Quasi-horizontal chemical plumes in the free troposphere can preserve their
concentrated structure for over a week, enabling transport on
intercontinental scales with important environmental impacts. Global Eulerian
chemical transport models (CTMs) fail to preserve these plumes due to fast
numerical dissipation. We examine the causes of this dissipation and how it
can be cured. Goddard Earth Observing
System (GEOS-5) meteorological data at
0.25

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

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

The theory presented by Rastigejev et al. (2010) for numerical diffusion of
stretched plumes begins with the advection equation (Eq. 1)

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

This implies an exponential decay in

The decay rate of the plume is proportional to the numerical diffusivity

Stretching in a divergent flow thins the plume, while numerical diffusion
thickens it. Under constant divergence (constant

Replacing into Eq. (6), we find

Thus the rate of decay in a stretched flow is less sensitive to

We also see from Eq. (9) that the decay rate increases with the rate of
stretching as measured by

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

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

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

We conduct 2-D (horizontal) and 3-D simulations at five different horizontal
resolutions: 0.25

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

Plume initialization locations overlaid on the mean 2-D Lyapunov
exponent

Exact solution of the advection equation translates mixing ratios downwind without altering them. In other words, initial mixing ratios in a plume remain constant as the plume is advected downwind. Any plume decay in our advection-only simulation must be the result of numerical diffusion. In the real atmosphere, plumes decay by molecular diffusion that operates on millimeter scales and is the end result of the turbulent eddy cascade that filaments the plume into finer and finer strands. This subgrid turbulence is particularly fast for vertical mixing in the boundary layer and is typically parameterized in models with a turbulent diffusion scheme. It is usually ignored in the horizontal direction or in the free troposphere, under the assumption that spurious numerical diffusion effectively carries out the mixing.

Numerical diffusion arises from finite differencing over grid cells when solving the advection equation. Odd-order schemes such as the PPM tend to introduce diffusion, artificially smoothing the solution in areas with sharp concentration gradients. Even-order schemes instead tend to be dispersive, producing spurious oscillations, and this artifact is even less desirable than numerical diffusion. Higher-order schemes also tend to produce spurious oscillations when there are discontinuities in the concentration field (Godunov, 1959; Brasseur and Jacob, 2017). To get around this, modern advection schemes such as FFSL-3 employ flux limiters that locally reduce the scheme to first order in the vicinity of discontinuities. This prevents spurious oscillations at the cost of increasing numerical diffusion.

Numerical diffusion in GEOS-Chem is illustrated in Fig. 2 with an example of
a 2-D plume at 1

Numerical diffusion of a 2-D (horizontal), inert plume in GEOS-Chem
for a 9-day simulation at 1

An alternate metric of numerical diffusion is the size of the plume. The
thick red contour in Fig. 2 shows the minimum area containing 90 % of the
total mass of tracer in the plume. As the simulation progresses, diffusion of
the plume increases this area. We define the square root of this area as the
characteristic size of the plume normalized by the value at plume
initialization. In 3-D, the plume size is taken as the cube root of the total
volume occupied by 90 % of the tracer mass after accounting for
differences in air density. Plume size is a more sensitive indicator of plume
diffusion, as shown in the bottom panel of Fig. 2, because it accounts for
the fraying at the plume edges and is a smoother function than

Plume stretching can be quantified by the local Lyapunov exponent of the
flow, as defined in Eq. (7), for horizontal stretching. Rastigejev et
al. (2010) calculated this Lyapunov exponent using a level-set approach
(Leung, 2011). Here we calculate an approximately equivalent quantity. If

Rearranging Eq. (10) gives

This can be directly applied in a Eulerian model framework, acknowledging
the separate treatment of winds in each dimension. For taxicab geometry, as
in the orthogonal latitude–longitude discretization, we measure the
separation

At each grid cell, the above equation is applied to yield an
estimator for the rate of horizontal flow stretching. Figure 1
displays the mean Lyapunov exponents at
0.25

Figure 3 shows the evolution of the plume peak VMR and decay constant in the
2-D simulations as a function of latitude for different grid resolutions.
The rate of plume decay increases with latitude. A tropical plume with
initial area of 12

Evolution with time of the maximum VMR and the decay constant

Rastigejev et al. (2010) presented a single example of a Chinese plume
transported over the Pacific in 2-D flow as an illustration of their theory.
Starting from the same initial 12

Following the theory of Rastigejev et al. (2010) as summarized in Sect. 2, we
examined the relationship between plume decay (as represented by the plume
decay constant

Dependence of plume decay on the stretching of the atmospheric flow
as measured by the Lyapunov exponent. The figure shows the average plume
decay constant

Results in Fig. 4 show in general a strong correlation between

Sensitivity of plume decay to grid resolution and its dependence on
plume age. At each resolution, the average value of the plume decay constant

The difference in the slope of the regression lines gives an approximate
measure of the improvement gained by increasing resolution by a factor of 4.
Increasing resolution yields a “delay” in terms of the minimum

The right-hand panel of Fig. 4 shows the response of the plume size to
diffusion after 200 h. At 4

The PPM advection scheme used in GEOS-Chem is third-order accurate, meaning
that the accuracy should improve as

Figure 5 shows the average improvement (reduction) in

Maximum VMR (top panel) and plume size (bottom panel) as a function of plume aging time. Values are global averages for the ensemble of 90 plumes in Fig. 1 and are shown for 2-D and 3-D simulations at the different horizontal grid resolutions indicated in the legend.

Outside the tropics where flow divergence is greater, the effective order
of accuracy is smaller and shows greater variability between grid
resolutions with plume age. It starts second-order (

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

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

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

Variation of the plume decay constant (

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

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

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 (

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

The GEOS-Chem model code used to derive these results is available from

Sebastian D. Eastham and Daniel J. Jacob designed the experiments. Sebastian D. Eastham developed model code and performed all experiments. Sebastian D. Eastham prepared the manuscript in collaboration with Daniel J. Jacob.

The authors declare that they have no conflict of interest.

The authors would like to thank Christopher Holmes for discussion and for advice on producing divergence-free, 2-D atmospheric flows. We would also like to thank Meiyun Lin for discussion on the significance of numerical diffusion in modern GCMs. This research was supported by the NOAA Climate and Global Change Postdoctoral Fellowship Program, administered by UCAR's Visiting Scientist Programs. Sebastian D. Eastham was also supported by a Harvard University Center for the Environment (HUCE) Fellowship. Daniel J. Jacob was supported by the NASA Earth Sciences Division. Edited by: F. Fierli Reviewed by: A. Stohl and one anonymous referee