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

Authors: Bin Mu; Jing Li; Shijin Yuan; Xiaodan Luo; Guokun Dai
Journal: Advances in Meteorology
 (2020)

---

## 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 theNAO+ 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 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 (ΔNAOI) caused by these two combinations achieve 2.12 for NAO+ and −2.72 for NAO−, respectively. In comparison, the maximum level of the NAOI variation resulting from single parameters reaches 1.45 for NAO+ and −1.70 for 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.


---

## Body

## 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 NAO+ (NAO−) leads to increased (reduced) temperature [3], and extremely cold weather along with snowstorms in Europe is closely associated with 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 ΔNAOI 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 NAO+ and 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: Section2 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 stateU satisfies the following partial differential equation:(1)∂U∂T=FU,P,Ut=0=U0,where F denotes the nonlinear propagator, and U0 is the initial state. P is a collection of parameters with default value in this model. Let Mt be the nonlinear propagator related to equation (1) from time 0 to t, then the solution at time t can be written as:(2)Ut=MtU0,P.The solution without perturbation (initial perturbationu0 or parameter perturbation p0) can be viewed as the reference state at time t. If we superimpose a parameter perturbation p0 on the initial parameter(s), the solution at time t becomes MtU0,P+p0 and may have difference with the reference state. The difference is described as(3)dt=MtU0,P+p0−MtU0,P.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:(4)Jp0∗=maxp0∈ΩJp0.(5)Jp0=dt=MtU0,P+p0−MtU0,P,where p0∗ is the optimal perturbation, and J is the objective function of this extreme value problem. Ω is the feasible domain of the parameter perturbation and satisfies p0≤βP,p0εΩ. β denotes the proportion of the default value and is set to 20% to ensure the rationality. In this work, the integral operator Mt 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:(6)NAOI=SLPNAO,SLPaSLPNAO,SLPNAO,where <> denotes the Euclidean inner product; SLPNAO is the NAO anomaly pattern extracted by empirical orthogonal function (EOF), and the SLPa is obtained by subtracting the climatological mean from the SLP output. Combining the equations (5) and (6), the objective function is described as(7)Jp0=ΔNAOI=NAOIpert−NAOIrefer=MtU0,P+p0−MtU0,P,SLPNAOSLPNAO,SLPNAO.Since the NAO has two types of phases, the positiveNAO+ and the negative 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 ΔNAOI>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 Figure1.Figure 1
The architecture of the CNOP-P solving framework.


### 2.3. PSO

At first, the parameters that are sensitive to NAO events need to be picked out from all model parameters using linear perturbation. Then the largest impact of each single-parameter and two-parameter combination is searched via the PSO algorithm. The PSO is a typical swarm intelligence algorithm for solving extreme optimization problems and is appropriate for solving CNOP.The main idea of the PSO is to search for the position which has the largest fitness value with multiple particles [49]. Each iteration records the global optimal particle and approaches the optimal position by updating the particles’ speed and position according to its optimal position and the optimal global position. Finally, the extreme value of the objective function and the parameter value are obtained with convergence to a stable state. For the kth iteration, the updating rules of the ith particle’s speed and position are(8)Vik+1=Vik+c1r1Xpbk−Xik+c2r2Xgbk−Xik,Xik+1=Xik+Vik+1,where c1 is the self-awareness coefficient for historical self-optimal position, and c2 is the social-awareness coefficient for global optimal position of all particles. The empirical value of c1 and c2 are both set to 2. r1 and r2 are random values with uniform distribution in 0,1. Xpbk denotes the self-optimal position of the current particle, and Xgbk denotes the global optimal position in the kth iteration.


### 2.4. CMA-ES

For multiparameter combination (parameter number≥3), we adopt CAM-ES, which is widely used for hyper-parameter adjustment in the field of machine learning, to investigate the most significant impact caused by multiple model parameters. CAM-ES is appropriate for complex optimization problems with a medium-scale number of parameters.The core concept of the CMA-ES is to adjust the covariance matrixC in the normal distribution Nmt,σt2Ct for updating the dependence between variables. The CMA-ES is performed in three main steps: (1) generate a new solution with sampling; (2) calculate the fitness value; (3) update the distribution parameters.In each iteration,λ solutions λ=xlogn,x∈N are generated from Nmt,σt2Ct:(9)xi=mt+σtyi,yi∼N0,I,where yi refers to the ith search direction and can be obtained from the eigen decomposition of the covariance matrix Ct=BDzi,zi∼N0,I:(10)yi=BDzi,zi∼N0,I.Then the fitness values of these solutions are calculated and are sorted in a descending (ascending) order forNAO+NAO−. After that, the distribution parameters mt and Ct are updated independently by a maximum likelihood estimate:(11)mt+1=∑i=1μωixi:λ=mt+∑i=1μωixi:λ−mt,where ωi satisfies ω1≥ωi≥ωμ>0 to highlight the solution with larger fitness value.The update of the covariance matrixC contains two parts: the evolution path and the natural gradient. The construction mode of the evolution path can be displayed as follows:(12)pt+1=1−cpt+c2−cμωmt+1−mtσt,where μω is set to 1/∑i=1μωi2 since μωmt+1−mt/σt=μω∑i=1μωiyi:λ∼N0,Ct. c denotes the learning rate, and the factor c2−c obeys the stationary condition 1−c2+c2−c2=1, which make pt+1∼N0,Ct.Combining with the formula (12), the update of the covariance matrix C can be written as(13)Ct+1=1−c1−cμCt+c1pt+1pt+1T+cμ∑i=1μωixi:λ−mtσtxi:λ−mtσtT,where the second item is to treat the evolution path as the search direction, and the third item is to evolve according to the natural gradient of the C. The learning rate c1 and cμ are inversely proportional to the number of parameters; that is, c1≈2/n2,cμ≈μω/n2.The adjust method used in CMA-ES is cumulative step size adaptation (CSA), which adjusts step size according to the angle between consecutive search directions. In other words, the consecutive search directions should be conjugated. The conjugate evolution path is(14)st+1=1−cσst+cσ2−cσμωC−1/2mt+1−mtσt,σt+1=σtecσ/dσst+1/EN0,I−1st+1/EN0,I,where st+1∼N0,I is in steady condition. Thus, st+1∼χn, and Est+1=n. Since Est+1=n, it proves the intuitive idea of the CSA. cσ and dσ are the coefficients of step size rangeability. In general, cσ∝1/n, dσ>1.


## 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 stateU satisfies the following partial differential equation:(1)∂U∂T=FU,P,Ut=0=U0,where F denotes the nonlinear propagator, and U0 is the initial state. P is a collection of parameters with default value in this model. Let Mt be the nonlinear propagator related to equation (1) from time 0 to t, then the solution at time t can be written as:(2)Ut=MtU0,P.The solution without perturbation (initial perturbationu0 or parameter perturbation p0) can be viewed as the reference state at time t. If we superimpose a parameter perturbation p0 on the initial parameter(s), the solution at time t becomes MtU0,P+p0 and may have difference with the reference state. The difference is described as(3)dt=MtU0,P+p0−MtU0,P.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:(4)Jp0∗=maxp0∈ΩJp0.(5)Jp0=dt=MtU0,P+p0−MtU0,P,where p0∗ is the optimal perturbation, and J is the objective function of this extreme value problem. Ω is the feasible domain of the parameter perturbation and satisfies p0≤βP,p0εΩ. β denotes the proportion of the default value and is set to 20% to ensure the rationality. In this work, the integral operator Mt 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:(6)NAOI=SLPNAO,SLPaSLPNAO,SLPNAO,where <> denotes the Euclidean inner product; SLPNAO is the NAO anomaly pattern extracted by empirical orthogonal function (EOF), and the SLPa is obtained by subtracting the climatological mean from the SLP output. Combining the equations (5) and (6), the objective function is described as(7)Jp0=ΔNAOI=NAOIpert−NAOIrefer=MtU0,P+p0−MtU0,P,SLPNAOSLPNAO,SLPNAO.Since the NAO has two types of phases, the positiveNAO+ and the negative 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 ΔNAOI>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 Figure1.Figure 1
The architecture of the CNOP-P solving framework.


## 2.3. PSO

At first, the parameters that are sensitive to NAO events need to be picked out from all model parameters using linear perturbation. Then the largest impact of each single-parameter and two-parameter combination is searched via the PSO algorithm. The PSO is a typical swarm intelligence algorithm for solving extreme optimization problems and is appropriate for solving CNOP.The main idea of the PSO is to search for the position which has the largest fitness value with multiple particles [49]. Each iteration records the global optimal particle and approaches the optimal position by updating the particles’ speed and position according to its optimal position and the optimal global position. Finally, the extreme value of the objective function and the parameter value are obtained with convergence to a stable state. For the kth iteration, the updating rules of the ith particle’s speed and position are(8)Vik+1=Vik+c1r1Xpbk−Xik+c2r2Xgbk−Xik,Xik+1=Xik+Vik+1,where c1 is the self-awareness coefficient for historical self-optimal position, and c2 is the social-awareness coefficient for global optimal position of all particles. The empirical value of c1 and c2 are both set to 2. r1 and r2 are random values with uniform distribution in 0,1. Xpbk denotes the self-optimal position of the current particle, and Xgbk denotes the global optimal position in the kth iteration.


## 2.4. CMA-ES

For multiparameter combination (parameter number≥3), we adopt CAM-ES, which is widely used for hyper-parameter adjustment in the field of machine learning, to investigate the most significant impact caused by multiple model parameters. CAM-ES is appropriate for complex optimization problems with a medium-scale number of parameters.The core concept of the CMA-ES is to adjust the covariance matrixC in the normal distribution Nmt,σt2Ct for updating the dependence between variables. The CMA-ES is performed in three main steps: (1) generate a new solution with sampling; (2) calculate the fitness value; (3) update the distribution parameters.In each iteration,λ solutions λ=xlogn,x∈N are generated from Nmt,σt2Ct:(9)xi=mt+σtyi,yi∼N0,I,where yi refers to the ith search direction and can be obtained from the eigen decomposition of the covariance matrix Ct=BDzi,zi∼N0,I:(10)yi=BDzi,zi∼N0,I.Then the fitness values of these solutions are calculated and are sorted in a descending (ascending) order forNAO+NAO−. After that, the distribution parameters mt and Ct are updated independently by a maximum likelihood estimate:(11)mt+1=∑i=1μωixi:λ=mt+∑i=1μωixi:λ−mt,where ωi satisfies ω1≥ωi≥ωμ>0 to highlight the solution with larger fitness value.The update of the covariance matrixC contains two parts: the evolution path and the natural gradient. The construction mode of the evolution path can be displayed as follows:(12)pt+1=1−cpt+c2−cμωmt+1−mtσt,where μω is set to 1/∑i=1μωi2 since μωmt+1−mt/σt=μω∑i=1μωiyi:λ∼N0,Ct. c denotes the learning rate, and the factor c2−c obeys the stationary condition 1−c2+c2−c2=1, which make pt+1∼N0,Ct.Combining with the formula (12), the update of the covariance matrix C can be written as(13)Ct+1=1−c1−cμCt+c1pt+1pt+1T+cμ∑i=1μωixi:λ−mtσtxi:λ−mtσtT,where the second item is to treat the evolution path as the search direction, and the third item is to evolve according to the natural gradient of the C. The learning rate c1 and cμ are inversely proportional to the number of parameters; that is, c1≈2/n2,cμ≈μω/n2.The adjust method used in CMA-ES is cumulative step size adaptation (CSA), which adjusts step size according to the angle between consecutive search directions. In other words, the consecutive search directions should be conjugated. The conjugate evolution path is(14)st+1=1−cσst+cσ2−cσμωC−1/2mt+1−mtσt,σt+1=σtecσ/dσst+1/EN0,I−1st+1/EN0,I,where st+1∼N0,I is in steady condition. Thus, st+1∼χn, and Est+1=n. Since Est+1=n, it proves the intuitive idea of the CSA. cσ and dσ are the coefficients of step size rangeability. In general, cσ∝1/n, dσ>1.


## 3. Experimental Settings

CESM 1.2.2, along with CAM5.3, are adopted for NAO simulation during winter (DJF). We selectF as the component set and conduct the experiments on a daily average. F integrates the atmospheric component, the land component, and the sea-ice component. The grids have 26 layers in the vertical under a resolution of 0.9°×1.25°. The region we mainly focus is located in the North Atlantic area between 20°N and 80°N and between 90°W and 40°E with 65×105 grids, as shown in Figure 2.Figure 2
The region of NAO events.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 from1% 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 ΔNAOI 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 NAO+ and NAO− are, respectively, combined into pairs, and 2∗C102=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∗C53=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.
NumberParameterDescriptionDefault valuep1ch4vmrCH4 volume Mixing ratio1.76∗10−6p2co2vmrCO2 volume Mixing ratio3.67∗10−4p3f11vmrCCl3F volume Mixing ratio6.53∗10−10p4f12vmrCCl2F2 volume Mixing ratio5.35∗10−10p5n2ovmrN2O volume Mixing ratio3.16∗10−7p6cldfrc_dp1Parameter for deep convection cloud fraction0.1p7cldfrc_dp2Parameter for deep convection cloud fraction500.0p8cldfrc_premibBottom height for midlevel liquid stratus fraction7.5∗104p9cldfrc_rhminhMinimum relative humidity for high stable clouds0.77p10cldfrc_rhminlMinimum relative humidity for low stable clouds0.92p11cldfrc_rhminl_adj_landAdjustment to rhminl for land without snow cover0.1p12cldfrc_sh1Parameter for shallow convection cloud fraction0.04p13cldfrc_sh2Parameter for shallow convection cloud fraction500.0p14cldsed_ice_stokes_facFactor applied to the ice fall velocity1.0p15cldwat_conkeTunable constant for evaporation of precip5.0∗10−6p16cldwat_icritcThreshold for autoconversion of cold ice1.8∗10−5p17cldwat_icritwThreshold for autoconversion of warm ice2.0∗10−4p18cldwat_r3lcritCritical radius at which autoconversion become efficient1.0∗10−5p19fcrit2Critical froude number squared1.0p20hkconv_c0Rain water autoconversion coefficient1.0∗10−4p21hkconv_cmftauCharacteristic adjustment time scale1.8∗103p22solar_constTotal solar irradiance (W/m2)1.36∗103p23zmconv_c0_lndAutoconversion coefficient over land3.5∗10−3p24zmconv_c0_ocnAutoconversion coefficient over ocean3.5∗10−3p25zmconv_keTunable evaporation efficiency1.0∗10−6p26dt_mlt_inMelt onset temperature tunable parameter1.5p27r_snwSnow grain radius tunable parameter1.5p28rsnw_melt_inMaximum melting snow grain radius tunable parameter1.5∗103Thereinto, 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 Table2.Table 2
The ranges of some parameters according to historical data.
NumberParameterParameter rangeUnitp1ch4vmr1.70−1.90ppmp2co2vmr370−430ppmp3f11vmr260−500pptp4f12vmr530−830pptp5n2ovmr311−328ppbp9cldfrc_rhminh33%−61%—p10cldfrc_rhminl33%−61%—p22solar_const1.356∗103−1.364∗103W/m2The 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 viaΔNAOI in equation (7). ΔNAOI 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 ΔNAOI and negative ΔNAOI triggered by single parameters is illustrated in Figures 3 and 4. The range of ΔNAOI+ is from 0.22 to 1.45, and p10 (Minimum relative humidity for low stable clouds) achieves the largest difference, while p1 (CH4 volume mixing ratio) has the least influence on the NAO+. In Figure 4, ΔNAOI− ranges from −0.54 to −1.70. The sensitivity extent of each parameter is approximately similar to the 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 ΔNAOI>1. As shown in Figure 4, some of the parameters can lead the NAO− events to happen, and the maximum value of ΔNAOI 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 3
The largest positiveΔNAOI triggered by 28 parameters.Figure 4
Similar to Figure3, but for negative ΔNAOI.For further experiments, the top 10 parameters that achieve higher positiveΔNAOI and lower negative ΔNAOI 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 forNAO+.
NumberΔNAOINAO+Parameter valuep101.450.61p81.068.81∗104p150.864.0∗10−6p90.850.55p60.820.12p220.811.36194∗103p70.79591.53p240.782.84∗10−3p200.768.27∗10−5p210.741.85∗103Table 4
Similar to Table2, but for NAO−.
NumberΔNAOINAO−Parameter valuep21−1.701.99∗103p22−1.491.36230∗103p20−1.371.20∗10−4p24−1.353.86∗10−3p15−1.325.54∗10−6p10−1.300.58p14−1.271.20p16−1.251.72∗10−5p28−1.171.64∗103p7−1.15574.87To 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 (Tables3 and 4) of NAO+ and 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) ΔNAOI due to these combinations. Figure 5 shows the results of 45 parameter sets related to the NAO+, and the diagonal refers to the largest uncertainties corresponding to single parameters. For instance, the first block at the upper left denotes the ΔNAOI 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 5
ΔNAOINAO+ for two-parameter combinations with parameters in Table 3.Figure6 depicts the results of two-parameter combinations for 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 NAO−. Compared to the other nine parameters, p28 (Maximum melting snow grain radius tunable parameter) has less contribution to the form of the NAO− events.Figure 6
ΔNAOINAO− for two-parameter combinations with parameters in Table 4.In the next group of experiments, 10 parameters (5 forNAO+, and the other 5 for 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 ΔNAOI 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 NAOI+. p9 (minimum relative humidity for high stable clouds) is another critical parameter that can promote the occurrence of the 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 ΔNAOI. In Table 5, the ΔNAOI are all greater than 1 and are significantly higher than the single-parameter results in Table 3. The top 10 parameter combinations for 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 ΔNAOI 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 forNAO+.
NumberParameterΔNAOINAO+Parameter valuec1p10, p201.780.38, 1.03∗10−4c2p10, p151.660.51, 4.78∗10−6c3p10, p221.640.43, 1.36222∗103c4p9, p241.620.53, 3.67∗10−3c5p7,p101.560.77, 0.61c6p9, p221.550.35,1.35875∗103c7p6, p101.520.097, 0.56c8p8,p101.517.25∗104, 0.55c9p9, p101.500.54, 0.41c10p6, p151.480.103, 4.85∗10−6Table 6
Similar to Table5, but for NAO−.
NumberParameterΔNAOINAO−Parameter valuec11p14,p22−2.650.95,1.35650∗103c12p10,p22−2.500.48,1.36269∗103c13p15, p21−2.474.98∗10−6,1.53∗103c14p7,p22−2.44504.14,1.35886∗103c15p15, p28−2.344.90∗10−6,1.53∗103c16p21, p24−2.181.99 ∗103, 4.20 ∗ 10−3c17p15, p20−2.009.78∗10−5,4.77∗10−6c18p20, p24−1.941.05∗10−4,3.67∗10−3c19p16,p22−1.921.72∗10−5,1.36207∗103c20p21, p22−1.831.83∗103,1.35805∗103The experiments with three-parameter combinations are performed with parameters that appear more frequently in Tables5 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 NAO−) by fitness values ΔNAOI, 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, 4.98∗10−6 , and 1.36206∗103, 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. ΔNAOI values are generally larger (smaller) than 1.5 (−1.5). In addition, a large gap is between these fitness values, especially for the NAO−, which is distributed between −1.5 and −3 (Table 8).Figure 7
ΔNAOINAO+ of three-parameter combinations that are combined with 5 parameters selected from Table 5. The red dot denotes the default value of the parameter.Figure 8
Similar to Figure7, but for NAO−.Table 7
The details of three-parameter combinations that cause the uncertainty inNAO+.
NumberParameterΔNAOINAO+Parameter valuec21p6, p10, p222.120.0997, 0.45, 1.35904∗103c22p10, p15, p221.940.48, 4.92∗10−6, 1.36202∗103c23p9, p10, p221.860.57, 0.46, 1.35735∗103c24p6, p10, p151.840.097, 0.47, 4.95∗10−6c25p6, p9, p101.680.102, 0.54,0.45c26p9, p15, p221.660.41,4.93∗10−6, 1.35801∗103c27p9, p10, p151.610.34, 0.41, 4.75∗10−6c28p6, p9, p151.540.098, 0.42,4.85∗10−6c29p6, p9, p221.440.097, 0.38, 1.36241∗103c30p6, p15, p221.430.102, 4.98∗10−6, 1.36206∗103Table 8
The details of three-parameter combinations that cause the uncertainty inNAO−.
NumberParameterΔNAOINAO−Parameter valuec31P15, p21, p22−2.724.78∗10−6, 1.82∗103, 1.36289∗103c32P15, p20, p22−2.594.76∗10−6, 1.01∗10−4, 1.35768∗103c33P20, p21, p22−2.511.00∗10−4, 1.87∗103, 1.35953∗103c34P15, p20, p24−2.365.01∗10−6, 9.77∗10−5, 3.55∗10−3c35p20, p21, p24−2.149.72∗10−5, 1.89∗103, 3.41∗10−3c36p20, p22, p24−2.081.03∗10−4, 1.36147∗103, 3.65∗10−3c37p15, p22, p24−1.814.77∗10−6, 1.35841∗103, 3.39∗10−3c38p15, p20, p21−1.756.00∗10−6, 1.14∗10−4, 1.44∗103c39p21, p22, p24−1.671.73∗103, 1.36138∗103, 3.51∗10−3c40p15, p21, p24−1.574.81∗10−6, 1.83∗103, 3.34∗10−3


### 4.2. The Evolution of NAOI with Parameter Perturbations

In this section, we aim to trace changing process of the NAOI triggered by parameter perturbations during the prediction time, and the variation trend of the NAOI is illustrated to observe the evolution of the NAO (see Figure9). The reference state is displayed with a red dotted line, and the perturbation states corresponding to the ten parameters mentioned in Tables 3 and 4 are shown as lines with different colors. The reference state denotes the final state without superposing perturbations. In Figure 9, different parameters appear different directions of variation for NAOI and fluctuate before converge to a stable state. In addition, the perturbation states and the reference state almost coincide in the first half of the prediction time. The nonlinear process takes effect in the final duration and develops into a diverse result.Figure 9
The NAOI variation flow of single parameters for two phases during the prediction time.
(a)(b)Figure10 shows the evolution of NAOI with two-parameter combinations for the two phases during the prediction time. The similarity with Figure 9 is that the fluctuation of the NAOI of each perturbation in the first-half phase is almost identical to the reference state. The trends of perturbations are diverse from each other in the last stage. Several perturbations tend to decrease in the later stage in spite of the higher NAOI. In the left chart of Figure 10, the perturbation flow remains on an upward trend. For NAO−, the perturbation flows of Figures 9 and 10 are basically in a state of continuous decline, and in Figure 10, the direction of the NAOI variation in the negative phase is more consistent than that of Figure 9.Figure 10
Similar to Figure9, but for two-parameter combinations.
(a)(b)Figure11 displays the variation of the three-parameter combinations for NAOI value, which has a slight enhancement than the two-parameter combinations.Figure 11
Similar to Figure9, but for three-parameter combinations.
(a)(b)


### 4.3. The SLP Anomaly Patterns

In general, NAO events occur with the abnormal high (low) NAOI as well as the dipole mode of anomaly SLP, and the spatial distribution of SLP reflects the variations of the NAO. To investigate the impact of parameter perturbations, the responses of the SLP field to model parameters are visualized in Figures12–17. The SLP field is presented in the form of the difference between perturbation state and reference state, and the critical region is selected as the area of North Atlantic Ocean. As it can be seen in Figure 12, although all of them have a relative high NAOI, the strong positive center and negative center only appear in patterns of p9, p10, and p22. The pressure structures of p9, p10, and p22 show a northeast-southwest tilt, with discontinuous high-pressure areas and messy pressure cores. In spite of the opposite pressure fields in the both sides of 60°N, no obvious pressure difference is generated in the rest of subplots. It is indicated that the perturbation state of most single parameters cannot evolve into NAO+ events. As for the mode of NAO− (see Figure 13), the patterns of p21, p22, p15, and p16 show the significant SLP difference with a range of −3500 Pa to 4500 Pa. The positive and negative pressure regions are respectively on either side of middle latitude, and a positive SLP anomaly occupies the south of the Greenland and Western Europe in these cases, resulting in the ΔNAOI less than −1. The west-positive patterns are formed with multiple pressure centers scattered around the 50°N. For most cases of single parameter, the SLP field is irregular and unstable or even no substantial difference. Specifically, p22 (Total solar irradiance) can exert marked influence on both NAO+ and NAO−. It is concluded that the solar irradiance is the common factor related to different physical mechanisms of NAO+ and NAO−.Figure 12
The SLP difference triggered by single parameters forNAO+.Figure 13
Similar to Figure12, but for NAO−.Figure 14
The SLP difference triggered by two-parameter combinations forNAO+.Figure 15
Similar to Figure14, but for NAO−.Figure 16
The SLP difference triggered by three-parameter combinations forNAO+.Figure 17
Similar to Figure16, but for NAO−.Similar work was carried out for two-parameter combinations (see Figures14 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 NAO+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 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.Figures16 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.


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

The maximal extent of uncertainty is quantified viaΔNAOI in equation (7). ΔNAOI 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 ΔNAOI and negative ΔNAOI triggered by single parameters is illustrated in Figures 3 and 4. The range of ΔNAOI+ is from 0.22 to 1.45, and p10 (Minimum relative humidity for low stable clouds) achieves the largest difference, while p1 (CH4 volume mixing ratio) has the least influence on the NAO+. In Figure 4, ΔNAOI− ranges from −0.54 to −1.70. The sensitivity extent of each parameter is approximately similar to the 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 ΔNAOI>1. As shown in Figure 4, some of the parameters can lead the NAO− events to happen, and the maximum value of ΔNAOI 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 3
The largest positiveΔNAOI triggered by 28 parameters.Figure 4
Similar to Figure3, but for negative ΔNAOI.For further experiments, the top 10 parameters that achieve higher positiveΔNAOI and lower negative ΔNAOI 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 forNAO+.
NumberΔNAOINAO+Parameter valuep101.450.61p81.068.81∗104p150.864.0∗10−6p90.850.55p60.820.12p220.811.36194∗103p70.79591.53p240.782.84∗10−3p200.768.27∗10−5p210.741.85∗103Table 4
Similar to Table2, but for NAO−.
NumberΔNAOINAO−Parameter valuep21−1.701.99∗103p22−1.491.36230∗103p20−1.371.20∗10−4p24−1.353.86∗10−3p15−1.325.54∗10−6p10−1.300.58p14−1.271.20p16−1.251.72∗10−5p28−1.171.64∗103p7−1.15574.87To 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 (Tables3 and 4) of NAO+ and 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) ΔNAOI due to these combinations. Figure 5 shows the results of 45 parameter sets related to the NAO+, and the diagonal refers to the largest uncertainties corresponding to single parameters. For instance, the first block at the upper left denotes the ΔNAOI 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 5
ΔNAOINAO+ for two-parameter combinations with parameters in Table 3.Figure6 depicts the results of two-parameter combinations for 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 NAO−. Compared to the other nine parameters, p28 (Maximum melting snow grain radius tunable parameter) has less contribution to the form of the NAO− events.Figure 6
ΔNAOINAO− for two-parameter combinations with parameters in Table 4.In the next group of experiments, 10 parameters (5 forNAO+, and the other 5 for 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 ΔNAOI 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 NAOI+. p9 (minimum relative humidity for high stable clouds) is another critical parameter that can promote the occurrence of the 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 ΔNAOI. In Table 5, the ΔNAOI are all greater than 1 and are significantly higher than the single-parameter results in Table 3. The top 10 parameter combinations for 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 ΔNAOI 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 forNAO+.
NumberParameterΔNAOINAO+Parameter valuec1p10, p201.780.38, 1.03∗10−4c2p10, p151.660.51, 4.78∗10−6c3p10, p221.640.43, 1.36222∗103c4p9, p241.620.53, 3.67∗10−3c5p7,p101.560.77, 0.61c6p9, p221.550.35,1.35875∗103c7p6, p101.520.097, 0.56c8p8,p101.517.25∗104, 0.55c9p9, p101.500.54, 0.41c10p6, p151.480.103, 4.85∗10−6Table 6
Similar to Table5, but for NAO−.
NumberParameterΔNAOINAO−Parameter valuec11p14,p22−2.650.95,1.35650∗103c12p10,p22−2.500.48,1.36269∗103c13p15, p21−2.474.98∗10−6,1.53∗103c14p7,p22−2.44504.14,1.35886∗103c15p15, p28−2.344.90∗10−6,1.53∗103c16p21, p24−2.181.99 ∗103, 4.20 ∗ 10−3c17p15, p20−2.009.78∗10−5,4.77∗10−6c18p20, p24−1.941.05∗10−4,3.67∗10−3c19p16,p22−1.921.72∗10−5,1.36207∗103c20p21, p22−1.831.83∗103,1.35805∗103The experiments with three-parameter combinations are performed with parameters that appear more frequently in Tables5 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 NAO−) by fitness values ΔNAOI, 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, 4.98∗10−6 , and 1.36206∗103, 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. ΔNAOI values are generally larger (smaller) than 1.5 (−1.5). In addition, a large gap is between these fitness values, especially for the NAO−, which is distributed between −1.5 and −3 (Table 8).Figure 7
ΔNAOINAO+ of three-parameter combinations that are combined with 5 parameters selected from Table 5. The red dot denotes the default value of the parameter.Figure 8
Similar to Figure7, but for NAO−.Table 7
The details of three-parameter combinations that cause the uncertainty inNAO+.
NumberParameterΔNAOINAO+Parameter valuec21p6, p10, p222.120.0997, 0.45, 1.35904∗103c22p10, p15, p221.940.48, 4.92∗10−6, 1.36202∗103c23p9, p10, p221.860.57, 0.46, 1.35735∗103c24p6, p10, p151.840.097, 0.47, 4.95∗10−6c25p6, p9, p101.680.102, 0.54,0.45c26p9, p15, p221.660.41,4.93∗10−6, 1.35801∗103c27p9, p10, p151.610.34, 0.41, 4.75∗10−6c28p6, p9, p151.540.098, 0.42,4.85∗10−6c29p6, p9, p221.440.097, 0.38, 1.36241∗103c30p6, p15, p221.430.102, 4.98∗10−6, 1.36206∗103Table 8
The details of three-parameter combinations that cause the uncertainty inNAO−.
NumberParameterΔNAOINAO−Parameter valuec31P15, p21, p22−2.724.78∗10−6, 1.82∗103, 1.36289∗103c32P15, p20, p22−2.594.76∗10−6, 1.01∗10−4, 1.35768∗103c33P20, p21, p22−2.511.00∗10−4, 1.87∗103, 1.35953∗103c34P15, p20, p24−2.365.01∗10−6, 9.77∗10−5, 3.55∗10−3c35p20, p21, p24−2.149.72∗10−5, 1.89∗103, 3.41∗10−3c36p20, p22, p24−2.081.03∗10−4, 1.36147∗103, 3.65∗10−3c37p15, p22, p24−1.814.77∗10−6, 1.35841∗103, 3.39∗10−3c38p15, p20, p21−1.756.00∗10−6, 1.14∗10−4, 1.44∗103c39p21, p22, p24−1.671.73∗103, 1.36138∗103, 3.51∗10−3c40p15, p21, p24−1.574.81∗10−6, 1.83∗103, 3.34∗10−3


## 4.2. The Evolution of NAOI with Parameter Perturbations

In this section, we aim to trace changing process of the NAOI triggered by parameter perturbations during the prediction time, and the variation trend of the NAOI is illustrated to observe the evolution of the NAO (see Figure9). The reference state is displayed with a red dotted line, and the perturbation states corresponding to the ten parameters mentioned in Tables 3 and 4 are shown as lines with different colors. The reference state denotes the final state without superposing perturbations. In Figure 9, different parameters appear different directions of variation for NAOI and fluctuate before converge to a stable state. In addition, the perturbation states and the reference state almost coincide in the first half of the prediction time. The nonlinear process takes effect in the final duration and develops into a diverse result.Figure 9
The NAOI variation flow of single parameters for two phases during the prediction time.
(a)(b)Figure10 shows the evolution of NAOI with two-parameter combinations for the two phases during the prediction time. The similarity with Figure 9 is that the fluctuation of the NAOI of each perturbation in the first-half phase is almost identical to the reference state. The trends of perturbations are diverse from each other in the last stage. Several perturbations tend to decrease in the later stage in spite of the higher NAOI. In the left chart of Figure 10, the perturbation flow remains on an upward trend. For NAO−, the perturbation flows of Figures 9 and 10 are basically in a state of continuous decline, and in Figure 10, the direction of the NAOI variation in the negative phase is more consistent than that of Figure 9.Figure 10
Similar to Figure9, but for two-parameter combinations.
(a)(b)Figure11 displays the variation of the three-parameter combinations for NAOI value, which has a slight enhancement than the two-parameter combinations.Figure 11
Similar to Figure9, but for three-parameter combinations.
(a)(b)


## 4.3. The SLP Anomaly Patterns

In general, NAO events occur with the abnormal high (low) NAOI as well as the dipole mode of anomaly SLP, and the spatial distribution of SLP reflects the variations of the NAO. To investigate the impact of parameter perturbations, the responses of the SLP field to model parameters are visualized in Figures12–17. The SLP field is presented in the form of the difference between perturbation state and reference state, and the critical region is selected as the area of North Atlantic Ocean. As it can be seen in Figure 12, although all of them have a relative high NAOI, the strong positive center and negative center only appear in patterns of p9, p10, and p22. The pressure structures of p9, p10, and p22 show a northeast-southwest tilt, with discontinuous high-pressure areas and messy pressure cores. In spite of the opposite pressure fields in the both sides of 60°N, no obvious pressure difference is generated in the rest of subplots. It is indicated that the perturbation state of most single parameters cannot evolve into NAO+ events. As for the mode of NAO− (see Figure 13), the patterns of p21, p22, p15, and p16 show the significant SLP difference with a range of −3500 Pa to 4500 Pa. The positive and negative pressure regions are respectively on either side of middle latitude, and a positive SLP anomaly occupies the south of the Greenland and Western Europe in these cases, resulting in the ΔNAOI less than −1. The west-positive patterns are formed with multiple pressure centers scattered around the 50°N. For most cases of single parameter, the SLP field is irregular and unstable or even no substantial difference. Specifically, p22 (Total solar irradiance) can exert marked influence on both NAO+ and NAO−. It is concluded that the solar irradiance is the common factor related to different physical mechanisms of NAO+ and NAO−.Figure 12
The SLP difference triggered by single parameters forNAO+.Figure 13
Similar to Figure12, but for NAO−.Figure 14
The SLP difference triggered by two-parameter combinations forNAO+.Figure 15
Similar to Figure14, but for NAO−.Figure 16
The SLP difference triggered by three-parameter combinations forNAO+.Figure 17
Similar to Figure16, but for NAO−.Similar work was carried out for two-parameter combinations (see Figures14 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 NAO+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 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.Figures16 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 bothNAO+ and 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 NAO+. Moreover, p10 is also an important factor in triggering the maximum uncertainty of 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 CO2 concentration is the large-scale exploitation of fossil fuels, while the bio-fermentation process increases CH4 [59]. Although N2O and CFC-11 CCl3F and CFC-12 CCl2F 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.



---

*Source: 6070789-2020-10-15.xml*