This study presents a novel method to estimate the performance of advection schemes in numerical experiments along with a semi-realistic, non-linear, stiff chemical system. This method is based on the examination of the “signature function”, an invariant of the advection equation. Apart from exposing this concept in a particular numerical test case, we show that a new numerical scheme based on a combination of the piecewise parabolic method (PPM) with the flux adjustments of Walcek outperforms both the PPM and the Walcek schemes for inert tracer advection as well as for advection of chemically active species. From a fundamental point of view, we think that our evaluation method, based on the invariance of the signature function under the effect of advection, offers a new way to evaluate objectively the performance of advection schemes in the presence of active chemistry. More immediately, we show that the new PPM

Chemistry-transport models are models that aim at representing the concentration of trace gases and particles in the atmosphere. Many such tools exist and are used for several purposes, including research and operational forecast. The core of such models consists of a chemical solver adapted to stiff ordinary differential equation (ODE) systems along with a framework for solving the advection equation for all the chemical species.

Among the possible strategies to solve the advection equation in chemistry-transport models are the flux-based advection schemes, based on the ideas of

In the past, many studies have focused on developing, improving and evaluating advection schemes (e.g.

Our goal is to provide such a test case for conditions more representative of tropospheric chemistry at the scale of an urban area and to deploy new tools to evaluate advection schemes in the presence of active chemistry. To meet this objective, apart from classical methods and metrics, we introduce a novel idea, the “signature function”, that permits giving a lower bound of model error compared to the exact solution for problems with inert tracer advection and isolating the error due to advection itself in problems including active chemistry. Even though this method is related to the area coordinate introduced by

The PPM

In Sect.

The chemical mechanism used here (Reactions

The reaction constants of Reactions (

Apart from the chemically active species defined in Reactions (R1)–(R12), we introduce an inert tracer denoted TRC, which undergoes no chemical reaction and is passively advected by the flow. In chemistry-transport models, Reactions (

Of course key processes like oxidation of methane and of other volatile organic compounds are not taken into account in the above mechanism, but it retains some key features of tropospheric chemistry, which we think important:

extreme stiffness;

OH production, which depends on the presence of ozone, water vapour and sunlight;

non-linear behaviour of ozone production (in this simplified system, ozone production depends on the simultaneous presence of nitrogen oxides, OH and available CO for oxidation).

The simulations are performed on a domain

The flow we use in this study is the swirling deformational flow introduced by

The time-dependent streamfunction for this flow is

Streamlines (black contours), wind vectors and wind module in m s

The numerical experiments will be conducted in the domain

Initial mixing ratio of

Finally, another species of inert tracer, TRCb, is introduced so that

The following existing advection schemes have been tested in the study:

PPM

These schemes are flux-based, upwind-biased, semi-Lagrangian schemes based on polynomial reconstructions of the average concentrations. These polynomial reconstructions are piecewise-constant for

The PPM scheme presents the same caveat as Van Leer in the vicinity of extrema, with a strong discontinuity on each side of the extremum (Fig.

We detail here the procedure applied for this scheme. Let

The procedure is as follows.

If

Otherwise we estimate the Walcek-adjusted flux as follows:

The

Reconstruction of a Gaussian mixing-ratio profile by

To evaluate the computational cost of these advection schemes, advection of a 1D vector composed of

Mean calculation time per cell and per time step for the five advection schemes retained for the present study. The calculation has been performed in Fortran, a programming language frequently used for operational chemistry-transport models, on a laptop with an Intel Core i7-1165G7 CPU.

A first comparison between the PPM

First, the PPM

Convergence rate of the five advection schemes used in the present study for

The stiff chemical system is integrated using an EBI method. As described in

Equation (

The advection time step

Integrate chemistry over

Integrate zonal advection over

Integrate meridional advection over

Integrate zonal advection over

Integrate chemistry over

Table

Summary of the main characteristics of the simulations that have been performed.

It is worth noting that, by construction, flux-based advection integration is mass-conservative since the mass flux out of a cell through a facet is compensated exactly by the mass flux into the neighbouring cell through the same facet. Equation (

However, due to the finite number of iterations in the iterative resolution of Eq. (

Mass calculations for C, active N, H and TRC have been performed between the beginning and the end of the simulations. The results of this calculation for the PPM

The imbalance results are similar for all simulations, including the base simulation that has no advection, which shows that the small mass imbalance for chemically active species (up to

Mass-conservation diagnostic for C, active N, H and TRC in the PPM

In summary, the integration strategy we introduce above permits the conservation of mass (up to numerical accuracy for inert species and up to an arbitrary numerical tolerance defined by the user, in our case

As the

In the present study, we will estimate model error in

Here we introduce a new idea to evaluate advection schemes. As far as we know, this idea has not been tested in the past literature but resembles the area-coordinate formulation used by

Let us imagine a fluid with density

With this definition, we always have

Equation (

Therefore, since we know that, for the exact solution,

In practice, in an Eulerian model discretized into

Figure

The concentrations of active species evolve not only under the effect of advection but also due to chemical reactions. Therefore, the time invariance of the signature function does not hold for these species. However, the signature function can still be used to compare and evaluate simulations if one remarks that, in the case we study here with no variations in air density and air temperature, and with no emissions, the chemistry that takes place in each Lagrangian air parcel is independent of its position. Therefore, for all species, the signature function at time

As an illustration of this, Fig.

Ozone mixing ratio at

The signature function for the

Figure

Regarding ozone extrema throughout the simulation (Fig.

Regarding the preservation of tracer maxima (Fig.

Figure

Time series for all the simulations for the

Unlike the partial metrics presented in Fig.

The first information we get from Fig.

Time series for

Normalized

While we have shown so far that the analysis of

First, in all cases we always have

Interestingly, the performance ranking between the five simulations obtained by analysing the signature error

Having verified this, it is useful to go back to Fig.

To test the impact of higher resolution on our results, we have performed the same test case as above but refining the resolution from 4 to 1 km and accordingly reducing the time step from 1800 to 450 s. The results for this higher-resolution simulation are shown in Table

Normalized

As shown by

The first is

The second is

Figure

Minimum and maximum values in the 4 km simulations (presented in the paper) for

We have introduced the signature function

In this context, we have shown that the signature function and its normalized

We have used this new invariant in order to evaluate a new advection scheme that we have designed for this study, based on the PPM scheme

Therefore, the conclusion of this study is twofold. First, regarding the signature function as an invariant of the advection equation, we feel that this invariant contains much more information than other invariants that have been typically used to check advection schemes, such as the minimum or maximum values of mixing ratios, while not requiring access to the exact solution. In the case of chemistry-transport models, generalizing this concept to more dimensions (by studying the mass-weighted probability distribution function of all species simultaneously rather than one signature function per species) could be promising. For the same reasons as exposed above, this multidimensional probability distribution function should be an invariant of the advection equation. This concept could be explored to quantify model error in a synthetic way across all variables, instead of separately for each variable. The approach introduced here with the

From a more applied point of view, the PPM

We have also observed (Table

This study has been performed using toyCTM v1.0.1 (

The Fortran code AdvBench v1.0.0 used to evaluate advection performance (Table

The model toyctm v1.0.1, AdvBench v1.0.0 and all the scripts used to launch the model and post-process its outputs for the present study are distributed under the GNU General Public License v2.0. No datasets were used for this study.

All the authors contributed to the design of the simulated cases; SM performed and analysed the simulations; SM developed the software with RP.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The numerical calculations in this study benefited from the IPSL Data and Computing Center ESPRI, which is supported by CNRS, SU, CNES and École Polytechnique.

This research has been supported by the Agence de l'Environnement et de la Maîtrise de l'Énergie (ESCALAIR).

This paper was edited by Sylwester Arabas and reviewed by Christopher Walcek, Hilary Weller, and one anonymous referee.