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Article

Quantifying the Role of Model Internal Year-to-Year Variability in Estimating Anthropogenic Aerosol Radiative Effects

by
Xiangjun Shi
* and
Yuxi Zeng
School of Atmospheric Sciences, Nanjing University of Information Science and Technology, Nanjing 210044, China
*
Author to whom correspondence should be addressed.
Atmosphere 2024, 15(1), 79; https://doi.org/10.3390/atmos15010079
Submission received: 7 November 2023 / Revised: 21 December 2023 / Accepted: 5 January 2024 / Published: 8 January 2024
(This article belongs to the Special Issue The Uncertainty of Estimating Aerosol Climate Effects Using Atmospheric Models)
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Abstract

:
The model internal year-to-year variability (hereafter, internal variability) is a significant source of uncertainty when estimating anthropogenic aerosol effective radiative forcing (ERF). In this study, we investigate the impact of internal variability using large ensemble simulations (600 years in total) with the same climate model under prescribed anthropogenic aerosol forcings. A comparison of the magnitudes (i.e., standard deviation, Std) of these influences confirms that internal variability has negligible impacts on the instantaneous radiative forcing (RF) diagnosed by double radiation calls but has considerable impacts on estimating ERF through rapid adjustments (ADJ). Approximately half of the model grids exhibit a strong internal variability influence on ERF (Std > 5 W m−2). These strong internal variabilities lead to a 50% probability that the 30-year linear change can reach 2 W m−2 and the 10-year linear change can reach 4 W m−2. A 50-year simulation can provide a relatively stable annual mean map of ERF (ERF = ADJ + RF), but it fails for ADJ. The statistically significant areas in the annual mean maps of both ERF and ADJ from a 10-year simulation exhibit instability with evident chaotic features. The insights derived from our analysis aid in assessing the stability of modeled ERF and contribute to the design of comparative experiments.

1. Introduction

Climate models are important tools for studying climate change. Compared to observation-based analyses, estimations based on models come with a significantly higher degree of uncertainty [1,2,3,4,5,6,7]. Uncertainty in model results can be attributed to the following two primary sources [8,9,10,11,12,13,14,15,16]. The first source is the systematic model bias when compared to the real world, which includes the discrepancies in the representation of physical mechanisms described within the model’s code and the climate forcers used as model input data (e.g., [17,18]). The second source is the model year-to-year internal variability (hereafter, internal variability), which represents the natural year-to-year fluctuations that occur during model simulations without year-to-year changes in external climate forcers (e.g., [19,20,21,22]).
In recent years, the role of internal variability in climate projections has gained substantial attention. This internal variability notably influences decadal and multi-decadal trends in surface temperature and precipitation, especially at regional scales [23,24,25,26,27]. Compared to model diversity (i.e., model bias), internal variability dominates climate uncertainty over the first decadal prediction (i.e., relatively shorter timescale) horizons at regional to local scales [28,29,30]. Fortunately, this kind of uncertainty can be significantly reduced through the application of large ensemble simulations [31,32,33,34]. In future climate projections, internal variability also significantly impacts extreme events and associated predictions such as droughts and crop production [35,36,37]. The influence of internal variability on climate risk assessment, adaptation management, and decision-making has increasingly come under scrutiny [2,38,39,40]. Exploring the random decadal and multi-decadal trends in net radiative fluxes at the top of the atmosphere (TOA) induced by internal variability can offer a more profound understanding of climate uncertainty. This constitutes one of the primary objectives of this study.
Climate models have also been extensively utilized to estimate Earth’s response to anthropogenic activities because climate models can efficiently provide comparative experimental results and explore the mechanisms behind their differences [41,42,43]. In such studies, effective radiative forcing (ERF) is widely used as a valuable metric for quantifying the impact of radiative perturbations on Earth’s energy imbalance. The ERF is defined as the difference in net radiative fluxes at TOA between pre-industrial (PI) and present-day (PD) experiments. Considering the influence of internal variability, significance tests must be performed to assess the statistical significance of values in annual mean maps [44,45,46,47,48]. Taking anthropogenic aerosol as an example, a simulation length of decades is required to estimate a global annual mean value with a precision of 0.1 W m−2, according to the significance test formula. If the estimation is performed on a regional scale instead of the global mean, the necessary simulation length might extend to several centuries [49]. Because century-long simulations are computationally expensive, the radiative effects of one anthropogenic climate forcer are typically estimated using comparative simulations over periods of 10 or 30 years. The annual mean maps of ERF are usually uncertain, even in statistically significant areas. This study tests this issue based on large ensemble simulations and tries to explore a few useful experiences.
Aerosols are tiny particles or liquid droplets suspended in the Earth’s atmosphere. They affect the climate via two prime ways. Aerosols can directly interact with solar radiation by scattering or absorbing it (direct effects). Aerosols also have an indirect impact through their role in cloud formation (indirect effects). For instance, high concentrations of aerosol particles can lead to more but smaller cloud droplets compared to clouds formed in clean air, a phenomenon known as the Twomey effect (i.e., the first indirect effect). Significant progress has been made in model simulations and analysis methods for a more accurate estimation of anthropogenic aerosol radiative effects [50,51,52,53]. An important example of progress is that the Coupled Model Intercomparison Project Phase 6 (CMIP6) focuses on how the Earth system responds to a specified anthropogenic aerosol forcing (i.e., a common climate forcer) [54]. After using the same prescribed anthropogenic aerosol forcing recommended by CMIP6, the difference among climate models is obviously reduced [55,56]. Additionally, with the advantage of prescribed anthropogenic aerosol forcing, anthropogenic aerosol ERF can be decomposed into robust instantaneous radiative forcing (RF) and unstable rapid adjustments (ADJ) [57,58]. Progressing a step beyond the previous study by Fiedler et al. (2017) [49], we investigate not only the influences of internal variability on ERF but also its influences on RF and ADJ. Furthermore, this study also performs significance tests under a non-null hypothesis, given that ERF encompasses a stable component (i.e., RF).
This study aims to comprehensively understand the role of internal variability in estimating anthropogenic aerosol radiative effects more effectively. Its influences are analyzed from three perspectives: different components of anthropogenic aerosol radiative effects, decadal trends of time series, and the sensitivity of the annual mean map to simulation lengths. The structure of this study is as follows: the calculation methods and experimental design are described in Section 2; the ensemble simulation results are presented and analyzed in Section 3; the discussion is presented in Section 4, and the conclusions are provided in Section 5.

2. Experiments and Methods

2.1. Modified CAM5 Model and Experiment Setup

CMIP6 provides a simple plume implementation of the second version of the Max Planck Institute Aerosol Climatology (MACv2-SP) for climate models. This implementation offers prescribed anthropogenic aerosol forcing [49,59]. The anthropogenic aerosol direct radiative effect is represented by prescribed anthropogenic aerosol optical properties (i.e., the aerosol optical depth, single-scattering albedo, and asymmetry factor). To represent the anthropogenic aerosol Twomey effect, MACv2-SP provides a normalized change in the cloud droplet number (rNc). The rNc is the ratio of the increase in the cloud droplet number (Nc) compared to the background Nc of the host model (only the natural aerosol contribution). The year-parameter in MACv2-SP can be set differently from that of the host model time system. In the year 1850 (i.e., PI), there was no anthropogenic aerosol forcing. More details about the MACv2-SP can be found in Stevens et al. (2017) [59].
The climate model used in this study is version 5.3 of the Community Atmosphere Model (CAM5; [60]) with CMIP6 anthropogenic aerosol forcing. In the default CAM5 model, a two-moment stratiform cloud microphysics scheme [61,62] is used and coupled with an aerosol module [63,64] for considering aerosol–cloud interactions. The RRTMG radiation package is used to improve the accuracy of aerosol and cloud effects calculations [65]. To estimate aerosol effects based on the CMIP6 protocol, the MACv2-SP is implemented into the CAM5 model. In the radiation scheme, total aerosol shortwave optical properties are calculated based on the anthropogenic aerosol optical properties from MACv2-SP and natural aerosol optical properties from the host model’s default mechanism. The optical properties of warm clouds are calculated based on the cloud droplet number (rNc × Nc). Meanwhile, the RF from the direct radiative effect (RFari) and Twomey effect (RFaci) can be calculated by double radiation calls at each model time step [57,58,66]. To achieve this, the radiation subroutine needs to be called many times to provide different net radiative fluxes at the top of the atmosphere (TOA). Table 1 lists the radiative variables used in this study. Here, the RFari can be calculated by two methods, denoted as RFari (Fari,aci − Faci) and RFari (Fari − F). Both of them are diagnosed by the updated radiation package. Meanwhile, model simulations also output two RFaci (i.e., Fari,aci − Fari and Faci − F). Compared to RFari (Fari,aci − Faci), RFari (Fari − F) is calculated in a background cloud without the Twomey effect. In other words, the difference between these two RFari indicates the impact of the Twomey effect. Similarly, the difference between these two RFaci indicates the impact of the anthropogenic aerosol direct radiative effect. These impacts will be shown through the model simulation results in Section 3.
A pair of experiments are carried out to estimate anthropogenic aerosol radiative effects, which use the PI (1850) and PD (2005) year-parameters of MACv2-SP. Both PD and PI experiments are atmosphere-only simulations (i.e., the sea surface temperature and sea ice are given) with a horizontal resolution of 1.9° latitude × 2.5° longitude and a model time step of 30 min. Considering the impact of internal variability, ensemble simulations are conducted. Each experiment comprised twelve simulations, each starting in a different month. The first ensemble member starts on 1 January, the second ensemble member starts on 1 February, and so on. All simulations run for approximately 51–52 years, and the analysis focuses on the results from the last 50 complete calendar years (from January to December). Both the PD and PI experiments have a simulation length of 600 years for analysis.

2.2. Estimating Methods and Significance Tests

Here, the anthropogenic aerosol ERF is the difference in the Fari,aci between PD and PI experiments (i.e., Fari,aciPD−PI). For clarity, we denote the experiment name as a superscript to indicate the source of the variable. Additionally, we calculate the contribution of rapid adjustments (ADJ) to ERF by subtracting RF from ERF. The equation, ERF (Fari,aciPD−PI) = RF (RFari + RFaci, Fari,aciPD − FPD) + ADJ, allows for the easy derivation of the calculation equation for ADJ: ADJ = FPD−PI. ADJ (FPD−PI) is associated with changes in the background atmosphere state (i.e., factors that affect radiation transfer, except for anthropogenic aerosol). In contrast to RF (RF = RFari + RFaci), ADJ (FPD−PI) cannot be decomposed into adjustments induced by a direct radiative effect and adjustments induced by a Twomey effect, because of the interactions among all adjustments at each model time step. It is necessary to point out that ERF (Fari,aciPD−PI) and ADJ (FPD−PI) are calculated based on a pair of corresponding simulations (e.g., the 11th ensemble member of the PD experiment and the 11th ensemble member of the PI experiment). A total of 600 annual estimates are included in this study. We analyze variability using standard deviations, abbreviated as Std, calculated from the averages of each year.
Significance tests are performed with a t-test, a commonly used method for assessing the statistical significance of differences in comparative experiments [67]. To perform a t-test, a t-value is calculated first:
t = n x 1 ¯ − x 2 ¯ − μ d S t d
S t d = 1 n − 1 ∑ i = 1 n x 1 − x 2 − x 1 − x 2 ¯ 2
where x 1 ¯ and x 2 ¯ are the sample means from two experiments, μd is the proposed population difference, Std is the standard deviation of these samples, and n is the sample size. A critical value, corresponding to the chosen significance level and the degree of freedom (n − 1), is also needed. Critical values for common significance levels can be easily obtained from tables in textbooks. For the convenience of discussion, this study only employs a 90% significance level. If the calculated t-value exceeds the critical value, there is a 90% probability that a difference (i.e., | x 1 ¯ − x 2 ¯ |> μd) indeed exists. Based on the t-value equation and critical value lookup table, passing a significance test requires that the difference (| x 1 ¯ − x 2 ¯ |− μd) exceeds 0.59, 0.24, and 0.07 times Std for a sample size of n = 10, 50, and 600, respectively. The required difference decreases as n increases. For example, for Std = 0.4, a sample size of n ≥ 50 is needed for estimating the difference from paired experiments with a precision of approximately 0.1 (0.4 × 0.24 = 0.096). Typically, μd is set to zero (i.e., null hypothesis), and the assessment focuses only on determining whether there is a significant difference (i.e., | x 1 ¯ − x 2 ¯ | ≠ 0).

3. Results

In this study, we only analyze shortwave radiative variables, consistent with the anthropogenic aerosol forcings provided by MACv2-SP. Furthermore, in this section, all radiative variables are analyzed under all-sky conditions, and all radiative forcings (i.e., ERF, RF, and ADJ) represent those induced by anthropogenic aerosols. Unless explicitly stated otherwise, annual means and corresponding standard deviations are calculated based on ensemble experiment results (i.e., 600 annual values), and the default assumption for statistical tests is the null hypothesis.

3.1. Impacts on Different Components of ERF

Based on model output results, ERF can be decomposed into three components: ERF (Fari,aciPD−PI) = RFari (Fari,aciPD − FaciPD) + RFaci (Faci PD − FPD) + ADJ (FPD−PI) or ERF (Fari,aciPD−PI) = RFari (FariPD − FPD) + RFaci (Fari,aciPD − FariPD) + ADJ (FPD−PI). Unlike ERF and ADJ, both RFari and RFaci are diagnosed during the PD model simulation (i.e., derived from one experiment). The year-to-year variability of the annual time series of the four radiative fluxes (Fari,aci, Fari, Faci, and F) diagnosed in a single simulation is nearly identical because they are calculated from the same background atmospheric state (natural aerosol and clouds without the Twomey effect, etc.). Clearly, both RFari and RFaci, representing differences among the above four radiative fluxes, are not significantly sensitive to their year-to-year variabilities (i.e., internal variability). However, a common radiative flux variable (e.g., Fari,aci and F) from two comparative experiments (i.e., PD and PI) exhibits different annual variation curves due to the chaotic year-to-year variability induced by internal variability. Accordingly, internal variability can significantly influence the estimates of ERF and ADJ. Figure 1 illustrates the annual mean of ERF and its decomposition. As expected, all regions of RFari and RFaci exhibit statistical significance. The RFari calculated using the two methods are almost the same. However, the global mean values show that the RFari (FariPD − FPD, −0.31 W m−2) is slightly stronger (more negative) than the RFari (Fari,aciPD − FaciPD, −0.30 W m−2). This is because the RFari (FariPD − FPD) is calculated based on the background cloud without the Twomey effect. The aerosol radiative effect would be decreased under dense cloud conditions, as shown in the comparison between clear-sky RFari and all-sky RFari in previous studies. RFaci calculated using two methods is also almost the same. The presence of small differences in global means (−0.28 W m−2 and −0.29 W m−2) has a similar explanation to that of RFari. In contrast to RFari and RFaci diagnosed from the radiation package, there are few statistically significant regions of ADJ. The global mean of ADJ is very small (0.02 W m−2). Generally speaking, RFari and RFaci are the primary contributors to ERF, especially over Europe, North America, East and South Asia, and their adjacent oceans (i.e., regions with a high anthropogenic aerosol burden). This is the main reason why ERF is statistically significant in the aforementioned regions. In other regions, ERF is statistically nonsignificant due to the uncertainty from ADJ. In summary, the magnitude of ERF mostly depends on RF, and the uncertainty in ERF mostly arises from ADJ. If feasible (e.g., MACv2-SP), decomposing ERF into robust RF and uncertain ADJ is advisable.
To better understand the uncertainties induced by internal variability, the magnitudes of year-to-year variabilities (i.e., Std) are shown in Figure 2. The Std of ERF is almost the same as that of ADJ. This analysis confirms that uncertainty in ERF almost entirely arises from ADJ. The Std of ERF and ADJ exceeds 1 W m−2 in most regions and exceeds 5 W m−2 in major anthropogenic aerosol source regions. Under the conditions of Std = 5 W m−2 and 600 samples, the minimum radiative forcing (absolute value) that passes statistical testing is 0.35 (i.e., 5 × 0.07) W m−2. This explains why most regions on the annual mean map of ADJ are statistically nonsignificant. Regionally, the radiative forcing thresholds for ERF and ADJ are also almost the same. Therefore, ERF, including robust RF, shows more areas where signals (i.e., the absolute value of radiation forcing) are statistically significant (Figure 1). In contrast to ERF and ADJ, the year-to-year variability of RFari depends not only on the background atmosphere state (e.g., clouds) but also on the magnitude of anthropogenic aerosol direct radiative forcing (i.e., RFari itself). This is likely the main reason why the Std of RFari is obvious (>0.1 W m−2) over Europe, North America, Central Africa, and East and South Asia (anthropogenic aerosol source regions). In contrast to the RFari, the Std of RFaci becomes much weaker. In summary, both RFari and RFaci exhibit much weaker Std values compared to the annual mean (Figure 1 and Figure 2). This suggests that the influence of internal variability on the diagnosis of RFari and RFaci is negligible. In the following analysis, we focus on the influence of internal variability on the estimation of ADJ, which is the source of uncertainty in ERF.

3.2. Impacts on the Time Series Trend

The time series of annual means provides an intuitive representation of internal variability. Figure 3 shows the annual ADJ time series in different spatial areas. It is necessary to point out that the annual ERF time series also shows similar year-to-year variation features (not displayed here). In all cases (i.e., global mean, region A, grid B, and grid C), the F curves from PD and PI simulations show different year-to-year variations. As a result, their difference (i.e., ADJ) also shows obvious year-to-year variation. Clearly, the amplitude of the time series is related to the spatial area of the research object. The larger the area, the more mutual cancellation occurs between positive and negative variations. As expected, the amplitude of the global mean ADJ time series is much smaller than that of other time series. This is why the simulation length required for estimating a global mean is much shorter than that needed for regional values. The amplitude of the time series for an individual model grid box depends on its location. Consistent with the spatial distribution of Std shown in Figure 2, grid B shows a much larger amplitude than grid C.
Internal variability is an intrinsic year-to-year fluctuation in the climate system that occurs during model simulation in the absence of external forcing. In theory, the annual ADJ time series should not exhibit long-term trends under constant forces. However, the time series may show obvious increasing or decreasing trends on decadal scales due to the chaotic feature of internal variability. In Figure 3, various 10-year ADJ time series windows with strong increasing or decreasing trends are displayed, such as the 33rd~42nd years in region A. There are also some 30-year ADJ time series windows with obvious trends, such as the 15th to 44th years in grid B. However, the strength of the trend over a longer period (i.e., 30 years) is hardly the same as the strength of the trend over a 10-year period. The following paragraphs will analyze the probability distributions of 10-year trends and 30-year trends.
In Figure 4, the probability distributions of ADJ 10-year and 30-year trends are shown. The trend is defined as the linear regressions slope for each 10-year or 30-year period in the time series in the 50-year simulation. Each ensemble member can produce 41 10-year trends and 21 30-year trends. Because the year-to-year fluctuations are chaotic, these trends possess an approximately normal distribution, especially the 10-year trends. This is consistent with previous studies on internal variability at decadal time scales (e.g., [22,23]). It is also necessary to point out that the 75th percentile of the trend distribution is not completely opposite to the 25th percentile because of the limited sample size. The trend distributions of ERF are almost the same as those of ADJ (not shown). In order to better understand these trends, 10-year and 30-year changes are compared with the annual mean ERF. The 25th and 75th percentiles of the 10-year trend distributions of the global mean are −0.16 W m−2 decade−1 and 0.13 W m−2 decade−1, respectively. The 10-year changes caused by these two trends equal about a quarter of the magnitude of the global annual mean ERF (−0.57 W m−2). Based on statistical theory, 30-year trends are usually much weaker than 10-year trends. The 25th and 75th percentiles of the 30-year trend distribution of the global mean are −0.03 W m−2 decade−1 and 0.02 W m−2 decade−1, respectively. In other words, there is a 50% probability that 30-year changes are outside the range of −0.09 (−0.03 × 3) to 0.06 (0.02 × 3) W m−2. In region A, half of the 30-year changes are outside the range of −0.96 (−0.32 × 3) to 1.47 (0.49 × 3) W m−2. Compared to the magnitude of spatial and temporal averaged ERF in region A (−2.44 W m−2), these 30-year changes cannot be neglected. On grid B, the annual mean ERF is −1.90 W m−2, and half of 30-year changes are outside the range of −2.94 (−0.96 × 3) to 2.58 (0.86 × 3) W m−2. On grid C, the annual mean ERF is −0.08 W m−2, and half of the 30-year changes are outside the range of −1.02 (−0.34 × 3) to 0.39 (0.13 × 3) W m−2. Figure 5 shows the 25th and 75th percentiles of the trend distribution at each model grid. Here, the 75th percentile of the trend distribution is generally opposite to the 25th percentile. There is a 50% probability that the 10-year trend is stronger than 4 W m−2 decade−1 (≤−4 and ≥4) over high Std areas (Std > 5 W m−2 in Figure 2). Over most high Std areas, the 25th and 75th percentiles of the 30-year trend are also relatively strong. In about half of the cases, 30-year changes can reach 2 (0.7 × 3 > 2) W m−2. In short, the influence of internal variability on 10-year and 30-year trends cannot be neglected, even on global scales.

3.3. Sensitivity to Simulation Lengths

Considering the computational expense of simulations, anthropogenic aerosol effects are usually estimated based on a pair of experiments with simulation lengths of decades (e.g., 10 and 50 years). Here, we investigate the stability of annual mean maps over 50-year and 10-year periods. First, the Std of 50-year and the Std of 10-year simulations are analyzed. Since the Std of ERF is almost the same as that of ADJ and the differences among the twelve ensemble members can be represented briefly by the difference among any three ensemble members (not shown), only the Stds of ADJ calculated from the first, second, and third ensemble members are shown in Figure 6. All three 50-year simulations show a spatial Std pattern similar to the entire experiment (i.e., 600 years, Figure 2). Meanwhile, the differences among these three maps are not obvious. Compared to 50-year simulations, the maps from 10-year simulations display more chaotic features, resulting in more noticeable differences among these three 10-year simulations. In summary, a 50-year simulation can provide a relatively reliable estimate of internal variables (i.e., Std), while achieving the same level of stability from a 10-year simulation is challenging. It is noteworthy that even under the condition of stable Std, the multi-year averages also have obvious fluctuations, which depend on the sample size (i.e., the number of years). This issue will be discussed in the following paragraph.
Figure 7 shows the annual mean maps of ADJ for different simulation lengths. The original (i.e., without significance tests) annual mean map from the entire experiment (i.e., 600 years) does not exhibit any obvious chaotic features. However, chaotic features become noticeable in the maps from 50-year simulations and worsen in the maps from 10-year simulations. Meanwhile, as compared to the entire experiment, the values on model grids become strong in the 50-year simulation maps and even stronger in the 10-year simulation maps. The distinctions among the three 50-year simulation maps are evident, and the 10-year simulation maps show almost different spatial patterns. These mentioned issues are more obvious in statistically significant areas (second and third columns in Figure 7). Two hypotheses are used for significance tests: the commonly used null hypothesis (i.e., ADJ ≠ 0) and a more rigorous hypothesis (|ADJ| > 0.5), which analyzes the existence of significant areas under a more rigorous significance test. Using the more rigorous significance test results in an expected reduction in statistically significant areas. Regarding statistical probability, 10-year and 50-year simulations might generate some significant areas due to low-probability events rather than prescribed aerosol forcing. These significant areas usually show chaotic features. This is the main reason that neither 50-year nor 10-year simulations are capable of generating large, contiguous, and reliable statistically significant areas. In contrast, statistically significant areas from the 600-year experiment are relatively large and contiguous, indicating that a simulation with a length of 600 years can provide reliable maps of statistically significant areas for the further analysis of ADJ. The significant areas under the commonly used null hypothesis do not display obvious common features among these 50-year simulations. This indicates that these significant areas are mostly unreliable. However, conducting a more rigorous significance test might provide significant areas with higher reliability. Using the significant areas from the entire experiment as a benchmark, under the more rigorous test, 50-year simulations could provide some useful information about significant areas, while 10-year simulations are nearly incapable of providing reliable significant areas. In summary, on regional scales, neither 50-year nor 10-year simulations can offer a stable estimate of the annual mean ADJ, and 50-year simulations might provide limited useful information about significant areas under more rigorous significance tests.
Most studies directly estimate ERF without decomposing it into RF and ADJ. The annual mean ERF map shows much more statistically significant areas than ADJ (Figure 7 and Figure 8). This is because ERF includes robust RF, which contributes most to ERF. All the three 50-year simulations can produce relatively large and contiguous statistically significant areas, and their common significant areas among these three maps are dominant. This suggests that a 50-year simulation can provide a relatively reliable estimate of ERF. The magnitudes of ADJ (the uncertain component of ERF) from 10-year simulations are generally much larger than those from 50-year simulations, as shown in Figure 7. Therefore, there are obvious differences among these three annual mean maps of ERF from 10-year simulations, especially for statistically significant areas. This indicates that a 10-year simulation cannot provide a stable estimate of ERF. The analyses above are consistent with the notion that the simulation lengths required for comparative experiments with atmosphere-only integrations are typically on the scale of decades, such as a ≥30-year simulation required in the protocol of the CMIP6 Radiative Forcing Model Intercomparison Project (RFMIP) [42].

4. Discussion

It should be noted that this study employs only one climate model due to the considerable computational cost of large ensemble experiments. Certain features of Std (i.e., internal variability) in the figures presented above may differ from those in other model simulations. For example, in the Arctic region, this study shows that the Std exhibits a high value (>5 W m−2) (Figure 2), while Fiedler et al. (2017) reported a relatively low Std (<2 W m−2) for the same region based on 30-year simulations using another atmospheric model [49]. Moreover, using the same model, different experiment settings (e.g., resolution, cloud scheme, atmosphere-only or atmosphere–ocean coupled simulations) can yield different results. One should keep in mind that the features of Std shown here (i.e., Figure 2) to some extent depend on the climate model used in this study. In this study, we did not explore the mechanisms underlying the magnitude and pattern of Std. However, we conducted a comparison of the Std under clear-sky and all-sky conditions and found that the year-to-year variability of simulated clouds is the main contributing factor (not shown). With the advantage of model experiments, the mechanisms of climate system internal variability can be deeply investigated in future studies. Note that model-based analysis cannot be confirmed by observations because observing climate system internal variability in the real natural world seems unfeasible. This may explain why there are few studies that discuss the mechanisms underlying internal variability.
Although this study focuses on anthropogenic aerosol radiative effects, it also offers some representative analyses for comparative experiment design. As an example, internal variability has a significant impact on estimates of ERF or ADJ derived from a pair of comparative experiments but does not have a significant impact on the RF diagnosed during the model simulation. This conclusion is also valuable for estimating the effects of other short-lived climate forcers, including transient responses and rapid adjustments. If feasible, it would be advantageous for climate models to supervise transient response processes and output diagnostic variables such as RF in the radiation package. These variables, diagnosed at each model time step, are seldom influenced by internal variability. It is relatively easy to figure out the mechanisms for these robust variables (i.e., transient response) and then facilitate a better understanding of the final estimate (i.e., the difference between a pair of comparative experiments). Another representative experience is that statistically significant areas with chaotic features, as illustrated by the ERF from 10-year simulations in Figure 8, are probably unreliable. If the aforementioned case occurs, it is better to further check the stability of the simulation results by comparing among three ensemble members.

5. Conclusions

This study explores the role of internal variability in estimating anthropogenic aerosol radiative effects. We conducted large ensemble simulations using the same climate model to quantify the impact of internal variability on regional scales (i.e., per model grid). Taking the advantage of prescribed aerosol forcing (i.e., CMIP6 protocol), it is possible to decompose ERF into robust RF and unstable ADJ.
The maps of the multi-year averages and corresponding Std from the large ensemble simulation (600 years) seem reliable since there are no obvious chaotic features in these maps. The Std, which represents the magnitude of internal variability influences, is negligible in the time series of RF (i.e., RFari and RFaci). This implies that RFari and RFaci diagnosed from the radiation package can be considered as offline simulation results, and a one-year simulation could provide a reliable estimate. In contrast to RF, the Stds of ERF and ADJ are both strong, and their values are almost identical. This comparison supports the previous studies confirming that the uncertainty in estimating ERF induced by internal variability almost comes from ADJ.
Internal variability can result in considerable decadal trends in ERF and ADJ time series. In theory, the probability of these trends follows a normal distribution, with the 75th percentile of the trend distribution being the negative counterpart of the 25th percentile. Here, the above features are generally reproduced by large ensemble simulations. The maps of the 25th and 75th percentiles exhibit similar spatial patterns to those of the corresponding Std. This confirms that the decadal trend distributions are primarily influenced by the magnitude of year-to-year variability. Over the model grids with a high Std (Std > 5 W m−2, about half of the world), there is a 50% probability that 30-year changes can reach 2 W m−2 and 10-year changes can reach 4 W m−2. These changes occurring over 10-year and 30-year periods induced by internal variability are typically much larger than the multi-year mean ERF. Therefore, it is crucial to be aware that conspicuous decadal trends might be a consequence of internal variability rather than increasing or decreasing climate forcers.
The uncertainty from internal variability can be reduced by extending simulation lengths. However, this solution requires additional computational resources. Therefore, estimates of anthropogenic aerosol effects typically rely on paired experiments with decades-long simulation periods, such as 10, 30, and 50 years. Compared to ADJ, providing a reliable annual mean map of ERF is relatively easy because the dominant contributor of ERF (i.e., RF) is robust. A 50-year simulation can provide a relatively stable annual mean map of ERF, while it fails for ADJ. The statistically significant areas in the annual mean maps of both ERF and ADJ from a 10-year simulation display instability and may exhibit noticeable chaotic features. Finally, it is necessary to point out that differences among the twelve ensemble members can be briefly represented by differences among any three members of the ensemble. This experience may explain why ensemble simulations usually comprise three members.

Author Contributions

X.S. designed this study and carried out simulations. X.S. and Y.Z. analyzed the simulation results and wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (grant nos. 41775095 and 42075145). The APC was supported by the same funders.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The modified CAM model code, the Fortran code for analyzing the model results, the NCL scripts for making plots, and corresponding data have been archived in a public repository (https://doi.org/10.5281/zenodo.10056701, accessed on 6 November 2023).

Acknowledgments

The authors would like to thank Bjorn Stevens for his guidance in using MACv2-SP. This study was conducted at the High-Performance Computing Center of Nanjing University of Information Science and Technology.

Conflicts of Interest

The authors declare no conflicts of interest.

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Atmosphere 15 00079 g001
Figure 1. Annual mean maps of annual anthropogenic aerosol effective radiative forcing (ERF, first row), instantaneous radiative forcing from a direct radiative effect (RFari, second row) and Twomey effect (RFaci, third row), and rapid adjustments (ADJ, fourth row). The global average is provided in the upper-right corner. Hatching represents the nonsignificant area at the 90% confidence level.
Figure 1. Annual mean maps of annual anthropogenic aerosol effective radiative forcing (ERF, first row), instantaneous radiative forcing from a direct radiative effect (RFari, second row) and Twomey effect (RFaci, third row), and rapid adjustments (ADJ, fourth row). The global average is provided in the upper-right corner. Hatching represents the nonsignificant area at the 90% confidence level.
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Figure 2. Similar to Figure 1, but for the corresponding standard deviations (Std) calculated from the averages of each year (600 values).
Figure 2. Similar to Figure 1, but for the corresponding standard deviations (Std) calculated from the averages of each year (600 values).
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Figure 3. Annual mean anthropogenic aerosol ADJ from the 12th ensemble member (first row) and the annual ADJ time series (second and third rows) calculated by the global mean, region A, and model grids B and C (marked in the annual mean map).
Figure 3. Annual mean anthropogenic aerosol ADJ from the 12th ensemble member (first row) and the annual ADJ time series (second and third rows) calculated by the global mean, region A, and model grids B and C (marked in the annual mean map).
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Figure 4. The probability distributions of the ADJ 10-year (left column) and 30-year (right column) trend from the global mean, region A, and model grids B and C. A total of 40 bins were used here. The vertical red lines indicate the 25th and 75th percentiles. Unlike Figure 3, all ensemble members were used for statistical analysis.
Figure 4. The probability distributions of the ADJ 10-year (left column) and 30-year (right column) trend from the global mean, region A, and model grids B and C. A total of 40 bins were used here. The vertical red lines indicate the 25th and 75th percentiles. Unlike Figure 3, all ensemble members were used for statistical analysis.
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Figure 5. Maps showing the 25th and 75th percentiles of ADJ 10-year (left column) and 30-year (right column) trend distributions.
Figure 5. Maps showing the 25th and 75th percentiles of ADJ 10-year (left column) and 30-year (right column) trend distributions.
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Figure 6. Maps for the Std of ADJ calculated from the first, second, and third ensemble members using the entire simulation lengths (50 years, first rows) and only the first 10 years (second rows). The global average is shown in the upper-right corner.
Figure 6. Maps for the Std of ADJ calculated from the first, second, and third ensemble members using the entire simulation lengths (50 years, first rows) and only the first 10 years (second rows). The global average is shown in the upper-right corner.
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Figure 7. Annual mean maps for the ADJ calculated from all ensemble experiment members (600 years, first row), the first, second, and third ensemble members with entire simulation lengths (50 years, second, third, and fourth rows) and only the first 10 years (fifth, sixth, and seventh rows). This first column shows the original multi-year averages without significance tests. The global average is given in the upper-right corner. The nonsignificant areas under the ADJ ≠ 0 hypothesis (second column) and |ADJ| > 0.5 hypothesis (third column) are denoted by a white color. Note that the color bar is different from that in Figure 1.
Figure 7. Annual mean maps for the ADJ calculated from all ensemble experiment members (600 years, first row), the first, second, and third ensemble members with entire simulation lengths (50 years, second, third, and fourth rows) and only the first 10 years (fifth, sixth, and seventh rows). This first column shows the original multi-year averages without significance tests. The global average is given in the upper-right corner. The nonsignificant areas under the ADJ ≠ 0 hypothesis (second column) and |ADJ| > 0.5 hypothesis (third column) are denoted by a white color. Note that the color bar is different from that in Figure 1.
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Figure 8. Similar to Figure 7, but for ERF.
Figure 8. Similar to Figure 7, but for ERF.
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Table 1. List of model output radiative variables used in this study.
Table 1. List of model output radiative variables used in this study.
NamesDescription
Fari,aciThe default shortwave net radiative fluxes, considering the anthropogenic aerosol direct radiative effect and Twomey effect
FaciSimilar to Fari,aci, but excluding the anthropogenic aerosol direct radiative effect
FariSimilar to Fari,aci, but excluding the anthropogenic aerosol Twomey effect
FSimilar to Fari,aci, but excluding the anthropogenic aerosol effects
RFariAnthropogenic aerosol instantaneous radiative forcing from the direct radiative effect, RFari = Fari,aci − Faci, RFari = Fari − F
RFaciAnthropogenic aerosol instantaneous radiative forcing from the Twomey effect, RFaci = Fari,aci − Fari, RFaci = Faci− F
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Shi, X.; Zeng, Y. Quantifying the Role of Model Internal Year-to-Year Variability in Estimating Anthropogenic Aerosol Radiative Effects. Atmosphere 2024, 15, 79. https://doi.org/10.3390/atmos15010079

AMA Style

Shi X, Zeng Y. Quantifying the Role of Model Internal Year-to-Year Variability in Estimating Anthropogenic Aerosol Radiative Effects. Atmosphere. 2024; 15(1):79. https://doi.org/10.3390/atmos15010079

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Shi, Xiangjun, and Yuxi Zeng. 2024. "Quantifying the Role of Model Internal Year-to-Year Variability in Estimating Anthropogenic Aerosol Radiative Effects" Atmosphere 15, no. 1: 79. https://doi.org/10.3390/atmos15010079

APA Style

Shi, X., & Zeng, Y. (2024). Quantifying the Role of Model Internal Year-to-Year Variability in Estimating Anthropogenic Aerosol Radiative Effects. Atmosphere, 15(1), 79. https://doi.org/10.3390/atmos15010079

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