The chemical mechanism used here (Reactions –) includes a subset of the main reactions of tropospheric gas-phase chemistry. Reactions ()–() are the three reactions that constitute the Leighton system, and Reactions () to () account for the formation of the hydroxyl radical OH through the photolysis of ozone in the presence of water vapour. Reactions () and () account for the production of the hydroperoxyl radical through oxidation of CO and the oxidation of NO into NO2 by HO2. Reactions ()–() are “termination reactions” that consume the radical species, and Reaction () describes the final consumption of the NOx species by formation of nitric acid. R1NO2+hν→NO+OR2O+O2+M→O3+MR3NO+O3→NO2+O2R4O3+hν→O(1D)+O2R5O(1D)+H2O→OH+OHR6O(1D)+N2→O+N2R7O(1D)+O2→O+O2R8CO+OH→CO2+HO2R9NO+HO2→NO2+OHR10HO2+HO2→H2O2+O2R11OH+HO2→H2O+O2R12NO2+OH→HNO3

The reaction constants of Reactions ()–() have been taken mostly from , with a temperature of 298 K and pressure of 101325 Pa. The photolysis rates have been set to typical midday values (e.g. ):

jR1=8×10-3s-1

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 ()–() are typically lumped in one single reaction, O3+hν→OH+OH, with a pseudo-reaction rate constant that depends on the concentration of air molecules and of water vapour molecules and on applying the quasi-steady-state approximation to O(1D). For this study, we have chosen to treat O(1D) as a prognostic species to preserve the full chemical stiffness of the problem, with lifetimes ranging from ≃4×10-9 s for O(1D) to several days for CO.

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;

The numerical experiments will be conducted in the domain D defined above, with the chemical scheme described above. To define the initial conditions, we introduce a concentration profile (between 0 and 1) as follows: 4φx,y=sin⁡22πxLsin⁡22πyLifx<L2andx<L2,5φx,y=0otherwise. The initial conditions are defined as follows (in terms of the mixing ratio):

αTRC=100ppb×φx,y (see Fig. )

The stiff chemical system is integrated using an EBI method. As described in and , we obtain the concentration vector ct+δtchem as the solution of 7ct+δtchem=ct+δtchemPt+δtchem1+δtchemLt+δtchem. For the present study, the focus is to test the performance of the advection scheme in articulation with active chemistry. Due to this focus, we limit errors in the resolution of the chemical system using a short time step for chemistry (δtchem=20 s).

Equation () is a non-linear, fixed-point equation and can be solved only numerically, usually with an iterative method. Formally, Eq. () guarantees exact mass conservation. However, this is true only if a very good convergence of the solution is reached . To limit violation of mass conservation in our study, we have set a very strict convergence criterion for the iterative resolution of Eq. (), stopping iteration when the estimate of ct+δtchem yields a relative difference less than ε for each species between ct+δtchem and ct+δtchemPt+δtchem1+δtchemLt+δtchem. The convergence parameter is set to ε=10-4 in the UKCA chemistry-transport model and ε=10-6 in the present study. This very strict convergence criterion is in line with the short chemical time step to obtain the best-possible numerical solution of the chemical evolution of the system, even at the cost of slow computations. Again, this choice is due to the purpose of this study to test the performance of the advection scheme, limiting as much as possible the errors in the chemical solver.

The advection time step δtadv has been set to δtadv=1800 s. With maximal wind module U≃1.8 m s-1 (Fig. ) and δx=4×103 m, this yields a maximal Courant number νmax≃0.8. The time-stepping strategy follows Strang-style time stepping , with the steps as follows:

Integrate chemistry over δtadv2 (45 chemical time steps).

Table  summarizes the six simulations that have been performed. The above-described case with Reactions ()–() has initial conditions as described in Sect. , a domain shape and discretization as described in Sect. , advection schemes as described in Sect. , and a chemical solver as described in Sect. . Along with the simulations performed with each of the five advection schemes, a “base” simulation has been performed with the same setup as the other simulations but without advection. As discussed later (Sect. ), this base simulation will serve as a benchmark to estimate advection errors in the other five simulations.

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 () also guarantees mass conservation in the Euler backward scheme as soon as the chemical reactions themselves are balanced (which is the case of Reactions – except for the imbalance of dioxygen in Reaction  due to integrating reaction H+O2→HO2 into the kinetically limiting time step CO+OH→CO2+H).

However, due to the finite number of iterations in the iterative resolution of Eq. (), mass conservation is only enforced with a finite precision of ε=10-6 (see Sect. ). Therefore, the relative mass imbalance in the outputs for C, active N (without taking into account N2) and H can be expected to be of the order of ε, since the Euler backward iterative scheme (Eq. ) is in principle mass-conservative, but mass conservation is obtained “only if a good convergence of the solution is reached” .

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 + W and base simulations are given in Table , showing that the relative mass imbalance at the end of simulation is ≃10-6 for active N, ≃10-12 for C and H, and ≃10-15 for TRC (which has no chemistry and is therefore affected only by advection which, as discussed above, ensures mass conservation up to numerical accuracy for the flux-based schemes we have implemented).

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 ≃10-6) is essentially due to the finite precision in the Euler backward iterative chemistry solver.

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 ε=10-6, for chemically active species) and conserves initially uniform mixing ratios up to numerical accuracy for the species with no active chemistry. All the advection schemes presented above are also built to respect monotonicity: they do not create new mixing-ratio extrema.

As the flow is designed so that at t=T every Lagrangian particle is back at its original location, it is possible to estimate the accuracy of numerical simulation by comparing the final simulated tracer concentration field to its initial value, therefore giving access to the magnitude of numerical error.

In the present study, we will estimate model error in ⋅1, introducing E1 as the normalized ‖⋅‖1 error in the mixing ratio: 8E1=∑i=1NρiViαit-αi0∑i=1NρiViαi0, where index i spans the entire domain. In the present case where density ρi and cell volume do not vary across cells, Eq. () boils down to 9E1=∑i=1Nαit-αi0∑i=1Nαi0.

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 ρx;y;z;t in a three-dimensional domain D, advecting a tracer having initially a mixing ratio αx;y;z;t=0∈0,1, following equations 10∂ρ∂t+∇ρu=0,11∂α∂t+u⋅∇α=0. For any given time t and any mixing ratio 0≤X≤1, we can define StX as the mass of fluid in the volume DtX defined as the set of all x;y;z where the tracer mixing ratio αx;y;z;t<X, divided by the entire mass of fluid in D: 12StX=∫DHX-αx;y;z;tρdV∫DρdV, where H is the Heaviside step function (H(u)=1 if u>0; H(u)=0 if u≤0). The St function can be, in some sense, interpreted as a mass-weighted cumulative probability density function of the tracer mixing ratio. If we reduce this definition to 2D flows with uniform density, St is related to the reciprocal function of the area-coordinate formulation of .

With this definition, we always have St0=0 (for all t values) and StX→1 when X→+∞ (more precisely, StX=1 as soon as X is larger than the maximum value of αx;y;z;t=0 over domain D).

Equation () makes clear that function St is invariant during the motion: for any given value of X, S0X is the (normalized) mass of the fluid parcel D0X that has a tracer mixing ratio α<X at t=0. We can observe that, since mixing ratio in Lagrangian parcels is preserved by pure advection (Eq. ), DtX and D0X represent the same Lagrangian fluid parcel at a different time. We also know that the mass of fluid in Lagrangian parcels is constant in time due to mass conservation for the carrier fluid (Eq. ). Since the total mass of fluid in D is also constant in time for the same reason, this implies that, for all t and X values, St(X)=S0(X). In other words, the signature function St is an invariant in time of the advection equation. The invariance of this function holds for both divergent and non-divergent flows.

Therefore, since we know that, for the exact solution, St=S0, the departure of the numerical evaluation of function St from S0 in numerical simulations can be used as an objective measurement of discretization errors.

In practice, in an Eulerian model discretized into N cells, each cell has an evaluation of the tracer mixing ratio αit. The evaluation of St is straightforward: 13StX≃∑i=1NHX-αiρiVi∑i=1NρiVi. Generally, the numerical evaluation of St in an Eulerian model will differ from S0: the signature of the initial tracer distribution will evolve under the effect of the errors of the advection scheme, and the magnitude of the signature modification can serve as a measure of the extent of advection error. In the particular case in which the flow is non-divergent and the carrier fluid mass ρiVi is the same in all model cells, the norm-1 difference between St and S0 can be calculated as 14∫0∞StX-S0XdX=1N∑i=1Nαt̃i-α0̃i, where αt̃i is the vector of all mixing ratios in model cells sorted in increasing order. This simplified formula holds only in the specific case in which the flow is invariant and the mass of fluid is distributed evenly between all the model cells. In this case, and only in this case, it is also convenient to introduce the normalized norm-1 difference E1 between St and S0 as 15E1=∑i=1Nαt̃i-α0̃i∑i=1Nα0̃i.

Figure  shows an example of how comparing the St signature function with S0 permits comparison of the accuracy of two simulations at a time when no analytic solution is easily accessible. Figure a–b show the mixing ratio for TRC at T2 in the Godunov and Van Leer simulations, respectively. Without access to the exact solution, it is hard to compare quantitatively the quality of these simulations at that stage, even though indicators such as the maximal tracer mixing ratio can give partial information. Figure c–d show the ST2 function compared with S0 for the Godunov and Van Leer simulations, respectively. This graphical representation permits giving an intuitive meaning to the normalized signature error E1 as the total area between the representative curves of St and S0 (shaded in Fig. c–d), divided by the total area left of the representative curve of S0 (hatched in Fig. c–d). This measure gives an indication of model error based on not only one particular point representative of a part of the tracer distribution (e.g. the maximal value, a very partial indicator), but also an integrated error indicator taking into account the maximal and minimal values as well as the entire tracer distribution in between these values. In the case presented in Fig. c–d, the area between S0 and St is smaller in the Van Leer simulation (Fig. d, with E1=0.218) than in the Godunov simulation (Fig. c, with E1=0.571).

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 t, St, should theoretically be the same as in the base simulation with no advection. If we note S‾t, the signature function at time t in the simulation without advection, for all t values we have St=S‾t: at any time, the signature of the distribution of every chemical species should be the same in the case with advection as in the case without. In other words, advection should only deform the map of all chemical species and not change the chemistry within each Lagrangian air parcel. With a non-linear chemical system such as Reactions ()–(), there is no easy access to the exact solution of the system even without advection: S‾t is not known exactly for the chemically active species. However, the base simulation resolves the chemical reactions with exactly the same chemical solver as the other simulations but without motion. Therefore, comparing the distribution of a chemical species at time t in a simulation with advection to its distribution in the base simulation without advection will permit having an estimate of how the numerical errors in advection affect the distribution of chemically active species.

As an illustration of this, Fig.  shows the ozone mixing ratio at T2, as simulated in three simulations with advection (Fig. a–c) and the base simulation without advection (Fig. d). This figure alone does not make it easy to discriminate between the three numerical simulations. Figure  shows the signature function for the same three simulations (Van Leer, PPM and PPM + W): this time, both the visualization of the agreement between St and S‾t and the objective calculation of E1 clearly show that the PPM + W simulation agrees better with the base simulation in terms of the signature function (E1=0.0622), followed by the PPM simulation (E1=0.0679) and the Van Leer simulation (E1=0.0787). Apart from this quantitative agreement, some qualitative and local conclusions can be drawn from the graphical comparisons between St and S‾t, such as the fact that the representation of the ozone minimum in the PPM + W simulation is more accurate than in the PPM simulation, which is visible in the initial part of the signature function (αO3<30 ppb) where the shaded area is smaller in PPM + W (Fig. c) than in PPM and Van Leer.

Figure  shows the evolution of some performance indicators along the course of the experiments. For the same reasons as for the signature function, these metrics should be the same in all simulations in the absence of numerical errors in the representation of advection, so the differences between the time series obtained in the simulations with advection and the base simulation reveal the effects of numerical errors in the representation of advection. It is interesting to note that all the metrics represented in Fig.  can be derived directly from the signature function.

Regarding ozone extrema throughout the simulation (Fig. a), we observe smaller differences between the different simulations. The PPM + W, PPM and Walcek simulations produce very comparable values of ozone maxima. Surprisingly, the simple Godunov schemes represents the ozone maximum slightly better than all other schemes except Walcek towards the end of the simulation. However, the representation of the ozone minimum by the Godunov simulation is very bad, failing to represent a minimum lower than the background value towards the end of the simulation. The PPM + W and Walcek simulations represent the smallest (and closest to the base simulation) minimum value for O3, followed by PPM and Van Leer.

Regarding the preservation of tracer maxima (Fig. b), the PPM + W simulation performs best, with a clear edge over the Walcek and PPM simulations (Fig. b). For this metric, the Walcek simulation gives better results than the PPM simulation, with a smaller half plume, closer to the base simulation, Walcek and PPM. The PPM and Walcek simulations perform similarly for this metric.

Figure c shows evolution of the non-dimensional half volume of the tracer plume during the simulation (defined as the smallest volume containing half of the mass of TRC divided by the total volume of domain D). As discussed in , this is a measure of the diffusivity of the advection schemes. With this metric, we see that the PPM + W simulation performs best, followed by Walcek and PPM.

Unlike the partial metrics presented in Fig. , the normalized ‖⋅‖1 signature error in the mixing ratios E1 is a performance diagnostic for the simulations that takes into account the entire distribution of mixing ratios and not just particular values such as the maximum or minimum. Figure  shows E1 for TRC, O3, NO and NO2 for all the simulations, with the smallest E1 indicating the best-performing simulation.

The first information we get from Fig.  is that, for all these compounds, the PPM + W simulation performs best in this regard. For the case of ozone (Fig. b), the E1 time series discriminates much more between the different simulations, with a clear superiority of the PPM + W simulation over the Walcek and PPM simulations, while the differences between these three simulations in the ozone min–max plot (Fig. b) appeared small. Analysing the evolution of E1 throughout the simulation shows that, for TRC, NO and NO2, the best-performing simulations are PPM + W, Walcek and PPM, in this order, but that for ozone, PPM performs better than Walcek, a conclusion that could not be drawn from the min–max plot (Fig. b), which indicated a better performance of Walcek regarding both the ozone maximum and the ozone minimum.

While we have shown so far that the analysis of E1 permits the drawing of clearer conclusions regarding the performance of the various simulations, in the present case we can confirm these results by comparing E1 to a more classical metric, the normalized ‖⋅‖1 error E1 (Eq. ) of the simulations with advection relative to the base simulation without advection. Unlike E1, E1 can only be calculated at the final time step (when all the Lagrangian particles are back at their initial location), since the exact solution for t<T is not accessible. Several observations can be made from the values of E1 and E1 in Table .

First, in all cases we always have E1<E1. Qualitatively, this can be interpreted as E1 being a weakened form of the ‖⋅‖1 error, retaining the differences in the distribution of mixing ratios but eliminating the differences in the location of the tracer plume.

Interestingly, the performance ranking between the five simulations obtained by analysing the signature error E1 is the same for all variables as with E1: for TRC, NO and NO2, PPM + W performs best, followed in this order by Walcek, PPM, Van Leer and Godunov, but for O3, PPM performs better than Walcek. In our study case, analysing E1 permits comparison of the performance of the various simulations without access to the exact solution and gives the same results as the analysis of E1 (which requires access to the exact solution). Even for simulations with comparable performance to the Walcek and PPM simulations, the signature error E1 permits diagnosing which of the two simulations performs better for each variable. Even though the differences between these two simulations are not drastic and depend on the species of interest, the conclusions drawn from the analysis of the signature error E1 are the same and correspond to E1. This gives confidence in the usability of the signature error E1 as a proxy for simulation accuracy when the exact solution is not available.

Having verified this, it is useful to go back to Fig.  and interpret the evolution of E1 in time as a hint of when error appears along the simulation. In this regard, we see two very different behaviours between the analysed variables. For TRC, NO and NO2, substantial errors appear almost instantly after 1800 s of simulation (one single time step). This is due to the action of the wind field on the initially very sharp peak of these species (Fig. ). On the contrary, for O3, having an initially uniform distribution, errors due to advection appear only when sufficient heterogeneity is introduced into the O3 map by chemical processes, since all the advection schemes are built to advect exactly a uniform mixing ratio, maintaining its uniformity. Therefore, the onset of advection errors in O3 is much slower than in the three species that have initially heterogeneous distributions.

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 . This table shows that, also at this resolution, the PPM + W offers the best performance of all the tested schemes, but, unlike in the 4 km resolution case, this does not hold for all species: at 1 km resolution, the PPM scheme performs comparably to PPM + W for the inert tracer TRC and slightly better for NO2. However, PPM + W clearly performs better for O3 and NO. These results suggest that the improvement brought by PPM + W over PPM may tend to become smaller for higher resolutions (contrary to what we observe in Fig. ).

As shown by , “among [linear] schemes of second order accuracy [for the advection equation], there is none which satisfies the monotonicity condition”. As a consequence of this result, known as the Godunov theorem, second-order, monotonic schemes for the advection equation are all non-linear. In particular, among all the schemes we study here, all are non-linear due to their specific treatment of the maxima to preserve monotonicity. However, these non-linear effects should be kept as small as possible. To test this property, we have tested the conservation of two sum species, which should stay uniform and constant along the run:

The first is αTRCb+αNO+αNO2+αHNO3, where TRCb is an inert tracer whose initial distribution is defined by Eq. (). The mixing ratio of this sum species should stay constant, uniform and equal to 110 ppb.

Figure  shows that, as imposed by the Godunov theorem, only the scheme preserves exactly the uniformity of sum species. Other schemes are higher-order and monotonic, but therefore non-linear, and do not preserve the uniformity of these sum species. The magnitude of the departure of the mixing ratio for these species from its theoretical value is a measure of the non-linearity of advection schemes, which is an undesirable property since the advection equation itself is linear. Figure  shows that non-linearities of up to 1 %–2 % appear for all schemes in the representation of CO+CO2 and that they tend to be strongest in the Walcek and the PPM + W schemes. Regarding TRCb+NO+NO2+HNO3, the non-linearities reach up to 10 % of the expected value and are strongest in the PPM + W scheme. These results suggest that non-conservation of sum species in the vicinity of mixing-ratio extrema, which is observed in all second-order monotonic schemes, tends to be stronger when the flux corrections are applied, which might represent a drawback of PPM + W compared to PPM. In spite of (or due to) this increasingly non-linear behaviour in the vicinity of the extrema, PPM + W preserves the values of extrema better than PPM (Fig. b).