As our observational reference for large-scale dynamics, we use three variables from the ERA5 reanalysis at 1° resolution: 500 hPa geopotential height (Z500) and 850 hPa zonal (U850) and meridional (V850) wind aggregated to daily means from 6 hourly instantaneous fields. Precursor patterns (see Sect. ) are computed using data over the period 1979–2024. While ERA5 has now been extended back to 1940, we limit ourselves to the satellite era where daily synoptic variability is more reliable. Precipitation data is taken from ERA5-land , at a spatial resolution of 0.25°. While precipitation in reanalyses is not as reliable as that from satellite- and gauge-derived products, reanalysis data has the advantage of being temporally and spatially homogeneous and is suitable for our focus on large-scale spatially-aggregated daily precipitation. As a check, we have made a qualitative comparison with 0.1°, 3 hourly precipitation data from MSWEPv3, which blends reanalysis with gauge and satellite data , and find consistent results.

We analyse two large-ensemble simulations, CESM2 LENS2 and MPI-GE , using 50 members from each of their historical and SSP3-7.0 scenario runs. While CESM2 has 100 members available, 50 are sufficient to constrain European precipitation statistics . For the historical runs we use the 1979–2014 period shared with ERA5, and for forced changes we use a “future” period covering 2060–2100. SSP3-7.0 represents the severe but increasingly plausible “rocky road” emissions pathway , characterised by global end-of-century warming of 2.8–4.6 °C. The precipitation response in both models is nearly linear across scenarios , so we expect our findings for SSP3-7.0 to be generally relevant to any SSP scenario. With only two models, our results will not span the full range of model uncertainty, but CESM2 and MPI-GE come from independent model families and have different bias patterns . The models have comparable transient climate responses, within the middle of the CMIP6 spread: 1.8 °C in MPI-GE and 2.0 °C in CESM2 . CESM2 has a higher horizontal resolution than MPI-GE, 1° vs. 1.8°, and a more realistic wintertime jet , but both models been found to reproduce observed mid-latitude mean and heavy precipitation with reasonable accuracy .

Precipitation regions are defined algorithmically based on shared precipitation variability. ERA5-land daily precipitation over 1979–2024 is taken over all land gridpoints within the domain [30–72° N 12° W–30° E], and subsampled by latitude to maintain a similar spacing between points. The Pearson correlation is computed between each pair of retained gridpoints, producing a correlation matrix, C, which was converted into a distance matrix D = 1-C. D is then used to cluster gridpoints agglomeratively using Scipy's linkage function, with method=”complete” . The clustering algorithm initially assigns each gridpoint as its own region, then iteratively reduces the number of regions by merging the two regions with the lowest maximum distance between the gridpoints they contain. Pragmatically, we choose the region number to be as small as possible while still ensuring that the average correlation between gridpoint precipitation within each region is at least 0.45. This gives the 38 regions shown in Fig. . Figure S1 in the Supplement shows the dependence of mean and minimum intra-regional correlation on region number. Precipitation from ERA5-land and from CESM2 and MPI-GE (bilinearly interpolated to the ERA5-land grid) were averaged over these regions using cosine-latitude weighting to obtain 38 scalar precipitation indices for each data set. We focus on categorical “heavy precipitation” events, defined separately for each season and regional precipitation index based on exceedance of the 95th ERA5-land percentile (given explicitly in Table 1 in the Supplement). We use this same ERA5-land threshold for model precipitation, a choice we justify in Sect. . For the two models we consider, biases and changes in heavy precipitation occurrence are closely related to intensity (cf. Fig. S3).

To analyse heavy precipitation occurrence we decompose it into dynamical contributions and conversion contributions. The dynamical contribution accounts for the occurrence of different synoptic flow conditions associated with heavy precipitation, based on precursor indices as described below. The conversion contribution accounts for the fine-scale processes that produce precipitation under a given synoptic condition, including microphysics, thermodynamics, boundary layer interactions, the land surface and mesoscale dynamics.

The mathematical formalism behind the decomposition is presented in full in Appendix A. In the main text, each equation is introduced as it becomes relevant alongside a corresponding concrete example so that each term can be linked directly to a meteorological reference. As a matter of notation, probabilities are always marked with a tilde if they correspond to a potentially biased quantity, and with an asterisk if they correspond to a future climate state.

The flow precursor framework developed in , is the basis of our decomposition. On a high level, the approach identifies the synoptic conditions corresponding to past heavy precipitation events using composite analysis, and defines time-evolving “precursor activity indices” based on those composites. While the flow precursor framework can be used for time-lagged dynamical fields, here we only use fields co-occurring with precipitation to form “lag-0 precursors”, but retain the terminology for consistency with . Our method for this paper is as follows:

For each region and season, composites of deseasonalised ERA5 Z500, U850 and V850 anomalies are computed for heavy precipitation days. Seasonal cycles are computed with a day-of-year climatology, smoothed with a 31 d Gaussian filter.

In combination with the regional precipitation aggregation described in Sect. , this process represents the interactions between synoptic dynamics and precipitation in each region as a functional relationship between 4 scalar indices: three precursor indices and one precipitation index. The time evolution of each precursor index is directly interpretable in terms of the evolution of its precursor pattern, just as, for example, the dipole variability of the NAO can be captured neatly within an NAO index. However, in contrast to the NAO, which represents the dominant variability mode of Atlantic geopotential height, the flow precursor framework identifies targeted patterns for each region-season combination that explain heavy precipitation variability. Thus, the framework captures stronger dynamical modulation of precipitation than traditional regime or analogue-based approaches. Figure S2 shows an exemplary comparison for DJF. We discuss only a small number of these precursor patterns explicitly in this paper, but the full set of 3 × 4 × 38 patterns is included in the Supplement and can be visualised at https://uib-precursors-cmip6-interactive.hf.space/app (last access: 15 April 2026) so that specific regional cases can be examined by the interested reader.

We quantify both the statistical sampling uncertainty and internal variability of our results. Bias terms for each model are computed using daily output of all ensemble members concatenated. Change terms are computed similarly by comparing the full future and historical ensembles. For both bias and change, sampling uncertainty is computed using 400 bootstrapped resamples of the daily ensemble data with replacement. Internal variability is quantified using the full ensemble spread, with all terms computed for each member separately.

Figure  visualises the decomposed terms of Eq. () for DJF southwestern Iberia in CESM2 and MPI-GE. In this case, both models have large and opposite dynamical biases (Fig. a): CESM2 generates too few days with strong precursors (i.e. δPS{8,9,10} < 0) while MPI-GE generates far too many, producing flows like Fig. Id almost a third of the time. The conversion terms from ERA5 clearly show that strong synoptic precursors (k = 10) are almost a prerequisite for heavy precipitation (Fig. b). CESM2 simulates the occurrence of heavy precipitation relatively well for moderately strong precursors (6 < k ≤ 9), with some underestimation for the strongest 10 % of flows, whereas MPI-GE exhibits markedly low heavy precipitation conversion for all of the strongest 40 % of precursor flows.

These flow-dependent biases can be aggregated into a three-term bias budget: 2model bias=P̃H-PH=∑kPH|SkξkPSk︸conversion bias+δPSk︸dynamical bias+ξk⋅δPSk︸non-linear bias These terms are shown in Fig. c, with the amplitudes of each term directly interpretable in units of altered heavy precipitation occurrence. CESM2 both undersamples strong precursor circulations (negative dynamical bias) and underestimates the conversion of strong precursors into heavy precipitation (negative conversion bias). The resulting dynamical and conversion biases compound, leading to a net bias of -2 %; heavy precipitation in the region only occurs 3 % of the time in CESM2 versus 5 % in ERA5. MPI-GE appears to perform better with a net bias of 1 %, but this is a misleading result that occurs due to compensation between dynamical and conversion biases. The positive dynamical bias of MPI-GE would imply a heavy precipitation occurrence of 8.5 %, but the model's severe under-conversion of dynamical forcing into heavy precipitation reduces this substantially. The two models' biases in this case are then not only of opposing sign, but of opposing type – compensating in MPI-GE and compounding in CESM2.

Different types of bias have different implications for model usability and interpretation. We propose five conceptual categories of bias which we discuss in detail in Sect. . To do this, we define a 2D bias space consisting of the conversion term, bconv, and the sum of the dynamical and non-linear terms

The non-linear term could also be grouped with the conversion term. However, as the most extreme conversion biases are much larger than the most extreme dynamical biases, a conversion+non-linear grouping can give bias terms with values lower than -100 %, which hurts interpretability.

The northern Adriatic in SON provides a second example with very different bias features. In this case heavy precipitation is not predominately driven by large-scale blocks or jet anomalies but by Rossby wave packets of zonal wavenumbers 6–7 , visible in the precursor patterns and synoptic composites of Fig. IIIa–h. Figure a reveals moderate dynamical biases: MPI-GE favours strong wave-trains in phases that both favour (k = 10) and suppress (k = 1) precipitation, while CESM2's bias towards negative precursors indicates a persistent Autumn ridge over central Europe (cf. Fig. IIIa). However these dynamical biases are minor overall in comparison to the conversion biases shown in Fig. b. Both models have severe challenges converting dynamical drivers into precipitation, a conversion which in the real world relies on factors these models cannot resolve: convection, diabatic processes over the warm, shallow Adriatic, and uplift and blocking from the Apennines and the Alps. As a result, conversion dominates the bias budget for both models and non-linear terms almost completely cancel the dynamical bias (Fig. c–d), representing that biases in rainfall-favouring dynamics are of little importance if a model would very rarely produce heavy rain anyway. In Sect.  we will discuss how usable information about future heavy precipitation can be obtained from models even in such cases of severe historical bias.

We now take a holistic view of model biases, considering each season and region, excepting extremely dry cases where the ERA5 heavy precipitation threshold is less than 2.5 mm. Figure  shows the impact of conversion biases to heavy precipitation, as given by Eq. (). The most coherent feature is the systematic under-estimation of JJA heavy precipitation conversion in both models, with occurrence biases of -60 % to -80 %. Similar, although typically weaker, conversion biases are seen in MAM and SON for central and southern Europe. Many of these cases correspond to wave-driven flows similar to the SON North Adriatic case or Mediterranean cutoff lows (see supplementary material), and it is plausible this low conversion captures poorly represented diabatic, convective and orographic processes.

In contrast, conversion biases in Northern Europe during MAM and SON can be positive, primarily in cases where large-scale zonal flows with fast eastward-extended jets form the main rainfall precursor. Conversion biases are most spatially diverse in DJF, perhaps surprisingly given the larger scale synoptic organisation. MPI-GE in particular struggles to convert dynamical precursors to heavy precipitation in the most mountainous domains. Regional downscaling over Norway increases DJF precipitation from convective sources and localises it on the coast , suggesting that resolution (possibly compounded by snow parameterisation errors) is likely the cause of these conversion biases.

Dynamical contributions to heavy precipitation bias are in general more spatially coherent than the conversion biases but are very different between the two models (Fig. ), and are partly linked to jet stream biases. CESM2 has positive dynamical biases in wintertime heavy precipitation over northern Europe (Fig. a) due to an overly strong and eastward extended jet , with negative dynamical biases over southern Europe due to a corresponding lack of wave-driven and negative NAO-like flows. Through the rest of the year, CESM2 shows dry biases in western Europe (Fig. b–d) due to a mispositioned jet and a persistent Atlantic ridge anomaly. MPI-GE has a very different dynamical bias pattern, with precipitation-causing weather patterns occurring excessively throughout SON, DJF and MAM. This is a signature of excessive wave activity and wintertime cutoff-lows in the East Atlantic, the latter of which may be related to MPI-ESM-LE's overly southward and zonal jet stream . This wet dynamical bias in MPI-GE is consistent with the “overshooting” precipitation found in higher resolution MPI-ESM models , since resolution increases may disproportionately reduce the model's negative conversion biases while leaving the positive dynamical bias relatively unchanged.

Figure  synthesises these biases using the 5-category bias categorisation introduced in Sect. . Results for all seasons and regions are also visualised in bias-space in Fig. S5, offering an alternate perspective. Firstly, we note that achieving minimal bias is possible for both models in some regions, even if fairly uncommon – the daily, regional, and near-extreme scales we are interested in are not inherently beyond the capabilities of CMIP6-class models. Secondly, we see that conversion biases are common, especially in JJA, as expected due to the relevance of summertime convection, and yet dominant dynamical biases can occur in any seasons. Perhaps most importantly, we emphasise that compensating biases – as we identified for MPI-GE during DJF over South West Iberia – are common, occurring in both models throughout the year. The two models tend to favour compensating biases of particular sign: CESM2 features excessive conversion for too-rare synoptic conditions, while MPI-GE counterbalances too-common synoptic conditions with weak conversion (cf. Fig. S5).

Decomposed precipitation changes can offer new insights into future scenarios, reveal sources of model disagreement, and uncover cases where model disagreement is greater than it initially appears. A good example of hidden model disagreement is given by JJA forced changes over Central UK and Ireland (Fig. ). Current large-scale summertime heavy precipitation in this region is driven by strong, localised cyclones (cf. Fig. II), and is dynamically suppressed by anticyclonic ridges over the east Atlantic (Fig. IIa, b, e, f).

Both climate models project a similar net decrease in heavy precipitation: a shift from 5 % to 4 % occurrence probability in CESM2, and to 4.5 % in MPI-GE. However, Fig. c shows that the decomposed budget of these forced changes is very different. MPI-GE's forced change is dominated by a negative dynamical contribution (Fig. a). CESM2's forced change by contrast is dominated by a negative conversion contribution (Fig. b). These are very different scenarios with differing implications: CESM2 describes a world with a similar number of strong summertime cyclones to today, but each less likely to cause heavy precipitation. MPI-GE describes a world with far fewer cyclonic anomalies but each more likely to produce heavy precipitation than those of today. Arguably the smaller net change in MPI-GE corresponds to the more volatile and impactful case.

Just as for biases in Sect. , we can construct a 2D space of conversion changes and summed dynamical and non-linear changes to define heuristic categories of forced changes. Figure d shows this change space, and highlights how ensemble mean signals from both models lie close to the same line of -20 % relative net change (dashed black line), therefore appearing in agreement on a statistically robust decrease in heavy precipitation events from a bulk perspective. However, Fig. c–d also demonstrates large internal variability in regional heavy precipitation changes: for any single 40-year period it is not a given that a decrease in heavy precipitation will be observed. The actual real-world implications of this are unclear as climate models have both insufficient low-frequency variability , including in European circulation features , and also suffer from a signal-to-noise problem which may lead to an overly-weak forced responses . Regardless, this example also demonstrates how decomposition of changes can help identify signals on which models are confident: while the MPI-GE absolute net change in individual members spans from -1.7 % to 1.5 %, the dynamical change is confidently negative for each member, -1.5 % to -0.3 %.

Figures  and  show the model conversion, and dynamical changes, respectively. Non-linear changes are small, as shown in Fig. S6. As our decomposition is based on the dominant precipitation driver in observations, it is conceivable that dynamical changes could be misattributed to the conversion term if new dynamics become important drivers of heavy precipitation in the future. Figures S7 and S8 indicate little evidence for this, with future conversion changes driven almost exclusively by changes in conversion during strong precursor situations (k = [8,9,10]).

Bulk forced changes, shown in Fig. , indicate a qualitative agreement between CESM2 and MPI-GE on future heavy precipitation occurrence. Broadly speaking, both models project a near zonally symmetric pattern of changes, with an increase in heavy precipitation in northern regions and a decrease in southern regions. The latitude at which the forced change reverses varies between seasons: decreased heavy precipitation below ≈ 55° N in JJA and 40° N in DJF for example. However, decomposing the sources of those changes reveals greater spatial asymmetry and considerable differences between the models.

In wintertime, conversion-driven increases in heavy precipitation occurrence of approx. 40 % occur in both models across all regions outside southwestern Europe (Fig. a, e). Notably, this includes the southeastern regions, where strong, negative dynamical changes drive decreased heavy precipitation overall (Fig. a, e). This implies a decline of eastern Mediterranean winter cyclones, each individually more likely to cause heavy precipitation but leading to a net negative change overall. This is consistent with CMIP6 results from a cyclone-focused analysis , although they found the forced signal changed dramatic dramatically in higher resolution simulations.

In northern Europe, the models disagree on sign and amplitude of dynamically-driven changes, with little signal in MPI-GE and 50 %–90 % increases in heavy precipitation in CESM2, caused by more frequent and stronger eastward jet extensions. CESM2's negative dynamical changes along the north Norwegian coast may be linked to the model's pronounced North Atlantic warming hole and the decline of cold-air outbreaks into the Greenland Sea . In MAM and SON, MPI-GE predicts a decline in heavy precipitation over southwestern regions due to compounding conversion and dynamical changes (Figs.  and f, h). In JJA, MPI-GE shows negative conversion changes in much of west and south-east Europe (Fig. e). CESM2 has similar conversion changes to MPI-GE (Fig. b–d), but different dynamical changes. CESM2 shows positive dynamical changes in MAM and SON in Northern Europe (Fig. b–d) again related to a projected eastward jet extension, while negative dynamical changes in JJA relate to a projected less wavy summertime-flow (the reliability of which remains contentious; ). It has recently been suggested that JJA extreme precipitation may be “shifting” to MAM and SON , as summer rainfall decreases and shoulder season rainfall increases, consistent with Fig. . We see here that this is not a coherent phenomena, but rather is dynamically driven in some regions and seasons, and driven by local-scale conversion changes in others.

The differences across space and between the two models are highlighted through the categorisation of heavy precipitation changes shown in Fig.  (and also in a 2D-space in Fig. S8). Predominately dynamical changes are rare, while changes in the relationship between synoptic dynamics and the surface (that is, a change in conversion) are nearly ubiquitous. However, dynamical changes cannot be neglected, as the majority of cases feature compensating and compounding changes where dynamical and conversion contributions are of similar magnitude.

If a model is very severely biased in a particular region we may not trust even a corrected estimate of the model's future forced changes as discussed above. However, flow decomposition can allow usable information to be extracted even in such cases. Considering again the northern Adriatic in SON, where both models showed severe conversion biases (cf. Sect. ) we find that both models project future increases in heavy precipitation probability (positive net change in Fig. c). We can use the additive reformulation of αk introduced in Sect.  to compute conversion changes of ≈ +2 %, but it is debatable whether this information should be used: the magnitude of the conversion biases indicates that the underlying real-world processes that will drive future conversion changes are simply not accounted for. The dynamical changes however can be considered quite independently. CESM2 projects negligible changes in the wave-driven dynamics that drive North Adriatic precipitation (Fig. a). MPI-GE shows signs of fewer mid-Mediterranean troughs and more ridges, resulting in a negative (-1 %) dynamical contribution to changes in heavy precipitation, and a negative dynamical change in every ensemble member (Fig. c). These dynamical changes are based on the model's dynamics and real-world precipitation conversion, and so can be considered a valid projection despite the model's conversion errors. Decomposing the overall change into these contributions allows each to be assessed and acted upon on its own merits, increasing the usability of the simulations overall.

As we have shown in Sects.  and , flow-dependent biases – especially compensating biases – can distort changes. In the worst case, these distortions can be of the same magnitude as the true change. This was the case for CESM2's projections of wintertime heavy precipitation: the model's projected strengthening of the wintertime jet would imply an increase in northern European heavy precipitation probability 30 %–60 % larger than direct model output indicates, given the observed synoptic-to-precipitation relationship.

Errors in a model's synoptic circulation often partially cancel out deficiencies in the conversion of dynamical forcing into heavy precipitation by mesoscale, boundary layer, land surface, thermodynamic and/or microphysical processes (cf. Fig. ). In these cases, models are less trustworthy than they initially appear, and improvements of model processes may worsen their bulk performance. This could explain the unexpectedly poor results from increasing model resolution without retuning, and warns that expensive regional downscaling efforts may prove disappointing when driven by models with compensating errors. While regional modellers select global models with an eye to dynamical biases , this focus is mostly limited to mean biases and dominant variability modes, which can hide compensating errors. With a flow-dependent decomposition of precipitation, we can identify, explain and quantify these and other cases, visualising them either in a 2D space (e.g. Fig. d) or on a map (e.g. Fig. ).

While the exact thresholds used to define categories should be tuned to different use cases, the practical interpretation of these conceptual categories remains the same:

Minimal bias. With both realistic dynamics and conversion processes, regional precipitation from the model can be used at face-value.

As demonstrated (cf. Fig. ), models may produce the same net change while representing very different futures, with different narratives for how weather and climate evolve. Different future scenarios can be efficiently distinguished with flow-dependent metrics and quickly visualised with a categorisation of forced changes, just as for biases:

Minimal change. Future heavy precipitation will resemble the present.

Decomposing model behaviour is not simply a way to critique existing climate simulations, but offers opportunities to improve their value. For moderate biases, we have shown how flow-dependent decomposition provides a flow-aware bias correction and recalibration of forced changes. Even when models arguably cannot represent regional conversion processes well enough to simulate their future change, as we suggest for the North Adriatic region (cf. Figs.  and ), the dynamical component of heavy precipitation change can still be isolated and understood.