Uncertain parameters in physical parameterizations of
general circulation models (GCMs) greatly impact model performance. In
recent years, automatic parameter optimization has been introduced for
tuning model performance of GCMs, but most of the optimization methods are
unconstrained optimization methods under a given performance indicator.
Therefore, the calibrated model may break through essential constraints that
models have to keep, such as the radiation balance at the top of the model. The
radiation balance is known for its importance in the conservation of model
energy. In this study, an automated and efficient parameter optimization
with the radiation balance constraint is presented and applied in the
Community Atmospheric Model (CAM5) in terms of a synthesized performance
metric using normalized mean square error of radiation, precipitation,
relative humidity, and temperature. The tuned parameters are from the
parameterization schemes of convection and cloud. The radiation constraint
is defined as the absolute difference of the net longwave flux at the top of the model (FLNT) and the net solar flux at the top of the model (FSNT) of less than 1 W m

The subgrid-scale physical processes in general circulation models (GCMs) are represented by parameterization schemes, which may exist with several uncertain parameters. Inappropriate parameters can seriously affect the overall performance of the model. The Intergovernmental Panel on Climate Change Fifth Assessment Report (IPCC AR5) pointed out that studies on parameter uncertainty are critical to improve climate simulation capabilities (Mastrandrea et al., 2011). Bauer et al. (2015) also indicated that small errors in the physical parameterization schemes could lead to large-scale systematic errors. Traditionally, to achieve better performance, the uncertain parameters are tuned based on the experience of model experts and statistical analysis. This is a labor-intensive job, and the tuning results make it difficult to achieve local or global optimality in complex climate models.

To efficiently reduce parameter-introduced uncertainty, quite a few automated parameter calibration methods have been proposed. These calibration methods can be categorized into two types. One attempts to obtain the probability distributions of the parameters by likelihood and Bayesian estimation methods. Cameron et al. (1999) exploited the generalized likelihood uncertain estimation to obtain parameter ranges with a specific confidence level. An adaptive Markov Chain Monte Carlo (MCMC) was used to calibrate the uncertain parameters in the fifth-generation atmospheric general circulation model (ECHAM5) (Järvinen et al., 2010). Edwards et al. (2011) proposed a simplified procedure of Bayesian calibration to make a quantification of uncertainty in climate forecasting. This type of method has also been successfully applied to the CAM3.1 model and the third Hadley Centre Climate Model (HadCM3) (Jackson et al., 2008; Williamson et al., 2013).

The other method is to adjust parameters using optimization methods to minimize the errors between model simulations and observations, which are formulated with a given performance indicator. Many intelligent evolutionary optimization algorithms were applied to model tuning. For example, both simulated stochastic approximation annealing (SSAA) (Yang et al., 2013) and multiple very fast simulated annealing (MVFSA) (Zou et al., 2014) were used for uncertainty quantification and parameter calibration.

Both methods can consider the interaction of parameters, achieve automatic optimization, and avoid the subjectivity and experientiality of manual calibration. However, they also share high computation cost challenges due to the hundreds and thousands of required simulations. This is usually unacceptable, especially for high-resolution climate models. To overcome the computational issues, the surrogate model, which is a way to replace the real climate model with a cheaper statistical model for faster optimization, has been recently introduced. Applications of these methods in climate models include the work presented by Neelin et al. (2010) and Wang et al. (2014). However, training a precise surrogate model for a complicated climate model such as the Community Earth System Model (CESM) is very challenging. Moreover, capturing the climatic characteristics of extreme events is difficult for the cheap statistical model. To make it possible to optimize parameters efficiently and quickly in the complex and highly nonlinear earth system models, an improved simplex algorithm was presented by Zhang et al. (2015). This method can overcome the shortcomings of the traditional simplex downhill method, and the computing efficiency of the algorithm is improved compared with evolutionary optimization algorithms.

The application of various automatic parameter optimization methods in
climate models has gradually received more attention; however, the optimization algorithms mentioned above are mostly unconstrained, and they lack
emphasis on the physical mechanisms of the model itself. This paper takes
radiation balance as an example. According to the Earth's energy
conservation theory, the absorbed solar radiation is approximately equal to
outgoing longwave radiation at the top of the model. Forster et al. (2007)
proposed that radiative balance is critical to the Earth's system, and the
bias of radiation has a big impact on the change in surface temperature.
Hourdin et al. (2017) pointed out that a 1 W m

Radiation balance is critical for GCMs, but its deviation can still exceed 1 W m

The purpose of this paper is to propose an effective constrained optimization method and demonstrate its feasibility in the calibration of uncertain parameters under the premise of ensuring the balance of radiation. This paper is organized as follows. Section 2 describes the details of the model and experimental design. Section 3 introduces the new constrained parameter calibration method. Evaluations and analysis of the optimization results are presented in Sect. 4. The last section contains the conclusion and discussion.

Parameters description of CAM5. The default, final optimal values by constrained and unconstrained calibrations, as well as the ranges of parameters. CAPE means the convective available potential energy.

The model used in this study is CAM5 (release v5.3), which is the atmospheric component of the CESM 1.2.1. The dynamical core uses the finite-volume method developed by Lin and Rood (1996) and Lin (2004).

More details on CAM5 can be found in the work of Neale et al. (2010). Deep convection is handled by a parameterization scheme developed by Zhang and McFarlane (1995) with the further modifications of Richter and Rasch (2008), as well as Neale et al. (2008). The original parameterization of stratiform cloud microphysics is handled by Morrison and Gettelman (2008). Modifications of ice nucleation and ice supersaturation can be found in Gettelman et al. (2010). The parameterization of fractional stratiform condensation is described by Zhang et al. (2003) as well as Park et al. (2014). Radiation scheme uses the rapid radiative transfer method for GCMs (RRTMG) (Mlawer et al., 1997; Iacono et al., 2008).

Table 1 shows the parameters to be adjusted, the ranges, and the default values. These parameters were identified as sensitive to cloud and convection processes in previous studies. Qian et al. (2018) showed that deep-convection precipitation efficiency zmconv_c0_lnd and zmconv_ c0_ocn have significant impact on the variance of shortwave cloud forcing (SWCF) over the land and ocean, respectively. Thresholds of relative humidity for high and low stable clouds (cldfrc_rhminh and cldfrc_rhminl) are regarded as the important parameters for cloud and radiation (Zhang et al., 2018). In addition, the relative humidity threshold for low clouds is also one of the strongest parameters impacting the global mean precipitation, and it makes a huge contribution to the TOA net radiative fluxes in CAM5 (Qian et al., 2015). The timescale for the consumption rate of deep convective available potential energy (CAPE) (zmconv_tau) is identified as the most sensitive parameter to the convective precipitation in the Zhang–McFarlance scheme by Yang et al. (2013). The cloud ice sedimentation velocity (cldsed_ai) has a significant effect on cloud radiative forcing (Mitchell et al., 2008), and it has been identified as a high-impact parameter in sensitivity experiments related to temperature, radiation, and precipitation, etc. (Sanderson et al., 2008). The ranges of these parameters are based on previous studies (Qian et al., 2015; Zhang et al., 2018).

The output variables used to evaluate performance metric index and the source of the corresponding observations.

The output variables used to synthesize a performance indicator are longwave cloud forcing (LWCF), SWCF, precipitation (PRECT), humidity at 850 hPa (Q850), and temperature at 850 hPa (T850), shown in Table 2. The observations of LWCF and SWCF are from CERES-EBAF (Clouds and the Earth's Radiant Energy System-Energy Balanced and Filled; Loeb et al., 2014). PRECT is from GPCP (Global Precipitation Climatology Project; Adler et al., 2003), and Q850 and T850 are from ERA-Interim, which was produced by the ECMWF (Dee et al., 2011).

In this study, we use 1.9

A constrained automatic optimization method is proposed based on Zhang et
al. (2015). The synthesized metrics used to evaluate the performance of
overall simulation skills are shown in Eq. (

The radiation balance is defined as the absolute value of the difference
between net solar flux (FSNT) and net longwave flux (FLNT) in climatology at
the top of the model of less than 1 W m

Coupled with the radiation balance constraint, the optimization problem of
this study can be expressed as Eqs. (

We use the improved simplex downhill method, proposed by Zhang et al. (2015), to optimize the augment function. Firstly, the single-parameter perturbation sample method (SPP) is used to obtain several better initial values, while ensuring that the initial geometry of simplex downhill is well-conditioned. The initial value preprocessing mechanism ensures that the algorithm starts from a good basis. This is important for the simplex algorithm, which may easily fall into local optimum. Next, the simplex downhill algorithm is applied to search for better performance.

The change in augmentation function

Comparison of results between the constrained optimization
algorithm and the unconstrained optimization algorithm. The 15 red squares
and 15 black triangles are optimized solutions found by the unconstrained
optimization algorithm and constrained algorithms, respectively. The blue
diamond is the result of the CNTL experiment. The

Taylor diagram of the climate mean state of each output variable from 2002 to 2004 between the model run with optimal parameters and the CNTL run. The number (1) in the diagram stands for EXP, and (2) stands for CNTL.

The best uncertainty parameters obtained by the unconstrained optimization
method optimize the overall performance of the simulation by 10.1 %, but
they have a radiation deviation of up to 3.8 W m

Synthesized performance metric index and radiation bias in the CNTL run and the optimal model run with unconstrained and constrained methods.

Meridional distribution of the difference between EXP/CNTL
and observed data of

Performance metric index of each variable in the optimal model run with unconstrained and constrained methods.

The optimization of each output variable is shown in Table 4. In addition, a Taylor diagram is used to estimate the model performance through the standard deviation and correlation (Fig. 3). By combining the results of Table 4 and Fig. 3, it can be concluded that SWCF and Q850 receive most optimization, as they achieve a better performance index. Also, compared to the default experiment, their standard deviations have improved. Table 5 shows the standard deviations of the variables, which are important for the model but not used as evaluation criteria. It is noteworthy that they are also close to the default experiment.

The percentage of standard deviation of the eight fields between the CNTL run and the optimal model run with constrained optimization according to the corresponding observations.

The spatial distribution of TOA SW cloud forcing of

For a more comprehensive analysis of the spatial variation of the output variables, the zonal distribution of the difference between the control (labeled as CNTL)/the optimized (labeled as EXP) simulations and observations of all metric variables are shown in Fig. 4. SWCF and Q850 have been obviously improved over low and middle latitudes, but the changes in PRECT and T850 are not particularly notable. Further, LWCF only showed significant improvement near the Equator, and it slightly deteriorated over the middle and high latitudes.

The optimized parameters values are provided in the “Constrained tune” column of Table 1. The deep-convection precipitation efficiency over land and ocean is reduced relative to the default values. The timescale for the consumption rate of CAPE for deep convection is smaller than the default value, and both relative humidity thresholds for high and low clouds are increased. Additionally, the sedimentation velocity of cloud ice is larger. Next, we will explain how the changes in these parameters are related to the results of the simulations.

The spatial distribution of specific humidity at
850 hPa of

The relative humidity threshold for low clouds is larger in optimization experiments than the default value, which will obviously lead to the decrease in low-cloud fraction. The decreased low-cloud fraction is consistent with the increase in SWCF. The CNTL experiment has excelled at simulating the spatial distribution of SWCF (Fig. 5c), but it has a negative bias over the ocean in the low latitudes, where the improvement is significant in the optimal experiment.

The zonal mean specific humidity at 850 hPa is significantly improved, and its spatial distribution is presented in Fig. 6. In the optimal experiment, the atmosphere is drier in the tropics and middle latitudes, which is closer to the observation than the CNTL experiment. Meanwhile, the middle to low troposphere is also slightly drier in these areas (Fig. 7), which may be related to the increased convective precipitation. A quasi-equilibrium closure is used in the deep-convection scheme in CAM5, which is based on CAPE. The adjustment timescale represents the denominator of the cloud bottom convective mass flux. When the timescale is shorter with less changed CAPE, the increased cloud-base mass flux would help to enhance the convective precipitation. Additionally, compared to the CNTL experiment, the lower troposphere gets warmer and the middle troposphere is colder, which exacerbates the instability of the temperature structure (Fig. 8) and leads to more convective precipitation. The spatial distribution of convective precipitation over the tropics where convection occurs most frequently can be seen in Fig. 9. The increase in convective precipitation may be related to the decrease in specific humidity at 850 hPa. However, the increase in total precipitation is not particularly significant and is dominated by the changes in convective precipitation. The main reason is likely associated with the decreased precipitation efficiency parameters, which could reduce the convective precipitation as compensation. Therefore, the decreases in precipitation efficiency partially offset the precipitation change caused by tau and temperature structure.

Pressure–latitude distributions of specific humidity of

Pressure–latitude distributions of temperature of

The spatial distribution of convective precipitation over
the tropics of EXP

Pressure–latitude distributions of cloud fraction of EXP

It is difficult for all variables to be optimized, due to the strong interaction among parameters and the complex relationship among output variables. The simulations of T850 between optimal and CNTL experiments are very similar. It is likely the result of the combined effects of all relevant parameterizations. In the optimal experiment, LWCF is closer to the observation in the tropics, but it becomes slightly smaller at middle to high latitudes compared to the CNTL experiment, which implies the larger bias. The relative humidity threshold for high clouds and the sedimentation velocity of ice crystals are correspondingly increased, and both of them would lead to the reduction in high clouds. High-cloud fraction changes compared to the CNTL experiment can be seen in Fig. 10c. The reduced high cloud is consistent with the reduction in LWCF. Cloud changes also inevitably affect SWCF. It can be seen that the middle cloud has increased relative to the default experiment (Fig. 10c), and the increase in the middle cloud may be related to the decrease in precipitation efficiency over the ocean.

Note that three of six parameters hit their lowest allowable limit with the TOA balance constraint. We found that the incoming shortwave radiation flux is more sensitive to tuning parameters than the outgoing longwave radiation flux. Thus, to reduce the TOA imbalance and keep the reasonable model performance, the shortwave radiation flux should be reduced largely via increasing low-cloud fraction and liquid water content. These three variables can help achieve this by setting to the lowest bounds. This suggests that getting both the TOA balance and reasonable model performance is a relatively complex and difficult problem due to model structure problems, as pointed out by Qian et al. (2018) and Yang et al. (2019). Meanwhile, finding out how to pick parameters with a similar sensitivity to both longwave and shortwave radiation flux might be a potential approach to overcome the bound limit and it warrants further studies.

In conclusion, the increase in SWCF is consistent with the decrease in cloud fraction for the sake of a larger relative humidity threshold of low clouds. Changes in the Q850 are related to increased convective precipitation. Precipitation only slightly increases in the tropics, and the global total precipitation has changed very little, which is related to the comprehensive effect of the changes in the convection adjustment timescale, the precipitation efficiency parameter, and the vertical temperature structure. T850 simulated by the optimization experiment is similar to the default experiment. The reduced LWCF is related to the decreased high clouds caused by the increased relative humidity threshold for high clouds and the increased sedimentation velocity of ice crystals.

Radiation balance is a crucial factor for the long-term energy balance of
GCMs, but it has not received enough attention in automatic parameter
optimization. First of all, this paper points out that the previous
parameter optimization algorithms do not consider radiation balance a
necessary condition, and the obtained optimization parameters are likely to
break this important physical constraint, which may lead to unacceptable
calibrated parameters. Thus we propose an efficient constrained automatic
optimization algorithm to calibrate the uncertainty parameters in CAM5 with
the constraint of the absolute value of the difference of net solar flux and
net longwave flux at the top of the model (less than 1 W m

The optimal parameters found by our method can increase the overall
performance of the model by 6.3 %, and the radiation imbalance is as low
as 0.1 W m

The unconstrained optimization methods calibrate the uncertain parameters in climate models without a consideration of the principles that model have to hold, this creates challenges in maintaining the physics constraints and improving the structure of models. Perhaps a more physics-guided optimization is a better way to understand the principles of climate systems and to best use these principles in tuning processes. In the future, we will apply this method to coupled models, where the radiation imbalance has a more significant impact on long-term simulation stability. In addition, we will also try to introduce more constraints, such as the surface energy balance, into automatic parameter calibration.

The code of our algorithm, the observations, and the related scripts can be found at

LW and TZ proposed the tuning method. LW, YQ, and WX designed the metrics and the constraint. YQ and LW evaluated the optimal results. LW, TZ, WX, and YQ wrote the paper.

The authors declare that they have no conflict of interest.

This work is partially supported by the National Key R&D Program of China (grant nos. 2017YFA0604500 and 2016YFA0602100) and the Center for High Performance Computing and System Simulation of Pilot National Laboratory for Marine Science and Technology (Qingdao).

This work is partially supported by the National Key R&D Program of China (grant nos. 2017YFA0604500 and 2016YFA0602100) and the Center for High Performance Computing and System Simulation of Pilot National Laboratory for Marine Science and Technology (Qingdao).

This paper was edited by Andrea Stenke and reviewed by two anonymous referees.