To enable ATTILA in a distributed memory parallel environment (e.g. applying a message-passing interface standard, MPI) we chose to follow a domain cloning approach. Whereas the base model EMAC follows a classical horizontal domain decomposition approach for distributed memory parallelisation, we distribute the global number N of ATTILA air parcels, which keep their identity throughout a simulation, (almost) equally among the parallel tasks (index i): 1N=∑i=0p-1ni, where p is the number of parallel tasks and ni the number of parcels bound to task physical and not by numerical i. Note that ni=nj for all (i,j), except for i=p-1, depending on whether N is divisible by p or not.

During the simulation, each parcel keeps being bound to its initial task. Since all parcels on each task move around the entire globe with time, it is necessary to provide the required input variables to drive ATTILA (such as the wind velocity vector from EMAC) as global fields (i.e. by cloning of the global domain of these variables). The subroutines for data transpositions between parallel decomposed grid points and corresponding cloned global variables have been added to the MESSy infrastructure submodel TRANSFORM.

To facilitate the exchange of Lagrangian objects between Lagrangian-enabled submodels as so-called channel objects (see , for a detailed explanation of the MESSy infrastructure submodel CHANNEL), we define a new representation (see , for a detailed explanation) of rank 1, global dimension length N, and local (i.e. task specific) dimension length ni. The corresponding MPI-based gather and scatter routines for serial NetCDF I/O have been added to the MESSy infrastructure submodel TRANSFORM. This new representation, named LG_ATTILA, is used by the Lagrangian submodels to define their specific Lagrangian objects.

For tracers we further define two additional tracer sets (see for a detailed explanation), one (“tracer_lg”) in the new Lagrangian representation to handle the Lagrangian tracers and one “tracer_lggp” in grid-point representation. The latter is solely used to transform the Lagrangian tracers into grid-point space for output and further analyses.

Subroutines to transform and transpose variables between Lagrangian representation and (parallel decomposed) grid-point representation, and vice versa, are collected in a specific toolbox module named ATTILA_TOOLS. This also comprises specific subroutines for the transformation of grid-point emission fluxes into Lagrangian tracer tendencies. For the latter, four options are implemented: the emitted mass from a grid cell is distributed in one of the following ways.

Evenly among all LG parcels in that grid box. In the case that there is no parcel at a given time in that grid box, the mass is stored and accumulated over time and eventually released into the next parcel(s) passing by.

ATTILA requires up to four series of pseudo-random numbers, one for the boundary layer turbulence parameterisation (Sect. ), one for the convection parameterisation (Sect. ), one for an envisaged (but not yet implemented) additional clear air turbulence parameterisation, and one for particle displacements parameterising a Monte Carlo diffusion approach. One additional pseudo-random number series is used for the initial distribution of the parcels in the model atmosphere. These pseudo-random number series are provided by the MESSy infrastructure submodel RND. RND provides uniformly distributed pseudo-random numbers between 0 and 1, calculated with the standard Fortran90 function, RANDOM_NUMBER, the Mersenne Twister algorithm , or the Luxury algorithm . Based on these, RND can also provide normally distributed random numbers centred around zero using the Marsaglia polar method

http://en.wikipedia.org/wiki/Marsaglia_polar_method(last access: 6 May 2019)

For every time step (in our simulations: Δt=600 s), the parcels are advected by the three-dimensional wind field using a fourth-order Runge–Kutta method. The wind field is interpolated on the parcel positions by linear interpolation horizontally (i.e. on the latitude–longitude grid) and by cubic Hermite interpolation vertically. The initialisation of the positions in the atmosphere is carried out randomly so that the number of parcels corresponds to the mass of the respective model layer.

In the vertical direction we may use either η-coordinate vertical velocities (η=pp0, η˙-kinematic velocity) calculated from the horizontal flux divergence using the continuity equation or isentropic coordinates ξ , where the vertical velocities ξ˙ are calculated from the EMAC diabatic heating rates (diabatic velocity). The kinematic velocity is provided by default from EMAC, whereas the diabatic velocity was newly implemented similar to and . In our notation, 3ξ=θf, with θ being the potential temperature and f being defined as 4Ifp>prf=sin⁡π21-pps1-prp0,5Ifp≤prf=1, with κ=Rv/cpp and p being the atmospheric pressure; pr is the atmospheric pressure of the climatological tropopause, and cp is the heat capacity at constant pressure. It characterises the transition from a pure θ-coordinate system to the ξ-coordinate system. The standard surface pressure is p0=1013.25 hPa, and ps is the actual surface pressure.

The vertical velocity in this coordinate system is defined as the time derivative of Eq. (): 6ξ˙=θ˙f+θf˙. The diabatic vertical velocity in the troposphere ξ˙ for p>pr appears as a mixed velocity between pure diabatic θ˙ and kinematic velocity in the troposphere according to Eq. (). Only in the stratosphere is ξ˙ a pure diabatic velocity.

Every parcel located within the planetary boundary layer (PBL) is randomly displaced in the vertical direction within the corresponding grid cell. This stochastic mixing represents the boundary layer convective mixing process. The boundary layer height is calculated outside of ATTILA within the submodel TROPOP.

The LG convection scheme uses the mass fluxes of the standard grid-box convection scheme in EMAC (submodel CONVECT) to calculate the convective parcel movement. Therefore, we will first shortly introduce the convection scheme of EMAC because the LG convection scheme is based on it. Convection in the standard convection scheme is initiated when the convergence of moisture in a vertical column of the atmosphere exceeds a certain threshold value and a convectively unstable layer exists. Three types of convection are distinguished: deep convection occurs if moisture convergence through advection and evaporation at the surface takes place. Shallow convection occurs if moisture convergence is only by evaporation at the surface, and mid-level convection occurs if the criteria of deep and shallow convection are not fulfilled but 90 % relative humidity is reached within the planetary boundary layer.

Convection is parameterised by dividing a vertical column into an area of updraft (superscript u), downdraft (superscript d), and an area of compensating motion in the environment (superscript e). Convective transport in EMAC is parameterised only in the vertical direction as a divergence of the tracer mass fluxes Fu=MuXu, Fd=MdXd, and Fe=MeXe: 7-1ρ‾∂(ρ‾w′X′‾)∂zconv=-1ρ‾∂Fu∂z+∂Fd∂z+∂Fe∂z, where X‾ is the tracer mass mixing ratio, M is the mass flux, ρ‾ is the air density, w‾ is the vertical wind component, and z the height. The quantities with an overbar are horizontal averages over the grid box, and the quantities marked with a prime are the horizontal deviations from the respective grid-box mean variables. Mu≥0 , Md≤0, and Me are the mass fluxes of air for updraft, downdraft, and the environment, respectively.

The change in mass fluxes with height is dependent on entrainment and detrainment fluxes. 8∂Mu∂z=Eu-Du9∂Md∂z=Ed-Dd10withMe=-(Mu+Md) Eu (Ed) comprises the entrainment (detrainment) rates due to turbulent exchange of mass through cloud edges, and for the updraft Eu only, it implies the organised inflow associated with large-scale moisture convergence in cases of deep or mid-level convection. Accordingly, the detrainment rates include the turbulent exchange in updraft, downdraft, and, for the updraft only, the organised outflow at cloud top.

The corresponding tracer mass fluxes are as follows. 11∂Fu∂z=∂MuXu∂z=EuXe-DuXu12∂Fd∂z=∂MdXd∂z=EdXe-DdXd13∂Fe∂z=∂(MeXe)∂z The calculation of the convective transport of tracer mass starts with the determination of the type of convection (deep, shallow, mid-level). According to the estimated convective available potential energy (CAPE) the mass flux at cloud base is calculated. Further details of the calculation of the mass fluxes are described by and .

In our LG convection scheme air parcels can follow the updraft, downdraft, or the compensating motion in the environment at a grid column with convection within one time step. The forcing used for the Lagrangian convection scheme is provided by the mass fluxes M of the convection scheme of EMAC for updraft and downdraft, respectively. Probabilities P for each level are calculated from the mass fluxes within a vertical column. Each LG parcel is equipped with a (precalculated) random number (see Sect. ). For each parcel, ascend (or descend) in an updraft (downdraft) is applied with probability P. The probability for an air parcel to follow the updraft P is equal to the ratio of the mass of the air parcel moving into the updraft to the mass of air at that level.

If (Mk-Mk+1)>0, which means that the mass flux increases with height, then 14Pk=memg=(Mk-Mk+1)Δtgpk+1-pk, with 15mg=(pk+1-pk)Agandme=(Mk-Mk+1)AΔt, where p is pressure (hPa), A is area (m2), M represents air mass fluxes (kgm-2s-1), g is gravity acceleration, and Δt is time step length (s).

If (Mk-Mk+1)<0, i.e the mass flux decreases with height, a negative probability is defined to reflect a situation in which a parcel may leave the updraft due to detrainment. The probability is equal to the ratio of the mass leaving the level to the mass entering the same level from below: 16P=(Mk-Mk+1)/Mk+1. The equations of the probability functions are analogous for the downdraft.

The LG convection scheme strictly conserves local mass because for every time step the number of parcels per grid box after convection equals the number before convection (see n= const. in Fig. ). Every updraft and downdraft forces a compensating large-scale motion of parcels. The probability P for subsidence is not estimated from the mass fluxes provided by EMAC. It is calculated for every layer, depending on the number of parcels that need to subside in order to fulfil the mass conservation for every layer.

We use 222Rn in situ measurements at the surface layer of 18 stations distributed worldwide as they were published by . The model results are long-term averages of 222Rn in the surface layer over the years 1960–2000. They are horizontally linearly interpolated to the respective location of the observations. Six stations (Crozet, Bermuda, Amsterdam Island, Kerguelen, Dumont, and Mauna Loa) are far away from the continents and show the effect of long-range transport (222Rn lower than 1000 mBq m-3; STP). A further six stations (Socorro, Cincinnati, Para, Puy de Dôme, Beijing, and Hohenpeißenberg) are located on continents in the vicinity of the 222Rn sources. Finally, six stations (Gosan, Hong Kong, Cape Grim, Livermore, Bombay, and Mace Head) are located at coastal sites influenced by the prevailing wind direction either from the sea or the continent. A detailed comparison between model results (horizontal axis) and measurements (vertical axis) is shown in Fig. . A total of 12 monthly values of the mean annual cycle of GP, LG(diab), and LG(kin) are presented as data points with different colours for each station. The results are similar to those of their Fig. 14. This cannot be taken for granted because used a nudged simulation and a model set-up with 90 vertical levels, whereas in our study the simulation is free running and our model set-up has 47 vertical levels. Two stations are for all 12 monthly mean values out of the thick dashed area in Fig. : Beijing and Kerguelen (southern Indian Ocean). Beijing's 222Rn values are too low and the Kerguelen values are too high as simulated with all models. In the measured 222Rn values for Kerguelen are not captured in the simulation. The simulated 222Rn values are strongly dependent on the local wind direction at the surface of the measurement sites, especially for the coast and the remote regions over sea.

The NARE campaign took place in the vicinity of Nova Scotia and Brunswick, Canada, in August 1993. Data were sampled over the ocean and over the continent. We used the simulation data of a climatological mean August (1960–2000) averaged over the region where the flights took place (60–70∘W and 41–46∘N). The triangles in Fig.  show the measured 222Rn in the atmosphere for several flights. The thick dashed curve is a spline interpolation on the grid of all flights. The 222Rn mixing ratio decreases with height up to about 3 km and remains around 10-24 molmol-1. Both LG simulation results agree with the observations and are close to the GP results. The measurements of the Moffett campaign (Fig. ) show a relatively large scatter of 11 single profiles in the free troposphere. The simulated 222Rn profiles in that region were selected as a climatological mean for the months June to August (1960–2000) and capture the observations quite well. Observed 222Rn emissions are highly variable due to the dependence on the physical characteristics of the soil . Therefore, a certain spread between observations and model results is expected and acceptable.

Mean AoA in the stratosphere is calculated from the SF6_AoA tracer. The transit time is estimated by comparing the tracer mixing ratio in the stratosphere with the SF6_AoA mixing ratio in the tropical boundary layer. Figure  shows a comparison of SF6_AoA at 20 km of height between the GP, LG(kin), and LG(diab) results along with AoA derived from satellite observations from MIPAS and in situ measurements of from the SF6 tracer, as well as from CO2 in situ data . GP and LG(diab) show realistic distributions of AoA, although slightly lower AoA than the in situ measurements. MIPAS data are known to overestimate AoA in the polar regions . This is attributed to a known sink of SF6 in the upper stratosphere that is not accounted for in our simulations. Noticeably, the LG(diab) simulation is closer to the observations and LG(kin) shows up with an age that is too low. For the analysis, we therefore restrict our further evaluation to LG(diab). The mean age distribution (1960-2010) of LG(diab) is shown in Fig. . It confirms the well-known characteristics of the stratospheric Brewer–Dobson circulation with younger air in the tropical pipe and older air over the poles. Furthermore, the simulated AoA is slightly older in the whole stratosphere compared to GP (Fig. ), which is most pronounced near the poles below 50 hPa. This difference is attributed to the Eulerian vertical velocity used in the flux-form semi-Lagrangian transport scheme for the GP simulations, which shows upwelling and downwelling at different high latitudes that are not related to the net tracer transport .

AoA spectra are calculated directly from the clock transit times. These LG clocks represent the actual time a parcel resides in the stratosphere after it has crossed the tropopause level. However, these clocks do not “mix their time” with other parcels. Therefore, the resulting spectrum might differ from age spectra calculated from so-called AoA “clock tracers” . Typical age spectra for the tropics (20∘ N–20∘ S) and the poles (70–90∘ N, 70–90∘ S) between 50 and 0.1 hPa are shown in Fig. . The frequency distribution of the LG(diab) simulation is calculated for every month for the years 1990–2010, binned into 0.5-year bins, and normalised. We find the largest frequency at a parcel age between 0 and 0.5 years in the tropics and an exponential shape of the spectrum. In contrast, the modal age is roughly 3 years at the poles. The shape of the spectrum is Gaussian with a positive skewness. The seasonal age spectra (selected between 400 and 500 K at 50–70∘ N; see Fig. ) show characteristic multiple local maxima along the time axis. The width of these maxima increases from spring (MAM) and summer (JJA) over autumn (SON) to winter (DJF). The distance of the maxima on the transit time axis is around 1 year. These maxima reflect the different contributions of air masses from the tropics and from high latitudes in the seasonal cycle . The seasonal age spectra look qualitatively similar as in Fig. 5 of , although our modal values are about 0.3 years younger. The different modal values between and our Fig.  are probably a consequence of the utilised concept in calculating the AoA. In LG(diab) we use our LG clocks to calculate the transit time in the stratosphere directly if the parcels cross the tropopause level (see Sect. ). calculated their seasonal spectrum from an AoA clock tracer and the Green's function and relate their stratospheric mixing ratios to the tracer mixing ratio in the boundary layer to calculate the AoA. Therefore, the larger modal values in refer to an additional transit time up to the tropopause level. The effect of these two different concepts on the calculation of AoA in the lower stratosphere is discussed further in Sect. .

AoA is influenced by the amount of mixing between adjacent parcels. Inter-parcel mixing can be regarded as a diffusion process leading to a reduction of local AoA gradients. The effect of inter-parcel mixing makes stratospheric air generally younger (Fig. ). Stratospheric AoA without inter-parcel mixing as represented in LG(diab), described as LG(diabnm), is mostly up to 2.5 months older compared to the AoA with standard mixing (see Sect. ) but slightly younger in the tropical lower stratosphere, where mixing with upper tropospheric air becomes important. This “younger air through inter-parcel mixing” should not be confused with the “ageing by mixing” concept of , , , and . They calculate AoA from the tracer budget equation of AoA and distinguish between the different terms: tendencies due to the residual stratospheric circulation and the tendencies of AoA due to eddy mixing and due to turbulent diffusion. Their concept allows for the separation of the contribution of mixing on the local AoA budget. In contrast, in our study we simply compare a simulated tracer with mixing to a tracer distribution without mixing (perturbation concept). Inter-parcel mixing in the troposphere (Fig. ) has only a small and statistically insignificant effect on the simulated AoA because the troposphere is a well-mixed region where parcels often have contact with the surface source. The surface is the only region (in the vertical) where the tracer mixing ratio for the tracer without inter-parcel mixing can be changed.