Process-based diagnostics using atmospheric budget analysis and nudging technique to identify sources of model systematic errors

Matsukawa, Chihiro; Rodríguez, José M.; Milton, Sean F.

doi:10.5194/egusphere-2025-1466

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https://doi.org/10.5194/egusphere-2025-1466

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17 Apr 2025

| 17 Apr 2025

Process-based diagnostics using atmospheric budget analysis and nudging technique to identify sources of model systematic errors

Chihiro Matsukawa, José M. Rodríguez, and Sean F. Milton

Abstract. Identifying sources of model systematic errors is a fundamental step to successfully reduce them in general circulation models by improving the representation of relevant physical processes. In this study, we examine model error sources in the Met Office Unified Model at numerical weather prediction timescale by the combined use of two diagnostics: 1) the relaxation or "nudging" in which wind and/or temperature fields are relaxed back towards analyses throughout the simulations, and 2) atmospheric zonal-mean zonal momentum and thermal budgets. The budget analysis quantifies resolved processes and subsequently estimates unresolved processes as a residual, corresponding to model dynamics and physics, respectively. This correspondence is demonstrated by a direct comparison between the budgets and the model tendencies. A systematic error addressed in this paper is the Northern Hemisphere mid-latitude zonal wind bias in the lower stratosphere in boreal winter, characterized by an initial easterly bias that subsequently develops as a westerly bias. The momentum and thermal budget analysis for control and nudging experiments indicates that a mechanical forcing predominantly from parametrized gravity wave drag causes the easterly error and an overly strong temperature gradient around the tropopause is one of the main sources of the westerly error through the Coriolis forcing. The relevant warm bias over the tropical tropopause is mainly attributed to the budget residual term that corresponds to a thermal forcing dominated by radiative processes. This is consistent with the experimental result that temperature nudging over the tropical tropopause significantly reduces the westerly wind bias.

Received: 30 Mar 2025 – Discussion started: 17 Apr 2025

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Journal article(s) based on this preprint

26 Nov 2025

Process-based diagnostics using atmospheric budget analysis and nudging technique to identify sources of model systematic errors

Chihiro Matsukawa, José M. Rodríguez, and Sean F. Milton

Weather Clim. Dynam., 6, 1539–1564, https://doi.org/10.5194/wcd-6-1539-2025,https://doi.org/10.5194/wcd-6-1539-2025, 2025

Short summary

Chihiro Matsukawa, José M. Rodríguez, and Sean F. Milton

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-1466', Anonymous Referee #1, 23 May 2025
General Comments:
This paper addresses the interesting questions regarding the development of model errors in temperature and zonal wind in the Met Office UM, using a nudging relaxation and a tendency budget decomposition. These methods are well described, and used to come to some interesting results. In the most part the paper is well written, the results well presented and the conclusions good, however I would suggest a substantial re-write of Section 4 and additional reference to related work before the paper is accepted for publication.
In several places the text would be easier to follow if each panel were given unique identifying letters. Figures 2 and 12 are good, figures 3-11 would benefit from additional labels.

Major comments:
L228-238, L430-432: As this is not the first article to identify temperature and wind biases in models, it would be appropriate to provide in the introduction at least a brief review or acknowledgment of the previous work on model temperature and wind biases (particularly the lower stratosphere zonal wind, and tropical tropopause temperature) (in addition to the brief mention of two examples on L555 at the end) and some summary of the existing knowledge. By providing a sufficient overview of the related work that has already been published you will be able to make it much clearer to readers the original contribution that this article makes in the context of the current state of knowledge.

Section 4:

As I understand it, the residual term for CNTL is the sum of the parameterised processes, any numerical integration errors and differences arising from the primitive budget equations being different from the actual equations of the UM. The CNTL-GLN difference in the residual is the difference of these in the CNTL and GLN plus the nudging, which is the very short timescale model error in the UM. However there is no single, early, concise explanation of this.

While various aspects of the above are written in various places in various ways, references to components of this and interpretations of the residual often seem confusing. For example lines 302-304 present the similarity as if it were not true by definition. The article could benefit from a concise re-write of lines 327-335 to clearly explain the interpretation of the residual, and moved earlier in the section. Then any subsequent discussion of the residual or any of its components can be re-considered in the context of the earlier explanation to make the section overall more cohesive and consistent.

A consequence of this is: taking the mean CNTL-GLN tendency from dynamics (4b, 6b, 8b, 10b), will this not be disproportionately dominated by the model dynamics bias in the first 6 hours being balanced by the nudging, therefore masking any information about differences between CNTL and GLN at timescales longer than this? If this is what you are trying to address in lines 320-324 it is not clear. With terms being so heavily dominated by the nudging, what information can be gotten from the dynamics/residual at timescales of longer than 24 hours in any quantities including data from GLN?
It is not clear that figure 4 provides any useful information not contained within figures 5 and 6 (and similarly that figure 8 does not show anything not in figures 9-11) other than the contour overlay which could be moved. It may be possible to make this section more concise by removing figures 4 & 8.
L323: This information not shown would perhaps benefit from being shown. With such an emphasis on the Coriolis term in the zonal wind budget, the article would benefit from the inclusion of further discussion of any biases in the meridional wind field.
L330-331: Is this conclusion necessarily true? Unsure if the results provided support this.
L367-370: The sign of the CNTL-GLN difference in physics is the opposite of the sign of the error in 10-b-3 and 10-b-1, which seems to suggest the opposite to what this sentence is saying.
L323-324: By “the error in Coriolis term against truth” do you mean the error in the Coriolis term in CNTL against truth? If so, state this.

Additional specific comments:
All figures: It would benefit the reader if each individual panel were assigned a letter identifier (a), (b), etc.
L33-34: There have been a wide variety of diagnostic methods… It would be appropriate to list some.
L124-128: I can discern what you mean, but it would be beneficial to re-write these sentences to make them clearer.
L191: This is not a general property of all partial differential equations, so remove these opening 4 words (or re-phrase the sentence to make it more accurate).
L253-254: I do not see evidence for this statement. Provide evidence, or clarify that this is not shown.
L262-263 & Figures 2d, 3b, 12a, 12b: The independence of GLN state with forecast lead time seems strange. Is the GLN state at lead times >24 hours truly identical? Can you explain this or comment further on why the differences are zero or imperceptibly small?
L292-293: a decrease in the tropics... State, a decrease in what in the tropics. Similarly for lines 293-295.
Figure 12-14: Are the differences between the dynamics and the sum of the resolved flow components, and the differences between the residual tern and the sum of the nudging + physics, in panels f compared to panels j, larger than one would expect resulting from the model level v.s. pressure level comparison? They seem quite large by eye. Can you comment on this please?
L479-480: This result seems important but it is not shown in this article (nor do you state in the text that it is not shown). With such an emphasis on the Coriolis term in the zonal wind budget, the article would benefit from the inclusion of further discussion of any biases in the meridional wind field.
L483-485: Returning to earlier comments on the interpretations of the residual term – this idea would benefit from further explanation and possibly inclusion of the “not shown” material.

Technical Corrections:
L29: within the first few days
L133-134: 10 degrees in the horizontal … two model levels in the vertical
L136: from December 2018 to February 2019 (or from the December of 2018 to the February of 2019)
L206: the word ‘definitely’ doesn’t need to be here
L242: we focus on three…
L262: but do not depend…
L538: “specially” is the wrong word to use here. One way to rewrite this would be for example “one promising way for operational …”
Citation: https://doi.org/10.5194/egusphere-2025-1466-RC1
- AC1: 'Reply on RC1', Chihiro Matsukawa, 04 Jul 2025
  
  We thank the referee #1 for their informative comments and constructive suggestions for improving the paper. We will make extensive corrections to a revised manuscript. More detailed responses to their comments are given in the attached pdf.
  
  Citation: https://doi.org/10.5194/egusphere-2025-1466-AC1
RC2:
'Comment on egusphere-2025-1466', Anonymous Referee #2, 02 Jun 2025

Review of "Process-based diagnostics using atmospheric budget analysis and nudging technique to identify sources of model systematic errors" by Matsukawa et al. submitted to WCD.
The reviewer would like to thank the authors for their comprehensive study on model-based diagnostics to identify sources of model systematic errors, which could be useful for any NWP and climate models. The method successfully detects sources and origins of model forecast errors in the Met Office Unified Model, which could be common in the state-of-the-art operational and research atmospheric GCMs. I think The paper is comprehensive and well organized as a WCD paper, so that I would already recommend minor revision, though I have several comments below before acceptance.
Specific comments:

1) The relaxation timescale of 6 hours could be practically useful to reduce/erase the initial errors that can grow up in subsequent forecasts. I am curious that how sensitive is "tau" (timescale) to your conclusion? For example, if you choose 12 or 24 hours as tau, you could obtain consistent results, that is, could detect the same origins causing the NH wind field errors?

The relaxation timescale might be similar with the evaluation time of the Forecast Sensitivity to Observations (Hotta et al. 2017, Prive et al. 2020). The shorter timescales might be more practical for forecasts.

(Hotta et al. 2017, MWR, DOI:10.1175/MWR-D-16-0290.1; Prive et al. 0220, QJRMS, DOI:10.1002/qj.3909)
2) Could you comment in your conclusion part about two topics that could be useful as future studies?

- Budget analysis based on conservation values, potential vorticity or angular momentum (TEM or MIM), could be useful for your further budget analysis. Currently I only refer to a paper by Kobayashi and Iwasaki (2016), though there may be other important studies to be cited.

- As a comparison with the experiment for 2018 boreal winter, additional experiment for an SH winter could be helpful to improve your conclusion (e.g., Yamazaki et al. 2023).

(Kobayashi and Iwasaki 2016, JGR-A, DOI:10.1002/2015JD023476; Yamazaki et al. 2023, WAF, DOI:10.1175/WAF-D-22-0159.1)

Citation: https://doi.org/10.5194/egusphere-2025-1466-RC2
- AC2: 'Reply on RC2', Chihiro Matsukawa, 04 Jul 2025
  
  We thank the referee #2 for their valuable comments. We will make substantial corrections to a revised manuscript based on their comments. More detailed responses to their comments are given in the attached pdf.
  
  Citation: https://doi.org/10.5194/egusphere-2025-1466-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-1466', Anonymous Referee #1, 23 May 2025
General Comments:
This paper addresses the interesting questions regarding the development of model errors in temperature and zonal wind in the Met Office UM, using a nudging relaxation and a tendency budget decomposition. These methods are well described, and used to come to some interesting results. In the most part the paper is well written, the results well presented and the conclusions good, however I would suggest a substantial re-write of Section 4 and additional reference to related work before the paper is accepted for publication.
In several places the text would be easier to follow if each panel were given unique identifying letters. Figures 2 and 12 are good, figures 3-11 would benefit from additional labels.

Major comments:
L228-238, L430-432: As this is not the first article to identify temperature and wind biases in models, it would be appropriate to provide in the introduction at least a brief review or acknowledgment of the previous work on model temperature and wind biases (particularly the lower stratosphere zonal wind, and tropical tropopause temperature) (in addition to the brief mention of two examples on L555 at the end) and some summary of the existing knowledge. By providing a sufficient overview of the related work that has already been published you will be able to make it much clearer to readers the original contribution that this article makes in the context of the current state of knowledge.

Section 4:

As I understand it, the residual term for CNTL is the sum of the parameterised processes, any numerical integration errors and differences arising from the primitive budget equations being different from the actual equations of the UM. The CNTL-GLN difference in the residual is the difference of these in the CNTL and GLN plus the nudging, which is the very short timescale model error in the UM. However there is no single, early, concise explanation of this.

While various aspects of the above are written in various places in various ways, references to components of this and interpretations of the residual often seem confusing. For example lines 302-304 present the similarity as if it were not true by definition. The article could benefit from a concise re-write of lines 327-335 to clearly explain the interpretation of the residual, and moved earlier in the section. Then any subsequent discussion of the residual or any of its components can be re-considered in the context of the earlier explanation to make the section overall more cohesive and consistent.

A consequence of this is: taking the mean CNTL-GLN tendency from dynamics (4b, 6b, 8b, 10b), will this not be disproportionately dominated by the model dynamics bias in the first 6 hours being balanced by the nudging, therefore masking any information about differences between CNTL and GLN at timescales longer than this? If this is what you are trying to address in lines 320-324 it is not clear. With terms being so heavily dominated by the nudging, what information can be gotten from the dynamics/residual at timescales of longer than 24 hours in any quantities including data from GLN?
It is not clear that figure 4 provides any useful information not contained within figures 5 and 6 (and similarly that figure 8 does not show anything not in figures 9-11) other than the contour overlay which could be moved. It may be possible to make this section more concise by removing figures 4 & 8.
L323: This information not shown would perhaps benefit from being shown. With such an emphasis on the Coriolis term in the zonal wind budget, the article would benefit from the inclusion of further discussion of any biases in the meridional wind field.
L330-331: Is this conclusion necessarily true? Unsure if the results provided support this.
L367-370: The sign of the CNTL-GLN difference in physics is the opposite of the sign of the error in 10-b-3 and 10-b-1, which seems to suggest the opposite to what this sentence is saying.
L323-324: By “the error in Coriolis term against truth” do you mean the error in the Coriolis term in CNTL against truth? If so, state this.

Additional specific comments:
All figures: It would benefit the reader if each individual panel were assigned a letter identifier (a), (b), etc.
L33-34: There have been a wide variety of diagnostic methods… It would be appropriate to list some.
L124-128: I can discern what you mean, but it would be beneficial to re-write these sentences to make them clearer.
L191: This is not a general property of all partial differential equations, so remove these opening 4 words (or re-phrase the sentence to make it more accurate).
L253-254: I do not see evidence for this statement. Provide evidence, or clarify that this is not shown.
L262-263 & Figures 2d, 3b, 12a, 12b: The independence of GLN state with forecast lead time seems strange. Is the GLN state at lead times >24 hours truly identical? Can you explain this or comment further on why the differences are zero or imperceptibly small?
L292-293: a decrease in the tropics... State, a decrease in what in the tropics. Similarly for lines 293-295.
Figure 12-14: Are the differences between the dynamics and the sum of the resolved flow components, and the differences between the residual tern and the sum of the nudging + physics, in panels f compared to panels j, larger than one would expect resulting from the model level v.s. pressure level comparison? They seem quite large by eye. Can you comment on this please?
L479-480: This result seems important but it is not shown in this article (nor do you state in the text that it is not shown). With such an emphasis on the Coriolis term in the zonal wind budget, the article would benefit from the inclusion of further discussion of any biases in the meridional wind field.
L483-485: Returning to earlier comments on the interpretations of the residual term – this idea would benefit from further explanation and possibly inclusion of the “not shown” material.

Technical Corrections:
L29: within the first few days
L133-134: 10 degrees in the horizontal … two model levels in the vertical
L136: from December 2018 to February 2019 (or from the December of 2018 to the February of 2019)
L206: the word ‘definitely’ doesn’t need to be here
L242: we focus on three…
L262: but do not depend…
L538: “specially” is the wrong word to use here. One way to rewrite this would be for example “one promising way for operational …”
Citation: https://doi.org/10.5194/egusphere-2025-1466-RC1
- AC1: 'Reply on RC1', Chihiro Matsukawa, 04 Jul 2025
  
  We thank the referee #1 for their informative comments and constructive suggestions for improving the paper. We will make extensive corrections to a revised manuscript. More detailed responses to their comments are given in the attached pdf.
  
  Citation: https://doi.org/10.5194/egusphere-2025-1466-AC1
RC2:
'Comment on egusphere-2025-1466', Anonymous Referee #2, 02 Jun 2025

Review of "Process-based diagnostics using atmospheric budget analysis and nudging technique to identify sources of model systematic errors" by Matsukawa et al. submitted to WCD.
The reviewer would like to thank the authors for their comprehensive study on model-based diagnostics to identify sources of model systematic errors, which could be useful for any NWP and climate models. The method successfully detects sources and origins of model forecast errors in the Met Office Unified Model, which could be common in the state-of-the-art operational and research atmospheric GCMs. I think The paper is comprehensive and well organized as a WCD paper, so that I would already recommend minor revision, though I have several comments below before acceptance.
Specific comments:

1) The relaxation timescale of 6 hours could be practically useful to reduce/erase the initial errors that can grow up in subsequent forecasts. I am curious that how sensitive is "tau" (timescale) to your conclusion? For example, if you choose 12 or 24 hours as tau, you could obtain consistent results, that is, could detect the same origins causing the NH wind field errors?

The relaxation timescale might be similar with the evaluation time of the Forecast Sensitivity to Observations (Hotta et al. 2017, Prive et al. 2020). The shorter timescales might be more practical for forecasts.

(Hotta et al. 2017, MWR, DOI:10.1175/MWR-D-16-0290.1; Prive et al. 0220, QJRMS, DOI:10.1002/qj.3909)
2) Could you comment in your conclusion part about two topics that could be useful as future studies?

- Budget analysis based on conservation values, potential vorticity or angular momentum (TEM or MIM), could be useful for your further budget analysis. Currently I only refer to a paper by Kobayashi and Iwasaki (2016), though there may be other important studies to be cited.

- As a comparison with the experiment for 2018 boreal winter, additional experiment for an SH winter could be helpful to improve your conclusion (e.g., Yamazaki et al. 2023).

(Kobayashi and Iwasaki 2016, JGR-A, DOI:10.1002/2015JD023476; Yamazaki et al. 2023, WAF, DOI:10.1175/WAF-D-22-0159.1)

Citation: https://doi.org/10.5194/egusphere-2025-1466-RC2
- AC2: 'Reply on RC2', Chihiro Matsukawa, 04 Jul 2025
  
  We thank the referee #2 for their valuable comments. We will make substantial corrections to a revised manuscript based on their comments. More detailed responses to their comments are given in the attached pdf.
  
  Citation: https://doi.org/10.5194/egusphere-2025-1466-AC2

Peer review completion

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

AR by Chihiro Matsukawa on behalf of the Authors (01 Aug 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (05 Aug 2025) by Thomas Birner

RR by Anonymous Referee #2 (22 Aug 2025)

RR by Anonymous Referee #1 (27 Aug 2025)

For final publication, the manuscript should be

accepted as is

accepted subject to technical corrections

accepted subject to minor revisions

reconsidered after major revisions

rejected

Were a revised manuscript to be sent for another round of reviews:

I would be willing to review the revised manuscript.

I would not be willing to review the revised manuscript.

Suggestions for revision or reasons for rejection

Thank you for reading and taking on my comments. I am glad you found them useful, and appreciate the improved clarity of the article. It is good to see that the majority of points raised by both reviewers have been addressed, however, unless I have missed something, it still seems that there is some inconsistency with regards to the comparison between the nudging, the dynamics quasi-error, and the total tendency quasi-error.
For brevity I will not reproduce previous responses in full here, but rather just section headings and opening words.

Under the category “Major comments”

As I understand it, the residual term for CNTL…

... However, they are implicitly linked through their governing equations, with different assumption and approximation as indicated by referee #1. We think it is inappropriate to consider the two diagnostics to be completely independent.
…
Thank you for clarifying the agreement on the points that the tendencies and budgets are linked due to the governing equations, and that it is inappropriate to consider them independent. This did not come across in the original text and is useful to have clarified.
The only further note I have on section 2.5 is for line 259, I might suggest changing “but is not necessarily true by definition” to “but they are not necessarily exactly equivalent”.
I assume based on lines 260-266 that this is the intended meaning, but as written I do not think the sentence means that. As written is seems something of a contradiction, as is the derivation of the budget not also the definition of the components?

A consequence of this is: taking the mean CNTL-GLN tendency from dynamics (4b, 6b, 8b, 10b), will this not be disproportionately dominated by the model dynamics bias in the first 6 hours being balanced by the nudging…

It may be the case that I was very unclear in my first review, and phrased this badly.
I do not think it is adequately addressed as to why the CNTL-GLN nudging in figure 6(h) appears to have a similar character but with opposite sign to the dynamics in figure 6(f), and whether this is important to the conclusions.

The similarity is noted in line 402, but in lines 413-414 you write that nudging increments are comparable to forecast errors. The forecast error in figure 1(c, e) has a similar character to the total tendency error in 4(d) and 6(e), however these do not have similar character to the resolved processes error (4e), the Coriolis error (5h) or the dynamics error (6f), which all instead are comparable to the nudging error (6h). Therefore it does not seem that nudging increments are comparable to forecast erros.

It is acknowledged in line 450 that the thermal budget and the momentum budgets are different. Here it is noted that there is similarity between the residual (8f) and the total (8d). This similarity is used to conclude deficiencies in physics parametrizations from the thermal budget. However by using the phrase “unlike the momentum budget”, this highlights my confusion that you also conclude for the momentum budget that it is a deficiency in the physics parametrizations in lines 413-417, with similar phrasing, despite the opposite result.
I interpret line 415-417 as saying that the model physics should be applying the same forcing that the nudging is, because you are using a nudged forecast as the truth.

A first look at figure 6 had me confused, as it seemed that the majority of the forcing of the nudging to move the GLN field towards the analysis was being directly un-done and opposed by the tendencies of the model dynamics. The majority of the nudging tendency being directly counteracted seemed to be an odd result, but as I have not personally used nudging in the same way as you are here in this study, it would be very useful if you could please provide your interpretation of what the cancellation of terms in figure 6(f) and 6(h) means, in terms of what the nudging is doing, how the model is responding, and how the differences between CNTL and GLN should be interpreted in light of this.
Furthermore, the nudging term in the thermal budget is not mentioned at all in the text of section 4.2, and as with the momentum is also of similar character and opposite sign to the resolved tendency, particularly at the lowest pressures.

L330-331: Is this conclusion necessarily true? … (now L413)

… Similarity between residual term error and total tendency error, which is proportional to the error of the corresponding variable itself as shown in the equation described above, suggests that the error of the corresponding variable itself stems from the residual term error …

I can agree that this would be true, however, as before, I do not see similarity between residual term error and total tendency error in figure 4. And, you also seem to acknowledge that this similarity is not there, in line 450.

Referee Report: PDF

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ED: Publish subject to minor revisions (review by editor) (29 Aug 2025) by Thomas Birner

AR by Chihiro Matsukawa on behalf of the Authors (18 Sep 2025) Author's response Author's tracked changes Manuscript

ED: Publish subject to technical corrections (14 Oct 2025) by Thomas Birner

AR by Chihiro Matsukawa on behalf of the Authors (22 Oct 2025) Manuscript

Journal article(s) based on this preprint

26 Nov 2025

Process-based diagnostics using atmospheric budget analysis and nudging technique to identify sources of model systematic errors

Chihiro Matsukawa, José M. Rodríguez, and Sean F. Milton

Weather Clim. Dynam., 6, 1539–1564, https://doi.org/10.5194/wcd-6-1539-2025,https://doi.org/10.5194/wcd-6-1539-2025, 2025

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Chihiro Matsukawa, José M. Rodríguez, and Sean F. Milton

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To identify sources of the model systematic errors, we investigate northern hemisphere mid-latitude wind errors at short- to medium-range timescale using the atmospheric zonal-mean budgets analysis and the relaxation technique. These process-based diagnostics specify to what extent the individual components in the budgets contribute to the total tendency of the corresponding variable. This study shows disentanglements of compensating errors caused by mechanical and thermal forcings.


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