The first crucial issue in a semi-Lagrangian scheme is to efficiently and accurately determine the trajectories in either backward schemes or forward schemes. Traditional backward schemes compute trajectories with an iterative algorithm. Several other economical schemes have been proposed using a truncated Taylor series approximation in backward schemes (McGregor, 1993; Hortal, 2002; Lauritzen et al., 2006) and forward schemes (Nair et al., 2003). This iterative procedure is a computationally expensive component of the semi-Lagrangian scheme and is worse in the polar zones where special treatment and approximations are required (Nair and Machenhauer, 2002; Zerroukat et al., 2009). There exists a non-interpolating semi-Lagrangian scheme (Ritchie, 1986). It decomposes the trajectory vector into the integer part and the fractional part. The fractional advection is done by the Eulerian upstream scheme. Lin and Rood (1996) extended it to a flux-form scheme for mass conservation.
In extending 1D schemes to higher dimensions, one choice is to get trajectories into higher-dimension space and use cascade remapping to split the high dimensions into two one-dimensional steps, because fully two-dimensional reconstruction is not easily monotone and positive. Cascade schemes are a little complicated because they apply one-dimensional remapping, first from the regular Eulerian cells to the intermediate grid cells along the Eulerian longitudes or latitudes, and second from the intermediate grid cells to the departure cells along the Lagrangian latitudes or longitudes (which is a curve), especially in the SLICE scheme (Figure 1 of Zerroukat et al., 2007). Another choice is operator splitting, which splits the advection process into a combination of operators in each of the two coordinate directions.
Several locally mass-conserved semi-Lagrangian schemes have been proposed. Kaas (2008) modifies the upstream interpolation coefficient area weights that transfer information from arrival Eulerian grid points to departure semi-Lagrangian points, which ensures that the total mass given off by a given Eulerian grid point to all the surrounding departure points is equal to the cell area represented by this grid point. The forecast densities in the arrival Eulerian grid points, including the effects of divergence, are equal to the modified upstream interpolated values divided by the cell area represented by the arrival Eulerian grid point.
Juang (2007, 2008) proposed a non-iteration dimen-sional-split semi-Lagrangian (NDSL) scheme, which we will describe in section 2. A two-dimensional NDSL will be presented in section 3. In section 4, we followed Kaas' test cases to compare the NDSL with other semi-Lagrangian schemes from the literature. Discussion and conclusions are given in section 5.
2. One-dimensional NDSL
The traditional backward/forward semi-Lagrangian scheme assumes the arrival/departure points at regular model grid points. It requires an initial guess and iterations to compute the trajectories, which means finding midpoint winds and transferring the fluid particles from the departure points to the arrival points. At the very least, it requires two iterations for three interpolations at the midpoint for wind and one interpolation for other variables at the departure/arrival point for the backward/forward scheme. Although some accelerated schemes have been constructed, the iteration is a computationally expensive component of the semi-Lagrangian scheme.
The proposed scheme NDSL (Figure 1) is a central scheme, which assumes the midpoint wind at the regular model grid point at time t in order to find the departure point at time t − Δt and the arrival point at time t + Δt, while the three-time-level scheme is selected. In other words, there is no initial guess and no iteration to find trajectories, but just one interpolation at the departure point and one remapping at the arrival point. Compared to the method which uses the upstream scheme, the wind of the central scheme may reduce numerical error because of no estimation and iteration; thus it is considered more precise.
For mass conservation, the divergent term should be considered and the 1D continuity equation can be expressed as
where ρ is a scalar quantity such as density and u is velocity quantity, so the divergence in the last term of (1) can be expressed as
where Δx is the cell length (volume). Substitute (2) into (1) and the continuity equation becomes
which is the Lagrangian form of the continuity equation, so local mass conservation can be obtained by
where subscripts D and A represent departure and arrival locations, from departure time n − 1 to arrival time n + 1, which is the same as with the traditional flux-form semi-Lagrangian computation but is simper and easier to implement. The relationship between the flux form and Lagrangian form has been illustrated in Juang and Hong (2010). To ensure mass conservation, we use conservative reconstruction and remapping schemes such as PPM, PSM and PQM.
Generally, the 1D mass-conserving NDSL (NDSL-MC) algorithm (Figure 2) is a finite-volume method (Toro, 2009). The computation details can be summarized as follows:
1.Compute the interface winds UL and UR using the Eulerian regular point winds at time n.
2.Locate the departure points at time n−1, the arrival points at time n+1 and get their cell respectively.
3.Interpolate density from the regular points to the departure points at time n − 1 using PPM or PSM.
4Transport mass from the departure points at time n − 1 to the arrival points at time n + 1 using (3).
5Remap density from the arrival points to the regular points at time n+1 using PPM or PSM.
Mass conservation can be proved as in the following. The density can be presented piecewise as ρ = S(x). Thus the total mass over the whole domain is conserved as
where subscript R is the regular grid, D is the departure grid, and A is the arrival grid. The first and third equations are guaranteed mass-conserving by finite volume-remapping methods and the second equation is mass conserving with (3).
3. Extending to two-dimensional NDSL
To extend to 2D advection we use a dimensional splitting method, which is a multidimensional problem that simply splits into a sequence of one-dimensional problems. The advantage is that only 1D interpolation and remapping are needed, so the mass conservation can easily be obtained, with the same monotone and positive as when the 1D monotone and positive limiter are used. Another important advantage is that the 1D method can be easily coded and most importantly is compatible with most numerical models. Our dimensional-split scheme will have no halo for parallel implementation and high compute efficiency because of 1D interpolation. Conversely, SLICE (Zerroukat et al., 2007) needs a halo because it needs a sweep along the Lagrangian direction, as does LMCSL (Kaas, 2008) because of 2D interpolation.
Following Toro (2009), consider the two-dimensional advection equation
where q is the tracer mixing ratio. The initial condition (IC) at time n is a function of x,y,tn and is denoted by qn.
Using dimensional splitting, (4) can be replaced by a pair of one-dimensional equations (5) and (6). The equation in the x-direction is
where the initial condition is the initial data for the origin equation (4) and its solution after a time Δt is denoted by . The equation in the y-direction is
where the initial condition is the solution of (5) and its solution after a time Δt is denoted by qn+1, which is regarded as the solution of (4) after a time Δt.
If X(t) and Y(t) are approximate solution operators for (5) and (6), then the splitting of the original two–dimensional (4) can be written thus:
Usually we use a second-order accurate splitting (Strang, 1968):
It is the 2D NDSL advection scheme when X and Y represent the 1D NDSL advection scheme.
For mass conservation, we need use tracer and continuity equations together as follows:
Then the density weighted tracer can be treated for conservation as
Combined with continuity equation, we can have conserved tracer advection
where Δ is the cell volume. Thus with the mass conservation scheme (5) and (6) will change to
We call this mass conservation scheme NDSL-MC1.
For a pure advection scheme which uses (5) and (6), there is no existing splitting error. When using a mass conservation scheme we need to consider the splitting error and generally the splitting error cannot be neglected in large-scale geophysical applications, especially for climate research, although the higher-order splitting methods will reduce the splitting error.
Several schemes to eliminate the splitting error have been proposed independently (Leonard et al., 1996; Lin and Rood, 1996; Clappier, 1998) and they essentially use the same concept. The splitting error comes from the fact that the flow deformation is evaluated at two different locations for their respective directions, so the divergence does not equal zero in the non- divergence case.
We use Lin and Rood's (1996) expression here. If we define F and G as the 1D mass conservation advection increment operator, which are the solutions for (9) and (10), and their corresponding pure advection increment operators f and g, which are solutions for (5) and (6), the correction scheme will be expressed as
This scheme is mass conserving since it only includes the conservative outer operators F and G. It should be noted that the mass conservation operators F/G and the pure advection operators f/g in NDSL use the semi-Lagrangian central scheme, which are different with the non-interpolating method in Lin and Rood (1996) scheme. The dispersive properties of the non-interpolating method for its Eulerian component are not generally as good as those of the interpolating semi-Lagrangian schemes (Staniforth and Cote, 1991; Lauritzen, 2007).
For eliminate the splitting error, we need to use a backward scheme instead of a centre scheme in the outer operator to make sure the contribution from flow deformations to the final forecast value is evaluated at the same spatial location for both coordinate directions. The correction scheme will be called NDSL-MC2.
The strong deformational non-divergence flow test (Clappier, 1998) is used to verify the result of the correction scheme. The whole field is initialized with a constant unit value and the flow is
After 100 time steps of 1 s (the Courant number is 1), the NDSL-MC1 scheme gets the field where the max is 1.0014 and the min is 0.9986, but the max of the NDSL-MC2 scheme is 1.0000000000000684 and the min is 0.99999999999993450.
4. Numerical results
The proposed NDSL scheme will be compared with other state-of-art semi-Lagrangian schemes (Zerroukat et al., 2007; Kaas, 2008) using several passive advection tests, including the slotted-cylinder rotation, the cosine hill rotation and the idealized cyclogenesis problem. For all the test problems, PSM-M (Zerroukat etal., 2006) is selected instead of PQM and QSM because of its inexpensive computational cost.
We follow the statistics of Williamson et al. (1992), which include root mean square error (RMSE), l1,l2,l∞,hmax and hmin as follows:
where N is the total number of grid cells and superscript a denotes the analytical solution.
The percentage relative error of the total mass (PDM) is defined as
where M0 is the initial total mass and Mt is the total mass at time t. Note that when calculating the PDM a larger computation domain is used to prevent the mass from disappearing outside the boundary.
4.1. Slotted-cylinder problem
The slotted-cylinder problem consists of a slotted cylinder rotating with constant angular velocity ω about a point (xc,yc). The analytical solution is
where ρ0 is a constant, σ is the radius of the cylinder, and sw and sl are the width and length of the slot, respectively. ξ and ζ are defined as relative coordinates with respect to the moving centre of the cylinder (xc − γcos(ωt),yc − γsin(ωt)):
where γ is the distance from the centre of the flow (xc,yc) to the centre of the cylinder.
Table 1 summarizes the statistics scores that verify the NDSL-MC2 scheme with different resolutions of the slotted-cylinder problem after one rotation. The Courant number C is from 1.64 (Res = 51) to 6.54 (Res = 201). The results with SLICE (Zerroukat et al., 2007) and LMCSL (Kaas, 2008) are also provided for comparison. It turns out that with the NDSL-MC2 scheme total mass is conserved and the errors are about the same size as with SLICE and LMCSL for a large Courant number. Table 2 and Figure 3 provide results after six full rotations for the slotted-cylinder problem.
Table 1. Statistics for the slotted-cylinder problem after one full rotation (Nt = 96).
Table 2. Statistics for the slotted-cylinder problem after six full rotations (Nt = 576).
4.2. Cosine hill problem
The cosine hill problem is similar to the slotted-cylinder one, except that the distribution is much smoother in space. The analytical solution is
where r is the radius of the cosine hill.
The results for the cosine hill problem are given in Tables 3 and 4. The Courant number C is from about 1.8 (Res = 33) to 7.23 (Res = 129). We find that the RMSE and most statistical errors of the proposed scheme are significantly smaller for the low-resolution case, which is chosen by most published schemes. The results for large courant number (7.23) are not good, perhaps because the method has trouble in predicting the trajectories correctly, which means the vector sum of two-direction advection may not be correct compared with a real trajectory with a large Courant number. In one direction, this method has no problem with a large Courant number. The results after two full rotations are shown in Figure 4.
Table 3. Statistics for the cosine hill problem after one full rotation (Nt = 71).
Table 4. Statistics for the cosine hill problem after two full rotations (Nt = 142).
Another error measure test case is that the time step is decreased with the grid resolution so as to keep a constant Courant number, which can show the rate of convergence. Table 5 shows good convergence results when the Courant number C is 3.62 with different resolutions.
Table 5. Statistics for the cosine hill problem after one full rotation (t = 628279,c = 3.62).
4.3. Idealized cyclogenesis problem
The idealized cyclogenesis problem consists of an initial circular vortex with a tangential velocity V(r) = v0 tanh(r)/cosh2(r), where r is the radial distance from the centre of the vortex (xc,yc), and v0 is a constant chosen such that the maximum value of V(r) is unity (i.e. Vmax = 1). The analytical solution is
where ω = V(r)/r is the angular velocity and δ is the characteristic width of the frontal zone.
Table 6 lists the statistics for the idealized cyclogenesis problem for different resolutions and Figure 5 shows the result after 16 time steps. The Courant number C is about 4.03. It is noted that the results with the NDSL-MC2 scheme are close to those with SLICE, but LMCSL the scheme achieves better scores than other schemes for this problem. The constant Courant number 2.01 results are shown in Table 7.
Table 6. Statistics for the idealized cyclogenesis problem (Nt = 16).
Ω = [0,10]2; Δt = 0.3125; δ = 0.05.
Table 7. Statistics for the idealized cyclogenesis problem (t = 5,c = 2.01).
A simple and efficient non-iterative dimensional-split semi-Lagrangian scheme with mass conservation has been tested in a limited area domain. Several tracer tests show that the proposed scheme is competitive with or better in accuracy than other state-of-the-art mass conservative semi-Lagrangian schemes without an iterative trajectory calculation and with reduced computational cost.
Note that LMCSL and SLICE used analytical upstream trajectories in these test problems—though that is not their default method—but the NDSL scheme used analytical trajectories in these test problems by default. Thus we believe that the additional advantages of the NDSL scheme will be revealed in shallow-water models and three-dimensional applications where the flow field is not steady and not analytical.
Furthermore, the scheme we proposed can benefit from any advance in the 1D conservative reconstruction and remapping scheme, including high-order schemes or better monotonic and extrema-preserving limiters.
We focus on the tracer advection with mass conservation in plane geometry in this paper. A paper on the 2D NDSL on the sphere has been submitted. The 2D NDSL shallow-water model and its extension to a global domain have been developed and will be published soon. A new dynamic core based on the NDSL scheme will be implemented in the NCEP GFS and RSM models.
Thanks to EMC global dynamics group for invaluable discussion, Dr Shrinivas Moorthi for his internal review, and Mary Hart for carefully editing the original manuscript. We also thank the editor and the anonymous reviewer for their constructive comments and suggestions, and especially for their detailed and helpful annotations in the manuscript.