The potential of the antidiffusive transport scheme proposed by
for resolving vertical transport in chemistry-transport
models is investigated in an idealized framework with very encouraging
results. We show that, compared to classical higher-order schemes, the
scheme reduces numerical diffusion and improves accuracy
in idealized cases that are typical of atmospheric transport of tracers in
chemistry-transport models. The increase in accuracy and the reduction in
diffusion are substantial when, and only when, vertical resolution is
insufficient to properly resolve vertical gradients, which is very frequent in
chemistry-transport models. Therefore, we think that this scheme is an
extremely promising solution for reducing numerical diffusion in
chemistry-transport models.
Introduction
Reducing numerical errors in chemistry-transport models (CTMs) is a necessary task for those wishing to improve these models' performance. Among the well-known errors in
Eulerian CTMs, excessive numerical diffusion in all directions is a well-known
drawback of many such models, and in the past decade, excessive or poorly
represented vertical transport and diffusion has been identified as a major
cause for numerical dispersion in these models
. This excessive vertical diffusion may
have a strong impact on the representation of ground-level ozone concentrations
due to spurious transport of stratospheric ozone into the troposphere
and also hinders the ability of Eulerian CTMs to represent
accurately intercontinental transport of densely polluted plumes such as
volcanic plumes . While CTMs
manage to represent such plumes in terms of general location, they typically
fail to maintain the fine-scale structure of the plumes and tend to dilute
them too much compared to observations .
Among other more model-specific solutions, suggest as the
main paths to solve this problem increasing vertical resolution and improving
the vertical advection scheme, in their case switching from a first-order to a
second-order advection scheme. In the same line,
and have discussed the need for increased vertical
resolution in order to adequately represent long-range advection of chemical
plumes.
In the line of improving the vertical advection scheme,
describe the implementation of the antidiffusive transport
scheme for vertical transport in the CHIMERE chemistry-transport model
() and its application to modeling the 18 March 2012
eruption of Mount Etna (Italy). They show that using this transport scheme
reduces numerical diffusion and permits a better representation of the
volcanic plume after long-range advection, thereby showing that it is possible
to strongly reduce numerical diffusion in CTMs without increasing the number
of vertical levels. However, that study, set in the framework of a fully fledged
chemistry-transport model fed by real-life atmospheric fields, makes it
difficult to fully disentangle the effect of the transport scheme itself from
other effects such as uncertainties in emission fluxes and mass–wind
inconsistencies in the forcing meteorological fields.
Therefore, the present study aims at answering the questions on the use of
that could not be addressed in the realistic framework of
. For that purpose, we have designed three idealized test
cases that permit comparison of the performance of the scheme
with the classical schemes of and , not
only in terms of diffusion but also in terms of accuracy compared to the exact
solution, which could not be done in . We also examine how
the performance of compares to the two above-cited schemes
and to the order-1 upwind Godunov scheme () when
resolution increases.
Section describes the numerical methods that have
been used, their implementation, the discretization strategies and the cases
that have been designed for the study. Section presents
simulation outputs and the diagnostics that have been designed to compare the
different transport strategies with each other. In Sect. the results are discussed, and Sect. shows our conclusions.
Numerical methods and case description
Continuity equation for the motion of air is as follows:
∂C∂t+∇Φ=0,
where u represents wind speed, C is air concentration (all species
together) in molecules per unit volume and
Φ=Cu
represents the air flux vector.
The continuity equation for species s is as follows:
∂Cs∂t+∇Φs=0,
where Cs is concentration of species s in molecules per unit volume and
Φs=Csu is the flux vector for species s. Equivalently,
Eq. () becomes
∂Cs∂t+∇αsΦ=0,
where αs is the mixing ratio for species s:
αs=CsC.
Chemistry-transport models try to solve Eq. () as
accurately as possible on their discretized grids, while keeping the cost of numerical resolution under control.
1D discretization of advection and advection schemes
In 1D, Eq. () becomes
∂Cs∂t+∂Csu∂x=0.
Here we follow a semi-Lagrangian approach by identifying the air parcels that
have entered cell i through its left boundary, and conversely the air
parcels that have left cell i through its right boundary (we suppose that
the wind is positive). Let Δ-x be so that the Lagrangian trajectory
starting at xi-12-Δ-x at time t passes through
xi-12 at time t+Δt. Δ-x is the distance
traveled by the last air particle entering grid cell i at time t+Δt. Let us define the wind speed representative of facet i-12
between t and t+Δt as u‾i-12=Δ-xΔt. If we define the wind speed representative of right facet
u‾i+12 in a similar way to Δ+xΔt where Δ+x is so that the Lagrangian trajectory starting at
xi+12-Δ+x at time t passes through
xi+12 at time t+Δt, then
Eq. () can be discretized over cell i as follows:
Cs,it+Δt-Cs,itΔt=u‾i-12C‾s,i-12-u‾i+12C‾s,i+12xi+12-xi-12,
where
C‾s,i-12=1Δ-x∫xi-12-Δ-xxi-12Cs(x,t)dx
and
C‾s,i+12=1Δ+x∫xi+12xi+12+Δ+xCs(x,t)dx.
Equation () is verified exactly with no
particular hypothesis on the wind speed u(x,t) nor the concentration field
Cs(x,t). For air concentration, the continuity equation can be discretized
in the same way:
Ci(t+Δt)-Ci(t)Δt=u‾i-12C‾i-12-u‾i+12C‾i+12xi+12-xi-12.
In order to permit monotonicity of the advection scheme in terms of mixing
ratios (i.e., the mixing ratio for species s stays within its initial range),
we reformulate Eq. () by using fluxes and
mixing ratios instead of winds and concentrations by introducing
α‾s,i±12=C‾s,i±12C‾i±12
and
F‾i±12=C‾i±12u‾s,i±12.
With these notations, Eqs. ()
and () become
Cs,i(t+Δt)-Cs,i(t)Δt=F‾i-12α‾s,i-12-F‾i+12α‾s,i+12xi+12-xi-12
and
Ci(t+Δt)-Ci(t)Δt=F‾i-12-F‾i+12xi+12-xi-12.
Equations () and ()
are a flux-form reformulation of semi-Lagrangian
Eqs. ()–().
The form of
Eqs. () and ()
makes it straightforward to verify that if
Eq. () is verified, and if
α‾s,i-12 lies between α‾s,i-1
and α‾s,i (and if α‾s,i+12
lies between α‾s,i and α‾s,i+1) then
the resulting advection scheme guarantees monotonicity of mixing ratios, which
is physically desirable and is the reason why chemistry-transport models
usually resolve advection of trace species using an approach based on
Eq. () rather than a straightforward
resolution of Eq. ().
Regarding chemistry-transport models, in practice, approximated values of
F‾i-12 are inferred from the wind and density fields
provided by the forcing meteorological dataset. Here we will avoid the typical
problem of mass-wind inconsistencies discussed in, for example,
, and , by working with
analytically defined non-divergent mass fluxes, and constant and uniform air
density, so that Eq. () is verified
exactly by construction.
This simplified framework will permit us to focus on the transport scheme of
the chemistry-transport model whose task, in flux-form, is to estimate the
values of α‾s,i±12 that are needed for
numerical resolution of Eq. ().
Advection schemes and tracer flux calculationThe Godunov donor-cell scheme
The most simple way of estimating α‾s,i±12 is
the Godunov donor-cell scheme (adapted from ), simply
evaluating α‾s,i,k+12 as follows:
14α‾s,i+12=αs,iif F‾i+12≥0,15α‾s,i+12=αs,i+1if F‾i+12<0.
This order-1 scheme is cheap, robust, linear, monotonous and mass-conservative
but extremely diffusive. It is therefore important to find more accurate ways
to estimate α‾s,i+12.
The scheme
The second-order slope-limited scheme of brought to our
notations and assuming uniform air concentration yields the following
expression of α‾s,i+12 (for
F‾i+12>0):
α‾s,i+12=αs,i+1-ν2signαs,i+1-αs,i×Min(12αs,i+1-αs,i-1,2αs,i+1-αs,i,162αs,i-αs,i-1),
where ν=Δ+xxi+12-xi-12 is the
Courant number for the donor cell. If ν>1, then more mass leaves the cell
than the mass that was initially present and the Courant–Friedrichs–Lewy
condition is violated, yielding numerical instability. If αs,i is a
local extremum of mixing ratio ((αs,k-αs,k-1)(αs,k+1-αs,k)≤0), no
interpolation is performed and α‾s,i+12=αs,i is imposed: in this case, the scheme falls back to the simple Godunov
donor-cell formulation (Eq. ). This order-2 scheme has been
used for decades in chemistry-transport modeling, being a good tradeoff
between reasonably weak diffusion, at least compared to more simple schemes
such as the Godunov donor-cell scheme, and small computational burden compared
to higher-order schemes such as the piecewise parabolic method
.
The piecewise parabolic method
The piecewise parabolic method (PPM) consists of
performing a parabolic reconstruction of the concentration field inside each
model cell using information from three upwind cells and two downwind cells,
and applying limiters to preserve the scheme's monotonicity and stability. The
detailed procedure is described in the seminal paper. Our
implementation of this method has been adapted from the CASTRO Compressible
Astrophysical Solver . While third-order by design,
application of limiters in the vicinity of extrema introduces first-order
truncation errors in their vicinity so that third-order convergence is not
expected with the PPM scheme as described in . However, this limitation does not prevent the
method giving much better results than simpler order-2
schemes such as , so that the PPM
scheme has been used successfully for a wide range of applications including
meteorological modeling , chemistry-transport modeling
, astrophysics etc.
The scheme
The scheme of is defined by their Eqs. (2) to (4). If
Fi+12>0, these equations brought to the notations of
Eq. (), give
α‾s,i12=αs,k+1-ν2×Max0,Min2ναs,i-αs,i-1αs,i+1-αs,i,21-ν17×αs,i+1-αs,i,
with the same notations as for the scheme (above). As
above, if ((αs,i-αs,i-1)(αs,i+1-αs,i)≤0, no interpolation is performed and the
scheme falls back to the simple Godunov donor-cell formulation
(Eq. ). As stated by its authors, this scheme is
antidiffusive. Unlike other schemes such as the scheme
described above, two unusual choices have been made by the authors in order to
minimize diffusion by the advection scheme:
Their scheme is accurate only to the first order.
The scheme is linearly unstable, but non-linearly stable (their Theorem 1).
The idea of the authors has been to make the interpolated value
α‾s,i+12 as close as possible to the downstream
value (αs,i+1 if the flux is positively oriented). This property is
desirable because it is the key property in order to reduce numerical
diffusion as much as mathematically possible while still maintaining the
scheme stability. The authors present 1D case studies with their scheme
obtaining extremely interesting results: fields that are initially
concentrated on one single cell do not occupy more than three cells even after a
long advection time (their Fig. 2), sharp gradients are very well preserved
(their Fig. 1), and, more unexpectedly due to its antidiffusive character, the
scheme also performs well in maintaining the shape of concentration fields
with an initially smooth concentration gradient. After extensive testing,
these authors also suggest (their Conjecture 1) that convergence of the
simulated values towards exact values occur even if the time step is reduced
before the space step: in simpler terms, this means that the scheme performs
very well even at small CFL values, a property that is not shared by most
advection schemes. Comparison of Eqs. () with () shows that
the numerical cost of the scheme is about the same as the
scheme.
2D discretization of advection and directional splitting
All the simulations in the present study are 2D x–z cases on a domain
discretized over a regular Cartesian mesh. In order to work with the usual
units (concentrations per unit volume and not per unit area), a third,
degenerate space dimension has been introduced, with one grid cell in the
y direction, δy=δx. Zero mass flux is prescribed in the
y direction. Index i is attributed to the x direction, index k to the
z direction, and no index is attributed to the degenerate
y dimension. Eq. () becomes
Cs,i,kt+Δt-Cs,i,ktΔt18=F‾i-12,kα‾s,i-12,k-F‾i+12,kα‾s,i+12,kxi+12-xi-12+F‾i,k-12α‾s,i,k-12-F‾i,k+12α‾s,i,k+12zk+12-zk-12.
Here, F‾i-12,k is the time-averaged mass flux of air
through the left boundary of cell i,k between t and t+Δt, and F‾i+12,k and
F‾i,k±12 have similar definitions. α‾s,i-12,k is the mixing ratio of species s in the air volume entering cell i,k
through its left boundary between t and t+Δt. If V is
the geometric volume containing at time t all the air parcels that are going
to cross the left boundary of cell i,k between t and t+Δt then
α‾s,i-12,k=∫VCsx,z,tdV∫VCx,z,tdV.
From a practical point of view, it is extremely difficult to actually find the
contours of V and to reconstruct and integrate the concentrations of
air and of tracer over this volume. This is why, in practice, Eulerian models
in Cartesian grids tend to split between the two (or three) space
directions. Integrating separately direction x and then direction z over
time Δt (generally called “Lie splitting”) gives order-1 error, and
applying the so-called Strang splitting by integrating
first in the x direction over Δt2, then in the
z direction over Δt, and finally once again in the x direction
over Δt2 reduces the splitting error to order 2.
Tested configurations
Since the present study is aimed at studying vertical transport only, we chose
to test the Godunov, , , and
schemes in the vertical direction with the same transport strategy over the
x axis, namely the PPM scheme. For all simulations,
splitting between the x and z direction has been performed using Strang
splitting as described above, in order to maintain second-order accuracy for
the Van Leer and PPM simulations. While it would have been possible to use
Lie splitting for simulations Upwind and DL99 (for which the vertical
advection scheme is only first-order accurate), the choice of using Strang
splitting for all four simulations guarantees that all simulations are strictly
identical except for their vertical advection scheme. The configurations that
have been tested are summarized in Table .
Summary of the different transport configurations that have been tested.
Domain resolution and size for Cases 1 and 2; set of increasing resolutions used for convergence tests (Cases 3 and 4). For all configurations, nz*Δz=12000m is the domain vertical extension.
nxΔx (m)L=nx×Δx (km)nzΔz (m)DurationCases 1 and 2 8025 0002000245002TCases 3 and 4 (convergence tests) 2050 0001000121000T4025 000100024500T8012 500100048250T1606 250100096125T3203 125100019262.5TTest case definition
We have defined three test cases designed to be representative of long-range
tracer transport situations in the atmosphere. The simulation domain covers
an x–z domain with length L=2000km from west to east and
thickness H=12km, with periodic boundary conditions at the
lateral boundaries and open boundaries at the top and at the bottom of the
simulation domain, with clean air (zero tracer concentration) entering from
these boundaries. We will use T=86400s, the length of a complete
day on Earth, as the timescale for the case studies, along with the corresponding
pulsation ω=2πT. The number density of the carrying fluid
(air) will be assumed uniform. These simplifying assumptions are designed to
ease the formalism and the formulation of exact solutions of the problem. Two
situations of relevance for atmospheric tracer transport have been
represented, along with another numerical experiment designed in order to
investigate the properties of the tested transport configurations in terms of
convergence rate. Case 1, presented in Sect. , aims to represent
the formation of a thin plume from an initially thicker tracer volume through
the action of zonal wind shear, a situation typical of long-range advection of
polluted plumes in the free troposphere. Case 2, presented in
Sect. , represents long-range advection of a thin plume under
the action of a strong zonal wind.
Except for tests of convergence rates for which increasingly fine
discretizations have to be tested, the domain is discretized into 80
evenly spaced cells from west to east (Δx=25km) and 24
evenly spaced cells from bottom to top (Δz=500m).
Case 1: thin layer formation under wind shear
In this case, we consider the evolution of an inert tracer initially distributed as follows:
αt=0,x,z=αm if H2-h1≤z≤H2+h1 and 20L2-δx1≤x≤L2+δx1,21αt=0,x,z=0 otherwise,
with h1=1500m the half-thickness of the initial layer, δx1=25km the half-length of the initial layer, H the height of
domain top (H=12km; see Table ) and L the
length of domain (L=2000km). This describes a uniform plume
initially confined vertically between z=4500m and
z=7500m and horizontally between x=975km and
x=1025km, with an initial (and arbitrary) mixing ratio of
100ppb.
Zonal wind is constant in time, zonally uniform and vertically sheared:
u(x,z,t)=U02zH,
with U0=L2T so that the horizontal motion of fluid at
z=H2 brings it back at its initial position after a time 2T,
which will be the duration of the numerical experiment.
We add a vertical wind defined as follows:
w(x,y,z,t)=w0cosωT.
The vertical wind speed scale is taken as 5×10-2ms-1, a
typical scale for synoptic-scale vertical motion in the troposphere. Since
∂u∂x=∂w∂z=0 and since the
density field is uniform, this mass flux is non-divergent.
This case can describe a plume that is initially vertical, covering a
50 km wide column (two grid cells), uniformly spanned over a
3 km altitude range (4500 to 7500 m, corresponding to six grid cells;
see Table ). This initially thick vertical column later
evolves under the action of wind shear into a thin layer.
Direct integration of Eq. () and then of
Eq. () give access immediately to the position of a
particle initially (ti=0) at position (xi;zi):
x(t)=xi+2U0Hzit+2U0W0Hω21-cos(ωt)
and
z(t)=zi+W0ωsin(ωt).
Equations () and ()
describe the superposition of a horizontal motion forced by the horizontal
wind speed defined in Eq. (), and an elliptic motion
with pulsation ω due to the oscillating vertical speed
(Eq. ) and its interaction with the horizontal wind
shear. The vertical semi-ax of this ellipse is w0ω≃688m, and the horizontal semi-ax is 2U0W0Hω2≃9.12×103m.
With (x(t);z(t)) being, for any given time t, affine functions of (xi;zi),
the wind field of Eqs. () and ()
advects straight lines into straight lines. In particular, the initial
rectangular zone containing the tracer will be advected, at any given time,
into a parallelogram, whose summits are readily given by applying
Eqs. () and ()
to the summits of the initial rectangle. Inside this moving parallelogram,
that is increasingly tilted with time, tracer mixing ratio is equal to
100ppb, zero outside, giving access to the exact solution of the
case at any time.
Case 2: long-range advection of thin layer
In this case, the initial tracer mixing ratio is as follows:
26α(t=0,x,z)=αm if -h2≤z-H2≤h2,27α(t=0,x,z)=0 otherwise,
with h2=500m. These equations describe a zonally infinite layer contained between z=5500m and z=6500m. As in Case 1, αm=100ppb.
Zonal wind is constant and uniform:
u(x,z,t)=U0=L2T
so that the horizontal motion of the fluid brings it back at its initial
x coordinate after time 2T, which will be the duration of the
experiment. Vertical wind is defined as follows:
w(x,y,z,t)=w0cos4πxL.
As for Case 1, w0=5×10-2ms-1. This vertical wind
speed has two maxima and two minima over the horizontal domain.
Integration of Eqs. () and () is
immediate and give the trajectory of a particle located at
(xi;zi) at time t=0:
x(t)=Lt2T
and
z(t)=zi+W0T2πsin4πxiL+2πtT-sin4πxiL.
In particular, after a time kT, k being an integer, all the fluid
particles are displaced by distance kL2 in the horizontal direction
and back to their initial altitude. At these times, since the initial tracer
plume is zonally uniform and infinite, the field of mixing ration will be
exactly back to its initial value everywhere.
This case is a simplified representation of long-range advection of a
1km thick layer of inert tracer under the action of a uniform zonal
wind and variable vertical wind, representing for example in an extremely
simplified way the advection of this layer through synoptic-scale
structures. In the atmosphere, such fine layers of tracers are frequently
formed by stretching of initially thicker polluted layers, as represented in
Case 1.
Case 3: fine layer advection and convergence rate test
This case has been designed to study the numerical convergence rate of the various
configurations that will be tested as a function of space resolution. The case
setup is the same as for Case 2, with wind speeds similar to
Eqs. () and ():
u(x,z,t)=U0=LT
and
w(x,y,z,t)=w0cos2πxL.
Due to the need for increasing resolution and therefore the numerical cost of
simulations, we simulate only one spatial and temporal period of this case
(instead of two spatial and temporal periods for Case 2), hence the differences
between Eqs. () and () and
() and ().
The initial tracer mixing ratio is prescribed as follows:
αi(x,z)=αm41+cosπ(z-H/2)h3234 if -h3≤z-H2≤h3,αi(x,z)=0 otherwise,
with h3=1500m. This initial mixing ratio distribution has been
designed to be C2 (and in fact C3)
everywhere. It is therefore smooth enough to permit a convergence experiment
with all the transport schemes which rely, at most, on the existence and
continuity of the second-order derivative of the transported field (for the
PPM scheme).
For convergence rate tests, five different resolutions have been tested
(Table ).
Implementation
Implementation of the idealized experiments and transport scheme have been
done within the under development ToyCTM code. ToyCTM is a Python code
targeting chemistry-transport studies in academic cases. So far, ToyCTM relies
on classical numpy arrays. Its object-oriented design provides a class
structure enabling extensibility; i.e., users can easily code new transport
schemes or define personal grid geometry. A basic chemistry module is present
and allows one to test chemical reactions on top of transport. See the Code
availability section at the end of the paper for access to the code
version used for the present study and to the current development version of
the code.
ResultsCase 1: thin layer formation under wind shear
Final state of the numerical simulation for Case 1 after simulations. Panels (a–d) show the results obtained with Upwind, VL, PPM and DL99, respectively, and (e) represents the exact solution discretized on model grid. In (a–e), the contour of the exact solution is materialized by a white parallelogram.
Figure shows the outputs of the four simulations
realized with different schemes for vertical advection. All four simulations
succeed in reproducing the tilted orientation of the final plume and its
location but differ greatly in their maximal value and spatial extension,
with the smallest maximal value of mixing ratio for the upwind scheme and the
maximal value for the DL99 scheme, with the Van Leer and PPM schemes ranging
in between. From a quantitative point of view (Table ),
the Upwind, VL and PPM schemes perform as could be expected from their order
of accuracy, with the third-order PPM scheme performing better than the
second-order Van Leer scheme and the first-order Upwind scheme in terms of all
the diagnostics that have been calculated. More surprisingly, the first-order
DL99 scheme performs better than all these schemes in terms of all these
diagnostics, by a wide margin: in this case study, the performance gain of
DL99 relative to PPM is similar to the gain of PPM relative to Godunov in
terms of maximal mixing ratio, percentage of mass in the envelope and
accuracy.
Performance of simulations performed with the Upwind, VL, PPM and DL99 vertical advection schemes relative to the discretized exact solution for Case 1: percent relative error in ‖⋅‖1 and ‖⋅‖2 and percent of total tracer mass contained in the correct envelope.
ExactUpwindVLPPMDL99Max. MR30.06.1110.311.718.2% error (norm 1)0.157.131.122.87.6% error (norm 2)0.86.176.973.860.4% mass in envelope100.023.339.044.664.9Case 2: long-range advection of thin layer
Figure shows the final state of Case 2 simulation
for the four advection schemes that have been tested, as compared to the exact
solution. Visually, the scheme has performed best in
bringing virtually all the tracer back into its original envelope after two
complete vertical oscillations. Its performance in this case is the best of all
the tested schemes (Table ), with only a 7.4 %
reduction in the maximal value of tracer mixing ratio (49 % with the
PPM method, even more with the Godunov and Van Leer schemes), and very small
error values in ‖⋅‖1 and ‖⋅‖2 compared to the other tested
schemes. 90.6 % of the mass is contained in the theoretical envelope
where it should be after the end of the numerical experiment (50.3 %
only with the PPM scheme).
Final state of the numerical simulation for Case 2 after simulations. Panels (a–d) show the results obtained with Upwind, VL, PPM and DL99, respectively; (e) represents the exact solution (which is strictly equal to the initial state due to periodicity in time of the case); and (f) represents the state of the DL99 simulation after 36 h simulation.
Performance of simulations performed with the Upwind, VL, PPM and DL99 vertical advection schemes relative to the exact solution for Case 2: percent relative error in ‖⋅‖1 and ‖⋅‖2 and percent of total tracer mass contained in the correct envelope.
ExactUpwindVLPPMDL99Max. MR100.24.742.250.892.6% error (norm 1)0.151.116.99.318.8% error (norm 2)0.82.669.863.214.2% mass in envelope100.24.742.050.390.6Case 3: fine layer advection and convergence rate test
The numerical convergence rates of all the tested advection configurations for
‖⋅‖1 and ‖⋅‖2 based on the last segment in
Fig. a–b (between nx=160 and nx=320) are
shown in Table . In ‖⋅‖1. These orders of
convergence are around 0.8 for the DL99 and Upwind schemes because vertical
resolution is still insufficient even at this high resolution to ensure
theoretical convergence for these schemes. The Van Leer scheme yields a
convergence rate of 1.80 in ‖⋅‖1, 2.43 for the PPM scheme. A
smoother test case has been performed to check that all schemes are able to
obtain convergence up to their theoretical order (see Appendix).
Convergence rate results for the four tested vertical advection schemes as a function of the number of horizontal points nx in the horizontal direction, in ‖⋅‖1(a) and ‖⋅‖2(b) for Case 3. The black dashed lines show the slope expected for order 1 (long dash) and order 2 (short dash).
Figure shows that, when resolution is too coarse
to appropriately resolve the thin layer, the scheme
strongly overperforms higher-order schemes and offers an accuracy that is
substantially better than both the scheme and the
scheme. This is consistent with the results obtained on
Cases 1 and 2 and explains why in these cases the scheme
permits to obtain excellent results in reproducing the thin layer of
tracer. On the other hand, when vertical resolution becomes sufficient to
appropriately resolve the smooth tracer layer, higher-order schemes perform
better than the schemes, which in turns fall back towards
an accuracy similar to the scheme, consistent with its
expected order or accuracy.
When vertical resolution becomes much too fine compared to the size of the
modeled object (polluted plume thicker that ≃50Δz), accuracy
of the simulations with the scheme stops improving with
resolution at an order-1 rate
(Fig. ). Examination of the simulation
outputs for these configurations reveal that progress in accuracy is hindered
by undesirable small-scale oscillations that degrade accuracy (not shown).
The numerical experiments that have been presented confirm the interest of
using the scheme for vertical transport in
chemistry-transport models, as already claimed by . The
idealized framework set up here permits examination of some of the questions that
were out of reach in the real case of due to
uncertainties in the forcing meteorological fields, the volcanic emissions and the lack of accurate measurements of plume structure.
The numerical experiments exposed here confirm that the
scheme is less diffusive than and , as
could be expected due to its antidiffusive design. They also reveal that, in
the presence of sharp vertical gradients that are not adequately resolved at
model resolution, using the scheme for vertical transport
also increases model accuracy compared to the exact solution. This improvement
is substantial for both Case 1 (Table and
Fig. ) representing the formation of a thin polluted
layer under the action of wind shear and Case 2 (Table
and Fig. ) representing long-range advection of a
thin polluted layer. The objective scores as well as the visual comparison of
the simulated final state with the exact solution show that, at this
resolution and for both these cases, using the scheme reduces
diffusion and increases accuracy compared to the schemes of
, , and . While reduction in
diffusion is in line with the results of and could be
expected because of the design of the schemes, improved accuracy in
presence of sharp gradients is a strong argument in favor of using the
schemes for vertical transport in chemistry-transport
models.
Improved accuracy of a low-order scheme compared to higher-order schemes for a
given resolution is not impossible from a theoretical point of view but still
counterintuitive since higher-order schemes are designed to reduce numerical
error at any given resolution compared to lower-order schemes due to
“smarter” reconstruction procedures. Theory imposes that, if model
resolution is fine enough and if the tracer field is smooth, higher-order
schemes should be more accurate than lower-order schemes. However, as shown by
, linear higher-order schemes cannot be monotonous, a
property usually known as Godunov's theorem. This is why, to ensure
monotonicity, the schemes of and
include non-linear slope-limiters which are activated in the vicinity of
extrema and discontinuities. In the vicinity of discontinuities, these
formulations introduce large inaccuracies: in these schemes, the use of
slope-limiters introduce large errors in the vicinity of discontinuities, and
these errors generate excessive numerical diffusion, which is visible in
Figs. and . On the other
hand, as discussed by its creators, the scheme is designed
to reduce numerical diffusion in these areas of steep gradients, which
explains why it performs better than and
in all respects for Cases 1 and 2, which describe
discontinuous tracer layers (Tables
and ).
To understand this surprisingly good behavior of compared
to the higher-order and schemes, we
have performed a convergence test for advection of a 3000 m thick
layer with a smooth (C3) initial profile for the tracer mixing ratio
(Eq. ). This convergence test
(Fig. ) shows that the scheme performs
better than these classical order-2 schemes if model vertical resolution
Δz is equal to 1000 or 500m, but that due to
their faster convergence rate, order-2 schemes perform better if Δz≤250m. At these coarse resolutions, gradients in the initial
tracer field (Eq. ) appear so sharp that the
scheme yields improved accuracy compared to
and due to its better ability to deal
with sharp gradients without introducing excessive numerical diffusion. On the
other hand, when resolution is fine enough, the tracer field and its
successive derivatives do not vary drastically from one model cell to the
next, so that the linear interpolations of and the
quadratic interpolations of work adequately and reduce
numerical errors compared to the first-order evaluation of concentration of
.
In more general words, this result suggests that the scheme
may be expected to perform better than classical schemes in
chemistry-transport models for the advection of polluted plumes thinner than
≃6Δz (Δz being the model's vertical resolution), while
higher-order schemes can be expected to perform better for the advection of
polluted plumes thicker than 6Δz if we suppose that the plume has a
smooth vertical profile. Under realistic conditions of wind shear, these
conditions of sufficient smoothness and thickness might actually be very
difficult to reach since, as described in Case 1, vertical wind shear tend to
the permanent thinning of atmospheric plumes (this question is discussed in
detail in ) so that the may frequently
overperform classical order-2 schemes in realistic wind conditions including
wind shear.
Conclusions
The first-order, antidiffusive advection scheme of has
been compared to classical second-order schemes of and
in three idealized test cases representing long-range
atmospheric transport of thin plumes in the troposphere. It is shown that the
generally overperforms these two schemes, offering both
improved accuracy and reduced diffusion. The scheme is
shown to allow a correct representation of long-range advection of fine tracer
layers when vertical resolution is so coarse that the classical Van Leer and
PPM schemes exhibit much weaker performance due to excessive vertical
diffusion, a feature that appears in the present idealized test cases but also
in realistic cases (e.g., ).
Convergence tests show that this improved performance exists only for tracer
layers that are represented by less than six grid cells in the vertical
direction, while for finer resolutions and smooth initial tracer fields,
higher-order schemes perform better than the first-order
scheme due to their faster convergence towards the exact solution. This
suggests that, if model resolution is fine enough to represent properly the
tracer plumes and their concentration gradients, higher-order schemes may
still be a better choice.
We think that these results are important because they explain under which
conditions the is able to reduce excessive vertical
diffusion in CTMs, as was observed in , and that this
reduced vertical diffusion comes together with an improved accuracy compared
to the exact solution. Since the vertical resolution of chemistry-transport
models is usually coarse in the free troposphere, while the advected plumes
tend to be extremely thin in this part of the atmosphere due to the action of
wind shear, the present study along with advocates for
using the transport scheme for chemistry-transport
modeling in the free troposphere, and probably even more in the stratosphere
where vertical diffusion needs to be extremely reduced. It is also worth
noting that have shown that their scheme maintains its
convergence and low-diffusion properties even if the CFL number becomes small,
which is very common for vertical advection in the free troposphere due to the
typically small vertical speed of air motion (typically a few
centimeters per second).
More investigation is needed in real and/or idealized cases to address several
questions:
Does the perform better in the boundary layer, where CTM resolution is typically finer than in the free troposphere?
If not, is it possible to use a traditional transport scheme in the boundary layer and the scheme in the free troposphere without introducing numerical artifacts in the buffer zone?
Atmospheric chemistry being a non-linear process, how does reduction in excessive numerical diffusion in the troposphere affect representation of chemistry inside chemically active air masses such as volcanic or biomass burning plumes?
Is it desirable to use antidiffusive transport schemes in the horizontal directions as well, and under which conditions should they be used?
Case 4: convergence test in a smooth caseCase definition
Convergence rate results for the four tested vertical advection schemes as a function of the number of horizontal points nx in the horizontal direction, in ‖⋅‖1(a) and ‖⋅‖2(b) for Case 4.
This case has been designed to study the numerical convergence rate of the various
configurations that will be tested as a function of space resolution. The case
setup is the same as for Cases 2 and 3, with wind speeds from
Eqs. () and (), but the initial tracer
mixing ratio is prescribed as follows:
αi(x,z)=αm41+cosπ(z-H/2)h4A1×1+cosπ(x-L/2)δ,
with h4=H2=6000m and δ=L2=5×105m. This initial tracer distribution represents a cosine bell
centered at (x=L2;z=H2), C∞
everywhere with extremely smooth variations in space. Domain resolution and
sizes are as shown in Table .
Results
Table and
Fig. show that all the tested
configurations exhibit the expected convergence order at least in ‖⋅‖1, but accuracy with the DL99 scheme stops improving when vertical
resolution becomes “too fine” compared to the thickness of the represented
layer. In the present case, this “saturation” of convergence occurs between
nx=80 (nz=48) and nx=160 (nz=96):
Convergence rates in ‖⋅‖1 and ‖⋅‖2 for Case 4. Convergence rates for the DL99 scheme, marked with a * symbol, are evaluated between nx=40 and nx=80, before numerical oscillations appear and begin to degrade the result.
ToyCTM is free software distributed under the GNU General Public License v2.
The exact version used for the present study is available
permanently in the HAL repository at
https://hal.archives-ouvertes.fr/hal-02933095. The latest stable version of ToyCTM is available from https://gitlab.in2p3.fr/ipsl/lmd/intro/toyctm/-/archive/master/toyctm-master.tar.gz (last access: 29 April 2021).
Author contributions
All the authors have contributed to the design of the simulated cases. SM performed and analyzed the simulations and developed the software with RP. LM, ML and RP contributed to writing and improving the paper.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
The simulations have been performed at the ESPRI/IPSL data center and at TGCC under GENCI A0070110274 allocation.
Financial support
This research has been supported by the Agence de l'Innovation de Défense (TROMPET grant).
Review statement
This paper was edited by Simone Marras and reviewed by two anonymous referees.
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