Diabatic transport schemes with hybrid zeta coordinates, which follow isentropes in the stratosphere, are known to greatly improve Lagrangian transport calculations compared to the kinematic approach. However, some Lagrangian transport calculations with a diabatic approach, such as the Chemical Lagrangian Transport Model of the Stratosphere (CLaMS), are not well prepared to run on modern high-performance computing (HPC) architectures. Here, we implemented and evaluated a new diabatic transport scheme in the Massive-Parallel Trajectory Calculations (MPTRAC) model. While MPTRAC can be used either with shared-memory multiprocessing on CPUs or with GPUs to offload computationally intensive calculations, making it flexible for many HPC applications, it has been limited to kinematic trajectories in pressure coordinates. The extended modelling approach now enables the use of either kinematic or diabatic vertical velocities and the coupling of different MPTRAC modules based on pressure or hybrid zeta coordinates.

This study focus on the accuracy of the implementation in comparison to the CLaMS model. The evaluation of the new transport scheme in MPTRAC shows that, after 90 d of forward calculations, distributions of air parcels in the upper troposphere and lower stratosphere (UTLS) are almost identical for MPTRAC and CLaMS. No significant bias between the two Lagrangian models was found. Furthermore, after 1 d, internal uncertainties (e.g. due to interpolation or the numerical integration method) in the Lagrangian transport calculations are at least 1 order of magnitude smaller than external uncertainties (e.g. from reanalysis selection or downsampling of ERA5). Differences between trajectories using either CLaMS or MPTRAC are on the order of the combined internal uncertainties within MPTRAC. Since the largest systematic differences are caused by the reanalysis and the vertical velocity (diabatic vs. kinematic), the results support the development efforts for trajectory codes that can access the full resolution of ERA5 in combination with diabatic vertical velocities. This work is part of a larger effort to adapt Lagrangian transport in state-of-the-art models such as CLaMS and MPTRAC to current and future HPC architectures and exascale applications.

The Massive-Parallel Trajectory Calculations (MPTRAC) model is a Lagrangian transport model that was developed with support for shared-memory multiprocessing on CPUs and offloading to GPUs to efficiently run on modern high-performance computing (HPC) architectures

However, unlike MPTRAC, CLaMS can be used with diabatic vertical velocities and a hybrid vertical coordinate (referred to as a hybrid zeta coordinate or zeta coordinate). Diabatic vertical velocities are calculated from the energy balance instead of the mass balance, as in the case of kinematic vertical velocities. The hybrid zeta coordinate was first introduced by

Earlier versions of MPTRAC were formulated in pressure coordinates only and were run with kinematic vertical velocities

In this study, we evaluate the accuracy of the newly implemented scheme in MPTRAC through a detailed intercomparison with results from the CLaMS model and by placing model differences in the context of the sources of uncertainty inherent in Lagrangian transport models. Uncertainty sources of Lagrangian transport models have been studied extensively in the past

External uncertainties of Lagrangian transport simulations due to differences between the used wind data are discussed frequently

Internal uncertainties related to different integration methods applied in MPTRAC have been investigated by

Differences between transport models have been studied as well. Differences in transport using different Lagrangian models (MPTRAC, CLaMS) driven by

To justify the fact that MPTRAC and CLaMS trajectory calculations can mutually substitute each other, the MPTRAC and CLaMS models do not to need to be bit-identical, but deviations must be much smaller than those from external uncertainty sources, e.g. reanalysis differences, vertical velocities, and sub-grid-scale diffusion, and must be on the order of combined internal uncertainties. In our study, we show that, after implementing hybrid zeta coordinates and diabatic vertical velocities in MPTRAC, MPTRAC, and CLaMS, results of forward-trajectory calculations differ only insignificantly. CLaMS and MPTRAC trajectory calculations can substitute each other, which bears a path forward for combined CLaMS–MPTRAC simulations in upcoming HPC systems. Further, we quantify and order in more detail the sources of transport uncertainties that are found in Lagrangian models and the driving data.

In Sect.

Diabatic transport calculations in hybrid zeta coordinates were implemented in MPTRAC, similarly to CLaMS. Lagrangian transport calculations rely on, first, the Lagrangian transport model itself and, second, the input wind fields that drive the model. In the following sections, the implementation of diabatic transport into MPTRAC and CLaMS, the used meteorological data, and the used diagnostic to evaluate diabatic transport in MPTRAC are described in detail.

CLaMS is a comprehensive chemical Lagrangian transport model including irreversible mixing and stratospheric chemistry

CLaMS applies the vertical hybrid zeta coordinate (

Equation (

However, the diabatic approach also has disadvantages, such as the need to smooth zeta profiles that are not monotonic with height and the fact that parameterizations developed for pressure coordinates are not accessible and would have to be reformulated. In our new implementation of diabatic vertical velocities in MPTRAC, we avoid the latter by performing the calculation of advection in zeta coordinates but transforming the zeta coordinates to pressure coordinates after advection and vice versa, from pressure to zeta coordinates before advection. In this way, other modules of MPTRAC (diffusion, convection, sedimentation, etc.) can still operate with pressure as the vertical coordinate, for which they were originally developed.

To compute Lagrangian trajectories, the ordinary differential equation

For an integration time step

For the integration of the diabatic transport scheme into MPTRAC, MPTRAC was equipped with functions to read the vertical velocities of the hybrid zeta coordinate (

For the Runge–Kutta method, wind fields must be interpolated four times to the given time, horizontal location, and zeta height. For the mid-point scheme, this is reduced to two interpolations. For MPTRAC and CLaMS, four-dimensional linear interpolation methods are performed, which are common for Lagrangian transport models

As a consequence of the mentioned differences between the models, the interpolations of CLaMS and MPTRAC follow two different concepts. Figure

To overcome this issue, MPTRAC instead starts with the horizontal interpolation of the zeta values and pressure values according to the horizontal air parcel position (

Concept of the interpolation illustrated in two dimensions for

Figure

Schematic steps during interpolation V0 of a quantity

The interpolation from pressure to zeta and from zeta to pressure is particularly important when coupling geophysical modules that operate with pressure as the vertical coordinate (e.g. convection, diffusion, and sedimentation), as is the case for MPTRAC. The precise and accurate inversion of the interpolation in CLaMS from pressure back to zeta coordinates is difficult because, during step (6), height indices can be found from the pressure that are inconsistent with height indices found using the zeta coordinate positions. Then, significant errors may occur, making this approach unsuitable for frequent transformations between zeta and pressure coordinates. Consequently, a fully reversible interpolation algorithm has been developed for MPTRAC to allow the coupling of pressure-based modules with the diabatic advection scheme, where frequent vertical coordinate inversions are required.

Figure

The algorithm in MPTRAC allows precise interpolation from zeta to pressure and back to zeta because the vertical column at the horizontal position of the air parcel gives a monotone relationship between zeta and pressure. In particular, the processing of pressure and zeta is analogous with opposite roles. The vertical 1D linear interpolation at the final step (9) can be performed accurately and unambiguously.

Schematic steps during interpolation V2 of a quantity

For comparison and error estimations, a third interpolation variant was implemented into MPTRAC, resembling more closely the interpolation in CLaMS (called interpolation “V3”). In this approach, the interpolation procedure follows the first steps (1) to (5) as defined in V2 and Fig.

However, note that all interpolations in MPTRAC are performed in Cartesian coordinates. That is, the line elements of the spherical-coordinate system are not applied to the air parcel positions during interpolation – rather, this is done so afterwards, assuming that the differences in the line elements within a grid box are negligible. The transformation from Cartesian coordinates to spherical coordinates is done separately from the interpolation process by applying the equations

Finally, pressure is interpolated logarithmically in CLaMS for zeta levels higher than 1000 K and linearly for levels below 500 K. In between those levels, the linear and logarithmic interpolations are combined. In contrast, MPTRAC uses linear interpolation for pressure on all hybrid zeta levels.

MPTRAC uses spherical coordinates to store the position of air parcels. CLaMS has a hybrid approach, with spherical coordinates for air parcels at latitudes between

In MPTRAC the spherical-coordinate singularity is handled differently. In MPTRAC, for air parcels very close to the pole (i.e. closer than 110 m or 0.001° latitude), the zonal transport is ignored. Horizontal coordinates are calculated with double precision to guarantee the required accuracy for this approach. The method has been shown to be reliable for different applications

Both models use the shallow-atmosphere approximation. This means that the horizontal plane is transformed from spherical to Cartesian coordinates, assuming that the height of the air parcel is negligible with respect to the Earth's radius. The two models have slight differences in the Earth's radius. In MPTRAC's default setting, the Earth's radius is assumed to be 6367.421 km, whereas in CLaMS, it is 6371.000 km. This has implications for transformations between the Cartesian and spherical-coordinate systems.

The full-resolution ERA5, downsampled ERA5, and ERA-Interim reanalyses were used to run the forward trajectory calculations with CLaMS and MPTRAC. ERA5 and ERA-Interim are provided by the ECMWF

The ERA5 reanalysis provides hourly meteorological data with 30 km horizontal grid resolution (sampled at 0.3°

The downsampled version of ERA5 (referred to as ERA5 1°

The implementation of the diabatic transport scheme in MPTRAC, used with the ERA5 reanalysis, is evaluated by a detailed intercomparison with CLaMS trajectory calculations for a global ensemble of air parcels. To put the differences found in the trajectory calculations between CLaMS and MPTRAC in a broader context, the effects of, first, external sources (using different reanalyses, resolutions, and vertical velocities) and, second, internal sources (e.g. interpolation and integration methods) were investigated. For the evaluation of the newly implemented diabatic scheme in MPTRAC, we use a model initialization with about 1.4 million globally distributed trajectory seeds. The forward calculations are calculated for the boreal summer (June, July, August). Short-term calculations of 1 d are initialized on 1 July 2016, while the long-term calculations of 90 d are started on 1 June 2016 to cover the entire boreal summer and austral winter. Seasonal differences are taken into account by separately analysing the Northern Hemisphere and the Southern Hemisphere. The air parcels are distributed horizontally quasi-homogeneously so that they have an average mutual distance of about 100 km. Vertically, they are distributed in specific layers. The layers are constructed such that each air parcel represents the same amount of entropy in the atmosphere, which is a product of density and the logarithm of the potential temperature

We employ different simulation scenarios to put the deviations of the two models into the perspective of known uncertainty sources. Table

Overview of different simulation scenarios for transport calculations with MPTRAC and CLaMS.

Scenario intercomparisons for the estimation of different uncertainties in the Lagrangian transport calculations. Two scenarios are compared (base and comparative scenario) for the estimation. In most cases, only one aspect of the model setup is varied. The first block focuses on internal uncertainties of CLaMS and MPTRAC separately. The second block focuses on the comparison of the two models. The third block focuses on the external uncertainties. The last block describes the “transport”, i.e. the difference between the start and the end point of a trajectory. Transport is not an uncertainty source, but it is a useful quantity for intercomparison with the uncertainties.

The sources of uncertainty are classified into internal, model, and external uncertainties. The first block of uncertainties in Table

Model differences (“model default”, “model default 1°

External uncertainties are used to show the significance of calculations with the fully resolved ERA5 and diabatic vertical velocities. Therefore, different reanalysis products such as ERA5, ERA-Interim, and ERA5 1°

Finally, we introduce the deviation between the initial position of the air parcels and their final position as a physical reference to compare with. This deviation is labelled “transport” in Table

For the intercomparison of the different model scenarios, a set of frequently used diagnostics was applied

The log-pressure altitude is defined as

To measure the conservation error of a quantity

The simulations are initialized globally and cover almost the entire height range of the free troposphere and stratosphere (about 1–50 km), allowing for the analysis of numerous meteorological conditions and different trajectories. Figure

The model deviations are significantly smaller than deviations from external sources such as downsampling of reanalysis data, different vertical velocities, variations in reanalysis data sets (here from ERA5 to ERA-Interim), or the influence of atmospheric diffusion. Trajectories with ERA5 1°

Examples of trajectories calculated 10 d forward from 1 July 2016. Different scenarios in three layers are shown: The

Different AVTDs in zeta coordinates after 1 d forward calculations for the entire ensemble of air parcels split into four height layers. The different uncertainty sources are defined in Table

AVTDs in log-pressure heights after 1 d forward calculations for the entire ensemble of air parcels split in four height layers. The boxplots indicate quantiles as defined in Fig.

Horizontal deviations quantified with the AHTDs after 1 d forward calculation for the entire ensemble of air parcels split in four height layers. The boxplots indicate quantiles as defined in Fig.

Figure

Separately assessed, the variation of the Earth radius, the time step variation from 180 to 1800 s in the Runge–Kutta method, and the interpolation variation of MPTRAC are estimated to cause median AVTD lower than

Only limited to trajectories in proximity to the poles, uncertainties due to the coordinate singularity must be considered. However, the transformation from spherical coordinates to the stereographic projection at high latitudes causes vertical deviations similar to deviations related to the selection of the integration method. If only air parcels that start at latitudes larger than 72° north or south are considered statistically, the median AVTD in zeta coordinates is on the order of

The CLaMS and MPTRAC models can also be configured to operate more similarly (i.e. using the same Earth radius and integration method and a similar interpolation method) so that the model uncertainty is substantially reduced (see Fig.

Already, with the default configuration, there are model deviations for CLaMS and MPTRAC that are 1 to 3 orders of magnitude smaller than deviations resulting from external factors (see Fig.

Figure

Moreover, in the stratosphere, the median AVTD in log-pressure coordinates between the initial and final positions after 1 d (see label “transport”) is larger than the deviation from vertical diffusion in the pressure coordinate, which is in contrast to the median AVTD in zeta coordinates, because the transport in the UTLS is mostly isentropic and hence might cross multiple isobars but fewer isentropes (see Figs.

When the AHTDs are considered, qualitatively similar results to the vertical transport deviations are obtained. The horizontal model differences and combined internal uncertainties are of the order of 0.1 to 1 km after 1 d of calculations, while external uncertainties lead to absolute horizontal deviations of the order of 1 to 100 km (see Fig.

From an overall statistical perspective, as depicted by the Figs.

The profiles of the transport uncertainties are similar in the two hemispheres. However, if hemispheres are compared in more detail, the strongest relative internal uncertainties are found in the winter hemisphere, i.e. the Southern Hemisphere (see Fig.

Moreover, the vertical profiles in Fig.

Smoothed vertical profiles of hemispheric average AVTD in zeta coordinates for different uncertainty sources (see Table

To investigate the uncertainty growth between the CLaMS and MPTRAC models and to better understand the model differences in the context of other uncertainties, trajectory calculations were performed for 90 d starting from 1 June 2016. Figure

The model deviations and other transport uncertainties vary with height. In the troposphere, the median AVTD of the external uncertainties remains below 1 K only for a short period (a few hours to days) due to the strong mixing and convection. Subsequently, external uncertainties in this region grow rapidly by up to 4.3 K per day. In particular, the selections of the reanalysis, the vertical velocity, and downsampling cause fast divergence in the troposphere. The median model AVTD is smaller than uncertainties related to changes in reanalysis data, downsampling of the data, or parameterized sub-grid-scale winds and diffusion. The median AVTD between the two models remains below approximately 1 K for the first week. Subsequently, there is also a sharp increase (up to 2.2 K per day), reaching a median AVTD of about 55 K at 40 d of simulation time, where the different uncertainties reach a similar magnitude.

In the lower and upper stratosphere, the AVTD remains smaller because air parcels mainly move isentropically. Additionally, horizontal mixing is much less in most regions of the lower and upper stratosphere in contrast to the troposphere. The median model AVTD is, again, much smaller than all other uncertainty sources but now for the entire 90 d integration period. In the lower stratosphere, 50 % of the air parcels have a model AVTD lower than 1 K for approximately 2 months, and, afterwards, the deviation still increases slowly (not more than 0.16 K per day). In the upper stratosphere, the same criterion is met after around 34 d, also with a slow to moderate increase afterwards (not more than 1.2 K per day).

Uncertainties from the selection of the vertical velocity and the reanalysis are of similar importance. In the UTLS and at higher altitudes, the variation of the vertical velocity first shows slightly larger median AVTD than the variation of the reanalysis. However, after a couple of weeks, the median AVTD from the reanalysis selection is higher because the choice of the vertical velocity does not affect the horizontal wind speeds, as is the case for the choice of the reanalysis. The smallest transport uncertainty from external sources throughout the atmosphere is given by the ERA5 1°

The differences between the two models have an impact on the horizontal distribution of the air parcels as well (Fig.

Evolution of the median AVTD in the zeta coordinate for different uncertainty sources for 90 d. The median AVTD between the two models is labelled “default”, as defined in Table

Evolution of the median AHTD of different uncertainty sources for 90 d. The median AHTD between the two models is labelled “default”, as defined in Table

Since individual trajectories are not expected to agree over time periods of several months, the statistical distribution of air parcels after a 90 d integration period is used to quantify the differences between the models and the uncertainty related to external sources. The air parcels were initialized on 1 June 2016. For reference, the initial density of the air parcels is shown in Fig.

Furthermore, more air parcels are leaving the Northern Hemisphere than entering it in the calculations, i.e. the cross-equatorial flow in the UTLS increases the air parcel density in the Southern Hemisphere in relation to the Northern Hemisphere. Since the air parcels were initialized on 01 June 2016, the simulation describes the boreal summer conditions. As indicated by averaged trajectories in Fig.

Initial and final air parcel distribution after 90 d when calculated with the CLaMS default setup. Black lines show box-wise averaged trajectories to indicate the average circulation of the trajectories. The dotted orange line indicates the 90 d average tropopause.

The global distribution of air parcels as simulated with MPTRAC is almost identical to the distribution as simulated with CLaMS, as can be seen in Fig.

When the diffusion module (see Fig.

The downsampling of the ERA5 data (see Fig.

With ERA-Interim, qualitatively very different results are found (see Fig.

The biases between simulations with diabatic and kinematic vertical velocities in ERA5 are of similar size compared to the biases between simulations with ERA-Interim and ERA5 (see Fig.

With kinematic velocities, increased air parcel numbers can be found closely above the tropopause as well in comparison to the diabatic calculations. This possibly indicates increased transport across the tropopause from below. However, for the kinematic velocities, higher numbers of air parcels are found in the troposphere because the applied criteria for excluding air parcels from further transport (reaching the level where the zeta coordinate is zero) are not fulfilled. Therefore, the increase in air parcels closely above the tropopause could be a consequence of higher air parcel numbers remaining in the troposphere as well.

Zonal mean bias of the air parcel distributions after 90 d between the default MPTRAC scenario and a selected scenario. Positive bias indicates lower frequency with the default MPTRAC scenario and higher frequency with the respective scenario. The dotted orange line is the 90 d average tropopause. The green contours show the 600, 1000, and 1400 air parcel numbers per cell contour of the air parcel distributions for intercomparison with the scenarios of

In the stratosphere, the potential temperature (

Evolution of the mean RTCE of

In this study, a diabatic transport scheme based on hybrid zeta coordinates was implemented in the MPTRAC Lagrangian transport model. This work's intention is to enable a transition from the CLaMS Lagrangian transport framework towards a code which enables shared-memory multiprocessing with CPUs or offloading to GPUs and which is, hence, more suitable for recent and upcoming HPC architectures. To evaluate the implemented scheme in MPTRAC, we conducted simulations using approximately 1.4 million globally distributed air parcels in the troposphere and stratosphere, following an initialization method commonly employed with CLaMS. Trajectory forward calculations were performed for the boreal summer of 2016. In the evaluation, the model differences were put in the context of various other uncertainty sources in Lagrangian transport calculations. Consequently, the model differences between CLaMS and MPTRAC were presented within a hierarchy of uncertainties associated with Lagrangian transport models.

The key differences between the two Lagrangian models relate to their approach for the interpolation of the driving meteorological data and the numerical integration scheme. Although both models apply four-dimensional linear interpolations, CLaMS performs them directly in spherical coordinates, while MPTRAC performs them in Cartesian coordinates. As a default, CLaMS uses the classical fourth-order Runge–Kutta scheme with 1800 s integration steps for numerical integration in order to run with feasible computational costs. MPTRAC employs the mid-point scheme with 180 s integration time steps. At a time step of 180 s, both integration schemes deliver very similar results. The residual differences between the models are likely caused by remaining differences in the interpolation. For improved agreement, CLaMS and MPTRAC should use the identical Earth radius. Further alignment of the interpolations could achieve even better agreement.

Despite the conceptual model differences, it was demonstrated that, for a period of 1 d, the discrepancy between CLaMS and MPTRAC air parcel vertical positions is comparable to the combined internal uncertainties associated with different Earth radii, interpolation methods, numerical integration schemes, and selected integration time steps. These deviations are, at a minimum, around 1 order of magnitude smaller than the uncertainties arising from external sources, such as differences between reanalysis data sets, downsampling of the ERA5 reanalysis data, and unresolved fluctuations of the wind fields. Thus, the analysis of the model differences indicates an excellent agreement of CLaMS and MPTRAC within the boundaries of known internal and external uncertainties. This also holds in the regions of the most notable differences, including the troposphere and the winter stratosphere with the polar vortex.

The uncertainty growth between the models and from external sources for 90 d was also estimated. The vertical transport uncertainty remains less than around 1 K for several weeks, in particular in the stratosphere. The transport deviation between the models is significantly smaller than the deviation caused by external sources of uncertainty for the entire 90 d time period. In particular, large uncertainty growth from variations of the vertical velocity (diabatic to kinematic) shows that the implementation of the diabatic transport scheme in MPTRAC has a significant impact on the transport of air parcels.

For a global, long-term study of trace gases, the statistical distribution of air parcels in the UTLS, as opposed to individual trajectory errors, becomes more important. In their present configurations, both models distribute air parcels very similarly even after 90 d, supporting the hypothesis that the models provide similar long-term tracer fields. Accordingly, no biases in the air parcel distributions were found between the two models. In contrast, known external uncertainties caused significant biases in the trajectory calculations over the 90 d integration period.

The diabatic transport calculations show that transport within the BDC is faster with ERA-Interim than with ERA5 between 400 and 600 K but slower for higher levels. This is in agreement with recent climatological and regional studies of vertical velocities and transport in the upper troposphere and stratosphere

Differences between calculations with diabatic and kinematic vertical velocities, even with ERA5, are still on the order of reanalysis differences, further corroborating the implementation of the diabatic scheme in MPTRAC. However, the difference between diabatic and kinematic calculations is significantly reduced with ERA5 in comparison to ERA-Interim with regard to the vertical transport in the circulation of the lower stratosphere but also concerning the cross-isentropic dispersion in the tropical lower stratosphere.

Furthermore, since model and internal uncertainties of the trajectory models are much smaller than uncertainties due to downsampling of ERA5 data, it can be concluded that using ERA5 1°

In conclusion, this evaluation demonstrates that MPTRAC can replace CLaMS' trajectory module with the newly implemented hybrid zeta coordinates and diabatic transport scheme. The evaluation found no significant biases or deviations between the models but highlights the significance of using the high-resolution ERA5 reanalysis combined with diabatic transport. Furthermore, now that the implementation has been validated, additional performance analyses and optimizations can be carried out.

To clarify the differences in the circulation patterns during the 90 d integration period, Fig.

Vector differences of end points of box-wise averaged trajectories for the 90 d transport calculations in the UTLS and troposphere. The arrows indicate how the final MPTRAC-default positions have to be adjusted to agree with the respective scenario. Accordingly, they show the trajectory biases.

The agreement between kinematic and diabatic trajectories might differ from reanalysis to reanalysis in terms of the general circulation and cross-isentropic dispersion of air

The cross-isentropic dispersion of air parcels can be quantified with the variance of the potential temperature

Intercomparison of air parcel distributions and biases inferred from diabatic (black contours) and kinematic (green contours) calculations for ERA5

The kinematic calculations with ERA5 still have a higher dispersion in comparison to the diabatic calculations, which supports the implementation of the diabatic scheme in MPTRAC. The variance is around 3 times higher with the kinematic calculations between 400 and 550 K. However, the dispersion by parameterized turbulent diffusion and sub-grid-scale wind fluctuations is much higher (see Fig.

Profiles of the zeta variance

The CLaMS code can be accessed from the Jülich GitLab server:

Conceptualization: JC, LH, BV, SG, and NT. Data curation: JC, LH, and NT. Formal analysis: JC. Funding acquisition: LH and BV. Methodology: JC, LH, BV, SG, and NT. Project administration: LH and BV. Resources: LH, BV, SG, and NT. Software: JC. Supervision: LH and BV. Validation: JC and LH. Visualization: JC. Writing – original draft: JC. Writing – review and editing: JC, LH, BV, and SG.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research has been supported by the Helmholtz Association of German Research Centres (HGF) through the Joint Lab Exascale Earth System Modelling (JL-ExaESM). We acknowledge the Jülich Supercomputing Centre for providing computing time and storage resources on the JUWELS supercomputer. Jan Clemens was partly funded by the Helmholtz Interdisciplinary Doctoral Training in Energy and Climate Research (HITEC). We also thank the ECMWF for providing access to the ERA5 and ERA-Interim reanalysis data. We also acknowledge modern, AI-based spelling software (DeepL, ChatGPT) which was limitedly used to check, correct, and improve the language of this paper. We thank Paul Konopka and Felix Plöger for the discussion of the results and methods.

The article processing charges for this open-access publication were covered by the Forschungszentrum Jülich.

This paper was edited by Volker Grewe and reviewed by two anonymous referees.