Using machine learning to build temperature-based ozone parameterizations for climate sensitivity simulations

A number of studies have demonstrated the importance of ozone in climate change simulations, for example concerning global warming projections and atmospheric dynamics. However, fully interactive atmospheric chemistry schemes needed for calculating changes in ozone are computationally expensive. Climate modelers therefore often use climatological ozone fields, which are typically neither consistent with the actual climate state simulated by each model nor with the specific climate change scenario. This limitation applies in particular to standard modeling experiments such as preindustrial control or abrupt 4xCO2 climate sensitivity simulations. Here we suggest a novel method using a simple linear machine learning regression algorithm to predict ozone distributions for preindustrial and abrupt 4xCO2 simulations. Using the atmospheric temperature field as the only input, the regression reliably predicts three-dimensional ozone distributions at monthly to daily time intervals. In particular, the representation of stratospheric ozone variability is much improved compared with a fixed climatology, which is important for interactions with dynamical phenomena such as the polar vortices and the Quasi-Biennial Oscillation. Our method requires training data covering only a fraction of the usual length of simulations and thus promises to be an important stepping stone towards a range of new computationally efficient methods to consider ozone changes in long climate simulations. We highlight key development steps to further improve and extend the scope of machine learning-based ozone parameterizations.


Introduction
The trace gas ozone plays multiple roles in the Earth system. Besides being an important greenhouse gas, it is the only absorber of harmful solar UV-B radiation which would otherwise make life on Earth impossible (WMO 2011). However, ozone's distribution in the atmosphere is subject to change. Anthropogenic and natural factors force variability and trends in its concentrations, mainly related to catalytic ozone depletion cycles (Hunt 1966, Crutzen 1970, Molina and Rowland 1974, Solomon et al 2016, the stratospheric circulation (SPARC 2010, Eyring et al 2013, background temperature (Fels et al 1980, Haigh andPyle 1982), the solar cycle (Haigh 1996, Ball et al 2016 as well as changes in two-way interactions with other constituents (Young et al 2013, Voulgarakis et al 2013. Ozone's importance for global radiative transfer, in turn, implies feedback effects on the Earth system by modulating temperature, dynamics and the biosphere (Lacis et al 1990, Thompson and Solomon 2002, Son et al 2008, Williamson et al 2014. Here we introduce a novel method for representing the time evolution of ozone (or ozone feedbacks) in state-of-the-art Earth system models. For this ozone parameterization we focus specifically on preindustrial control and abrupt 4xCO 2 climate simulations where ozone feedbacks are typically not included. Our work is timely due to a number of recent studies that have demonstrated the importance of ozone in this context. Preindustrial control and abrupt 4xCO 2 simulations are core experiments in many climate modeling intercomparsion projects , Kravitz et al 2013, Eyring et al 2016. Preindustrial simulations help us understand climate variability in the absence of human influences (e.g. Bellenger et al 2013). Abrupt 4xCO 2 simulations, in which atmospheric carbon dioxide (CO 2 ) mixing ratios are quadrupled starting from preindustrial conditions, are most frequently used to study climate sensitivity (e.g. Ceppi andGregory 2017, Knutti et al 2017); in the broadest sense the global warming response to increased CO 2 . They also provide insights on how climate dynamics might change in high-CO 2 climates, for example in the form of major climate modes (Guilyardi et al 2012, Rashid et al 2016.
Many studies have highlighted that the representation of ozone can impact the results of such climate sensitivity simulations. The use of non-adaptive ozone climatologies in abrupt 4xCO 2 simulations can affect climate sensitivity estimates (Li et al 2013, Muthers et al 2014, Dietmüller et al 2014, Nowack et al 2015, 2018, the position of the jet streams Polvani 2017, Nowack et al 2018) as well as the response of the Walker circulation and ENSO (Nowack et al 2017). Similar effects have been found for paleo-climates and solar forcing scenarios (Haigh 1996, Heinemann 2009, Chiodo and Polvani 2016, Muthers et al 2016, Noda et al 2017. Already the use of zonally averaged ozone climatologies, even if otherwise consistent with the climate state, can affect atmospheric variability due to the lack of ozone-dynamics interactions (Gabriel et al 2007, Crook et al 2008, Gillett et al2009, McCormack et al 2011, Albers and Nathan 2012, Rind et al 2014. A key issue is the high computational cost associated with calculating ozone feedbacks. For example, the climate model used here is slowed down by a factor of around three by including a comprehensive atmospheric chemistry component. While this number will depend on the chemistry scheme itself and the underlying climate model, the slow-down is expected to be substantial for any similar case. There are two main contributors to this slow-down (Esentürk et al 2018). Firstly, atmospheric chemistry is represented by a large system (>100) of coupled partial differential chemical rate equations. Secondly, chemical species are subject to transport as part of the atmospheric circulation. Representing either requires significant computing power to numerically solve the respective chemical and dynamical equations at each timestep.
The chemistry-climate modeling community has thus provided an ozone dataset for the Representative Concentration Pathways scenarios defined for the Coupled Model Intercomparison Project Phase 5 (CMIP5; Cionni et al 2011, Eyring et al 2013, which has been widely used in climate models that lack fully interactive chemistry schemes. While being an improvement over neglecting changes in ozone altogether (Son et al 2008, these climatologies were neither consistent with individual model responses nor the actual forcing scenario. In comparison, there has been no coordinated effort to provide similar, or even better datasets, for preindustrial control or abrupt 4xCO 2 simulations. Consequently, it is imperative to test alternative, computationally less expensive approaches with the specific application to preindustrial and 4xCO 2 simulations in mind.
Here, we show that ozone parameterizations using machine learning (ML) could be a powerful alternative to fully interactive chemistry schemes for climate studies that do not specifically focus on atmospheric chemical mechanisms. ML is a subset of artificial intelligence and the two terms are colloquially often used interchangeably. More specifically, ML describes a collection of statistical methods that allow computers to learn relationships from data without being explicitly programmed (e.g. Samuel 1959). As such, ML has the potential to advance scientific understanding and modeling efforts in climate science (Monteleoni et al 2013). For example, ML has been used to study coupling mechanisms in the climate system (Boers et al 2014, Runge et al 2015, for statistical downscaling in regional climate studies (Vandal et al 2017, Anderson andLucas 2018) and has been suggested as a new tool to parameterize convection and clouds (Schneider et al 2017.
To model ozone, the resulting parameterization requires atmospheric temperature as the only input variable, which will be available from any climate model simulation, thus putting virtually no constraint on the general applicability of the method. In section 2, we describe our method, the climate model and its data output used to fit the ML model. In section 3, we first demonstrate that our parameterization can well predict the general ozone climatology and trends. We then show that it does also capture large parts of variability in the ozone distribution, in particular in the stratosphere. Finally, we discuss our results as well as future challenges in section 4.

General approach
Our goal is to predict the distribution of ozone without calculating the underlying system of coupled partial differential chemical rate equations (Pyle 1980, Jonsson et al 2004. Ideally, we aim to only use variables in this parameterization that are readily available in climate models without atmospheric chemistry component. As demonstrated below and discussed in section 4, variations and trends in atmospheric temperatures are reflective of processes affecting the production, depletion and transport of ozone and can thus be used as a good predictor for ozone trends and variability, in particular in the slowly-evolving stratosphere. Figure 1 shows the general work-flow of the method suggested here. We approach this task as a typical multi-output regression problem, with the predicted variables being ozone mass mixing ratios at timestep t in each grid cell and the input features being the modeled temperature at the previous timestep in all grid cells, giving rise to individual regressions for each grid cell. The time resolution (t versust 1) is a question of choice; we tested climatologically relevant time-scales of daily-mean to monthly-mean ozone. For the regression itself, we tested a number of linear and non-linear ML methods from Ridge regression to neural networks (see supplementary materials stacks. iop.org/erl/13/104016/mmedia). We found that the relationship between temperatures and ozone is highly linear, thus making the linear Ridge regression a computationally and conceptually favorable approach. As shown below, this applies in particular to the 4xCO 2 scenario, where a linear method allows for extrapolation outside the training domain. In order to speed up the algorithm, we apply principal component analysis (PCA, Bishop 2006) to the temperature input, thereby reducing the number of input regression variables under the condition of maintaining >98% of variance. The PCA dimension reduction significantly reduces the number of input variables on which the algorithm has to be trained, which was essential in order for us being able to carry out the large number of tests presented in this paper and its supplementary materials. We found the loss in predictive skill due to using PCA to be very small as compared to using the entire temperature field as input, whereas the time needed to train the algorithm decreased by several orders of magnitude. However, once the model is fit to the data, the prediction of ozone fields for new temperature input is fast with or without prior PCA step. A final version of the algorithm presented here, for example to be implemented in a climate model, might as well be trained directly on the entire temperature field.

Climate model and data
The data for this study was produced using the atmosphere-ocean coupled configuration of the Hadley Centre Global Environment Model version 3 (HadGEM3-AO) from the UK Met Office (Hewitt et al 2011), coupled to the United Kingdom Chemistry and Aerosols atmospheric chemistry scheme (Morgenstern et al 2009, Nowack et al 2015. The model, which has a self-contained Quasi-Biennial-Oscillation (QBO), is described in detail in supplementary section 1.
We use data from two types of simulations: (1) a preindustrial simulation with atmospheric CO 2 at 285 ppmv that has been spun up for >1000 years.
(2) an abrupt 4xCO 2 simulation with CO 2 quadrupled to 1140 ppmv, which was initialized from this preindustrial run. As shown in figure 2(a), we observe the characteristic rapid increase in global mean surface Figure 1. Sketch of method. For each climate model grid cell the ozone mass mixing ratio is predicted at a given time using the atmospheric temperature distribution at the previous timestep. Here we focus our discussion on Ridge regression for the mapping step, but more complex algorithms could be used instead. temperature following the abrupt increase in CO 2 (Andrews et al 2012, Nowack et al 2015, 2018. In order to train the ML model, the data was split into training and test datasets (Bishop 2006). The training datasets were used to optimize the model concerning bias and overfitting (section 2.3), whereas the independent test datasets served to evaluate the learned model. The 4xCO 2 simulation was run for 200 years in total (figure 2(a)). In section 3, we explore various ways of splitting this dataset into training and test datasets. For the preindustrial run, we randomly picked a 50-year-long interval for training and another 13-year-long independent extract for testing. For all regressions we used daily-mean model output for temperature and ozone mass mixing ratios, for the former on pressure levels, for the latter on model altitude levels (measured in kilometres above the surface). We selected a subset of the actual model vertical resolution (14 pressure, 22 altitude levels) covering the entire troposphere and stratosphere. Choosing vertical subsets allowed us to run a larger number of tests, without impacting our general results or conclusions.

Regression model
We use Ridge regression, also referred to as Tikhonov regularization, for the mapping between the temperature state of the atmosphere and ozone's distribution. Ridge regression is a linear least squares regression augmented by L2-regularization to address the biasvariance trade-off (Hoerl andKennard 1970, James et al 2013). For the regression, we minimize the cost function for each model grid cell k over N timesteps t given time series training data for ozone (Y) and the first p principal component scores of temperature variability (X), with the latter preceding the former by one timestep in order to predict ozone. The ozone prediction for each grid cell is thus (potentially) a function of temperature variability in all grid cells, as characterized by the time series weights of the leading p principal components of temperature variability. The cost function determines the optimization goal. Its first term is the ordinary least squares regression error, the second penalty term avoids overfitting by nudging the regression towards small regression coefficients c kj . In general, smaller (larger) values of α put weaker (stronger) constraints on the size of the coefficients, thus favoring overfitting (high bias). We use a standard time series cross validation method to find the best value for α, in which the time-ordered training data is split into five subsets of equal size. Preceding subsets are then sequentially used as training data for each subsequent subset (i.e. set 1 for 2, set 1+2 for 3 etc), which is referred to as validation set. α is found according to the average generalization error on these validation sets. Prior to the regression and PCA step, both X and Y were preprocessed by removing each grid cell's mean and by scaling to unit variance. Therefore, the final ozone predictions for Y on a separate testset ¢ t have to be re-scaled to mass mixing ratios. The number of components chosen for X and the optimized regularization parameters α for each fit are documented in supplementary section 2. For all regression, normalization and dimension reduction tasks, we used Python's scikit-learn package (Pedregosa et al 2011).
For the ML models presented in this main part of the paper, we choose a prediction timestep of one day, that is, we predict daily average ozone distributions using the temperature state of the atmosphere one day earlier. We also implemented models for 5-day-mean (predicting 5-day-average ozone fields five days in advance) and monthly-mean data (predicting one month in advance). Increasing the lead time results in the expected decline in the quality of the predictions (supplementary figure S1).
For the preindustrial and the 4xCO 2 scenario, we used the first 2000 and 2500 PCA temperature components as input, respectively, as compared to around 100000 spatial grid cells. The components explain >99% of variance for the preindustrial training set and >98% for the 25-year-long 4xCO 2 training set (introduced below). For daily-mean data, these choices represent a good compromise between minimizing the number of input variables and the prediction error (figures S1a-d). For the 4xCO 2 case, we also dimension-reduced the ozone field prior to the regression fit, which, after the necessary additional transformations, yields essentially identical results to grid-cell-wise regressions (see supplementary section 2).

Results
3.1. Average ozone fields Any ozone parameterization should be able to reproduce the average ozone climatology of an interactive chemistry model with high fidelity. For 4xCO 2 simulations, this requires that the method also captures ozone trends under forcing (Nowack et al 2015). Figure 2(c) shows percentage differences in ozone mixing ratios under 4xCO 2 relative to the preindustrial climatology ( figure 2(b)), averaged over years 50-200 after the CO 2 forcing. We find ozone decreases in the lower stratosphere, where ozone is long-lived enough to be influenced by transport, most notably in the tropical lower stratosphere, which is an ubiquitous feature in chemistry-climate model simulations (SPARC 2010). In the upper stratosphere, we find ozone increases, which are also well understood based on the temperature-dependency of catalytic ozone depletion cycles (Haigh andPyle 1982, Meul et al 2014). All of these changes are not included in simulations with fixed ozone climatologies (Nowack et al 2015).
After minimizing the cost function according to equation (1), the ML model predicts the time-mean ozone climatology for both preindustrial and 4xCO 2 conditions well (figures 2(d), (e) and S2/S3). Specifically, the prediction error on the preindustrial testset for the zonal mean ozone climatology is smaller than 5% everywhere. A model trained separately on the initial 25 years of the 4xCO 2 run can also predict the average ozone field for the remaining 175 years to within 5% almost everywhere ( figure 2(e)). For this, the regression necessarily captures long-term trends, which are key to the evolution of ozone in the dynamically-driven lower stratosphere where there are significant changes over the entire period (Dietmüller et al 2014, Nowack et al 2015 figures S4/S5). We found that the same extrapolation would not work using the model fit on preindustrial data; using the preindustrial regression model leads to a catastrophic prediction failure for the 4xCO 2 scenario, implying that the co-variability in temperature and ozone for the preindustrial state is not a valid predictor of ozone's response to an external forcing.
However, further tests indicate that the length of the training dataset could be reduced for most purposes. The bulk of the prediction error vanishes using 10-15 years of training data (figures S1e,f and S5). In comparison, taking 50 years of training data would lead to mean squared errors (MSE) almost as small as for the preindustrial testset (figure 2(d)), despite the additional complexity of underlying trends under 4xCO 2 (figures S1d,f and S5). Here we focus on the model trained on 25 years of data as an intermediate case. Next, we show that the resulting ML model also poses a significant improvement over a fixed ozone climatology in terms of resolving temporal and spatial ozone variability. Figure 3 shows six representative cases of ozone times series data produced by the chemistry-climate model (blue lines) and the corresponding predictions of the ML model (red lines) on the 13-year-long preindustrial testset for specific grid cells. Figure 4 shows similar data for the 4xCO 2 testset. Ideally, the time series for the climate model and the ML model should be in perfect sync. In each figure, the ozone field as given by a 'standard' fixed ozone climatology (gray lines) is indicated. Such a field varies on a seasonal basis, but not interannually and does also not adapt to the CO 2 forcing. Extensive samples for other grid cells and for other time resolutions can be found in supplementary figures S8-S12.

Representation of variability
For the preindustrial testset, the ML model generally captures key aspects of ozone variability across the entire atmosphere. At all levels the MSE is reduced, reaching improvement factors greater than seven in the tropical stratosphere ( figure S6). This is also evident from figures 3(a)-(c) where the predictions and climate model match well from locations in the lower to the upper tropical stratosphere. In comparison, a fixed ozone climatology misses many undulations and sometimes even the periodicity is completely out of sync. Figure 3(b) exhibits just one prominent example for the tropical mid-stratosphere, where the QBO is a key driver of ozone variability ( figure 3(b)). The good representation of ozone variability there is crucial because ozone feedbacks are well-known for two-way interactions with the QBO as well as its remote impacts on the extratropics (e.g. Rind et al 2014, Silverman et al 2018. Major performance gains are also found around the positions of the jet streams and the polar vortices ( figure S6). Our method also represents the zonal structure in the ozone distribution at any given time well (figures S12.1-S12.5), both in the troposphere (e.g. differences between land and oceanic regions) and in the stratosphere (e.g. due to wavy polar vortices).
In some areas the MSE improvement over a fixed climatology is not as large. For the example shown in figure 3(d), the climatology is already a very good approximation due to little interannual variability. More pronounced are the differences in the troposphere (e.g. Figure 3(f)), but even in the least favorable cases, the ML model still performs better than the climatology and captures the seasonal cycle. We argue that this should be sufficient for typical climate sensitivity simulations, where tropospheric ozone abundances are particularly low (Chiodo et al 2018) so that any variability around their mean state will have little impact on the modeled climate. Instead, we argue that it is more important to capture tropospheric ozone trends under 4xCO 2 (figure 2(c)), which as we showed are modeled well by the parameterization (figure 2(e)). As discussed in section 4, improving the representation of tropospheric ozone variability could be an exciting aspect of future work, especially with regard to other forcing scenarios.
Results for the abrupt 4xCO 2 scenario are similar ( figure 4). However, there the ML model does not only reproduce well the internal ozone variability but also ozone trends, again with limited ability to pick up on very short-term tropospheric fluctuations. The ability to reproduce trends is crucial for the 4xCO 2 case, because the climate continues to warm over the entire 175 years prediction period following the initial 25-year training period ( figure 2(a)). As a result, temperatures will increase beyond those encountered in the training dataset. Leaving the training domain can lead to catastrophic failure of ML models, because they usually perform well on interpolation tasks, but perform poorly for extrapolations. This is particularly true for highly non-linear regression functions such as neural networks. In contrast, the linear ML algorithm used here remains stable and robust in its predictions. This is probably not surprising given that it simply assumes the linear relationships between temperature inputs and ozone outputs to hold even outside the training domain, therefore acting as a better extrapolation tool than many popular non-linear algorithms.
The time series of the predictions in figure 4 remain highly correlated with the ozone field produced by the chemistry-climate model, reaching almost perfect correlations in the stratosphere. Improvements compared to a fixed climatology in terms of the MSE are much larger, which is not surprising given the large differences not just in variability but also in the mean values for ozone; an important factor that is missed in models using climatologies. As a result, improvement factors >150 are attained in the stratosphere ( figure S7). Clearly, the ML model outperforms the use of a preindustrial climatology by far. We further note that climatologies taken over the training period would only be able to capture mean changes in the quickly responding upper stratosphere ( figure 4(a)), but not the slower, surface-temperature correlated trends in the climatologically so important lower stratosphere (figures 4(c) and S4, Dietmüller et al 2014, Nowack et al 2015, or the interannual variability in ozone.

Physical intuition
Our parameterization represents well the climatologically important variability and trends in ozone for preindustrial and 4xCO 2 scenarios. The only input variable is the atmospheric temperature state at the previous timestep. While temperature does affect ozone production and depletion, it is not the only driver of changes in ozone. This raises an obvious question: why does the regression work so well?
The simple answer is that the regression does not just pick up on direct causal relationships, but can also gain predictive skill from systematic indirect correlations. For example, a stable polar vortex characterized by cold, isolated polar stratospheric air will be tantamount to hindered poleward transport of ozone across the vortex edge. A faster stratospheric Brewer-Dobson circulation will be reflected in a colder tropical lower stratosphere and increased poleward heat transport. The resulting temperature structure in turn is indicative of enhanced poleward transport of ozone. Long-term changes in the circulation, which are key for ozone trends under 4xCO 2 forcing, are also strongly correlated with the degree of tropospheric warming found in climate models McLandress 2011, Chiodo et al 2018). The seasonal cycle in ozone, which is primarily driven by changes in transport and photochemical ozone production, will equally be reflected in the atmospheric temperature field. Atmospheric humidity, an important factor in tropospheric ozone production and loss mechanisms and future ozone trends, will also be correlated with temperature (Held andSoden 2006, Young et al 2018). In the upper stratosphere, where ozone is short-lived, catalytic ozone depletion mechanisms are directly dependent on temperature (Haigh andPyle 1982, Jonsson et al 2004).
All these factors will interact in complex ways at any time and altogether give rise to a temperature state of the atmosphere that can be related to a corresponding, representative ozone distribution. ML can identify these relationships automatically where each process (for example the vortex state) could be characterized by the time-dependent weights of a large number of principal components in temperature (or alternatively, temperatures in specific grid cells). Each temperature component in turn can be important for predicting a number of different processes needed to . Color code and labels as in figure 3; however, the daily-mean predictions and model output were smoothed by calculating their monthly-means to aid visualization. The time series remain in sync long after the training period has ended, despite remaining trends in the system, see (c) and (d). Actual daily-mean time series are documented in supplementary figures S8-S11. The correlation coefficients and MSE ratios were calculated using daily-mean data. model ozone trends and variability. For example, a temperature component reflecting the seasonal cycle might characterize certain seasonal aspects of both the atmospheric circulation and regional changes in solar radiation.
In summary, the main processes driving ozone variability and trends will in some way be correlated with atmospheric temperature. Even though this does not not always represent direct cause-effect relationships, it implies that once a certain temperature state of the atmosphere has been associated with a specific ozone distribution, such information can be used to predict future ozone distributions.

Outlook
Despite promising results, we have only conducted a first feasibility study, which we hope will motivate further work in this direction. While our method could immediately reduce the necessary length of interactive chemistry simulations to periods as short as 10-15 years (whereas preindustrial and 4xCO 2 simulations are usually run on centennial to millennial time-scales), it still asks for significant effort on the model developer side. However, since we find that linear models provide good approximations, there is hope that our scheme could become transferable across models by using scaling arguments involving the strength of the circulation response, surface temperature change etc, without the need for repeated training episodes. If used across a number of climate models, such a parameterization could result in another significant cut in necessary computing and model development effort.
The next logical step is the actual implementation of our approach in a (fully coupled) climate model. In particular, this would allow us to evaluate its performance in terms of climate statistics. Here, we demonstrated for the first time that ozone can be modeled solely as a function of atmospheric temperatures. However, the fit is necessarily an approximation so that the climate model would see a slightly different ozone field compared to using a fully interactive chemistry scheme (see our validation), even though the error would probably be within the range produced by different state-of-the-art chemistry schemes. As a result, there might also be small differences in simulated temperatures, which in turn would feedback onto the ozone again. With our approach, climatic differences should be significantly smaller compared to using a fixed ozone climatology instead of interactive chemistry. We hope future studies will investigate these two-way interactions in detail.
Additional opportunities lie in the inclusion of other input variables. In particular, a better representation of short-term tropospheric ozone variability should be aimed for, which is not yet captured in great detail by the current parameterization (figures 3(f) and 4e). This is not surprising given our prediction lead time of one day, which is long in terms of tropospheric time-scales and processes such as convection or turbulent transport. Increasing the time resolution might help, but also the inclusion of emissions of ozone precursors, for example of nitrogen oxides by lightning. Ultimately, similar methods could be applied to other, more complex forcing scenarios. The resulting advanced regression models could be a useful tool for comparing chemistry-climate models to observations in terms of their representation of co-variability in temperature, or any other possible input features, and ozone. Such a tool could open up new pathways for detecting and removing significant model biases or for identifying missing processes. More immediately, we see scope for using similar methods in solar forcing or paleoclimate scenarios. We also highlight the potential significance of taking lagged variables into account, ideally in combination with other regression methods (Runge et al 2015, Kretschmer et al 2017. Having actual implementations of ozone parameterizations in climate models in mind, we also point towards the intrinsic limitations of semi-statistical models. For example, we found that in tropospheric regions of relatively high ozone variability, the parameterization could sometimes 'overshoot' slightly, resulting in rare cases of small negative predictions for ozone mass mixing ratios (figures S8-S11). While not necessarily of great importance for radiative transfer, such unphysical negative values should be prevented by setting specific boundary conditions (Schneider et al 2017), e.g. by imposing the value from the previous timestep. Finally, we note that trace gases other than ozone, in the form of methane and nitrous oxide, are also typically prescribed as climatologies.
In conclusion, we have demonstrated that machine learning regression models could be a useful tool for representing ozone more consistently in climate simulations. We hope that our research will be a starting point for an informed discussion on possible ML parameterizations of ozone, moving beyond the current binary framework of fixed ozone climatologies and interactive chemistry schemes. In this context, we wish to acknowledge relaxation schemes as a simplified option to implement interactive ozone chemistry (e.g. Cariolle and Déqué 1986, McLinden et al 2000, Cariolle and Teyssèdre 2007. These calculate purely chemical tendencies and therefore still require advection of ozone. They are ideal to evaluate interannual variability and small forcings (Braesicke and Pyle 2003), but have limited scope under extreme climate change scenarios. In comparison, our ML method directly accounts for both local and non-local (e.g. circulation) effects and is easily extendible to extreme scenarios. Another difference is that relaxation schemes linearize the chemical equation system, whereas we use a regularized linear regression approach to model ozone as a function of temperature. Therefore, ozone can still vary non-linearly in time given non-linear variations in temperature. We invite data collaborations for climate sensitivity simulations with interactive chemistry schemes, which will facilitate the process of building robust and transferable ML-based ozone parameterizations.