The model error in climate models depends on mesh resolution, among
other factors. While global refinement of the computational mesh is
often not feasible computationally, adaptive mesh refinement (AMR)
can be an option for spatially localized features. Creating a
climate model with AMR has been prohibitive so far. We use AMR in
one single-model component, namely the tracer transport scheme.
Particularly, we integrate AMR into the tracer transport module of
the atmospheric model ECHAM6 and test our implementation in several
idealized scenarios and in a realistic application scenario (dust
transport). To achieve this goal, we modify the flux-form
semi-Lagrangian (FFSL) transport scheme in ECHAM6 such that we can
use it on adaptive meshes while retaining all important properties (such as mass conservation) of the original FFSL implementation. Our
proposed AMR scheme is dimensionally split and ensures that
high-resolution information is always propagated on (locally) highly
resolved meshes. We utilize a data structure that can
accommodate an adaptive Gaussian grid.
We demonstrate that our AMR scheme improves both accuracy and
efficiency compared to the original FFSL scheme. More importantly,
our approach improves the representation of transport processes in
ECHAM6 for coarse-resolution simulations. Hence, this
paper suggests that we can overcome the overhead of developing a
fully adaptive Earth system model by integrating AMR into single
components while leaving data structures of the dynamical core
untouched. This enables studies to retain well-tested and complex
legacy code of existing models while still improving the
accuracy of specific components without sacrificing efficiency.
Introduction
The climate system is inherently multi-scale. In climate models,
various processes are under-resolved because the resolution cannot
represent details of these processes. One of the most straightforward
approaches to better accuracy is increasing spatial
resolution. However, high-resolution climate simulations are still
computationally expensive, especially for long-term climate
simulations like paleoclimate simulation. Adaptive mesh refinement
(AMR) is an attractive alternative for global high-resolution climate
models. The AMR technique refines and coarsens grid cells locally during
runtime based on designated refinement criteria.
There is active research on AMR applications in the climate community
dating back to the 1980s. For example, proposed an early
non-hydrostatic model using AMR. More recently
constructed a finite-volume general circulation model on a reduced
latitude–longitude (lat–long) grid. constructed an atmospheric model using a
Galerkin method on a cubed sphere. These efforts focus on the
dynamical cores of atmospheric models. Utilizing these methods for
realistic climate simulations needs further research and development.
We propose an alternative pathway towards adaptivity in climate models
to address difficulties applying AMR in operational climate models ranging from
properties of numerical schemes to the coupling between dynamical core
and physics packages .
Constructing a complete model from scratch usually takes decades of
research. Instead, we propose integrating AMR into single components
of existing models (here ECHAM6), which could bring about immediate
benefits. It is not uncommon to apply different resolutions for
different components of a numerical model. For example,
showed that a high-resolution dynamical core using
low-resolution parameterizations generates satisfactory results.
Enabling AMR in the passive tracer transport module of a climate model
can improve the representation of such transport processes and
can potentially improve the general quality of its host climate
simulation. The tracer transport module controls advective passive
tracer transport processes in climate models. Because tracers interact
with many other processes in the climate system and generate
feedback to the radiative balance or cloud formations, their accurate representation affects the state of the climate system.
Despite potential benefits of integrating AMR into the tracer transport
module of an existing model, there are difficulties in achieving
this goal.
How does the tracer transport scheme perform with
non-conforming adaptive meshes?
How much improvement can we gain from an adaptive
tracer transport scheme without refining other components?
We introduce AMR into the tracer transport module of ECHAM6. ECHAM6 is
the atmospheric model component of the MPI-ESM . The
first part, “EC”, indicates that the model was derived from the
European Center's model, while “HAM” means it was developed mainly in
Hamburg, Germany. ECHAM6 solves the hydrostatic primitive equations
using a spectral transform method. The tracer transport module uses
the flux-form semi-Lagrangian (FFSL) scheme . The FFSL
scheme has two essential properties: mass conservation and
semi-Lagrangian time stepping. Semi-Lagrangian schemes are
particularly useful for the Gaussian grid in ECHAM6. The Gaussian grid
is a variation of the lat–long grid, where the longitude is equally
spaced in the longitudinal dimension, and the latitude grid
corresponds to Gaussian quadrature points for numerical
integration. The Gaussian grid leads to smaller grid intervals around poles, which poses a limit on the time step size due to the Courant–Friedrichs–Lewy (CFL) criterion. If the time step size is large, the numerical scheme can become unstable.
However, on the adaptive mesh ECHAM's existing transport scheme does not retain all desired properties when hanging nodes are
present. Hanging nodes lie at the interface between high-resolution
and low-resolution areas. So-called ghost cells are commonly used to
treat hanging nodes. Such scheme creates high-resolution ghost cells in
low-resolution areas along the interface to high resolution, such that the discretization stencil of the
numerical scheme relies on a (virtual) uniform resolution. For example,
used ghost cells for the FFSL scheme but their
implementation does not maintain the semi-Lagrangian time-stepping.
adopted the FFSL scheme for shallow water equations
on a block-structured AMR scheme that also did not retain the
large Courant number.
Another approach to deal with the interface between high- and low-resolution areas is to substitute the existing transport
scheme by a mass conservative semi-Lagrangian scheme, which can handle
irregular meshes. For example, proposed a
cell-integrated semi-Lagrangian scheme; proposed a more
efficient mass conservative semi-Lagrangian scheme using Stokes'
theorem. However, the comparison between the original climate model
and the climate model with adaptive tracer transport would be
difficult if we used two different transport schemes.
We propose a modified version of the existing tracer transport scheme
that retains essential properties of the original scheme. By keeping
the numerical properties of our AMR-enabled transport scheme as close
to the original as possible, we state that our transport module has
the same numerical properties as the original module. Furthermore, our
modified tracer transport scheme allows us to reuse the code for
vertical tracer transport and a class of limiters in the existing
model without further investigation. As a hydrostatic model, ECHAM6
uses a 1-D finite-volume method for the vertical transport. The
vertical transport is independent from the horizontal transport. This
treatment of the vertical tracer transport is similar to the original
FFSL scheme in but differs from it due to the use of
hybrid η coordinates. The reuse of the vertical tracer transport
of ECHAM6 also allows the reuse of the grid-to-grid transformation in
ECHAM6 described by . The grid-to-grid
transformation alleviates the wind–mass inconsistency issue due to
different numerical schemes for continuity and tracer transport
equations in hybrid vertical coordinate systems. As we adopt the
treatment of the wind–mass inconsistency in the existing ECHAM6 setup
directly, the paper focuses on the effect of AMR and does not further
address the wind–mass inconsistency.
Utilizing idealized test cases, we quantitatively investigate the
properties of our modified scheme on adaptive meshes and
non-adaptive meshes even though many other tracer transport schemes
using AMR are well studied . In
particular, we examine the effect of using coarse-grid initial condition
and wind field using idealized test cases as we only apply AMR
to a single component of the climate model.
We further validate our proposed AMR approach simulating the
prototypical but realistic example of dust transport in ECHAM6. Dust
is particularly suitable to demonstrate the effect of AMR since it has
local sources and is transported around the entire globe. The global
distribution of dust develops pronounced local features, which can be
represented more accurately by local refinements.
The paper is organized as follows. We introduce our adaptive tracer
transport scheme in Sect. . In order to
quantitatively demonstrate the properties of the modified AMR-enabled
scheme, we show results of idealized tests in Sect. . We further demonstrate the idea of integrating AMR
into more realistic single-component tracer transport of the existing
ECHAM6 model in Sect. and conclude with a discussion of our results and
future work in Sect. .
The adaptive transport scheme
In order to ensure a fair examination of the partial introduction of
AMR into the existing model ECHAM6, we use the original FFSL scheme in
ECHAM6. The FFSL scheme is particularly suitable for climate models
because it is accurate, efficient, mass conservative and
semi-Lagrangian. The FFSL scheme is a combination of a dimensionally
split technique, 1-D finite-volume transport scheme, and
semi-Lagrangian extension for finite-volume schemes.
The dimensional splitting within the FFSL scheme is of the second order in
time. The overall order of accuracy of the FFSL scheme therefore also depends on
the 1-D solver of the transport equation. In our idealized tests, we
use the piecewise parabolic method (PPM) in space, which is formally
fourth and third order in space for equidistant and non-equidistant grids,
respectively. The operational code ECHAM6 uses a mixture of first-order
forward Euler time-stepping and PPM space discretization, a practice
we adopt in the realistic test. In order to deal with large Courant
numbers, we use a first-order Euler method to compute the departure
cells.
Our aim is to use the FFSL scheme on adaptive meshes. However, we
cannot extend the FFSL scheme to adaptive meshes while retaining all
its properties without modification. We will explain details of the FFSL
scheme, the problem of applying it to adaptive meshes, and our modification in this section.
The flux-form semi-Lagrangian scheme
We present the flux-form semi-Lagrangian (FFSL) transport scheme
proposed by . The FFSL scheme solves the 2-D transport
equation. Climate models often rely on the transport equation in
spherical coordinates:
∂ρc∂t+1acosθ∂ρcu∂λ+∂ρcvcosθ∂θ=0,
where a is the radius of the sphere, (λ,θ) is the
longitude and latitude on the sphere, (u,v) is the horizontal
velocity, ρ is the air density, c is the tracer
mixing ratio. For convenience of introducing the scheme, we set
c≡1.
The dimensionally split technique of the FFSL scheme is second-order
accurate in time. The method splits the 2-D transport equation in
Eq. () into two 1-D transport equations:
2∂ρ∂t+∂ρuacosθ∂λ=0,3∂ρ∂t+∂ρvcosθacosθ∂θ=0.
The dimensionally split technique eases the difficulty in extending
1-D methods into higher dimensions and enables the application of
various 1-D limiters to 2-D problems.
This method is equivalent to the COSMIC splitting proposed in
. The advantage of the FFSL scheme is that it leads
to a mass-conservative and consistent dimensionally split technique
since the Strang splitting cannot preserve both mass conservation and
consistency condition for tracer transport problems.
The FFSL scheme defines a 1-D conservative operator for the flux
difference of two cell edges FC(ρ):
FCλ(ρ)=-1acosθΔλ∫(ρu)i+12-(ρu)i-12dt,FCθ(ρ)=-1aΔsinθ∫(ρvcosθ)i+12-(ρvcosθ)i-12dt.
Here, the subscript “C” means that the operator is conservative and the
superscript represents the coordinate direction of the 1-D
operator; the subscript i±12 represents the cell boundaries of cell i.
The conservative operator is the flux differences of the
cell in one time step. The dimensionally split technique allows any 1-D finite-volume transport scheme to solve the 1-D operator FC(ρ). The
finite-volume scheme ensures mass conservation of the FFSL scheme.
In order to achieve the consistency condition of the FFSL scheme, the
scheme also uses an advective operator with the assumption of non-divergent flows,
which is a variation of the FC(ρ):
FAλ(ρ)=FCλ(ρ)+Δtρ∂uacosθ∂λ,FAθ(ρ)=FCθ(ρ)+Δtρ∂vcosθacosθ∂θ,
where “A” means the operator only solves the advective part of the
transport equation and Δt is the time interval. The second term of Eq. () is computed by a second-order finite-difference scheme .
Schematic illustration of the dimensionally split
scheme. ρn, ρn+1, ρA(λ), and
ρA(θ) are tracer mixing ratios corresponding to
Eqs. (), (), and
(); the Greek letters α, β, γ, and
ζ represent the individual cells.
Similar to the Strang splitting, the FFSL scheme alternates the
direction sequentially. The dimensionally split scheme first solves
the 1-D equation in λ or θ dimension:
ρA(λ)=ρn+FAλ(ρn),ρA(θ)=ρn+FAθ(ρn),
where the superscript n denotes the current time step. The scheme uses the
advective operator FA(ρ) as the inner operator, which guarantees
the consistency condition.
Using ρA as the initial condition, the scheme subsequently
solves the 1-D equation in the other direction:
ρ(ρA(λ),ρn)=ρn+FCλ(ρn)+FCθ(ρA(λ)),ρ(ρA(θ),ρn)=ρn+FCθ(ρn)+FCλ(ρA(θ)),
where the mass conservation is guaranteed by the conservative outer
operator. Results of ρ(ρA(λ),ρn) and
ρ(ρA(θ),ρn) tilt to different directions. Hence,
the final solution for the next time step, n+1 is the average of the
outer operator in each direction:
ρn+1=12(ρ(ρA(λ),ρn)+ρ(ρA(θ),ρn)).
We illustrate the scheme in Fig. . If the cell
ζ is the departure cell corresponding to the arrival cell α, the
scheme transports information dimensionally from cell ζ to cells
β and γ. The process of transport from cell ζ to cells
β and γ corresponds to the advective operator in Eq. ().
After the intermediate step, cell β and γ are the departure
cells of the arrival cell α in each dimension, which is updated by
Eq. ().
Therefore, ρn+1 is based on ρA(λ) and ρA(θ) as intermediate step.
Semi-Lagrangian extension on adaptive meshes
The FFSL scheme attains long time steps by a semi-Lagrangian extension
from 1-D finite-volume schemes . Similar to
traditional semi-Lagrangian schemes, the extension requires
computation of trajectories described by the flow field. However, by
construction, the extension also requires the mass flux of each cell
edge during one time step, which is a sweep of mass along
trajectories. This semi-Lagrangian computation accounts for the
exact integration of mass flux across an edge, similar to a finite-volume scheme, and thus yields mass conservation. In order to improve
the efficiency of the implementation, the FFSL scheme employs the
widely used idea of cumulative mass first described in
. The cumulative mass of a cell is the mass from
the beginning of the domain to the cell. Thus, the mass along the
trajectory is the difference between the arrival cell and the
departure cell and the finite-volume flux at the departure
cell. Using cumulative mass significantly reduces the computational
cost.
Illustration of the
semi-Lagrangian extension for finite-volume schemes on adaptive
meshes. The marks, α, β, and γ, represent their
underlying cells. Cell α is the arrival cell with high
resolution, while cells β and γ are coarse cells. The
dashed red cells are ghost cells. The shaded domain represents the
departure area determining the mass flux into the arrival
cell.
However, when using the semi-Lagrangian extension on adaptive meshes,
problems arise. The FFSL scheme assumes a structured rectangular grid,
where the cell centers align with each other in each dimension such
that the dimensionally split scheme can use 1-D solvers for each
dimension. For example, the cell center always lies at the same
latitude when the scheme computes for longitudinal direction. However,
hanging nodes on adaptive meshes cannot guarantee an alignment as
shown in Fig. . Breaking the
alignment assumption leads to inconsistency and violates mass
conservation. For example, if a 1-D finite-volume scheme computes the
value of the next time step at the arrival cell α in Fig. , the 1-D scheme would include the mass
at the entire cell β, while a consistent treatment needs only the
mass at the lower shaded area of cell β.
In order to satisfy the alignment assumption, we could use ghost cells,
illustrated as the red cells in Fig. . However, using ghost cells for large
Courant numbers prevents the scheme from using cumulative mass since
it is difficult to define the cumulative mass for high-resolution
cells. Without cumulative mass, the semi-Lagrangian extension may lead
to multiple computations of the mass because the departure trajectory
of different edges may overlap, leading to an inefficient scheme.
Modified flux-form semi-Lagrangian
scheme
As described in Sect. , the original FFSL
scheme cannot handle hanging nodes efficiently because it uses a
finite-volume scheme with a semi-Lagrangian extension to solve 1-D
problems, where it is computationally expensive to obtain the mass
along the trajectory. We expect that a mass conservative
semi-Lagrangian scheme without the sweep along trajectories can solve
the problem arising with hanging nodes. The cell-integrated
semi-Lagrangian (CISL) scheme by is a good candidate.
Instead of adding up the mass along the whole trajectory of cell
edges, the CISL scheme updates values from the mass at departure
cells. In particular, shows that the CISL
scheme is an alternative point of view of Godunov-type finite-volume
schemes with a semi-Lagrangian extension. Hence, we can safely
substitute the finite-volume scheme with the CISL scheme and expect
similar numerical results on adaptive and non-adaptive meshes.
Here, we present a brief description of the CISL scheme under
reference coordinates instead of spherical coordinates. The numerical
results can easily be mapped between reference and spherical
coordinates. Similar to finite-volume schemes, in a 1-D setting the
CISL scheme assumes the cell center value as the cell average:
ρic=1Δxi∫Δxiρdx,
where x∈[-12,12] and Δxi is the
width of cell i. The integrand is a sub-cell reconstruction
function based on the cell center value. For example, the Godunov
scheme assumes the sub-cell reconstruction function to be constant.
Illustration of the CISL scheme in 1-D and 2-D settings;
α, β, and γ are labels of cells. u denotes
the longitudinal velocity at cell edges. We set cell α as
arrival cell in both the 1-D and 2-D cases, and hence the subscript
i=α in Eqs. () and
(). The dashed line in the 1-D scheme is the
departure interval, and the shaded area is the departure cell in
the 2-D scheme. The 1-D CISL scheme follows Eq. () using a 1-D integral, while the 2-D CISL uses
Eq. () with an area integral that uses a
2-D sub-grid distribution as a reconstruction
function.
In the CISL scheme, the departure cell is formed by the departure
position of the cell edges of the arrival cell and the 1-D scheme
updates values from the departure cell:
ρin+1(x)=1Δxi∫Δxdρndx,
where Δxd=xd,i+12-xd,i-12 is the interval of departure cells in each dimension
and i±12 corresponds to cell edges.
As shown in Fig. , the dashed line is the departure cell in 1-D. The
scheme gets new values from the mass at the departure cells, which is
an integral of the sub-cell reconstruction function over the interval
of departure cells. The CISL
scheme avoids the computation of mass along the trajectory while
keeping the advantage of long time steps on adaptive meshes.
On the sphere, the departure position of cell edges in each
dimension is described by
acosθdλdt=uadμdt=vcosθ,
where μ=sinθ.
Here, we follow ECHAM6 and use a first-order Euler method to solve the ODE:
λd,i+12=λi+12-ui+12acosθaΔtμd,i+12=μi+12-(vcosθ)i+12Δt,
where θa is the latitude of the cell center. Similar to Arakawa
C-staggering, the velocity (u,v) is defined on
cell edges and the first-order Euler method assumes constant velocity
along the trajectory. This practice can provide a fair comparison
between our AMR method and the original scheme used in ECHAM6.
The staggering of the velocity means that vcosθ=0 at
poles. Hence, the cross-pole advection is controlled by the velocity
u in the λ direction restricted by the deformational
Courant number,
|∂uΔtacosθ∂λ|, which
is less restrictive than the Courant number. When the deformational
Courant number is less than 1, trajectories do not cross, which
ensures the stability of the semi-Lagrangian scheme. This restriction
holds on adaptive meshes and we disable mesh refinement in case
interpolated wind would lead to trajectory crossing. We will also
discuss the restriction of the deformational Courant number on mesh
refinement in Sect. .
Illustration of the use of different reconstruction
function in our modified scheme. The shaded area
ΔAλ,d is the departure cell of the arrival cell
α. When the departure cell overlaps with the underlying
Eulerian cell β, the size (refinement level) of the
departure cell and Eulerian cell are the same and a 1-D
reconstruction function suffices. When the departure cell overlaps
with the underlying Eulerian cell γ, the size (refinement
level) of the departure cell is smaller (higher) than the Eulerian
cell and a 2-D reconstruction function is
required.
On an adaptive mesh with hanging nodes, the 1-D integral in Eq. () does not consider the sub-grid distribution in the
other dimension, which breaks the 2-D mass conservation as discussed
in Sect. . Therefore, we must use a 2-D
integral:
ρn+1(λ)=1ΔAi∬ΔAλ,dρndλdμρn+1(θ)=1ΔAi∬ΔAθ,dρndλdμ,
where ΔAi=ΔμiΔλi is the area of the
arrival cell. The definition of the cell area follows
. The area of the departure cell is
ΔAd=ΔμdΔλd, and the dimensionally
split scheme uses the fractional area of the departure cell in each
dimension:
ΔAλ,d=ΔAdΔμiΔμdΔAθ,d=ΔAdΔλiΔλd.
Here, we make use of the benefits of the dimensionally split
technique. The scheme only needs to compute 1-D departure positions of
the cell while the scheme performs a 2-D integral to compute the
mass. Equation () can be reduced to Eq. () when the departure cell is aligned with the arrival
cell. As shown in Fig. , 1-D CISL is sufficient when
the arrival cell α aligns with the departure cell in the
Eulerian cell γ, where Δμγ=Δμα. However, 2-D CISL is necessary as
Δμα≠Δμγ.
The resemblance between Eqs. () and
() allows us to use 1-D and 2-D reconstructions for
different conditions. As shown in Fig. , we apply a 2-D reconstruction
function on adaptive meshes when a departure cell has a lower
refinement level than the arrival cell. Otherwise, we apply a 1-D
reconstruction function. For example, in Fig. , a 1-D reconstruction function is
used for an integral over the shaded area in the cell β as
Δμα=Δμβ and Eq. () can be reduced to Eq. (), while
a 2-D reconstruction function is used for an integral over the shaded
area in the cell γ.
In order to be consistent with the original implementation, we choose
the same reconstruction function as the one used by the FFSL scheme in
ECHAM6 such that we can make a fair comparison between the AMR scheme
and the original scheme in the following sections, and thus our idealized
tests can provide insight for realistic simulations. The default
option of the FFSL scheme in ECHAM6 uses the piecewise parabolic
method (PPM) as 1-D finite-volume solver. The PPM is a finite-volume
Godunov-type method, which assumes a quadratic subcell distribution
function. Interested readers can refer to for a
detailed description of the PPM. Here, we use a 1-D second-order polynomial
and a quasi-2-D reconstruction as in in a reference coordinate:
ρ(λ,μ)=ρc+axx+bx(112-x2)ld>=lρc+axx+bx(112-x2)+ayy+by(112-y2)ld<l,
where x∈(-12,12) is either λ or μ
in 1-D case, the condition l represents the refinement level of the
Eulerian cell, ld represents the refinement level of the departure
cell, the coefficients a and b are computed following
:
a=ρi+12-ρi-12b=6ρc-3(ρi+12+ρi-12),
where ρi-12 and ρi+12 are interpolated by a quartic polynomial based on .
The limiters are applied to the coefficients a and b. We do not use any limiters
in the idealized tests in Sect. , but we apply the default relaxed limiters
in ECHAM6 as described in Appendix B of for dust simulations in Sect. .
Because a and b are computed by 1-D
interpolations, we remap the coarse-cell values to refined cells by
recursively using Eq. () to form the
interpolation stencil. The 2-D reconstruction function can also be
used in the fully 2-D schemes, as in the original work of
. The dimensionally split scheme benefits from the
simplicity of the implementation in that the computation of the
departure cell's position is still 1-D and the departure cell's shape
is more regular than in a fully 2-D scheme.
Using our modified 1-D operator in the FFSL scheme, the original
FCd(ρ) in Sect. becomes
FCλ(ρ)=ρn+1(λ)-ρn+1,FCθ(ρ)=ρn+1(θ)-ρn+1,
where ρn+1 is the updated value in Eq. ().
Our modified operator for the dimensionally split scheme retains the
semi-Lagrangian time stepping. Moreover, the efficiency of the CISL
scheme is similar to the original finite-volume scheme with a
semi-Lagrangian extension. Finally, the scheme is mass conserving as
is the original scheme.
Wind interpolation for tracer transport
In our targeted applications, our integrated adaptive transport scheme
uses information from the non-adaptive low-resolution dynamical core and
parameterizations. For each time step, in the one-way coupling the AMR
method obtains wind information and surface pressure from the coarse-resolution ECHAM6
model.
The coarse-resolution model (dynamical core and parameterization) runs independently from the AMR method, and the
refined tracer distribution is not averaged back into the coarse-resolution host model.
As the momentum equations – from which the wind data are obtained – are still solved on a coarse resolution by the
spectral dynamical core, our AMR scheme needs to interpolate the wind
field from the coarse mesh to the AMR mesh. To prevent numerical oscillations and maintain
monotonicity, we use first-order bilinear interpolation. The wind
interpolation can lead to trajectory crossing around poles, especially
when the resolution around the poles is higher than other regions
on the lat–long grid. We need to avoid mesh refinement when the interpolated
wind leads to trajectory crossing on refined mesh. Hence,
we do not refine cells around the poles when wind interpolation is
necessary (e.g., in the realistic test case). For most cases, it is
sufficient to avoid refinement at a distance of only one grid cell from the poles.
The wind interpolation is
not applied when we use analytical wind fields in idealized test
cases in Sect. .
Compared to the high-resolution simulations, our AMR experiments lead
to two sources of error: the error from coarse initial conditions and
the error from wind interpolations.
investigated the sensitivity of wind interpolation on tracer fields,
indicating that even with interpolated wind, local refinement can improve the numerical
accuracy of passive tracer transport schemes. Hence, wind
interpolation should be an effective method when a high-resolution
wind field is not available. We further investigate the numerical
error in an idealized test case in Sect. .
Refinement strategy
Our refinement procedure follows the description in .
AMR requires flexible data structures, and thus
the original mostly array-oriented data structure needed to be replaced by a forest of trees data structure.
A forest of trees is used for example in the parallel p4est
library . However, as our targeted application
has a simpler geometry, we use the simplified
data structure in . While the forest of trees data structure can be
readily parallelized , we do not consider this here and run it in serial,
since it is not the focus of our study.
The data structure allows drastic spatial resolution changes. However,
to alleviate numerical oscillations due to sudden spatial resolution
variations, we restrict our simulations to a 1:2 refinement ratio such
that it is locally quasi-uniform. In our idealized tests, we present
results with up to two refinement levels.
Based on the data structure, our mesh can be refined or coarsened at
each time step. To predict the tracer distribution in the next time step,
we use a first-order non-conservative semi-Lagrangian
scheme. We
refine the mesh using refinement criteria based on the predicted
tracer distribution and then perform the modified FFSL scheme
described in Sect. .
To select refinement criteria one can either choose mathematically
rigorous error estimators, based on the convergence theory of the
underlying equation and on the consistency of the numerical scheme, or
one can choose more ad hoc physics-based refinement indicators
. The investigation of appropriate refinement criteria is
an active research field that is outside the scope of this study. In climate
models, it is often not possible to use mathematical error estimators
because rigorous convergence is hard to achieve for such complex
multi-physics systems.
In our experiments, we use two different refinement criteria: a
gradient-based and a value-based criterion. Both criteria are used in
non-normalized versions and are calibrated to the specific test case.
We acknowledge that this is an ad hoc approach and refer to the
literature e.g., for a
more concise description of such criteria.
In order to use the refinement criteria, we assign each cell a
quantity: ϑi,j. Based on the targeted applications, we
set ϑr as the threshold for the refinement and ϑc
as the threshold for the coarsening of the cell. We refine a cell when
ϑi,j>ϑr and coarsen a cell when
ϑi,j<ϑc. The refinement criterion and the
threshold determines whether a cell is refined or coarsened. As we
use ad hoc refinement criteria instead of an error estimator, we need
to set a maximum number of refinement levels to prevent the AMR from
excessive refinement. In this paper, we test the AMR scheme with a one-level refinement and a two-level refinement.
For dimensionally split schemes, we need to consider an additional
refinement criterion. While in multi-dimensional transport the information
propagates directly from the departure area to the arrival area and refinement
is applied to both, the tracer is always represented by refined grid cells.
In contrast, dimensionally split schemes propagate information in each
coordinate direction independently.
As indicated in Fig. , using the advective (inner) operators
in Eq. (), the scheme moves the information from the
departure point, cell ξ, to intermediate positions, cell γ and β,
before moving the information to the arrival point, cell α, using the
final update in Eq. (). Therefore, the AMR scheme needs to track this
information and needs to refine intermediate steps corresponding
to Eq. ().
Idealized tests
In order to test the implementation and verify our design choices for the AMR scheme, we conduct a number of idealized tests.
Idealized tests can expose the accuracy and efficiency of
the AMR scheme under various conditions. We design our
experiments to mimic the behavior of the
intended application to prepare for the integration of the adaptive tracer
transport scheme into an existing model while keeping other components
unchanged.
The idealized tests are intended to demonstrate three essential aspects of
our AMR scheme. Firstly, we show that the dimensionally split scheme
needs a special refinement strategy in the AMR applications. Secondly, we
examine various properties of our AMR scheme, including accuracy,
efficiency and mass conservation. Thirdly, we explore the accuracy of
the solution on adaptive meshes in situations where the AMR scheme
interpolates low-resolution wind fields to high-resolution meshes.
We utilize three test cases: a solid body rotation test case
, a divergent test case , and a moving
vortices test case . Each test case poses different
challenges to the transport scheme. Hence, we can demonstrate that our
AMR scheme possesses all numerical properties essential to the
purpose of application.
The solid body rotation test case has a discretely divergence-free
wind field, and in the theoretical absence of diffusion the shape of
the tracer distribution should not change during the run time. In the
solid body rotation test case, the flow orientation can be controlled
by the parameter α, where α is the angle between the
flow orientation and the equator. This test case is challenging when
the tracer moves around the poles due to the convergence of coordinate
lines. It is a useful test case to explore accuracy and efficiency of
our numerical scheme under idealized conditions.
The divergent test case deforms the tracer distribution with a
divergent wind field. Divergent wind is especially challenging for
large time steps since the transport scheme needs to correctly move
the tracer when the divergent wind leads to a high gradient in the
tracer mixing ratio.
Different from the solid body rotation test case and the divergent
test case, the moving vortices test case distributes tracer over the
entire globe. The moving vortices test case also severely deforms the
tracer, and the vortices form filaments in the tracer
mixing ratio. Strong deformation leads to steep gradients and
furthermore poses challenges for the AMR scheme because improper
refinement criteria may result in refinement of the entire domain.
Here we use a gradient-based refinement criterion:
ϑi,j=max|ci,j-ci-1,jacosθΔλ,ci+1,j-ci,jacosθΔλ,ci,j-ci,j-1aΔμ,ci,j+1-ci,jaΔμ|,
where c is the tracer mixing ratio and the subscript i,j is the
index of the grid cell. We use the same refinement criterion for all
idealized test cases and apply different thresholds for refinement,
ϑr, and coarsening, ϑc, for different test cases. Our
implementation of the gradient criterion is a way to measure the
changes between the cell and its adjacent cells. By this we ensure
capturing steep slopes, which in turn lead to the largest error in
reconstructing the upstream integrals in the CISL scheme. We note that
in atmospheric modeling, wind-based refinement criteria are sometimes
preferred, but these would not capture those sensitive regions where
the tracer needs to be represented accurately.
We use a Gaussian grid in the idealized test cases. To provide straightforward
information, we denote the spatial resolution in degrees. The idealized test cases
are run in a stand-alone application independently from ECHAM6, while the dust transport test in Sect. uses the AMR scheme incorporated as a module into ECHAM6.
In these idealized tests, we measure the numerical results
quantitatively in the ℓ2 and ℓ∞ error
norms:
19ℓ2=∑intcell(qi-qiexact)2dAi∑intcell(qiexact)2dAi,20ℓ∞=max|qi-qiexact|max|qiexact|,
where qi is the tracer mixing ratio in the ith cell,
qiexact is the exact solution in the ith cell, and dAi is
the cell area of the ith cell. In order to test the performance of
our AMR scheme, we do not apply any limiters to the scheme in
idealized tests. Hence, in the idealized tests, we do not preserve
positive tracer mixing ratio.
In many tests, we need to investigate the number of cells in a
simulation. The number of cells changes with time on adaptive
meshes. In order to show the overall number of cells in each test, we
average the number of cells over time:
N:=number of cells=∑tntntcellnt,
where nt is the number of time steps, ntcell is the number of
cells at time step t. The cell number can effectively and
objectively reflect the efficiency of the AMR scheme regardless of the
optimizations applied to the rest of the code, since the number of floating point operations in the transport scheme is directly proportional to the number of cells.
We use Δx→0 when we focus on the numerical
accuracy of the numerical scheme, while it is helpful to also look at the
efficiency of the numerical scheme using a plot with
N→∞.
Grid refinement for intermediate steps
As mentioned in Sect. , the dimensionally split
scheme requires the refinement of intermediate steps. Here, using the
solid body rotation test case as an example, we compare numerical
errors between two refinement strategies. One strategy refines
intermediate steps, whereas the other does not. The flow transports the
tracer around the globe with an angle of α=0 and
α=320π with respect to the equator. These two
settings lead to different maximum Courant numbers
|u|ΔxΔt, i.e., the speed of information
propagation in one time step. Here, |u| is the wind speed in the
longitudinal direction, Δx is the grid space in the
longitudinal direction, and Δt is the time step size.
In dimensionally split schemes, large Courant numbers can highlight
the displacement between intermediate steps and final results because
the information propagation is far away from the departure cell. When
α=0, there is no divergence in each dimension in the wind
field, and the AMR scheme allows arbitrarily large Courant numbers. We
use a Courant number of around 6 over the globe corresponding to a total
number of 13 time steps on a roughly 5∘× 5∘ Gaussian grid.
The total number of time steps doubles with doubled spatial resolution.
The dimensionally split scheme poses a limit to the time step interval
even if the two-dimensional wind field is divergence free, which is
given analytically on both AMR and non-AMR meshes. The
dimensionally split scheme essentially performs 1-D
semi-Lagrangian steps. The divergence-free wind field in 2-D can be
a result of the cancellation of 1-D divergence wind, where the 1-D divergence
wind field leads to crossing of trajectories in 1-D and limits the
time step interval.
When
α=320π, the maximum Courant number around poles
is 12 in the longitudinal direction, which is the largest Courant
number without the crossing of trajectories in 1-D. However, the tracer
does not cross poles. The maximum Courant number for the local tracer
is around 1.8, which is far smaller than the maximum Courant number
on the domain. This setup corresponds to a total
of 55 time steps on a roughly 5∘× 5∘ Gaussian grid.
In order to expose the difference in these two refinement strategies,
we use different spatial resolutions and keep the Courant number
roughly fixed. Note that the Courant number is not exactly the same on
different resolutions as the grid spacing changes with the
latitude. The AMR scheme uses a gradient-based refinement criterion.
When α=320π, the threshold for mesh refinement is
ϑr=10-3 and the threshold for coarsening is
ϑc=5×10-3. When α=0,
ϑr=5×10-6 and ϑc=5×10-5.
Illustration of the displacement of the numerical solution
between the intermediate step after update in latitudinal
direction and final results. The red distribution is the
intermediate step, and the black distribution is the final
result. When α=0, the flow
orientation is parallel to the equator and the Courant number is
around 6. When α=320π, tracer is affected by
a Courant number around 1.8.
In Fig. , we illustrate how both flow
orientations induce displacements between intermediate steps and final
results under both flow orientations on a mesh with
1.25∘× 1.25∘ spatial resolution. The displacement is
more visible when the tracer rotates along the equators due to
different Courant numbers.
Comparison of the
error of the solid body rotation test case after 12 d between
refinement with intermediate step and refinement without
intermediate step. Filled markers show results with refinement at
intermediate steps, and empty markers show results without
refinement at intermediate steps. The x axis is the maximum
resolution in the domain. Hence, one-level refinement and
two-level refinement has the same maximum resolution (the cosine bell
is covered by the same resolution), and only the coarsest
resolution is lower when using two-level
refinement.
Figure shows the numerical errors of
these two refinement strategies. The AMR results use the same maximum
resolution as the non-adaptive results. Hence, the base resolution of
AMR mesh is lower than the maximum resolution. When α=320π,
numerical errors and the convergence rate of these two refinement
strategies are comparable. Similar results arise from small
displacements between intermediate steps and final results as shown in
Fig. . Our local high-resolution areas
cover intermediate steps due to our sensitive refinement criterion.
Numerical errors show a significant
difference between these two
refinement strategies when α=0. Without refining intermediate
steps, the numerical error is higher on adaptive meshes than on
non-adaptive meshes because high-resolution information (the same
resolution as the non-adaptive meshes) is contaminated
on the low-resolution base mesh during the intermediate step. The AMR
scheme leads
to similar accuracy on adaptive meshes and non-adaptive meshes when
the numerical scheme refines intermediate steps. Our implementation
exposes the difference as the AMR scheme transports information from
the mesh for the previous time step to the mesh for the new time
step. Computations for both intermediate and final time step exist on
the mesh for the new time step.
Percentage of cell difference of cell numbers between
refinement of intermediate
steps and without intermediate steps when they use the same maximum
resolution with one-level refinement.
We show the difference of cell numbers between these two refinement
strategies in Fig. . Due to the large Courant number for
the case of α=0 the number of additional cells for intermediate refinement
is larger than for the case α=320π. Refinement
of intermediate steps leads to larger numbers of cells in general, but the
overhead of additional cells amounts to less than 10 %. Furthermore, the
additional cost of intermediate refinement is less significant or even negligible on
high-resolution meshes.
Our results demonstrate that dimensionally split schemes require
refinement of intermediate steps for better accuracy when the Courant
number is large. Although it is unlikely that the numerical model uses
an extremely large Courant number away from the poles, we refine
intermediate steps to ensure accuracy.
Numerical accuracy and efficiency
The transport scheme behaves differently under different initial
conditions and flow features. We examine the accuracy, efficiency and
mass conservation of our AMR scheme using three different test cases.
Non-divergent flow with local tracer distribution: the
solid body rotation test case
We examine our adaptive transport scheme in the solid body rotation
test case. The solid body rotation test case has discretely
non-divergent flow given analytically on both adaptive and non-adaptive
meshes. The non-divergent flow also does not severely
distort the tracer distribution and the gradient of the tracer does
not change during the test. Hence, we can test the numerical
properties in an ideal condition.
The test case uses a local tracer distribution with a radius of a
third of the Earth's radius. The test case allows us to initialize the
tracer distribution on high-resolution adaptive meshes. The AMR scheme
should result in very local high-resolution areas.
Snapshots of the solid body rotation test case when
α=0 and α=0.5π at each day with one-level
refinement. The coarse mesh has a resolution of
5∘× 5∘, and high-resolution areas have a
resolution of 2.5∘× 2.5∘.
We set the flow orientation as α=0, α=π4, and α=π2. When α=0, the tracer rotates around the globe
parallel to the equator. When α=π4, the flow leads
to a solid body rotation along the line, which is 45∘ with respect
to the equator. When α=π2, the flow leads
to cross-pole transport, which suffers from the geometrical problem of
Gaussian grids at poles.
We test these three flow orientations with a maximum Courant number
around 1 and 6. When α=0, the total number
of time steps is 84 for a maximum Courant number around 1. We use 13
time steps for a maximum Courant number around 6 on a spatial
resolution of 5∘× 5∘. When α=π4,
the total number of time steps is 1320 for a maximum Courant number
around 1 and 240 time steps for a maximum Courant number around 6
on a spatial resolution of 5∘× 5∘. When
α=π2. The total number of time steps is 1800 for
a maximum Courant number around 1 and 240 for a maximum Courant
number around 6 on a spatial resolution of 5∘× 5∘.
The AMR scheme utilizes a
gradient-based criterion. Our threshold for cell refinement is
ϑr=0.02 and the threshold for cell coarsening is
ϑc=0.015 when α=π4 and
α=π2, while the threshold
for α=0 is the same as in Sect. .
As shown in Fig. , the cosine bell is located in
the high-resolution area throughout the simulation, showing the
ability of the refinement criterion to detect the significant
regions. The large high-resolution areas are a result of the strategy
to refine intermediate steps.
The distribution of mesh cells explains the numerical accuracy of our
transport scheme on adaptive meshes. The discrete representation of
the non-zero tracer components is similar on high-resolution areas of
adaptive meshes and on the uniformly refined grid in case of equal
maximum resolution. This is illustrated in Fig. for α=0 and α=π2.
Convergence rate of the numerical results with respect to the
number of cells in the solid body rotation test
case.
Figure also shows that the AMR scheme demands
fewer cells than non-adaptive schemes to achieve similar accuracy. We
also note that higher-order refinement does not necessarily result in
fewer cells on the mesh. The solid body rotation test case uses a
cosine bell, which is not infinitely differentiable around the boundary of
the tracer, and we observe a second-order convergence rate. Hence, we
cannot observe the optimal convergence rate of the third order even if the
splitting error diminishes and the exact departure position is
computed when the cosine bell is transported along the equator.
Convergence rate of the numerical results with respect to the
number of cells in the solid body rotation test
case with α=π4.
Figure additionally shows the numerical
efficiency when α=π4. The 45∘ solid body
rotation test case poses a challenge to the dimensionally split scheme
as the FFSL scheme introduces splitting errors compared to the case
when α=0. The convergence rate is not severely affected since
the FFSL scheme has a second-order splitting error in time. Due to
the refinement of the intermediate time steps, the numerical errors on
adaptive meshes are comparable to the results on non-adaptive meshes.
Evolution of the cell number rotating around the equator
(left) and cross-pole transport (right) in the solid body rotation
test case with a resolution of 2.5∘× 2.5∘. The
solid line shows the cell number evolution with time when the
Courant number is small, and the dashed line shows the cell number
evolution with time when the Courant number is
large.
The Gaussian grid accumulates cells around poles. Since the refinement
area at the pole covers a larger number of cells, refinement generates
proportionally more refined cells when passing the poles. Figure illustrates this with maxima of the cell number
at times when the tracer passes the pole.
Evolution of the normalized numerical error for α=0
and α=π2 on two different resolutions
in the solid body rotation test case. The resolution for each
figure represents the highest spatial resolution on the mesh.
Figure shows the time evolution of the
numerical error in the solid body rotation test case. The numerical
error gradually grows with time. When the tracer crosses poles, the
ℓ∞ error clearly grows due to strong deformation on the
mesh. On high-resolution meshes, the ℓ∞ error is higher
than ℓ2 error since oscillations in the numerical solutions can
only be shown in a more sensitive metric. On low-resolution meshes,
the ℓ2 error is comparable to or larger than the ℓ∞
error, mainly because of larger numerical oscillations, which can be
captured by the ℓ2 error. There is no observable difference
between non-adaptive meshes and adaptive meshes. The results are
consistent with Fig. .
CPU time per time step compared to the cell number. The left
figure indicates the CPU time per time step for the transport scheme,
while the right figure shows the percentage of the CPU time per time
step used for mesh refinement compared to the total CPU time.
To demonstrate the efficiency of the AMR, we also present a CPU
time per time step in serial runs in Fig. . The
code is run on one CPU of a Dual-Core Intel Xeon E5-2697A, 2.6 GHz machine.
Even though our
current transport scheme implementation is not fully optimized,
the CPU time per time step is nearly
linear with respect to the number of cells. Figure also
shows that
the CPU time per time step for mesh refinement is relatively fixed
compared to the total CPU time per time step and that the higher refinement
level consumes more time. We need to note that the CPU time for the
numerical scheme can be further reduced with better implementation (e.g.,
avoiding frequent memory (de)allocation.). We note further that with an overhead for the refinement of currently
approx. 30 %–40 % of the total computing time of the transport scheme,
the refined features need to be local to gain computational benefit from AMR (as indicated
in Fig. ).
In summary, we explored the numerical accuracy, efficiency, and
convergence rate of the adaptive transport scheme in an ideal context,
where we use a high-resolution initial condition and a non-divergent
wind field. Our adaptive transport scheme, using reduced numbers of
cells, achieves similar accuracy to the original scheme on
non-adaptive meshes.
Divergent flow with local tracer distribution: the
divergent test case
We test our AMR scheme in the divergent test case. The magnitude and
the direction of the wind change swiftly in a divergent flow. The
swift change in wind challenges the accuracy of our semi-Lagrangian
scheme, which needs the correct departure position. Furthermore it may
reveal inexact mass conservation, since the tracer mixing ratio will
change to compensate for converging or diverging trajectories.
In this test case, background flow transports two cosine bells along
the equator, while the divergent flow stretches them. From day 6 on,
the test case reverses its direction and the tracer theoretically restores
to its initial state. The final tracer distribution at
day 12 is the same as the initial condition. There is no analytical
solution for the test case, but we can compare the final state with the
initial condition to obtain a quantitative error.
Similar to the solid body rotation test case, the tracer distribution does not cover the entire domain but only limited areas. However, the size of the tracer is larger in the divergent test
case than in the solid body rotation test case. The AMR scheme might
need more grid cells to cover the whole tracer. To compare numerical
properties of the AMR scheme and non-AMR scheme, we assign a given
wind field on adaptive meshes exactly instead of using wind
interpolation.
We initialize the tracer distribution on the high-resolution areas and
use a gradient-based refinement criterion. Our threshold for the
refinement is ϑr=0.2, and the threshold for the coarsening
is ϑc=0.15.
Numerical results of the divergence test case with a
resolution of 5∘× 5∘ in the left panel and one-level refinement in the right panel. The maximum resolution is
2.5∘× 2.5∘. The Courant number is around 1.
In the divergent test case, we take three steps to verify the
performance of our AMR scheme.
We first run the test case with and without one-level refinement
using a Courant number around 1 and a resolution of
5∘× 5∘ and investigate the representation of the
tracer on a high-resolution mesh. This test requires 120 time steps on
non-adaptive meshes and 240 time steps on adaptive meshes.
As shown in Fig. , the refinement criterion
captures the tracer completely. The asymmetry in the high-resolution
area at day 0 is a manifestation of the refinement of intermediate
steps based on the initial wind field. As the tracer gets stretched
during the runtime, the high-resolution area leads to a better
representation of filaments. The final tracer distribution is not
completely the same as the initial condition, which is a result of
numerical damping and distortion.
Secondly, we use multiple levels of refinement to verify the
sensitivity of the refinement level to the numerical accuracy and
efficiency. The AMR scheme runs with an initial resolution of
20∘× 20∘. The refinement on adaptive meshes ranges
from two-level refinement up to five-level refinement, resulting in a
resolution of up to 0.625∘× 0.625∘ using a Courant
number around 5, which corresponds to 24 time steps in
a 5∘× 5∘ mesh.
As shown in Fig. , we observe a
similar convergence rate between uniformly refined meshes and
locally refined meshes. Our results show that the AMR scheme and the
non-AMR scheme generate numerical results with similar accuracy
where the AMR scheme requires only a reduced number of cells in the
divergent flow.
Thirdly, we inspect another aspect of numerical accuracy: mass
conservation. We show the evolution of relative mass change in the
divergent test case when the maximum resolution is
0.625∘× 0.625∘ with no adaptive refinement and one-level refinement with a coarse resolution of
1.25∘× 1.25∘. We define the relative mass change as follows:relative mass change=mass-massmeanmassmean,where mass is the mass at individual time step and
massmean is the temporal average of the mass in all
time steps. The relative mass shows the deviation of the
mass at one time step compared to the time-averaged mass.
We observe that mass is conserved without AMR in Fig. . However, mass declines with AMR experiments. After
960 time steps, the loss of relative mass change is on the order of
10-12 and the mass is greater than time-averaged mass initially.
The loss of mass arises from the accumulation of
rounding error of floating-point calculation with time in the
computation of geometrical information in AMR
procedures. Nevertheless, the mass variation in each time step is at
machine precision, which is on the order of 10-16.
Convergence rate of the numerical results with respect to
the number of cells in the divergent test case using the same
initial spatial resolution with multiple refinement
levels.
Evolution of mass change on both non-adaptive (a) and
adaptive (b) meshes. Note that we do not plot the mass error but the
mass with respect to the average, which explains the initially non-zero
value for the adaptive run. The loss of mass arises from the accumulated
floating point rounding error with time on adaptive meshes. The mass variation in each
time step is at machine precision (on the order of
10-16).
Summing up, our adaptive transport scheme is capable of accurately
handling the divergent flow on adaptive meshes. The numerical error is
nearly the same on non-adaptive meshes as on adaptive meshes, and the
scheme conserves mass in each time step. The heuristic gradient-based
refinement criterion controls the mesh distribution by capturing the
relevant tracer field and improves the efficiency of the numerical
simulation. Better error estimators may further improve computational
efficiency. The test case demonstrates that our adaptive transport
scheme is able to be used in realistic simulations.
Non-divergent flow with global tracer distribution: the
moving vortices test case
The moving vortices test case is a challenging test case for
AMR. Numerical accuracy on adaptive meshes and globally refined meshes
is similar regardless of the feature of the flow when we use local
tracer distributions as shown in Sect. and
. The moving vortices test case utilizes a
global tracer distribution. To avoid global refinement in our AMR
runs, the goal of our AMR scheme is to improve the local
representation of the tracer distribution in vortices instead of
improving the numerical accuracy globally.
As the vortices in this test case develop with time, local refinement
is not present at initial time steps. Our numerical experiments use
low-resolution initial condition, which is different from experiments
in Sect. and . The
moving vortices test case allows us to mimic the setting in our
targeted applications in ECHAM6 as described in Sect. . Figure shows the effect
of omitting grid refinement around poles due to the wind
interpolation.
Numerical results of the moving vortices test case at the
final time step on a lat–long plane, indicating that the cells around
poles are not refined. The numerical results have the resolution of a
5∘× 5∘ coarse grid with one-level refinement
and an interpolated wind field.
Numerical results of the moving
vortices test case. The left column shows the numerical results on the
a resolution of a 5∘× 5∘ coarse grid. The right
column shows the numerical results on the resolution of a
5∘× 5∘ coarse grid with one-level refinement
and an interpolated wind field.
To investigate errors from coarse initial conditions and wind fields, we
examine three different settings. (1) We set up numerical experiments,
where the initial condition and wind field is defined analytically on
grid cells. (2) We run AMR experiments with one-level and two-level
adaptive refinement, where coarse initial
condition and interpolated wind field from initial refinement levels
are used. (3) We also set up experiments using uniform refinement with
coarse initial condition and wind interpolation. Here, uniform
refinement refines all cells on the mesh, leading to a higher global
resolution than the coarse mesh, such that the third experiment
setting can be used as a reference solution to experiment 2 because both
experiment 2 and 3 use the interpolated wind field from coarse
meshes.
In all experiment settings, we set α=π4 and test
the numerical scheme with both large and small Courant numbers on
various resolutions. On a mesh of 5∘× 5∘, the test
requires 1320 time steps for a small Courant number and 240 for large Courant numbers.
On adaptive meshes, the refinement threshold for the gradient-based
refinement criterion is ϑr=0.8 and the coarsening threshold is
ϑc=0.4. The threshold in this test case is more relaxed than
in the solid body rotation test case due to the strong deformation
arising from the vortices. We use the same gradient-based criterion
with different thresholds for all idealized test cases. This avoids
focusing on the choice of the refinement criterion in this study and
focuses on the effect of AMR in the transport module of an existing
model. We expect that the choice of a refinement criterion requires
further investigations, especially in operational settings, to
maximize computational efficiency and accuracy.
We show snapshots of the numerical solution at
5∘× 5∘ coarse resolution and one-level refinement in
Fig. . The refinement criterion captures
the development of the vortices. Finer grids reduce the error around the
steep gradient induced by the vortices. The filaments of the tracer
are not identifiable in low-resolution simulations, but high-resolution
simulations can capture the fine-scale feature in the tracer field
such that we resolve finer filaments. The adaptive transport scheme
refines the regions where vortices appear.
The large refinement area in Fig. is a result
of the gradient-based refinement criterion, which is sensitive to the
accumulation of grid cells around the poles. The less tailored
refinement criterion still shows improved efficiency for the idealized
test cases.
Our results indicate that AMR can improve local accuracy of numerical
results even if the scheme can only access coarse grid information,
which is consistent with the results from .
Convergence rate of the numerical results in the moving
vortices test case on adaptive meshes using a coarse initial
condition and interpolated wind except for zero-level
refinement.
As shown in Fig. , errors from the
initial condition and wind interpolation do indeed contribute to the
overall error. While the results with the initial condition and wind on the same resolution behave similar when refined adaptively or
uniformly, using a high-resolution initial condition and wind with
uniform mesh shows better accuracy. A higher level of refinement means
a lower-resolution initial condition and thus a larger contribution of
the interpolation error. On the other hand, even with low-resolution
initial conditions and wind, higher adaptive resolution improves the
results due to the improved ability to resolve filamentation.
The convergence rate of the numerical scheme using zero-level refinement
is as expected. The numerical scheme can be third order, as shown in
Fig. in the solid body rotation for
optimal conditions, i.e., smooth tracer distribution and constant
wind field. In low-resolution runs, the scheme shows a convergence
rate between the first and second order due to the sharp gradient arising
from the vortices, which is consistent with the results from
, who used basically the same scheme with a Courant number of
less than 1. Although tested the scheme with
α=0, our results also show similar numerical accuracy using
zero-level refinement. The curved convergence rate toward its best
performance in this test case is also observed by
using a different numerical scheme.
Differences of numerical errors between non-adaptive meshes
using exact initial conditions and exact wind fields and
adaptive or uniformly refined mesh using a coarse initial condition and
interpolated wind field in the moving vortices test
case.
To highlight the effect of wind interpolation, we present the
difference of numerical errors between the standard test case, where
data (wind and initial conditions) are given at finest grid resolution,
and tests using coarse data interpolated to the finest grid level in
Fig. . Uniform refinement using coarse
data leads to additional errors where two-level refinement, which uses
data that is 2 times more coarsely resolved than the exact initial
condition, shows larger errors than one-level refinement. AMR and
uniform refinement expose similar behavior with a slight advantage in
some situations for uniform refinement. The error due to wind
interpolation is generally one to two orders of magnitude smaller
than the solution error (cf. Fig. ),
indicating that even with interpolated data AMR leads to accurate
results with low computational effort.
Although the coarse initial distribution reduces the effect of
refinement, using the high-resolution mesh still results in better
numerical accuracy than only using the low-resolution mesh. Coarse
input wind reduces the numerical accuracy. However, we still observe
convergent and accurate numerical results using the AMR scheme. Our
AMR scheme can improve the numerical accuracy using fewer grid cells
than uniformly refined mesh when we integrate it into the tracer
transport module of an existing coarse resolution model.
A realistic test case: simulation of dust
transport
The tracer transport process exhibits multi-scale features in climate
simulations. As indicated in Sect. ,
low-resolution simulations cannot represent fine-scale features of the
tracer transport processes. Improving the local representation of the
tracer transport scheme can therefore reduce at least one source of
error in climate models. On the other hand, the tracer transport
process plays an important role in climate systems. The transported
gases and aerosols have a significant impact on the state of climate
through solar radiation . For example, carbon dioxide
is one of the major driving factors of anthropogenic climate
change. Volcanic ashes have a cooling effect on the global
temperature. Hence, better tracer transport simulations can improve
overall results in climate simulations.
We select dust to test our adaptive transport scheme in realistic
settings. Dust has evident local origins like the Sahara and it
can traverse across long distances while retaining local features because
the atmospheric flow can lift dust to higher levels
. Emission and deposition parameterizations have less
impact on higher-level aerosols. Hence, dust simulations are suitable
to demonstrate the advantages of using AMR.
We test our AMR scheme while maintaining a non-adaptive coarse climate
model to which our AMR scheme is coupled in a one-way fashion. The
one-way coupling prevents our tracer from interacting with other
components of the climate model such that we can compare the
difference between our adaptive tracer transport scheme and the
original scheme using our conclusions from Sect. .
The host model: ECHAM-HAMMOZ
We integrate our adaptive tracer transport scheme into ECHAM6 without
breaking its current code structure. Further, the structure of ECHAM6
can also provide insight into numerical results of our simulation of
dust transport. Hence, it is necessary to understand the model.
ECHAM6 is the atmospheric component of the Earth system model MPI-ESM
. It is composed of several components: the dynamical
core, the physical parameterizations, and a land surface model (JSBACH).
The dynamical core solves hydrostatic primitive equations of the
atmosphere, which describe the motion of air and assume absence of
acceleration in the vertical. The dynamical core in ECHAM6 was
originally derived from an early version of the atmospheric model
developed at the European Center for Medium-Range Weather Forecast
. ECHAM6 also applies a terrain-following coordinate to
accommodate the varying orography at the bottom of the atmosphere. The
terrain-following coordinate is a hybrid coordinate . Both
the passive tracer transport scheme and the parameterizations in
ECHAM6 are computed on a Gaussian grid using the flux-form
semi-Lagrangian scheme, which we discussed in detail in Sect. . ECHAM6 also includes various
parameterization schemes, including convection, cloud, radiation, and
vertical diffusion. The land surface model comprises a class of
parameterizations that provides the properties of land surface for
other components of the climate model.
ECHAM-HAMMOZ is a coupled model that combines ECHAM6 and HAMMOZ, where
ECHAM6 is flexible enough to host various sub-models. The sub-model HAMMOZ
provides a class of aerosol and atmospheric chemistry modules
that predict the evolution of aerosols and trace
gases. In our application, we focus on the evolution of the dust
mixing ratio. ECHAM-HAMMOZ divides tracers into seven different modes
. These modes are dependent on the size and solubility
of the particles. There are four different modes for dust:
accumulation mode mixed (DU_AS), coarse mode mixed (DU_CS),
accumulation mode insoluble (DU_AI), and coarse mode insoluble
(DU_CI). HAMMOZ describes the emission, diffusion, dry deposition,
wet deposition, cloud scavenging, and sedimentation of these tracers.
Tendency equation of dust
concentration
We replace the 2-D tracer transport scheme in ECHAM6 with our proposed
AMR scheme. However, the evolution of the dust mixing ratio in a climate
model is more complicated than a 2-D tracer transport equation. The
large-scale temporal changes in dust mixing ratio are not only
controlled by tracer transport but also affected by various other
parameterizations. The large-scale temporal changes in the tracer
mixing ratio are also referred to as the tendency of the tracer mixing
ratio.
In this section, we present the tendency equation of the dust mixing
ratio in ECHAM6. In addition, we also present our implementation when
integrating our adaptive transport scheme to ECHAM6.
Numerical treatment of tendency equation in ECHAM6
ECHAM6 describes the tendency equation of the tracer mixing ratio
using the following equation:
∂ρc∂t+∇⋅(ρcu)=F.
Here ρ is the air density, c is the tracer mixing ratio, the
combination of ρc is the density of the tracer in the air,
∂ρc∂t is the tendency of the tracer
density, ∇⋅ is the three-dimensional divergence operator, and F
represents external forcings. In climate models, the tracer mixing
ratio c represents the mixing ratio, which is the mass of the aerosol
or gas relative to the mass of dry air. The unit of the mixing ratio is
kgkg-1.
The forcing term includes the vertical diffusion, dust emission, dry
deposition, wet deposition, sedimentation, and cloud scavenging
process. The wet deposition process also involves the convective and
cloud processes. Hence, the forcing term is a collection of
parameterizations.
ECHAM6 uses η coordinates as follows:
ηk+12=Ak+12p0+Bk+12pk+12=Ak+12+Bk+12ps.
where k is the kth vertical layer, A and B are constant
coefficients, and ps is the surface pressure.
The transport equation under hybrid η coordinate is as follows:
∂∂t∂p∂ηc+∇⋅∂p∂ηcu+∂∂η(η˙∂p∂ηc)=0,
where the velocity vector u is the horizontal velocity vector,
the vertical velocity is η˙, and η∈[0,1]. The
boundary condition for the equation is η˙=0 at η=0
and η=1.
Integrating both sides of Eq. () over
η and using the finite-difference method, the tendency equation
in hybrid coordinates is as follows:
26∂Δpkck∂t+∇⋅(Δpkckuk)=F,27∂Δpk∂t+∇⋅(Δpkuk)=F,
where Δpk is the pressure at the kth layer, ck is the
tracer mixing ratio at the kth layer, and uk is the
horizontal wind vector at the kth layer.
The FFSL scheme solves the vertical transport separately in the
hydrostatic model . As our mesh refinement runs on a 2-D
mesh and keeps the vertical mesh fixed, the vertical transport
subroutine of ECHAM6 is reused. In ECHAM6, the surface pressure is
ps=∑kΔpk and the pressure at each layer is
pk=ps-∑kΔpk. This leads to an inconsistency
between pk and the definition of pressure levels in Eq. (). To solve the problem and the vertical
transport, ECHAM6 uses the technique introduced in
and PPM remapping. We reuse the vertical remapping subroutine in the
original ECHAM6 without any modifications in the AMR scheme.
The FFSL scheme actually used in ECHAM6 leads to more diffusive
results due to some modifications making it computationally less
expensive than the scheme presented in Sect. . For
example, the FFSL scheme in ECHAM6 uses a first-order Godunov scheme
as the inner operator and a third-order piecewise parabolic method
(PPM) as the outer operator instead of the third-order PPM for both
inner and outer operators. In ECHAM6, the scheme includes limiters to
ensure the positivity of the numerical results and averages over the
longitude bands around the poles to avoid pole problem. We reuse these
limiters in our experiment in this section for the realistic dust
simulations. Note that we do not apply any limiters or special
treatment around the poles in Sect. .
Refinement strategy
One of the benefits of integrating AMR into an existing model is that
we do not need to implement and design a new model with the AMR
technique. Rather, we can reuse most components of the existing
model. In realistic dust simulations, we only need to replace the
horizontal tracer transport scheme by our adaptive scheme.
The hydrostatic primitive equations require the vertical integration
of a column over each cell. Hence, for simplicity, instead of refining
the mesh in 3-D, we only refine the horizontal 2-D mesh, obtaining
locally smaller columns. Using 2-D refinement enables us to reuse the
vertical tracer transport scheme without any modification.
As we integrate AMR into the passive tracer transport module without
any modification in other components, the passive tracer transport
module always gets wind, pressure, and passive tracer mixing ratio on a
coarse grid. High-resolution wind can therefore only be obtained by
interpolation from a coarse grid. Similar to the treatment of wind in
Sect. , we use a bilinear interpolation. As our
aim is to demonstrate the applicability of AMR for a single tracer
transport module, we apply an absolute value refinement criterion
instead of a gradient-based criterion here to enforce the generation
of high-resolution regions even when dust mixing ratio are low (but
present). Therefore, we use the absolute value of ρc as a
refinement criterion. When N tracers are simulated in ECHAM6, the
refinement criterion is mini(∑lρci), where l is the
vertical level and i=1,…, N corresponds to the tracer
components. Thus, for each column we first take the sum of the density
of each tracer for all vertical levels in a single column, and then we
take the minimum value of the N tracers as the refinement
criterion. We apply a refinement threshold of
ϑr=10-11kgkg-1=10-5mgkg-1 and a coarsening threshold of
ϑc=10-12kgkg-1=10-6mgkg-1.
Results of one-way coupling dust
simulation
We test our adaptive tracer transport scheme with realistic dust
mixing ratio data using one-way coupling; i.e., we get coarse
resolution wind and pressure as input data at each time step. During
the simulations, we do not map the dust mixing ratio back to the
coarse resolution mesh used by other components. Therefore, the dust
mixing ratio does not affect other components of the climate model,
especially pressure and wind field. This corresponds to the situation
in the idealized simulations of Sect. with
realistic data.
The dust mixing ratio is always simulated on adaptive meshes. Since
the parameterizations compute the tendency of tracer mixing ratio in
columns, our adaptive scheme can accommodate the use of the existing
parameterizations.
Experiment setting
In our one-way coupling experiments, parameterization schemes running
on coarse-resolution meshes should affect the dust mixing ratio on
adaptive meshes. Our implementation (refining columns) is aware of the
original ECHAM6 parameterizations and is a positivity-preserving method,
leading to a compatible dust transport.
We can illustrate our treatment using a differential equation:
DcAMRDt=F(Xcoarse,cAMR),
where DDt is the material derivative, cAMR is
the tracer mixing ratio of the AMR scheme, F is a parameterization
scheme, and Xcoarse is a vector of variables involved in
the parameterization scheme other than the tracer mixing
ratio. Therefore, our one-way coupling always uses coarse-resolution
parameters for parameterization schemes even if our tracer mixing
ratio is at a higher resolution. We can achieve such an implementation
since parameterization schemes run within each column of the
horizontal mesh. The flowchart in Fig.
illustrates this approach.
Illustration of our setting for the one-way coupling
experiment. c is the tracer mixing ratio on the coarse
resolution, Xcoarse is a vector of variables other
than the tracer mixing ratio in the model at a coarse
resolution, and cAMR is the tracer mixing ratio of the
AMR scheme. The rectangles include modules and processes in the model,
ellipses are the output of each module or process, and arrows indicate the
input variables in each module or process.
ECHAM6 provides a variety of options for the parameterization
schemes. Although there are default settings for most
parameterizations, we use some non-default options to simplify our
experiment. In our experiment we use a vertical resolution of 31
layers, (L31), corresponding to a model top at 10 hPa. Hence,
ECHAM6 does not compute the mid-atmosphere in our experiments.
In order to perform dust emission, we turn on the ECHAM-HAM submodel
while muting the chemistry and MOZ1.0 (Schultz et al., 2018) submodel for simplicity. In our
experiment, we also use the dust scheme proposed by
and omit the additional Saharan and East Asian dust sources in the
default settings.
We also set all agricultural and biogenic emissions as inactive, including
forest fire and volcanic ashes. Hence, we only have emissions of dust
species from the dust emission parameterizations. With this setting we
simulate the dust evolution during the period of 1 to 31 October 2006 as there were dust emission events in the Sahara during this
month.
Comparison between low-resolution and high-resolution
simulations
We expect that high-resolution simulations can represent climate
states with higher quality. High-resolution climate models better represent not only
the initial conditions but also the boundary
conditions, such as the topography and different types of land
surface.
Our AMR scheme increases the resolution of the passive tracer
transport scheme. However, our scheme can improve neither the initial
condition nor the representation of the boundary
conditions. Nevertheless, it is still of interest to compare the dust
mixing ratio on a low spectral resolution of T31L31
(3.75∘× 3.75∘ in degrees) and a higher resolution of
T63L31 (1.875∘× 1.875∘ in degrees) configuration
such that we can understand the difference between high- and
low-resolution simulations.
We adopt the default time step setting in ECHAM6. In the T31
resolution, the time step length is 1800 s, while in the T63
resolution it is 450 s. In the following
experiments, we use the time step configuration based on the coarsest
component of the model.
Dust mixing ratio of DU_AI
(mgkg-1) at 800 hPa on 3, 6, 12,
and 15 October using model resolutions of T31L31 (left) and
T63L31 (right). The dust mixing ratio is masked due to high
altitude in areas such as the Tibetan Plateau.
We present the dust mixing ratio of DU_AI in Fig. . The Saharan air layer as a large-scale system
is assumed to lift and transport dust up to a height of 5 km
. In order to capture the transport of dust without
interference from the emission in lower levels, we show the dust
mixing ratio of DU_AI at 800 hPa.
The simulation at a uniform resolution of T31L31 shows dust appearing in the
800 hPa layer after 3 October. The wind field transports dust
westward toward the Atlantic Ocean. After day 9, the dust
mixing ratio increases in East Asia and gradually moves
southwestward. The uniform high-resolution T63L31 simulation shows quite different patterns.
There is a high dust mixing ratio at the east and west of
North Africa on 6 October, and we cannot observe such high
dust mixing ratios using low-resolution simulations. Although both dust
simulations show a westward transport, the pattern of the dust
distribution differs significantly. For example, hardly any dust
disperses in East Asia in the high-resolution simulations.
These simulations show an important fact of multi-physics simulations: there exist
sub-grid-scale parameterizations that inhibit convergence in a classical
mathematical sense. The differences between T31 and T63 horizontal resolution
simulations are caused by both the increased resolution in the dynamical core and the necessary change in parameterizations due to the increased resolution.
In particular,
showed that the dust emission scheme is sensitive to different
horizontal resolutions. The observed dust mixing ratio is also affected
by wet and dry deposition, which itself is affected by cloud and
convection parameterizations. These results indicate that we cannot use a
high-resolution simulation as a converged-state quasi-reference solution.
Our analysis of accuracy will therefore be more subtle.
Since we will add AMR only to the tracer transport, our comparison will
be focused on differences in filamentation of tracer clouds and the
resolution of sharp gradients. Our scheme cannot compensate for insufficient
scale-awareness of the parameterization, and we will rely on the given
parameterization schemes.
Comparison between low-resolution and adaptive meshes
There are multiple sources of uncertainties in low-resolution
simulations. The coarse initial condition and boundary condition can
lead to less accurate results, while the coarse resolution dynamical
core and parameterizations cannot resolve the finer features of the
atmosphere.
The results from our idealized tests in Sect.
show that using AMR in the tracer transport module can effectively
reduce the numerical error of the tracer transport process. Using an
interpolated wind field with a coarse-resolution initial condition can
still improve the numerical accuracy of passive tracer transport
schemes. It is promising that we can treat one source of error by
using AMR in coarse resolution climate simulations.
Since we observed in the previous paragraph that uniform refinement of
the whole atmosphere model does not yield a converged solution that is usable as
a reference, we adopt the following approach. We will use a dust transport
scheme run on a uniform high-resolution T63 grid, coupled to a coarse
T31 dynamical core with corresponding low-resolution parameterizations.
This solution, shown in the left panel of Fig. , will serve as a reference for our adaptive mesh simulations.
Compared to low-resolution simulations, we observe that uniformly
refined meshes show less diffusive results. Dust mixing ratio is
higher than in low-resolution simulations, while the filaments of the
dust distribution are more obvious. Even with a low-resolution dynamical
core and parameterization, the higher-resolution tracer
transport leads to reduced numerical diffusion and thus better-quality
simulation results.
Dust mixing ratio of DU_AI
(mgkg-1) at 800 hPa on 3, 6, 12,
and 15 October based on a coarse model resolution of T31L31.
The entire model runs on T31L31 with the
tracer transport module at a doubled resolution in the left panel, while the dust transport
is on adaptive meshes in the right panel.
Now, we take the uniformly refined transport module mesh as the benchmark for our adaptive
mesh refinement. Our results in Fig. show that
AMR captures the appearance of dust very well. The results for
uniformly refined meshes and adaptive meshes are very similar,
indicating that using AMR for only one component can improve the
accuracy of the simulation.
We also observe large refined regions in Fig. .
The size of the refined regions is a result of the thresholds used in the
refinement criterion. Further optimization of refinement criteria could
potentially alleviate this in future applications.
However, a more important reason is that the mesh is refined only horizontally.
Therefore, even if a significant amount of tracer concentration is only present in
a lower (or higher) level of the atmosphere, the refinement is performed on all levels.
Finally, another reason for such large refined regions is that four different
dust tracers share the same adaptive mesh. Using different adaptive
meshes can be desirable when the number of tracers is high, but it can
affect the reuse of the departure point computations. One of the
benefits of multi-tracer efficiency in the semi-Lagrangian scheme
arises from its capability to reuse departure points of
trajectories. As a compromise, putting tracers into groups sharing the
same (adaptive) mesh may achieve a better balance between the individual
adaptivity of meshes and the multi-tracer efficiency in
semi-Lagrangian schemes.
We note that even with the non-optimal refinement criterion the
one-way coupled dust simulation on an adaptive mesh requires 9062
cells on average over the 30 d simulation, while the uniformly high-resolution transport
mesh requires 17 280 cells. This difference highlights the potential
efficiency gain from adaptive mesh refinement.
Dust mixing ratio of DU_AI (mgkg-1) at
800 hPa on 3 and 6 October at a model resolution of
T31L31 using our modified transport scheme in the region of
[10∘ S, 50∘ N] × [30∘ W, 90∘ E]. In the left panel, the entire model runs on T31L31. In the right panel, the dust transport is on adaptive meshes and the rest of the model is on T31L31. The insets in the right column
show the mesh distribution.
In order to show the difference between the local-resolution runs and
adaptive runs, we show a local tracer distribution in North Africa in
Fig. , which highlights the less diffusive
and more pronounced tracer mixing ratio in high-resolution regions.
Our results show that integrating AMR into a passive tracer transport
scheme can effectively reduce errors even if we do not use high-resolution data for other components.
Conclusions
We propose a new approach toward adaptivity in climate models. Our
method is different from the traditional AMR approach, which
constructs a completely new climate model using AMR. Our approach
overcomes the difficulty of integrating AMR into operational climate
models. We integrate an AMR passive tracer transport module into the
existing atmospheric model ECHAM6. Partially integrating AMR into the
existing climate model improves accuracy and efficiency in operational
climate simulations.
We demonstrate the effectiveness of our approach by simulating dust
transport processes in ECHAM6.
In a first step, we find that running the tracer transport module on a
uniformly refined mesh improves the quality of the results. Adding adaptive mesh
refinement yields similar high-resolution accuracy with improved
efficiency, since our AMR approach avoids mesh refinement
of the entire globe and successfully captures regions where
high-resolution meshes are necessary.
Since we apply only one-way
coupling, high-resolution simulations improve the accuracy of dust
transport processes, but the general accuracy of the climate simulation
remains limited by the coarse spatial resolution of other components, such as
the dynamical core and parameterizations. This approach
allows us to rely on the general model infrastructure, such as parameterization schemes and
vertical convection schemes.
Our idealized tests indicate that the AMR approach can potentially be
as accurate as global high-resolution simulations when the tracer is
present at local areas and the AMR scheme can access the exact wind
field. Reducing local numerical errors can improve the overall
accuracy of numerical solutions. Our AMR scheme leads to superior
accuracy and efficiency compared to non-adaptive schemes.
Enabling AMR in existing climate models relies on several techniques
proposed here: adequate AMR enabled transport schemes, refinement
strategies, and transparent data structures, which were described in
. These techniques can be applied in a wider context than
the applications shown here.
Our modification to the widely used flux-form semi-Lagrangian (FFSL)
scheme in ECHAM6 allows the transport scheme to be used on adaptive
meshes while retaining its important properties, i.e., being dimensionally split and
mass conserving and featuring semi-Lagrangian time stepping. Preserving the
dimensionally split property results in efficiency and numerical
compatibility between the new AMR and the original scheme. Mass conservation
is essential for climate models as an unphysical numerically induced mass
variation in transport processes could accumulate over the long simulation
cycles of climate models. The semi-Lagrangian time stepping is
particularly useful for AMR because it can use a uniform
time step on multi-resolution meshes without any stability
issues. Hence, similar to the original FFSL scheme, our AMR scheme is
a candidate for more complex systems .
We also demonstrate the effectiveness of the proposed refinement strategy for
dimensionally split schemes. Our AMR strategy ensures that
high-resolution information remains highly resolved over the whole
propagation cycle from departure cell to target cell, which in turn
guarantees the accuracy of numerical
results. Thus, our AMR strategy results in accurate simulations, as
discussed in Sect. . The mentioned properties of
our AMR-enabled FFSL transport allow for a transparent replacement of
existing non-adaptive transport modules in climate models.
We expect that our results from dust simulations are applicable to
other aerosols and gases as well. However, more rigorous
investigation is needed. It is still of interest to explore
two-way coupling, where aerosols on adaptive meshes have an impact on
processes such as cloud formation, radiation, or pressure. The
development of two-way coupling would require the retention of
high-resolution information on the low-resolution mesh, i.e.,
effective upscaling. Averaging can lead to the loss of some fine-scale
features, so more sophisticated multi-scale methods to upscale
high-resolution information to low-resolution meshes need to be applied
e.g.. These upscaling methods are in a sense the reverse of
AMR.
While two-way coupling is still not available, this study provides a
first step towards full functionality of AMR
approaches in climate models.
Our method may also be extended to more components of climate
models. To achieve full operability our AMR scheme requires
additional work on code optimization and parallelization.
An alternative possible use of AMR
could be dynamical coarsening of the mesh for a single
component. Dynamical coarsening can circumvent the limitation of
coarse initial conditions and parameterizations. However, this may
require extended data structures.
Our approach provides an AMR-enabled transport module with transparent data structures and numerical properties
similar to the original scheme, which allows us to include component-wise AMR into existing climate models.
This reduces the time of development significantly compared to constructing
a complete new AMR climate model and opens an evolutionary path towards
AMR-enabled climate modeling.
Code and data availability
The code for running and plotting idealized
tests in Sect. is available from
10.5281/zenodo.4013277 under the GNU
General Public License v3.0. The results from realistic test cases
in Sect. are generated from our modified
version of ECHAM-HAMMOZ. The code for the realistic test cases can be
made available per individual request, and the source code has been
made available to the editor. Our modified ECHAM-HAMMOZ model and
the input data are both under the ECHAM-HAMMOZ
license
(https://redmine.hammoz.ethz.ch/projects/hammoz/wiki/1_Licencing_conditions, last access: 29 April 2021). The original model of echam630-ham23-moz10
is also available
(https://redmine.hammoz.ethz.ch/projects/hammoz, last access: 29 April 2021). The input data are available at
https://redmine.hammoz.ethz.ch/projects/hammoz/wiki/V0002 (last access: 6 July 2020).
Author contributions
YC developed the model code and
performed the simulations. This article is mainly derived from parts
of his PhD thesis titled “A New Approach toward Adaptivity in
Climate Models” at Universität Hamburg, Germany, where the
co-authors supervised the PhD work. The thesis is available at
https://ediss.sub.uni-hamburg.de/volltexte/2020/10266/pdf/Dissertation.pdf (last access: 28 April 2021). JB and KS contributed scientific guidance
and prepared the manuscript with all co-authors.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This work was supported by German Federal Ministry of Education and
Research (BMBF) as part of the Research for Sustainability initiative (FONA);
http://www.fona.de (last access: 28 April 2021) through Palmod project (FKZ: 01LP1513A). We
also acknowledge support by the Cluster of Excellence CliSAP
(EXC177), Universität Hamburg, and Germany's Excellence Strategy
– EXC 2037 “CLICCS – Climate, Climatic Change, and Society” –
(project no. 390683824), a contribution to the Center for Earth
System Research and Sustainability (CEN) of the Universität Hamburg,
both funded by the German Science Foundation (DFG). This
work has also been partially supported by the completion scholarship at
Universität Hamburg. Yumeng Chen has also been funded by the UK Natural
Environment Research Council award NCEO02004.
The ECHAM-HAMMOZ model is developed by a
consortium composed of ETH Zurich, Max-Planck Institut für
Meteorologie, Forschungszentrum Jülich, the University of Oxford,
and the Finnish Meteorological Institute and managed by the Center
for Climate Systems Modeling (C2SM) at ETH Zurich.
Financial support
This research has been supported by the Bundesministerium für Bildung und Forschung (grant no. FKZ01LP1515D), the Cluster of Excellence CliSAP
(EXC177), Universität Hamburg, and Germany's Excellence Strat45
egy – EXC 2037 “CLICCS – Climate, Climatic Change, and Society”
(project no. 390683824), and
the UK Natural Environment Research Council award (project no. NCEO02004).
Review statement
This paper was edited by Andrea Stenke and reviewed by two anonymous referees.
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