A novel split-explicit (SE) external mode solver for the Finite volumE Sea ice–Ocean Model (FESOM2) and its sub-versions is presented. It is compared with the semi-implicit (SI) solver currently used in FESOM2. The split-explicit solver for the external mode utilizes a dissipative asynchronous (forward–backward) time-stepping scheme that does not require additional filtering during the barotropic sub-cycling. Its implementation with arbitrary Lagrangian–Eulerian vertical coordinates like

All version 2 iterations of the Finite volumE Sea ice–Ocean Model (FESOM2;

This section provides a detailed description of the proposed asynchronous time-stepping scheme for FESOM2 that incorporates a split-explicit barotropic solver whose schematic can be found in Appendix

The standard set of equations under the Boussinesq and standard approximations is solved. The equations are taken in a layer-integrated form, and the placement of the variables on the mesh is explained in

First, we estimate the transport

Next is the barotropic step where

As a default time stepping for the barotropic part, the forward–backward dissipative time stepping by

Note that in FESOM,

Note that for the SE case, we integrate transport

Note that the use of dissipative time stepping in Eq. (

The estimate of the thickness at

While

Note that alternative formulations of re-trimming are also possible, for example, setting

Also note that despite being found to be redundant in practice, the re-trimming

This section provides a detailed numerical analysis of the new external mode solver when using SE or SESM time stepping vs. the current semi-implicit solver (

We begin with the semi-implicit method of

The explicit method of

Finally, the SE method by

Depending on the control parameters, the schemes above may lead to different dissipation and phase errors. Let the characteristic matrices of Eqs. (

Here, we are exploring only the physical solution that corresponds to a wave propagating in a negative direction.

, we can expand it for smallComparison of amplitude and phase error for different schemes using the same dissipation

This section compares measurements from the new external mode solver to the existing one in FESOM2. The tests are done for both an idealized case and a realistic global setup. The idealized case is expected to highlight threshold performance of the new solver compared to the global case, where its impact will also be governed by mesh non-uniformity, the presence of external forcing, complicated boundaries, and bottom topography. The global case will, however, crucially assess the practicality of this new solver.

In Sect.

Comparison of area-averaged mean depth profiles for eddy kinetic energy (m

In this test, we further show how the SE solver for the barotropic mode being proposed is less dissipative and how the differences between it and the current SI solver of FESOM2 become more obvious when the barotropicity of the flow is dominant. For that, we use a simple surface gravity wave (SGW) setup where we simulate a channel (of the same geometry as in Sect.

Comparison of elevation (m) distribution snapshots for both solvers (after 3 d) and their total available potential energy density (m

For this case, we now test a more complicated case of a global ocean–sea ice simulation similar to the one used by

Comparison of biases in sea surface heights (m), temperatures (°C), and eddy kinetic energies (m

In summary, no significant difference in terms of time-averaged measurements from the new SE external mode solver was observed. For both the idealized and the global test cases, the new SE external mode solver maintained mean dynamics close to those reported by the current SI solver.

This section compares the parallel scalability of the new external mode solver to the existing one of FESOM2. As in Sect.

Scaling results for the idealized test case with a 2 km uniform mesh on the Ollie HPC cluster and the global test case with a 60–25 km unstructured mesh on the Albedo HPC cluster. The black line indicates linear scaling and the coloured lines give the mean computing time over the parallel partitions for the solver part of the code. Here, the wall-clock time measured corresponds to the model runtime per baroclinic step.

A preliminary implementation of the new split-explicit external mode solver within the sea ice model FESOM2 as proposed in this paper, including the test cases, can be found in the public repository at

The new split-explicit external mode solver proposed in this paper is more phase accurate, faster, and more scalable than the SI solver used in FESOM (

The new SE solver is one part of the adjusted time stepping of FESOM that facilitates the use of the arbitrary Lagrangian–Eulerian vertical coordinate. As a demonstration, we extended

We note that on unstructured meshes, a semi-implicit method can be more forgiving than a split-explicit one toward the size of mesh elements. A small element in deep water will hardly affect the solution of the semi-implicit solver but may require an increased number of barotropic substeps in a split-explicit method. This is why the semi-implicit option will be maintained in FESOM alongside the novel split-explicit option. It will however, be modified to allow more general ALE options as described in Appendix

Schematic diagram of the control flow for the split-explicit asynchronous time stepping proposed in this paper for FESOM2.

Comparison of area-averaged depth profiles for eddy kinetics (m

The main difference from the split-explicit method is that the elevation

The predictor step is as follows

The corrector step is as follows

We write the elevation step as

The corrector step is used to compute

We write the thickness equation for the ALE step as

The tracers are as follows

By virtue of the thickness equation above,

Both split-explicit and semi-implicit asynchronous schemes are relatively straightforward to implement. The semi-implicit method with

In the case of the

A preliminary implementation of the new split-explicit external mode solver within the sea ice model FESOM2 as proposed in this paper, including the test cases, can be found in the public repository at

TB, SD, and KK developed the algorithm. TB, SD, DS, and PS implemented the FESOM prototype and the main FESOM branch. All authors contributed to the writing and discussions.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This paper is a contribution to the projects M5 (reducing spurious mixing and energetic inconsistencies in realistic ocean modelling applications) and S2 (improved parameterizations and numerics in climate models) of the Collaborative Research Centre TRR 181 “Energy Transfer in Atmosphere and Ocean”.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 274762653).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 Christopher Horvat and reviewed by Mark R. Petersen and one anonymous referee.