In the study of climate change, it is crucial to explore the response of the Earth system to a variety of possible future greenhouse gas emission scenarios and quantify the uncertainties associated with future projections. Despite their limitations, state-of-the-art Earth System Models (ESMs), such as those participating in the Climate Model Intercomparison Project CMIP, , remain our most reliable tools for estimating the Earth system response to increased greenhouse gas concentrations. These large-scale models aim to represent as many aspects of the climate system as faithfully as possible. However, because of the high computational and material costs of running ESMs, these models can only simulate the Earth system response to a few potential future scenarios . On the other hand, studies of climate mitigation and adaptation strategies often seek to explore a wide range of possible solutions, creating a need for methods to compare localized impacts across a wide range of emissions scenarios . Our work provides a framework for extending projections of future climate scenarios beyond those simulated with ESMs. The method captures spatially coherent changes at any scale in both the mean and variance of climate variables in response to user-defined warming scenarios.

Emulators of ESMs have been gaining popularity as a way to facilitate access to climate projections. Here, we consider the class of emulators trained on the output of ESMs to cheaply and quickly predict the climate response to arbitrary climate change scenarios. We will not consider other types of emulators that are being actively pursued for other applications, including climate model downscaling, parametrization of subgrid-scale processes, and model parameter calibration e.g,. From here on, the term emulator will refer exclusively to the class of emulators trained to extend ESMs to arbitrary future scenarios. The importance of these emulators is likely to rise due to the increasing computational demands of ESMs: their ever refining spatial resolution, their growing complexity (as embodied by the number of model components and their sophistication), the use of more accurate numerical methods , and the push to run larger ensembles of simulations differing in initial conditions . Emulators are necessary to both compress existing information into a more manageable form as well as to bridge the gap between the computational demand of running a full ESM and the computational hardware available to everyday consumers of climate information. This need is best illustrated by the evidence that projections of future emissions are updated every year, while a new deck of ESM simulations by CMIP takes close to a decade to generate and organize for community sharing.

The simplest and most common emulation technique is pattern scaling . Pattern scaling estimates spatially resolved changes in climate variables by linearly regressing local variables on global mean surface temperature or cumulative emissions. While pattern scaling has been shown to have remarkable skill in emulating projected changes in local mean temperatures , it has no inherent probabilistic component and is less accurate for other climate variables . Recent work has been dedicated to constructing machine learning-based climate emulators e.g. to move beyond the linear assumption inherent in pattern scaling. However, questions remain about their overall skill compared to linear pattern scaling . Additionally, their lack of interpretability and the computational cost of model training may limit their applications. The emulator discussed in this work is rooted in the paradigm of statistical emulation as a function of globally averaged surface temperature (e.g., pattern scaling) rather than in the alternative paradigm of autoregressive emulation that predicts the evolution of climate variables or of their statistics as a function of initial conditions (e.g., autoregressive machine learning models). In particular, we aim to emulate spatially averaged second-order statistics of variables of interest given a specified future value of globally averaged surface temperature.

Our focus on emulating variability stems from the recognition that a shift in a climate variable is significant only if it lies outside the bounds of the internal variability inherent in the system. Internal variability has been shown to dominate uncertainty in climate projections over a decade for regional temperature and over much longer periods for other variables, like regional precipitation . The concept of time of emergence (ToE) is used to quantify when a climate signal becomes statistically distinguishable from internal variability . In its simplest form, ToE is defined as the first time a climate variable (e.g., ensemble mean temperature globally or at a point) exceeds a threshold derived from internal variability (often the ensemble variance), which may itself evolve under climate change. The interest in ToE and related questions (such as climate attribution and projection uncertainty) underpins the need for emulators to capture both mean trends and internal variability.

Significant work has been dedicated to the extension of linear pattern scaling to account for both internal variability in the climate system and structural uncertainty in ESMs. A common approach is to take a large ensemble of climate simulations, either from a single model or from a multi-model set, and apply pattern scaling separately to each simulation, deriving a measure of uncertainty from the resulting ensemble spread . Generally, examining ensemble spread within a single model is among the most reliable ways of quantifying the internal variability of the climate system, while inter-model spread includes structural uncertainty . Other statistical emulators have gone beyond pattern scaling to explicitly include stochastic components to mimic the climate system's inherent variability . For example, researchers have augmented pattern-scaling frameworks with residual resampling or autoregressive processes, ensuring that emulator outputs exhibit the same spatio-temporal covariance structure and ensemble spread as the original ESM runs . The MESMER emulators have applied approximate methods (Gaussian copulas) to compute the covariances among different climate variables and locations.

Our emulator design for averaged climate fields marries the pattern scaling approach with a statistical model of internal variability that can account for spatial and cross-variable correlations. This is achieved with a generative data-driven emulation method for spatially resolved climate fields, applied in this work to monthly temperature and relative humidity. As a first step, the ESM training data is projected into a reduced-dimensional space using empirical orthogonal function (EOF) decomposition . We then model the evolution of the means and covariances of the data in this reduced-order space as a function of global mean surface temperature. To recover projected statistics, a reverse transformation is applied that maps the means and covariances of EOF amplitudes into the means and variances of climate variables in physical space. In designing our emulation approach, we prioritized computational efficiency in order to improve the accessibility of the emulator for users and applications without access to significant computational resources.

The main innovation of our method is the balance of simplicity alongside robustness in modeling the lower-order statistics of the system. This includes modeling the full EOF covariance matrices, which are necessary to compute changes in variance and correlations of climate variables over arbitrary geographical regions. Covariances between locations are required to estimate variability on a region beyond the size of an original model grid cell (e.g., the variability over Sub-Saharan Africa is much larger if all regions experience the same temperature fluctuations than if they are all independent). Additionally, covariances between different variables are essential for quantifying the impact of internal variability on climate projections and to assess compound risks (e.g., high temperature and relative humidity are more problematic than high temperature and low relative humidity).

The climate emulator literature includes models on many timescales, ranging from yearly to monthly to daily. The temporal resolution required depends on the specific application of the emulator. In this work, we use monthly data for illustrative purposes; our approach to emulation could be applied to other temporal resolutions as well, as in . Projections of monthly climate statistics are important for understanding a number of impacts of climate change, such as changes in the seasonal cycle, water management, and other phenomena of agricultural relevance . Furthermore, some emulators with finer temporal resolution, such as the one in , require monthly projections as input; our emulator could thus be used as pre-conditioning information. Importantly, by emulating statistics beyond the mean, our emulator provides additional information that may be used to condition downstream models or aid in decision-making (e.g., via uncertainty quantification through emulated ensemble spread). Such a modular emulation pipeline is common, as illustrated by the MESMER family of emulators .

We condition our emulator on the global, ensemble, and yearly mean surface temperature, which we refer to as “global mean temperature” for short. This procedure is common among other emulators of spatially-resolved climate variables e.g.,. Global mean temperature has been found to scale approximately linearly with cumulative emissions , at least for smoothly-growing emissions and ignoring, e.g., the time-lagged response to radiative forcing, or the impact of short-lived aerosols and nonlinear feedbacks like those from melting ice. This simple relationship underpins the utility of a global mean temperature-based emulator. However, one could easily go further by relying on Simple Climate Models e.g. to translate arbitrary emission pathways into novel trajectories of global mean temperature. Our emulator would then use these trajectories to generate realizations of spatially resolved monthly variables under those emission scenarios, continuing in line with the modular emulation framework. We comment that often the global mean temperature anomaly from a preindustrial period is used rather than the actual global mean temperature, but here we will use the absolute global mean temperature.

A sufficiently large ensemble of initial conditions is necessary to infer non-stationary statistical information. For this reason, we choose to emulate the evolution of climate variables generated by a CMIP-class model, specifically MPI-ESM1.2 LR (v1.2.01p7) , that ran a large ensemble (50 members) of simulations for a number of emissions scenarios (see Sect.  for details). We emphasize that throughout this work, when referring to an ensemble or ensemble mean, we are considering only a single-model ensemble (e.g., an initial-condition ensemble). We do not intend for these methods to be applied to multi-model ensembles. We illustrate our approach for two variables: surface temperature and surface relative humidity. Still, the approach is agnostic to the variables being emulated. It can easily be applied to any variables from a single ESM ensemble, so long as their EOF amplitude statistics are mostly described by their means and covariances. While in this work we emulate each of the two variables separately, our approach is also applicable to the joint emulation of several variables with minimal modification, which we discuss in Sect. .

Our paper is organized as follows: in Sect. , we introduce the dataset used in this work. Sect.  discusses the EOF-based coarse-graining procedure applied to the data. In Sect. , we discuss the details of the emulator design and the regression problem. In Sect. , we show the emulator's ability to replicate the data's statistics under climate change. In Sect. , we illustrate one possible application of the emulator using a case study. Finally, in Sect. , we discuss the broader implications of this work and propose future directions for constructing complementary emulator models.

To train our emulator, we use the output from the MPI-ESM1.2-LR (v1.2.01p7) ESM model , which contributed to CMIP6. We chose this model because of the large number of simulations (ensemble members) it ran for each emission scenario: 50 simulations are run for each scenario, differing only in their initial conditions. A large ensemble is necessary to separate a model's internal variability from the anthropogenic signal . In the CMIP6 model set, only three ESMs submitted ensembles of 30 or more members: MPI-ESM1.2-LR, EC-Earth3, and CanESM5. Among these, the MPI model is the only one with an equilibrium climate sensitivity to greenhouse gas emissions within the “likely” range determined by multiple lines of evidence and with the entire ensemble available for open download.

Analysis performed in on the same MPI dataset suggests that an emulation method that uses polynomial fitting for projecting changes in climate variables as a function of space, such as ours, captures the evolution of the ensemble mean variables even if trained on a few ensemble members (in terms of emulation error). However, an ESM with many ensemble members is necessary to train the emulation of internal variability . How many members constitute a “large enough” ensemble will depend on the particular variable(s) being targeted and the application being considered. For present purposes, the ESM training data must contain sufficient ensemble members to accurately represent the climate internal variability, i.e., the variance and cross-correlation of the climate variables of interest. finds that an ensemble size of at least 10 members is necessary to capture the internal variability of temperature and precipitation. This is consistent with our finding that 30 ensemble members are sufficient for the training of our emulator.

Each MPI-ESM1.2-LR ensemble member is run for the historical period, spanning 165 years between 1850–2014, and for various future warming scenarios spanning the 86 year future period 2015–2100. We consider output from simulations of three future scenarios from the ScenarioMIP experiments: SSP5-8.5, SSP2-4.5, and SSP1-1.9. The ScenarioMIP experiments are plausible futures corresponding to different climate mitigation and cooperation narratives . Figure  reports the global mean temperature profiles of the historical period and the three future scenarios considered in this work for each of the 50 MPI-ESM1.2-LR ensemble members.

Our goal is to develop an emulator that predicts changes in the mean and covariance of climate fields as a function of emissions. The first step in the process is to represent the statistics as a function of (or, more appropriately, conditioned on) global, ensemble, and yearly mean temperature and hence cumulative emissions (see ). As mentioned above, the ensemble mean is calculated over a single model with multiple realizations. Following standard practice, we will refer to the global, ensemble, and yearly mean surface temperature as “global mean temperature” throughout the text, denoted T‾g.

We select the historical experiment and the SSP5-8.5 high-warming scenario for training the emulator because together they span the widest range of global mean temperatures. This leaves SSP2-4.5 and SSP1-1.9 as the validation sets for regression, see Sect. . SSP2-4.5 features milder monotonic warming that levels off at the end of the century (through the elimination of emissions), while SSP1-1.9 features a peak in global mean temperature around mid-century followed by a decrease to end-of-historical-period temperatures by 2100 (as a consequence of negative emissions), see Fig. . Because we are developing an emulator conditioned only on global mean temperature from a scenario with monotonically increasing emissions (without accounting for emissions history or other memory effects), it is important to test its performance in scenarios with non-monotonic emissions. Such scenarios are of ever-increasing interest to mitigation and adaptation studies and policy. We emphasize that our approach is purely data-driven and should not be used to extrapolate statistics in climates outside of its global mean temperature training range.

To presume that time-dependent climate statistics for different emissions scenarios can be parameterized by a state-dependent (i.e., time-independent or history-independent) scalar quantity is a strong assumption but one that is justified a posteriori for the cases considered in this work, see Fig.  and, later on, Figs.  and . Figure  shows histograms of surface temperature data in the MPI model for selected locations (top row) and hemisphere averages (bottom row) in January (left) and July (right). Two 5 year windows in the SSP1-1.9 future scenario are compared: 2018–2022 (in blue) and 2093–2097 (in red). The SSP-1.1.9 scenario is non-monotonic, and these two periods correspond to similar global mean temperatures of around T‾g = 289.3 K, separated by an intermediate 70 year period of higher temperatures. We observe that, for the most part, the histograms for the two periods overlap. We thus conclude that a state-dependent parametrization of climate statistics is not unreasonable for our purposes.

Before delving into the specific details of the emulator formulation, it is crucial to first understand the underlying rationale guiding our approach. Our key assumption is that the statistics of climate system fields, in our case the EOF modal amplitudes a (see Sect. ), can be represented by a probability density for every possible state, with time and emissions history replaced by the global mean temperature T‾g and seasonal information such as the month m: 1ρ(a,t| emissions)→ρa|T‾g,m.

This assumes that conditioning on global mean temperature serves as an informative parametric form to characterize the changing distribution of the climate-relevant quantities. In the application described in the following sections, we do not consider the full probability distribution and instead assume that only the first two moments – means and covariances – can be represented as functions of global mean temperature.

We illustrate our approach for monthly mean surface (2 m) temperature and monthly mean surface (2 m) relative humidity. These are the “tas” and “hurs” variables, respectively, in the CMIP6 nomenclature. The MPI-ESM1.2-LR model has a horizontal resolution in the atmosphere of 1.8°. We use the model output on its original 192 × 96 lat-lon grid. We calculate the global mean temperature (used to condition the emulator) by globally averaging the 2 m temperature variable.

We coarse-grain the representation of the temperature and relative humidity fields, using Empirical Orthogonal Functions (EOFs) as a dimensionality reduction technique ; see for a comprehensive review of the technique's history. EOF modes have been shown to effectively capture the dominant patterns of variability of the Earth system . EOF decomposition has been used in previous emulator work for both dimensionality reduction and more generally as a method of generating an uncorrelated projection basis . More recently, and have demonstrated how modal amplitudes of large-scale patterns (under the additional assumption that they can be approximated as multivariate Gaussian distributions) can be used to interpret patterns of variability and teleconnections through data-driven approaches.

The EOF modes are ranked according to the fraction of overall variability they capture. The leading EOF modes represent patterns that span large geographical regions. While EOF modes can, with some limitations , be interpreted physically, here they are only used as a dimensionality reduction technique. Using a subset of leading EOF modes as a fixed-in-time orthogonal basis for the projection of ESM data, we shall model the statistics (means and variances) of EOF amplitude coefficients as a function of global mean temperature.

We compute the EOF basis ϕ through a singular value decomposition of our data in the historical period of one of the ensemble members. The resulting basis constitutes 165 × 12 = 1980 EOFs ordered by the magnitude of the singular values. We discard the last 980 basis functions, leading to a total of 1000 basis functions ϕi(x),i∈[1,1000]. The notation (x), which we use throughout this work, indicates that EOF basis functions are spatially resolved, i.e., functions of the location x. We use the same basis set for all months and compute EOFs separately for each variable of interest. We project data from every scenario and every ensemble member onto our constant basis. Because we are using the EOF modes as a basis for dimensionality reduction, the choice of using only one ensemble member to generate the basis and keeping the basis fixed throughout the emulation process significantly reduces the computational cost of the emulator. The specific ensemble member used was found to not impact the performance of the resulting emulator. We further emphasize that because the EOFs are used as basis onto which data are projected, we do not require any assumptions about the structure of the EOFs or their stationarity over time. Indeed, the method described in the following section is agnostic to the specific choice of dimensionality reduction technique.

Although in this work we emulate each variable separately, our method may also be used to emulate the joint evolution of two or more climate variables. To do so, one would only need to concatenate the multiple variables into one super-variable (after, perhaps, properly normalizing them) before proceeding with the emulation technique as described for a single variable, now on the super-variable. The only difference between the single- and multiple-variable cases is the need for extra care in constructing the EOF basis. When performing a singular value decomposition, the resulting modes are ranked in order of decreasing percent of variance captured. For a single variable, the leading EOFs are thus easily interpretable as a sort of set of dominant modes. In contrast, in the multiple-variable case, the modes will capture the dominant modes of joint-variable variability, but not necessarily those of the individual variables. Depending on the particular set of variables and intended applications, one must assess whether the resulting EOF basis is capturing the desired variability and how many leading modes one should accordingly retain. We emphasize, however, that the purpose of the EOF decomposition is dimensionality reduction, and any set of basis functions is suitable for the purpose of emulation, but a poor choice of EOFs may result in a useless, albeit accurate, emulator.

The choice to work with the leading EOFs is not only one of convenience to reduce the data size. It is also motivated by evidence that the projections of climate models are more reliable at larger scales . Finer-scale details captured by high-order EOFs–such as full temperature distributions at a point–are better modeled using different approaches, such as downscaling from coarse-grained information. In probabilistic jargon, we divide the set of modal amplitudes a into modes corresponding to large-scale coarse structures aC and “fine-scale modes” aF. The probability distribution for climate variables (for a fixed global mean temperature T‾g and month m) can then be decomposed as the product of two conditional distributions: 2ρaC,aF|T‾g,m=ρaF|aC,T‾g,mρaC|T‾g,m.

Our work focuses on approximating the means μ and covariances C of ρ(aC|T‾g,m), the distribution of aC as a function of the global mean temperature T‾g and month m. Armed with the means and covariances of the modes aC, one can then reconstruct means and variances of the original fields both locally or averaged over some area, as we will see in Sect. . In addition, if the statistics of the variables of interest are Gaussian, means and covariances are sufficient to reconstruct the whole probability density functions (see Appendix ).

Approximating fine-scale structures conditioned on coarse-grained variables, i.e., approximating ρ(aF|aC,T‾g,m), is left to future work. In downscaling exercises, it is typically assumed that 3ρaF|aC,T‾g,m≈ρ(aF|aC), i.e., information about the coarse scales may be sufficiently informative to parameterize the distribution of the fine scales.

After projecting the ESM data into the EOF space, we model the EOF coefficients as a function of global mean temperature see also and proceed to emulate their means and covariances. The method can be applied to any variables regardless of their distributions insofar as one is only interested in the means and covariances of any linear functional of the variable field, see Sect. . Despite not requiring a Gaussian approximation, we will often overlay histograms of variables from the MPI model with Gaussian distributions with the means and covariances generated by our emulator. This is an arbitrary, yet somewhat constrained choice because a Gaussian distribution is among the most natural ways to visualize a distribution parameterized solely by its mean and variance. It so happens, however, that the EOF amplitudes of monthly average temperature and relative humidity have approximately Gaussian distributions. This is to be expected from the multivariate version of the central limit theorem, which states that any variable that is the average of many independent realizations of the same process has a Gaussian distribution. In our case, we average in time, over a month, and in space by the coarse-graining implied by the EOF decomposition. See Appendix  for further discussion.

We are now ready to describe our emulation approach in detail. Section  describes the procedure for fitting to data, and Sect.  describes how to utilize the emulator and its relation to pattern scaling.

Now that we have described the design of our emulator, it is time to validate its performance. The emulator is trained on the historical period and the SSP5-8.5 future scenario of the MPI ESM-1.2-LR model. The testing is performed by (a) applying it to other scenarios run by the MPI model that were not used in training and (b) using it to compute various statistics of interest. We use the SSP2-4.5 and SSP1-1.9 as our “test” scenarios.

Thus far, we have only applied the emulator to reproduce SSP scenarios to test its skill in generalizing to scenarios not used in the training. In practice, however, the main application of an emulator is in quickly and cheaply inferring the behavior of an ESM in arbitrary scenarios. Figures  and illustrate a case study for such an application, motivated by the idea of the time of emergence (ToE) of a climate signal. We define an arbitrary new climate change scenario in terms of the global mean temperature. This scenario, which is shown in the first column of Fig. , grows exponentially at a rate significantly slower than that of SSP5-8.5, with an end-of-century global mean temperature 3 °C above the 1850–1900 average. One could imagine that this scenario is the result of a slight reduction of emissions, driven by changing economic incentives, but without strong international cooperation on climate goals. If we follow the course set out by this scenario, how soon would we be able to see the effects of emission reduction relative to the worst-case path of SSP5-8.5? In other words, when will the diverging scenarios yield significantly different climate statistics, and how will this vary across different locations on Earth? Questions such as these are relevant as they can help quantify the regional impact of climate policy over time. Similar questions have been explored in the BRACE project , but focusing only on two pre-defined scenarios.

To answer these questions, we examine the statistics for temperature and relative humidity at two geographic locations: (40.1° N, 88.1° W) in the Midwest United States, near Chicago, and (23.3° N, 84.4° E) in Jharkand, in northeastern India. Each region spans four cells on the ESM numerical grid. In the second and third columns of Fig.  we show the respective distributions of temperature (top) and relative humidity (bottom) at the end of the century (in 2100) for all months. We plot, for both SSP5-8.5 and the new scenario, the mean monthly values ± 2 standard deviations as estimated by the emulator. For both locations, the variance provides critical context for assessing whether a change in mean is significant. Note that, for both variables, the difference between the two scenarios is much more prominent in NE India. In contrast, in the Midwest US, the differences between the scenarios are largely within the range of internal variability of the model; e.g., although the mean relative humidity in August appears to change, accounting for variance suggests that this change is not statistically significant. The emulator additionally resolves the impact of the two scenarios by seasons; e.g., for relative humidity in NE India, there is a significant difference between the scenarios only in the winter months November-April. It would be impossible to resolve such nuances without the emulated covariances, which allow us to calculate the standard deviations of variables over arbitrarily sized regions.

Our emulator can further be used to quantify how soon the effects of a reduction in emissions will be felt, which we illustrate in Fig. . The left and right columns of Fig.  show, for NE India and the Midwest US, respectively, the evolution of projected January mean temperature (top) and relative humidity (bottom) under SSP5-8.5 (in red) and the new scenario defined in Fig.  (in blue). The bands correspond to ± 2σ over the regions, as projected by the emulator. The figure illustrates that towards the end of the century, the two scenarios begin to significantly diverge beyond what is naturally expected from the local climate variability in NE India, but not in the Midwest US. We might consider the time when one observes a divergence beyond natural variability as the time of emergence of the effects of climate mitigation in the new scenario. In the middle column, we quantify this notion using the Kullback-Leibler divergence, which we proceed to describe below.

We are interested in the change over time (under a particular climate change scenario) of some distribution d1 of a climate variable, computed over a region of interest. In particular, we would like to measure the difference between this distribution in the given scenario and the same distribution d2 in a reference scenario (e.g., business-as-usual, or SSP5-8.5). At which point in the future would we be able to say that the two scenarios are significantly different? This question is essentially a generalization of the question of the ToE of a climate change signal. To answer it, one might look at when the mean of d1 is one or two standard deviations away from the mean of d2. We instead use the Kullback-Leibler (KL) divergence to quantify at every point in time the difference between the two distributions. The KL divergence is a widely used information-theoretic measure related to Shannon entropy . For two general continuous distributions P and Q, the KL divergence D is defined as 19D(P,Q)=∫-∞∞p(x)log⁡p(x)q(x)dx.

In order to apply the divergence to our case study, we thus must assume a form of the distribution parameterized by the means and covariances given by the emulator. Following the plotting convention adopted throughout this work, we assume a Gaussian form for the distributions. For two Gaussian distributions d1 and d2, the KL divergence simplifies to 20D(d1,d2)=ln⁡σ2σ1+σ12+(μ1-μ2)22σ22-12 where d1∼N(μ1,σ12) and d2∼N(μ2,σ22). Note that the KL divergence is an asymmetric measure, and Eq. () quantifies the divergence of d1 from d2. Although D is dimensionless, one can use 0.5 and 2 as approximate benchmarks for 1σ and 2σ significance, respectively. We obtain these values by setting σ1=σ2 and μ1=μ2-σ2 to yield D=0.5. Similarly, setting σ1=σ2 and μ1=μ2-2σ2 yields D=2.

The middle column in Fig.  shows, for both locations, the time evolution of the KL divergence between the new scenario and SSP-8.5 distributions for the January surface temperature (top) and relative humidity (bottom). In both locations, the temperature differences between scenarios are more significant than the relative humidity ones. For both temperature and relative humidity, the effects of the emission reduction would be more significantly felt in NE India, as evidenced by higher values of D, in agreement with visual assessment of the distributions. These effects would also be felt sooner: if we consider D=0.5 to be the approximate significance threshold for a 1σ change, a significant change in NE India would be felt by 2068, while in the Midwest US it would still not be approached by 2010 when the divergence reaches a maximum value of D=0.44. The same is true for relative humidity: in NE India, the effect of emissions reduction on relative humidity would be felt both sooner and more significantly. We would observe a significant change around 2080 in NE India, and not before the end of the century for the Midwest US, with a maximum end-of-century value of D=0.22.

In this case study, we have demonstrated how our emulator may be used to assess the effects of a reduction in emissions on regional climate outcomes. In a similar vein, our emulator could also be used to assess the ToE of a particular climate signal by comparing a possible future scenario with a baseline no-warming scenario. Our emulator is well-suited to such policy and impact assessment problems. The modeling of EOF covariances allows for the seamless calculation of variances across spatial scales, providing crucial information to quantify significance and uncertainty. Furthermore, our model is lightweight and computationally efficient, meaning it can be run almost instantaneously to iterate on different scenarios. These two features make it an ideal tool for policymakers, educators, and scientists looking to quickly explore outcomes of widely varying future climates.

We have introduced an emulator for the evolution of the means and covariances of climate variables and applied it to spatially resolved monthly averaged temperature and relative humidity. This emulator provides a simple and computationally efficient method for extending the MPI-ESM1.2-LR global climate model to arbitrary warming scenarios while retaining the ability to quantify the significance of trends by modeling covariances. Computational efficiency is achieved by limiting the emulation to means and covariances and by compressing the data onto the leading 1000 EOF modes of variability, reducing its memory footprint by two orders of magnitude compared to the training dataset. This dimensionality reduction is not a significant limitation, as Earth System Models have substantially greater fidelity at large scales than at fine spatial resolution.

We have focused on accurately capturing the internal variability of the original ESM and flexibly generalizing it to previously unseen warming scenarios. The internal variability provides critical context for the assessment of projected changes in the climate. Thanks to our modeling of the full EOF covariance structure, the design of our emulator allows for the seamless calculation of second-order statistics across regions of any size, allowing for easy calculation of regional-scale variability. We assume that EOF covariances vary smoothly with global mean surface temperature. However, this assumption may break down in regions or timescales where localized processes – such as abrupt sea ice loss or monsoon regime shifts – introduce discontinuities in the covariance structure. Capturing such transitions may require future extensions that incorporate non-smooth, regime-dependent components.

Several straightforward extensions of the proposed emulator are possible. These include applying the method to additional climate fields, adopting alternative basis functions, using higher-order regression schemes, or incorporating additional predictors beyond global mean temperature. The framework can also be used to capture correlations between different variables (e.g., temperature and relative humidity) by modeling joint EOF amplitudes or post hoc correlations. Temporal coherence could similarly be introduced, for example, by estimating auto-covariances of EOF amplitudes across months.

The emulator framework we introduced cannot be easily extended to statistics beyond means and covariances. Estimating even the three-point correlation of a high-dimensional distribution becomes cumbersome, requiring the computation and storage of a tensor with N3 points if one uses a basis of N EOF amplitudes. Generally, a multivariate distribution of size N necessitates the storage of Nn points for an n-point correlation, rapidly becoming intractable for large N or n. To emulate higher-order statistics, like weather extremes, generative models like diffusion-based emulators may be better suited. These approaches are well-equipped to capture higher-order correlations. However, this added flexibility comes with increased cost and training requirements, which may limit its use in applications that demand fast inference and interpretability on limited hardware. Our emulator is, on the other hand, well suited for computation-limited applications.

More generally, the present work lays the foundation for specialized downstream emulators. For example, one could condition a high-resolution or extreme-event emulator on monthly temperature fields produced by our framework. This would integrate well with existing techniques, including Generalized Extreme Value theory or machine-learning-based generative models. Recent work, such as and illustrates how such hybrid frameworks might be developed. Figure  sketches a potential ecosystem of interconnected emulators. A simple climate model (e.g., FaIR-GP, ) translates emissions scenarios into global mean temperature trajectories. These are passed to an emulator of monthly-averaged spatial fields such as the one developed here. The final stage would use a downscaling model to reconstruct finer-grained or daily fields. For example, the DiffESM emulator uses monthly climate values as input to generate daily values; an ensemble of different values, which can be generated with out probabilistic emulator, can be used to condition different daily outcomes. Limitations of this pipeline framework include the assumption that all regional statistics are determined by the global mean surface temperature and the challenge of capturing spatially heterogeneous forcing agents such as aerosols.

Projections accounting for model spread, such as those emulated in this work, are vital to designing effective solutions to climate adaptation . The low computational cost of our emulator makes it well suited for applications that benefit from quick iteration on the warming scenario being explored, such as policy and economic analysis, climate communication, and education. Once trained, our emulator can be run many times at little additional cost on modest hardware such as single-core CPUs. We have experimented with using the emulator in a prototype graphical climate education tool that visualizes changes in regional climate impacts from various warming mitigation policies. While the present emulator is trained on a particular Earth System Model (MPI-ESM1.2-LR), the methodology is broadly applicable to any model ensemble with sufficient realizations. Moreover, such model-trained emulators can serve as physically consistent priors that are further constrained with observational data to reduce structural model bias, as demonstrated in .