Numerical diffusion for a given species in a Eulerian chemical transport model is caused by the error when numerically solving the 3-D advection equation: ∂C∂t+u∂C∂x+v∂C∂y+w∂C∂z=0, where C is the species mixing ratio (kilogram of species per kilogram of air), and (u, v, w) are the horizontal and vertical wind components. Exact solution of the advection equation translates the mixing ratio downwind while conserving its magnitude, even in divergent flow (Chapter 7.2 of Brasseur and Jacob, 2017). However, the discretization of the model grid requires a numerical solution. Numerical schemes in models typically use high-order approximations to the upstream derivatives. But a first-order scheme allows here a simple analysis, and is relevant to our problem because higher-order schemes degrade to first order as a plume gets stretched to be resolved by only a few grid cells (Huynh, 1997; Rastigejev et al., 2010).

Using a 3-D first-order upwind scheme with no cross terms, applied to a grid cell (i, j, k) with time level n and wind vector components (u, v, w), Eq. (3) is approximated by Cn+1,i,j,k-Cn,i,j,kΔt+uCn,i,j,k-Cn,i-1,j,kΔx+vCn,i,j,k-Cn,i,j-1,kΔy+wCn,i,j,k-Cn,i,j,k-1Δz=0. Here, we have assumed that the wind components are positive so that the first-order approximation of the upwind derivatives is given by backward finite difference. Let us apply the Taylor expansion to each term in Eq. (4); for example, Cn,i-1,j,k=Cn,i,j,k-Δx∂C∂x+Δx22∂2C∂x2+oΔx2, which yields for that term u∂C∂x-uCn,i,j,k-Cn,i-1,j,kΔx=uΔx2∂2C∂x2+oΔx. The right-hand side of Eq. (6) is the truncation error between the u(∂C / ∂x) term in the true equation (Eq. 3) and its numerical approximation in Eq. (4). Adding up the error for each term in Eq. (3), we obtain the total truncation error ε: ε=-Δt2∂2C∂t2+uΔx2∂2C∂x2+vΔy2∂2C∂y2+wΔz2∂2C∂z2+oΔt+Δx+Δy+Δz. In typical truncation error analysis, terms like Δx are important as indicators of the order of accuracy of the scheme, while terms like ∂2C / ∂x2 are just coefficients. The scheme here is first order because the error decreases linearly with Δx. The modified equation approach (Chapter 3.3.2 of Durran, 2010; Warming and Hyett, 1974) provides a different view. We can modify the advection equation (Eq. 3) to add the error terms from Eq. (7) on the right-hand side: ∂C∂t+u∂C∂x+v∂C∂y+w∂C∂z=-Δt2∂2C∂t2+uΔx2∂2C∂x2+vΔy2∂2C∂y2+wΔz2∂2C∂z2. Using the same scheme (Eq. 4) to solve this modified equation, the error becomes second order, i.e., decreases quadratically (Δx2): ε′=oΔt+Δx+Δy+Δz. Thus, we can say that, instead of representing the original advection equation (Eq. 3), the numerical scheme (Eq. 4) better represents the advection–diffusion equation (Eq. 8) with the diffusion term D=-Δt2∂2C∂t2+uΔx2∂2C∂x2+vΔy2∂2C∂y2+wΔz2∂2C∂z2. This view is different from Eq. (7) in that terms like ∂2C / ∂x2 can now be interpreted as explicit diffusion in the differential equation while terms like Δx become just coefficients. The magnitude of the diffusion term decreases as the resolution increases, i.e., as the grid spacing Δx decreases, bringing the numerical scheme closer to the original equation (Eq. 3).

The time derivative (Δt/2)∂2C / ∂t2 in Eq. (10) is not a standard diffusion term and does not have a clear physical meaning, but we can show following Odman (1997) that in the 1-D upwind scheme it is approximated by the spatial derivative (see Appendix A for proof): Δt2∂2C∂t2≈αuΔx2∂2C∂x2, where α=uΔt/Δx is the Courant–Friedrichs–Lewy (CFL) number. Eulerian advection schemes require α≤1 for stability (CFL condition). This means, as long as the CFL condition is satisfied, that the time discretization error will not be larger than the spatial discretization error and will not limit the overall accuracy. We omit it in what follows and only consider D≈uΔx2∂2C∂x2+vΔy2∂2C∂y2+wΔz2∂2C∂z2. If the horizontal grid spacings (Δx, Δy) decrease while the vertical grid spacing (Δz) remains the same, we will eventually reach a point where uΔx2∂2C∂x2+vΔy2∂2C∂y2≪wΔz2∂2C∂z2, which implies D≈wΔz2∂2C∂z2. Under this condition, the numerical diffusion is independent of the horizontal resolution and only depends on the vertical resolution. Equation (14) explains why Eastham and Jacob (2017) found that increasing the horizontal resolution beyond 1∘ × 1∘ in their model did not lead to further improvement in plume preservation. Similarly, if the vertical resolution increases, the numerical diffusion will eventually be determined by the horizontal resolution.

To avoid being limited by one dimension, the horizontal and vertical diffusion terms in Eq. (12) should have similar magnitude: uΔx2∂2C∂x2+vΔy2∂2C∂y2≈wΔz2∂2C∂z2. There is general isotropy in the horizontal dimensions; thus, we have uΔx2∂2C∂x2≈vΔy2∂2C∂y2. The optimal grid spacing can then be obtained by equating horizontal and vertical diffusion: 2uΔx2∂2C∂x2=wΔz2∂2C∂z2. Rearranging, we obtain an expression for the optimal ratio between horizontal and vertical grid resolution to balance the effect of numerical diffusion: ΔxΔzopt=|w∂2C∂z2|2|u∂2C∂x2|. Let L and H be the horizontal and vertical extents of the plume and ΔC be the change in mixing ratio from the center of the plume to the background. We have ∂2C∂x2≈ΔCL2,∂2C∂z2≈ΔCH2. The above approximation can be obtained by either scale analysis (Chapter 2.4 of Holton, 2004) or a finite-difference approximation at the center of the plume (x=x0): ∂2C∂x2|x=x0≈C|x=x0+L+C|x=x0-L-2C|x=x0L2=-2ΔCL2. Placing Eq. (19) into Eq. (18), we get ΔxΔzopt=w2uLH2. Equation (21) means that the optimal ratio of horizontal and vertical grid resolutions depends on the wind velocity and the plume aspect ratio. To get an intuition for this, consider two extreme cases: (1) if w=0, i.e., the 3-D advection problem degrades to 2-D, there will be no vertical diffusion and thus no requirement on the vertical resolution (Δzopt→∞); (2) if H→ 0, i.e., the plume is infinitely thin, we will need infinitely small vertical grids to resolve it (Δzopt→ 0). In the real atmosphere with typical large-scale wind speeds u=10 m s-1, w=1 cm s-1 and typical plume sizes L=1000 km, H=1 km, we get (Δx / Δz)opt=500, larger than the dynamical criteria reviewed in the introduction (Eqs. 1 and 2). Although this numerical value is little more than an order-of-magnitude estimate, considering the uncertainty in the individual terms, it suggests that numerical diffusion of chemical plumes may place greater restriction on model vertical resolution than atmospheric dynamics. The estimated plume aspect ratio L / H=1000 applies to the bulk of the plume but might not be appropriate for small filaments. Later in this paper (Sect. 4.3), we will use numerical simulations to derive (Δx / Δz)opt and compare it to the result of this simple theoretical analysis.

In practice, the trade-off between horizontal resolution (Δx) and vertical resolution (Δz) must be considered in the context of a given allocation of computational resources. Increasing horizontal resolution by a factor m increases the number of grid cells by m2, since the increase is applied to both the x and y dimensions. In addition, the time step must generally be decreased by a factor m to satisfy the CFL condition, so that the computation cost scales as m3. Increasing vertical resolution does not generally affect the CFL condition because vertical winds are weak relative to Δz. A fixed amount of computation can thus be expressed by ΔzΔx2=P (where P is a constant), ignoring the CFL condition, or by ΔzΔx3=P, accounting for the CFL condition.

Here, we consider the general problem of minimizing the numerical diffusion for a given allocation of computational resources and with a trade-off parameter k, where k=1 represents equal costs for decreasing Δx and Δz, k=2 represents a quadratic cost of decreasing Δx (because of corresponding decrease in Δy), and k=3 represents a cubic cost of decreasing Δx (factoring in the CFL condition): ΔxkΔz=P. From Eqs. (12) and (16), the magnitude of the numerical diffusion term can be written as D=AΔx+BΔz, where A and B are coefficients: A=u∂2C∂x2,B=w2∂2C∂z2. In Sect. 2.2, and following Eq. (21), we estimated B/A≈500 for typical atmospheric conditions.

For a given amount of computing P, the optimal Δx / Δz ratio is the one that minimizes the numerical diffusion term D. This minimum is readily found graphically, as illustrated in Fig. 1. In this figure, the filled contours are isolines of D as given by Eq. (23) with B / A=500. The solid lines are the computational trade-offs ΔzΔx2=P and ΔzΔx3=P. For a given value of P, the numerical diffusion is minimized when the contour lines of D(Δx, Δz) and P(Δx, Δz) are parallel, i.e., when their gradients have the same direction: ∇PΔx,Δz∝∇DΔx,Δz. From Eq. (22), ∇P(Δx,Δz)=(kΔxk-1Δz,Δxk)=Δxk-1 (kΔz,Δx). From Eq. (23), ∇D(Δx, Δz)=(A,B). Thus, Eq. (25) becomes kΔz,Δx∝(A,B), which yields ΔxΔzopt=kBA. In Sect. 2.2, we implicitly assumed that the computational costs of adjusting Δx or Δz would be the same, i.e., k=1 in Eq. (22). Equation (27) is then the same as Eq. (18), and using the same estimate B/A=500 as in Sect. 2.2 yields (Δx/Δz)opt=500. Accounting for higher computational cost when increasing horizontal resolution (k > 1) results in a higher optimal ratio. The dashed lines in Fig. 1 show the optimal Δx / Δz ratios derived for k=2 and k=3. For k=2, we find (Δx/Δz)opt=1000, and for k=3 we find (Δx / Δz)opt=1500. It is actually remarkable that the dependence of this optimal ratio on k is linear rather than exponential. The reason is that it is based on the relative contributions of numerical diffusion in the horizontal vs. vertical directions; if numerical diffusion is caused by a coarse horizontal grid, then increasing vertical resolution (even if cheap) will not provide benefit.

Figure 2 shows the surface pressures and 700 hPa wind fields on day 8 of the plume simulation, at C48L20 and C384L20 resolutions. The simulation describes a typical quasi-geostrophic system at midlatitudes with low and high pressure centers and the associated geostrophic winds. We find that increasing the horizontal resolution intensifies the cyclones, as shown in previous studies (Jablonowski and Williamson, 2006; Lauritzen et al., 2010), while increasing vertical resolution from L20 to L160 has almost no effect. Hence, the GCM emphasis on increasing horizontal resolution.

Also shown in Fig. 2 is the local Lyapunov exponent λ of the wind field. λ is the rate constant at which nearby air parcels separate in the direction of the flow, i.e., the intensity of flow stretching. Rastigejev et al. (2010) showed theoretically that the Lyapunov exponent should be a predictor of numerical diffusion in Eulerian models, and Eastham and Jacob (2017) confirmed this in GEOS-Chem model simulations. We calculate the Lyapunov exponent locally, using the formula derived by Eastham and Jacob (2017): λ≈Δu+ΔvΔx+Δy, where Δu and Δv are the changes in wind speed between the local grid cell and the grid cell downwind, and Δx and Δy are the corresponding grid spacings. Between 30∘ N and 60∘ N where the plume transport takes place, the Lyapunov exponents are of order 10-5 s-1, consistent with values derived from the GEOS-FP wind data (Eastham and Jacob, 2017). Higher horizontal resolution increases stretching because small-scale eddies are better resolved, which offsets some of the reduction in numerical diffusion (Rastigejev et al., 2010). We find the 700 hPa vertical wind speed w at 30–60∘ N of -0.1 ± 1.0 cm s-1 (mean ±1 SD), typical of the range of large-scale vertical wind speeds in the real atmosphere. Thus, the FV3 simulation provides a realistic environment to investigate how global-scale transport of chemical plumes is sensitive to model grid resolution. It does not account for deep convection or boundary layer turbulence, but our focus here is on the free troposphere under the prevailing stable conditions that allow for plume preservation during global-scale transport. The plume may originate from a convective updraft, and it may be eventually entrained and mixed in the turbulent boundary layer, but convection and boundary layer turbulence are out of scope.

Figure 3 illustrates the evolution of the plume over the 8-day period in the C384L160 case (≈ 25 km, 6 hPa resolution). The plume is initialized over Alaska, reaches eastern North America by day 4, and Eurasia by day 8, with strong filamentation along the way due to wind shear. Such rapid transport and filamentation are typical of free tropospheric plumes at northern midlatitudes (Stohl et al., 2002). The plume gradually subsides and dilutes vertically over the 8-day period, with a subsidence rate typical of observations (Crawford et al., 2004). The spreading and dilution of the plume apparent in Fig. 3 are due in part to the plotting of column and meridional average mixing ratios for visualization purposes; the actual numerical diffusion is less and can be quantified by the mixing ratio decay and entropy increase for the actual plume, as described in Sect. 4.2.

Exact solution to the advection equation conserves the mixing ratio, even for divergent or sheared flow (Chapter 7.2 of Brasseur and Jacob, 2017). Our simulation includes advection as the only process. It follows that any mixing ratio decay in the model plume must be due solely to numerical diffusion and provides a metric for this diffusion.

Figure 4 shows the rate of decay of the maximum mixing ratio in the plume for the different horizontal and vertical resolutions of our simulations. The timescale for this decay diagnoses the rate of plume dissipation from numerical diffusion and can be used to compare different grid resolutions (Rastigejev et al., 2010; Eastham and Jacob, 2017).

At the lowest resolution (C48L20), the maximum mixing ratio in the plume drops from 1.0 to 0.1 after 8 days. Such rapid diffusion is consistent with the midlatitude results of Eastham and Jacob (2017) using GEOS-FP winds. Starting from C48L20, solely increasing the vertical resolution has no benefit in reducing numerical diffusion (Fig. 4, top left panel). Solely increasing horizontal resolution has some benefit for the first 4 days of aging, but by day 5 the benefit is gone (Fig. 4, bottom left panel). This is consistent with the theory in Sect. 2.1 that inadequate resolution in one direction will limit the overall accuracy, making grid refinement in the other direction useless.

However, once the resolution of one dimension is high enough that it is no longer a limiting factor, grid refinement in the other direction becomes effective. This is illustrated in the right panels of Fig. 4. Increasing vertical resolution in a C384 simulation has sustained benefit from L20 to L160, and increasing horizontal resolution in a L160 simulation has sustained benefit from C48 to C384. At the highest resolution (C384L160), the decay in the maximum mixing ratio is only 35 % after 8 days of transport, a drastic improvement over the simulation cases presented by Rastigejev et al. (2010) and Eastham and Jacob (2017).

The behavior of decay rates in Fig. 4 lends further insights into numerical diffusion. We see that the decay rates are initially slow and then abruptly increase. This is because the plume is initially well resolved on the grid, but as the plume gradually filaments and becomes poorly resolved, fast numerical diffusion takes over. Increasing horizontal resolution delays the onset of this fast numerical diffusion, as seen most dramatically in the bottom left panel of Fig. 4. Thus, a factor in the choice of resolution should be the extent of time over which the model plumes must be preserved, considering that molecular diffusion will eventually dissipate the plumes in the actual atmosphere as they filament down to the millimeter Kolmogorov scale (Chapter 8 of Brasseur and Jacob, 2017). Observations show that intercontinental free tropospheric plumes can retain their structure for at least a week (Heald et al., 2003; Zhang et al., 2008), so there is benefit in the highest range of resolutions investigated in our simulations.

Figure 5 shows the vertical profile of maximum mixing ratios for each model level after 8 days of simulation, at the lowest model resolution (C48L20), the highest model resolution (C384L160) and intermediate cases where only horizontal or vertical resolution is increased from the low-resolution case (C384L20, C48L160). Starting from C48L20, solely increasing either the horizontal resolution (to C384L20) or the vertical resolution (to C48L160) has limited improvement on the vertical profile. This is the familiar picture of models being unable to preserve the vertical structure of pollution plumes on intercontinental scales (Heald et al., 2003). Increasing both horizontal and vertical resolutions (to C384L160) drastically improves the preservation of the vertical profile and largely fixes the problem. The surface concentrations are close to zero in all cases but this is because the FV3 dynamical core does not include boundary layer physics. From the concentrations at 900–950 hPa, we can conclude that the high-resolution simulation when implemented in a full GCM would lead to much stronger localized impact of the subsiding plume on surface concentrations.

Maximum mixing ratio in the plume is an extreme value diagnostic that is relevant for plume observation and impact but is an imperfect measure of plume dilution (Eastham and Jacob, 2017). As shown in Fig. 3, the plume shears into multiple filaments as it ages but the maximum mixing ratio diagnoses just one of these filaments. Also, numerical diffusion will first erode the plume as its edges while preserving the maximum mixing ratio at the center. Eastham and Jacob (2017) used the expanding size of the plume as an alternate diagnostic but this relies on an arbitrary concentration threshold.

As a more general diagnostic of plume preservation, we calculate the entropy that takes into account all grid cells in the global domain (Lauritzen and Thuburn, 2012). The entropy S of a 3-D mixing ratio field can be calculated by S=-k∑i=1nmiCilog⁡Ci, where n is the total number of grid cells of index i, Ci is the mixing ratio, mi is the mass of air in the grid cell, and k is a scaling factor such that the initial entropy is unity. Pure advection conserves entropy but diffusion increases it, and S is maximized when the mixing ratio field C becomes uniform (complete mixing). A non-monotonic advection scheme can unphysically decrease entropy, but here we use strictly monotonic schemes in both horizontal and vertical dimensions so this would not happen. Entropy would increase in a real-world plume as filamentation leads to eventual plume dissipation by molecular diffusion, but numerical diffusion is much faster on our scales of relevance (D'Isidoro et al., 2010). Thus, our goal here is to minimize entropy, i.e., minimize numerical diffusion.

Figure 6 shows the increase in entropy as the plume dilutes at different model grid resolutions. Results are similar to the maximum mixing ratio diagnostic (Fig. 4) in showing the limiting effects of either horizontal or vertical resolution, and the benefit of coupling the two to improve the simulation. One difference is the absence of a time lag for plume dilution. Whereas the maximum mixing ratio is initially sheltered from numerical diffusion if the plume is resolved by a number of grid cells, numerical diffusion erodes the plume edges and the thinner filaments, and this is captured by the entropy diagnostic. The entropy diagnostic also shows a slowdown of plume dilution with time, particularly at coarse resolution, and this is due to the smoothing of the plume that allows concentration gradients to be better represented by the numerical schemes. Nevertheless, the entropy continues to increase even as plume edges become smoother. Ultimately, the choice of maximum mixing ratio or entropy as a diagnostic of plume dissipation may depend on the application, but the implied requirements for grid resolution are similar. This is discussed further below (Sect. 4.3).

The results from Sect. 4.2, following on the theoretical analysis of Sect. 2, show that preserving plumes in global models may be limited by either horizontal or vertical resolution. It follows that there must be an optimal ratio of horizontal to vertical grid spacing (Δx / Δz)opt for simulating the global-scale transport of plumes, as there is for the dynamical criteria reviewed in the introduction. We derived such a ratio from theoretical analysis in Sect. 2, and here we derive it from the FV3 plume simulations.

Figure 7 illustrates the trade-offs between horizontal and vertical resolution in the FV3 plume simulations, presented in a similar manner to the results of the theoretical analysis in Fig. 1. The contours measure the preservation of the plume after 8 days, as diagnosed by either the maximum mixing ratio or the entropy, using the day 8 data from Figs. 4 and 6 with additional simulations at intermediate resolutions to better define the contours. As in Sect. 2.3, we aim to preserve the maximum mixing ratio and/or minimize entropy under the computational trade-offs ΔzΔx3=P and ΔzΔx2=P. The actual computational costs of our GFDL-FV3 simulations follow the ΔzΔx3=P curve, since the time step is reduced proportionally with the horizontal resolution. The solid lines show the computational trade-offs. Along each trade-off line, it is generally beneficial to move away from Δx / Δz < 100 (the upper left region of Fig. 7, Δx=25–50 km, Δz=0.6 km) and toward Δx / Δz ∼ 1000 (the bottom region of Fig. 7, Δx=50–200 km, Δz=0.08 km), since it leads to better preservation of the plume without incurring more computational cost. Thus, we already see that the current generation of models (Δx / Δz ∼ 20) is out of balance in privileging horizontal over vertical resolution.

As in Sect. 2.3, the optimal ratio (Δx / Δz)opt is defined by the point where the computational trade-off line parallels the contour line. Different ratios Δx / Δz are shown as yellow dashed lines in Fig. 7. For the ΔzΔx3=P trade-off (the red solid lines in Fig. 7), the optimal Δx / Δz is in the range 400–700. For the ΔzΔx2=P trade-off (white solid lines), the optimal Δx / Δz is around 400. This is consistent with the theoretical derivation in Sect. 2.3 where the optimal Δx / Δz is of order 1000.

We conducted sensitivity tests with plumes of different initial vertical thicknesses and horizontal extents, and found similar results. Thicker plumes have better initial preservation of the maximum mixing ratio but this advantage is rapidly lost as the plume filaments. Although the theoretical analysis of Sect. 2 implies that (Δx / Δz)opt should depend on the plume size, this applies to the stretched rather than to the initial plume. During model transport, plumes of different initial thicknesses tend to be stretched to similar steady-state thicknesses where the stretching rate (thinning the plume) is balanced by the numerical diffusion rate (thickening the plume) (Rastigejev et al., 2010).

The estimated (Δx / Δz)opt should not greatly depend on the advection scheme used, since fast numerical diffusion occurs when the plume has filamented to the point where gradients cannot be resolved and any advection scheme collapses to first-order accurate. One concern is whether FV3 represents realistically the ratio of vertical-to-horizontal shear that would occur in a wet atmosphere; this should be tested in simulations in an actual GCM. Nevertheless, it appears that the vertical resolution requirements for global simulation of chemical plumes are larger than the ratio (Δx / Δz)opt ∼ 100 derived from dynamical concerns of resolving fronts and gravity waves, and that current models have inadequate vertical resolutions.

Our recommendation to increase vertical resolution in the free troposphere is specific to global models, and our emphasis on an optimal Δx / Δz is to make the point that the simulation of global-scale plumes with large horizontal/vertical aspect ratios (reflecting the strong horizontal wind shear and vertically stable conditions of the free troposphere) is currently limited by vertical rather than by horizontal model resolution. The situation would be very different in cloud-resolving models (Δx < 1 km) where turbulence is much more isotropic.