In order to enable trajectory calculations, MPTRAC requires an input file providing the initial positions of the air parcels. The initial positions are defined in terms of time, log-pressure height, longitude, and latitude. Internally, MPTRAC applies pressure as its vertical coordinate. For convenience, the initial pressure p is specified in terms of a log-pressure height z obtained from the barometric formula, p(z)=p0exp⁡-zH, with a fixed reference pressure p0=1013.25hPa and scale height H=7km. Longitude and latitude refer to a spherical Earth approximation, with a constant radius of curvature of RE=6367.421km.

Internally, MPTRAC uses the time in seconds since 1 January 2000, 00:00 UTC as its time coordinate. Tools are provided along with the model to convert the internal time from and to UTC time format. The internal time step Δt of the model simulations can be adjusted via a control flag. MPTRAC allows for arbitrary start and end times of the trajectories by adjusting (shortening) the initial and the final time steps to synchronize with the internal model time steps Δt. A control flag is provided to specify whether the model should conduct the trajectory calculations forward or backward in time.

Unless the initial positions of the trajectories are derived from measurements, a set of tools provided with MPTRAC can be used to create initial air parcel files. The tool atm_init creates a regular grid of air parcel positions in time, log-pressure height, longitude, and latitude. Around each grid point and time, the tool can create random distributions of multiple particles by applying either uniform distributions with given widths or normal distributions with given standard deviations. The total mass of all air parcels can be specified, which is distributed equally over all air parcels. Alternatively, a fixed volume mixing ratio for each air parcel can be specified. With this tool, initializations for a wide range of sources (e.g., point source, vertical column, surface, or volume source) with spontaneous or continuous emissions can be created.

A tool to modify air parcel positions is atm_split. It can be used to create a larger set of air parcel positions based on randomized resampling from a given, smaller set of air parcel positions. It splits the initial small set into a larger set of air parcels. The total number of air parcels that is desired for the large set can be specified, which is helpful because it provides control over the computational time needed for a simulation. The probability by which an air parcel is selected during the resampling can be weighted by its mass. The total mass of the initial air parcels can be retained or adjusted during the resampling process. During the resampling process, uniform or Gaussian stochastic perturbations in space and time can be applied to the air parcel positions. These stochastic perturbations can be used to spread the air parcels over a larger volume, which might represent the measurement volume covered by an instrument, for example. Finally, a vertical weighting function can be applied in the resampling process in order to redistribute the air parcel positions to follow the vertical sensitivity of a satellite sounder, for instance.

Next to the particle positions, the other main input data required by MPTRAC are meteorological data from a global forecasting or reanalysis system. At a minimum, MPTRAC requires 3-D fields of temperature T, zonal wind u, and meridional wind v. The vertical velocity ω, specific humidity q, ozone mass mixing ratio xO3, cloud liquid water content (CLWC), cloud rain water content (CRWC), cloud ice water content (CIWC), and cloud snow water content (CSWC) are optional but often needed for specific applications. The model also requires the surface pressure ps or log-surface pressure (ln⁡ps) and the geopotential height at the surface Zg to be provided as 2-D fields. Optionally, near-surface temperature T2 m and near-surface winds (u10 m,v10 m) can be ingested from the input files.

The model requires the input data in terms of separate NetCDF files at regular time steps. The meteorological data have to be provided on a regular horizontal latitude × longitude grid. The model internally uses pressure p as its vertical coordinate; i.e., the meteorological data should be provided on pressure levels. However, the model can also ingest meteorological data on other types of vertical levels, e.g., hybrid sigma vertical coordinates η as used by ECMWF's Integrated Forecast System (IFS). In this case, the model requires p as a 3-D coordinate on the input model levels in order to facilitate vertical interpolation of the data to a given list of pressure levels for further use in the model.

Once the mandatory and optional meteorological input variables have been read in from disk, MPTRAC provides the capability to calculate additional meteorological variables from the input data. This includes the options to calculate geopotential heights, potential vorticity, the tropopause height, cloud properties such as cloud layer depth and total column cloud water, the convective available potential energy (CAPE), and the planetary boundary layer (PBL). Having the option to calculate these additional meteorological data directly in the model helps to reduce the disk space needed to save them as input data. This is particularly relevant for large input data sets such as ERA5. It is also an advantage in terms of consistency, if the same algorithms can be applied to infer the additional meteorological variables from different meteorological data sets. The algorithms and selected examples on the meteorological data preprocessing are presented in the Supplement of this paper.

The upper boundary of the model is defined by the lowest pressure level of the meteorological input data. The lower boundary of the model is defined by the 2-D surface pressure field ps at a given time. If an air parcel leaves the pressure range covered by the model during the trajectory calculations, it will be pushed back to the lower or upper boundary of the model, respectively. Therefore, the particles will follow the horizontal wind and slide along the surface or the upper boundary until they encounter an up- or downdraft, respectively. For the surface, this approach follows . The next release of the MPTRAC model will provide a new option to alternatively reflect particles at the top and bottom boundaries, in order to avoid accumulation of particles at the boundaries.

We recognize that pressure might not be the best choice for the vertical coordinate of a Lagrangian model, in particular for the boundary layer, because a broad range of surface pressure variations needs to be represented appropriately by a fixed set of pressure levels. However, as the scope of MPTRAC is on applications covering the free troposphere and the stratosphere, we consider the limitation of having reduced vertical resolution in terms of pressure levels near the surface acceptable for now. The next release of MPTRAC will allow the user to choose between pressure and the isentropic-sigma hybrid coordinate (ζ) of as vertical coordinate.

For global meteorological data sets, MPTRAC applies periodic boundary conditions in longitude. During the trajectory calculations, air parcels may sometimes cross either the pole or the longitudinal boundary of the meteorological data. If an air parcel crosses the North or South Pole, its longitude will be shifted by 180∘ and the latitude will be recalculated to match the ±90∘ latitude range, accordingly. If an air parcel leaves the longitude range, it will be repositioned by a shift of 360∘ to fit the range again. The model works with both meteorological data provided at -180∘…+180∘ or 0∘… 360∘ of longitude. Following , we mirror data from the western boundary at the eastern boundary of the model, in order to achieve full coverage of the 360∘ longitude range and to avoid extrapolation of the meteorological input data.

In MPTRAC, 4-D linear interpolation is applied to sample the meteorological data at any given position and time. Although higher-order interpolation schemes may achieve better accuracy, linear interpolation is considered a standard choice in Lagrangian particle dispersion models . The approach requires that two time steps of the meteorological input data are kept in memory at each time. Higher-order interpolation schemes in time would require more time steps of the input data and would therefore require more memory.

Following the memory layout of the data structures used for the meteorological input data in our code and to make efficient use of memory caches, the linear interpolations are conducted first in pressure, followed by latitude and longitude, and finally in time. Interpolations can be conducted for a single variable or all the meteorological data at once. Two separate index look-up functions are implemented for regularly gridded data (longitude and latitude) and for irregularly gridded data (pressure). Interpolation weights are kept in caches to improve efficiency of the calculations.

For various studies, it is interesting to investigate the sensitivity of the results with respect to the spatial and temporal resolution of the meteorological input data. For instance, assessed the sensitivity of Lagrangian transport simulations for the free troposphere and stratosphere with respect to downsampled versions of the ERA5 reanalysis data set. To enable such studies, we implemented an option for spatial downsampling of the meteorological input data. The downsampling is conducted in two steps. In the first step, the full resolution data are smoothed by applying triangular filters in the respective dimensions. Smoothing of the data is required in order to avoid aliasing effects. In the second step, the data are downsampled by selecting only each (l,m,n)th grid point in longitude, latitude, and pressure, respectively.

Note that downsampling has not been implemented rigorously for the time domain. Nevertheless, the user can specify the time interval at which the meteorological input data should be ingested. If filtering of the downsampled data is required, downsampling in time needs to be done externally by the user by applying tools such as the Climate Data Operators (CDO) to preprocess the input data of different time steps accordingly. In any case, it needs to be carefully considered whether the meteorological input data represent instantaneous or time-averaged diagnostics at the given time steps of the data.

The advection of an air parcel, i.e., the position x(t) at time t for a given wind and velocity field v(x,t), is given by the trajectory equation, 1dxdt=v(x,t). Here, the position x is provided in a meteorological coordinate system, x=(λ,ϕ,p), with longitude λ, latitude ϕ, and pressure p. The velocity vector v=(u,v,ω) is composed of the zonal wind component u, the meridional wind component v, and the vertical velocity ω, respectively. Based on its accuracy and computational efficiency for trajectory calculations , we apply the explicit midpoint method to solve the trajectory equation, 2x(t+Δt)=x(t)+Δtvx(t)+Δt2vx(t),t,t+Δt2. The model time step Δt controls the trade-off between the accuracy of the solution and the computational effort. Similar to the Courant–Friedrichs–Lewy (CFL) condition, Δt should, in general, be selected such that the steps Δx=x(t+Δt)-x(t) remain smaller than the grid box size of the meteorological data. MPTRAC will issue a warning, if Δt>ΔλmetRE/umax for a longitudinal grid spacing Δλmet of the meteorological data, the mean radius of Earth RE, and a maximum zonal wind speed, umax=150ms-1. A default time step Δt=180s has been selected to match the effective horizontal resolution of Δx≈31km of ECMWF's ERA5 reanalysis.

Calculations of the numerical solution of the trajectory equation require coordinate transformations between Cartesian coordinate distances (Δx,Δy,Δz) and spherical coordinate distances (Δλ,Δϕ,Δp), 3Δλ=ΔxREcos⁡(ϕ),4Δϕ=ΔyRE,5Δp=-pH0Δz. The vertical coordinate transformation between Δp and Δz uses an average scale height of H0=7km. However, note that the vertical transformation is currently not needed to solve Eq. (), because the vertical velocity is already provided in terms of pressure change, ω=dt/dp, in the meteorological data. The transformation is reported here for completeness, as it is required in other parts of the model. For convenience, we assume λ and ϕ to be given in radians; i.e., scaling factors of π/180∘ and 180∘ /π to convert from degrees to radians, and vice versa, do not need to be introduced here.

The coordinate transformation between Δλ and Δx in Eq. () bears numerical problems due to the singularities near the poles. Although this problem might be solved by means of a change to a more suitable coordinate system at high latitudes, we implemented a rather simple solution. Near the poles, for |ϕ|>ϕmax, we discard the longitudinal distance Δλ calculated from Eq. () and use Δλ=0 instead. The high-latitude threshold was set to ϕmax=89.999∘, a distance of about 110 m from the poles, following an assessment of its effect on trajectory calculations conducted by .

As an example, Fig.  shows the results of kinematic trajectory calculations for particles located in the vicinity of the Northern Hemisphere subtropical jet (case A) and the Southern Hemisphere polar jet (case B) in January 2017 using ERA5 data for input. The trajectories were launched at (33.5∘ N, 165∘ W) at 11.25 km log-pressure altitude for the subtropical jet and at (57.5∘ S, 165∘ W) at 8.5 km of log-pressure altitude for the polar jet on 1 January 2017, 00:00 UTC, respectively. The trajectory seeds are located close to the horizontal wind speed maxima at 165∘ W in the upper troposphere and lower stratosphere at the given time. In both cases, the trajectories nearly circumvent the Earth during the time period of 8 d covered by the trajectory calculations. Significant horizontal meandering and height oscillations with the jet streams are revealed. The trajectories were calculated with the explicit midpoint method and a default integration time step of 180 s. Varying the time step between 45, 90, 180, 360, and 720 s did not reveal any significant deviations of the results (not shown).

Rather complex parametrizations of atmospheric diffusivity are available for the planetary boundary layer e.g.,, which have not been implemented in MPTRAC, so far. Much less is known about the diffusivities in the free troposphere and in the stratosphere, which are in the scope of our model. In MPTRAC, the effects of atmospheric diffusion are simulated by adding stochastic perturbations to the positions x of the air parcels at each time step Δt of the model, 6Δxi(t+Δt)=xi(t)+2DiΔtξi. This method requires a vector ξ=(ξx,ξy,ξz) of random variates to be drawn from the standard normal distribution for each air parcel at each time step. The vector of diffusivities D=(Dx,Dx,Dz) is composed of the horizontal diffusivity Dx and the vertical diffusivity Dz. The model allows us to specify Dx and Dz separately for the troposphere and stratosphere as control parameters. The following choices are made for the FLEXPART model : default values of Dx=50m2s-1 and Dz=0 are selected for the troposphere and Dx=0 and Dz=0.1m2s-1 are selected for the stratosphere. Diffusivities will therefore change, if an air parcel intersects the tropopause. A smooth transition between tropospheric and stratospheric diffusivities is created by linear interpolation of Dx and Dz within a ±1km log-pressure altitude range around the tropopause.

In addition to turbulent diffusion, the effects of unresolved subgrid-scale winds, also referred to as mesoscale wind perturbations, are considered. The starting point for this approach is the separation of the horizontal wind and vertical velocity vector v into the grid-scale mean v¯ and the subgrid-scale perturbations v′, 7v=v¯+v′. It is further assumed that the mean wind v¯ is given by the coarse-grid meteorological data, whereas the wind perturbations v′ need to be parameterized. The subgrid-scale wind perturbations are calculated by means of the Langevin equation, 8vi′(t+Δt)=rvi′(t)+1-r2(fσi)2ξi,9r=1-2ΔtΔtmet. From a mathematical point of view, this is a Markov chain or a random walk, which adds temporally correlated stochastic perturbations to the winds over time. The degree of correlation depends on the correlation coefficient r, and therefore on the time step Δt of the model and the time step Δtmet of the meteorological data. The variance (fσi)2 of the random component added at each time step depends on the grid-scale variance σi2 of the horizontal wind and vertical velocity data and a scaling factor f, which is used for downscaling to the subgrid scale. The scaling factor f needs to be specified as a control parameter of the model. The default value is f=40%, following a choice made for the FLEXPART model . For each air parcel, the grid-scale variance σi2 is calculated from the neighboring eight grid boxes and two time steps of the meteorological data. To make computations more efficient, the grid-scale variance of each parcel is kept in a cache. As before, ξ is a vector of random variates to be drawn from the standard normal distribution.

Figure  shows zonal means of the grid-scale standard deviations of the horizontal wind and the vertical velocity of ERA5 data for 1 January 2017. Although they are treated separately in the parametrization, we here combined the zonal and meridional wind components for convenience, noting that the standard deviations of the horizontal wind are largely dominated by the zonal wind component. The grid-scale standard deviations of the horizontal wind become as large 20 m s-1 in the tropical mesosphere. The vertical velocity standard deviations are largest (up to 0.08 m s-1) at the Northern Hemisphere midlatitude stratopause. Standard deviations are generally much lower in the troposphere and the lower stratosphere. However, some pronounced features of the horizontal wind (up to 3 m s-1) are found at the tropopause in the tropics and at northern and southern midlatitudes and for the vertical velocities (up to 0.02 m s-1) in the tropical and midlatitude upper troposphere. The parameterized subgrid-scale wind fluctuations vary significantly with time and location and the atmospheric background conditions.

Figure  illustrates the effects of the simulated turbulent diffusion and the subgrid-scale winds on the trajectories calculated for the jet streams. Sets of 1000 particles were released at the same starting point for both cases. The trajectory sets reveal significant horizontal and vertical spread over time. The spread mostly increases in regions where the jets are meandering and where notable horizontal and vertical wind shears are present. We calculated absolute horizontal and vertical transport deviations (AHTDs and AVTDs; see Sect. ) in order to quantify the effects of simulated diffusion and subgrid-scale wind fluctuations on the jet stream trajectory calculations. Results for case A, the Northern Hemisphere subtropical jet, are shown in Fig. . The analysis of the transport deviations showed that the subgrid-scale wind fluctuations were the major source of the spread of the particles. After 8 d of trajectory time, the AHTDs and AVTDs due to the subgrid-scale winds alone (i.e., Dx=Dz=0 and f=40 %) were about 550 km and 450 m, respectively. In contrast, the corresponding transport deviations due to the constant diffusivities only (i.e., f=0, Dx=50m2s-1 in the troposphere, and Dz=0.1m2s-1 in the stratosphere) were about 150 km for the AHTDs and 200 m for the AVTDs.

Note that several studies with Lagrangian models provided estimates of atmospheric diffusivities , but the data required for the free troposphere and stratosphere for the parametrizations are typically still not well constrained. In addition, in earlier work, we found that the parametrizations of diffusion and subgrid-scale winds applied here may yield different particle spread for different meteorological data sets, even if the same parameter choices are applied . As an example, Fig.  shows results of sensitivity tests on the parameter choices for the diffusion and subgrid-scale wind parametrizations. The parameter choices may need careful tuning for each individual meteorological data set. However, this also opens the possibility for potentially interesting considerations regarding sensitivities to the relative importance of diffusion and subgrid wind fluctuations.

The spatial resolution of global meteorological input data is often too coarse to allow for explicit representation of convective up- and downdrafts. Although the downdrafts may occur on larger horizontal scales, the updrafts are usually confined to horizontal scales below a few kilometers. Furthermore, convection occurs on timescales of a few hours; i.e., hourly ERA5 data may better represent convection than 6-hourly ERA-Interim data. A parametrization to better represent unresolved convective up- and downdrafts in global simulations was implemented in MPTRAC, which is similar to the convection parametrization implemented in the Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) and Stochastic Time-Inverted Lagrangian Transport (STILT) models . In the present study, the convection parametrization was applied for synthetic tracer simulations as discussed in Sect. .

The convection parametrization requires as input the convective available potential energy (CAPE) and the equilibrium level from the meteorological data. These data need to be interpolated to the horizontal position of each air parcel. If the interpolated CAPE value is larger than a threshold CAPE0, which is an important control parameter of the parametrization, it is assumed that the up- and downdrafts within the convective cloud are strong enough to yield a well-mixed vertical column. The parametrization will randomly redistribute the air parcels in the vertical column between the surface and the equilibrium level, which is taken as an estimate of the cloud-top height. In order to achieve a well-mixed column, the random distribution of the particles needs to be vertically weighted by air density.

The globally applied threshold CAPE0 can be set to a value near zero, implying that convection will always take place everywhere below the equilibrium level. This approach is referred to as the “extreme convection method”. It will provide an upper limit to the effects of unresolved convection. On the contrary, completely switching off the convection parametrization will provide a lower limit for the effects of unresolved convection. Intermediate states can be simulated by selecting other specific values of the threshold CAPE0, which is an important tuning parameter of the parametrization. The Supplement of this paper provides some guidance on how different choices of CAPE0 can affect the simulated convection.

The extreme convection parametrization implemented here is arguably a rather simple approach, as it relies only on CAPE and the equilibrium level from the meteorological input data. Nevertheless, first tests showed that this parametrization significantly improves transport patterns in the free troposphere. We also conducted experiments to further improve the parametrization by considering additional parameters such as the convective inhibition (CIN) to improve the onset of convection. More sophisticated convection parametrizations have been developed and implemented in Lagrangian transport models during recent years . These parametrizations consider additional meteorological variables such as convective mass fluxes and detrainment rates that originate from external convective parameterizations applied in state-of-the-art forecasting models. Future work may look at evaluating the extreme convection parametrization and implementing more advanced convection parametrizations in MPTRAC.

In order to take into account the gravitational settling of particles, the sedimentation velocity vs needs to be calculated. Once vs is known, it can be used to calculate the change of the vertical position of the particles over the model time step Δt. In MPTRAC, vs is calculated for spherical particles following the method described by . In the first step, we calculate the density ρ of dry air, 10ρ=pRT, from pressure p, temperature T, and the specific gas constant R of dry air. Next, the dynamic viscosity of air, 11η=1.8325×10-5kgms416.16KT+120KT296.16K1.5, and the thermal velocity of an air molecule, 12v=8RT, are used to calculate the mean free path of an air molecule, 13λ=2ηρv, as well as the Knudsen number for air, 14K=λrp, where rp refers to the particle radius. The Cunningham slip-flow correction is calculated from 15G=1+K1.249+0.42exp⁡-0.87K. Finally, the sedimentation velocity is obtained by means of Stokes law and from the slip-flow correction, 16vs=2rp2gρp-ρ9ηG, with particle density ρp and conventional standard gravitational acceleration g. Note that rp and ρp can be specified individually for each air parcel. A larger set of parcels can be used to represent a size distribution of aerosol or cloud particles.

Figure  shows sedimentation velocities calculated by Eq. () for different particle diameters as well as pressure levels and mean temperatures at midlatitude atmospheric conditions. A standard particle density of ρp=1000 kg m-3 was considered. The sedimentation velocity vs scales nearly linearly with ρp; i.e., increasing ρp by a factor of 2 will also increase vs by the same factor. Note that the overall sensitivity on temperature T is much weaker than the sensitivity on pressure p (not shown). Sedimentation velocities are shown here up to particle Reynolds number Rep≤1, for which vs is expected to have accuracies better than 10 % . Larger particles will require additional corrections, which have not been implemented.

Wet deposition causes the removal of trace gases and aerosol particles from the atmosphere within or below clouds by mixing with suspended water and following washout through rain, snow, or fog. In the first step, it is determined whether an air parcel is located below a cloud top. The cloud-top pressure pct is determined from the meteorological data as the highest vertical level where cloud water or ice (i.e., CLWC, CRWC, CIWC, or CSWC) is existent. For numerical efficiency, the level pct is inferred at the time when the meteorological input data are processed. During the model run, pct can therefore be determined quickly by interpolation from the preprocessed input data. Likewise, the cloud-bottom pressure pcb is determined quickly from precalculated data.

In the second step, the wet deposition parametrization determines an estimate of the subgrid-scale precipitation rate Is, which is needed to calculate the scavenging coefficient Λ. We estimated Is (in units of mm h-1) from the total column cloud water cl (in units of kg m-2) by means of a correlation function reported by , 17Is=(2cl)1/0.36, which is based on a regression analysis using existing cloud and precipitation data. At the same time as determining pct and pcb, the total column cloud water cl is calculated by vertically integrating the cloud water content data when the meteorological input data are processed. During the model run, cl can therefore also be determined quickly by horizontal interpolation of the preprocessed input data to the locations of the air parcels. For efficiency, the parametrization of wet deposition is not further evaluated for a lower limit of Is<0.01mmh-1.

In the third step, it is inferred whether the air parcel is located within or below the cloud because scavenging coefficients will be different under these conditions. The position of the air parcel within or below the cloud is determined by interpolating the cloud water content to the position of the air parcel and by testing whether the interpolated values are larger than zero.

In the fourth step, the scavenging coefficient Λ is calculated based on the precipitation rate Is, 18Λ=aIsb. For aerosol particles, the constants a and b need to be specified as control parameters for the “within-cloud” and “below-cloud” cases, respectively. Typical values of a=4…8×10-5 and b=0.79 are used in the HYSPLIT and Next-Generation Atmospheric Dispersion Model (NAME) models . For gases, wet deposition depends upon their solubility and the scavenging coefficient can be calculated from 19Λ=HRTIsΔzc-1, where H is Henry's law constant, R is the universal gas constant, and Δzc is the depth of the cloud layer, which we calculate from pct and pcb. Henry's law constant is obtained from 20H(T)=H⊖exp⁡-ΔsolHR1T-1T⊖. The constants H⊖ and ΔsolHR with enthalpy of dissolution ΔsolH at the reference temperature T⊖=298.15 K need to be specified as control parameters. Values for a wide range of species are tabulated by . The values of selected species of interest are summarized in Table  and included as default parameters in MPTRAC.

Finally, once the scavenging coefficient Λ for the gases or aerosol particles within or below a cloud is determined, an exponential loss of mass m over the time step Δt is calculated, 21m(t+Δt)=m(t)exp⁡-ΔtΛ.

Dry deposition leads to a loss of mass of aerosol particles or trace gases by gravitational settling or chemical and physical interactions with the surface in the dry phase. In the parametrization implemented in MPTRAC, dry deposition is calculated for air parcels located in the lowermost Δp=30 hPa layer above the surface. This corresponds to a layer width of Δz≈200 m at standard conditions.

For aerosol particles, the deposition velocity vdep will be calculated as described in Sect.  as a function of surface pressure p and temperature T as well as particle radius rp and particle density ρp. For trace gases, the deposition velocity vdep needs to be specified as a control parameter. Currently, this parameter is set to a constant value across the globe for each trace gas. For future applications with a stronger focus on the boundary layer, vdep will need to vary geographically to account for dependence on the surface characteristics and atmospheric conditions.

For both particles and gases, we calculated the loss of mass based on the deposition velocity vdep, the model time step Δt, and the surface layer width Δz from 22m(t+Δt)=m(t)exp⁡-ΔtvdepΔz. Note that both the dry deposition and the wet deposition parametrization were implemented only very recently and need further testing and evaluation in future work. They will also have to be compared to more advanced parametrizations e.g.,. The parametrizations are described here for completeness, but they were not included in the model verification and performance analysis in Sect. .

In this section, we discuss the MPTRAC module that is used to simulate the loss of mass of a chemical species by means of reaction with the hydroxyl radical. The hydroxyl radical (OH) is an important oxidant in the atmosphere, causing the decomposition of many gas-phase species. The oxidation of different gas-phase species with OH can be classified into two main categories, bimolecular reactions (e.g., reactions of CH4 or NH3), and termolecular reactions (e.g., CO or SO2).

For bimolecular reactions, the rate constant is calculated from Arrhenius law, 23k(T)=A×exp⁡-ERT, where A refers to the Arrhenius factor, E to the activation energy, and R to the universal gas constant. For bimolecular reactions, the Arrhenius factor A and the ratio E/R need to be specified as control parameters. The reaction rate coefficients for various gas-phase species can be found in the Jet Propulsion Laboratory (JPL) data evaluation . Table  lists bimolecular reaction rate coefficients for some species of interest, which were also implemented as default values directly into MPTRAC.

Termolecular reactions require an inert component M (e.g., N2 or O2) to stabilize the excited intermediate state. Therefore, the reaction rate shows a temperature and pressure dependence and changes smoothly between the low-pressure-limiting rate k0 and high-pressure-limiting rate k∞. Based on the molecular density of air, 24[M]=NApRT. with Avogadro constant NA, the effective second-order rate constant k is calculated from 25k(T,[M])=k0(T)[M]1+k0(T)[M]k∞(T)0.61+log⁡10k0(T)[M]k∞(T)2-1. The low- and high-pressure limits of the reaction rate constant are given by 26k0(T)=k0298T298-n,27k∞(T)=k∞298T298-m. The constants k0298 and k∞298 at the reference temperature of 298 K and the exponents n and m need to be specified as control parameters. The exponents can be set to zero in order to neglect the temperature dependence of the low- or high-pressure limits of k0 and k∞. The termolecular reaction rate coefficients implemented directly into MPTRAC are listed in Table .

Based on the bimolecular reaction rate k=k(T) from Eq. () or the termolecular reaction rate k=k(T,[M]) from Eq. (), the loss of mass of the gas-phase species over time is calculated from 28m(t+Δt)=m(t)exp⁡-k[OH]Δt. The hydroxyl radical concentrations [OH] are obtained by bilinear interpolation from the monthly mean zonal mean climatology of . This approach is suitable for global simulations covering at least several days, as hydroxyl concentrations may vary significantly between day- and nighttime as well as the local atmospheric composition.

A rather generic module was implemented in MPTRAC, to simulate the loss of mass of an air parcel over a model time step Δt due to any kind of exponential decay process, e.g., chemical loss or radioactivity, 29m(t+Δt)=m(t)exp⁡-Δtte. The e-folding lifetime te of the species needs to be specified as a control parameter. As typical lifetimes may differ, we implemented an option to specify separate lifetimes for the troposphere and stratosphere. A smooth transition between the tropospheric and stratospheric lifetime is created within a ±1km log-pressure altitude range around the tropopause.

Finally, a module was implemented to impose constant boundary conditions on particle mass or volume mixing ratio. At each time step of the model, all particles that are located within a given latitude and pressure range can be assigned a constant mass or volume mixing ratio as defined by a control parameter of the model. Next to specifying fixed pressure ranges in the free troposphere, stratosphere, or mesosphere to define the boundary conditions, it is also possible to specify a fixed pressure range with respect to the surface pressure, in order to define a near-surface layer.

The boundary condition module can be used to conduct synthetic tracer simulations. For example, the synthetic tracer E90 is emitted uniformly at the surface and has a constant e-folding lifetime of 90 d throughout the entire atmosphere. In our study, we implemented this by assigning a constant volume mixing ratio of 150 ppb to all air parcels residing in the lowermost 30 hPa pressure range above the surface. With its 90 d lifetime, the tracer E90 quickly becomes well mixed in the troposphere. However, the lifetime is much shorter than typical timescales of stratospheric transport. The tracer E90 therefore exhibits sharp gradients across the tropopause. Being a passive tracer in the upper troposphere and lower stratosphere region, E90 is evaluated in studies related to the chemical tropopause and stratosphere–troposphere exchange.

Another example of a synthetic tracer is ST80, which is defined by a constant volume mixing ratio of 200 ppb above 80 hPa and a uniform, fixed 25 d e-folding lifetime in the troposphere . Like E90, the tracer ST80 is of interest in investigating stratosphere–troposphere exchange. The Northern Hemisphere synthetic tracer (NH50) is considered to assess interhemispheric gradients in the troposphere. It is defined by a surface layer volume mixing ratio of 100 ppb over 30 to 50∘ N and a uniform 50 d e-folding lifetime . Simulation results of E90, ST80, and NH50 obtained with the CPU and GPU code of MPTRAC will be discussed in Sect. .

For diagnostic purposes, it is often necessary to obtain meteorological data along the trajectories. However, typically only a limited subset of the meteorological variables is of interest. Therefore, we introduced the concept of “quantities”, allowing the user to select, which variables should be calculated and stored in the output data files. Next to selecting the specific types of model output that are desired, also a list of the specific quantities that should be provided as output needs to be specified via control parameters. In total, about 50 output variables are implemented in MPTRAC (Table ). In addition to the predefined output variables, it is possible to include user-defined variables. The user-defined variables are typically not modified during the simulations, but they are useful to store information that are required for further analyses. For example, the starting time and position of the particles can be stored as user-defined variables, and they can later be used to calculate the trajectory time or the distance from the origin.

A distinct module was implemented into MPTRAC that is used to sample the meteorological data along the trajectories. This module can become demanding in terms of computing time, as it requires the interpolation of up to 10 3-D variables and 16 2-D variables of meteorological data that are either read in from the meteorological input data files or calculated during the preprocessing of the meteorological data. However, the computing time needs of this module are typically strongly reduced by the fact that meteorological data along the trajectories are usually not required at every model time step but only at user-defined output intervals.

Next to the option of sampling of the meteorological data along the trajectories, we also implemented tools to provide direct output of the meteorological data. This includes tools to extract maps, zonal mean cross sections, and vertical profiles on pressure or isentropic levels. These tools allow for time averaging over multiple meteorological data files. Another tool is available to sample the meteorological data based on a list of given times and positions, without involving any trajectory calculations. Altogether, these tools provide a great flexibility in exploiting meteorological data in many applications.

At present, MPTRAC offers seven output options, referred to as “atmospheric output”, “grid output”, “CSI output”, “ensemble output”, “profile output”, “sample output”, and “station output”. The most comprehensive output of MPTRAC is the atmospheric output. Atmospheric output files can be generated at user-defined time intervals, which need to be integer multiples of the model time step Δt. The atmospheric output files are the most comprehensive type of output because they contain the time, location, and the values of all user-defined quantities of each individual air parcel.

As the atmospheric output can easily become too large for further analyses, in particular if many air parcels are involved, the output of gridded data was implemented. This output will be generated by integrating over the mass of all parcels in regular longitude × latitude × log-pressure height grid boxes. From the total mass per grid box and the air density, the column density and the volume mixing ratio of the tracer are calculated. Alternatively, if the volume mixing ratio per air parcel is specified, the mean volume mixing ratio per grid box is reported. In the vertical, it is possible to select only a single layer for the grid output, in order to obtain total column data. Similarly, by selecting only one grid box in longitude, it is possible to calculate zonal means.

Another type of output that we used in several studies is the critical success index (CSI) output. This output is produced by analyzing model and observational data on a regular grid. The analysis is based on a 2×2 contingency table of model predictions and observations. Here, predictions and observations are counted as yes, if the model column density or the observed variable exceed user-defined thresholds. Otherwise, they would be counted as no. Next to the CSI, the counts allow us to calculate the probability of detection (POD) and the false alarm rate (FAR), which are additional skill scores that are often considered in model verification. More recently, the CSI output was extended to also include the equitable threat score (ETS), the linear and rank-order correlation coefficients, the bias, the root mean square (rms) difference, and the mean absolute error. A more detailed discussion of the skill scores is provided by .

Another option to condense comprehensive particle data is provided by means of the ensemble output. This type of output requires a user-defined specific ensemble index value to be assigned to each air parcel. Instead of the individual air parcel data, the ensemble output will contain the mean positions as well as the means and standard deviations of the quantities selected for output for each set of air parcels having the same ensemble index. The ensemble output if of interest, if tracer dispersion from multiple point sources needs to be quantified by means of a single model run, for instance.

The profile output of MPTRAC is similar to the grid output as it creates vertical profiles from the model data on a regular longitude × latitude horizontal grid. However, the vertical profiles contain not only volume mixing ratios of the species of interest but also profiles of pressure, temperature, water vapor, and ozone as inferred from the meteorological input data. This output is compiled with the intention to be used as input for a radiative transfer model, in order to simulate satellite observations for the given model output. In combination with real satellite observations, this output can be used for model validation but also as a basis for radiance data assimilation.

The sample output of MPTRAC was implemented most recently. It allows the user to extract model information on a list of given locations and times, by calculating the column density and volume mixing ratio of all parcels located within a user-specified horizontal search radius and vertical height range. For large numbers of sampling locations and air parcels, this type of output can become rather time-consuming. It requires an efficient implementation and parallelization because it needs to be tested at each time step of the model whether an air parcel is located within a sampling volume or not. The numerical effort scales linearly with both the number of air parcels and the number of sampling volumes. The sample output was first applied in the study of to sample MPTRAC data directly on TROPOspheric Monitoring Instrument (TROPOMI) satellite observations.

Finally, the station output is collecting the data of air parcels that are located within a search radius around a given location (latitude, longitude). The vertical position is not considered here; i.e., the information of all air parcels within the vertical column over the station is collected. In order to avoid double counting of air parcels over multiple time steps, the quantity STAT (Table ) has been introduced that keeps track on whether an air parcel has already been accounted for in the station output or not. We used this type of output in studies estimating volcanic emissions from satellite observations using the backward-trajectory method .

By default, all output functions of MPTRAC create data files in an ASCII table format. This type of output is usually simple to understand and usable with many tools for data analysis and visualization. However, in the case of large-scale simulations, it is desirable to use more efficient file formats. Therefore, an option was implemented to write particle data to binary output files. Likewise, reading particle data from a binary file is much more efficient than from an ASCII file. The binary input and output is an efficient way to save or restore the state of the model during intermediate steps of a workflow. Another interesting option of output is to pipe the data directly from the model to a visualization tool. This will keep the output data in memory and directly forward it from MPTRAC to the visualization tool. This option has been successfully tested for the particle and the grid output in combination with the graphing utility gnuplot .

For diagnostic purposes, it is often helpful to calculate trajectory statistics. The tool atm_stat can calculate the mean, standard deviation, skewness, kurtosis, median, absolute deviation, median absolute deviation, minimum, and maximum of the positions and variables of air parcel data sets. The calculations can cover the entire set or a selection of air parcels, based on the ensemble variable or extracted by means of the tool atm_select. The tool atm_select allows the user to select air parcels based on a range of parcels, a log-pressure height × longitude × latitude box, and a search radius around a given geolocation. The trajectory statistics are calculated by applying the algorithms of the GNU Scientific Library .

The tool atm_dist allows us to calculate statistics of the deviations between two sets of trajectories. Foremost, this includes the AHTDs and AVTDs over time, 30AHTD(tn)=1N∑i=1N[xi(tn)-Xi(tn)]2+[yi(tn)-Yi(tn)]2,31AVTD(tn)=1N∑i=1Nzi(tn)-Zi(tn). Here, the horizontal distances are calculated approximately by converting the geographic longitudes and latitudes of the particles to Cartesian coordinates, [xi(tn),yi(tn),zi(tn)] and [Xi(tn),Yi(tn),Zi(tn)], followed by calculation of the Euclidean distance of the Cartesian coordinates. Vertical distances are calculated based on conversion of particle pressure to log-pressure altitude using the barometric formula.

Next to AHTDs and AVTDs, the tool atm_dist can calculate the relative horizontal and vertical transport deviations (RHTDs and RVTDs), which relate the absolute transport deviations to the horizontal and vertical path lengths of individual trajectories. It can also be used to quantify the mean absolute and relative deviations as well as the absolute and relative bias of meteorological variables or the relative tracer conservation errors of dynamical tracers over time. The definitions of these quantities and more detailed descriptions are provided by . In this study, we mostly present absolute transport deviations to evaluate the differences between CPU and GPU trajectory calculations (Sect. ).