The potential of the antidiffusive transport scheme proposed by

Reducing numerical errors in chemistry-transport models (CTMs) is a necessary task for those wishing to improve these models' performance. Among the well-known errors in
Eulerian CTMs, excessive numerical diffusion in all directions is a well-known
drawback of many such models, and in the past decade, excessive or poorly
represented vertical transport and diffusion has been identified as a major
cause for numerical dispersion in these models

Among other more model-specific solutions,

In the line of improving the vertical advection scheme,

Therefore, the present study aims at answering the questions on the use of

Section

Continuity equation for the motion of air is as follows:

The continuity equation for species

Chemistry-transport models try to solve Eq. (

In 1D, Eq. (

Here we follow a semi-Lagrangian approach by identifying the air parcels that
have entered cell

Equation (

In order to permit monotonicity of the advection scheme in terms of mixing
ratios (i.e., the mixing ratio for species

With these notations, Eqs. (

Equations (

Regarding chemistry-transport models, in practice, approximated values of

This simplified framework will permit us to focus on the transport scheme of
the chemistry-transport model whose task, in flux-form, is to estimate the
values of

The most simple way of estimating

This order-1 scheme is cheap, robust, linear, monotonous and mass-conservative
but extremely diffusive. It is therefore important to find more accurate ways
to estimate

The second-order slope-limited scheme of

The

The scheme of

Their scheme is accurate only to the first order.

The scheme is linearly unstable, but non-linearly stable (their Theorem 1).

All the simulations in the present study are 2D

Here,

From a practical point of view, it is extremely difficult to actually find the
contours of

Since the present study is aimed at studying vertical transport only, we chose
to test the Godunov,

Summary of the different transport configurations that have been tested.

Domain resolution and size for Cases 1 and 2; set of increasing resolutions used for convergence tests (Cases 3 and 4). For all configurations,

We have defined three test cases designed to be representative of long-range
tracer transport situations in the atmosphere. The simulation domain covers
an

Except for tests of convergence rates for which increasingly fine
discretizations have to be tested, the domain is discretized into 80
evenly spaced cells from west to east (

In this case, we consider the evolution of an inert tracer initially distributed as follows:

Zonal wind is constant in time, zonally uniform and vertically sheared:

We add a vertical wind defined as follows:

The vertical wind speed scale is taken as

This case can describe a plume that is initially vertical, covering a
50

Direct integration of Eq. (

Equations (

With

In this case, the initial tracer mixing ratio is as follows:

Zonal wind is constant and uniform:

As for Case 1,

Integration of Eqs. (

This case is a simplified representation of long-range advection of a

This case has been designed to study the numerical convergence rate of the various
configurations that will be tested as a function of space resolution. The case
setup is the same as for Case 2, with wind speeds similar to
Eqs. (

Due to the need for increasing resolution and therefore the numerical cost of
simulations, we simulate only one spatial and temporal period of this case
(instead of two spatial and temporal periods for Case 2), hence the differences
between Eqs. (

The initial tracer mixing ratio is prescribed as follows:

For convergence rate tests, five different resolutions have been tested
(Table

Implementation of the idealized experiments and transport scheme have been done within the under development ToyCTM code. ToyCTM is a Python code targeting chemistry-transport studies in academic cases. So far, ToyCTM relies on classical numpy arrays. Its object-oriented design provides a class structure enabling extensibility; i.e., users can easily code new transport schemes or define personal grid geometry. A basic chemistry module is present and allows one to test chemical reactions on top of transport. See the Code availability section at the end of the paper for access to the code version used for the present study and to the current development version of the code.

Final state of the numerical simulation for Case 1 after simulations. Panels

Figure

Performance of simulations performed with the Upwind, VL, PPM and DL99 vertical advection schemes relative to the discretized exact solution for Case 1: percent relative error in

Figure

Final state of the numerical simulation for Case 2 after simulations. Panels

Performance of simulations performed with the Upwind, VL, PPM and DL99 vertical advection schemes relative to the exact solution for Case 2: percent relative error in

The numerical convergence rates of all the tested advection configurations for

Convergence rate results for the four tested vertical advection schemes as a function of the number of horizontal points

Figure

When vertical resolution becomes much too fine compared to the size of the
modeled object (polluted plume thicker that

Convergence rates in

The numerical experiments that have been presented confirm the interest of
using the

The numerical experiments exposed here confirm that the

Improved accuracy of a low-order scheme compared to higher-order schemes for a
given resolution is not impossible from a theoretical point of view but still
counterintuitive since higher-order schemes are designed to reduce numerical
error at any given resolution compared to lower-order schemes due to
“smarter” reconstruction procedures. Theory imposes that, if model
resolution is fine enough and if the tracer field is smooth, higher-order
schemes should be more accurate than lower-order schemes. However, as shown by

To understand this surprisingly good behavior of

In more general words, this result suggests that the

The first-order, antidiffusive advection scheme of

Convergence tests show that this improved performance exists only for tracer
layers that are represented by less than six grid cells in the vertical
direction, while for finer resolutions and smooth initial tracer fields,
higher-order schemes perform better than the

We think that these results are important because they explain under which
conditions the

More investigation is needed in real and/or idealized cases to address several
questions:

Does the

If not, is it possible to use a traditional transport scheme in the boundary layer and the

Atmospheric chemistry being a non-linear process, how does reduction in excessive numerical diffusion in the troposphere affect representation of chemistry inside chemically active air masses such as volcanic or biomass burning plumes?

Is it desirable to use antidiffusive transport schemes in the horizontal directions as well, and under which conditions should they be used?

Convergence rate results for the four tested vertical advection schemes as a function of the number of horizontal points

This case has been designed to study the numerical convergence rate of the various
configurations that will be tested as a function of space resolution. The case
setup is the same as for Cases 2 and 3, with wind speeds from
Eqs. (

Table

Convergence rates in

ToyCTM is free software distributed under the GNU General Public License v2.
The exact version used for the present study

All the authors have contributed to the design of the simulated cases. SM performed and analyzed the simulations and developed the software with RP. LM, ML and RP contributed to writing and improving the paper.

The authors declare that they have no conflict of interest.

The simulations have been performed at the ESPRI/IPSL data center and at TGCC under GENCI A0070110274 allocation.

This research has been supported by the Agence de l'Innovation de Défense (TROMPET grant).

This paper was edited by Simone Marras and reviewed by two anonymous referees.