the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A mixed finite element discretisation of the shallow water equations
- 1Dynamics Research, Met Office, Exeter, UK
- 2Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
- 1Dynamics Research, Met Office, Exeter, UK
- 2Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Abstract. This paper introduces a mixed finite-element shallow water model on the sphere. The mixed finite-element approach is used as it has been shown to be both accurate and highly scalable for parallel architecture. Key features of the model are an iterated semi-implicit time stepping scheme, a finite-volume transport scheme, and the cubed sphere grid. The model is tested on a number of standard spherical shallow water test cases. Results show that the model produces similar results to other shallow water models in the literature.
James Kent et al.
Status: final response (author comments only)
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CEC1: 'Comment on gmd-2022-225', Juan Antonio Añel, 25 Oct 2022
Dear authors,Unfortunately, after checking your manuscript, it has come to our attention that it does not comply with our "Code and Data Policy".https://www.geoscientific-model-development.net/policies/code_and_data_policy.htmlYou have archived the code of your shallow water model on GitHub. However, GitHub is not a suitable repository. GitHub itself instructs authors to use other long-term archival and publishing alternatives, such as Zenodo. Therefore, please, publish your code in one of the appropriate repositories according to our policy, and reply to this comment with the relevant information (link and DOI) as soon as possible, as it should be available for the Discussions stage. Also, in a potential reviewed version of your manuscript, you must include the modified 'Code and Data Availability' section with the DOI of the new repository.Moreover, for the MetOffice code and data, you must be much more specific. You must provide the exact version number for the code and its DOI. For the data, if it can not be stored outside the MetOffice servers, you should provide a DOI, too, to ensure that it is easy to identify and obtain.Please, be aware that failing to comply promptly with this request could result in rejecting your manuscript for publication.Regards,Juan A. AñelGeosci. Model Dev. Exec. Editor
- AC1: 'Reply on CEC1', James Kent, 21 Dec 2022
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RC1: 'Comment on gmd-2022-225', Anonymous Referee #1, 26 Oct 2022
Summary
The article presents a new numerical method for solving the shallow water equations on the sphere that employs mimetic finite element methods on a cubed-sphere grid.
It represents a significant advance from previous work, extending the methods to spherical geometry as a step toward development of an accurate and computationally efficient atmospheric dynamical core for operational use.
As a replacement for an operational model that uses a latitude-longitude grid, the article correctly suggests that this method is capable of significant improvements in parallel scalability on the latest computing architectures.
Results from the standard test cases are presented, and demonstrate that the model performs appropriately.Advanced numerical techniques such as mimetic finite element methods preserve significant properties of the continuous equations in their discrete form and are capable of achieving high efficiency in the massively parallel simulations required by high resolution atmospheric models.
The article represents an important step toward development of such a method, and I recommend it for publication with minor revisions, as suggested below.General comments
While mimetic methods are becoming more common, it seems a lot to ask of the interdisciplinary GMD audience to follow a discussion of function spaces and de Rham complexes without some assistance. An illustration such as Figure 4 from Melvin et al. (2018) that corresponds to the specific cases mentioned by equations (7) and (8) would be most helpful.
Similarly, the article could benefit from a quick reminder of why mixed finite element methods are useful, especially since the lowest order formulation is used here. Differences between finite element methods, finite volume methods, and finite difference methods often disappear when used with low order discretizations. What is gained here that would not be present in a staggered finite volume method such as the one presented by Thuburn et al. (2014)?
Given the emphasis given to computational efficiency in the introduction, I expected more discussion of the method's computational performance. Detailed scaling studies are likely unrealistic this stage of development, but some general discussion would be helpful. Are the expected gains strictly due to the choice of grid, i.e., cubed sphere vs.~latitude-longitude? Or is the numerical method helpful, too, for example, are its stencils for field reconstruction (e.g., Figure 1) smaller than other methods, implying less communication is required during runtime?
Specific comments
- Equations (5) and (6) suggest that the advecting velocity is $\overline{\vec{u}}^{1/2}$ even in cases where $\alpha \ne 1/2$; is this true? Wood et al. (2014) suggest that off-centering by setting $\alpha > 1/2$ is important in the context of a 3D deep atmosphere model with orography. Is that concern relevant here, given that the method is presented as a step toward a full 3d atmosphere model?
- How many iterations of GMRES are typically required to solve (33)? How sensitive is this number to the resolution?
- The description in Section 6 of a ``finite-element representation of the sphere within a cell with polynomials'' is difficult to follow. I assume that the four vertices of an element lie on the sphere; for the case of a quadratic elements, are the nodes that are not vertices also on the sphere?
- Is the fact that some internal points of a cell may not exactly lie on the spherical surface related to the fact that different function spaces are used for different variables? It doesn't seem to be an issue with other finite-element dynamical cores such as Guba et al. (2014), that also rely on mappings to and from a reference quadrilateral.
- I found the discussion of error at the beginning of Section 7.1 confusing; it states that the method is second-order in both space and time, but immediately preceding this remark at the end of Section 6, fourth-order convergence is cited as the reason for choosing quadratic elements. I agree that the method should be overall second order, so I presume that the fourth-order convergence refers to reconstructing the spherical surface itself, rather than an arbitrary scalar function?
References
- T. Melvin, T. Benacchio, J. Thuburn, and C. Cotter, 2018, Choice of function spaces for thermodynamic variables in mixed finite element methods, Q. J. Roy. Met. Soc. 144:900--916.
- J. Thuburn, C. J. Cotter, and T. Dubos, 2014, A mimetic, semi-implicit forward-in-time, finite volume shallow water model: comparison of hexagonal-icosahedral and cubed-sphere grids, Geosci. Model Dev. 7:909--929.
- N. Wood, A. Staniforth, A. White, et al., 2014, An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations, Q.~J.~Roy.~Met.~Soc. 140:1505--1520.
- O. Guba, M.A. Taylor, P.A. Ullrich, J.R. Overfelt, and M.N. Levy, The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity, Geosci. Model Dev. 7:2803--2816.
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AC2: 'Reply on RC1', James Kent, 21 Dec 2022
Thank you for your review.
In response to your general comments:
We have produced a figure that shows the mixed finite element spaces used in the model and have included it in section 3.2 of the manuscript.
We agree that at lowest order these types of methods often become very similar. However, in Thuburn and Cotter, JCP, 2015, it is shown
that the lowest order FE discretization has more benefit when it comes to consistency of the Coriols on non-orthogonal meshes than the
FV model of Thuburn et al 2014. Another benefit of FE is the flexibilty to go to a higher-order element model. We have included this discussion
in the introduction of our manuscript.We've added some discussion to the conclusions. We highlight that the cubed sphere grid has
fewer cells than a corresponding lat-lon grid, and that the cubed sphere removes the pole and associated issues with parallel
computing. Regarding the stencil size, the MoL transport scheme uses a small stencil for each reconstruction, but it must
compute a reconstruction for each stage of the RK scheme. It is not clear at this stage whether this improves communication
cost when compared to a scheme with a large stencil that is only called once.In response to your specific comments:
1) This is true, even if alpha $\neq 1/2$. This is consistent with Wood et al. 2014, and is used to get the second-order time
discretization. Currently we use alpha=0.5 in the model configuration. For shallow water we have not seen the need to off-centre.
We agree that this is important for a full 3D model.2) It seems to take around 2-3 iterations for GMRES to converge to a tolerance of 10^-4 on the C24 and C48 grids
for both the mountain and Galewsky test. We have stated this at the end of section 5.3) We have rewritten parts of this section, including adding a sentence describing a linear element to make things clearer. For the quadratic elements all the nodes lie on the sphere.
4) The different function spaces is not why we use the sphere parameterisation. Representing the sphere with elements removes the need for analytic transformations. This means we can use an arbitrary grid (although in this paper we only consider the cubed sphere grid). A down side is we are parameterizing the sphere, but as shown using quadratic elements on the C96 grid gives a maximum error of 0.0018 m.
5) You are correct that the fourth-order is for the spherical surface, and the second-order is for the Williamson 2 test case. We have edited the text here to make the distinction clearer.
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RC2: 'Comment on gmd-2022-225', Hilary Weller, 30 Oct 2022
Generally, a referee comment should be structured as follows: an initial paragraph or section evaluating the overall quality of the preprint ("general comments"), followed by a section addressing individual scientific questions/issues ("specific comments"), and by a compact listing of purely technical corrections at the very end ("technical corrections": typing errors, etc.).
General Comments
This paper clearly presents the shallow water model which uses some of the numerical methods that will be used in the next Met Office dynamical core. It is therefore an important model description paper. It brings together mixed-finite element modelling of the second-order wave equations, finite-volume modelling of transport and semi-implicit time stepping. The paper is concise and easy to follow, drawing on other published work where needed in order to define the model, although some clarifications are still needed. The results are clearly presented and, at this stage, nearly comprehensive.
Specific Comments
The motivation for this new model could be a lot stronger. Much of the motivation provided could have been written last century, for example the need for parallisation and the need to go beyond finite differences, finite volume and semi-Lagrangian. The motivation for mixed finite elements is easy and has already been written about. The motivation needs in involve massive parallelisation, wave dispersion, spectral elements and DG.
Section 4 needs to define the order of accuracy in space of the transport scheme. I think it must be limited to two because you do not define how you fit a polynomial using cell average values.
Figure 4 and the related discussion (lines 239-243) are weak. Figure 4 only really shows that your model works. It doesn't, as you say, show that the "results are comparable to other shallow water models" or demonstrate "the model's ability to correctly simulate flow over orography". I would plot errors rather than figure 4 (in comparison to STSWM) and convergence with resolution. It is also informative to show the vorticity after 50 days which is a good indicator of conservation, balance and a lack of spurious artefacts in the solution. Eg see:
Fig 11 of "A uniï¬ed approach to energy conservation and potential vorticity
dynamics for arbitrarily-structured C-grids", Journal of Computational Physics 229 (2010) 3065–3090
or fig 5 of
"Computational Modes and Grid Imprinting on Five Quasi-Uniform Spherical C-Grids", Weller, Thuburn and Cotter.Technical Corrections
Try to make your writing more concise. For example, delete phrases like "and the interested reader is referred there for more information".
Please also see
Shaw, J., Weller, H., Methven, J. and Davies, T. (2017) Multidimensional method-of-lines transport for atmospheric flows over steep terrain using arbitrary meshes. Journal of Computational Physics, 344. pp. 86-107. ISSN 0021-9991
for a description of the creation of stencils and polynomials for this type of transport scheme.In table 1, use scientific notation rather than exponents.
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AC3: 'Reply on RC2', James Kent, 21 Dec 2022
Thank you for your review.
Regarding motivation, we have rewritten parts of the introduction to takes these points into account. We have stressed the need for massively parallel models for the future of weather and climate forecasting, and have discussed the benfits of mixed finite-element methods over finite-volume.
The transport scheme in section 4 is actually 3rd order in space and time. The temporal order comes from the SSPRK3 algorithm.
The spatial order comes from the quadratic reconstruction of the field at flux points. The fitting of the polynomial is
such that the integral of the polynomial is equal to the integral of the variable within each cell. We have made this clearer in the text in section 4.We have significantly rewritten large parts of the mountain test case section. We use a high-resolution semi-implicit semi-Lagrangian scheme as a reference to produce error plots, which we then compare with other models in the literature. We also look at the error convergence with
resolution. We have extended the energy and potential enstrophy statistics to 50 days, and provided a plot of the day 50 potential vorticity.We have removed the text "and the interested reader is referred there for more information" and have used scientific notation in the error norm table.
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AC3: 'Reply on RC2', James Kent, 21 Dec 2022
James Kent et al.
James Kent et al.
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