Large-eddy simulation (LES) and Lagrangian stochastic modeling of passive particle dispersion were applied to the scalar flux footprint determination in the stable atmospheric boundary layer. The sensitivity of the LES results to the spatial resolution and to the parameterizations of small-scale turbulence was investigated. It was shown that the resolved and partially resolved (“subfilter-scale”) eddies are mainly responsible for particle dispersion in LES, implying that substantial improvement may be achieved by using recovering of small-scale velocity fluctuations. In LES with the explicit filtering, this recovering consists of the application of the known inverse filter operator. The footprint functions obtained in LES were compared with the functions calculated with the use of first-order single-particle Lagrangian stochastic models (LSMs) and zeroth-order Lagrangian stochastic models – the random displacement models (RDMs). According to the presented LES, the source area and footprints in the stable boundary layer can be substantially more extended than those predicted by the modern LSMs.

Micrometeorological measurements of vertical turbulent scalar fluxes in the
atmospheric boundary layer (ABL) are usually carried out at altitudes

Schematic representation of the footprint evaluation algorithm.

Traditionally, footprint functions

The measurements of the scalar flux footprint functions in natural
environment are restricted

Modeling approaches used for footprint calculation include stochastic models,
such as single-particle first-order Lagrangian stochastic models based on the
generalized Langevin equation (LSMs) and zeroth-order stochastic models (also
known as the random displacement models, RDMs) (see the reviews listed in the
papers

Large-eddy simulation (LES), employing the Eulerian approach for the
transport of scalars, was applied for the first time for a footprint
calculation in

Lagrangian transport in LES is complicated by the problem of the description
of small-scale (unresolved) fluctuations of the particle velocity, which is
similar to the problem of subgrid modeling of Eulerian dynamics. A common
approach for Lagrangian subgrid modeling in LES is the application of subgrid
LSMs

The statistics of simulated turbulence in LES may significantly differ from
the statistics of real turbulence. For example, the use of dissipative
numerical schemes or low-order finite-difference schemes usually results in a
suppression of fluctuations over almost the entire resolved spectral ranges
of discrete models

Numerical simulations of Lagrangian transport in LES are also limited by the low scalability of parallel algorithms. This is due to the impossibility of uniform loading of processors in a joint solution to the Euler and Lagrangian equations, a large number of interprocessor exchanges and an unstructured distribution of characteristics required for Lagrangian advection in the computer RAM memory.

Thus, all methods of numerical and analytical determination of the functions

According to the need for computational cost reduction, one of the objectives of this study is to establish the role of stochastic subgrid modeling in the correct description of the particle dispersion in LES. Is it possible to simplify the calculation and to avoid the introduction of stochastic terms without the loss of accuracy in some integral characteristics, such as the footprints or the concentration of pollutants emitted from the point sources? The role of subgrid fluctuations is reduced with an increase in spatial LES resolution. Therefore, the independence of results from the mesh size is used as a criterion for checking the quality of Lagrangian transport procedures in LES. It will be demonstrated that the subgrid stochastic modeling in LES can be omitted in most cases. Instead, we propose a “computationally cheap” procedure of inverse filtering supplemented by divergent correction of Eulerian velocity to replace the subgrid stochastic modeling in LES (see the description below).

Subgrid transport is especially significant near the surface and/or under
stable stratification – all are the cases associated with small eddy size.
That is why the stable ABL was selected as the key test scenario in this
study. We slightly modified the setup of the GABLS

LES results are used as the input data for the stochastic models (LSMs and RDMs). These data are pre-adjusted using known universal dependencies and taking into account an incomplete representation of turbulent energy in LES. The comparison of results of different stochastic models and the results from LES allows one to specify the parameters for the LSMs and permits one to identify the differences between LSMs and RDMs under the conditions that have not been tested previously.

The paper is organized as follows. Section

In addition to the basic calculation, we carried out a series of tests (see
Supplement Sect. S1) under unstable stratification in the ABL with different
grid steps in the LES model. This allows us to compare the results presented
here with similar results obtained in previous studies

Computational methods for determination of footprints often reduce to the
implementation of Lagrangian transport of marked particles. Each particle can
contain a number of attributes, including its initial coordinate

In the horizontally homogeneous case, one can calculate the footprint

The nonuniform Cartesian grid

Lagrangian particle velocity

There is a great practical interest in the calculation of footprints, as well
as of spatial and temporal characteristics of pollution transport from
localized sources above heterogeneous surfaces and in the areas with complex
geometry (in the urban environment, over the surfaces with complex terrain or
over the alternating types of vegetation). LES of such flows becomes a
routine procedure with increasing performance of computers. However, the
calculation of statistical characteristics of Lagrangian trajectories is
complicated in this case by the need of transport of huge number of tracers

On the other hand, a large number of particles (see, e.g., Supplement
Fig. S2.1b) allows one to estimate the local instantaneous spatially filtered
concentration of the scalar:

Another approach (more widespread due to a lower computational cost) is the
replacement of the entire turbulent component of velocity by a random process
(Lagrangian stochastic models (LSMs)):

A single-particle first-order LSM is formulated as follows. Velocity

The function

Lagrangian models are very sensitive to the choice of universal functions
that define the normalized root mean square (RMS) of the vertical velocity

The data of measurements in the ABL demonstrate large variation. For example,
the values of

There is no consensus on the value of

The simplest approach for development of the models of particle dispersion
entails replacement of the Eulerian advection–diffusion equation

Probability density of particle position

Using the Fokker–Planck equation, it can be shown that Eq. (

An RDM has some major disadvantages. First, it shares the limitation of
Eulerian eddy-diffusion treatment of turbulent dispersion, i.e.,
“K-theory”. Correspondingly, it is not able to describe the non-diffusive
near field of a source. Also, an RDM can not be applied for the convective
ABL, where the counter-gradient transport is observed. Besides, it requires
the exact values of diffusion coefficient

A system of equations (

A mixed model

The procedure of calculation of the coefficients

The generalized solution to the discrete analog of Eq. (

Eddy-diffusion models are used for subgrid heat and concentration transfer:

A distinctive feature of this model is that the discrete spatial filter operator

Below, the subgrid and subfilter modeling methods used for the simulations in
the current study are listed. These methods will also be used in combinations
as defined in Sect.

First, we will use the recovering of “subfilter” fluctuations
(Eqs.

Note that for the use of such a procedure, LES models should exhibit the
properties of a model with an explicit filtering. A similar approach was
recently applied by

Second, we will apply the subgrid stochastic model proposed in

The parameter

Thus, the total unresolved kinetic energy was calculated as the sum of
“subfilter” energy

To evaluate the value

All the values required for a application of this model were linearly
interpolated into the particle position everywhere except at heights

Third, the RDM specified in Sect.

This model does not contains the arbitrary specified parameters except those
which were already used in the Eulerian LES. The coefficient

Finally, in order to find out whether the subgrid mixing is one of the key
processes in the dispersion of Lagrangian tracers, we introduced an
additional correction to the particle velocities:

Correction given by Eqs. (

The idea of such a correction was based on the assumption that details of the
mechanism of subgrid mixing have a little influence on the statistics of
trajectories at sufficiently large distances from the source and at long
enough time

In preliminary tests it became clear that trilinear interpolation of each
velocity component provides no advantages for footprint calculation in
comparison with the following simplified linear interpolation on a staggered
grid:

Stable boundary layer at the latitude 73

This setup is based on the observation data (see

Mean wind velocity

The mean wind velocity and the potential temperature, calculated with the
different spatial steps

Passive Lagrangian tracers were transported simultaneously with the
calculations of dynamics. Each particle, when reaching a lateral boundary of
domain, is returned from the opposite boundary in accordance with periodic
conditions. The reflection condition is used at the ground. The particles are
ejected at the height

To limit the number of particles involved in the calculation, the absorption
condition is applied at the height of 100 m within the ABL. It was verified
previously that the upper boundary condition does not have a large impact on
the results of calculations of footprints for the heights

In the last hour of simulation, the averaged number of particles in each cell
of the grid near the surface was approximately equal to 700–800, 350–400,
180–200 and 110–130 for grid steps

Total

Figure

The unresolved fraction of the flux

Crosswind-integrated scalar flux footprints

Figure

If the particles are advected by the filtered velocity

The inclusion of stochastics within the first layer improves the result
(dashed curves in Fig.

The advection of particles by the velocity

While the particles were advected by the “defiltered” flow, we have also
used the correction (Eqs.

In the inertial range of three-dimensional turbulence along with the kinetic
energy the variance of a passive scalar concentration is transferred from
large scales to small scales with the formation of the spatial spectrum

Figure

One can expect that in more complicated cases (e.g., the turbulent flow
around geometric objects and the formation of quasi-periodic eddies), the
accumulation of small-scale noise in the concentration field may lead to the
incorrect advection of concentration by the resolved eddies. This effect may
also be important for inertial particles when the nonphysical variance of
concentration can directly affect dynamics. In additional tests it was found
that the correction given by Eqs. (

Crosswind-integrated scalar flux footprints

One can obtain footprints close to those presented in Fig.

Crosswind-integrated scalar flux footprints

In Fig.

Generally, results are in close agreement with the results of LES with the
fine grid, except for some details. One can see the intrinsic defect of the
RDM when it is applied to the dispersion of particles in a near field of a
source. That is, as the RDM is the approximation of the diffusion process
with the infinite speed of the signal prorogation, this model overestimates
values of

In contrast to the subgrid LSM and to the methods of velocity correction
proposed above, the advantage of the subgrid RDM consists in the absence of
the arbitrary prescribed parameters and in the absence of the need to involve
the additional suppositions. In terms of Eulerian statistics, this model is
identical to Eq. (

The impact from the subgrid RDM is reduced when it is applied within the first grid layer only. In this case, the footprints are approximately the same as the footprints computed using the other approaches.

The trajectories of a large number of particles (

Two-dimensional footprints

The contribution of the negative part of the flux to the “measured” flux is
significant, as shown in Fig.

The LES results with grid step

Mean wind velocity at the height

The fluxes

The variances of velocity components

The final estimations of the variances of velocity components are shown in
Fig.

Usual interpolation is not applicable to the calculation of dissipation
rate near the surface, where

Additional analysis showed that, if

Discrete values of non-dimensional dissipation

A random displacement model (Eq.

The horizontal eddy diffusivities

One can see that the formula (Eq.

The following stochastic models were tested using the data prepared as described above.

RDM0 is the random displacement model with uncorrelated components.
Particle position is computed by the formula similar to Eq. (

RDM1 differs from RDM0 by using the noise with
inter-component correlations:

LSM0 is the Lagrangian
stochastic model without a WMC:

LSM1 is based on the one-dimensional well-mixed model:

LSMT is a three-dimensional Lagrangian stochastic model satisfying a WMC,
which is proposed by

The setups of numerical experiments with RDMs and LSMs were close to particle
advection conditions in LES (absorbtion at altitude 100 m, ejection at

Crosswind-integrated footprints

Figure

Models RDM0, RDM1 and LSM1 provide very similar results. Faster mixing is
observed in stochastic models below altitude

The substantial disagreements with LES were obtained using the
three-dimensional Thomson model (Eq.

LSMT (Eq.

Taking into account this expression and Eq. (

One can see that Thomson's multi-dimensional model with

Finally, it can be seen from Fig.

Turbulent Prandtl

Two-dimensional footprints

The crosswind mixing can be characterized by an RMS of transversal
coordinates of the particles depending on the mean distance from the source:

Wind direction rotation leads to widening of a concentration trace from the
point source (see the thin dashed line in Fig.

Scalar dispersion and flux footprint functions within the stable atmospheric boundary layer were studied by means of LES and stochastic particle dispersion modeling. It follows from LES results that the main impact on the particle dispersion can be attributed to the advection of particles by resolved and partially resolved “subfilter-scale” eddies. It ensures the possibility of improving the results of particle advection in discrete LES by the use of recovering of small-scale partially resolved velocity fluctuations. If one uses the LES model with the explicit filtering, then this recovering is straightforward and consists of application of the known inverse filter operator. Apparently, a similar method can be implemented for other LES when the spatial filter is not specified in an explicit form. This would require, however, the prior analysis of the modeled spectra to identify an effective spatial resolution and the actual shape of the implicit filter. For substantial improvement of particle transport statistics, it is enough to use a subgrid Lagrangian stochastic model within the first computational layer only, where the LES model becomes equivalent to the simplified RANS model.

When the particles are advected by a divergence-free turbulent velocity field, then the variance of the particle concentration can be accumulated at small spatial scales. In the considered case, it does not affect directly the particle advection by the large eddies and has no significant influence on the results of footprint calculations. In those cases, when the instantaneous characteristics of the scalar field of a particle concentration are important, additional correction to particle velocities may be required. It can be done both through the introduction of stochastics, resulting in the diffusion of concentration, and through the “computationally inexpensive” divergent correction of the Eulerian velocity field.

Under the stable stratification, to calculate the flux footprint, it is
preferable to use stochastic models, which describe the particle dispersion
close to the process of scalar concentration diffusion with the effective
coefficient

One-dimensional stochastic models can be supplemented by the horizontal
particle dispersion in a simple way. Introduction of the correlation between
particle displacement components in RDM does not improve or change results
substantially. However, the coefficients of horizontal diffusion

Model LSM1, constructed as a combination of independent stochastic models in
each direction (well mixed in the vertical direction only), gives reasonable
results, although this model does not satisfy a WMC in general. In contrast,
the three-dimensional Thomson model with a WMC and

According to the presented LES, the source area and footprints in the stable
ABL can be substantially more extended than those predicted by the modern
LSMs and footprint parameterizations based on their results (e.g., the
parameterization by

We emphasize that a very simple case of the moderately stratified stable ABL in almost steady-state conditions was considered here. This setup of numerical experiments permits the detailed intercomparison of different approaches for the particle dispersion modeling, which utilize identical simplifications. On the other hand, in a real environment the scalar flux footprint functions can be greatly influenced by the meteorological non-stationarity, the peculiarities of mixing inside the roughness layer, internal radiative heating or cooling in the ABL, and so on. Also, a wider investigation of different stability regimes from neutrality to strong stratification must be undertaken in future studies to confirm the universality of the findings.

The code of the LES model is available on request for scientific researches in cooperation with the first author (and.glas@gmail.com). The data from LES are attached to the Supplement. These data were prepared as was discussed in Sect. 5.1 and can be used for the stochastic models' evaluation. Besides, the Supplement contains the data for crosswind-integrated footprints and two-dimensional footprints obtained in LES (see Figs. 6 and 9).

To confirm the small impact of the top boundary condition on the results
presented above, an additional run was performed (LES with

The footprint functions

The functions

Finally, we want to mention that the stable ABL case considered here is specific. The findings described above may not be valid for different types of ABL. We select this setup of the numerical experiment intentionally for the sake of convenience of the comparisons of statistics obtained by the Eulerian and Lagrangian methods. This provides additional ability for the testing of Lagrangian particle transport routines implemented in the LES model code.

This research is implemented in the framework of Russian–Finnish collaboration, funded within the CarLac (Academy of Finland, 1281196) and GHG-Lake projects. The Russian co-authors are partially supported by the Russian Foundation for Basic Research (RFBR 14-05-91752, 15-05-03911 and 16-05-01094). The Finnish co-authors acknowledge EU project InGOS, the National Centre of Excellence (272041), ICOS-FINLAND (281255), and Academy professor projects (1284701 and 1282842) of the Academy of Finland.Edited by: S. Unterstrasser Reviewed by: three anonymous referees