This is a nice paper that explores the use, benefits and pitfalls of variable-resolution atmospheric simulations based on unstructured dycore formulations a'la MPAS. Using a simplified (moist) shallow water formulation, the authors compare a range of quasi-uniform and variable resolution configurations, detailing differences in simulation quality/skill, as well as identifying various numerical oscillations/artefacts. 

While I am definitely supportive of publication, there are a number of questions and potential revisions that I feel could be addressed: 

- It appears an unlimited, centred-difference scheme is employed for tracer fluxes --- using the 'standard TRSK' discretisation of the divergence operator when solving for advective transport. This approach can be expected to lead to significant grid-scale noise and oscillation in the tracer fields, as per all results shown for the moist SWE system. As noted by the authors, the full MPAS model does not use centred-differencing for tracer transport for this reason, instead using flux-limited upwind-biased advection schemes. I feel that a better quasi-monotone advection scheme should be implemented and these runs re-analysed. With such significant grid-scale noise present, it is not clear the current comparisons between the UR and VR results for the moist SWE are necessarily as fair as they could be. 

- Is PV stabilisation (eg. APVM or LUST [3]) employed in the SWE dycore? TRSK supports an unstable-mode/grid-scale oscillation in PV dynamics that upwinded PV fluxes are needed to control. The PV contours in Fig. 10b show very significant grid-scale checker-boarding, and it would be interesting to understand the nature of PV stabilisation applied here, and to re-run with upwinding active if needed.

- The need to apply momentum dissipation/damping, especially for longer runs, seems consistent with practical experience. The nature/magnitude of the dissipation required is of interest though. In MPAS-O, dissipation is applied as a combination of del^2 and del^4 operators [2], with mesh scale dependent coefficients. Such higher-order dissipation could be expected to offer more selective damping of grid-scale noise, and potentially less impact on overall energy conservation.

- The barotropic jet case is known to be a challenging problem, with the nature of the turbulent vortex roll-up a strong function of mesh resolution/alignment [1,4]. I think it is interesting to use the jet as a test case, though I don't think close agreement between the UR and VR approaches should necessarily be expected.

- I feel that additional discussion/explanation on constraints on mesh generation for TRSK could be of interest, detailing, for example, why generating eg. 'well-centred' meshes (triangles contain their own circumcentres) is important for the TRSK scheme in particular (ie. ensuring primal and dual edges intersect, 'kite' areas associated with PV remapping are positive, etc). There are many good references that could be relevant here --- previous work of mine [5] is one option. 

[1] Exploring a Multiresolution Modeling Approach within the Shallow-Water Equations

Todd Ringler, Doug Jacobsen, Max Gunzburger, Lili Ju, Michael Duda, and William Skamarock (2011)

MWR, doi.org/10.1175/MWR-D-10-05049.1

[2] A multi-resolution approach to global ocean modeling

Todd Ringler, Mark Petersen, Robert Higdon, Doug Jacobsen, Philip Jones, Mathew Maltrud (2013)

OM, doi.org/10.1016/j.ocemod.2013.04.010

[3] Controlling the Computational Modes of the Arbitrarily Structured C Grid

Hilary Weller (2012)

MWR, doi.org/10.1175/MWR-D-11-00221.1

[4] Computational Modes and Grid Imprinting on Five Quasi-Uniform Spherical C Grids

Hilary Weller, John Thuburn, and Colin Cotter (2012)

MWR, doi.org/10.1175/MWR-D-11-00193.1

[5] JIGSAW-GEO (1.0): locally orthogonal staggered unstructured grid generation for general circulation modelling on the sphere

Darren Engwirda (2017)

GMD, doi.org/10.5194/gmd-10-2117-2017

Summary This paper investigates the advantages and limitations of local grid refinement to better resolve topography features with very steep gradients and other small scale structure.  The goal is to smoothly refine the computational grid, based on geometric characteristics of the local topography, separately treating orography land mass.  

The authors focus on Spherical Centroidal Voronoi Tesselations (SCVT), refined using a density function that depends on four parameters and on the topography.  The dry and moist shallow water equations are solved using the TRSK discretization of the MPAS dynamical core and applied to a set of standard test cases.  The main result is that this local refinement may generate numerical inertia-gravity waves that produce both qualitative and quantitative errors, but that these waves can be easily removed by increasing Laplacian horizontal diffusion of both the divergent and rotational modes.  Local refinement over topography sometimes improves cloud representation, although overall it is not clear it improves precipitation results.

General Comments The question of how to appropriately refine computational grids is vital to avoid unacceptably large errors generated by small scale features that cannot be resolved on a uniform grid.  These features may be topographical (e.g. mountains), dynamical (e.g. localized fronts or hurricanes), or related to user-defined criteria (e.g. need for more accuracy in urban areas). Topography details should be easier to deal with than dynamical features since their location is fixed and the grid refinement can be static.  In this case, one may decide that the topography geometry itself is a sufficient “error indicator” to determine grid resolution. Nevertheless, the problems are not independent since topography features may generate localized waves whose influence extends beyond the topography.  It is also not obvious that a grid refinement criterion based exclusively on orography is sufficient to control the error of the shallow water equations. However, this paper shows that even this relatively simply case presents challenges

The choice of the four parameters and the sensitivity of the computations to these choices needs more justification and investigation.  How sensitive are the results/errors to each of the parameters? Could their choice be optimized in some way? (Please also see questions below.)

This investigation considers only one example of local refinement on the Andes range (a long narrow structure with multiple steep valleys), so it is not obvious how much the results can be generalized to other cases of local refinement (e.g. around cities or topography with different gradient structures).  Could you please comment on how generalizable you think the current results are?

Recommendation This is an interesting and well written paper that advances our understanding of how well-designed local static grid stretching can be used to make the representation of small scale orographic features more accurate.  I will be happy to recommend publication in GMD after my general comments and specific questions have been addressed.

Questions

1. P 3 Nested grids and static AMR type refinement requires a bit more discussion of their advantages and disadvantages compared with the grid “stretching” approach used here.  Nested grids are used extensively in numerical weather prediction, and they do not not always suffer from wave reflection at refinement boundaries.  AMR techniques could also be used to provide static grid refinement and it has been used successful for dynamical adaptivity. Multi-scale or multigrid approaches, such as wavelet based adaptive refinement used in WAVETRISK, also do not show spurious effects at refinement boundaries because of the use of prolongation/restriction between scales to ensure consistency. 
2. Is it necessary for the refined region to extend beyond the topography to ensure proper representation of dynamical features, such as waves?  Because your grids are refined very smoothly, the refined regions actually do have this property (extending well beyond the Andes region).  Is this necessary for dynamical accuracy as well as to guarantee good grid properties? 
3. P 6 Please give more explanation and justification for the precise form of the density function in equation (7).  It is clearly not the unique density function consistent with relation (4).   For example, the density function depends on the topography b(x), but not (directly) on  its gradient (which is presumably an important factor).
4. P 6 The density function (7) is based purely on the topography and will therefore not necessarily reduce errors in the solutions of the shallow water equations. However, it is also possible to optimise the choice of parameters using optimal control to minimize the height errors for certain shallow water test cases (e.g. the Williamson suite and the Galewsky jet).  Have you considered a more systematic way of choosing these parameters?  An optimized set of parameters may avoid the need for numerical diffusion and make better use of computational resources by refining the grid only where it is needed for accurate representation of the dynamics.
5. Figure 3 How did you choose the parameters for the refined grid shown here? Are these the choices you will use for all simulations? Given the shape of the Andes, would an elliptical (rather than circular) refinement region be a better choice?  It would be helpful to include the grid refinement parameters in the caption for table 1.
6. P 6 What is the actual resolution in km of the ETOPO data used? How is this choice related to the resolution of the computational grids?
7. P 12 Since numerical diffusion is entirely for stabilizing purposes, why did you not use hyper diffusion, which limits diffusion to larger wavenumbers? Laplacian diffusion smooths over more wavenumbers and is often not the best choice to stabilize a computation.
8. P 12 Have you tried a local diffusion coefficient that is zero outside the refined grid area?  
9. How sensitive are the results to the precise choice of each of the grid refinement parameters?  Have you done any sensitivity analysis (e.g. using Sobol’ indices or Global Sensitivity Analysis GSA) on the parameters?  It would be helpful to know which parameters are most influential.
10. Conclusions. Please comment on the extent to which the results can be generalized to a 3D hydrostatic atmosphere model, which presumably is even more sensitive to topography gradients and structure than the shallow water models considered here because of vertical fluxes.

This paper investigates the usage of topography based local refinement of a Voronoi grid using the Model for Prediction Across Scales (MPAS) with the TRSK method for the classical and moist shallow-water equations.

The main motivation is a better representation of the Andes mountain topography in the South American continent, which plays a key role in the weather patterns of the region, while maintaining a coarser grid for the rest of the globe to reduce the computational cost of the model.

The authors propose a density function with parameters that can be tuned to specify the approximate ratio between the coarse and fine grid resolution areas; the refinement level specifically to the Andes region; and the refinement level of the larger South America region.

This density function is used to generate a series of locally refined grids with an increasing number of Voronoi cells, which are used in different test cases where the results are compared to solutions using uniform resolution grids.

The test cases show that when the grid with local refinement is used spurious numerical waves are produced, an artifact not present with the uniform grids, and that adding artificial diffusion to the simulation mitigates this issue.

Although the issues produced by the local refinement can be mitigated, the classical shallow water model test cases show that the results obtained are not improved when compared to a solution using a uniform grid with the same number of Voronoi cells.

With the moist-shallow water model using the locally refined grid lead to improved cloud representation, nevertheless, spurious precipitation was also present.

The paper is well written and well organized and provides useful information for the scientific community.

I recommend publication but also suggest a few potential revisions to further improve the paper:

• One addition that I believe would greatly improve the paper is the inclusion of a set of grids with a greater coarse to fine cell ratio, that is, a grid generated with a greater value for γ in the density function (For example, γ=6). How the results would compare to those with γ=1 and γ=3? Are the spurious waves magnified? If so, linearly with gamma or following a higher order? Would errors start developing at earlier times? Would the inclusion of the same amount of artificial diffusion still mitigate the problem?

Some minor suggestions/questions to be discussed in the paper are:

1. In the steady geostrophic flow analysis, do all VR1 to VR7 results follow the same pattern of developing spurious waves? Figure 6 might suggest that is happing only for the VR6 and VR7 grids.
2. Figure 7c shows the alignment index for the VR7 grid. What it looks like for a uniform grid, such as UR7 or UR8?
3. Still regarding figure 7c, I'd expect the bad alignments would happen in the transition of refined/coarse cells region. Is there an explanation for the bad indices in the mostly uniform region of the grid?
4. For the Flow over the Andes mountain test case, besides the figures with the error, it would be interesting to also include figures of the actual final solution for the velocity magnitude and height fields, since there is no similar simulation in the literature. It would also give the reader a better sense of how big the error is compared to the actual solution.
5. On page 15, line 300 of the manuscript, it is written "Despite that, the magnitude of the does not increase...", I believe it should be "Despite that, the magnitude of the error...".
6. In figure 10, for comparison purposes, it would be better if the same colormaps ranges were used for all plots, with colors cropped at a minimum of -1.8e-8 and a maximum of 1.8e8 and different colors could be used for data outside that range.