Monotonicity is an important property of remapping operators for coupled weather and climate models. However, it is often challenging to design highly accurate operators that avoid the generation of new extrema or keep a remapped field between physically prescribed bounds. To that end, this paper explores several traditional and novel approaches for both conservative and non-conservative monotone remapping on the sphere. The accuracy and effectiveness of these algorithms are evaluated in the context of several different real and idealized fields and meshes.

An important operation in global climate models is the transferring, or remapping, of data between different component grids. For example, information needs to be exchanged at the interface between the atmosphere and ocean models, when both are typically defined on different grids. Atmospheric models often use icosahedral or cubed sphere grids, while ocean models have relied on unstructured meshes

There are a number of desirable properties of remapping operators, in addition to accuracy. These properties include consistency, conservation, and monotonicity and correspond, respectively, to the mapping of the constant field to the constant field, preservation of total mass, and no generation of new extrema

The main property of remapping schemes that we are concerned with in this paper is monotonicity. In the case of conservative remapping, monotonicity is often achieved by way of limiters

TempestRemap

This paper consists of three main sections. First, we will describe the basic setup of remapping problems, the test cases that are used in our numerical experiments, and the metrics used to assess the accuracy of the remapping schemes. In the next section, we will look at monotone conservative remapping. In general, it is difficult to construct remapping operators that satisfy conservation and monotonicity, while still maintaining high-order accuracy. So one of the main purposes of this section is to examine the extent to which a conservative and monotone remapping operator can maintain the accuracy of its non-monotone counterpart. We will also analyze the effectiveness of this conservative and monotone operator in minimizing the errors associated with the remapping of discontinuous source fields, as well its ability to remap real data fields accurately. The subject of the next section is non-conservative monotone remapping, and it is divided into two main parts. The first part focuses on traditional approaches to monotone remapping and includes a description of the bilinear method used in the Earth System Modeling Framework (ESMF)

Let

Following

We will use several idealized test cases for our numerical experiments, including a low-frequency harmonic denoted by

The focus of this section is on monotone conservative remapping and assessing potential improvements in accuracy that arise from employing a nonlinear remapping technique to enforce bounds preservation. We consider fields whose total mass needs to be conserved across the remapping process and that need to remain between specified bounds. This form of “bounds preservation” is important for fields such as mixing ratios, which are required to remain between zero and unity, and it corresponds to a global form of monotonicity wherein no new global extrema are generated. We also consider local forms of bounds preservation, which are stronger than global monotonicity in the sense that they will not introduce any new local extrema.

High-order remapping methods can lead to overshoots or undershoots of the remapped field, which is problematic for several reasons. For instance, high-order remapping of discontinuous source fields may lead to oscillatory behavior of the remapped field similar to the Gibbs phenomenon

Conservative and bounds-preserving schemes have been used in semi-Lagrangian schemes

In this section, our goal is to examine the utility of the CAAS algorithm as a way of ensuring bounds preservation and reducing the Gibbs phenomena while still ensuring accuracy and conservation. In particular, we are interested in documenting the effect of CAAS on standard error norms, as implemented in TempestRemap.

Here, we look at the case in which the source and target meshes are both finite-volume. In particular, we are interested in applying the CAAS algorithm with two different types of local bounds preservation, which we now describe.

We let

Convergence test for finite-volume to finite-volume remapping from cubed spheres to a 1

Convergence test for finite-volume to finite-volume remapping from a cubed sphere to a 1

In all cases, the

Here, we examine bounds preservation in the case in which the source mesh is finite-element. Local bounds preservation is defined similarly to how it was for finite-volume source meshes, but now the minimum and maximum in Eq. (

Convergence test for finite-element to finite-volume remapping from a cubed sphere to latitude–longitude mesh using local bounds preservation for three different test cases. The setup is the same as it was for finite-volume to finite-volume remapping, with the dashed lines showing the results using CAAS with local bounds preservation and the solid lines showing the results without CAAS.

The

In this section, we examine the effectiveness of CAAS in reducing overshoots and undershoots associated with remapping a discontinuous source field. To that end, we modify the vortex test case by defined by Eq. (

In Fig.

The Gibbs oscillations for a finite-volume to finite-volume remapping from a resolution 60 cubed sphere to a 1

One-dimensional cross sections for remapping of a discontinuous field at the Equator. Panel

To test the performance of the CAAS algorithm on real data, we use the cloud fraction data generated from the MIRA real data emulator

The cloud fraction field used in evaluating the effectiveness of CAAS on a real data field.

Error norms for remapping from an ne90 cubed sphere to a 1

Error norms for remapping from an ne360 cubed sphere to a 0.25

In this section, we describe several different approaches to monotone remapping that are consistent but non-conservative. In general, traditional approaches to monotone remapping perform poorly when the source mesh is significantly finer than the target mesh. To correct this, we propose what we call

In brief, for consistent and monotone remapping operators, we express the value of

Here, we describe the non-integrated approach to monotone bilinear remapping found in the ESMF. Suppose we are given a point on the target mesh onto which we are remapping. Call this point

Once this polygon is found, and assuming it has to be further triangulated, we solve the following equation:

An overview of the Delaunay triangulation remapping scheme, whereby the images of the source face centers are triangulated.

The weights used in the weighting based on the Delaunay triangulation.

In this section, we describe an alternative to the remapping scheme described in the previous section. We obviate the need to triangulate an arbitrary polygon by constructing the Delaunay triangulation of the face centroids of the source mesh. We outline our approach as follows. We seek a triangle on the source mesh whose vertices are source face centroids that contains a given point on the target mesh. To that end, we divide the sphere into six panels: call them

The final scheme we describe is based on what are called generalized barycentric coordinates

We generalize to the sphere by interpreting the areas in Eq. (

As was the case with bilinear interpolation outlined in Sect.

Here we show the results of two different numerical tests. In the first case, the remapping is done from cubed spheres to a fixed 1

Convergence results for several non-integrated monotone remapping schemes for a fixed latitude–longitude target mesh and cubed sphere source meshes.

For the second test, the target is still a fixed 1

Convergence results for several non-integrated monotone remapping schemes for a fixed latitude–longitude target mesh and triangular source meshes.

Convergence results for several integrated monotone remapping schemes for a fixed latitude–longitude target mesh and cubed sphere source meshes.

The remapping schemes described in the previous sections work well when the source mesh is not too much finer than the target mesh. However, when the resolution of the source mesh is greater than that of the target mesh, pointwise sampling of the source mesh to determine a field value on the target mesh is inappropriate and inaccurate. In this case, a large number of points on the source mesh contribute no weights to the remapping operator. To combat this under-sampling, we now describe an approach that ensures all points on the source mesh are sampled via construction of the overlap mesh or supermesh. Approaches of this type are called integrated because of their analog to numerical quadrature and are distinguished from the non-integrated approaches described in Sect.

Convergence results for the integrated and non-integrated bilinear remapping schemes from cubed spheres to a fixed latitude–longitude mesh.

This section again consists of two tests. The first test is to establish second-order convergence of the integrated schemes, and it is identical to the setup of the first test shown in Sect.

In the next test, we consider a setup wherein the source meshes are refined beyond the resolution of the target mesh. The source meshes are cubed spheres with

In this paper we have examined a number of different schemes for conservative and non-conservative monotone remapping. For monotone conservative remapping, we showed that the “clip and assured sum” method provides an accurate way of remapping conservative fields that are required to stay bounded and is effective at reducing the Gibbs-like oscillations associated with discontinuous source fields.

We then described several different approaches to non-conservative remapping. Two of these have, to the best of our knowledge, never been applied to remapping problems on the sphere. These methods have what are referred to as non-integrated and integrated versions, and it was shown that the integrated versions are capable of maintaining second-order accuracy across a wide range of source mesh resolutions by systematically sampling the degrees of freedom on the source mesh, albeit at higher computational costs.

As discussed in the Introduction, the methods described in this paper have been implemented as part of v2.1.6 of the TempestRemap software package

The code used in this paper is part of the TempestRemap software package and is available on Zenodo

The data used in this paper are available on Zenodo at

DHM developed the functionality in TempestRemap and wrote the paper; PAU advised and edited the paper.

At least one of the (co-)authors is a member of the editorial board of

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Primary support for this work was provided by the SciDAC Coupling Approaches for Next Generation Architectures (CANGA) project, which is also funded by the DOE Office of Advanced Scientific Computing Research.

This research has been supported by the Department of Energy, Office of Science (grant no. DE-AC52-06NA25396).

This paper was edited by Min-Hui Lo and reviewed by Robert Oehmke, Sang-Wook Kim, and one anonymous referee.