The interparcel mixing algorithm in the Lagrangian advection scheme with shape matrix (LASM) is updated to make the scheme more robust. The linear degeneration criterion is replaced by the maximum deviation of the skeleton points so that the new algorithm is more effective in controlling the shape of parcels, which is vital for long time simulation. LASM is inherently shape-preserving without any complicated filter or limiter, and it is linear. This fact contributes to the ability to preserve the sum of multiple tracers exactly on the parcels in LASM. A newly proposed terminator “toy”-chemistry test is used to test LASM, which shows that LASM can preserve the weighted sum of two reactive species precisely. The physics–dynamics coupling (i.e., tendency evaluation type) is also discussed. A flow generated by a WRF large-eddy simulation is also used to test the 3-D extension of LASM.

Lagrangian modeling approaches have long been recognized as a better
alternative to the Eulerian or semi-Lagrangian ones for modeling the
advection of tracers, due to their ability to reduce the numerical diffusion
significantly, though some roadblocks need to be cleared. Therefore, more and
more Lagrangian advection (or transport) schemes were proposed during the
last two decades from different research communities. Based on the
application objects, the research directions of the Lagrangian methods can be
divided into four categories: (1) atmospheric dispersion modeling,
(2) atmospheric chemical transport modeling, (3) cloud-resolving modeling,
and (4) general circulation modeling.

In the first category, the turbulent diffusion needs to be tackled
thoroughly, because the temporal and spatial scales are relatively short and
small respectively. These models are called Lagrangian particle dispersion
models (LPDMs). The flow is decomposed into resolved mean and unresolved
turbulent parts (the molecular diffusion is regularly ignored), where the
first part comes from the reanalysis or model output, and the latter part is
described by the stochastic process

The chemical transport models (CTMs) consider a large number of tracer species
and chemical reactions in the second category. One of the widely used
Lagrangian CTMs is STOCHEM

The third category includes the works on the cloud simulation in Lagrangian
way. The clouds are divided into warm clouds and ice clouds.

The fourth research effort comes from the general circulation modeling (GCM)
community. The current research topic is the Lagrangian advection scheme in
the dynamical core, which is to replace the Eulerian or semi-Lagrangian
schemes by the Lagrangian ones. The turbulent diffusion is not considered,
which is left to other physical parameterizations. The pioneer works were
done by

The major roadblock of the Lagrangian schemes is the aliasing error,
which manifests itself as noises in the tracer density
distribution. During the advection, the discrete parcels (or called
particles, mass packets in other works) are assumed to be isolated
from each other. When the flow is nonlinearly deformative, the shape
of parcels will be deformed or elongated into filaments, or even be
split into parts. Most Lagrangian advection schemes do not explicitly
simulate the parcel shape and assume the parcel represents the mean
state of a compact volume around it. This assumption of the parcel
shape is generally not valid, so large aliasing error will occur when
remapping the tracer density onto the mesh of a GCM, if no interparcel
mixing is considered. Different researchers designed different
interparcel mixing algorithms to reduce or eliminate such errors. For
example, ATTILA redefines the parcel boundaries by simply bringing the mass
mixing ratio

This work also discusses the problems when considering physical or chemical
tendencies of tracers. A new terminator “toy”-chemistry test

The basic formulation of LASM (e.g., linear deformation matrix, skeleton
points) was introduced in

The continuous fluid is discretized into parcels in finite number

Parcel

After the calculation of volume, the densities of all the species on parcel

Since

At the initial time step, the density is remapped from grids to parcels by
using Eq. (

The mass fixer will affect some properties on the grids. For example, without
the mass fixer, the constant sum of three discontinuous tracers in the
deformation test case

By now, the trajectories of parcels and densities of tracers are calculated, and the densities are remapped onto the grids. The next important ingredient of LASM is the interparcel mixing algorithm, which is updated as follows.

Due to the discretization of the continuous fluid, an initial compact parcel cannot keep its integrity under the deformation of the flow. Some part of it needs to be exchanged with other parcels to form some kind of interparcel mixing, which will improve the degree of linear approximation of the parcel shape.

Previously, a parcel with index

The parcel shape is approximated by a linear deformation matrix

A case of linear approximation degeneration of a parcel. The red ellipse is the current shape of the parcel, and the green points are the skeleton points of the parcel. The green points not on the red ellipse indicate that the parcel shape is not approximated well by the linear deformation matrix.

Another difference from the old mixing algorithm is that the neighbor parcel
shapes are not changed, so those parcels will not be disturbed by the mixing.
This is similar to Eulerian schemes, where the mesh is fixed. Parcel

After the density mixing calculation, the shape of parcel

The mixing in the Lagrangian schemes is driven by the flow deformation,
whereas the inherent mixing in the Eulerian and semi-Lagrangian schemes is
driven by the tracer density gradient, which is similar to the molecular
diffusion. Currently, the above interparcel mixing is only for the
computational aspect, but when the physical mixing is required, the similar
form can be utilized with different parameters

In the real applications, some other processes will calculate the tendencies for different tracer species. There are two types of tendency, which are both implemented in LASM.

The tendency in the first type resides on the
Eulerian mesh of an existing model, such as the physics
parameterizations (e.g., convection, microphysics) in an AGCM. The
gridded tendencies need to be remapped onto the parcels, and the mass tendency is chosen to be remapped to ensure mass
conservation:

Deformation test case results with different

The tendency in the second type is computed directly on the parcels, which is
more natural and no inconsistency will occur. In the CTMs, it is easier to
compute the tendencies in this type, because the chemical reactions can be
described by the box model

The effects of the two tendency types will be compared in a newly proposed
terminator “toy”-chemistry test in Sect.

LASM can be extended to 3-D by handling the following three factors: (1) the rigid boundary condition in the vertical direction (the horizontal boundary conditions are periodic), (2) the initial shape of the parcels, and (3) the reshaping rule after mixing.

The rigid boundary condition is implemented by ensuring the vertical velocity
is zero on the boundary. This is trivial in this study, since the flow is
given by the external sources (e.g., WRF-LES simulation). The parcel
centroids will never move outside the domain, if the time step size is not
too large, but the parcel ellipsoid may penetrate through the boundary. To
avoid potential problems, all the parcels within the boundary grid cells will
be reset to a sphere, and mixed accordingly. Similarly, ATTILA relocates air
parcels in the boundary layer after one time step, due to the rapid turbulent
mixing

The interparcel mixing algorithm described above checks the parcel filament
degree

When reshaping the mixed parcels, there is one more element

Deformation test case results with different mixing coefficient

We utilize the 3-D large-eddy simulation (LES) in WRF to construct a
turbulent flow to verify this extension. The results are analyzed in
Sect.

In this section, four tests are used to verify the updates of LASM. The first
two cases are the same as used in

The conventional parameters of LASM used in the tests.

The deformation flow test was first proposed in

The effects of changing

The correlation results evaluated on the parcels for different
values of

Comparison of the parcel shapes between the old and new interparcel mixing algorithms in a barotropic test case. Two extremely small and large parcels in the old algorithm are spotted by the red ellipses. In the new algorithm, there is no such ill-shaped parcels.

Comparison between minimum mixing and all mixing at day 4. All
mixing means the interparcel mixing is triggered each time step for each
parcel. The mixing coefficient

The effects of changing

The barotropic test provides more realistic flow, though no analytical
solution is available. The same finite difference barotropic model is used to
drive LASM as in

Firstly, we check the validity of the new criterion that judges the linear
degeneration. The comparison between the old and new criterion is shown in
Fig.

Contour plots of

Contour plots of

The results at day 4 are in Fig.

Cross section of

This test

3-D volume-rendered advection results driven by the flow outputted by WRF-LES. The left figure is simulated by LASM, and the right one is by the advection scheme in WRF.

Evolution of the parcel distributions during the WRF-LES simulation. Shown is the horizontally averaged number of parcels contained in the grid cells. Each line represents a level.

Except the scheme itself, the physics–dynamics (or chemistry transport in
this test) coupling also affects the results, such as the tendency evaluation
type. In LASM, the tendencies of

In summary, it is better to evaluate the tendencies on the parcels in LASM.
For CTM application, the chemical reaction can be easily calculated on the
parcels, and the interparcel mixing in LASM will also improve the
representation of the parcels relative to the earlier Lagrangian models

All the previous tests are in 2-D, so there are still concerns about the
behavior of LASM in 3-D. This test will use the 3-D flow outputted from
WRF-LES, which is extremely turbulent. The parameters for the LES are the
standard ones in WRF, with

Since WRF uses a terrain-following hydrostatic-pressure vertical coordinate

The results in the 3-D Cartesian domain after 1.5 h for both LASM and WRF
are depicted in Fig.

One major concern about LASM is the cluster and rarefaction of parcels. For
example, is it possible that a parcel has no or only distant neighbors? It
should be noted that the distribution of the parcels is controlled by the
flow, which is constrained by the fluid dynamics in turn. In

The interparcel mixing algorithm in LASM is updated by replacing the criterion of the linear degeneration of the parcel shape and the mixing rule. The new criterion is a more direct indicator of the degeneration symptom, so the parcel shape is better controlled. In the new mixing rule, the densities of the involved parcels are restored to a mean density.

Several tests are utilized to verify the updates, including a newly proposed terminator “toy”-chemistry test. The results reveal that the new algorithm is effective and less arbitrary. It is noteworthy that LASM is a linear scheme, so it can preserve the sum of multiple tracer species and also the weighted sum of two reactive species exactly on the parcels and accurately on the grids. Two triggers for the interparcel mixing are compared, and LASM shows great flexibility so that different triggers can be devised in the future to address different needs.

The two tendency evaluation types are discussed and tested. The first one, which evaluates the tendency on the parcels, is the most natural type. Some processes are trivial to be calculated in this type, such as the chemical reaction and microphysics. The other one, which evaluates the tendency on the grids, can cause negative values on the parcels, which should be carefully handled.

LASM is also extended to 3-D by handling the rigid boundary condition, adjusting the initial parcel shape, and changing the reshaping rule after mixing. The flow outputted by the WRF-LES is used to verify this extension of LASM, and the results of LASM and WRF indicate that LASM has far less numerical diffusion. The parcel distribution also shows that there is no cluster and rarefaction of parcels.

The codes of LASM are managed by using GIT and hosted on GitHub. The
repository URL is

Five external libraries (Udunits, Boost, Armadillo, NetCDF and MLPACK) also
need to be installed. You can install them by using a package
manager called PACKMAN (

This work is supported by the National Natural Science Foundation of China (grant no. 41305094) and the National Grand Fundamental Research 973 Program of China (grant no. 2014CB441302). Edited by: S. Unterstrasser