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

1. 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?
2. How many iterations of GMRES are typically required to solve (33)?  How sensitive is this number to the resolution?
3. 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? 
4. 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.
5. 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.

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 unified 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.