In this study we explore a new way to model sub-grid turbulence using particle systems. The ability of particle systems to model small-scale turbulence is evaluated using high-resolution numerical simulations. These high-resolution data are averaged to produce a coarse-grid velocity field, which is then used to drive a complete particle-system-based downscaling. Wind fluctuations and turbulent kinetic energy are compared between the particle simulations and the high-resolution simulation. Despite the simplicity of the physical model used to drive the particles, the results show that the particle system is able to represent the average field. It is shown that this system is able to reproduce much finer turbulent structures than the numerical high-resolution simulations. In addition, this study provides an estimate of the effective spatial and temporal resolution of the numerical models. This highlights the need for higher-resolution simulations in order to evaluate the very fine turbulent structures predicted by the particle systems. Finally, a study of the influence of the forcing scale on the particle system is presented.

Following the increase in computing power, the resolutions of meteorological models have increased steadily over the past years. The refinement of the temporal and spatial resolution of atmospheric models requires a finer and finer representation of physical phenomena. The current weather forecast models have resolutions of approximately 1 km. However, the small processes, which have local effects, are still sub-grid processes in such models. Thus, they are subject to physical parametrization.

The issue of downscaling concerns many meteorological research fields, from
snowpack modeling to cloud-cover modeling. A particularly difficult matter is
the modeling of the turbulence in the atmospheric boundary layer (ABL). In
the ABL, there is a transfer of energy from scales of the order of 1 km down to sub-meter scales. This transfer is called the energy
cascade. Thus, whatever the model resolution, some turbulent processes are
sub-grid processes. For numerical weather forecast models, the processes
associated with sub-kilometer scales are not resolved yet. For instance, a
recent study shows that these processes are not resolved on the scale of
Applications of Research to Operations at Mesoscale (AROME) Airport, which has a horizontal resolution of 500 m

Because of their variability and their sensitivity to local conditions, these
turbulent phenomena are especially difficult to model. Instead of a reduction
in grid size, we suggest here another way to model sub-grid turbulence. In
this paper, we present a stochastic downscaling approach. Our method is based
on particle systems that are driven by a local turbulence model. Those
particles are embedded in grid cells (Fig.

In order to keep the average particle's behavior consistent with the grid-point model, some grid-point fields are used as external forcing on the particle system. The grid-point fields provide the values of the control parameters of the particle evolution model. This forcing is constant during the grid-point model time step and is applied every time new values are available. However, the particle evolution is performed at a shorter time step. Thus, the suggested downscaling method enables the refinement of both time and space scales.

In this work, the French Mesoscale Non-Hydrostatic (Meso-NH) research model is used to obtain
high-resolution grid-point fields. The chosen simulations have been performed
for the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) experiment

First, the framework is presented. The BLLAST field experiment and the
particle system and grid-point model coupling scheme are introduced. A
description of the models follows in the Sect.

The BLLAST (Boundary Layer Late Afternoon and Sunset Turbulence) field
campaign was conducted from 14 June to 8 July 2011 in southern France, in an
area of complex and heterogeneous terrain. The BLLAST experiment resulted
from a collaboration of several European laboratories spearheaded by the
Laboratoire d'Aérologie. The experiment aim is to study the turbulence in
the boundary layer during the late afternoon transition

To perform this study, all turbulence sources were investigated. A wide range
of integrated instrument platforms including light aircraft, remotely
piloted aircraft systems (RPAS), remote-sensing instruments, radiosoundings,
tethered balloons, surface flux stations and various meteorological towers
were deployed over different surface types

The BLLAST experiment addresses a wide range of scientific issues such as the
turbulence decrease

In this section, we introduce the two models used in this work. First, the grid-point model is presented, and the coupling scheme is described. Then, the focus is put on the particle system and its evolution model.

To perform the grid-point simulation of the ABL, we have used the research
model Meso-NH. It is a mesoscale atmospheric model jointly developed by the
Laboratoire d'Aérologie and by the French national center for meteorological research CNRM

The equation system solved using the Meso-NH model is an approximated form of
the

For our simulations, the 3-D turbulence scheme is a one and a half order
closure scheme

To introduce the presentation of the Meso-NH simulation which has been used, the framework of the coupling scheme is described.

Our scheme is a first step in coupling particle systems and grid-point models. Here, we work on the downscaling from the grid-point model to the particle system in one way, so the information flow goes only from the grid-point model to the particles. We have used simulations of the convective boundary layer to force a particle system which models sub-grid turbulent phenomena.

The particle systems are forced with a large-scale grid-point meteorological fields. The large scale used in our work is later described in detail. The forced particles are used to model the sub-grid fields for the large-scale model. To validate the downscaling, a higher-resolution model is used. In theory, the turbulent fields represented by the particles should be compared to the same fields simulated by a high-resolution grid-point simulation. For computational reasons we do not have access to different high-resolution simulations. Thus, the large-scale simulations have been built from the available simulations.

The process consists of three steps as outlined in Fig.

The downscaling scheme. Coarse fields are used to force a sub-grid particle system. The sub-grid fields are compared to a reference Meso-NH simulation.

Now that we have a reference simulation, a coarser simulation is built in
order to force the particle system. To this end, we have chosen to average
the grid-point fields of several cells. To be consistent, we have also
applied a temporal average. The resulting coarse Meso-NH fields thus have lower spatial and temporal resolutions than the reference simulation while
being consistent with it. However, due to the average, the coarse fields
include not only the components resolved on the grid, but also the average of
the sub-grid components. This limitation will be discussed in
Sect.

In each cell of the coarse grid and during each coarse time step, the particle average behavior is forced. If the forcing method works, the average of the particles should be in good agreement with the coarse fields.

To assess the method, we have worked on wind fields and turbulent kinetic
energy (TKE) fields. As explained in Sect.

In order to model the sub-grid processes, an applied mathematical technique
is used: the probability density functions are described using a particle
system. In this study, the particles sample the wind probability density
function. The particle technique is widespread in research fields such as
mechanical system modeling (automotive, aeronautics), but it is not yet
currently used for atmospheric modeling. However, Lagrangian particle models
for dispersion have been discussed for quite a long time. Guidelines for
evaluating the relevance of a stochastic model have been given by

Our study framework. We use a high-resolution model, which is averaged to obtain a coarse field. Particles are forced with coarse fields, and then fine fields are compared to particle fields.

Here, a particle is a realization of the surrounding atmosphere. Depending on the complexity of the evolution model, the particles carry physical properties, such as fluid velocity, temperature or humidity rate. In our study, the evolution model is simple, and each particle is a position–velocity couple. Using the particle approximation of the probability density function and the physical properties of the particles, the statistics of the turbulence are computed. In particular, the wind variance can be computed. Thus, using the particle wind, the TKE is directly available.

The suggested stochastic downscaling method completes the long list of
downscaling techniques developed to improve geophysical model resolutions.
Among them, we find the adaptive mesh refinement for oceanic and atmospheric
models

Previous studies have already explored stochastic downscaling methods for the
fifth-generation Penn State–NCAR mesoscale model MM5 (see

Different choices have been made to develop the presented downscaling method. We have chosen to force the particles cell by cell using the grid-point fields, but there is no imposed condition to the edge of the domain. The particles move freely in the domain and may go from one cell to another. When some particles go outside the domain, they are deleted and replaced by new particles inside the domain. For each new particle, the particle position is randomly chosen. Then, the new particle velocity is computed using the velocities of the particles which are in the same cell. Thus, the particle system contains information relative to different scales, including local components of the fields associated with sub-grid scale and large or mean components coming from the forcing, associated with the grid scale.

To compare the fields represented by the particles to the grid-point fields
modeled with Meso-NH, average particle velocities are computed cell by cell.
For instance, the wind

For the particles to be realizations of the surrounding atmosphere, the
particle evolution is driven by a local turbulence model. It is a stochastic
Lagrangian model (SLM) inspired by

In the evolution equations, the large-scale influence is given by the
pressure gradient

The buoyancy effect term

The stochastic Lagrangian model has two control parameters for the velocity
equation. The EDR,

As we have seen, the SLM equations contain some locally averaged terms,
denoted by

To validate the downscaling method, the particle fields are compared to the
fine Meso-NH simulation. In a grid-point model, the characteristic length of
the modeled processes is twice as large than the grid size according to the
Nyquist's frequency

The average operator

To compare the particle TKE to the TKE simulated by Meso-NH, the particle
values are averaged cell by cell as explained in
Sect.

As Meso-NH is a research model, its grid size and its time step may differ
from one simulation to another. For the simulation used here, the horizontal
grid size is

To force the particles, averaged fields are used. They are deduced from the
reference fields. The averaged fields, called coarse fields, are obtained by
averaging several grid points, on 12 time steps (

We have selected data from 13:55 to 14:10 h during the convective period, i.e., the 20 June 2011. Among all the available variables in Meso-NH, we have extracted the atmospheric pressure, the three components of the wind, the TKE and the EDR for the reference configuration. The pressure gradient is computed using the pressure field. The previously described average has been applied to these reference fields to obtain the coarse fields. We underline that only the coarse fields are used to force the particle system. The reference fields are simply used to evaluate the fields obtained from the particles.

To ensure its consistency with the real case, the high-resolution Meso-NH
simulation has been compared to another LES simulation performed with the LES
model of the National Center for Atmospheric Research (NCAR) for the same
case and shows similar results

The aim of this work is to study the ability of a particle system forced by grid-point data to model the sub-grid processes. To do so, we use two grid-point simulations: a coarse one to force the particle system and a fine one to assess the fields reconstructed by the particles.

The starting point of our downscaling scheme requires having the large
Meso-NH simulations on a 3-D domain including

initializing the particles in each cell using velocities given by the coarse Meso-NH simulations

performing particle evolution with the SLM model and the time step

calculating the sub-grid wind and the sub-grid turbulent parameters using the local average operator

updating the values of

On the scale of the particles, the coarse grid-point data represent an averaged forcing. Note that the particle horizontal velocities are not directly forced with coarse winds. The horizontal velocities are forced with the pressure gradient and the dissipation rate. For the vertical velocities, the forcing is slightly different: it uses the vertical velocity coarse fields instead of the pressure fields. This choice has been made because horizontal velocities are driven by pressure gradients, whereas vertical velocities are driven by the buoyancy. To improve the downscaling method, temperature gradients computed from Meso-NH simulations could be taken into account. Note that in this work, the EDR used to force the particles is considered isotropic.

During 12 time steps, the values of the control parameters remain constant.
To compute the particle simulation, steps 2 and 3 are repeated in a
continuous loop until the time

The domain of simulation: 8

In this procedure, the particle management is hidden. In our simulations, the
particle number is constant. In practice, we have to ensure that all the
particles are in the simulation domain. In our work, the particles follow the
simulated air-flow. So, at each time step some particles leave the domain.
The outside particles are replaced by new particles with consistent positions
and velocities as explained in Sect.

As the particles evolve freely in the domain, we also have to ensure a
homogeneous repartition of the particles inside the domain. To do so, for
each cell of the fine grid we keep the particle number between a minimal
value and a maximal value which is given at the beginning of the simulation.
By displacing particles, this method of particle management limits trajectory
length and prevents rogue trajectories as described by

In this section we first review how TKE is computed in Meso-NH. Then, we present the TKE computation using the particle system.

To characterize turbulence, the TKE and the EDR are the two parameters usually used. The TKE is the turbulent kinetic energy associated with the small-scale turbulent structures, while the EDR quantifies the energy transfer from the large-scale structures to the small-scale structures.

In this work, the two turbulent parameters play different roles. The EDR is used to force the particle system, whereas the TKE is used to assess the particle representation. For the EDR fields, we use the Meso-NH variable directly. It is computed from the TKE using a mixing-length closure hypothesis. We now give details about how the TKE is computed.

The TKE modeled by Meso-NH is made of two terms: the resolved TKE and the
sub-grid TKE. The resolved TKE is a diagnostic variable. It is calculated
using the grid-point 3-D wind field

The sub-grid TKE,

In the Meso-NH simulations, the grid size is fine and the resolved TKE is the major contribution to the total TKE, as expected far from the surface layer.

The different timescales used in this work. The forcing is applied
at each coarse time step

The particle system is used here to model the wind inside the grid Meso-NH
model. As detailed in Sect.

In the previous sections, the downscaling algorithm has been described in detail. In this section, the downscaling results are presented. To assess the behavior of the particle system, we compare the 3-D wind given by the particle and by Meso-NH. First, results on the coarse grid are shown. Then, we present the comparison between the particle fields and the fine Meso-NH fields. Wind power spectral densities are then presented. Finally, results for sub-grid TKE are presented. These results are presented separately because the sub-grid TKE is computed from the particle TKE and not directly from the wind.

To illustrate the results of the downscaling scheme, the wind results on the coarse grid and on the fine grid are presented for one cell and for the three dimensions.

To model the sub-grid fields, the particles are forced by coarse grid-point fields of pressure gradient, EDR and mean vertical velocity increment.

To assess the downscaling method, the first thing to look at is the agreement between the coarse wind and the average of the sub-grid wind modeled by the particle system. The aim is to assess the particle behavior on the forcing scale. This verification is important, especially for the horizontal velocity, which is not directly forced by the coarse horizontal velocity fields. Recall that to compare the particle wind to the coarse wind fields, the particle values are averaged coarse cell by coarse cell.

Figure

We are now interested in the particle behaviors on the fine scale. Here, the particle values are average fine cell by fine cell to obtain wind fields at the fine Meso-NH resolution. The particles are forced using only coarse Meso-NH fields. Thus, the particle fields could differ from the reference fine Meso-NH fields.

In Fig.

The reference fields are represented with a 5 s time step. At this
frequency, it appears that the Meso-NH wind is smoother than wind usually
observed in the boundary layer. By comparison, the fluctuating profile of the
particle wind seems consistent with wind observations obtained by a 3-D sonic
anemometer mounted below a tethered balloon (see

To explain the smoothness of the Meso-NH wind, two comparisons have been done. First, we have compared the wind fields from the studied area to fields from different areas at the same vertical level. Subsiding or ascending areas have been chosen. It appears that Meso-NH produces a smooth wind in both ascending and subsiding areas. Next, a simulation without numerical diffusion was performed, and the modeled winds with and without diffusion have been compared. This comparison shows that the smoothness is not due to the numerical diffusion used in the simulations.

Coming back to Fig.

Time evolution of the three wind components obtained from the coarse Meso-NH simulation (black) and by the particle model (red) for one cell of the coarse grid.

Time evolution of the three wind components obtained with the fine Meso-NH simulation (black) and with the particle model (red) for one cell of the fine grid.

To check that the particle wind mainly follows the Meso-NH wind fluctuations,
we apply a low-pass filter to the particle wind. The aim is to suppress the
fast fluctuations and then to assess the low-frequency component of the
particle wind. A second-order low-pass filter with a cutoff frequency of

The results are presented in Fig.

The particle wind fields contain the same low-frequency information as the fine Meso-NH wind fields. Thus, the suggested downscaling method and the model coupling have worked. Comparing to the Meso-NH wind, the particle wind has a faster fluctuating component. The question is now to determine if the fast fluctuations are due to smaller turbulent structures than those modeled by Meso-NH or if they are only a noise added to the low-frequency signal.

To begin answering this question, we present the study of the power
spectral densities (PSDs) of the wind and its low- and high-frequency
components in Sect.

To further the comparison between the Meso-NH wind and the particle wind, we have computed the wind PSDs. First, spectra of time series have been studied. Then, the PSD of the wind anomalies – differences between the particle wind and its low-frequency component – are shown. Finally, we discuss briefly the effective resolution of the Meso-NH model by comparison with the LES model of the NCAR.

To assess the temporal variability of the particle wind, time PSDs are
computed. For each vertical level, the PSDs are computed using groups of
4

Components of the fine Meso-NH wind in black and components of the particle wind in red, for one cell on the second level of the fine grid.

Figure

Contrary to the particle wind, the Meso-NH wind spectra have a “spoon shape”:
they follow the

The spectral analysis of the time series has clarified the validity domain – in terms of temporal resolution – of the two simulations. It also shows the limit of the grid-point model for high-frequency wind fluctuation modeling.

Power spectral densities of the components of the fine Meso-NH wind
in black and of the components of the particle wind in red, calculated on a
4

The spatial resolution of the particle simulations is trickier to estimate. A
first estimation may be given by the Lagrangian lengths associated with the
wind components. The lengths can be evaluated using the power spectral
densities and the mean velocities. Looking at the spectrum of the first
component of the wind, we can see that the spectrum is flat for frequencies
higher than

To get an idea of the Meso-NH's effective spatial resolution, we have
compared the Meso-NH simulations to other simulations performed for the
BLLAST experiment. Spatial PSDs are computed for a given time step on the
whole domain – 256

“Row-average”

In this section, the particle wind PSDs are not available. Indeed, the downscaling scheme has been applied to a restricted domain which is far too small to compute the spatial PSDs.

Thus, to have a comparative element, simulations of the NCAR LES are used

At high frequencies, their shapes differ. From their formulation, the NCAR LES
model spectra show a clear cutoff frequency. This frequency is around
8.10

Fine Meso-NH TKE in black and particle TKE in red, for one cell of the fine grid. The gray area represents the standard deviation of the particle TKE for the particles in the cell.

We may also notice the asymmetry of the spectra. It shows that the structures in the boundary layer are organized following preferential directions.

The study of the spatial spectra has shown that Meso-NH is able to model the
spatial variability of the wind with a

To complete the validation of the particle wind, we have studied the wind anomaly PSDs. The wind anomalies are defined here as the difference between the wind and its low-frequency component. The study shows that the time PSDs of the anomalies follow the energy cascade. So, the anomalies are not a white noise. Thus, the particles do not add a simple noise to the coarse wind. The added information is in good agreement with the Kolmogorov K41 theory. It illustrates the effectiveness of the suggested downscaling method.

In this section, the TKE simulated using the particle system is presented. As
explained in Sect.

The results are presented in Fig.

In

Five-minute time series of TKE depending on the forcing scale: the TKE
obtained with 160 m

To force the particle system, we have used data from 360 and 400 m high from 13:55 to 14:10 UTC. There is no TKE observation at this precise height
during this period, but sonic anemometer and tethered-balloon observations
are available at several heights from 30 to 550 m depending on the time. The
TKE observations obtained using these instruments for the
20 June 2011 are given in

Comparing the particle TKE to the TKE observations, we note that the particle
TKE has the same order of magnitude as the TKE observed during the afternoon from 0.6 to 1.7 m

The comparison of the particle TKE with the observations shows encouraging results. These results are a first step to demonstrate the ability of the particle system to model very small-scale turbulence. However, to end the validation, the suggested downscaling method will be applied to a larger domain and to other field experiment cases.

In the previous section, all the presented results have been obtained using the same forcing scale. Here, we suggest briefly looking at the influence of the forcing scale on the fields modeled with the particle system. As a first approach to qualify this influence, the particle system has been forced by two different scales.

The previously used grid was 160 m

First, the particle winds are compared to the fine Meso-NH wind. The root
mean square error (RMSE) between each particle wind and the Meso-NH wind is presented in Table

Wind RMSE depending on the forcing scale.

As expected, the particle wind obtained with the finest forcing grid is the closest to the Meso-NH wind. However, the difference between the two forcing methods is rather small. Using the finest grid reduces the RMSE by 12 % for the total wind and by 20 % for the low-frequency component.

The influence of the forcing scale on the TKE is illustrated in
Fig.

According to these results, the two particle simulations are consistent. Reducing the forcing scale reduces the difference between the particle fields and the model fields. However, for the two forcing scales, the particle fields are more turbulent than the Meso-NH fields.

To complete the work on the sensitivity to the forcing scale, a forcing grid
of 40 m

This article presents a first work on a new way to model sub-grid processes using particle systems. One of the major improvements is the use of a simple turbulence model instead of complex model such as LES or direct numerical simulation (DNS). However, to fully validate the method, one of the first steps should be to use a DNS or to apply the downscaling method to a toy model with known sub-grid fields. Unfortunately, such a validation has not been done yet.

For the work presented here, tests have been conducted on a small domain, with a reduced number of particles in each cell. These two constraints were related to computational time restrictions. Extending the domain and the duration of the simulation should be one of the next steps. It would improve the PSD quality and limit the influence of the edges. Then, supplementary work on the spatial resolution of the particle simulations should be done and the robustness of the results should be tested. To give a preliminary answer to the robustness issue, recall that we have compared the studied fields to fields from different areas at the same vertical level. The comparison has shown that Meso-NH fields are similar in the different areas. Thus, the downscaling method should provide similar results when applied in these areas.

Related to the question of the spatial resolution of particle simulations, there is also the fundamental question of the scale of the turbulence represented in the particle fields. So far, only a first estimation of the scale has been given, and specific work still needs to be done to work out the scales represented by the particle model.

In this work, the coarse fields were computed by averaging the fine Meso-NH fields. In a more advanced exercise, the coarse fields would be real Meso-NH fields computed with a coarse grid. To further this study, we could also add to the SLM an equation to model the temperature evolution. Therefore, the sub-grid buoyancy effect could be modeled and compared to a high-resolution Meso-NH simulation.

Concerning the SLM, another point has to be discussed. The Wiener processes used for the dispersion terms involve a locally Gaussian assumption of the pdf described by the particles. In our work, the Gaussian assumption is not valid on the grid-cell scale. Indeed, at a given time in a given cell, particles with different characteristics are mixed. This is partially due to the free evolution of the particles in the domain. Thus, the velocity pdf described by the particles in one fine cell is obtained by a mixing Gaussian pdf, but it is not necessarily Gaussian.

We would like to underline an important point about the EDR fields used to force the particle system. Here, the chosen EDR is the Meso-NH variable. The advantage of this choice is that the EDR is directly available. However, as it is a diagnostic variable of the Meso-NH model, it is computed using a closure scheme. The closure scheme may induce errors in the EDR modeling due to the underlying assumptions. To control the assumptions which are made, we could compute the EDR from the grid-point wind field, and compare it to the EDR calculated using different closure schemes. As the EDR controls the particle dispersion, an improvement in the EDR modeling will directly lead to an improvement in the sub-grid turbulence modeling.

Therefore, in future work, the downscaling method should be applied to a larger domain and sub-grid fields should be compared to observations. In addition, an in-depth comparison of TKE parametrization used in Meso-NH and TKE modeled with the particle method should be conducted. Despite the computational time, from a long-term perspective, future exercises should include replacing the sub-grid parametrization used in Meso-NH by sub-grid particle modeling. Indeed, for research purposes, the downscaling method may be an alternative solution to common turbulence closures, which often assume isotropic and homogeneous turbulence.

We present here a new downscaling method based on the coupling of a grid-point model and a particle model. The downscaling method has been applied to a simulation performed for the BLLAST experiment.

The particle system was forced by a coarse model. Then, the particle fields were assessed against a high-resolution simulation. The particle winds seem in good agreement with the high-resolution winds, but higher-resolution simulations should be performed. The same conclusions are given for the TKE simulations.

Even if the domain size is a limitation of the present study, the presented results are very encouraging. They prove the relevance of the suggested forcing method. Forcing a particle system is quite a simple process, and the sub-grid fields seem consistent with observations. Therefore, the first step to couple the SLM model and the Meso-NH model is achieved.

In the longer term, this work may be used to compare and to test the different turbulent schemes, parameterizations or closure hypotheses available in the research models and in the operational weather forecast models.

The data used in this study are freely available from the
BLLAST database:

The authors declare that they have no conflict of interest.

The BLLAST field experiment was made possible thanks to the contribution of several institutions and supporters: INSU-CNRS (Institut National des Sciences de l'Univers, Centre national de la Recherche Scientifique, LEFE-IDAO program), Météo-France, Observatoire Midi-Pyrénées (University of Toulouse), EUFAR (EUropean Facility for Airborne Research) and COST ES0802 (European Cooperation in Science and Technology). The field experiment would not have been possible without the contribution of all participating European and American research groups, which have all contributed a significant amount. The BLLAST field experiment was hosted by the instrumented site of the Centre de Recherches Atmosphériques, Lannemezan, France (Observatoire Midi-Pyrénées, Laboratoire d'Aérologie).The 60 m tower is partly supported by the POCTEFA/FLUXPYR European program. BLLAST data are managed by SEDOO, from Observatoire Midi-Pyrénées. The French ANR (Agence Nationale de la Recherche) supports the BLLAST analysis in the 2013–2015 BLLAST_A project. The authors particularly thank Clara Dardieu for providing the NCAR LES simulations and Bruno Piguet for the tethered-balloon observations. The authors also thank the coeditor and the reviewers for their comments, which have improved the quality of this article. Edited by: E. Pardyjak Reviewed by: two anonymous referees