The starting point of our development was the trajectory module developed by . The software package consists of two files: data_part and src_part. The first file provides information on parameters, global variables and namelist variables; the file src_part contains the LPDM code. There is only a weak link between Itpas and COSMO. We only added a switch to activate the particle module and call statement in the COSMO main routine.

The particle model is called at the end of every COSMO time step and reads in the most recent wind and turbulence field arrays. Every particle is handled individually by the LPDM, so, as illustrated in Fig. , the overall structure is a loop over every active particle. A typical call performs the following steps: first, the particle transport by the mean wind is investigated. For describing the particle's vertical motion, the settling velocity is taken into account for particles with a mass. Afterwards the turbulent disturbance is calculated. Therefore, it is important to know if the particle resides inside the ABL or above:

Inside the ABL, the turbulent part of the wind velocity is calculated with a general approach which is presented in detail in the next section. The turbulent wind velocity is integrated with an Euler forward step. This leads to the turbulent part of the particle motion which is added to the mean particle motion. Since the particle is not allowed to leave the boundary layer directly, the model checks if the particle is still inside the boundary layer after the motion. If not, the particle is reflected back into the ABL. If the particle hits the surface, it is decided whether the particle is removed by dry deposition or whether it is reflected back into the atmosphere, based on a probability function (see Eq. ).

Itpas is controlled by a namelist and a start file. In the namelist, important parameters can be defined, like the maximum number of particles, paths for input and output, the seed for the random number generator and the output increment. The start file defines the particle start positions and attributes like size and density. It is possible to define continuously emitting sources and start points that release multiple (thousands of) particles. More details on the model set-up can be found in the ReadMe file which is included in the Supplement.

The output is written in NetCDF format and contains the trajectory positions for every output step as well as the attributes size and density. One of the biggest advantages of the LPDM compared to the Eulerian system is the ability to produce strong gradients and sharp edges of a particle plume. Therefore we avoided transforming the trajectories into gridded data (e.g. trajectory-density). But if necessary this can be done in the post-processing.

The Reynolds average of the wind velocity u=u‾+u′ describes the true wind velocity u as the superposition of the mean wind velocity u‾ with a turbulent disturbing u′. Also in the turbulent boundary layer, the major part of the transport is due to advection as a function of the mean wind. For the advective transport, we use the iterative integration scheme from the online trajectory module of .

To determine the turbulent proportion of the wind velocity we use a generalised form of Langevin's equation . 1u′=a(x,u,t)dt+b(x,u,t)ξ The first term of this equation describes the conservation of momentum. The coefficient a is a memory function that depends on the particle's position (x) and velocity (u) at a specific time (t). The second term gives a random displacement. The coefficient b contains the potential strength of the turbulence; ξ is a normal distributed random number with a mean value of zero and a standard deviation of one. In that way, every particle gets a random push with the mean strength of the turbulence. The general solution for this selection problem was established by . Looking at u′ for the discrete time step t+1, Langevin's equation is set up as follows: 2ut+1′=aut′+bξ. The parameter a is defined as the correlation function R : 3a=R=exp⁡-dtτL. It describes the connection of the model time step dt and τL, the characteristic timescale for the Lagrangian dispersion process. The correlation function expresses the following: if dt>>τL;u,v,w then there is no further dependency between the velocities ut+1′ and ut′. The parameter b is defined as follows: 4b=σ1-R2, with σ the standard deviation of the wind .

Now parametrisations for σ and τL are necessary to solve the equation of motion. A common approach uses the TKE (e‾), which describes how much kinetic energy is available to generate atmospheric turbulence. Kinetic energy is well known as Ekinm=12v2. Based on this the kinetic energy of the turbulent wind fraction is defined as follows: 5e‾=12u′2‾+v′2‾+w′2‾, with the turbulent wind u′, v′ and w′ of the spacial directions. The variance of a variable is defined as the average of the squared fluctuation σu2=u′2‾ . This allows the standard deviation of the wind velocity to be parametrised with the TKE, 6σ=2me‾. The mi factors (mi∈R with 0≥mi≥1 and ∑mi=1; i={u,v,w}) describe the weighting of the TKE in each spatial direction, which has a strong dependence on the ambient wind velocity and the atmospheric stratification. In our approach we define the mi values as the fractions of the mean wind on every model time step. 7mi=|i||u‾|

For the parametrisation of the characteristic timescale, the diffusion coefficient of momentum (Km) is used, 8Km,x=12σx2u‾x. Here, σx2 is the variance of a spatially dispersed tracer. showed that σx2 can be expressed as follows: 9σx2=2σu2τL,ut, under the condition that time t≫τL,u. The combination of Eqs. () and () with t=xu‾ leads directly to the parametrisation of the characteristic timescale, 10τL=Kmσ2.

Now, all parts of the turbulent particle motion are parametrised with available variables of the NWP model. This allows Eqs. (), () and () to be combined into the general solution (), which can be solved with the parametrisations () and (): 11u′(t)=Ru′(t-Δt)+σ1-R2ξ+(1-Rw)τL,w∂σw2∂z+σw2ρ∂ρ∂z001. For the vertical turbulence, there is a correction necessary that takes into account the vertical variability of the density and the profile of the standard deviation .

At the top of the boundary layer, there is a hard transition of the wind regime. Inside the boundary layer, there is a turbulent wind regime with possibly strong vertical velocities. Above the boundary layer, the turbulent part of the particle motion is strongly reduced. Therefore a simplified approach can be used for which the standard deviation of the turbulent wind is described by the diffusion coefficient per time step σ=Kmdt , which leads to a turbulent wind velocity of 12u′=2Kmdtξ.

The model is designed to describe the airborne transport of aerosol particles like mineral dust. As these particles settle by gravity, we need to introduce the gravitational settling velocity (vg), which depends on the particle size and therefore on the Reynolds number (Re). For small Reynolds numbers, vg can be calculated directly : 13vg=ρpdp2gCc18ηfor Re≤0.4, with the density and diameter of the particle ρp and dp, the gravity g, the dynamic viscosity of air η, and slip correction factor (Cunningham correction) Cc. For particle settling under conditions of high Reynolds numbers (Re>0.4), an empirical approach is used : 14vg=ηρairdpexp⁡-3.07+0.9935J-0.0178J2, where 15J=ln⁡4ρpρairdp3g3η2.

Particles near the surface may be removed by dry deposition. Thereby the dry deposition velocity vd is important, which describes the velocity of the diffusion near the surface. Following it is defined as follows: 16vd=vg+1Ra+Rs, with the atmospheric resistance Ra and the surface resistance Rs. Since motion is already calculated individually for each particle, it is useful to handle the deposition individually as well. Therefore we use the deposition probability Wd. It describes how likely it is that a particle remains at the ground once it has hit the surface. An approach for this can be found in : 17Wd=2πvdσ0Fg+π2vdσ0. Here σ0=2.5u⋆mw is the standard deviation of the vertical wind right at the surface, where u⋆ is the friction velocity. Fg describes the gravitational part of the dry deposition with γ=vg2σ0-1 and erf(γ) the error function of γ, 18Fg=π2vgσ0+e-γ21+erf(γ).

The experiment took place at an agricultural field near Müncheberg, Germany, approximately 50 km east of Berlin, on 31 May 2017 . During this day, the weather over eastern Germany and western Poland was dry with some shallow cumulus clouds. During the experiment a gentle breeze (∼ 5 ms-1) blew from the northwest.

The experiment was carried out as follows: a tractor drove in parallel lines across the field, orientated perpendicular to the ambient wind (Fig. ). Downwind, beside the field, the particle number concentration was measured with two vertically stacked Environmental Dust Monitors (EDM 164, GRIMM-Aerosol Technique) at two different inlet heights (1.5, 3.8 m). The EDMs measured particle number concentrations at 31 size bins ranging from 0.25 to 32 µm.

The experiment was divided into two parts: EXP1

Dried chicken manure was spread on the field as fertiliser from 08:50 to 09:45 UTC.

During both experiments, a particle plume rose behind the tractor; although the tilling entrained far more particles into the atmospheric boundary layer than the fertiliser spreading. While the measurements provide information on the particle number and size distribution and thus characterise the particle plume developing, the question remains of how long these particles will remain in the atmosphere and how they will be dispersed. As it can be assumed that the turbulent nature of the convective daytime boundary layer is relevant to the accurate simulation of the dispersion, these two experiments were simulated in order to showcase the strength of our online coupled LPDM Itpas driven by prognostic TKE.

In the framework of this study, dust entrainment was mechanically driven by the tools pulled behind the tractor and not by wind. A source function was obtained based on the measured particle concentrations in order to describe the dust particle uplift. Therefore we created an idealised version of the particle plume for which we use the EDM data to define a vertical profile of particle concentration. The profile was then multiplied by a half-sphere in order to get a 3D concentration field with which we can define the particle start points for the Itpas simulation. Thereby, the particle concentration determines the number of the trajectories starting at the respective point. The particle size is used to assign the corresponding particle properties such as the diameter, which is relevant for determining the particle's residence time in the atmosphere.

In particular, we defined a vertical profile that represents the centre of the tractors particle plume (Fig. a). Therefore we used the observation data of the moment when the tractor passed by the measurement device as close as possible. Besides the values of the two measurement heights (inlet heights) two additional assumptions are necessary. First, the particle concentration at ground level corresponds to that at 1.5 m. In general, it seems to be more realistic that the particle concentration is higher at ground level than at 1.5 m, especially for coarse particles. But since there is no information about the near-surface concentration available, this conservative assumption is the best choice. Second, the particle concentration becomes zero at a height of 5 m. The plume height also depends on multiple properties such as atmospheric conditions, soil texture, soil moisture and the kind of tool pulled by the tractor. But, based on observations and photo documentation of the experiment, 5 m seems to be a good estimate, at least for this specific case. With these four initial values (two measured and two assumed), the vertical particle number concentration profile of the plume centre was created with a polynomial fit (Fig. a).

Then a normalised spherical concentration field with a radius of 5 m was multiplied with the vertical profile, resulting in a three-dimensional field of the particle concentration (Fig. b). From this, the total number of particles inside the plume is determined. For the measurements in the case study that were some billion particles. This number of particles cannot be handled individually within the model simulation and thus needed to be downscaled. Here we chose a scaling factor of 2×10-7. So, one particle in the model represents five million measured particles. In total, we used ∼ 100 000 particles in EXP1 and ∼ 270 000 particles in EXP2. All particles are assigned to random positions in the idealised cloud under the condition that particle distribution matches the concentration field (Fig. c). This analysis was performed for each bin provided by the EDM data, so the particle size distribution in the model corresponds to the measurements. An extended version of Fig.  with profiles and concentration fields for the individual bins can be found in the Supplement (Figs. S1–S4). With this method we created a bell-shaped cloud of individual particles that we used as model input. Please note that the exact shape of the particle source becomes less important with increasing transport distance.

For EXP1 and EXP2 individual simulations were performed. As input data for the NWP model, we used reanalysis data from COSMO, provided by the German weather service. The model domain covers an area from eastern Germany to central Poland including the cities of Berlin and Poznań (lower left corner: 51∘ N, 12∘ E; upper right corner: 54.4∘ N, 19.3∘ E) and has a size of 167×150 grid cells with a spatial grid spacing of 2.8 km and 50 vertical levels.

In the first simulation EXP1 (the spreading of the dry fertiliser), the particles were emitted during the 55 min time span from 08:50 to 09:45 UTC at the position 52.477∘ N, 14.177∘ E. In EXP2 (incorporation of the fertiliser with a cultivator), the particles were emitted at the same position for 50 min from 11:40 to 12:30 UTC. Local time is UTC+2. For both experiments, a continuous release of particles was assumed during the time of emission.

Analysing the simulation output, we focused in particular on the horizontal dispersion, the deposition and the vertical mixing as these three aspects may provide best insights into the model's capability to capture the dispersion of particles within a turbulent boundary layer. These parameters are presented in Figs.  and , each with panel (a) showing the trajectories indicating the dispersion from a bird's eye perspective. Each line represents the path of one simulated particle trajectory. Panel (b) shows the same map, but with points indicating where particles were deposited. Panel (c) shows the altitude of the particles as a function of time, so it visualises the vertical dispersion and mixing of the particles. The colour indicates the particle size ranging from sub-micron particles in light blue to the coarse particles in orange. Please note that smaller particles are overlaid by the larger ones.

In the first experiment, the range of the particle transport was limited to the local surrounding (Fig. a). The vast majority of particles have not left this area so that the longest trajectories have a length of about 10 km. The particles spread only a few hundreds of metres in the direction perpendicular to the mean wind . The deposition map (Fig. b) shows a colour gradient from west to east, which indicates the coarse particles depositing close to the source while smaller particles are able to travel longer distances. This becomes even clearer from time series of the vertical dispersion (Fig. c). Please note that the altitude axis is restricted to 6 m. The particles were not lifted into higher levels and fell down more or less quickly after entrainment depending on their size. At the upper edge of the plume, it is visible that the smallest particles, shown in light blue, form a kind of wave-like structure, which is linked to the random fluctuation that every particle experiences. Because of gravity forces, the amplitude of these waves becomes smaller with increasing particle size and disappears totally for the heavy particles that fall down directly. At the end of the displayed time range, individual particles experience an upward movement and travel further on. These particles perform a similar movement to the particles in EXP2.

Only a brief look at Fig.  is necessary to notice that the situation changes drastically in EXP2. Beginning again with the horizontal dispersion (Fig. a) it is evident that the particles are going into long-range transport mode. The particles drift evenly apart as they cross the German–Polish border. Above western Poland (∼ 15.5∘ E), the spread of the trajectories widens and thus the area covered by the particle plume (sum of the trajectories) increases significantly. Particularly affected hereof are the small particles (shown in light blue colour shades) which partly deviate strongly from the centre of the plume. The trajectories terminate above central Poland where the edge of the simulation domain is located. The most obvious feature on the deposition map (Fig. b) is the tendency of the small particles to spread further away from the core plume than the large particles do. But more interesting are the locations where the large particles deposit. It is apparent that these particles tend to fall down early during their journey. An indication for this is the dark orange stripe from the source to the German–Polish border (Fig. b). But also further on over Poland, a remarkable amount of dark orange dots can be found on the deposition map. This is an indication of an efficient vertical transport that was able to uplift the coarse-mode particle fraction. The particle uplift (Fig. c) can be described by three phases: the first phase is characterised by the emission of the dust particles (11:40–12:30 UTC), during which the particles slowly rise up to a couple of hundreds of metres above ground level. Then, at around 12:20 UTC, the particles experienced a strong uplift onto a level of 1200 m (second phase), visible as a shoulder in the uplifting trajectory pattern (cf. Fig. c). Finally, the particles experience a second strong lifting up to 2500 m above ground at around 13:00 UTC (third phase). The upper edge of the plume can clearly be identified as the top of the boundary layer. During the afternoon, the particles were mixed over the whole depth of the ABL. At 18:00 UTC (local time: UTC+2) again a transition in the distribution pattern is visible. Below 800 m altitude the particles slowly sink to the ground while the ones above this height start to travel on constant levels with strongly reduced vertical mixing. This process forms the overhanging lip of the vertical pattern in the last third of the plot.

To completely understand the behaviour of Itpas, it is necessary to have a look into the atmosphere that was produced by COSMO, as illustrated in Fig. . It shows a temporal cross section through the atmosphere or, more precisely, the air column at the mean particle position. During the morning, this is the start point of the plume (the short-range transport from EXP1 is ignored). During the afternoon, the displayed air column follows a mean trajectory from the plume of EXP2. The mean trajectory travels in the centre of the particle plume and reaches the eastern edge of the model domain around 21:30 UTC. For the remaining time it rests at its terminal position. The altitude of the mean trajectory is marked in the plot as a red line together with the standard deviation in pale red. The main variable displayed in the cross section is the virtual potential temperature θv in colour shades from deep blue to dark orange. θv is a combined measure of temperature and moisture, weighted with the level height. Therefore it is well suitable to classify the atmospheric stratification. Vertically increasing values of θv indicate stable decreasing values unstable stratification. Figure  in the Appendix section shows the θv profile for the consecutive hours of the day and may help to interpret Fig. .

A closer look into the near-surface atmospheric conditions helps to understand the model behaviour in the morning hours (EXP1), during which the particles were not able to rise. A detailed view on the bottom-most 50 m can be found in the upper left corner of Fig. . Here, it is visible that θv increases at the levels above the surface and forms a small stripe of stable stratification. In a stable atmosphere, the vertical motion is inhibited, so no momentum is available to uplift the particles. During the morning hours, the vertical θv gradient becomes weaker and disappears until noon. But not only the atmospheric stratification is important for the vertical mixing, the overall driver of turbulent momentum is still the TKE. Now, focusing on the main plot of Fig. , the available TKE is drawn in dark grey isolines. The hours of emission during the different experiments are marked in green on the time axis. Besides the limited vertical motion, there is no TKE available at the beginning of EXP1, although a slight increase in the TKE during the end of EXP1's emission time occurs, which may provide enough momentum to push single particles upwards out of the stable surface layer (cf. Fig. c). However, even if the vertical particle motion in EXP1 behaves reasonably, the horizontal movement of the particles should be interpreted carefully. In the simulation, within a radius of up to 10 km around the source, the particles are uplifted less than 5 m above the surface. This close to the ground level, the COSMO model cannot provide reliable information of the horizontal wind do to its vertical resolution. Furthermore, the spatial grid resolution of 2.8 km is relatively coarse compared to the travel range (10 km). Additionally, small surface structures like buildings, trees or forests are not included in the simulation. Thus, EXP1 should be considered as a case without particle transport. In essence, the dust particles emitted by the tractor are whirled up but then deposited soon after emission.

Later during the day for EXP2, we can see changes in both the near-surface TKE and the θv profile. At noon, a decrease in θv up to a height of at least 500 m is notable, which indicates unstable conditions in this layer. This allows the particles to rise, whereby the TKE induces the vertical mixing. During the afternoon, the unstable layer extends up to 2000 m. Above 2500 m a strongly increasing gradient of θv indicates the top of the ABL. This gradual development of the ABL can also be observed from the trajectories (cf. Fig. c between 12:00 and 13:00 UTC).

Between 13:00 and 14:00 UTC, the lower half of the particles (below the mean trajectory) passes a region with strongly increased TKE and the particles build momentum which enables them to shear out strongly from the main plume. This situation corresponds spatially and temporally to the enhanced spreading of the trajectory plume above western Poland (cf. Fig. a).

After 16:00 UTC the temperature at the surface is decreasing and the ABL collapses forming the transition into a nocturnal boundary layer structure: a stable surface layer forms which deepens during the evening into the typical stable nocturnal boundary layer. The particle plume gets divided into two parts, one remaining within the stable ABL and one above. Similar to the particles in EXP1 the part of the plume inside the stable nocturnal boundary layer lacks all turbulent momentum keeping the particles aloft, and, ultimately, the particles settle down. Only these particles within the residual boundary layer above the nocturnal boundary layer remain airborne. In general, the residual layer is neutrally stratified, the turbulent mixing is strongly reduced and the model uses the short approach (cf. Fig. ). Thus, the particles are trapped in this layer overnight and propagate within the general air flow at almost constant altitude.

Our results above showed that the representation of the diurnal cycle of the ABL is vital to an accurate trajectory calculation. The approach presented here includes a TKE-based approach to account for the ABL's turbulent features. A closer look into the role of turbulence to the particle transport is provided in Fig. : for a reduced version of EXP2 with ∼ 30 000, we ran three different set-ups aiming at representing the role of the different components of the wind ultimately determining the particle's trajectory. In the first set-up (Fig. a) we suppressed all vertical particle motion – those of the mean wind as well as those of the turbulence. It is considered as a control case. Comparable to EXP1 all particles deposit directly after the emission. Please note that the altitude is limited to 6 m.

The second set-up shown in Fig. b focuses on the role of the mean wind and its ability to mix the emitted particles in the ABL. Therefore we suppressed the vertical turbulence for this set-up. We observe that the mean wind of the model alone is not able to induce the particles into the atmosphere. Only 7 particles (out of 30 000) get lifted up to 2000 m. Nevertheless, a glimpse of the role of the mean wind mixing to the particle trajectories can be caught. The particles perform waves with a length up to 1 h and large amplitudes. The origin of these waves are the changing up- and downwind regions in the convective ABL. We also observed this kind of wave in EXP2 (cf. Fig. c) and assign them to the mean wind. In the convective ABL, turbulence is skewed towards the upward transport since updrafts and downdrafts are not evenly distributed, which may result in an underestimation of the near-surface concentration . However, we estimate the influence of this effect to be small in our model due to the high-frequency wind information.

Figure c focused on the turbulent portion of the vertical mixing; the vertical mean wind is suppressed in this third set-up. From this figure it becomes evident that turbulence is essential for the entrainment of particles into the boundary layer. But it also becomes visible that the heavier particles accumulate in the lower ABL, since the turbulent fluctuation is not as strong as the convective updrafts. Hence, it results in a boundary layer that is not well-mixed. In sum, the artificial set-ups emphasise that both components are significant for a mixing of particles from the near surface in the ABL that conforms with the criteria for well-mixed from . Our approach provides a good balance between these components and therefore allows a realistic representation of the boundary layer dynamic in the particle trajectories.

COSMO uses the Message Passing Interface (MPI) for parallel computing on multiple processors, which divides the model domain into subdomains with each assigned to one computing process on a singe core. Thereby, a single process exclusively holds the information of its own subdomain, so it can only calculate a particle motion that is located in its own subdomain, and if a particle leaves a subdomain, it has to be passed to the next one. To handle this, we apply the strategy proposed by : during the iteration the process gathers all particles that leave the subdomain. At the end of the time step all particles leaving the subdomain get passed to their new subdomain in a single communication step. For an even number of particles in each subdomain, the model system would be able to process them in parallel. However, when the particles travel as a plume they are most probably assigned to a few or even only one subdomain. As a consequence, the calculation is performed quasi-serially. On the other hand, if we distribute the particles independently of their subdomain, we would also need to pass the atmospheric data for the calculation. This may result in a communication overhead. The efficient calculation of online trajectories remains a dilemma between communication and data availability. So further research is necessary on this technical aspect.

Our simulations were performed multiple times on different machines with 36 processors. For the simulation of EXP1 during which (nearly) all particles are deposited within the first minutes, the runtime was about 40 % longer than for a test case without particles. The simulation of EXP2 took roughly 10 times longer than the simulation without particles. On our most recent system (Intel(R) Xeon(R) Platinum 8160 CPUs; 2.10 GHz) the runtime of EXP2 was 260 min compared to 36 min for EXP1 and 25 min without particles.