Improved double Fourier series on a sphere and its application to  a semi-implicit semi-Lagrangian shallow-water model

Yoshimura, Hiromasa

doi:https://doi.org/10.5194/gmd-15-2561-2022

Articles | Volume 15, issue 6

https://doi.org/10.5194/gmd-15-2561-2022

© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/gmd-15-2561-2022

© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 15, issue 6

Development and technical paper

|

28 Mar 2022

Development and technical paper |

| 28 Mar 2022

Improved double Fourier series on a sphere and its application to a semi-implicit semi-Lagrangian shallow-water model

Hiromasa Yoshimura

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Final revised paper (published on 28 Mar 2022)
Supplement to the final revised paper
Preprint (discussion started on 08 Jul 2021)
Supplement to the preprint

Interactive discussion

Status: closed

RC1:
'Comment on gmd-2021-168', Anonymous Referee #1, 30 Jul 2021

Comments are provided in a PDF file.

Citation: https://doi.org/10.5194/gmd-2021-168-RC1
- AC1: 'Reply on RC1', Hiromasa Yoshimura, 27 Aug 2021
  
  The comment was uploaded in the form of a supplement: https://gmd.copernicus.org/preprints/gmd-2021-168/gmd-2021-168-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/gmd-2021-168-AC1
RC2:
'Comment on gmd-2021-168', Anonymous Referee #2, 02 Aug 2021

The comment was uploaded in the form of a supplement: https://gmd.copernicus.org/preprints/gmd-2021-168/gmd-2021-168-RC2-supplement.pdf

Citation: https://doi.org/10.5194/gmd-2021-168-RC2
- AC2: 'Reply on RC2', Hiromasa Yoshimura, 27 Aug 2021
  
  The comment was uploaded in the form of a supplement: https://gmd.copernicus.org/preprints/gmd-2021-168/gmd-2021-168-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/gmd-2021-168-AC2
RC3:
'Comment on gmd-2021-168', Anonymous Referee #3, 10 Sep 2021
Comments on “Improved double Fourier series on a sphere and its application to a semi-implicit semi-Lagrangian shallow water model” by Hiromasa Yoshimura.

General comments

The author proposes an alternative formulation to alleviate the pole problem used in a shallow water model using double Fourier series. The paper presents specific forms for the gradient of a scalar, vectors, and Laplacian and Helmholtz operators. The comparison against spherical harmonics and the new double Fourier series provides insights into the properties of the former. I recommend the publication of the paper provided that the following concern is appropriately addressed.

Major comments

As the other reviewers noted, the tests seem to be randomly chosen. I recommend the author to investigate at least the convergence (error vs horizontal resolution) and conservation (energy, vorticity, etc.). Avoid visual comparison where possible and conduct quantitative evaluation. The errors should be given in standard error norms by numerical values.

The presentation of the paper can be improved. For example, Subsection 5.2 and 5.4 are short and 5.3 is of two long paragraphs. Author’s intention for the tests should be provided. The author discusses the pole problem, but Figures 4, 7 and 9 are shown in longitude–latitude; Figure 9 omits the polar region. It would be nice to add a diagram to show the differences of expansion visually.

Minor comments

Page 1, Line 21: more accurate rather than good

Page 1, Line 25: O(N³) memory usage, unless calculated on-the-fly

Page 2, Line 4: Alternatively (avoid repeating the same word, another).
Citation: https://doi.org/10.5194/gmd-2021-168-RC3
- AC3:
  'Reply on RC3', Hiromasa Yoshimura, 18 Sep 2021
  [Reply on RC3]
  We are very grateful to the referee #3 for the useful comments. They help to improve the paper quality. Below, we will reply to the specific comments.
  
  Major comments
  As the other reviewers noted, the tests seem to be randomly chosen. I recommend the author to investigate at least the convergence (error vs horizontal resolution) and conservation (energy, vorticity, etc.). Avoid visual comparison where possible and conduct quantitative evaluation. The errors should be given in standard error norms by numerical values.
  Thank you for the advice. We will investigate the convergence and conservation and give the errors by numerical values.
  
  The presentation of the paper can be improved. For example, Subsection 5.2 and 5.4 are short and 5.3 is of two long paragraphs. Author’s intention for the tests should be provided. The author discusses the pole problem, but Figures 4, 7 and 9 are shown in longitude–latitude; Figure 9 omits the polar region. It would be nice to add a diagram to show the differences of expansion visually.
  Thank you for the advice. We will modify the paper according to the comments. We will show the global region including the polar region in Figure 9. Figure 2 shows the differences of the DFS expansions. We will also show the same figure for the SH expansion for comparison.
  
  Minor comments
  Page 1, Line 21: more accurate rather than good
  Page 1, Line 25: O(N3) memory usage, unless calculated on-the-fly
  Page 2, Line 4: Alternatively (avoid repeating the same word, another).
  Thank you for the advice. We will modify the texts according to the comments.
  
  Citation: https://doi.org/10.5194/gmd-2021-168-AC3

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Hiromasa Yoshimura on behalf of the Authors (28 Oct 2021) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (01 Nov 2021) by Ignacio Pisso

RR by Anonymous Referee #3 (20 Nov 2021)

RR by Anonymous Referee #2 (05 Dec 2021)

Suggestions for revision or reasons for rejection

The author made a good effort to make the paper more accessible and I appreciate the responses to my comments. Except one perhaps, if they already have a parallelised version, why not comment on the connectivity needed between processors (need all zonal values for zonal expansion but parallel over m, need xx for the meridional, parallel over xx ?).
But I appreciate this may be the subject of future work to refine, so looking forward to more information on this in the future.

I find some of the notation too much (e.g. upper case N,M and equation 15 (needed?), many of these could be dropped in my view, just stating that the prognostic variables are expressed as truncated series, this would make many equations a lot less busy.

I think what is missing is a quick summary "cook book", which could take the form of (if I understood correctly):

(1) Define expansion in double Fourier series, with zonal expansion [equation (8)] and meridional expansion [equation (10)]

(2) Apply recursive formulas for the inverse transform from spectral to gridpoint [equations 13-14)

(3) For the direct transform (grid to spectral) using the new DFS method:

a) define new expansion (tilde values, equation 23), and evaluate coefficients by forward transform directly from the values at the gridpoints (see appendix B)
b) minimise (with variation of the unknown spectral expansion coefficients of 10) over the sphere the squared residual R defined in equation 25: R is the distance between the series approximations defined in 10 and 23 leading to conditions 26 and with 27 to equation 28, noting the orthogonality condition with respect to R.
c) Evaluate the unknown spectral coefficients using the resulting matching conditions with the known expansion coefficients (tilde), derived by inserting 10 (in lhs of 28) and 23 (in rhs of 28) leading to equations 30 and the evaluation procedure 31.

Notably, above is slightly different to what is in the current manuscript, in particular the derivation of 29 with appendix D and the current definition of R I find confusing even if it leads to the same result. Is my interpretation above incorrect ?

Other comments:

The new title is very long, I am wondering if "Improved double Fourier series for the shallow water equations on the sphere" would be more fitting. The rest are details that belong in the abstract.

In the abstract: "These small oscillations..."

I would also remove the sentence in the conclusions : "The Eulerian shallow water model using the new DFS method without horizontal diffusion is also likely to be stable, although we have not tested it yet."

Overall the text has changed/moved a lot, and it is difficult to verify all the revised cross references, but I assume editorial work will check this once more.

Otherwise I found the appendices helpful, albeit many. In the end it is justified as this serves as a documentation of a new dynamical core and a reference in comparison to the spectral transform models based on spherical harmonics.

I would like to thank the author for his hard work on addressing the different test cases and questions related to the accuracy of the wave operator.

Hide

ED: Publish subject to minor revisions (review by editor) (21 Dec 2021) by Ignacio Pisso

Dear Author,

Please consider the following points raised by the reviewer.

"The author made a good effort to make the paper more accessible and I appreciate the responses to my comments. Except one perhaps, if they already have a parallelised version, why not comment on the connectivity needed between processors (need all zonal values for zonal expansion but parallel over m, need xx for the meridional, parallel over xx ?).
But I appreciate this may be the subject of future work to refine, so looking forward to more information on this in the future.

I find some of the notation too much (e.g. upper case N,M and equation 15 (needed?), many of these could be dropped in my view, just stating that the prognostic variables are expressed as truncated series, this would make many equations a lot less busy.

I think what is missing is a quick summary "cook book", which could take the form of (if I understood correctly):

(1) Define expansion in double Fourier series, with zonal expansion [equation (8)] and meridional expansion [equation (10)]

(2) Apply recursive formulas for the inverse transform from spectral to gridpoint [equations 13-14)

(3) For the direct transform (grid to spectral) using the new DFS method:

a) define new expansion (tilde values, equation 23), and evaluate coefficients by forward transform directly from the values at the gridpoints (see appendix B)
b) minimise (with variation of the unknown spectral expansion coefficients of 10) over the sphere the squared residual R defined in equation 25: R is the distance between the series approximations defined in 10 and 23 leading to conditions 26 and with 27 to equation 28, noting the orthogonality condition with respect to R.
c) Evaluate the unknown spectral coefficients using the resulting matching conditions with the known expansion coefficients (tilde), derived by inserting 10 (in lhs of 28) and 23 (in rhs of 28) leading to equations 30 and the evaluation procedure 31.

Notably, above is slightly different to what is in the current manuscript, in particular the derivation of 29 with appendix D and the current definition of R I find confusing even if it leads to the same result. Is my interpretation above incorrect ?

Other comments:

The new title is very long, I am wondering if "Improved double Fourier series for the shallow water equations on the sphere" would be more fitting. The rest are details that belong in the abstract.

In the abstract: "These small oscillations..."

I would also remove the sentence in the conclusions : "The Eulerian shallow water model using the new DFS method without horizontal diffusion is also likely to be stable, although we have not tested it yet."

Overall the text has changed/moved a lot, and it is difficult to verify all the revised cross references, but I assume editorial work will check this once more.

Otherwise I found the appendices helpful, albeit many. In the end it is justified as this serves as a documentation of a new dynamical core and a reference in comparison to the spectral transform models based on spherical harmonics.

I would like to thank the author for his hard work on addressing the different test cases and questions related to the accuracy of the wave operator."

Hide

AR by Hiromasa Yoshimura on behalf of the Authors (15 Jan 2022) Author's response Author's tracked changes Manuscript

ED: Publish subject to technical corrections (17 Feb 2022) by Ignacio Pisso

AR by Hiromasa Yoshimura on behalf of the Authors (23 Feb 2022) Author's response Manuscript

Short summary

This paper proposes a new double Fourier series (DFS) method on a sphere that improves the numerical stability of a model compared with conventional DFS methods. The shallow-water model and the advection model using the new DFS method give stable results without the appearance of high-wavenumber noise near the poles. The model using the new DFS method is faster than the model using spherical harmonics (especially at high resolutions) and gives almost the same results.