1. Introduction

The conservation laws that describe tracer transport of a scalar variable such as aerosols, hydrometeors, and other chemical species in chemistry climate models are given by

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The above system implies that

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where ∂/∂t is the local time derivative and d/dt denotes the Lagrangian derivative, V is velocity field, and q represents the tracer concentration (mass). Given the velocity field V(t) at time t, the distribution of tracer as time progresses from t to t + Δt can be computed from a stable numerical advection scheme in a uniform flow, guaranteeing the initial shape of the tracer field is preserved in the absence of numerical diffusion (Rasch and Williamson 1989). Indeed, numerical diffusion is introduced via interpolation errors when a field variable is approximated at intermediate positions in a given domain; hence, accuracy of spatial estimation is limited by the order of interpolation.

Computationally efficient and accurate advection schemes have been designed by Thuburn (1997), Lipscomb and Ringler (2005), and Nair et al. (2005) for high-resolution chemistry–climate models using spherical geodesic grids. Besides correctly approximating the steep flow and tracer gradients, the high-resolution models have an additional advantage that they can be directly validated with the satellite data. Geodesic grids were introduced more than 30 yr ago for their suitability to meteorological applications (Sadourny et al. 1968; Williamson 1968). For a complete description of an icosahedral–hexagonal grid, one may also refer to Baumgardner and Frederickson (1985), Heikes and Randall (1995), and Majewski et al. (2002). Moreover, the isotropic structure of this grid system and its easy adaptability to higher resolutions are the key features that underscore the need of designing an efficient advection scheme to solve the flow equations on a sphere so that fine-resolution models are viable. Many numerical schemes are available for simulating advection on a regular latitude–longitude grid (Robert et al. 1985; Rasch and Williamson 1989; Staniforth and Côté 1991; Hourdin and Armengaud 1999; Amato and Carfora 2000; Nair and Machenhauer 2002; Nair et al. 1999; and references cited therein). Indeed, these schemes are either monotonic-conservative Eulerian or nonconservative semi-Lagrangian (SL) in nature (Smolarkiewicz and Rasch 1991). In Eulerian schemes, tracer evolution across a fixed computational geometry is considered in terms of fluxes, but this may not maintain the positivity of positive-definite scalar fields. However, it could be resolved by enforcing an additional constraint of monotonicity of the scalar. Moreover, the Eulerian schemes have a restriction of relying on small time steps for advancing model integration. Both of these factors—monotonicity and small time steps—add to undesirable additional computational costs. Several improvements have been suggested to enhance the efficiency and accuracy of such schemes: upwind, donor cells scheme, flux-corrected transport (FCT), and other similar schemes. On the other hand, SL schemes have been popular since the last decade with most of the present GCMs using them.

Much earlier, Richardson (1926) described that the diffusion processes in eddy-dominant atmospheric flows, where the scale of the flow was recognized as the key parameter, influence the distribution of a tracer. In this elegant work, he devised a mathematical framework to comprehend diffusivity on a range of scales in one coherent manner. This suggests that the advection–diffusion schemes designed for climate models should be applicable to a wide range of scales and should not be restricted to step size or resolution. The SL scheme offers this property and hence a natural choice for global models. However, a severe problem arises from its inherently nonconservative nature—a key issue in dealing with the meteorological and chemistry variables in climate models. To achieve conservation with an SL scheme, higher-order interpolation formulas are needed. On the contrary, pure Lagrangian schemes are conservative, follow accurate trajectory, and inhibit interpolation error growth. Yet, its continuous use for long-term model integration may not be desirable as regularly spaced information could evolve into a highly irregular spatial structure. The numerical method utilized in this study, takes the advantage of Lagrangian time integration to minimize spatial and temporal discretization errors, to render the semi-Lagrangian scheme accurate and nearly conservative.

The paper has been organized as follows: geodesic grids are described in section 2, followed by the methodology of advective transport described in section 3. Two-dimensional test cases (viz., solid-body rotation and a deformational flow) are presented in section 4 along with a discussion on error norms in section 5. Finally, the results of this study are summarized in section 6.

2. The spherical geodesic grid system

In the literature, the system is referred to as either icosahedral–hexagonal or spherical geodesic grid systems. Many researchers have utilized geodesic grids in hydrodynamical atmospheric circulation modeling (Sadourny et al. 1968; Sadourny and Morel 1969; Williamson 1968; Cullen 1974; Masuda and Ohnishi 1986; Heikes and Randall 1995; Thuburn 1997; Stuhne and Peltier 1999; Majewski et al. 2002; Ringler and Randall 2002a; Bonaventura and Ringler 2005; Lipscomb and Ringler 2005). In this system, the Pole is surrounded by a fixed number of grid points that are at a fixed distance Δλ, unlike the latitude–longitude grid. The main advantage of this grid system is that the grid distance between neighboring points is constant over the entire globe. The grid is generated by subdividing the great circle arcs of 20 spherical triangles, constituting the spherical icosahedron (sphere). The icosahedron is formed with 12 fixed points on the earth including the two Poles. This grid divides the sphere into triangular elements of equal area and shape. The resolution level n implies that an edge of a spherical triangle is subdivided into n number of parts. The generation procedure suggested by Sadourny et al. (1968) has been followed here. It may be remarked that at the 12 fixed points, five triangles meet; whereas, at all other points, six triangles meet. The basic geometric cell structure, common to all geodesic grids, is therefore a triangle.

To arrive at a relevant mathematical formulation, some mathematical objects are first defined. The finite-dimensional approximation space Ω^h is given as

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where Ω is surface area of the spherical earth, Ω_T represents an area of single grid cell (planer triangle), and nt is the total number of grid cells constituting Ω^h. Evidently, Ω^h → Ω as resolution becomes finer. One can notice the increase in percentage area coverage of the sphere from n = 1 to n = 40 (Table 1). However, if the triangles are allowed to be spherical shells then the sum of the areas of these shells will be exactly equal to the area of spherical earth at all resolutions. At a given level n, the total number of triangles in Ω^h is given as

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and the total number of grid points in Ω is

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The average area of each triangular grid cell is 4πa²/20n² with a as the radius of earth. The mean grid length at resolution level n is Δl = ωa/n and ω is given as

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The comparative statistics of latitude–longitude and geodesic grids available in the literature have been utilized to check the accuracy of grid generation procedure. The motivating factor concerning the suitability of this grid over the regular latitude–longitude geometry is the complete avoidance of the well-known Pole problem. Moreover, the uniform coverage of the sphere by triangles and their isotropic structure completely obliterates filtering in the polar latitudes. Filtering is necessary in polar regions in order to maintain a unique time step for the entire globe if a latitude–longitude grid is used. The geodesic grid geometry structure has some further advantages: for example, it provides flexibility in designing discrete operators with their dual hexagonal–triangular geometry. Atmospheric models on this geometry may be viewed as the future-generation high-resolution AGCMs of twenty-first century. The forecasting system consisting of an icosahedral–hexagonal model together with its adjoint has the capability to handle different temporal and spatial scales with easy upgradation to higher resolutions. Furthermore, natural parallelization is evident at the problem level itself to speed up computations without overheads, which in turn allows the exploitation of parallelization strategies available at the algorithm and hardware levels (Majewski et al. 2002).

c. Discrete operators

In designing a full-fledged GCM, discrete representations of operators like gradient, divergence, curl, Laplacian, Jacobian, etc., are required. Here, the discrete operators are designed for a two-dimensional coordinate system, as a finite number of layers are assumed to be introduced in the vertical. Two local coordinate systems have been utilized, one with VPs as the origin and the other centered on DPs. The interconnection between the local and global coordinates has been maintained via an absolute three-dimensional (3D) coordinate system with center of the sphere as the coordinate origin. The grid generation is done using this 3D system by assigning a location to each grid point on the sphere, which is an ordered pair of latitude and longitude (λ, ϕ). The basic grid geometry is therefore a hexagon on which several staggering methodologies and discrete analogs for differential operators have been suggested in the literature (Williamson 1969; Cullen 1974; Nickovic et al. 2002; Ringler and Randall 2002b; Satoh 2003; Bonaventura and Ringler 2005). The discrete analog of gradient operator, which appears in the solution of transport equations [(1) and (2)], has been obtained from a weak formulation. Let f be a piecewise continuous function and S be a test function, defined on a polygonal domain H (Wachspress 1975). The values of the shape function S on H are shown in Fig. 3. Thus, to estimate the gradient at node k, one may write

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If S_k is the shape function defined at the node k and H = H_k constitutes the support of the node k, the above relation in discrete form can be written as

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In a similar manner ∂f /∂y may be estimated. The support H_k consists of nb triangles and

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Here the shape functions are defined in hexagonal cells and the gradients of the shape functions are estimated linearly on triangular elements. The other basic operators like divergence, curl, Laplacian, and Jacobian can be obtained similarly on the hexagonal cells. These operators are required to compute divergence in time-varying flow fields. In a divergent flow, one can use the operator-splitting technique to solve (1) and (2), by computing advection terms using the semi-Lagrangian method and updating the tracer tendency by computing divergence separately. It is worthwhile to remark further that the updating of mass and tracer fields required in a shallow-water model can be done in a manner suggested in Lipscomb and Ringler (2005) even for a semi-Lagrangian scheme. At a later date, we hope to report on the results of this kind of updating for the shallow-water model, which is already in an advanced stage of preparation.

3. Advection scheme

Semi-Lagrangian advection on a discrete domain may be understood as the movement of a parcel from one point in the direction of wind to some other point on the sphere. The departure point may or may not be a grid point. But the isotropy of the icosahedral–hexagonal grid system suggests that at very high resolution, every advected air parcel will be in close proximity to some grid point on the sphere if it does not occupy a node position. The validity of this assumption is evident from Fig. 4, where one can notice increased matching of the computed field with the analytical field as resolution is refined from level 10 to 40. The absolute error at level 10 is about 20%, which reduces to 1% at level 40. Thus, it seems that at sufficiently higher resolutions one could obliterate interpolation (Ritchie 1986, 1991) by applying the near-neighbor concept (see the appendix), which is inherently mass conserving. But, for slow-moving flows as well as strongly deforming flows, this method could cause nonnegligible phase errors. Besides this, there can be accumulation of extra information at some grid points thereby creating information data voids at neighboring grid points. In such a situation, one has to rely on interpolation procedures to achieve accuracy. But, for practical applications there is a need to investigate how one could achieve a good accuracy without using higher-order interpolations in an advective scheme. Moreover, the accuracy of a semi-Lagrangian scheme depends on exact trajectory approximations besides spatial approximations. Traditional semi-Lagrangian schemes are based on higher-order accurate integrations that use Taylor series expansions (Nair et al. 2003; McGregor 1993; Giraldo 1999; Amato and Carfora 2000). Like Dupont and Liu (2003), here also the error is evaluated during the course of integration in order to increase the accuracy of the results. The process has been explained below in a stepwise manner, where (uⁿ, υⁿ) denote the velocity field and qⁿ is the tracer distribution at time tⁿ. Also, q̂ and q̃ represent the tracer values at Lagrangian mesh and q* and q** represent the tracer values at Eulerian grid points (DPs).

• Step 1: Estimate the intermediate Lagrangian mesh (λ^n+1/2, ϕ^n+1/2) at time t^n+1/2 using

  

  

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• Step 2: Compute the tracer distribution (q*^n+1/2) as

  

  

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  where L is a linear area interpolation (Wachspress 1975) operator that transfers tracer values from the Lagrangian to the Eulerian grid point.

• Step 3: Compute (u^n+1/2, υ^n+1/2) at (λ^n+1/2, ϕ^n+1/2).

• Step 4: Estimate new Lagrangian mesh points (λ^n+1/2, ϕ^n+1/2) using (u^n+1/2, υ^n+1/2).

• Step 5: Estimate tracer distribution q**ⁿ⁺¹ at DPs using the value of q at (λ^n+1/2, ϕ^n+1/2) as

  

  

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• Step 6: Repeat steps 1 and 2 to estimate tracer distribution q̃ⁿ⁺¹ at the next time level q*ⁿ⁺¹ as

  

  

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• Step 7: Because the q*ⁿ⁺¹ and q**ⁿ⁺¹ both give an estimation of the tracer concentration at the same time level on Eulerian grid points, the difference (E = q**ⁿ⁺¹ − q*ⁿ⁺¹) between these two estimates could be used for correctly estimating qⁿ⁺¹_corr at tⁿ⁺¹ as

  

  

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  where 0 < γ < 1. In this work, we take γ = 0.5, which essentially removes the principal component of error.

However, these operations involve repeated interpolation steps, that cause loss in the efficiency of computations, which could be minimized by applying the corrective step [(10)] only after a few iterations (i.e., whenever the information is needed at grid points). Otherwise, integration could be advanced in the Lagrangian framework with a single linear interpolation per step. Since in a few time steps, Lagrangian mesh will never distort to a highly irregular pattern, it is possible to combine Lagrangian and semi-Lagrangian time stepping in a manner similar to the leapfrog–Matsuno combination used in gridpoint Eulerian models. In this context, the corrective step needs to be applied only at an instant when information is needed at grid points for further calculations (i.e., for “physics” in a GCM). Now the only point that remains to be explained with regard to the proposed numerical scheme is the way interpolation is carried out on the triangular grid for applying the forward trajectory method. To this end, one needs to identify a particular grid point that is confined in a particular Lagrangian grid triangle. The procedure to identify this Lagrangian grid triangle is as follows:

• Step 1: Find the nearest grid point to every particle departure location.

• Step 2: For every fixed grid point P, collect the set of all departure points that lie in its neighborhood (as that is done in objective analysis).

• Step 3: Identify the smallest triangle among these points containing the grid point P.

Note that there is no search involved in the execution of the above algorithm. Once the triangle has been identified, the values on the fixed grid are interpolated from departure gridpoint values. At this point, one can use inverse weighting or area interpolation or Waschpress functions to obtain the desired interpolation value.

4. Two-dimensional test cases

The forward semi-Lagrangian method [(10)] described in the preceding section has been applied to the following two test cases and the global errors for each case are given.

a. Test case I: Solid-body rotation

The solid-body rotation is one the most common experiments for testing an advection scheme (Rasch and Williamson 1989). Here, the velocity components of the advecting wind field are given by

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Because atmospheric flows can be decomposed into zonal and meridional components, two flow situations have been considered: (i) west–east purely zonal flow along a latitude circle with α = 0, and (ii) transpolar flow along the meridional great circle by taking α = π/2. The initial tracer distribution is prescribed as

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where r represents the great circle distance from the center of the cosine bell, (λ₀, ϕ₀) = (3π/2, 0), and the bell radius R is 7π/64. The numerical experiments are done at resolutions: level 32 (np = 10 242), level 48 (np = 23 042), and level 64 (np = 40 962), where np is the total number of grid points on the globe. The time step is chosen so that a complete revolution around the globe takes 288 iterations.

b. Deformational flow

Solid-body rotation is a nondivergent, nondeforming flow. To further test the advection scheme representing tracer evolution in a nondivergent but deforming flow, a second test problem of idealized cyclogenesis (Nair et al. 1999) is considered. In this problem two steady vortices are located in the global flow field. The flow field is obtained by transforming the coordinate system (λ, θ) such that the Poles coincide with the vortex centers in rotated system (λ′, θ′) with a constant angular velocity ω′ = ω(θ′) such that

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The normalized tangential velocity of the vortex field is given by

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where ρ′ = ρ cosϕ is the radius of the vortex. The exact solution at time t is given by

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where the parameter δ defines the characteristic width. In the numerical experiment reported in this study, the new location of the North Pole in (λ′, θ′) system is (π − 0.8, π/4.8), δ = 5, and ρ = 3 as taken in Nair et al. (2005). The model is integrated on the level-32 grid for 6 time units with different time steps of Δt = 6/16, Δt = 6/64, and Δt = 6/128.

5. Results and discussion

The most important aspect of a numerical method is that the numerical approximation is consistent (i.e., the method converges to the continuous operator as the grid spacing goes to zero). This has been assessed via a convergence study of the spatial truncation error. To do this, Δt is held fixed and the solution error is assessed over a range of grid resolutions during integration. The solution error is the difference between the numerical solution and the analytic solution at the end of the integration. The L₂, L₁, and L_∝ norms of the solution error for pure zonal flow (solid-body rotation) have been given here. However, on the isotropic grid, the rotation of fluid along any great circle arc can be transformed to a flow along equator by the coordinate transformation given in McDonald and Bates (1989). Experiments with different values of α (=0, 0.05, π/2, π/2 − 0.05), as suggested in Williamson et al. (1992), have been performed and found to give similar results. Figure 5 shows the shape of the cosine bell after one complete rotation at three different spatial resolutions along with their differences from the analytical field. It may be noticed that at the highest resolution (level 64) considered here, the simulated cosine bell is very close to the analytical one. The order of accuracy of the method is diagnosed by computing global error norms after one complete rotation. It may also be noted from Table 2 that the error norm values and total tracer mass variations diminish consistently as the grid spacing is reduced. For assessing the accuracy of the numerical scheme, the error is represented as a truncation of the discretization parameter (δ) following Nair et al. (1999). The magnitude of δ is same as the spacing between the grid points. Thus, an estimate for the order of the scheme can be obtained as {p_est = log₂[E(δ)/E(δ/2)]}. From Table 2 it follows that the scheme is approximately first-order accurate. At lower resolutions, norms are comparable to those mentioned in Table 5 of Nair et al. (1999). Because the slope of the error norm is considered much more important than its values, particularly at low resolutions, the gradient of the error norm is an important quantity. It has been found to be constant for this scheme. Thuburn (1997), in particular, reported an erosion of about 150 m in the original cosine bell height of 1000 m. A similar erosion is noted in the results of this study. The maximum value of the bell is simulated at the location that almost coincides with the analytical case; it is clearly illustrated by the L_∝ norm values. Furthermore, it has been observed that the corrective step is very effective for this test case in maintaining the bell amplitude even at coarser resolutions. However, when a simple linear interpolation is employed, the cosine bell amplitude is affected adversely as a result of diffusion.

While applying the near-neighbor concept (as explained in the appendix) to the cosine bell case, it was observed that the shape and amplitude of the cosine bell remained intact, but constant phase errors accumulated with every time step. The estimated one step maximum phase error could be Δl/2 (Δl is the length of the edge in triangular grid, which depends on the spatial resolution only). However, this constant error decreases linearly with increasing resolution, but accumulation of error could cause large deflections in the position of the cosine bell and the scheme may not be acceptable in operational models. Furthermore, one may have to also confront a situation when small Courant–Friedrichs–Lewy (CFL) numbers are encountered during the integration. A small CFL number will result in never moving parcels with the near-neighbor approach. This situation will occur if for a fixed resolution level and a fixed time step Δt, the magnitude of velocity field is less than threshold velocity U^t, which is given as

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However, these two problems—never moving parcels and accumulation of phase errors—can be tackled by using a combination of Lagrangian and semi-Lagrangian time stepping. In this design, a parcel of tracer-like water vapor will remain fixed and follow the Lagrangian trajectory, until the physics is called. At the physics step, the water vapor parcels can be transferred from Lagrangian grid to fixed grid (Eulerian) by utilizing the near-neighbor concept. Thus, parcels reaching a particular triangle will belong to that triangle and if the physics calculations are performed at the centroids of the grid cells it will not affect the accuracy of the forecast. In view of this, at a very high resolution (of the order of a few kilometers), one can find immense utility of this concept. However, this needs to be examined in a shallow-water model by avoiding the other complexities of a comprehensive GCM.

In the second test case simulations, the numerical scheme is assessed via keeping the resolution fixed and changing Δt. For this test case, much better error norms have been obtained even at a low resolution. To elucidate the impact of the corrective step during integration, a comparison of error norms with and without the corrective step (Table 3) has been given. This study of refining the time step demonstrates the error dominance due to spatial terms, as there are slight changes in the norm values. The corrective step gives large improvements by reducing the error in tracer mass. The other norm values also improve with the corrective step. The maximum and minimum values in the simulated and analytical fields remain exact, and small errors of O(10⁻³) are observed near the vortex cores. The simulated and analytical tracer fields after 6 time units with the time step of 6/64 are shown in Fig. 6, where an appreciable matching is evident. In view of the above results from the two test cases, it is expected that this type of dynamic correction might be very successful for semi-Lagrangian advection in a GCM.

6. Conclusions

A forward trajectory semi-Lagrangian scheme has been described for icosahedral–hexagonal grids. It has been tested for achieving less diffusive results without using higher-order interpolations. Two test cases presented here seem to favor the implementation of this advection scheme in a more comprehensive and elaborate atmospheric model. The computation of the error norm shows that the gradient of the error norm is constant and that the scheme is approximately first-order accurate. The idea of combining semi-Lagrangian and Lagrangian time stepping seems to be promising in its utilization for tracer transport in full GCMs. As tracer mass conservation is straightforward from the near-neighbor concept, its application in conjunction with the time stepping presented in this paper may find its utility in very high-resolution models. One significant achievement of this development work is that an efficient search algorithm has been devised, which reduces the complexity of search from O(n²) to O(n) on geodesic grids. The consistency of the gradient operator has been tested, by carrying out an 8-day integration of the barotropic vorticity equation, but the results are not included here and would form a part of a future paper on the shallow-water model using the icosahedral–hexagonal grid.