A programme () for regular grids (i.e. with independent dimensions) is available. To handle this type of grid, a classical multilinear interpolation is performed. Figure  shows the variables for N=3 defined hereafter in the section.

For the particular case of a regular grid with independent dimensions, the result Ỹ of the multilinear interpolation of the 2N identified neighbours can be expressed as 2Ỹ=wN0…wi0…w10×Y0(0…0…0)+wN0…wi0…w11×Y1(0…0…1)+…+wNδN…wiδi…w1δ1×Yk(δN…δi…δ1)+…+wN1…wi1…w11×Y2N-1(1…1…1), with δi the binary digit equal to 0 or 1, and the weights wiδi for i∈[1,N] defined as 3wi0=θi1-xiθi1-θi0wi1=1-wi0.

Variable Θi is the list of interval edges on each dimension i and does not depend on other dimensions. θiδi indicates the bottom (δi=0) and top (δi=1) edges on each dimension i∈[1…N] so that xi∈]θi0,θi1]. Yk is a one-dimensional array with 2N elements storing the value Y of the function at the identified neighbours Ψ on each dimension: 4Yk(δN…δi…δ1)=fΨ(θNδN,…,θiδi,…,θ1δ1), with k∈[0,2N-1].

The tuple (δN…δi…δ1) is the binary transformation of integer k defined as ∑i=0N-1(δi×2i). The coefficients Γk=wNδN…wiδi…w1δ1 as a product of weighting factors on each direction can be seen as a binary suite that is convenient to handle in a compacted and optimized Fortran programming strategy for the regular grid version of the code (Appendix ).

Considering the general programme called , the coordinates of edge points are stored in a one-dimensional array of n=∏i=1NIi elements with Ii the number of edges on each dimension i. The tuple of coordinates (j1,…,jN) of an interval edge θki, with ji the indexed coordinate on dimension i, is transformed in a one-dimensional array indexed on k∈[1,n] by 5k=∑j=1Nij-1∏l=0j-1Il+1, with I0=1 for initialization.

Once the nearest neighbour is found, the result Ỹ of the interpolation is a weighting procedure of the 2N closest vertex using a Shepard interpolation based on the inverse distance calculations: 6Ỹ=∑k=02N-1(Γk×Yk), with Yk=f(Υk) the value of the function f at neighbour Υk of coordinates (θk1,…,θkN) and 7Γk=1/dk∑k=02N-1(1/dk). The distance dk between the point of interest of coordinates (x1,…,xN) to the neighbour k∈[1,n] is calculated as 8dk=∑i=1N∣xi-θki∣p1p. The previous formulas are valid for dk≠0; in the case of dk=0, the procedure stops and exits, returning the exact value of the corresponding data of the nearest neighbour. For a distorted mesh or matrix, or dimensions with different units (e.g. mixing time with length), a hard-coded option (norm=.true. or .false.) is also available to normalize the intervals with an average interval Δi value for the calculation of distances, so that 9dk=∑i=1N∣xi-θki∣∣Δi∣p1p.

In order to test this new interpolation programme, it is implemented in a back-trajectory model called Backplumes. This model was already used in some studies such as and . Backplumes is open source and is available on the CHIMERE website. Backplumes calculates back trajectories from a starting point and a starting date. It is different from other back-trajectory models, such as the Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) , Stochastic Time-Inverted Lagrangian Transport (STILT) and FLEXible PARTicle dispersion model (FLEXPART) , because it launches hundreds of particles and plots all trajectories as outputs. Thus, the answer is complementary compared to the other models: the output result is all possible trajectories and not only the most likely.

An advantage of Backplumes for the WRF and CHIMERE users is that the code is dedicated to directly read output results of these models. Being developed by the CHIMERE developer teams, the code is completely homogeneous with CHIMERE in terms of numerical libraries. Another advantage is that the code is very fast and calculates hundreds of trajectories in a few minutes. Using the wind fields of WRF or CHIMERE, and running on the same grid, the results of back trajectories are fully consistent with the simulations done by the models.

Backplumes is dedicated to calculate transport but not chemistry: only passive air particles (or tracers) are released. But a distinction could be made between gaseous or particulate tracers: for the latter, a settling velocity is calculated to have a more realistic trajectory. The model is easy to use and light because a small set of meteorological parameters is required. These meteorological parameters are described in Table  for WRF and CHIMERE.

The first step of the calculation is to choose a target location as a starting point. The user must select a date, longitude, latitude and altitude, obviously included in the modelled domain and during the modelled period. From this starting point, the model will calculate trajectories back in time. The number of trajectories is up to the user and may vary from one to several hundreds of tracers.

At each time step and for each trajectory, the position of the air mass is estimated by subtracting its pathway travelled as longitude Δλ, latitude Δϕ and altitude Δz to the current position. To do so, all necessary variables are interpolated with the NterGeo.v2020a interpolation programme described in the previous section. The calculation is described in Appendix .

In order to respect the Courant–Friedrichs–Lewy (CFL) number, a sub-time step may be calculated. If the input data are provided hourly (as in many regional models), the meteorological variables are interpolated between the two consecutive hours to obtain refined input data.

The goal of Backplumes is to estimate all possible back trajectories. Then, starting from one unique point, it is necessary to add a pseudo-turbulence in the calculation of the altitude. Depending on the vertical position of the tracer, several hypotheses are made. Two parameters are checked for each tracer and each time step: (i) the boundary layer height enables us to know if the tracer is in the boundary layer or above in the free troposphere, (ii) the surface sensible heat fluxes enable to know if the atmosphere is stable or unstable.

When the tracer is diagnosed in the boundary layer, there are two cases: the boundary layer is stable or unstable. If the boundary layer is stable, Q0<0, the tracer stays in the boundary layer at the same altitude. The new vertical position of the tracer is 14zt-1=zt.

If the boundary layer is unstable, Q0>0, the tracer is considered in the convective boundary layer and may be located at every level in this boundary layer for the time before this. Therefore, a random function is applied to reproduce a potential vertical mixing.

15zt-1=Rand×h‾

The random function “Rand” calculates a coefficient between 0 and 1 to represent stochastic vertical transport of the tracer.

It is considered that 15 mn is representative of a well-mixed convective layer . If the time step is larger than 15 mn, the random function is applied. But if the time step is less than 15 mn, the vertical mixing is reduced to the vicinity of the current position of the tracer. In this case, we have 16zt-1=Rand×Δz×[zt], where Δz=12ztk-1+ztk+1 and k is the vertical model level corresponding to zt.

In the free troposphere, the evolution of the tracer is considered to be influenced by the vertical wind component. A random function is applied to estimate its possible vertical motion with values between 0 and w/2 m s-1, representative of all possible values of vertical wind speed in the troposphere, . The vertical variability of the tracer's position in the free troposphere is calculated by diagnosing the vertical velocity as 17zt-1=zt-(0.5+Rand)w3600Δt.

An example is presented for the same case and the WRF and CHIMERE models. The difference between the two models is the number of vertical levels (35 for WRF and 20 for CHIMERE, from the surface to 200 hPa). The online modelling system WRF-CHIMERE is used, meaning that the horizontal grid is the same (a large domain including Europe and Africa and with Δx=Δy=60 km). The wind field is the same for both models: CHIMERE directly uses the wind field calculated by WRF. The boundary layer height is different between the two models, with WRF using the schemes and CHIMERE using the scheme. The surface sensible heat flux is the same between the two models, with CHIMERE using the flux calculated by WRF. WRF has more vertical model levels than CHIMERE; thus, meteorological fields are interpolated from WRF to CHIMERE. It impacts the horizontal and vertical wind fields.

Figure  presents the results of back trajectories launched on 10 August 2013 at 12:00 UTC. The location is at longitude 10∘ E and latitude 25∘ N, with an altitude of 0 m a.g.l. This location is of no scientific interest but is in the middle of the domain, in order to have the longer trajectories. The complete duration of trajectories represents 10 d back in time. Overall, 120 trajectories are launched at the same position and time. They are randomly mixed when they are in the boundary layer to represent the mixing and the diffusion.

The most important part of the plume comes from the north of the starting point. For this main plume, the calculation is similar between the two models. Another large part of Backplumes is modelled at the east of the starting point. However, this fraction is mainly modelled with WRF but not with CHIMERE, where only a few trajectories are diagnosed. One possible explanation may be found by analysing the vertical transport of the trajectories.

Figure  presents all plumes displayed in the previous figure but projected along the same time–altitude axis. The differences between the two Backplumes results are mainly due to the calculation of the boundary layer height. When WRF diagnoses an altitude of ≈ 3000 m, CHIMERE diagnoses ≈ 2000 m, leading to different direction and wind speed. Then, this implies a split of the plumes with WRF but not with CHIMERE. This illustrates the sensitivity of the result to the driver model. But, in both cases, the answer in our case is clearly that the main contributions of the air masses located at the starting point are mainly coming from the north-east, crossing Tunisia and then the Mediterranean Sea and Europe. The main difference between the two calculations is the eastern part of the plume, which is more intense with WRF than CHIMERE.