The paper presents the transport module of the System for Integrated modeLling of Atmospheric coMposition SILAM v.5 based on the advection algorithm of Michael Galperin. This advection routine, so far weakly presented in the international literature, is positively defined, stable at any Courant number, and efficient computationally. We present the rigorous description of its original version, along with several updates that improve its monotonicity and shape preservation, allowing for applications to long-living species in conditions of complex atmospheric flows. The scheme is connected with other parts of the model in a way that preserves the sub-grid mass distribution information that is a cornerstone of the advection algorithm. The other parts include the previously developed vertical diffusion algorithm combined with dry deposition, a meteorological pre-processor, and chemical transformation modules.

The quality of the advection routine is evaluated using a large set of tests. The original approach has been previously compared with several classic algorithms widely used in operational dispersion models. The basic tests were repeated for the updated scheme and extended with real-wind simulations and demanding global 2-D tests recently suggested in the literature, which allowed one to position the scheme with regard to sophisticated state-of-the-art approaches. The advection scheme performance was fully comparable with other algorithms, with a modest computational cost.

This work was the last project of Dr. Sci. Michael Galperin, who passed away on 18 March 2008.

One of the key problems in atmospheric composition modelling is the accuracy and reliability of numerical schemes. A less appreciated but important issue is the consistency of the approaches applied in different modules of the modelling system. Usually, model construction follows process-wise split (Yanenko, 1971; Marchuk, 1995; Seinfeld and Pandis, 2006), thus considering separately the advection and diffusion algorithms, physical and chemical transformations, and dry and wet deposition. In practical model developments, features of the transport algorithms, first of all, the advection scheme, largely shape the model and determine its area of application.

There are numerous types of advection schemes currently employed in atmospheric dispersion models. Two major categories refer to Lagrangian or Eulerian treatment of tracers: as small-size masses (Lagrangian particles) or as the concentration fields discretised in a prescribed grid. The Eulerian schemes, the primary subject of this paper, can be divided into flux-form finite volume, semi-Lagrangian, or expansion-function schemes. The expansion-function schemes approximate the solution with a given set of basis functions and, in turn, can be divided into spectral, pseudospectral and finite-element approaches. Some classic schemes are also based on finite-difference approximations of the advective term of the transport equation. The basic principles of all these approaches were formulated several decades ago and, with certain modifications, are still in use. Many modern schemes combine several approaches.

The large diversity of the advection algorithms is explained by a long list of requirements for such schemes. The most important ones are positive definition, minimal numerical diffusion, limited non-monotonicity and non-linearity, stability with regard to high Courant numbers (the number of the model grid cells passed within one advection time step), small phase error, local and global mass conservation, and high numerical efficiency. Some of these requirements contradict each other. For example, numerical diffusion “blurs” the resulting patterns but also makes them smoother, thus improving the monotonicity.

The finite-difference schemes involve direct discretisation of the dispersion equation and various interpolation functions to evaluate derivatives of the concentration fields (see the reviews of Richtmyer, 1962; Leith, 1965; Roach, 1980, as well as Sect. 3.1 in Rood, 1987); specific examples are, for instance, Russell and Lerner (1981), and Van Leer (1974, 1977, 1979). These schemes, being once popular, usually suffer from large numerical diffusion and limited stability, which sets stringent limitations on the Courant number, usually requiring it to be substantially less than 1. Therefore, the interest has gradually shifted towards flux, finite-element, and semi-Lagrangian schemes.

The flux schemes represent the transport via mass fluxes at the grid cell borders, which are calculated from concentrations in the neighbouring cells and wind characteristics. They are inherently mass conservative and have become popular in atmospheric chemistry transport models (Kukkonen et al., 2012). Probably the most widely used flux-type scheme is the one made by A. Bott (Bott, 1989, 1992, 1993), especially if one would count the numerous Bott-type schemes (see examples in Syrakov, 1996; Syrakov and Galperin, 1997; Syrakov and Galperin, 2000; Walcek and Aleksic, 1998; Walcek, 2000; Yamartino, 1993), which are based on the same principle but involve different approximation functions, monotonicity and normalisation procedures, etc.

The semi-Lagrangian schemes have been among the most widely used approaches for decades, with numerous algorithms using its basic concept (Crowley, 1968; Egan and Mahoney, 1972; Pedersen and Prahm, 1974; Pepper and Long, 1978; Prather, 1986; Smolarkiewicz, 1982; Staniforth and Cote, 1991, and references therein; Lowe et al., 2003; Sofiev, 2000, etc.). In the forward form, these schemes consider the transport of mass starting from the grid mesh points (departure points) and calculate the masses at the grid points closest to the arrival point. Backward algorithms start from arrival grid points and find the grid points near the departure point. The schemes can be based on tracking either grid points or grid cells along their trajectories. The grid-point-based schemes are not inherently mass-conserving, whereas the volume-based schemes achieve mass conservation by integrating the mass over the departure volume. They can sometimes be described as a combination of finite-volume and semi-Lagrangian methods (Lin and Rood 1996, 1997). Stability of these schemes can be ensured for a wide range of Courant numbers (Leonard, 2002). A general review can be found in Lauritzen et al. (2011), whereas a comparison of 19 modern schemes is discussed in Lauritzen et al. (2014), hereinafter referred to as L14.

Modelling in spectral space is another approach with a long history (Ritchie, 1988; Kreiss and Oliger, 1972; Zlatev and Berkowicz, 1988; Prahm and Christensen, 1977), but is not widely used today.

One of the main problems of the existing schemes is substantial numerical diffusion originating from the finite-step discretisation along space and time. Seemingly inevitable in an Eulerian context, this phenomenon, however, does not exist in Lagrangian advection schemes, which do not contain explicit discretisation of particle movement. The Lagrangian domain is a continuous space rather than a set of pre-defined grid meshes, and the position of the particles can be calculated precisely. As a result, numerical diffusion of purely Lagrangian schemes is always zero – at a cost of strongly non-monotonous concentration fields due to limited spatial representativeness of a single Lagrangian particle and a limited number of particles.

One of the ways to reduce the diffusivity of an Eulerian scheme is to store additional prognostic variables to describe the state of each grid cell with higher spatial resolution than the formal cell size: a sub-grid mass distribution. This can take the form of extra conservation equation(s) for e.g. first- or higher-order moments (Egan and Mahoney, 1972; Prather, 1986). There are other approaches that use different kinds of extra information. For instance, the conservative semi-Lagrangian schemes (Yabe and Aoki, 1991; Yabe et al., 2001) use a semi-Lagrangian step to evaluate the mixing ratio at cell interfaces, and then use the interface values along with the cell integrals to derive an interpolant representing the sub-grid distribution.

In a series of works, Michael Galperin developed a semi-Lagrangian scheme that used the sub-grid information on mass centre location inside the cell. The scheme was made fully non-diffusive for isolated plumes, positively defined, and very efficient computationally (Galperin et al., 1994, 1995, 1997; Galperin, 1999; Galperin and Sofiev, 1998; Galperin and Sofiev, 1995; Galperin, 2000). The early version of this scheme applied in the large-scale modelling by Sofiev (2000) resembled the “moving-centre” approach widely used in aerosol dynamics models (Kokkola et al. 2008) and shared its characteristic weakness – high non-monotonicity. The later developments substantially reduced it without damaging other features (Galperin, 1999, 2000). Further development of this scheme is the subject of the current paper.

Diffusion algorithms are less diverse than advection schemes. The physical reason for one of the common diffusion parameterisations is described in detail by Smagorinsky (1963), who suggested a formula for grid-scale scalar eddy diffusivity based on the model resolution and wind speed derivatives, thus connecting the numerical features of the simulations and hydrodynamics. It is widely used in chemical transport models (Kukkonen et al., 2012).

The dry deposition is usually accounted for as a boundary condition for the vertical advection–diffusion equation. Computation of dry deposition for gases practically always follows the electrical analogy, for which one of the first comprehensive parameterisations was suggested by Wesely (1989). Among the extensions of this approach, one was suggested by Sofiev (2002), who combined it with vertical diffusion and connected it with the Galperin advection scheme. The algorithm used an effective mean diffusion coefficient over thick layers calculated from high-resolution meteorological vertical profiles, the direction also recommended by Venkatram and Pleim (1999). These thick layers were determined using the subgrid information available from the advection scheme, which increased the accuracy of both algorithms (Sofiev, 2002).

For aerosol species, the electrical analogy is not correct (Venkatram and Pleim 1999). Compromising approaches suggested by Slinn (1982) and Zhang et al. (2001) and updated by Petroff and Zhang (2010) involve numerous empirical relations, sometimes on thin ground. A more rigorous approach unifying the gas and aerosol deposition parameterisations into a single solution was developed by Kouznetsov and Sofiev (2012).

The current paper describes the Eulerian transport algorithm of the System for Integrated modeLling of Atmospheric coMposition SILAM v.5, which is based on the advection scheme of Michael Galperin with several updates.

The paper is organised as follows. Section 2 describes the original algorithm of Michael Galperin and positions the scheme among other approaches. Section 3 presents the improvements made during its implementation and testing in SILAM. Section 4 outlines the scheme interconnections with other model parts. Standard and advanced model tests are shown in Sect. 5. Finally, the discussion in Sect. 6 includes an analysis of the scheme performance in the tests, as well as comparison with other algorithms.

Following the SILAM standards, all units throughout the paper are the basic
SI: (mole) for chemicals, (kg) for aerosols without chemical speciation, (m)
for distance and size, (s) for time, etc. The model operates with
concentrations (mol m

We consider the forward dispersion equation with the first-order
K-theory-based closure for diffusion:

Here

Equation (1) is considered on the time interval

Boundary conditions depend on the type of the simulations. In a general form,
they constrain concentration and/or its first derivative:

Here the values of

The current section presents the advection algorithm suggested by Michael Galperin in the 2000s as a contribution to the Eulerian transport scheme of SILAM. The idea of the scheme can be found in a short methodological publication of Galperin (2000) (in Russian) and conference materials (Galperin, 1999; Sofiev et al., 2008). It is very briefly outlined by Petrova et al. (2008) (hereinafter referred to as P08), but no systematic description exists so far.

For the 1-D case, let us define the simulation grid,

Let us represent the mass distribution in each grid cell via the rectangular
slab:

Advection step of the scheme of Michael Galperin.

The advection scheme consists of a transport step and a re-projection step.
At the transport step, each slab

The original Galperin scheme employed wind at the cell mid-point

Following the transport step (7), the masses

The coefficients

In multiple dimensions, the slabs are described by the total mass in
multidimensional cells and centres of mass along each dimension. In two
dimensions, an analogue of Eq. (5) will be

However, a simpler procedure used in the original scheme is obtained with
dimensional splitting, where the transport in each dimension is processed
sequentially with the grid projection applied in between. For an

The

The Galperin scheme shares some features with other approaches (see the reviews of Rood, 1987, and Lauritzen et al., 2011). Arguably the closest existing scheme is the finite-volume approach of Egan and Mahoney (1972), hereinafter referred to as EM72, and Prather (1986), hereinafter P86. The main similarity between these schemes is the representation of the mass distribution as a set of slabs (rectangular in EM72 and continuous polynomial distributions in P86), one per grid cell, with the mass centre identified via the slab first moment, plus additional constraints. During the EM72 and P86 advection steps, mass and the first moment are conserved, similarly to the reprojection step (8). However, the similarity ends here.

There are several principal differences between the EM72/P86 and Galperin algorithms.

Firstly, in EM72 the slab width is computed via the second moment (variance) of the mass distribution in the grid cell. P86 uses the second moments to constrain the shape of the polynomials. As a result, this moment has to be computed and stored for the whole grid, and the corresponding conservation equation has to be added, on top of those for the mass and the first moment. Galperin's approach does not require the second moment, instead positioning the slab against the cell wall. In fact, EM72 pointed out that the second moment can be omitted, but did not use the wall-based constraint in such a “degenerated” scheme, which severely affected its accuracy.

Secondly, EM72 uses the movements of the slabs in adjacent grid cells to calculate the mass flows across the border. Such local consideration requires the Courant number to be less than 1: the so-called “portioning parameter” (a close analogy to the Courant number in the scheme) is limited between 0 and 1. The same limitation is valid for the P86 approach. Galperin's scheme can be applied at any Courant number and its re-projection step can rather be related to Lin and Rood (1996).

There are several features of the original scheme which make it difficult to
use in large-scale chemical transport simulations. These are listed here and
the corresponding improvements are introduced in the following sub-sections.

The scheme is formulated with zero inflow boundary conditions.

Real-wind tests have shown that the scheme has difficulties in complex-wind and complex-terrain conditions, similar to EM72 (Ghods et al. 2000).

The explicit forward-in-time advection (Eq. 7) is inaccurate.

The scheme, being very good with individual sharp plumes over zero background, noticeably distorts the smoother fields with a non-zero background – see P08.

The open boundary for the outgoing masses is kept in SILAM regional
simulations. The inflow into a limited-area domain is defined via prescribed
concentration at the boundary (Eq. 3),

Here

The top boundary follows the same rules as the lateral boundaries. At the surface, the vertical wind component is zero, which is equivalent to closure of the domain with regard to advection.

Global-domain calculations require certain care: SILAM operates in
longitude–latitude grids; that is, it has singularity points at the poles
and a cut along the 180

Vertical motion in the pole cylinders is calculated separately using the vertical wind component diagnosed from the global non-divergence requirement.

The original Galperin scheme tends to accumulate mass at stagnation points where one of the wind components is small. Similar problems were reported by Ghods et al. (2000) for the EM72. Ghods et al. (2000) also suggested an explanation and a generic principle for solving the problem: increasing the number of points at which the wind is evaluated inside the grid cell. For application in the Galperin scheme, it can be achieved by separate advection of each slab edge instead of advecting the slab as a whole. This allows for shrinking and stretching of the slab following the gradient of the velocity field. Formally, this can be written as follows.

Let us again consider the 1-D slab that has been formed according to Eq. (5).
Its edges are

The advection step takes the wind velocity at the left and right slab edges
and transports them in a way similar to Eq. (6) with
the corresponding wind velocities. The new slab is formed as a uniform
distribution between the new positions of the edges:

The explicit advection step (Eq. 7) is inaccurate and can be improved under
the assumption of linear change of wind inside each grid cell, with values at
the borders coming from the meteo input:

If

Applying sequentially Eqs. (18) and (19) until completing the model time step

This approach neglects the change of wind with time. However, the
integration method is robust, since the linearly interpolated wind field is
Lipschitz-continuous everywhere, which in turn guarantees the uniqueness of
the trajectories of

The original scheme tends to artificially sharpen the plume edges and to gradually redistribute the background mass in the vicinity of the plume towards it (Fig. 2, blue shapes). Similar “antidiffusive” distortions were also reported by P08 and by EM72 – for their scheme.

Shape preservation tests:

The reason for the feature can be seen from Eq. (8): if a large mass is concentrated at one of the grid cell sides, the centre of mass becomes insensitive to the low-mass part of the cell; that is, a small mass that appears there from the neighbouring cell is just added to the big slab with little effect on its position and width.

A cheap, albeit not rigorous, way to confront the effect is to compensate for
it via an additional small pull of the mass centre towards the cell mid-point
before forming the slab for advection:

The way this smoother works becomes clearer if one notices that the Galperin
scheme becomes similar to the upwind algorithm if the locations of the
centres of masses are always forced to the middle of the grid cells at the
beginning of the time step. The upwind scheme is very diffusive, and
relaxation towards it confronts the antidiffusive features of the Galperin
approach. The actual value of

The non-linearity introduced by the coupling of cell mass and centre of mass in Eq. (8) makes formal stability and convergence analysis after Charney et al. (1950) difficult. However, the features of the scheme can be investigated numerically following the approach of Kaas and Nielsen (2010).

The scheme was run for 200 time steps in a 1-D domain with 100 grid points.
For each wave number up to 25, the scheme was initialised with the
corresponding sine function and run with Courant numbers ranging from 0.05 to
0.95 in steps of 0.05. This allowed evaluation of the spectrally resolved
root mean squared error (RMSE) and, after a Fourier transform, the spectral
amplification factor (cumulative for the 200 steps) for each wave number. The
amplification factor quantifies the scheme's ability to resolve the
corresponding harmonics, while the RMSE additionally includes the effect of
phase errors and possible spurious modes. Since the scheme is formulated for
nonnegative concentrations, a constant background

Spectral analysis for 1-D. Panels

Figure 3 presents the amplification factor and RMSE for the Galperin scheme
without the smoother (panels a, d) and with it,

Introducing the smoothing

To further investigate the spectral response of the scheme, it was evaluated
with a broadband input:

Example of input and output spectra for broadband input to the
advection schemes with zero and nonzero background levels. Left panels: exact
and numerical solutions. Right panels: power spectrum densities initially and
after one revolution. Top:

With

Similarly to the single-harmonic tests, the situation changes in the presence
of a significant background (

Construction of the dispersion model using the Galperin advection scheme as its transport core is not trivial because all other modules should support the use of the sub-grid information on positions of the mass centres. In some cases it is straightforward, but in others one can only make the module to return them undamaged back to advection.

For the vertical axis, SILAM combines the Galperin advection with the
vertical diffusion algorithm following the extended resistance analogy
(Sofiev, 2002), which considers the air column as a sequence of thick layers.
Vertical slabs within these layers are controlled by the same 1-D advection,
which is performed in absolute coordinates – either

The centres of masses are used but not modified by diffusion: the effective
diffusion coefficient between the neighbouring thick layers is taken as an
inverse of aerodynamic resistance between the centres of mass of these
layers (Sofiev, 2002):

The effective thickness

In the lowest layer, the dry deposition velocity is calculated at the height
of the centre of mass

The advantages of using the mass centres as the vertical diffusion meshes are discussed in detail by Sofiev (2002), where it is shown that the effect can be substantial if an inversion layer appears inside the thick dispersion layer. Then the location of the mass centre above/below the inversion can change the up/down diffusion fluxes by a factor of several times.

Emission data are the only source of sub-grid information apart from the advection itself: location of the sources is transformed into the mass centre positions of their emission.

For point sources, the mass is added to the corresponding grid cell and
centres of masses are updated:

For area sources, the approach depends on the source grid. If it is the same
as the computational one, the mass centre is put to the middle of the cell
(no extra information can be obtained). If the grids are different, the
source is re-projected. For each computational grid cell

Modifications described in Sect. 3 require staggered wind fields, which have to be provided by the meteo pre-processor (unless they are directly available from the input data). Moreover, the pre-processor needs to ensure consistency between the flow and air density fields (Prather et al. 1987; Rotman et al., 2004; Robertson and Langner, 1999). This is particularly important with the present advection scheme, since mixing ratio perturbations caused by the mass-flow inconsistency are not suppressed by numerical diffusion.

The wind pre-processing follows the idea of a “pressure fixer”, which means
adding a correction

Substituting Eq. (26) into (25) yields a Poisson equation for

This interface is implemented in a very simple manner: the mass centres are not affected by the transformations. The chemical module deals exclusively with concentrations in the grid cells. The newly created mass is added to the existing one, thus accepting its centre position in the cell. If some species did not exist before the transformation, the new mass centre is put to the middle point of the cell.

A set of basic tests and comparison with some classical approaches has been presented by Galperin (1999) and P08 for the original scheme, along with Bott, Holmgren, and several other schemes. Their main conclusions were that the scheme is very good for sharp-edge patterns: in particular, it transports delta functions without any distortions. It had, however, issues with long slopes, smooth shapes, etc., where the tendency to gradually convert them to a collection of rectangles was noticeable.

Addressing these concerns, tests used during the scheme improvements and implementation in SILAM included puff-over-background, conical and sine-shaped peaks and dips, etc. (some examples are shown in Fig. 2), a divergent 1-D high-Courant wind test in the 1-D divergent wind field (Fig. 5), a constant-level background field in eight vortices with stagnation points (Fig. 6), and rotation tests for various shapes (Fig. 7).

Linear-motion tests with a constant-release point source at

Test with eight non-divergent 2-D vortices. Left panel: test of the original scheme (5)–(7), time step 8; right panel: improved scheme (15)–(16), time step 50. Both tasks were initialised with constant value 0.4, also used as boundary conditions. Without smoother.

Double-vortex rotation tests for a rectangular split between the
vortices (upper panels); three single-cell peaks and two connected rectangles
(middle panels); and sin- and cone-shaped surfaces (lower panels). A series
of time steps is shown in the left panels, except for the low panel (shown

The scheme stays stable at arbitrarily high Courant numbers and handles the convergence and divergence of the flows (Fig. 5).

Transport and rotation tests of the improved scheme maintain low distortions
of the shapes: the

The test with eight vortices was difficult for the original scheme (Fig. 6a) due to its insufficient sub-grid resolution, but the improvement Eqs. (15)–(16), Sect. 3.2, resolved the problem (Fig. 6b). This refinement is instrumental in complex-topography domains.

Performance of Galperin's advection scheme in the global spherical domain was
assessed with the collection of demanding tests of Lauritzen et al. (2012).
The cases are designed to evaluate the accuracy of transport schemes at a
wide range of resolutions and Courant numbers. The tests used a prescribed
non-divergent 2-D velocity field defined on a sphere and consisting of
deformation and rotation, so that the initial concentration pattern is
reconstructed at the end of the test,

Four initial concentration distributions were used (Fig. 8): “Gaussian
hills” with unity maximum value, “cosine bells” with a background of 0.1
and maxima of 1, “slotted cylinders” – a rough pattern with a 0.1
background and 1 maximum level, and “correlated cosine bells” – the
distribution obtained from “cosine bells” with a function

Initial shapes of the puffs for the 2-D global test on the sphere.

Half-period (

Final shapes (

The error fields for the final shapes of Fig. 10 as compared with the slotted cylinder initial shape in Fig. 8. Without smoother.

Examples of the most challenging runs with slotted cylinders at

Deviation of the resulting field

Dependence of the performance metrics l1, l2, and

Advection should also keep the local ratio of the tracer's concentrations.
Such a ratio between “cosine bells” and “correlated cosine bells” was
calculated at

Mixing preservation test for cosine bells and correlated cosine
bells Eq. (27) at

A histogram of the mixing diagnostic (stacked) for the same
resolutions, Courant number and smoother factor as in Fig. 13. Metrics are
the following (see text and Lauritzen et al. (2012) for more details):

Constant-vmr test with real-wind conditions after 122 h.

Testing the scheme with real-wind conditions has one major difficulty: there is no accurate solution that can be used as a reference. An exception is simulations of the constant-mixing-ratio 3-D field, which, once initialised, must stay constant throughout the run. Deviation from this constant is then the measure of the model quality. Such a test verifies both the scheme and the meteo-to-dispersion interface, which has to provide the consistent wind fields.

The constant-vmr test was set with winds taken from the ERA-Interim archive
of ECMWF, for the arbitrarily selected month of January 1991 (Fig. 15). The
model was initialised with vmr

The results of the test are shown in Fig. 15, which depicts the model state after 240 h of the run, panel a) showing the boundary-layer vmr, and panel b) presenting it in the stratosphere. The zonally averaged vertical cross section is shown in panel c). Green colours in the pictures correspond to less than 1 % of the instant-field error.

An important message is that the limited distortions about 1–2 % are visible in a few places, but they are not related to topography, rather being associated with the frontal zones and cyclones. The comparatively coarse spatial and temporal resolution of the test makes the associated changes of the wind quite sharp, so that the dimension-split errors start manifesting themselves. Smoother flows in the stratosphere posed minor challenges for the scheme. The L2 error (not shown) is approximately proportional to the model time step.

The presented SILAM v.5 transport module is based on the semi-Lagrangian advection scheme of Michael Galperin with subgrid information available through the positions of centres of masses. It poses certain challenges in implementation. Firstly, one has to organise the sub-grid information use and transmission between the advection and other model units. Secondly, the scheme requires storage of four full fields for each transported species (mass and moments) and care should be taken to maintain an efficient exchange between the processors and the computer memory. Thirdly, the possibility to run with high Courant numbers and MPI parallelisation via horizontal domain split can be utilised only if the MPI split allows for sufficient buffer zones. Finally, the better performance of the advection at a Courant number greater than 1 challenges the implementation of other modules, first of all, chemistry and emission. Indeed, introduction of emitted mass once per long time step would result in a broken plume unless the mass is spread downwind over the corresponding distance. Similar problems show up in chemical transformation calculations. At present, the actual SILAM applications are performed with Courant close to but mostly smaller than 1 to avoid such problems.

The above challenges are mostly technical and their solution allows the scheme to demonstrate strong performance with low computational costs.

In particular, by attributing the release from point source to its actual location, one can reduce the impact of the common problem of Eulerian models: point release is immediately diluted over the model grid cell. This substantially improves the transport but does not solve the problem completely: (i) the chemical module still receives the diluted plume concentration, and (ii) the slab size in the case of the source near the centre of the grid cell will still be as large as the grid cell itself. A more accurate solution would be the plume-in-grid or similar approaches, which is being built in SILAM. Another example of the sub-grid information usage is utilisation of full meteorological vertical resolution to calculate effective values of meteo variables for thick dispersion layers (Sofiev, 2002).

The model can operate at any Courant number (Fig. 5). Its time step is limited not by grid cell size, but by a spatial scale of the wind-shear field; that is, it has to satisfy a much less restrictive Lipschitz criterion, which relates spatial and temporal truncation errors (Pudykiewicz et al., 1985). It follows from the advection step Eq. (6) and the reprojection step Eq. (8), which do not restrict new positions of the slabs: they can find themselves anywhere in the grid or outside it after the time step is made.

SILAM heavily relies on such features of Galperin scheme as mass conservation and accountability: the scheme provides complete mass budget including transport across the domain boundaries. In particular, nesting of the calculations is straightforward and does not need the relaxation buffer at the edges of the inner domain: the inflow through the boundaries is described by the same slabs as the main advection. The scheme is also shape-preserving – in the sense this term is used by L14 – that is, it does not result in unphysical solutions, such as a negative mixing ratio. Some distortions are still possible (Fig. 2), which can be reduced by the smoother described in Sect. 3.4, Eq. (20).

Evaluating the Galperin scheme with the simple tests (Figs. 2–7), one can point out the known issues of the classical schemes resolved in the Galperin approach: high-order algorithms suffer from numerical diffusion, oscillations at sharp gradients (require special efforts for limiting their amplitude), high computational costs and stringent limits to Courant number. None of these affects the Galperin scheme.

The main issue noticed during the implementation of the original scheme was the unrealistically high concentrations near the wind stagnation points. Thus, the concentration pattern at the test Fig. 6a resembles the situation of a divergent wind field. However, it is not the case: the 2-D wind pattern is strictly solenoidal. The actual reason is insufficient resolution of the advection grid: one centre of mass point is not enough if the spatial scale of the wind variation is comparable with the grid cell size. Tracking the edges of the slab rather than its centre resolves the problem (Fig. 6b).

The other challenging tasks for Galperin algorithm were those with smooth background and soft gradients, a frequent issue for semi-Lagrangian schemes, which is easily handled by more diffusive approaches. This feature was visible in the P08 tests where the scheme noticeably distorts the Gaussian and conical plumes. For the puff-over-background pattern, the scheme makes a single low-mass dip in the vicinity of the puff, which receives this mass (Fig. 2). From a formal point of view, the scheme does not conserve the higher moments inside the grid cell, which becomes a problem when the pattern changes at a spatial scale shorter than the grid cell size. The smoothing step (20) may be advised despite it having no rigorous basis and, as in L14 evaluation of other schemes, may damage some formal quality scores (adding this step introduces numerical viscosity – Fig. 2).

The application of the scheme to the highly challenging tests of Lauritzen et al. (2012) allowed its evaluation in a global 2-D case and comparison with the state-of-the-art schemes evaluated by L14 and Kaas et al. (2013).

Performing these tests with different spatial and temporal resolutions, as well as Courant numbers, suggested that the scheme has an “optimal” Courant number for each spatial resolution where the error metrics reach their minimum, so that the increase in temporal resolution is not beneficial. Indeed, in Fig. 12 the low Courant runs are by no means the most accurate. This is not surprising: for an ideal scheme, increasing the grid resolution and reducing the time step should both lead to gradual convergence of the algorithm; that is, the error metrics should reduce. For real schemes, higher temporal resolution competes with accumulation of the scheme errors with increasing number of steps. Convergence in L14 tests was still solid for all fixed Courant number series (Fig. 12), but excessive temporal resolution (specific to each particular grid cell size) was penalised by higher errors. Similarly, the most accurate representation of the correlated patterns is obtained from the runs with the intermediate Courant numbers (Fig. 13). This seems to be a common feature: the same behaviour was noticed by L14 for several schemes.

High optimal Courant numbers, however, should be taken with care. For L14, the smooth wind fields reduced the dimension-split error and made the long time steps particularly beneficial.

It is also seen (Fig. 11) that the best performance, in case of a near-optimal Courant, is demonstrated by the high-spatial-resolution simulations, which have reproduced both the sharp edges of the slotted cylinders, the flat background and the cylinder's top planes.

The scheme demonstrated a convergence rate higher than 1 for all metrics and
all tests with smooth initial patterns. Even for the most stringent test with
the slotted cylinders, the scheme showed the first-order convergence rate in
the

Among the other features of the solution, one can notice a certain
inhomogeneity of the background field away from the transported bodies. The
error is very small (

Similar non-monotonicity of background is visible for some schemes tested by L14. Unfortunately, no error fields are given there, but Figs. 7–10 there are comparable with our Fig. 9 (results without a smoother). With few exceptions (schemes TTS-I and LPM, notations of L14), all algorithms manifested such patterns unless filters are applied. For some schemes (SFF-CSLAM3, SFF-CSLAM4, UCISOM-CS, CLAW, and CAM), these inhomogeneities are visible also for the tests with shape-preserving filters. One should note however that the 0.1 level, which distinguishes between the two violet colours in Figs. 9 and 7–10 of L14, corresponds to the background level in the slotted-cylinder test. As a result, even a very small deviation leads to the appearance of such shapes in the plots (note the stripes in the background of Fig. 8).

Comparing the so-called “minimal resolution” threshold for

Another criterion can be the optimal convergence of

Interestingly, the L14 tests were limited with 3

The scheme demonstrated limited distortion of pre-existing functional
dependence – see the cosine bells and correlated cosine bells tests in
Eq. (27) (Fig. 13). Formal scores suggested by Lauritzen et al. (2012)
calculated for the Galperin scheme are shown in Fig. 14. Notations are the
following.

A peculiarity of the mixing diagnostic scores is that they are significantly affected by the background areas far from the advected bells, which occupy only a small fraction of the domain (Fig. 8). As a result, small background fluctuations discussed above in application to slotted cylinders (see the error field in Fig. 11) contribute significantly to the mixing diagnostic scores too. In particular, the high Courant simulations, which accurately reproduce the bells themselves (the dots are close to the curve in the scatter plots in Fig. 13), still show poor formal scores due to non-zero width of the cloud near the location (0.1, 1), where all background dots should arrive. This issue contributes most significantly to the “overshooting” part of the error, but also to the other two components.

Expectedly, the smoother improves the mixing diagnostic scores, mainly affecting the representation of the bells themselves (Fig. 13). This is in contrast with the schemes tested in L14, where the shape-preservation filters mostly removed the penalty for overshooting the background but rarely improved the other two components, sometimes worsening them.

Following the conclusions of Sect. 3.4 and the 1-D tests, we used the smoothing factor of 0.08, which is a compromise between the scheme diffusivity and distortion reduction. As a result, some non-linearity exists also in the smoothed solution. The test showed that a simple increase in temporal resolution leads to an increase in the number of steps and related re-projections, which then worsen the representation of the bells – but improve the background field by reducing the dimension-split errors. A synchronous rise of the resolution in time and space with the same Courant number (columns in Fig. 13) showed better results for higher-resolving set-ups.

Further investigating the flat-field behaviour in complex wind patterns, the
simulations with the constant-vmr initial conditions (Fig. 15) were
performed, showing that the model has no major problem in keeping the
homogeneous distribution: deviations do not exceed a few %, with no relation
to topography. The existing ups and downs of the vmr are related to cyclones
and atmospheric fronts, which challenge the dimension-splitting algorithm
rather than the core 1-D advection (it transports the homogeneous field
perfectly – no distortion was found after 10

When deciding whether to apply the smoother Eq. (20), one has to keep in mind that the Galperin scheme is always positively defined and does not need a shape-preserving filter to provide a “physically meaningful” solution, i.e. without negative values. It is free from this caveat. The purpose of the smoother is only to reduce the non-linear distortions of fields.

The smoother has both a positive and negative impact on the scheme performance. Among the positive ones are that (i) it damps the distortions of smooth shapes and gradients (Sect. 3.4), (ii) it reduces the amplification factor, precluding it from exceeding 1 even for a few time steps (Sect. 3.5), and that (iii) it reduces the unmixing problem (Fig. 14). Its negative features are that (i) the obtained solution is diffusive (Sect. 3.4), (ii) moderate and high frequencies in the solution spectrum are damped (Sect. 3.5), and that (iii) formal scores and convergence rates are lower in some tests (Sects. 5.2 and 6.2). The smoother has little impact on background inhomogeneity.

Most of the positive and negative features coincide with the impact of shape-preserving filters (e.g. L14), despite the different idea and formulations.

Since the smoother computational cost is negligible, one can decide whether to apply it depending only on the problem at hand. Strict interconnections between the species, smooth patterns and tolerance to diffusion form a case for the smoother. Conversely, sharp plumes over zero background (e.g. the accidental release case) argue against it.

The smoother impact grows monotonically with its parameter

Evaluation of the scheme efficiency is always very difficult as it strongly
depends on the algorithm implementation, but also on computers,
parallelisation, compiler options, etc. Nevertheless, basic characteristics
of the scheme can be deduced from comparison of its original version with
several classical schemes made by Galperin (2000). It included, in
particular, EM72 and Bott, which appeared

In this section, the efficiency of the updated Galperin scheme is evaluated from several points of view: (i) the scalability with regard to the number of transported species, spatial and temporal resolution, specific to the problem at hand, (ii) comparison with “standard implementation” of the Bott algorithm and the semi-Lagrangian scheme, and (iii) comparison of the run time in the L14 tests with the HEL and CSLAM schemes.

The scalability of the scheme and the whole SILAM model was tested in
real-wind global simulations for an arbitrarily taken 3 days
(15–17 May 2012). The reference run was set with 0.5

Scalability of the Galperin advection scheme and the SILAM model.
Panel

The model was run in a single-processor mode but compiled with O3 optimisation and OMP code pre-processing. Runs were made in a notebook with an Intel Core i7 processor and repeated in a workstation with an Intel Xeon E5. The scaling differed by 10–20 %, which was considered to be negligible.

The results (Fig. 16) highlight the scalability of the scheme and its
implementation in SILAM. The species-unrelated time of horizontal 2-D
advection (Fig. 16a, offset in regression line) is

With the chemical module turned off, advection constitutes

Since the scheme operates with the source grid cells, it can check that

A faster-than-proportional growth of the horizontal advection time with
increasing spatial resolution (Fig. 16c, normalised run time) is a result of
a growing Courant number: for a 4-times smaller grid cell (0.25

The time spent by advection is practically proportional to the temporal resolution (Fig. 16c); that is, it follows the number of times the advection is computed in the run.

Comparison with other schemes is arguably the most uncertain part of the
exercise: the scheme efficiency is strongly dependent on the quality of the
implementation (note the different results for the Bott scheme obtained by
Galperin, 2000, and Petrova et al., 2008). To obtain reproducible results, we
made this comparison against the “standard implementation” of the Bott code
available from the Internet
(

The test with 10

The above numbers should be considered as indicative only since the environment for the tests was completely artificial: the schemes were used as a stand-alone code applied in 1-D space. The Galperin scheme needed only one moment instead of three, which would be the case for 3-D advection. Despite very limited extra computations, this would still raise the memory exchange. The Bott scheme was taken without a shape-preservation filter, which would be needed for any real-life applications.

The tests were also made for our own implementation of the semi-Lagrangian
scheme (took

The L14 tests allowed rough benchmarking of the SILAM implementation of the
scheme in 2-D tasks. In particular, the run with 0.75

Formal benchmarks of the computers, the main uncertainty in this comparison,
are essentially the same: Kaas et al. (2013) used an Intel Core2 Duo E6550
processor (Intel Linpack 20 GFlops,

In SILAM applications, advection is parallelised using the shared-memory OMP
technology, whereas the MPI-based domain split is being developed. The OMP
parallelisation is readily applicable along each dimension, thus exploiting
the dimensional split of the advection scheme. For MPI, care should be taken
to allow for a sufficient width of the buffer areas to handle the Courant

The original scheme was formulated for the bulk mass of all transported tracers, thus performing the advection step for all species at once: the tracer's mass in the slab definition Eq. (5) was the sum of masses of all species. This is much faster than the species-wise advection and reduces the number of the moments per dimension down to 1 regardless of the number of tracers. It is also useful in the case of strong chemical interconnections between the species because the bulk advection keeps all pre-existing relations between the species. However, transport accuracy diminishes if the species have substantially different lifetimes in the atmosphere, are emitted from substantially different sources, or otherwise decorrelated in space.

The current paper presents the transport module of the System for Integrated modeLling of Atmospheric coMposition SILAM v.5, which is based on the improved advection routine of Michael Galperin combined with separate developments for vertical diffusion and dry deposition.

The cornerstone of the advection scheme is the subgrid information on distribution of masses inside the grid cells, which is generated at the emission calculation stage and maintained in a consistent way throughout the whole model, including chemical transformation, deposition, and transport itself. This information, albeit requiring substantial storage for handling, allows for accurate representation of transport.

The scheme is shown to be particularly efficient for point sources and sharp gradients of the concentration fields, still showing solid performance for smooth patterns. The most challenging task was found to be the puff-over-plain test, where the scheme showed noticeable distortions of the concentration pattern. Application of a simple smoother efficiently reduces the problem at a cost of non-zero viscosity of the resulting scheme.

Advanced tests and comparison with state-of-the-art algorithms confirmed the compromise between the efficiency and accuracy. SILAM performance was fully comparable with the other algorithms, outperforming some of them.

Among the future developments, implementation of the scheme in 2-D space and replacement of the smoother with extensions of the core advection algorithm are probably the most pressing ones.

SILAM is a publicly available model. Our experience shows however that its
successful application critically depends on the user's modelling skills and
understanding of the model concepts. Therefore, SILAM is available on an
on-request basis from the authors of this paper, who also provide support in
the initial model installation and set-up. The model description, operational
and research products, as well as reference documentation, are presented at

The stand-alone code of the Galperin advection scheme used in the above 1-D
tests is available at

The development of the SILAM model was supported by the ASTREX project of the Academy of Finland, as well as by the ESA-ATILA and FP7-MACC projects. The authors thank the late M. Galperin for the original version of the scheme, A. Bott for the publicly available version of his scheme, and E. Kaas and an anonymous reviewer for detailed comments on the manuscript.Edited by: V. Grewe