Chemical plumes in the free troposphere can preserve their identity for more
than a week as they are transported on intercontinental scales. Current
global models cannot reproduce this transport. The plumes dilute far too
rapidly due to numerical diffusion in sheared flow. We show how model
accuracy can be limited by either horizontal resolution (

Global transport of pollution mainly takes place in the free troposphere
where winds are strong and pollutant lifetimes are long. The free troposphere
extends from the top of the planetary boundary layer (PBL, typically 2 km
altitude) up to the tropopause. It is a prevailingly stable environment with
strong wind shear. Much of pollution transport in the free troposphere takes
place as plumes, typically

Preserving the structure of chemical plumes during global-scale transport is important for representing non-linear chemical and aerosol processes (Wild and Prather, 2006) and for quantifying intercontinental influences on surface air (Lin et al., 2012; Zhang et al., 2014). For example, models are unable to capture the plumes of Asian ozone pollution frequently observed at 2–5 km altitude over California (Hudman et al., 2004; Nowak et al., 2004). Air quality agencies in California have claimed that they cannot meet the current surface ozone standard because of this Asian pollution influence (Neuman et al., 2012). Models find Asian pollution influence in surface air over California to be only a few ppb (Goldstein et al., 2004; Zhang et al., 2008), but since they cannot resolve the structure of Asian pollution plumes crossing the Pacific they have little credibility.

General circulation models (GCMs) used for global simulations of atmospheric
dynamics including meteorological data assimilation have increased their
resolutions 1000-fold over the past 50 years, buoyed by the growth of
computing power (Balaji, 2015). Increasing horizontal resolution has been
privileged, and attention to vertical resolution has mainly focused on the
PBL. For example, assimilated meteorological data produced operationally by
the NASA Goddard Earth Observing System (GEOS) started in the 1990s with
2

There are important reasons why horizontal resolution is a priority in GCMs,
as reviewed by Haarsma et al. (2016). Increasing horizontal resolution
improves the simulation of large-scale features such as the El
Niño–Southern Oscillation (ENSO), as well as small-scale features such
as tropical cyclones. It has been argued that increasing vertical resolution
should follow suit. Pecnick and Keyser (1989) recommend an optimal
relationship between horizontal and vertical grid spacing to resolve fronts:

Preserving chemical plume gradients in the free troposphere may have its own
resolution requirements. In idealized tests by Kent et al. (2012), where
plumes were advected by a solid-body rotation flow coupled with vertical
oscillation, doubling the vertical resolution brought down the numerical
diffusion error by more than half. Numerical diffusion of plumes is
considerably more severe in realistic sheared/stretched atmospheric flows
(Rastigejev et al., 2010). Eastham and Jacob (2017) used GEOS-FP
meteorological data with 0.25

Solution of the advection equation in models should conserve the mixing ratio
for the transported species (mass of species per unit mass of air) but
should also account for the filamentation of plumes down to the millimeter
Kolmogorov scale where molecular diffusion takes over to complete the
dissipation process. An Eulerian model computing advection with no error
would underestimate the actual diffusion process if it did not account for
subgrid filamentation, which is often parameterized by adding a turbulent
diffusion term to the advection equation. D'Isidoro et al. (2010) examined
the relative importance of numerical and actual (physical) turbulent
horizontal diffusion in air quality models and found that numerical diffusion
dominates for grid cell sizes larger than

Increasing free tropospheric vertical resolution in GCMs would also have meteorological implications for the transport of water vapor, similar to chemical plumes (Tompkins and Emanuel, 2000; Pope et al., 2001). Water vapor in the free troposphere is layered in the same way as other chemical species (Newell et al., 1999; Thouret et al., 2000). An early intercomparison of GCMs found that the radiative effect of water vapor is relatively insensitive to model vertical resolution (Ingram, 2002), which might explain the lack of attention to this issue. However, all GCMs in that intercomparison had coarse resolution that would make them inadequate for addressing the problem properly.

Here, we use the GFDL-FV3 dynamical core as a computationally flexible framework to explore the horizontal and vertical resolution requirements for free tropospheric plume transport. The dynamical core solves the atmospheric dynamics equations with no complications from physical parameterizations such as boundary layer mixing or deep convection. In a full GCM, one would need to account for the vertical resolution dependence of physical parameterization schemes (Lane et al., 2000; Kent et al., 2012). In a dry dynamical core, we are free to choose any horizontal and vertical resolutions to solve the dynamics equations. A realistic sheared/stretched atmospheric flow can be simulated in a dry dynamical core by triggering baroclinic instability (Jablonowski and Williamson, 2006).

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:

Using a 3-D first-order upwind scheme with no cross terms, applied to a grid
cell (

The time derivative (

To avoid being limited by one dimension, the horizontal and vertical
diffusion terms in Eq. (12) should have similar magnitude:

In practice, the trade-off between horizontal resolution (

Here, we consider the general problem of minimizing the numerical diffusion
for a given allocation of computational resources and with a trade-off
parameter

For a given amount of computing

Optimal combination of horizontal and vertical grid resolutions
(

We conduct an 8-day simulation of a chemically inert plume in the GFDL-FV3
(

Numerical diffusion takes place in FV3 during Eulerian horizontal advection (due to finite differencing of the spatial derivatives) and during vertical remapping of the Lagrangian surfaces to the model grid (due to interpolation error). Vertical remapping can use a larger time step than horizontal advection, but the interpolation scheme can be very diffusive if monotonicity is required. Our own comparisons of the vertically Lagrangian scheme to a high-order Eulerian scheme show that they have similar vertical diffusion (Appendix B).

Atmospheric flow generated by the FV3 dynamical core in a baroclinic
instability test, 16 days after initialization of the test and 8 days after
the release of the chemical plume. Shown are surface pressures, 700 hPa flow
streamlines and Lyapunov exponents

An effective way to emulate realistic turbulent atmospheric flows in a dynamical core is the baroclinic instability test, originally developed by Jablonowski and Williamson (2006) as a dynamical core benchmark and subsequently used in tracer transport simulations (Jablonowski et al., 2008; Ullrich et al., 2016). Baroclinic instability is the main mechanism for cyclogenesis in midlatitudes. Instability can be triggered by applying a small perturbation to an initial reference state in geostrophic and hydrostatic balance. Starting from the initial perturbation, the baroclinic wave typically becomes observable around model day 4 and generates strong cyclones by day 8 (Jablonowski and Williamson, 2006).

The 8-day simulation of plume transport in the FV3 dynamical core at
C384L160 resolution (

Plume dilution due to numerical diffusion at different model grid
resolutions. The plume is released in the free troposphere at northern
midlatitudes with an initial mixing ratio of unity. Plume dilution is
measured by the decrease in the maximum mixing ratio as a function of time.
Model horizontal resolution is defined by a cubed-sphere grid ranging from
C48 (

Here, we first run the baroclinic instability simulation for 8 days so that cyclones become intense enough for realistic flow shearing/stretching. We then initialize an inert tracer plume with uniform mixing ratio at the location where flow stretching is the strongest. This initial plume extends horizontally and vertically over a number of grid cells depending on the grid resolution, as detailed below. We continue the simulation for 8 days and diagnose the transport of the plume. Tracer transport involves solely advection. There is no subgrid turbulent diffusion or convection.

We conduct simulations at horizontal cubed-sphere resolutions ranging from
C48 (

The time step for the Lagrangian remapping is 30 min for the lowest horizontal resolution case (C48) and is reduced proportionally at higher horizontal resolutions. Within this time step are eight sub-steps for horizontal dynamics calculations. The frequency of horizontal tracer advection calculations is determined on the fly based on the CFL criterion.

The plume is initialized with a uniform mixing ratio normalized to unity over
a horizontal area corresponding to 6

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

Figure 3 illustrates the evolution of the plume over the 8-day period in the
C384L160 case (

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).

Vertical profile of the maximum mixing ratio for each model vertical level after 8 days of simulation, at low model resolution (C48L20), high model resolution (C384L160) and intermediate cases where only horizontal resolution or vertical resolution is increased from the low-resolution case (C384L20, C48L160).

Same as Fig. 4 but with entropy instead of maximum mixing ratio as a diagnostic for numerical diffusion. The entropy is initialized on day 0 with a value of 1. Pure advection conserves entropy but diffusion increases it.

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.

Optimal combination of horizontal and vertical grid resolutions
(

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

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 (

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

As in Sect. 2.3, the optimal ratio (

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 (

The estimated (

Our recommendation to increase vertical resolution in the free troposphere is
specific to global models, and our emphasis on an optimal

Current global models are unable to simulate the observed persistence of
chemical plumes in the free troposphere on intercontinental scales. The
plumes dilute too rapidly due to numerical diffusion in sheared flow. This is
a major problem for global simulations of atmospheric composition and for
diagnosing intercontinental pollution influences on surface air quality. We
investigated how this problem could be solved through increasing horizontal
and vertical grid resolutions, and in what optimal combination. We used for
this purpose the GFDL-FV3 global dynamical core to perform plume transport
simulations, driven by flow with realistic shear as generated from a
baroclinic instability test. The flexibility of this dynamical core allowed
us to conduct simulations over cubed-sphere horizontal resolutions ranging
from C48 (

We began with a theoretical analysis of the plume advection problem to show
that numerical diffusion may be limited by either horizontal grid resolution
(

We applied the FV3 dynamical core to simulate the transport over 8 days of a
chemically inert free tropospheric plume at northern midlatitudes. Transport
in the dynamical core is solely by advection, and exact solution should
therefore preserve the initial mixing ratio in the plume. We diagnosed
numerical diffusion over the 8-day simulation by the decay of the maximum
mixing ratio and the increase in entropy. We demonstrated how improvements in
preserving the plume during transport can be limited by either horizontal or
vertical resolution, in a manner consistent with the theoretical analysis.
Our highest-resolution simulation (C384L160) preserved the maximum mixing
ratio in the plume to within 35 % after 8 days in strongly sheared flow,
retained the vertical structure of the plume and led to much larger local
intercontinental impacts on surface air than the coarser-resolution
simulations. The required vertical resolution in the free troposphere is
6 hPa (

There are strong reasons for GCMs to focus on increasing horizontal resolution, as this allows better representation of cyclogenesis and other aspects of the meteorological simulation. However, simulations of global chemical transport require higher vertical resolution in the free troposphere. Considering that the free troposphere accounts for only about a third of all vertical levels in the current generation of models, adding vertical resolution only to that part of the atmosphere would not be expensive. A proper vertical resolution in the free troposphere would also benefit the simulation of water vapor with implications for the radiative budget and for cloud formation. Within the framework of current GCMs, it may be possible to improve chemical transport by conducting offline CTM simulations with high vertical resolution, interpolating the meteorological archive from the parent GCM. The feasibility of such hybrid-resolution simulations has been studied by Methven and Hoskins (1999). Adaptive mesh refinement (AMR) is also a computationally efficient approach to improve plume simulations (Semakin and Rastigejev, 2016), but implementing AMR in existing CTMs requires significant engineering efforts especially on parallelization.

The FV3 source code was obtained from

Following Odman (1997), we start with the 1-D advection equation:

To further eliminate this mixed derivative, we apply the operation (

Here, we use the GEOS-Chem CTM to compare vertical numerical diffusion in
FV3's advection scheme to that in TPCORE, a 3-D Eulerian advection scheme
(Lin and Rood, 1996). TPCORE is the standard advection scheme in the
“classic” version of the GEOS-Chem CTM (Bey et al., 2001), while FV3 is
used in the high-performance version of GEOS-Chem (GCHP; Long et al., 2017;
Yu et al., 2018). Unlike FV3, TPCORE uses a regular latitude–longitude
geometry and a vertically Eulerian discretization. When the CFL number is
less than 1, the horizontal tracer transport uses the monotonic PPM as in
FV3; otherwise, a semi-Lagrangian method is used. The vertical tracer
transport uses PPM with Huynh's second constraint (Huynh, 1997). We use a C48
horizontal resolution for GEOS-Chem with FV3 and a corresponding
2

Comparing vertical diffusion in the GEOS-Chem CTM using either the
TPCORE Eulerian advection scheme

We use the idealized Hadley-like circulation test in the 2012 Dynamical Core
Model Intercomparison Project (Kent et al., 2014) to benchmark the vertical
diffusion in both models. The simulation is illustrated in Fig. B1. The
initial tracer layer (Fig. B1, left panel) is advected in the vertical by a
Hadley-like flow (Fig. B1, middle panels) and then gets reverted to the
original state by a reverse flow (Fig. B1, right panels). The true solution
at the final state should be the same as the initial condition, and the
deviation from the initial condition is due to numerical error. The error
norm can be calculated by

There are many equivalences between remapping schemes and advection schemes. For example, both higher-order remapping and higher-order advection schemes are not monotonic by default and need additional limiters or constraints to prevent overshoots. If gradients are sharp, monotonic limiters will degrade higher-order schemes to first order, at the expense of making the schemes more diffusive. Increasing the grid resolution will make both remapping and advection schemes more accurate and less diffusive. Due to these similarities between advection and remapping, our Eulerian-based theoretical analysis in Sect. 2 should also apply to vertically Lagrangian schemes.

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

The authors would like to thank Lucas Harris, Xi Chen and Shian-Jiann Lin at GFDL for general guidance on the GFDL-FV3 model. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. This work was supported by the NASA Modeling, Analysis, and Prediction (MAP) Program.Edited by: Anja Schmidt Reviewed by: three anonymous referees