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.

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,

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

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?

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,

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,

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

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

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 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.

We present the flux-form semi-Lagrangian (FFSL) transport scheme
proposed by

The dimensionally split technique of the FFSL scheme is second-order
accurate in time. The method splits the 2-D transport equation in
Eq. (

This method is equivalent to the COSMIC splitting proposed in

The FFSL scheme defines a 1-D conservative operator for the flux
difference of two cell edges

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

Schematic illustration of the dimensionally split
scheme.

Similar to the Strang splitting, the FFSL scheme alternates the
direction sequentially. The dimensionally split scheme first solves
the 1-D equation in

Using

The FFSL scheme attains long time steps by a semi-Lagrangian extension
from 1-D finite-volume schemes

Illustration of the
semi-Lagrangian extension for finite-volume schemes on adaptive
meshes. The marks,

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.

In order to satisfy the alignment assumption, we could use ghost cells,
illustrated as the red cells in Fig.

As described in Sect.

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:

Illustration of the CISL scheme in 1-D and 2-D settings;

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:

On the sphere, the departure position of cell edges in each
dimension is described by

The staggering of the velocity means that

Illustration of the use of different reconstruction
function in our modified scheme. The shaded area

On an adaptive mesh with hanging nodes, the 1-D integral in Eq. (

The resemblance between Eqs. (

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

Because

Using our modified 1-D operator in the FFSL scheme, the original

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.

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.

Our refinement procedure follows the description in

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

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

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

In order to use the refinement criteria, we assign each cell a
quantity:

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.

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

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

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:

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.

In these idealized tests, we measure the numerical results
quantitatively in the

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:

We use

As mentioned in Sect.

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

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

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

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

In Fig.

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

Figure

Numerical errors show a significant
difference between these two
refinement strategies when

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.

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.

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.

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

We set the flow orientation as

We test these three flow orientations with a maximum Courant number
around 1 and 6. When

The AMR scheme utilizes a
gradient-based criterion. Our threshold for cell refinement is

As shown in Fig.

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.

Convergence rate of the numerical results with respect to the number of cells in the solid body rotation test case.

Figure

Convergence rate of the numerical results with respect to the
number of cells in the solid body rotation test
case with

Figure

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

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

Evolution of the normalized numerical error for

Figure

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.

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.

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

Numerical results of the divergence test case with a
resolution of 5

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

As shown in Fig.

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

As shown in Fig.

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

We observe that mass is conserved without AMR in Fig.

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

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.

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.

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.

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

Numerical results of the moving
vortices test case. The left column shows the numerical results on the
a resolution of a 5

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

On adaptive meshes, the refinement threshold for the gradient-based
refinement criterion is

We show snapshots of the numerical solution at
5

The large refinement area in Fig.

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.

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.

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.

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.

The tracer transport process exhibits multi-scale features in climate
simulations. As indicated in Sect.

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

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.

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

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

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

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.

ECHAM6 describes the tendency equation of the tracer mixing ratio
using the following equation:

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

The transport equation under hybrid

Integrating both sides of Eq. (

The FFSL scheme solves the vertical transport separately in the
hydrostatic model

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.

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 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.

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.

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:

Illustration of our setting for the one-way coupling
experiment.

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, (

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

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.

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

We adopt the default time step setting in ECHAM6. In the

Dust mixing ratio of DU_AI
(

We present the dust mixing ratio of DU_AI in Fig.

The simulation at a uniform resolution of

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

In particular,

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.

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.

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

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
(

Now, we take the uniformly refined transport module mesh as the benchmark for our adaptive
mesh refinement. Our results in Fig.

We also observe large refined regions in Fig.

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 (

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.

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.

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

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.

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

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.

The code for running and plotting idealized
tests in Sect.

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

The authors declare that they have no conflict of interest.

This work was supported by German Federal Ministry of Education and
Research (BMBF) as part of the Research for Sustainability initiative (FONA);

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).

This paper was edited by Andrea Stenke and reviewed by two anonymous referees.