Two recently proposed variants of CD-type discretizations of sea ice dynamics on triangular meshes are implemented in the Finite-VolumE Sea ice–Ocean Model (FESOM version 2). The implementations use the finite element method in spherical geometry with longitude–latitude coordinates. Both are based on the edge-based sea ice velocity vectors but differ in the basis functions used to represent the velocities. The first one uses nonconforming linear (Crouzeix–Raviart) basis functions, and the second one uses continuous linear basis functions on sub-triangles obtained by splitting parent triangles into four smaller triangles. Test simulations are run to show how the performance of the new discretizations compares with the A-grid discretization using linear basis functions. Both CD discretizations are found to simulate a finer structure of linear kinematic features (LKFs). Both show some sensitivity to the representation of scalar fields (sea ice concentration and thickness). Cell-based scalars lead to a finer LKF structure for the first CD discretization, but the vertex-based scalars may be advantageous in the second case.

The emergence of several global ocean models formulated on unstructured (triangular or hexagonal) meshes, such as the Finite-VolumE Sea ice–Ocean Model (FESOM)

On triangular meshes, the A, B and CD placements of the discrete sea ice velocity result in different numbers of discrete degrees of freedom (DOF), with a ratio of

This paper presents the new implementation of two CD-grid discretizations in the sea ice component of FESOM. Both are based on the standard Hibler viscous–plastic (VP) rheology

The second CD-grid discretization, referred to as CD2, is similar to that used by

We use the test case proposed by

The following sections describe main equations (Sect. 2), the implementation and the Fourier analysis (Sect. 3), and test simulations (Sect. 4). They are followed by Discussion (Sect. 5) and Conclusions (Sect. 6).

The sea ice momentum equation is written as

The modified elastic–viscous–plastic method (mEVP) is used to solve for the sea ice dynamics in the same form as in

In the discretization of the sea ice momentum equation, spherical geometry is taken into account similarly to

For brevity, we will begin with the VP momentum, Eq. (

The stresses are also considered to be constant on triangles, which requires that the ice strength

The next step is to obtain the Galerkin approximation. The above polynomial approximations are inserted in Eq. (

The nonconforming functions are orthogonal on elements, so

Computations of

The integration over edges in the stabilization term involves

The extension of this discretization to the mEVP method requires empirical adjustment of the strength of the stabilization. The point is that stresses in this method are iterative EVP approximations to the VP stresses. The stabilization term is the contribution to the stress divergence, and it does not appear in the iterative sub-cycling of stresses in the mEVP method. It has been empirically found that the stabilization pre-factor has to be essentially smoother than

As mentioned above, both vertex (

The difference from the previous (CD1) case lies in the selection of basis and test functions. Consider triangle

CD1: the nonconforming linear function

The available degrees of freedom are associated with edge velocities, same as in the previous section. The edge velocity is interpreted as a mid-edge value. Values of velocity at mesh vertices are reconstructed as a weighted mean of edge velocities,

The weights are normalized so that

In terms of

To obtain the Galerkin approximation, the test function is taken to be any of

In the terms with the time derivative and

The computations of the third and fourth terms on the left-hand side of Eq. (

It is instructive to perform the Fourier analysis of CD2. It will provide an independent argument on the accuracy of this discretization, similarly to the analysis in

Consider an infinite triangular mesh made of equilateral triangles with side length

A vertex velocity is reconstructed from the edge velocities. The amplitude of vertex velocity is

In the same way as in

We first compute the strain rates on each of the four sub-triangles of a primary mesh triangle (see Fig.

These have to be compared with the eigenvalues of CD1 and other discretizations given in

Since this work relies on the already existing discretizations, we only compare the performance of the two new CD methods in FESOM with respect to their numerical efficiency and ability to represent linear kinematic features (LKFs) based on the test case proposed by

The test case is run on a triangular mesh occupying a rectangular domain of 512 by 512 km. Except for western and eastern boundaries, the triangles are equilateral. Smaller rectangular triangles are added along the western and eastern boundaries to make these boundaries straight. The sides of equilateral triangles are 2 km for CD discretizations. The A-grid simulation is run on the mesh with a triangle side of

Sea ice concentration

Figure

Figure

Same as in Fig.

The third column in Fig.

The higher resolving capability of the CD (edge) placement of velocity compared to the vertex (A-grid) placement is related to its number of discrete velocities being 3 times as large. The larger number of degrees of freedom implies shorter distances between their locations and may potentially lead to a more accurate approximation of differential operators. The gain in accuracy depends on the discretization, and an elementary Fourier analysis of the eigenvalues of the linearized stress divergence operator in

A caveat of the CD1 discretization is that it needs stabilization to remove kernels in differential operators, as discussed by

We also note that CD1 shows a tendency to simulate very close LKFs, separated by several mesh cells. They are well seen in Fig.

In our implementation, CD1 is approximately 2 times and CD2 approximately 4 times more expensive than the A-grid code on the same mesh. In all cases there are two basic cycles over triangles. The stresses are computed in the first cycle, and the divergence of stresses is computed in the second cycle. For the A-grid discretization these two cycles take most of the CPU time. In CD1, there are two additional cycles over triangles to compute the contribution of stabilization, which largely explains the CPU time doubling. The cycles over triangles in CD2 include an inner cycle over four sub-triangles, which is the main reason for the observed increase in the computational load in this case. We speculate that some optimization is still possible, so these numbers can only be treated as preliminary estimates. In addition to the increase in the time needed for computations, the number of halo exchanges in parallel implementation also increases in the CD cases. As compared to the A-grid code, in our implementation the CD1 discretization requires an additional halo exchange for edge velocity differences. The CD2 case needs additional exchanges for vertex velocities and for the contributions to the divergence of stresses that are assembled at vertices. As demonstrated by

The A-grid run on a

There is a possible extension of CD2 discretization. Instead of considering vertex velocities as dependent variables one treats them as independent ones, in addition to the edge velocities, and uses

We describe the implementation of two CD-type discretizations of sea ice mEVP dynamics in FESOM2. They are based on the finite element method and the use of longitude–latitude coordinates. Both discretizations have been proposed earlier by

Both CD1 and CD2 demonstrate higher LKF-resolving capability than the A-grid discretization. Although CD2 shows lower resolving capacity than CD1, it may be more robust in (m)EVP dynamics as it does not need much additional adjustment. The new discretizations can be sensitive to particular implementation details. It also remains to be seen how these new discretizations behave in realistic global climate simulations compared to the standard A-grid discretization of FESOM, which is the subject of our future work.

The exact version of the model used to produce the results used in this paper is archived on Zenodo (

SD worked on the implementation. All the authors contributed to writing and discussion.

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

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We acknowledge the use of the METIS (

The work of Carolin Mehlmann is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; project no. 463061012). The article processing charges for this open-access publication were covered by the Alfred-Wegener-Institut Helmholtz-Zentrum für Polar- und Meeresforschung.

This paper was edited by Philippe Huybrechts and reviewed by Giacomo Capodaglio and one anonymous referee.