomputer Center in Guangzhou (NSCC-GZ). The authors also thank the help of Dai as an expert in the area of atmosphere. This study was supported by the Fundamental Research Funds for the Central Universities (grant no. 22120190207), the Key Project Fund of Shanghai 2020 “Science and Technology Innovation Action Plan” for Social Development (grant no. 20dz1200702), and National Natural Science Foundation of China (grant no. 42075141).

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