considered whether to use all or only part of the control time series. They recommended using the entire control time series to estimate the drift “to minimize the contamination of the drift estimate by internal variability” p. 8597. To derive a single best estimate of drift, we should use the entire control time series.

also considered different drift correction methods, including linear, quadratic, and cubic drift correction approaches. Quadratic and cubic drift correction both risk overfitting any curvature in the control time series . On the other hand, such curvature may contain valuable information about non-linearity in the drift. Among recent studies focusing on the Earth's energy budget or sea level change, a few consider the possibility that results may be sensitive to alternative linear, quadratic, and/or cubic models of drift e.g.. However, most studies use only a single statistical model of drift: either linear , quadratic e.g., or cubic e.g.. We observe that researchers generally select a statistical model a priori before analysing any data. They do not generally compare the a posteriori performance of alternative statistical models.

To quantify drift uncertainty, we propose a Monte Carlo drift correction (MCDC) framework. Following , MCDC estimates the drift using the entire control time series. MCDC also samples the standard error associated with the estimated drift. MCDC proceeds as follows.

Select a statistical model of drift, such as a linear trend or quadratic polynomial. Alternative models of drift correspond to alternative MCDC methods: for example, a linear trend corresponds to linear-method MCDC (see below).

Different statistical models of drift (step 1) correspond to different MCDC methods. In this paper, we apply three MCDC methods: integrated-bias-method MCDC, linear-method MCDC, and agnostic-method MCDC.

Integrated-bias-method MCDC can be applied to ΔE and ΔH, both of which are derived by cumulatively integrating a flux (E′ or H′). MCDC is first applied to the flux by modelling the “drift” as a constant: in other words, we assume that the flux contains a constant bias. This application of MCDC to the flux could be referred to as Monte Carlo bias correction. Each bias-corrected E′ (or H′) time series is subsequently integrated cumulatively to produce an integrated bias-corrected ΔE (or ΔH) time series.

Linear-method MCDC models drift in a state variable (e.g. ΔE, ΔH, or ΔZ) using a linear trend. In other words, we assume that the underlying drift is linear. For ΔE and ΔH, the assumption of linearity is consistent conceptually with the assumption that the underlying flux contains a constant bias. The assumption of linearity will likely contribute to an underestimation of drift uncertainty (Sect. 5.2).

Agnostic-method MCDC relaxes the assumption of linearity. In addition to sampling the uncertainty associated with the parameters of a given statistical model, agnostic-method MCDC also samples the uncertainty associated with the choice between alternative statistical models: we assume that linear, quadratic, and cubic models of drift are equally valid. This corresponds to the common practice of selecting one of these alternative statistical models a priori (Sect. 3.1). For each of the three models of drift, we draw 500 drift samples. We then combine these samples, producing 1500 drift samples in total.

When applying a quadratic or cubic model of drift, the branch time must be known: which part of the control time series parallels the historical time series? For earlier generations of global climate model ensembles, this branch-time metadata may have been either unavailable or unreliable . To address this, proposed an alternative method to identify the branch time. Here, we assume that the branch-time metadata that accompany the CMIP6 data are reliable. When defining and fitting each statistical model of drift, we use the year 1850 as the reference year that corresponds to the origin.

An example of drift is illustrated in Fig. 1. For UKESM1's control simulation, the top-of-atmosphere radiative flux (E′) is close to zero (Fig. 1a), consistent with quasi-equilibrium radiative balance at the top of the atmosphere. The UKESM1 team have achieved their stated goal of keeping E′ close to zero when tuning the model . Therefore, the drift in excess system energy (ΔE) is small (Fig. 1c).

However, the sea surface heat flux (H′) has a negative bias (Fig. 1b), showing that the ocean loses heat spuriously. When we integrate the flux cumulatively, the negative bias in H′ integrates into a negative trend in excess ocean heat (ΔH; Fig. 1d). This illustrates the connection between bias and drift: a constant bias in a flux will drive a linear trend in the cumulatively integrated flux . In this example, the drift in ΔH also dominates the historical time series, obscuring the anthropogenic warming signal during the 20th century (Fig. 1d). Anthropogenic forcing only becomes strong enough to offset the negative bias in H′ towards the end of the 20th century (Fig. 1b).

If energy is conserved within the modelled climate system, the ΔE–ΔH relationship should be approximately linear (Eq. A3). In contrast, for UKESM1's historical simulation, the drift in ΔH drives a strange ΔE–ΔH relationship (Fig. 1f): for most of the historical period, ΔE remains close to zero, but ΔH decreases. This reveals energy leakage. Furthermore, the inconsistency between ΔE and ΔH reveals that the energy leakage occurs somewhere between the top of the atmosphere and the sea surface i.e. outside the ocean;.

In this example, thermosteric sea level rise (ΔZ) also exhibits drift (Fig. 1e). The drift in ΔZ obscures the anthropogenic sea level rise signal during the 20th century. The drift will also contaminate future projections. Furthermore, the ΔH–ΔZ relationship (Fig. 1g) is inconsistent with Eq. (A4).

Figure 2 illustrates the application of MCDC to UKESM1's ΔE time series. For each MCDC method, we produce 1500 drift samples using the uncorrected control time series (Fig. 2a, d, and g). For UKESM1, these drift samples all indicate positive drift during the historical period. We subtract these drift samples from the uncorrected control time series to produce drift-corrected control time series (Fig. 2b, e, and h). In comparison to the uncorrected time series, the trends of these drift-corrected control time series are much closer to zero, as expected. We also subtract the drift samples from the uncorrected historical time series to produce drift-corrected historical time series (Fig. 2c, f, and i).

The spread among the drift samples and the drift-corrected time series indicates drift uncertainty. For UKESM1, the drift uncertainty is largest when integrated-bias-method MCDC is used (Fig. 2a–c). The linear-method drift uncertainty is much smaller (Fig. 2d–f). This demonstrates that integrated-bias-method MCDC and linear-method MCDC produce different estimates of drift uncertainty, even though these two methods have similar underlying assumptions – namely, a constant bias in E′ drives a linear trend in ΔE. We ascribe the difference between the two methods to the size of the standard error: even if autocorrelation is accounted for, the standard error of the trend of ΔE is smaller than the standard error of the mean of E′ because integrating cumulatively effectively averages over the substantial inter-annual variability in E′. Therefore, integrated-bias-method MCDC provides a much weaker constraint on the drift correction. To minimise drift uncertainty, it is preferable to use linear-method MCDC.

However, the underlying drift may be non-linear. If so, a linear model of drift may be inappropriate. Therefore, linear-method MCDC will likely underestimate drift uncertainty. By partially accounting for the possibility of non-linearity, agnostic-method MCDC should provide a more accurate estimate of drift uncertainty. For UKESM1, agnostic-method drift uncertainty is larger than linear-method drift uncertainty yet smaller than integrated-bias drift uncertainty (Fig. 2; Table S2).

We extend our analysis to other global climate models within the CMIP6 ensemble by quantifying the drift uncertainty associated with ΔE during the historical period (2000s relative to 1850s; Fig. 3; Table S2). For all models, the linear-method drift uncertainty is substantially smaller than the integrated-bias-method drift uncertainty. The agnostic-method drift uncertainty is always larger than the linear-method drift uncertainty and is often larger than the integrated-bias-method drift uncertainty. When averaged across the ensemble, agnostic-method MCDC produces the largest estimate of drift uncertainty for ΔE, with a median of 0.09 YJ (Table 1). The ensemble maximum drift uncertainty is 0.17 YJ, approximately twice as large as the ensemble median (Table 1; Fig. 3h).

Differences between global climate models also drive differences in ΔE (Fig. 3). We refer to the inter-model range as model uncertainty . For ΔE, the model uncertainty of 0.56 YJ is more than 3 times as large as the ensemble maximum drift uncertainty (Table 1).

For each CMIP6 model, the drift uncertainty associated with ΔH (Fig. S1; Table S3) is similar to the drift uncertainty associated with ΔE. The ensemble statistics are very similar to those of ΔE (Table 1). Therefore, we can draw very similar conclusions for the ΔH results as we have for the ΔE results. This is not surprising because ΔH is closely related to ΔE: both consist of cumulatively integrated fluxes, and ΔH should track ΔE closely if the fraction of excess energy absorbed by the ocean (η) is close to 1.0 (Sect. 4.5; Appendix A).

For ΔZ over the historical period (2000s relative to 1850s), the agnostic-method drift uncertainty is always larger than the linear-method drift uncertainty, as expected (Fig. S2; Table S4). (Integrated-bias-method MCDC is not applicable to ΔZ, which does not consist of a cumulatively integrated flux.) The agnostic-method drift uncertainty ranges from 3 to 24 mm, with an ensemble median of 10 mm (Table 1). This is smaller than the model uncertainty of 64 mm (Table 1). On the other hand, the drift uncertainty is of comparable magnitude to the impact of omitting volcanic forcing in control simulations: found that neglecting pre-industrial volcanic forcing leads to an underestimate of 5–30 mm over the period 1850–2000. Furthermore, for models with relatively large drift uncertainty, the drift uncertainty may influence the extent to which the historical time series agrees with estimates of ΔZ derived from observations, such as the reconstructions of and . For two CMIP6 models, the agnostic-method drift uncertainty is large enough to obscure the positive ΔZ signal during the historical period: the 2nd–98th percentile range includes negative values of ΔZ (Fig. S2a and n).

If energy is conserved and if the fraction of excess energy absorbed by the ocean (η) is constant, then excess ocean heat (ΔH) should be a linear function of excess system energy (ΔE): ΔH=ηΔE (Eq. A3). However, as noted in Sect. 4.1 above, the uncorrected ΔE–ΔH relationship for UKESM1's historical time series is non-linear and reveals energy leakage (Fig. 1f). Can drift correction address this problem? After we apply agnostic-method MCDC, the trend-corrected ΔE–ΔH relationships are linear (Fig. 4a–b). Estimates of the coefficient η are positive and less than 1.0 (Fig. 4c). Therefore, for UKESM1, the drift-corrected ΔE–ΔH relationships are generally consistent with Eq. (A3).

When applying MCDC, we produce 1500 drift-corrected time series for each variable, scenario, model, and MCDC method. Therefore, we also calculate 1500 estimates of η, so we can quantify drift uncertainty. For all three MCDC methods, the drift uncertainty in η is relatively large during the historical period (Figs. 4c and 5): the relatively weak global warming signal in the ΔE and ΔH historical time series leaves them susceptible to drift uncertainty, which in turn influences the η estimates. The drift uncertainty in η is much smaller during the 21st century projection period (Figs. 4c and 5): global warming drives clear trends in the ΔE and ΔH projection time series, constraining the η estimates. We average across the projection simulations to summarise the drift uncertainty in η (Table S5; Table 1). For UKESM1, the agnostic-method drift uncertainty is approximately 0.01 (Table S5). Across the CMIP6 ensemble, the agnostic-method drift uncertainty ranges from 0.01 to 0.14, with a median of 0.06 (Table 1).

The η estimates differ slightly between the different projection simulations (Figs. 4c and 5). The differences arise from the response of the climate system to different forcing scenarios. We refer to the inter-scenario range as the scenario uncertainty. For the UKESM1 projection simulations, the scenario uncertainty is approximately 0.01, similar to the drift uncertainty (Table S5). For most models in the CMIP6 ensemble, the scenario uncertainty is similarly small (Fig. 5; Table S5), with an ensemble median of 0.01 (Table 1): in general, η does not depend strongly on the scenario. However, for two models, the scenario uncertainty is 0.08 and 0.07, respectively (Table S5; Fig. 5c–d): for these models, η is sensitive to the choice of scenario.

Estimates of η vary between different models within the ensemble (Fig. 5). The model uncertainty of 0.17 is the largest source of uncertainty (Table 1). For most of the CMIP6 models, the η estimates lie in the range 0.9–1.0, consistent with our understanding of the climate system Appendix A;: even if the ocean were to absorb all excess energy entering the climate system, this would lead to a theoretical maximum of 1.0. However, some of the η estimates are larger than 1.0. For four models (25 % of the ensemble), the median η estimates are consistently greater than 1.0 for all projection scenarios (Fig. 5h, k, l, and m): even after drift correction, the energy balance of these models remains problematic.

As the ocean warms, it expands, leading to thermosteric sea level rise. We expect the relationship between ΔH and ΔZ to be approximately linear: ΔZ=ϵΔH (Eq. A4). As noted in Sect. 4.1 above, the uncorrected data from UKESM1's historical simulation are inconsistent with Eq. (A4) (Fig. 1g). However, after we apply agnostic-method MCDC, the drift-corrected ΔH–ΔZ relationships are approximately linear (Fig. 4d and e). Drift correction successfully establishes meaningful ΔH–ΔZ relationships that can be interpreted using Eq. (A4).

However, in disagreement with Eq. (A4), the drift-corrected ΔH–ΔZ relationships during the historical period exhibit hysteresis-like behaviour (Fig. 4d). ΔH and ΔZ are referenced to the 1850s decadal mean, so the ΔH–ΔZ relationships begin near the origin. ΔH and ΔZ then become negative for much of the historical period, driving the relationships towards the lower left of Fig. 4d. ΔH and ΔZ become positive later in the historical period, driving the relationships towards the upper right. However, ΔH and ΔZ do not necessarily both become positive at the same time, so the relationships do not necessarily pass through the origin, as demonstrated by the “max intercept” and “min intercept” lines. It is possible that the ΔE–ΔH–ΔZ relationships may depend on the sign of the forcing see. However, the hysteresis-like behaviour in Fig. 4d – and also in Fig. 4a – does not reveal systemic hysteresis: for different MCDC samples, the intercept can be either negative or positive. Regardless of the explanations for this hysteresis-like behaviour, we use ordinary least squares with an intercept – not regression through the origin – to estimate the linear regression coefficient (Appendix B).

The linear regression coefficient is ϵ, the expansion efficiency of heat. As was the case for η, the drift uncertainty in ϵ is much larger during the historical period than the 21st century (Figs. 4f and S3) due to the comparatively weak forcing signal during the historical period. When drift uncertainty is considered, the historical period provides only a weak constraint on the value of ϵ.

For the UKESM1 projection simulations, the estimates of ϵ depend on the scenario, ranging from approximately 117 mmYJ-1 (SSP1-2.6) to 126 mmYJ-1 (SSP5-8.5; Fig. 4f). This scenario uncertainty of 9 mmYJ-1 is much larger than the agnostic-method drift uncertainty of 1 mmYJ-1 (Table S6), leading to distinct non-overlapping box plots for the projection scenarios (Fig. 4f). For all models in the CMIP6 ensemble, the ϵ estimates are smallest for SSP1-2.6 and largest for SSP5-8.5 (Fig. S3). This suggests that the thermal expansion of the ocean is sensitive to the forcing scenario and is not merely a linear function of ΔH see also. However, the strength of the dependence on scenario varies: across the ensemble, the scenario uncertainty varies from 4 to 10 mmYJ-1, with a median of 7 mmYJ-1 (Table 1).

The agnostic-method drift uncertainty varies over a wider range, from 1 to 22 mmYJ-1, with a median of 7 mmYJ-1 (Table 1). The linear-method drift uncertainty is much smaller, suggesting that linear-method MCDC underestimates the drift uncertainty. The model uncertainty is 12 mmYJ-1. When analysing ϵ across the CMIP6 ensemble, drift uncertainty, scenario uncertainty, and model uncertainty all constitute substantial sources of uncertainty.

Following and , we have considered the energy budgets of global climate models. Inconsistent relationships between excess system energy (ΔE) and excess ocean heat (ΔH; Fig. 1f) are symptomatic of energy leakage outside the ocean domain . In agreement with and , we have shown that drift correction can partially address this problem of energy leakage. Drift-corrected ΔE–ΔH relationships are approximately linear (Fig. 4a and b). For all 16 models analysed here, the coefficient η – the fraction of excess energy absorbed by the ocean – is positive (Fig. 5). For most of these models, η is also less than 1.0, consistent with energy conservation. However, even after drift correction, the energy balance of several models remains suspect: energy leakage is revealed by η estimates greater than 1.0. For four models, the median η estimates are greater than 1.0 for all projection scenarios (Fig. 5).

When evaluating the performance of global climate models, the coefficient η is a useful diagnostic. This diagnostic could be used alongside other diagnostics relating to energy balance, such as the diagnostics considered by and .

previously explored the contribution of drift to uncertainty in global climate sensitivity indices. To do this, added drift (via spurious forcing) and inter-annual variability (via noise) to a simple two-timescale response model. In contrast, we have quantified drift uncertainty directly from global climate model data using Monte Carlo drift correction (MCDC).

We have compared alternative methods of correcting drift in cumulatively integrated fluxes (such as ΔE and ΔH). When applying linear-method MCDC, the flux is first integrated cumulatively, and then the spurious trend is quantified and subtracted. When applying integrated-bias-method MCDC, the bias in the flux is first quantified and subtracted, and then the bias-corrected flux is integrated cumulatively. Conceptually, these two alternative approaches are similar: a constant bias in flux (e.g. E′) will drive a spurious trend in cumulatively integrated flux (e.g. ΔE) . In practice, however, we find that these two alternative methods produce differing results: the linear-method drift uncertainty is smaller than the integrated-bias-method drift uncertainty because integrating cumulatively effectively averages over the substantial inter-annual variability, resulting in a smaller standard error. In other words, compared with integrated-bias-method MCDC, linear-method MCDC provides a much stronger constraint on the appropriate drift correction to apply. When correcting drift in an integrated variable, it is generally preferable to integrate the variable first and then apply the drift correction afterward. Nevertheless, we recognise that there may be cases for which it makes sense to correct the bias before integrating: for example, when performing an analysis that involves both E′ and ΔE, we may prefer to use the integrated bias method for consistency. Researchers should clarify whether they correct drift before or after integration. Although we primarily refer to cumulative integration across time here, such clarity should also extend to other calculations: for example, clarify that they calculate spatial integrals before correcting drift.

When estimating drift uncertainty, our preferred method is agnostic-method MCDC. If the drift is non-linear, then linear-method MCDC will likely underestimate the drift uncertainty. Therefore, agnostic-method MCDC – which assigns equal prior probabilities to linear, quadratic, and cubic models of drift – should provide a more accurate estimate of drift uncertainty. For the variables analysed in this paper, we have found that agnostic-method drift uncertainty is larger than linear-method drift uncertainty, as expected. We emphasise that agnostic-method corresponds to an assumption that linear, quadratic, and cubic models of drift are all equally plausible. This assumption corresponds to the common practice of selecting a statistical model of drift a priori before fitting the model to data (Sect. 3.1).

A possible further step would be to fit and compare alternative statistical models of drift a posteriori using measures such as the Bayesian information criterion. We could then select the best statistical model(s) for a specific time series. When applying this “best-fit” approach, we could also consider additional statistical models of drift, including signal processing filters e.g.. This best-fit approach would be a sensible way to correct drift. Application of the best-fit approach should lead to a reduction in drift uncertainty.

It may be important to quantify drift uncertainty when analysing time series with a relatively weak trend. In particular, historical time series may be susceptible to drift uncertainty. For the historical period, the agnostic-method drift uncertainty in ΔZ can be as large as 24 mm, which is of comparable magnitude to the impact of omitting volcanic forcing in control simulations . Therefore, drift uncertainty deserves similar attention to that given to volcanic forcing.

We hypothesise that drift uncertainty may influence the extent to which a historical simulation agrees with an observation-based estimate. Drift uncertainty should be considered when comparing historical simulations with observation-based estimates of thermosteric sea level change such as the reconstruction of. Drift uncertainty should also be considered when using ocean-related variables – such as ocean heat content  – as an emergent constraint for global climate models.

We further hypothesise that drift uncertainty may be large for drift-susceptible time series that exhibit large internal variability, such as regional dynamic sea level . Large variability will lead to a large standard error when fitting the statistical model of drift. Therefore, drift uncertainty should be considered when analysing time series with large internal variability, provided that the time series is known to be susceptible to drift.

On the other hand, drift is often weak for time series unrelated to the deep ocean . Therefore, many analyses – such as multi-model mean hemispheric partitioning of upper-ocean heat – may be insensitive to drift . In such cases, drift correction may be unnecessary and drift uncertainty may be ignored.

In this study, we have focused on global variables. When correcting drift in multivariate regional contexts – such as might be done in preparation for dynamical downscaling – drift correction approaches may be informed by additional considerations to ensure consistency . Furthermore, we have focused on correcting drift in long-term climate simulations, which are dominated by external forcing. In contrast, shorter-term decadal simulations are further complicated by observation-based initialisation; hence, decadal simulations require different drift correction methods . The quantification of drift uncertainty in such contexts remains an open question.