Since no single satellite instrument provides all the required variables with sufficient temporal and spatial resolution, we integrate data from multiple sources into this study. The following observational or reanalysis-based datasets were used:

w*: Derived Transformed Eulerian Mean (TEM) momentum terms v0.1.1,, based on European Centre for Medium-Range Weather Forecasts (ECMWF) ERA5 reanalysis using diagnostics from ;

TOMCAT is a three-dimensional off-line CTM , driven here by winds and temperatures from the ERA5 reanalysis . Given prescribed atmospheric transport and temperature fields, TOMCAT calculates the distributions of chemical species in the troposphere and stratosphere. A stratospheric full-chemistry simulation, including all of the NO_y chemistry discussed in this paper, was run at a horizontal resolution of 2.8° × 2.8° with approximately 1.5 km vertical resolution in the stratosphere . The model uses time varying sulfate aerosol surface area density , solar fluxes , and lower boundary concentrations of GHGs and ODSs , recommended for CMIP6 simulations. We used the monthly average output in our analysis. The TOMCAT CTM simulation was chosen for its ability to provide a continuous time series without spatial or temporal gaps, making it ideal for robust comparison with the observational datasets.

Monthly anomalies of w*, N₂O, NO₂, and O₃ in the tropical (10° S–10° N) middle (10 hPa) stratosphere were calculated from observations and the TOMCAT CTM simulation relative to the climatological mean over August 2004–December 2021. The analysis begins in August 2004, due to the availability of MLS N₂O observations. However, for simplicity, we refer to the period as 2004–2021. To maintain a robust comparison against temporal sampling discrepancies, TOMCAT CTM data were masked to align with observations, omitting any dates with missing observational data. For causal inference, all time series were standardized. Since the application of causality requires the stationarity of the time series (see Sect. ), we remove the linear trends from all analyzed time series. For more details about preprocessed timeseries from observations and the TOMCAT CTM simulation and their further comparison, see Appendix .

We apply the latent Peter-Clark momentary conditional independence (LPCMCI) algorithm , which is an extension of the PCMCI+ algorithm specifically designed to deal with latent (i.e. unobserved) variables . LPCMCI employs ideas from the Fast Causal Inference (FCI) algorithm to learn not only directed causal relationships but also infer the presence of latent confounders . LPCMCI benefits from the same ideas underlying PCMCI+ by increasing the effect size of conditional independence (CI) tests through including causal parents in conditioning sets. The LPCMCI method seeks to learn a Directed Partially Ancestral Graph (DPAG), which captures the causal relationships among the observed variables. In contrast to Maximal Ancestral Graphs (MAGs) that contain directed arrows (→) and bidirected edges (in other words, double-headed arrows ↔), PAGs can include additional edge types. These edges, drawn as ∘→ and/or ∘-∘, indicate the presence of hidden variables or uncertainty about the exact causal direction.

To understand the causal structure of the underlying complex dynamical system, the observed time series Xt=(Xt1,…,XtN), where N stands for the different variables represented by time series, is assumed to follow the following causal process: Xtj=fjP(Xtj),ηtj, where fj is a measurable function that depends on all its inputs, ηtj represents dynamical noise (independent across t′≠t). Here, P(Xtj)⊂Xt+1-=(Xt,Xt-1,…)∖{Xtj} denotes the set of parent variables of Xtj since the value of Xtj is determined from the variables in P(Xtj) and the dynamical noise ηtj. Bidirected links between Xti and Xtj in this model imply that the associated noise terms ηti and ηtj are dependent through unobserved confounding. We assume causal stationarity, meaning Xt-τi∈P(Xtj) if and only if Xt-τ-Δti∈P(Xt-Δtj). In practice, causal discovery algorithms, including LPCMCI, require approximately stationary time series, meaning time series whose statistical properties, such as mean and variance, remain approximately constant over time. Nonstationary behavior, such as long-term trends, can introduce spurious statistical dependencies and bias CI tests, leading to incorrect causal links. Therefore, removing or accounting for such trends (via masking or sliding window) is a methodological necessity to ensure that the algorithm identifies causal relationships associated with the internal dynamics of the system rather than coincidental alignment of long-term shifts in the variables .

Additionally, we also assume the absence of cyclic causal relationships, which, due to the temporal order, limits interactions to contemporaneous cases only when τ=0. Furthermore, we assume the standard assumptions of constraint-based causal discovery, the Markov condition and the faithfulness condition , which implies that conditional independence in the observed distribution generated by the structural causal model above implies directional separation (i.e. separation of variables by conditioning on appropriate sets of other variables) in the associated time series graph and vice versa.

We conducted a series of sensitivity tests with various settings of significance level αpc and the maximum time delay τmax, but only used the causal graphs with αpc=0.02 and τmax=1 for the toy model (see Sect. ), and with αpc=0.05 and τmax=2 for the observations and the TOMCAT CTM simulation (see Sect. ). Since some of the analyzed variables have non-gaussian distributions, we use the RobustParCorr conditional independence test, which transforms the data to a normal distribution before the partial correlation test. This usage implies that we assume the functions fj in the model above to be linear. As a trade-off to its ability to also deal with latent confounding, LPCMCI suffers from lower recall compared to, e.g., PCMCI+ . Despite this, LPCMCI successfully identified causal connections in observational data that align closely with expert knowledge and the literature review. However, LPCMCI did not robustly detect anticipated connections in the TOMCAT CTM simulation. To estimate the direct causal effects for the period 2004–2021 in the TOMCAT CTM simulation, we refined the causal graphs by incorporating expert knowledge and insights from the literature review (as further discussed in Sect.  and depicted in Fig. ).

Over a hundred years ago, suggested a method to estimate causal effects in linear models. This approach estimates the so-called path coefficients for all links in causal paths and then sums the products of these path coefficients over all causal paths. This method applies only to DAGs; therefore, in the case of DPAG, we ensure that all edges are directed before applying causal effect estimation. Causal effect estimation consists of the following steps:

For all causal links i→j that belong to causal paths from X to Y (where X is a parent and Y is a child), estimate the path coefficient βi→j by regressing j on all its parents in a multivariate regression and taking the coefficient corresponding to parent i see e.g..

To quantitatively characterize O₃ variability in the tropical middle stratosphere during 2004–2021, we employ a process-oriented causal inference approach depicted in Fig. . Although causal analysis has already gained significant application in atmospheric sciences, including the analysis of Arctic processes and their links to middle latitudes , teleconnections , atmosphere-biosphere interactions , and evaluating climate models and their sensitivities , it has not yet been applied to the study of stratospheric chemical-dynamical interactions.

Figure  outlines the process-oriented causal inference framework, which is built upon three essential components (blue squares): (1) a causal graph that contains information about qualitative cause-and-effect relationships , (2) observational and/or modeled data, and (3) a method for estimating causal effects. To construct (1) the causal graph, we employ a triangulated approach that integrates (i) expert knowledge, (ii) a comprehensive literature review, and (iii) a data-driven causal discovery algorithm. While each of these components can independently contribute to the creation of the causal graph, we recommend employing the triangulated approach to ensure a more robust and reliable framework. To assess the performance of the selected (iii) causal discovery algorithm before applying it to real-world data, we first construct a (iv) “toy model” using synthetic data. This synthetic dataset is designed to replicate the properties and challenges of the real system while incorporating known underlying ground truth from (i) expert knowledge and (ii) a literature review. The toy model is then used to evaluate the performance of the causal discovery method in realistic, finite sample scenarios . To ensure the robustness of the results, it is recommended to further perform sensitivity tests on free algorithm parameters, such as αpc and τmax for (iii) the causal discovery algorithm and (iv) the toy model.

It is important to note that if (iii) the causal discovery algorithm does not robustly detect anticipated relationships in the analyzed (2) real-world or modelled data, the user can integrate physical knowledge into the algorithm. Alternatively, the causal graph may be constrained purely based on (i) expert knowledge and (ii) a comprehensive literature review, including previous successful applications of causal discovery to related research topics. The final (1) causal graph, derived from (2) real-world data, serves as a foundation for estimating (3) direct and total causal effects. Causal effect estimation can be further refined through a process-oriented analysis, such as, for example, masking on different atmospheric regimes. Additionally, (v) total causal effects can be assessed across different time lags. This complex approach outlined in Fig.  ensures robust and reliable causal inference, particularly for complex systems such as the stratospheric chemical-dynamical interactions investigated here.

Confidence intervals for direct causal effect estimates, computed using Wright's path coefficient (Sect. ), were obtained via bootstrapping with 500 repetitions. Only significant direct causal effects are shown, defined as those for which the bootstrap confidence interval does not include 0. For both direct and total causal effects across different time lags (Sect. ), confidence intervals were also obtained from 500-member bootstrapping. Because the causal effect estimates were conducted on the same data as the prior causal discovery step, the confidence intervals do not cover the uncertainty from this prior model-selection step. In our case, this issue is less pronounced because the causal graph was based on a triangulation and not fully data-driven.

For the regime-oriented analysis of causal effects during different QBO phases (Sect. ), we first calculate the QBO wind shear as the vertical gradient of the zonal mean zonal wind between 10 and 30 hPa. For observations, the shear is derived directly from radiosonde-based zonal wind at the two pressure levels . For the TOMCAT CTM simulation, the zonal wind is first averaged over 10° S–10° N before computing the vertical gradient between 10 and 30 hPa. The resulting shear time series is subsequently standardized for use in the process-oriented causal analysis. Positive (negative) values correspond to a westerly (easterly) shear zone, which plays a key role in modulating secondary circulation and stratospheric transport. We focus on the 10–30 hPa shear layer since no data is available above 10 hPa in the observational record used here.

Before applying causal discovery to analyze chemical–dynamical interactions using observations and the TOMCAT CTM simulation, we first summarize the relationships of O₃ variability in this region in a shape of DAG-based on expert knowledge and literature review as discussed and interpreted by e.g., following the procedure discussed in Sect. . Figure a depicts a simple linear chain from the cause w* to the outcome O₃ (grey nodes), with mediating variables N₂O and NO₂ (magenta nodes). The inferred DAG, therefore, represents an effective causal structure that emerges under the influence of dynamical variability, rather than a representation of isolated chemical relationships. In particular, a positive relationship from w* to N₂O (labelled A) indicates that an increase in residual vertical velocity leads to enhanced N₂O concentrations. The relationship from N₂O to NO_x (labelled B) is negative, despite N₂O being a source of NO_x. This apparent contradiction is an example of Simpson's paradox and arises because tropical residual velocity w* acts as a confounding dynamical process, leading to an anti-correlation between N₂O and NO₂. Namely, slower (faster) upwelling results in lower (higher) N₂O concentrations and consequently longer (shorter) N₂O residence time in this region, which allows more (less) time for the photochemical production of NO_x from N₂O. Consequently, higher (lower) NO_x levels lead to lower (higher) O₃ concentrations via the NO_x-catalyzed O₃ destruction cycle, resulting in a negative relationship labelled C, see. Table  summarizes the discussed chemical-dynamical relationships in the tropical middle stratosphere as depicted in Fig. a.

We further justify the assumed causal DAG in Fig. a and validate the reliability of the causal inference method. For this, we require a benchmark dataset with known causal ground truth for validation as depicted in Fig. a. We consider the linear structural causal process with four time series as an example that comes from a data-generating process using the following model: 1Xt0=0.3Xt-10+ηt0Xt1=0.3Xt-11+0.4Xt-10+ηt1Xt2=0.3Xt-12-0.94Xt1+ηt2Xt3=0.3Xt-13-0.95Xt2+ηt3 where ηt⋅ stands for the independent Gaussian white noise processes with variances σ⋅2, X0 depicts residual vertical velocity w*, X1 – the concentration of N₂O, X2 – NO_x, X3 – O₃. This set of variables and their dependencies define a toy model. Although this toy model is designed to replicate causal dependencies in the tropical middle stratosphere, it is important to emphasize that there are multiple approaches to constructing such a model. While additional variables could be further introduced based on expert knowledge and a thorough literature review, the goal of the analysis here is not to maximize the number of variables but to create an intuitive system that can simply and effectively replicate the processes under study.

Figure b–e illustrate the causal graphs inferred from the causal discovery algorithm (see Sect. ) when applied to the toy model from Eq. (). To evaluate the performance of the causal discovery algorithm on time series of different lengths, Fig. b, c show causal graphs for generated time series of 120 time steps (equivalent to 10 years of monthly data) and 240 time steps (equivalent to 20 years), respectively. When applied to the shorter time series (Fig. b), causal discovery fails to detect several expected connections as anticipated from Fig. a, likely due to limitations in the toy model, such as weaker causal signals or higher noise. In contrast, despite the plausible limitations of the toy model, most expected connections are recovered in the 20-year time series depicted in Fig. c. However, the inferred graph lacks directionality between N₂O and NO₂ (shown as ∘-∘ edge), resulting in a DPAG. Since causal effect estimation requires a fully directed DAG, we applied the triangulation approach to resolve ambiguities, yielding the DAG in Fig. d. This final causal graph (Fig. d) serves as the basis for causal effect estimation, shown in Fig. e. Given approximately linear relationships among analyzed variables (see Fig. b–d), we assume linear causal effects and apply Wright’s method . This approach is akin to linear regression slopes between two variables X and Y, with the critical distinction that the graph is used to detect and eliminate confounding influences before regression see Sect.  and.

Figure  presents the magnitude and sign of direct causal effects computed using Wright's approach on causal graphs identified by the triangulation based on observations (a) and the TOMCAT CTM simulation (b) during 2004–2021. The original graphs inferred by the causal discovery algorithm (see Sect. ) without the application of the triangulated approach are shown in Appendix . The causal discovery algorithm successfully identifies the anticipated connections in the observations (Fig. a) since the signs of the direct causal effects align well with the expected processes outlined in the Introduction and as discussed in Sect. .

Notably, the positive lagged link in observations from w* to N₂O indicates that an increase in residual vertical velocity intensifies N₂O transport. This enhanced transport, in turn, reduces the residence time of N₂O, leading to less time for NO₂ production via the reaction N2O+O(1D)→NO+NO (also labelled B in Fig. a). Consequently, the causal contemporaneous link from N₂O to NO₂ reflects the negative relationship confounded by upwelling. The negative contemporaneous link from NO₂ to O₃ indicates that lower/higher NO₂ levels are associated with higher/lower O₃ concentrations, as O₃ loss in the tropical middle stratosphere is largely driven by catalytic destruction by NO_x. Causal discovery further detects a bidirected connection between w* and O₃ in the observations, indicating the presence of a latent common driver of w* and O₃ and that neither variable is an ancestor of the other (see Appendix , Fig. a). As causal effect estimation requires a DAG, we define the direction from w* to O₃ to quantify the strength of this link. This choice allows us to estimate the direct dynamical influence of w* on O₃, which is sometimes identified by a causal discovery algorithm in sensitivity tests. Based on Fig. , the direct influence of w* on O₃ is much weaker, and a mediated pathway via N₂O and NO₂ dominates. However, temperature-mediated effects could amplify the apparent strength of the connection from NO₂ to O₃, since enhanced upwelling induces both cooling (increasing O₃) and reduced NO_x species. To assess this, we performed additional analysis, including temperature anomalies in the tropical middle stratosphere. The inferred graph structure was not robust, likely because w* is a derived diagnostic that depends on thermodynamic fields. Removing w* altered the parent structure and prevented a direct comparison of direct causal effects shown in Fig. a. We therefore interpret the identified NO₂-mediated pathway as the dominant mechanism, while acknowledging that temperature-related effects may potentially project onto this link.

Unlike the observations, in the TOMCAT CTM simulation, the causal discovery algorithm does not fully reproduce the expected chemical-dynamical coupling as outlined in the Introduction, Sect.  and as shown in Fig. a. In particular, the anticipated one-month lagged link from w* to N₂O is not robustly detected. We found that this occurs because O₃ exhibits very strong contemporaneous coupling with both N₂O and NO₂, such that conditioning on O₃ makes the one-month lagged w* to N₂O link statistically insignificant. As further demonstrated and discussed in Appendix , this does not indicate a lack of dynamical coupling in the TOMCAT CTM simulation. Instead, it reflects the strong shared variability among the chemical tracers as the model uses a chemically consistent scheme for all the variables. In order to still assess the strength of the processes represented in TOMCAT, we therefore did not rely on the TOMCAT-derived graph. Instead, by adopting a fixed graph structure derived from observational ground truth and expert knowledge, we can accurately estimate direct causal effects with the TOMCAT data (Fig. b), providing a valid and pragmatic solution for quantifying model sensitivities. Similar to observations, the direct one-month lagged w*-N₂O connection is estimated as significant in the TOMCAT CTM simulation for the analyzed period. Additionally, the N₂O-NO₂ negative contemporaneous link is slightly stronger in the TOMCAT CTM simulation (Fig. b) compared to those in the observations (Fig. a), with a similar causal pattern observed for the NO₂-O₃ link.

To ensure the robustness of the connections detected in Fig. a in observations in the tropical middle stratosphere during the period 2004–2021, Fig.  demonstrates the results of the application of the causal discovery algorithm with different setups of τmax (depicted on the x axis) and αpc (depicted with markers). τmin is set to zero to account for the contemporaneous connections. It should be noted that choosing a τmax that is too low risks missing causal links with longer delays, which violates the assumption of causal sufficiency. However, choosing τmax too high without mitigation can dilute the detection power of the causal algorithm. Specifically, a larger τmax expands the search, as the algorithm tests more possible lagged pairs. This leads to larger conditioning sets, which can further reduce the effect size and detection power. Therefore, it is important to condition only on a few relevant variables that actually explain the relationship . Sensitivity tests from Fig.  show very similar results for different configurations of τmax and αpc. The chemical connections related to N₂O-NO₂ and NO₂-O₃ pairs are robustly detected as contemporaneous across all experiments. The dynamical connections w*-N₂O and w*-O₃ are robustly detected with a one-month lag.

A further analysis of the sensitivity experiments reveals an additional feature in the N₂O-NO₂ relationship. A positive one-month lagged link from N₂O to NO₂ is detected across all tested τmax, but primarily at relaxed significance thresholds (αpc=0.2). For τmax=1, the link is also identified at αpc=0.1 and 0.05. Such a positive lagged connection is physically plausible, as NO₂ is produced from N₂O, as discussed in Table , and a delayed response may emerge at the monthly timeseries. However, given its sensitivity to the choice of αpc, this link cannot be considered a robust pathway.

We adapt the process-oriented causal analysis to major drivers of tropical middle stratospheric O₃ variability to further understand the robustness of the connections during different regimes. Given that N₂O, NO₂, and O₃ exhibit a strong QBO signal , we mask the data to easterly and westerly shear zones (see Sect. ). This approach makes it possible to explore the regime-dependent robustness of causal relationships. A robust connection between specific variables indicates that the particular link is consistently detected across multiple resampled datasets. A robust connection also suggests that the relationship is less sensitive to variations in the data, providing higher confidence that the detected connection is not a result of random fluctuations or sampling variability. It is also important to note that the analyzed period includes an unprecedented QBO disruption in 2016 , which stalled the descent of the easterly shear toward 10 hPa and caused anomalous upwelling and wind patterns, temporarily altering transport and mixing in the tropical middle stratosphere. To assess how QBO phase affects the strength of the identified relationships, we keep the causal graph inferred from the full period and re-estimate the link strengths separately for easterly and westerly QBO conditions. This approach isolates regime-oriented changes in coupling strength without altering the underlying network structure. Figure  illustrates the estimated direct causal effects for both observations (panel a) and the TOMCAT CTM simulation (panel b) for the full period 2004–2021 (blue), which also corresponds to direct causal effects shown in Fig. a,b, easterly (orange), and westerly phase of the QBO.

The dynamical positive link from w* to N₂O at a lag of one month shows slightly reduced magnitude in observations when separating into QBO phases compared to the full period. In the TOMCAT CTM simulation, the link strength is similar to observations and also slightly reduces across QBO phases. The negative contemporaneous chemical coupling between N₂O and NO₂ is stronger in the TOMCAT CTM simulation compared to observations and of similar strength across phases. In observations, the absolute magnitude is slightly larger during the westerly phase than during the easterly phase. The contemporaneous negative link from NO₂ to O₃ is likewise stronger in the TOMCAT CTM simulation compared to the observations. In the observations, this link weakens during the easterly QBO phase, whereas in the TOMCAT CTM simulation its magnitude is comparatively stable across QBO phases.

The imposed one-month lagged connection from w* to O₃ is the weakest among the analyzed pathways and substantially smaller than the chemically mediated pathway. In the TOMCAT CTM simulation, no link is detected for the easterly QBO phase. This phase-dependent absence of this connection points to its limited robustness. This also indicates that the direct influence of the residual circulation on O₃ variability is predominantly mediated through N₂O and NO₂, rather than through a direct dynamical effect. Overall, the sign of all connections is robust across datasets and QBO phases. In the observations, the chemical links (from N₂O to NO₂ and from NO₂ to O₃) tend to strengthen during the westerly QBO phase.

While the causal discovery algorithm, along with causal effects estimation of the direct links, offers a general overview of whether the methodology captures the expected dependencies, one can further study the propagation of causal effects through the causal graph across different time lags. The total causal effects (for further discussions, see Appendix ), which are not necessarily just depicted by a single arrow in the causal graph, can be derived from direct causal effects using Wright's path analysis for specific relationships across different time lags. Figure  shows total (lines) and direct (labels) causal effects across different time lags for a selection of different (X, Y) pairs of variables, where a positive connection, similar to Fig. , indicates that an increase in X leads to an increase in Y after a time lag (τ). Due to the limited number of variables in the causal graphs, we do not analyze the role of mediators for specific connections, as their influence is straightforward. For example, the total causal effect from w* to NO₂ is mediated solely by N₂O, as shown in Fig. . However, for the analyses that involve more complex causal graphs, similar to , we recommend estimating the contribution of various mediators on a specific set of connections.

Figure  demonstrates that the total causal effects across different time lags in the TOMCAT CTM simulation (dashed lines) closely align with observations (solid lines) across analyzed regimes, indicating that the model reproduces not only the sign but also the temporal structure of the effects. However, the amplitudes in the TOMCAT CTM simulation are slightly larger, suggesting a somewhat stronger coupling between analyzed variables in the simulation. The direct causal effect of w* on N₂O (Fig. a) exhibits a clear maximum at a lag of around three months for the full period 2004–2021. The total causal effects of w* on NO₂ (Fig. b) and on O₃ (Fig. c and Appendix ) also show similar lagged maxima for the full period 2004–2021. This lagged behavior likely indicates the spatiotemporal structure of the covariability between w* anomalies at the analyzed 10 hPa level and below. Upwelling anomalies are primarily governed by the QBO and descend over time with the associated shear zones. As a result, an anomaly in w* at 10 hPa, originating from higher altitudes, produces an instantaneous but modest local effect on composition, while over subsequent months, it persists and increasingly affects lower altitudes. This leads to a cumulative impact on N₂O, NO₂, and consequently O₃, with a delayed maximum in the total causal effect. The timing of this maximum is therefore linked to the vertical extent of the w* anomaly and the rate at which QBO shear zones descend.

We can further decompose the total causal effect of w* on O₃ into direct and indirect contributions. As shown by Figs. a and a, the direct one-month lagged w* on the O₃ pathway is approximately 0.10 in the observations and 0.14 in the TOMCAT CTM simulation. The indirect contribution, mediated via N₂O and subsequently NO₂, reaches 0.16 in the observations and 0.23 in the simulation (not shown here). Therefore, the total causal effect of w* on O₃ shown in Fig. c at a lag of one month therefore represents the sum of both pathways, yielding approximately 0.26 for the observations and 0.37 for the TOMCAT CTM simulation over the full analyzed period. The same analysis is performed for the easterly and westerly QBO phases, but an analogous description is not discussed further. Conditioning on both N₂O and NO₂ does not alter the magnitude of the mediated effect, indicating that the dominant indirect pathway proceeds sequentially through w*, N₂O, NO₂, and O₃.

The N₂O on NO₂ relationship (Fig. d) strengthens gradually with increasing lag, with close agreement between observations and the TOMCAT CTM simulation after two to three months. Differences are largest at short lags, where the observational estimate shows stronger variability. The observations suggest that this link appears weaker during easterly QBO shear, linked to vertical ascent due to the strengthening of tropical upwelling . The total N₂O to O₃ effect (Fig. e) increases from lag zero to lag one (in the observations) and then gradually decreases toward longer lags. Here, observations also show a weaker total causal effect during easterly QBO phase. These results are also consistent with direct causal effects analyzed for different QBO regimes (see Fig. ). The NO₂ on O₃ effect (Fig. f) peaks at lag zero and rapidly decreases, approaching zero after about one month. This behavior is consistent with fast NO_x-driven O₃ chemistry. A slightly longer persistence during the earlier subperiod suggests a more sustained influence of NO₂ on O₃ during that time.