Global transport of pollution mainly takes place in the free troposphere where winds are strong and pollutant lifetimes are long. Much of this transport takes place in well-defined, concentrated layers or plumes that can remain coherent for a week or more while traveling over distances of intercontinental scale. Models fail to reproduce these persistent plumes due to rapid dissipation by numerical diffusion. Here we use the GEOS-Chem chemical transport model (CTM) to understand this problem.

The free troposphere, defined as the region between the turbulent planetary boundary layer and the quiescent stratosphere, experiences strong wind shear (divergence) in a convectively stable environment. Stability allows the formation of persistent laminae (layers) or plumes, first detected by early radiosonde measurements (Danielsen, 1959) and later shown to be ubiquitous throughout the free troposphere (Newell et al., 1999; Thouret et al., 2000). The plumes are quasi-horizontal, typically fanning out over hundreds of kilometers with a vertical thickness on the order of 1 km (Mauzerall et al., 1998; Stoller et al., 1999; Heald et al., 2003; Jaffe et al., 2003; Hudman et al., 2004; Colette et al., 2005; Liang et al., 2007). Free tropospheric plumes resulting from stratospheric intrusions can retain 150 ppb of ozone over a period of weeks (Trickl et al., 2011). Such global-scale transport with little dilution has important implications for environmental impacts, interactions with weather, and chemical aging.

Eulerian models used for simulating global atmospheric transport fail to reproduce this persistent layered structure. The modeled plumes dissipate within days by mixing with the background (Heald et al., 2003; Vuolo et al., 2009). Eulerian models simulate transport as a flux divergence for fixed grid cells, and the rapid dissipation implies a large transportation error from numerical diffusion even when a highly accurate advection algorithm is used (Rastigejev et al., 2010). A Lagrangian approach, in which transport is calculated for individual air parcels carried by the flow with no interaction between neighbors, would avoid this problem (Khosrawi et al., 2005). Global Lagrangian models have been used with success, for example, to describe the sharp gradients at the edge of the polar vortex (Fairlie et al., 1999; Hoppe et al., 2014; Konopka et al., 2003). However, they are rarely used for comprehensive calculations of global atmospheric composition because of their inhomogeneous coverage and difficulties in dealing with nonlinear chemistry (Brasseur and Jacob, 2017). An intermediate approach, using Lagrangian surfaces to represent vertical transport but conventional Eulerian techniques to represent horizontal transport, has had greater success in capturing these layers (Lin et al., 2012a and b) but still exhibits excessive diffusion compared to observations (Lin et al., 2015). Adaptive mesh refinement techniques have shown promise in addressing this issue in Eulerian models (Semakin and Rastigejev, 2016) but are computationally complex. There is a need to understand why persistent free tropospheric plumes are so rapidly dissipated in Eulerian models and how this behavior can be corrected.

A theoretical study by Rastigejev et al. (2010) examined the causes of the fast numerical dissipation of intercontinental plumes in Eulerian models. Numerical diffusion is due to finite differencing of the advection equation on the model grid such that the gradients between grid cells are imperfectly described. Rastigejev et al. (2010) showed that a highly accurate, third-order, finite-volume advection scheme such as is used in GEOS-Chem (Lin and Rood, 1996) successfully preserves plume structures for over 10 days in a uniform flow, but it fails rapidly when real-world divergence is applied. Flow divergence acts to filament, stretch, and thin the plume until it is resolved by only a few grid cells. At that point the gradient can no longer be represented with a high-order scheme; the scheme collapses to first order, resulting in very fast numerical dissipation. Increasing grid resolution only delays the onset of this effect. Rastigejev et al. (2010) presented a theoretical argument that the plume dissipation rate in a stretched flow should be set by the Lyapunov exponent λ=∂u/∂x of the flow, defined as the exponential rate at which adjacent trajectories (aligned with a flow of wind speed u in direction x) diverge from each other. Increasing the grid resolution Δx of the model would then slow down the dissipation rate only as Δx0.5, rather than Δx3 as might be expected from a third-order advection scheme. Increasing grid resolution is computationally expensive, and the analysis of Rastigeyev et al. (2010) implies that it offers only marginal improvement for resolving plumes. Data assimilation in meteorological models, done to enable the simulation of particular events, may exacerbate flow divergence and hence numerical diffusion (Stohl et al., 2004).

In this paper, we examine whether the theory of Rastigejev et al. (2010) can explain the fast numerical decay of free tropospheric plumes in Eulerian models, and what the implications are for curing this problem through increasing grid resolution. We use for this purpose global 2-D (horizontal) and 3-D versions of GEOS-Chem to simulate atmospheric flow at horizontal grid resolutions ranging from 0.25∘ × 0.3125∘ (∼ 25 × 30 km2) to 4∘ × 5∘ (∼ 400 × 500 km2) and with a vertical resolution of ∼ 0.5 km in the free troposphere. We quantify the decay rate for plumes originating from different locations around the world, relate this decay rate to flow stretching, and conclude as to the potential to preserve the plumes through the use of improved grids.

The theory presented by Rastigejev et al. (2010) for numerical diffusion of stretched plumes begins with the advection equation (Eq. 1) ∂n∂t+∇⋅nu=0, where n is the number density of an inert chemical (“tracer”) and u is the wind vector. Rastigejev et al. (2010) expressed this equation in its advective form (Eq. 2) and included a numerical diffusion term with diffusivity D (cm2 s-1): ∂C∂t+u⋅∇C=D∇2C. Here C=n/na is the volume mixing ratio (VMR) and na is the number density of air. This form explicitly accounts for the effect that numerical diffusion has on the modeled flow. Without this numerical diffusion, the advection equation would strictly conserve the VMR. When numerical diffusion is included, the VMR decays over time as the plume dissipates.

Let us consider now the conceptual picture of a model plume with uniform VMR diluting by numerical diffusion into a background atmosphere with a VMR of zero. The plume has surface area S and volume V. Mass balance for the plume is given by VdCdt=-DSk⋅∇C, where k is a unit vector normal to the plume surface. Rastigejev et al. (2010) defined a characteristic length rb for decay across the edge of the plume so that ∇C ∼ C/rb. They further defined a characteristic width of the plume as W=V/S. Thus Eq. (3) becomes dCdt=-DrbWC.

This implies an exponential decay in C, such that Ct+Δt=Ctexp⁡-αΔt, where the decay constant α is given by α=DrbW and Δt is some time interval.

The decay rate of the plume is proportional to the numerical diffusivity D, which is dictated by the order f of accuracy of the numerical advection scheme D ∼ Δxf. However, the decay rate also depends on rb. In a divergent flow, stretching of an initially broad plume (W ≫ rb) causes rb to decrease and hence the decay rate to increase. This stretching can be represented by the Lyapunov exponent, defined as λ=-∂v∂y=∂u∂x, where (u,v) are the (x,y) components of the velocity and x is taken as the direction of stretching. The Lyapunov exponent defines the exponential rate constant at which initially adjacent trajectories diverge.

Stretching in a divergent flow thins the plume, while numerical diffusion thickens it. Under constant divergence (constant λ), an equilibrium size for rb is reached when these two processes proceed at the same rate. Since the rate constant for diffusion is ∼D/rb2, and the rate constant for stretching is λ, equilibrium is reached when Drb2=λ.

Replacing into Eq. (6), we find α=DλW.

Thus the rate of decay in a stretched flow is less sensitive to D than expected. Furthermore, if the plume has stretched to be only a few grid boxes thick, then the gradient across the plume boundary cannot be resolved by a high-order scheme anymore and any numerical advection scheme collapses to first order (Godunov, 1959). Under these conditions D ∼ Δx and thus α ∼ Δx0.5; the decay rate improves only as the square root of the grid resolution.

We also see from Eq. (9) that the decay rate increases with the rate of stretching as measured by λ. Regions of divergent flow are expected to experience faster plume decay. Eventually, for a fully stretched plume we have W = rb. Under these conditions, Eqs. (8) and (9) yield α=λ and the decay rate is independent of the grid resolution – a remarkable result.

The Rastigejev et al. (2010) theory thus paints the following picture for the model decay of a free tropospheric plume in a divergent flow (as is realistically found in the atmosphere) and its dependence on grid resolution Δx. A plume that is initially well resolved on the model grid will decay with a rate constant proportional to Δxf, where f is the order of accuracy of the numerical advection scheme. As the plume stretches, shears, and filaments, it becomes poorly resolved on the model grid; at that point the decay rate becomes proportional to (λΔx)0.5, i.e. only weakly responsive to increasing grid resolution and dependent on the stretching rate. In fact, increasing grid resolution may increase the stretching of the flow by introducing additional convergence–divergence zones that are averaged out at coarser resolutions. Under those conditions Rastigejev et al. (2010) find that the decay rate may be proportional to Δx0.25, an even weaker grid resolution dependence. Eventually, the filamented plume decays with a rate constant defined by λ and at that point very fast dissipation takes place that is resolution independent. This theory, if correct, has major implications for understanding the decay of free tropospheric plumes in Eulerian models and the value of increasing grid resolution. In the following we test the theory using global simulations in actual atmospheric flow with the GEOS-Chem CTM.

We simulate transport of free tropospheric plumes in v11-01e of the global Eulerian GEOS-Chem CTM originally described by Bey et al. (2001). The model is driven by winds and other meteorological data archived every 3 h from the Data Assimilation System of the NASA Goddard Earth Observing System (GEOS-5) with 0.25∘ × 0.3125∘ horizontal resolution on 72 vertical levels. The vertical grid resolution in the free troposphere between 4 and 8 km altitude is about 0.6 km. We apply the model to an inert chemical tracer with only advection enabled. Subgrid transport processes, including convection and boundary-layer mixing, are disabled. Thus the model only solves for advection (Eq. 1), using the 3 h GEOS-5 forward processing (FP) archive of mean horizontal winds and instantaneous surface pressure. Horizontal winds are adjusted with a “pressure fixer” (Horowitz et al., 2003) to ensure consistency with the 3 h pressure change. Vertical winds are derived from divergence of the horizontal winds and the change in surface pressure.

Horizontal advection is calculated using the flux-form semi-Lagrangian (FFSL)-3 finite volume scheme developed by Lin and Rood (1996) and commonly called “tpcore”. This scheme uses the monotonic piecewise parabolic method (PPM) when the Courant–Friedrichs–Lewy number (CFL) is less than or equal to one, and a semi-Lagrangian method for CFL > 1. A semi-monotonic PPM is used in the vertical direction with the enforcement of Hyunh's second monotonicity constraint. The FFSL-3 scheme is formally third-order accurate in space, such that increasing the grid resolution Δx by a factor of 2 should reduce numerical errors by a factor of 8.

We conduct 2-D (horizontal) and 3-D simulations at five different horizontal resolutions: 0.25∘ × 0.3125∘ (native), 0.5∘ × 0.625∘, 1∘ × 1.25∘, 2∘ × 2.5∘, and 4∘ × 5∘. Two-dimensional simulations allow an analysis of the effect of horizontal resolution over a factor of 16 (from 0.25∘ × 0.3125∘ to 4∘ × 5∘) to test the theoretical dependences of plume dissipation on grid resolution (Sect. 2). We cannot carry out a similar analysis in the vertical because the ∼ 0.5 km native vertical resolution of the GEOS-5 data in the free troposphere is too coarse. However, we will comment on the effect of vertical resolution, drawing on the results of sensitivity to horizontal resolution. In all cases, winds and pressures are retrieved from the GEOS-5 FP archive at the 0.25∘ × 0.3125∘ native resolution and subsequently averaged spatially to drive coarser-resolution simulations as is routinely done in GEOS-Chem (Bey et al., 2001; Philip et al., 2016). A dynamical timestep of 5, 5, 10, 15, and 30 min is used for each resolution. Decreasing the timestep to 5 min for all simulations had no significant effect. All concentrations shown and discussed are based on instantaneous VMRs stored at one hour intervals.

Two-dimensional simulations are performed by taking the pressure-weighted average of the wind velocity in each atmospheric column and setting the surface pressure tendency to zero. Although clearly idealized, there is some realism to the 2-D simulations in that free tropospheric layers are vertically stratified and most of the shearing and dissipation can be expected to take place in the horizontal. Most relevantly, the 2-D simulations allow us to test the theory of Rastigejev et al. (2010) for the sensitivity of plume dissipation to grid resolution.

We conduct simulations of the first 9 days of July 2015 for plumes initialized in different locations around the world with a homogeneous unit mixing ratio over a cuboid 12∘ in latitude by 15∘ in longitude and zero outside. This size is chosen so that the initial plume is coarsely resolved at 4∘ × 5∘ but finely resolved at 0.25∘ × 0.3125∘. Ninety non-interacting plumes are initialized over the global domain (Fig. 1) to examine how location affects numerical diffusion. In our 2-D simulations, only horizontal advection is enabled, and the model domain is restricted to a single vertical layer. In our 3-D simulations, vertical advection is enabled, and each plume is initialized at 3.9 km pressure altitude over a single GEOS-5 model level that is 470 m thick.

We now turn to 3-D simulations for a more practical evaluation of the gains that could be made from increasing model resolution. An important distinction here is that we cannot explore a wide range of grid resolutions in the vertical; the ∼ 0.5 km native vertical resolution of the GEOS-5 data in the free troposphere is comparable to the observed thickness of free tropospheric plumes (Thouret et al., 2000). Such coarse resolution in the free troposphere seems typical of the current generation of models, which have emphasized improving horizontal resolution more than vertical resolution. For example, the ERA-Interim reanalysis, produced by the European Centre for Medium-range Weather Forecasts (ECMWF), has a similar mean vertical resolution of 570 m in the free troposphere (Dee et al., 2011).

Global mean plume decay rates and plume sizes in 3-D are shown in Fig. 6 as a function of plume aging times and compared to the 2-D cases discussed previously. In the 2-D simulations, each doubling of resolution yields a 10–20 % improvement in the final maximum VMR after 9 days, up to a value of 89 % at a resolution of 0.25∘ × 0.3125∘. The plume size improvement with increasing resolution is smaller but equally consistent. After 9 days, the plume is double its initial size at 0.25∘ × 0.3125∘. In 3-D the numerical diffusion is considerably larger. At the 0.25∘ × 0.3125∘ grid resolution, the maximum VMR after 9 days drops to 13 % of its original value, and the plume size increases by a factor of 6 from its original value. The reason is that the ability to preserve the plume is limited by numerical diffusion in the vertical, where the plume is initially poorly resolved in all cases because of the low native vertical resolution in the GEOS-5 fields.

There is also a counterproductive aspect to increasing horizontal resolution in a 3-D simulation. As the horizontal resolution is increased, fine-scale vertical eddies are resolved that increase the vertical stretching of the plume, compromising the advantages gained from the slower horizontal diffusion. This is highlighted by the negligible improvement in plume size after 9 days between 3-D simulations at grid resolutions of 0.5∘ × 0.625∘ and 0.25∘ × 0.3125∘. The increased vertical diffusion almost completely offsets the improvement provided by reducing spurious horizontal diffusion.

Figure 7 summarizes the differences between the 2-D and 3-D results in the rate of plume decay as a function of grid resolution, latitude, and the flow divergence as measured by the Lyapunov exponent. The rate of improvement of the solution as the grid resolution increases is indicated in the figure by the vertical separation of points along each line relative to the maximum value. In 2-D, increasing resolution yields consistent benefits, such that simulations at or finer than 1∘ × 1.25∘ achieve near-zero diffusion in the tropics. In 3-D, high rates of flow stretching remain correlated with higher rates of diffusion, but increasing the horizontal resolution yields a smaller relative decrease in the rate of diffusion and the overall rate of convergence is of the order of Δx0.25 or worse. We find that extratropical regions still consistently experience greater rates of numerical diffusion. However, even in the tropics where flow stretching is slow and 2-D simulations performed well, vertical diffusion due to poor vertical resolution provides a lower bound of 3×10-6 s-1 for α, corresponding to a decay timescale of 3 days. This shows that the ability of current global Eulerian models to resolve free tropospheric plumes is limited by the vertical grid resolution. In purely 2-D simulations, increasing horizontal resolution to 1∘ × 1.25∘ was sufficient to reduce α to effectively zero in tropical plumes, even after 10 days of transport. Assuming isotropy in requirements between the horizontal resolution (which we were able to investigate in detail) and the vertical resolution, we suggest that a 100 m vertical resolution is required to preserve the transport of free tropospheric plumes. This needs further investigation with a suitable meteorological model. Increasing the vertical grid resolution would likely also improve the ability of global models to resolve other processes such as cloud–radiation interactions and transport of water vapor (Lane et al., 2000; Tompkins and Emmanuel, 2000).

We examined why global models are unable to simulate the intercontinental-scale transport of quasi-horizontal plumes in the free troposphere, dissipating them in a few days instead of preserving their coherence. Our focus was to test theoretical results by Rastigejev et al. (2010) that this dissipation is due to fast numerical diffusion in the divergent flow typical of the free troposphere. Divergence (shear, stretching) causes the plume to filament rapidly to the point when it is not properly resolved on the model grid. At that point, fast dissipation takes place in the model regardless of the accuracy of the advection scheme and only weakly dependent on grid resolution.

We conducted a large worldwide ensemble of simulations of free tropospheric plumes with the GEOS-Chem chemical transport model driven by NASA GEOS-5 assimilated meteorological data. The simulations used horizontal resolutions ranging from 0.25∘ × 0.3125∘ (native GEOS-5) to 4∘ × 5∘, including only advection (no subgrid turbulence or chemistry). Restriction to advection allowed us to focus on numerical diffusion – in a purely advective problem the plume should not dilute, even in a divergent flow. We diagnosed plume decay caused by numerical diffusion in the model by the decrease in maximum volume mixing ratio in the plume (decay rate constant α) and by the increase in plume size. Native vertical resolution in GEOS-5 (and other current meteorological models) is ∼ 0.5 km in the free troposphere, too coarse to adequately resolve vertical gradients in plumes (typically ∼ 1 km thick). Thus we conducted both 2-D (horizontal) and 3-D simulations. Restriction to 2-D allowed us to investigate in detail the sensitivity to grid resolution for initially well-resolved plumes (12∘ × 15∘). Extension to 3-D allowed us to examine numerical diffusion in realistic model situations.

We find that extratropical plumes decay much faster than tropical plumes and that this can be explained by stronger flow divergence measured by the Lyapunov exponent of the flow (λ). Under strongly divergent flow typical of the extratropics, the rate of plume decay varies linearly with λ and improves only as the square root of the grid resolution Δx. We find that the sensitivity of the plume decay to grid resolution decreases as the plume ages, initially improving as Δx3 (the order of accuracy of the GEOS-Chem advection scheme) and eventually decaying after a few days to Δx2 in the tropics and Δx0-1 in the extratropics.

Three-dimensional plume decay in our simulations is much faster than in 2-D and consistent with the general inability of models to preserve the coherence of free tropospheric plumes. The plume decay rate in 3-D still depends on horizontal flow divergence, but the sensitivity to horizontal grid resolution is weaker and the decay is instead limited by the coarse vertical resolution. Vertical numerical diffusion is very fast and is amplified at finer horizontal resolution by vertical eddies that would be smoothed out at coarser horizontal resolution. Even tropical plumes decay with a time constant of about 3 days.

Our work suggests that increasing the vertical grid resolution in the free troposphere is essential for models to resolve the intercontinental-scale transport of chemical plumes. Although further testing is required to quantify this requirement, extrapolation of our findings with respect to horizontal resolution suggests that a vertical resolution of ∼ 100 m is necessary. Increasing horizontal resolution beyond 1∘ is futile. More advanced solutions might involve adaptive grids designed to resolve local conditions of large chemical gradients and large divergence, or embedding Lagrangian plumes in the global Eulerian modeling framework.

The GEOS-Chem model code used to derive these results is available from http://www.geos-chem.org. The GEOS-5 FP meteorological data used in this study have been provided by the Global Modeling and Assimilation Office (GMAO) at NASA Goddard Space Flight Center.