Adaptive time step algorithms for the simulation of marine ecosystem models using the transport matrix method implementation Metos3D (v0.5.0)
 KMS  Centre for Interdisciplinary Marine Science, Dept. of Computer Science, Kiel University, 24098 Kiel, Germany
 KMS  Centre for Interdisciplinary Marine Science, Dept. of Computer Science, Kiel University, 24098 Kiel, Germany
Abstract. The reduction of the computational effort is desirable for the simulation of marine ecosystem models. Using a marine ecosystem model, the assessment and the validation of annual periodic solutions (i.e., steady annual cycles) against observational data are crucial to identify biogeochemical processes, which, for example, influence the global carbon cycle. For marine ecosystem models, the transport matrix method (TMM) already lowers the runtime of the simulation significantly and enables the application of larger time steps straightforwardly. However, the selection of an appropriate time step is a challenging compromise between accuracy and shortening the runtime. Using an automatic time step adjustment during the computation of a steady annual cycle with the TMM, we present in this paper different algorithms applying either an adaptive step size control or decreasing time steps in order to use the time step always as large as possible without any manual selection. For these methods and a variety of marine ecosystem models of different complexity, the accuracy of the computed steady annual cycle achieved the same accuracy as solutions obtained with a fixed time step. Depending on the complexity of the marine ecosystem model, the application of the methods shortened the runtime significantly. Due to the certain overhead of the adaptive method, the computational effort may be higher in special cases using the adaptive step size control. The presented methods represent computational efficient methods for the simulation of marine ecosystem models using the TMM but without any manual selection of the time step.
Markus Pfeil and Thomas Slawig
Status: closed

CEC1: 'Comment on gmd2021392', Juan Antonio Añel, 22 Feb 2022
Dear authors,
After checking your manuscript, it has come to our attention that it does not comply with our Code and Data Policy.
https://www.geoscientificmodeldevelopment.net/policies/code_and_data_policy.html
You have archived your code in GitHub. However, GitHub is not a suitable repository. GitHub itself instructs authors to use other alternatives for longterm archival and publishing, such as Zenodo. Therefore, please, publish your code in one of the appropriate repositories, and reply to this comment with the information about it.In this way, you must include in a potential reviewed version of your manuscript the modified 'Code and Data Availability' section, including the DOI of the code.
Juan A. Añel
Geosci. Model Dev. Exec. Editor AC1: 'Reply on CEC1', Markus Pfeil, 30 Mar 2022

RC1: 'Comment on gmd2021392', Anonymous Referee #1, 13 Apr 2022
General Comments:
I went through the manuscript from the beginning to the end twice carefully. As an ocean biogeochemistry modeler, I found the manuscript is very hard to understand and follow. It was not like a model technique paper, but more like a bunch of model testing results report. I reject the paper and provide my comments below for authors to improve in the future.
Specific Comments:
 The model technique that was intend to address
The prepared manuscript tried to improve the time step selection in the transport matrix method (TMM). It argued the used time step affects both the computational effort and accuracy of the steady annual cycle computation. I think both the “computational effort” and “steady annual cycle” should be well explained in the introduction part. What are they? Are they being recognized as a common problem in previous research? How that impact the follow modelling results? It was very hard to feel the importance of the technique problem presents in so far writing.
 Experiment design and way to present results
“The experiments are designed to shorten the running time of the computed steady annual cycle. The accuracy and cost save the calculated approximation. It was very hard to understand the present results from so far figures and tables.” Why not have a figure to show what is the nonsteady state annual cycle globally look like and what a steady state annual cycle looks like? Why not have comparisons between on and off line methods to show how they impact the steady annual cycle and the accuracy. It was also very hard to tell how much accuracy has been improved and cost saving was necessary from so far results.
It will be good to read and learn the cited manuscript “Accelerated simulation of passive tracers in ocean circulation models” carefully and learn how to present results like this to guide the reader. Another good example is “Performance of offline passive tracer advection in the Regional Ocean Modelling System” by Thyng et al. on GMD
 Mathematical forms
` I felt there was too much details about how to get A from B, which are very annoying to follow the final results. It should be provided in the supplementary material. It should also provide some math in matrix form, which will be easier to follow
Technical Corrections:
L68L71:
“Due to the fully coupling of the ocean circulation with the tracers, ……, the tracer concentrations affect the circulation, the simulation of a fully coupled model … to single model evaluations.”
It was not correct to say this. Commons like this for online and offline methods should be careful. Online coupling is a mature technique and widely used. With the development of the computational source, the computational cost is not that high. It was true the offline model will be more efficient, but it was also difficulty in accurately representing vertical fluxes due to deep convection.
L83:
“No fluxes on the boundary”
Need to state all experiments are for global run.

AC4: 'Reply on RC1', Thomas Slawig, 19 May 2022
*** First of all and although you reject the paper, we would like to thank you foryour report. It shows us that at least parts of the manuscript are not wellunderstandable. And since you write you are a modeler, we would of course alsowould like that also modelers understand our presentations.We provide our answers to your commenst in italics.General Comments:I went through the manuscript from the beginning to the end twice carefully.As an ocean biogeochemistry modeler, I found the manuscript is very hard tounderstand and follow. It was not like a model technique paper, but morelike a bunch of model testing results report. I reject the paper and providemy comments below for authors to improve in the future.*** Our paper is an algorithm development and evaluation paper. But we think that thiscan be alos regarded as "model technique".Specific Comments:The model technique that was intend to addressThe prepared manuscript tried to improve the time step selection in thetransport matrix method (TMM). It argued the used time step affects both thecomputational effort and accuracy of the steady annual cycle computation.I think both the “computational effort” and “steady annual cycle” shouldbe well explained in the introduction part. What are they? Are they beingrecognized as a common problem in previous research? How that impact thefollow modelling results? It was very hard to feel the importance of thetechnique problem presents in so far writing.*** We have to clarify the term "computational effort" as already mentioned bythe other reviewer. We will also clarify the term steady annual cycle, we thoughtthis was clear by equation (11) and the text before.Experiment design and way to present results“The experiments are designed to shorten the running time of the computedsteady annual cycle. The accuracy and cost save the calculated approximation.It was very hard to understand the present results from so far figures andtables.” Why not have a figure to show what is the nonsteady state annualcycle globally look like and what a steady state annual cycle looks like?Why not have comparisons between on and off line methods to show how theyimpact the steady annual cycle and the accuracy. It was also very hard totell how much accuracy has been improved and cost saving was necessary fromso far results.*** It was not our aim to compare off and online methods, and we did not want toargue against any of the two. We also do not see what benefit it would be for the readerto show a nonsteady solution. We are trying to reduce the runtime necessary to computea steady annual cycle. This is one form of the process usually called spinup of a model.We thought that we made clear in the introduction that runtime reduction is of importancein these kind of simulations. We will try to emphasize this point even more.It will be good to read and learn the cited manuscript “Acceleratedsimulation of passive tracers in ocean circulation models” carefully andlearn how to present results like this to guide the reader. Another goodexample is “Performance of offline passive tracer advection in the RegionalOcean Modelling System” by Thyng et al. on GMD*** Thank you for the recommendation. We will look into the seocnd paper you mentioned as well.Mathematical formsI felt there was too much details about how to get A from B, which arevery annoying to follow the final results. It should be provided in thesupplementary material. It should also provide some math in matrix form,which will be easier to follow*** Here, it would be nice to have more details. We are aware of the fact thatthere is some mathematics in the paper that may be a little bit lengthy.We will try to reduce some of the formulas in a revised version.Technical Corrections:L68L71:“Due to the fully coupling of the ocean circulation with the tracers, ……, thetracer concentrations affect the circulation, the simulation of a fully coupledmodel … to single model evaluations.”It was not correct to say this. Commons like this for online and offline methodsshould be careful. Online coupling is a mature technique and widely used. Withthe development of the computational source, the computational cost is not thathigh. It was true the offline model will be more efficient, but it was alsodifficulty in accurately representing vertical fluxes due to deep convection.*** We agree, this statement was definitely to strict. It will be reformulated andthe last part "to single model evaluation" will be skipped in a revised version.L83:“No fluxes on the boundary”Need to state all experiments are for global run.*** This is correct, we will add this point.

RC2: 'Comment on gmd2021392', Anonymous Referee #2, 22 Apr 2022
First, I have not read the other reviews, so my opinion is given independently.
This manuscript presents novel algorithms for adapting the time step in biogeochemical models driven by the Transport Matrix Method (TMM). The authors present 3 different ways of controlling the time step, and show that while Algorithms 1 and 3 can potentially speed up the spinup of the biogeochemical models in some (but not all cases), Algorithm 2 generally does not. This work seems worthy of publication after some moderate revisions and further critical evaluation of how well each method works. Some aspects of the data presentation and writing could also be improved.
General Comments
It is striking that the step size control method (Algorithm 1) does not actually do much continuous adjustment of the time step most of the time. While the authors go to great lengths to enable the time step to change responsively, in reality the step size quickly expands out to 32x the original time step, or it stays anchored at 1x the original time step. The adjustment happens very early in the simulations, meaning that for most of the 10,000 years the algorithm simply creates more overhead for the simulation without any additional benefit. This suggests that the whole process of finding the right time step could be restricted to an appropriate initiation period (perhaps 100 years, or whatever minimum envelope is needed), after which the time step is held fixed.
On a related note, I thought the design of the Algorithm 1 could be significantly improved by only running the error checks on a subset of time steps (not every single time step). Presently, the steps 11 and 12 are computed for every time step of the simulation to verify the accuracy of the chosen delta T. This error check could easily be made on a subset of timesteps. Perhaps on every 10^{th} time step, an error check could be run, or if there needs to be a continuous block of time steps, only do this for a limited window at periodic intervals. It seems wasteful to me to run the error check on every time step, when most of the time it will have no influence.
Algorithm 3 appears to have the most utility, since it clearly saves time in the spinup, and achieves a reasonable approximation to the reference case (correct me if I’ve got that wrong). Algorithm 2 fails because it appears to me that the exclusion of negative tracers is too stringent a condition. The authors mention that negative tracer concentrations sometimes occur in the reference case… so this algorithm should be a nonstarter, shouldn’t it?
Overall, I think there could be more critical evaluation in the discussion and conclusions as to which algorithms actually performed well, which are recommended or not, and why.
Line Comments
L36: “parallelization… lowers the computational effort”. Not really, it just speeds up the result. Parallelization results in more computational resources being used not less… the benefit is in human time.
L4345: There are a lot of different timesaving methods listed here, but there is no evaluation of which methods are pertinent to the current study. I think there needs to be a discussion of why one needs an explicit timestepping method for the present study. The NewtonKrylov method is briefly mentioned, but the authors don’t explain why they are not using that method (does it prevent the biogeochemistry models from working properly?)
L56: “ignoring and avoiding negative tracer concentrations”. I did not understand this sentence until I had read the whole manuscript. I think this should be rephrased for clarity, to state more simply that a stepsize control method was implemented with and without a condition to exclude negative tracer concentrations.
L68: “Due to the fully coupling”. Grammatically this should be “full coupling”
L8081: In these equations, the terms A, D ,qi and dn are shown without any explanation (until later in the manuscript). These new terms should be briefly labelled here for clarify.
L84: You could write here: “advection (A) and diffusion (D)” to partially address the point above.
L84: “in marine water” sounds strange. Why not say “in the ocean”?
L9899: “is called marine ecosystem model”. Here an article (“a” or “the”) is needed in front of “marine”
L109: “above equations”: please specify which equations you mean
L116: “refer to Kriest…”: I think you mean “refer the reader to Kriest…”
L151: “for the biogeochemistry tutorial”: this is confusing. What is “the biogeochemistry tutorial”? Do you mean this model was created for teaching purposes?
L152: “As the NDOP model” > “As in the…”
L171: Why have the equation numbers suddenly disappeared here?
L175176: Equation numbers?L188 and L190: Equation numbers?
There are many more examples where equation numbers are not included… this seems confusing to me.
L251: “Despite of such” > “Despite such”
L245257: This paragraph could use further discussion on why excluding negative concentrations is justified in the algorithm, given that negative values can occur in the reference case regardless of the accelerated time steps. On balance, it seems to me that this is a poor choice of criterion (the results are not good for Algorithm 2).
L284: I don’t understand the backslash here.
Figure 2: These 6 panels really need titles. It is cumbersome to have to refer backwards and forwards to the caption for the meaning of them.
Figure 3: As in Figure 2, the subplots need titles.
Table 3: I think an extra table, analogous to Table 3, should be added which shows the computational cost saving factor for each model, and the time step multiplication factor m at the end of the simulation.
L458: “Only in half of the simulation runs decreased the algorithm”: Grammar is wrong.
L460: “… applied the entire spinup large time steps”: grammar is wrong.
L461463: The sentence starting with “Although the algorithm…” is confusing to read. Please rephrase and clarify.
L466: “an reasonable”: typo
L492: “local error always needed two evaluations of the same time interval”: This highlights my general comment that the algorithm should not be checking the error every single time step.
L493494: “Due to negative concentrations in the approximations, the algorithm then used nearly always the smallest time step.” This suggests to me that this algorithm should not be recommended in the future.
L515524: Finishing the paper with a list of dot points is not a good way to conclude. Please rewrite this as a normal paragraph, or if you want to list these points like this, don’t make it the final statement of the paper.
P.S. Sorry for the delay in submitting my review.

AC2: 'Reply on RC2', Thomas Slawig, 19 May 2022
***First of all: Thank you very much for your report. We repeat your commets here and add our responses in italics below.
General Comments
It is striking that the step size control method (Algorithm 1) does not actually
do much continuous adjustment of the time step most of the time. While the
authors go to great lengths to enable the time step to change responsively, in
reality the step size quickly expands out to 32x the original time step, or it
stays anchored at 1x the original time step. The adjustment happens very early
in the simulations, meaning that for most of the 10,000 years the algorithm
simply creates more overhead for the simulation without any additional benefit.
This suggests that the whole process of finding the right time step could be
restricted to an appropriate initiation period (perhaps 100 years, or whatever
minimum envelope is needed), after which the time step is held fixed.
*** This was also some kind of surprise to us, maybe it should be stressed a
little bit more in our conclusions. And your recommendation is definitely right.On a related note, I thought the design of the Algorithm 1 could be significantly
improved by only running the error checks on a subset of time steps (not every
single time step). Presently, the steps 11 and 12 are computed for every time
step of the simulation to verify the accuracy of the chosen delta T. This error
check could easily be made on a subset of timesteps. Perhaps on every 10th time
step, an error check could be run, or if there needs to be a continuous block of
time steps, only do this for a limited window at periodic intervals. It seems
wasteful to me to run the error check on every time step, when most of the time
it will have no influence.
*** In fact this is controlled by the parameter n_s in the algorithm (see Alg. 1,
5th input parameter and used in the forloop in line 10). We will stress this point
even more since it seemed not to be clear enough.Algorithm 3 appears to have the most utility, since it clearly saves time in the
spinup, and achieves a reasonable approximation to the reference case (correct
me if I’ve got that wrong). Algorithm 2 fails because it appears to me that the
exclusion of negative tracers is too stringent a condition. The authors mention
that negative tracer concentrations sometimes occur in the reference case… so
this algorithm should be a nonstarter, shouldn’t it?
*** We agree and draw the conclusion that this alg. has no effect on time reduction.
However, we just wanted to show what happens if this strict criterion is matched.Overall, I think there could be more critical evaluation in the discussion and
conclusions as to which algorithms actually performed well, which are recommended
or not, and why.
*** We will put the points that you mentioned (see also below) in the discussion
section.Line Comments
L36: “parallelization… lowers the computational effort”. Not really, it just
speeds up the result. Parallelization results in more computational resources
being used not less… the benefit is in human time.
*** You are right. We will replace the term computational effort or cost by
runtime throughout the manuscript.L4345: There are a lot of different timesaving methods listed here, but there
is no evaluation of which methods are pertinent to the current study. I think
there needs to be a discussion of why one needs an explicit timestepping method
for the present study. The NewtonKrylov method is briefly mentioned, but the
authors don’t explain why they are not using that method (does it prevent the
biogeochemistry models from working properly?)
*** We will extend the discussion of this point. Besides the mentioned work of
Khatiwala on Newton's method, we have our own experience (ref. Piwonski Slawig 2016a)
with Newton's method: It was much more sensitive to the choice of the initial values
than the standard spinup. That is the reason why we concentrated on the spinup
in this paper. It might also be that choosing a different or varying time step in
Newton's method will affect its convergence behavior. Moreover, in Newton's method
only one year is computed in each iteration, whereas the spinup is more similar to the
solution of an initial value problem, for which adaptive timestepping
was designed originally. The GPU implementation will not be affected by an automatic
choice of the stepsize, its main improvement comes form the fact that the considered
biogeochemical models are watercolumn models. This structure allows effective acceleration
on GPUs, but only for the biogeochemical part, not that much for the ocean transport part.L56: “ignoring and avoiding negative tracer concentrations”. I did not understand
this sentence until I had read the whole manuscript. I think this should be
rephrased for clarity, to state more simply that a stepsize control method was
implemented with and without a condition to exclude negative tracer concentrations.
*** This is a misleading formulation, we will clarify it already at this point of the manuscript.L68: “Due to the fully coupling”. Grammatically this should be “full coupling”
*** will be corrected
L8081: In these equations, the terms A, D ,qi and dn are shown without any
explanation (until later in the manuscript). These new terms should be briefly
labelled here for clarify.
*** This will be added directly below the equations.L84: You could write here: “advection (A) and diffusion (D)” to partially address
the point above.
*** We will follow the suggestion.L84: “in marine water” sounds strange. Why not say “in the ocean”?
*** will be changed.L9899: “is called marine ecosystem model”. Here an article (“a” or “the”) is
needed in front of “marine”
*** will be added.L109: “above equations”: please specify which equations you mean
*** eqns. (1),(2) were meant, will be added.
L116: “refer to Kriest…”: I think you mean “refer the reader to Kriest…”
*** yes, will be corrected.L151: “for the biogeochemistry tutorial”: this is confusing. What is
“the biogeochemistry tutorial”? Do you mean this model was created for teaching
purposes?
*** I think this part of the sentence can be skipped.L152: “As the NDOP model” > “As in the…”
*** Will be corrected.
L171: Why have the equation numbers suddenly disappeared here?
L175176: Equation numbers?
L188 and L190: Equation numbers?
There are many more examples where equation numbers are not included… this seems
confusing to me.
*** We will add equation numbers throughout the manuscript.L251: “Despite of such” > “Despite such”
*** Will be changed.L245257: This paragraph could use further discussion on why excluding negative
concentrations is justified in the algorithm, given that negative values can
occur in the reference case regardless of the accelerated time steps. On balance,
it seems to me that this is a poor choice of criterion (the results are not good
for Algorithm 2).
*** It is definitely a very strict criterion because of the reasons you mentioned.
We included it anyway to show what effect this criterion has if it is applied
(it basically destroys the benefit of the stepsize choice, as you write).
On the other hand, we wanted to show that violation of the nonnegativity in Alg.
1 and 3 had no negative effects.L284: I don’t understand the backslash here.
*** This is a math notation for a difference of sets, we will use a clearer notation.Figure 2: These 6 panels really need titles. It is cumbersome to have to refer
backwards and forwards to the caption for the meaning of them.
Figure 3: As in Figure 2, the subplots need titles.
*** We will include titles.
Table 3: I think an extra table, analogous to Table 3, should be added which
shows the computational cost saving factor for each model, and the time step
multiplication factor m at the end of the simulation.
*** Will be added in a revised version.
L458: “Only in half of the simulation runs decreased the algorithm”: Grammar is
wrong.
L460: “… applied the entire spinup large time steps”: grammar is wrong.
*** will be corrected.
L461463: The sentence starting with “Although the algorithm…” is confusing to read.
Please rephrase and clarify.
*** Meant was: In some of these cases, the algorithm temporarily decreased the time step.
However, this hardly effected the accuracy of the approximation.
L466: “an reasonable”: typo
*** will be corrected.L492: “local error always needed two evaluations of the same time interval”:
This highlights my general comment that the algorithm should not be checking the
error every single time step.
*** In fact this is controlled by the parameter n_s in the algorithm (see Alg. 1,
5th input parameter and used in the forloop in line 10). We will emphasize this point
even more.
L493494: “Due to negative concentrations in the approximations, the algorithm
then used nearly always the smallest time step.” This suggests to me that this
algorithm should not be recommended in the future.
*** This definitely is a reasonable recommendation. We will clarify this in the
conclusion section, together with the remark that already the reference runs
sometimes produce small negative values.L515524: Finishing the paper with a list of dot points is not a good way to
conclude. Please rewrite this as a normal paragraph, or if you want to list
these points like this, don’t make it the final statement of the paper.*** We will change this and remove the bullet points.

AC3: 'Reply on RC2', Thomas Slawig, 19 May 2022
Small correction to my last response: Computational effort and runtime are both affected by the step size reduction, whereas parallelization only affects runtime, not effort, as you mentioned. We will make this difference clearer by the choice of the two terms "Computational effort" and "runtime" in the whole manuscript.

AC2: 'Reply on RC2', Thomas Slawig, 19 May 2022
Status: closed

CEC1: 'Comment on gmd2021392', Juan Antonio Añel, 22 Feb 2022
Dear authors,
After checking your manuscript, it has come to our attention that it does not comply with our Code and Data Policy.
https://www.geoscientificmodeldevelopment.net/policies/code_and_data_policy.html
You have archived your code in GitHub. However, GitHub is not a suitable repository. GitHub itself instructs authors to use other alternatives for longterm archival and publishing, such as Zenodo. Therefore, please, publish your code in one of the appropriate repositories, and reply to this comment with the information about it.In this way, you must include in a potential reviewed version of your manuscript the modified 'Code and Data Availability' section, including the DOI of the code.
Juan A. Añel
Geosci. Model Dev. Exec. Editor AC1: 'Reply on CEC1', Markus Pfeil, 30 Mar 2022

RC1: 'Comment on gmd2021392', Anonymous Referee #1, 13 Apr 2022
General Comments:
I went through the manuscript from the beginning to the end twice carefully. As an ocean biogeochemistry modeler, I found the manuscript is very hard to understand and follow. It was not like a model technique paper, but more like a bunch of model testing results report. I reject the paper and provide my comments below for authors to improve in the future.
Specific Comments:
 The model technique that was intend to address
The prepared manuscript tried to improve the time step selection in the transport matrix method (TMM). It argued the used time step affects both the computational effort and accuracy of the steady annual cycle computation. I think both the “computational effort” and “steady annual cycle” should be well explained in the introduction part. What are they? Are they being recognized as a common problem in previous research? How that impact the follow modelling results? It was very hard to feel the importance of the technique problem presents in so far writing.
 Experiment design and way to present results
“The experiments are designed to shorten the running time of the computed steady annual cycle. The accuracy and cost save the calculated approximation. It was very hard to understand the present results from so far figures and tables.” Why not have a figure to show what is the nonsteady state annual cycle globally look like and what a steady state annual cycle looks like? Why not have comparisons between on and off line methods to show how they impact the steady annual cycle and the accuracy. It was also very hard to tell how much accuracy has been improved and cost saving was necessary from so far results.
It will be good to read and learn the cited manuscript “Accelerated simulation of passive tracers in ocean circulation models” carefully and learn how to present results like this to guide the reader. Another good example is “Performance of offline passive tracer advection in the Regional Ocean Modelling System” by Thyng et al. on GMD
 Mathematical forms
` I felt there was too much details about how to get A from B, which are very annoying to follow the final results. It should be provided in the supplementary material. It should also provide some math in matrix form, which will be easier to follow
Technical Corrections:
L68L71:
“Due to the fully coupling of the ocean circulation with the tracers, ……, the tracer concentrations affect the circulation, the simulation of a fully coupled model … to single model evaluations.”
It was not correct to say this. Commons like this for online and offline methods should be careful. Online coupling is a mature technique and widely used. With the development of the computational source, the computational cost is not that high. It was true the offline model will be more efficient, but it was also difficulty in accurately representing vertical fluxes due to deep convection.
L83:
“No fluxes on the boundary”
Need to state all experiments are for global run.

AC4: 'Reply on RC1', Thomas Slawig, 19 May 2022
*** First of all and although you reject the paper, we would like to thank you foryour report. It shows us that at least parts of the manuscript are not wellunderstandable. And since you write you are a modeler, we would of course alsowould like that also modelers understand our presentations.We provide our answers to your commenst in italics.General Comments:I went through the manuscript from the beginning to the end twice carefully.As an ocean biogeochemistry modeler, I found the manuscript is very hard tounderstand and follow. It was not like a model technique paper, but morelike a bunch of model testing results report. I reject the paper and providemy comments below for authors to improve in the future.*** Our paper is an algorithm development and evaluation paper. But we think that thiscan be alos regarded as "model technique".Specific Comments:The model technique that was intend to addressThe prepared manuscript tried to improve the time step selection in thetransport matrix method (TMM). It argued the used time step affects both thecomputational effort and accuracy of the steady annual cycle computation.I think both the “computational effort” and “steady annual cycle” shouldbe well explained in the introduction part. What are they? Are they beingrecognized as a common problem in previous research? How that impact thefollow modelling results? It was very hard to feel the importance of thetechnique problem presents in so far writing.*** We have to clarify the term "computational effort" as already mentioned bythe other reviewer. We will also clarify the term steady annual cycle, we thoughtthis was clear by equation (11) and the text before.Experiment design and way to present results“The experiments are designed to shorten the running time of the computedsteady annual cycle. The accuracy and cost save the calculated approximation.It was very hard to understand the present results from so far figures andtables.” Why not have a figure to show what is the nonsteady state annualcycle globally look like and what a steady state annual cycle looks like?Why not have comparisons between on and off line methods to show how theyimpact the steady annual cycle and the accuracy. It was also very hard totell how much accuracy has been improved and cost saving was necessary fromso far results.*** It was not our aim to compare off and online methods, and we did not want toargue against any of the two. We also do not see what benefit it would be for the readerto show a nonsteady solution. We are trying to reduce the runtime necessary to computea steady annual cycle. This is one form of the process usually called spinup of a model.We thought that we made clear in the introduction that runtime reduction is of importancein these kind of simulations. We will try to emphasize this point even more.It will be good to read and learn the cited manuscript “Acceleratedsimulation of passive tracers in ocean circulation models” carefully andlearn how to present results like this to guide the reader. Another goodexample is “Performance of offline passive tracer advection in the RegionalOcean Modelling System” by Thyng et al. on GMD*** Thank you for the recommendation. We will look into the seocnd paper you mentioned as well.Mathematical formsI felt there was too much details about how to get A from B, which arevery annoying to follow the final results. It should be provided in thesupplementary material. It should also provide some math in matrix form,which will be easier to follow*** Here, it would be nice to have more details. We are aware of the fact thatthere is some mathematics in the paper that may be a little bit lengthy.We will try to reduce some of the formulas in a revised version.Technical Corrections:L68L71:“Due to the fully coupling of the ocean circulation with the tracers, ……, thetracer concentrations affect the circulation, the simulation of a fully coupledmodel … to single model evaluations.”It was not correct to say this. Commons like this for online and offline methodsshould be careful. Online coupling is a mature technique and widely used. Withthe development of the computational source, the computational cost is not thathigh. It was true the offline model will be more efficient, but it was alsodifficulty in accurately representing vertical fluxes due to deep convection.*** We agree, this statement was definitely to strict. It will be reformulated andthe last part "to single model evaluation" will be skipped in a revised version.L83:“No fluxes on the boundary”Need to state all experiments are for global run.*** This is correct, we will add this point.

RC2: 'Comment on gmd2021392', Anonymous Referee #2, 22 Apr 2022
First, I have not read the other reviews, so my opinion is given independently.
This manuscript presents novel algorithms for adapting the time step in biogeochemical models driven by the Transport Matrix Method (TMM). The authors present 3 different ways of controlling the time step, and show that while Algorithms 1 and 3 can potentially speed up the spinup of the biogeochemical models in some (but not all cases), Algorithm 2 generally does not. This work seems worthy of publication after some moderate revisions and further critical evaluation of how well each method works. Some aspects of the data presentation and writing could also be improved.
General Comments
It is striking that the step size control method (Algorithm 1) does not actually do much continuous adjustment of the time step most of the time. While the authors go to great lengths to enable the time step to change responsively, in reality the step size quickly expands out to 32x the original time step, or it stays anchored at 1x the original time step. The adjustment happens very early in the simulations, meaning that for most of the 10,000 years the algorithm simply creates more overhead for the simulation without any additional benefit. This suggests that the whole process of finding the right time step could be restricted to an appropriate initiation period (perhaps 100 years, or whatever minimum envelope is needed), after which the time step is held fixed.
On a related note, I thought the design of the Algorithm 1 could be significantly improved by only running the error checks on a subset of time steps (not every single time step). Presently, the steps 11 and 12 are computed for every time step of the simulation to verify the accuracy of the chosen delta T. This error check could easily be made on a subset of timesteps. Perhaps on every 10^{th} time step, an error check could be run, or if there needs to be a continuous block of time steps, only do this for a limited window at periodic intervals. It seems wasteful to me to run the error check on every time step, when most of the time it will have no influence.
Algorithm 3 appears to have the most utility, since it clearly saves time in the spinup, and achieves a reasonable approximation to the reference case (correct me if I’ve got that wrong). Algorithm 2 fails because it appears to me that the exclusion of negative tracers is too stringent a condition. The authors mention that negative tracer concentrations sometimes occur in the reference case… so this algorithm should be a nonstarter, shouldn’t it?
Overall, I think there could be more critical evaluation in the discussion and conclusions as to which algorithms actually performed well, which are recommended or not, and why.
Line Comments
L36: “parallelization… lowers the computational effort”. Not really, it just speeds up the result. Parallelization results in more computational resources being used not less… the benefit is in human time.
L4345: There are a lot of different timesaving methods listed here, but there is no evaluation of which methods are pertinent to the current study. I think there needs to be a discussion of why one needs an explicit timestepping method for the present study. The NewtonKrylov method is briefly mentioned, but the authors don’t explain why they are not using that method (does it prevent the biogeochemistry models from working properly?)
L56: “ignoring and avoiding negative tracer concentrations”. I did not understand this sentence until I had read the whole manuscript. I think this should be rephrased for clarity, to state more simply that a stepsize control method was implemented with and without a condition to exclude negative tracer concentrations.
L68: “Due to the fully coupling”. Grammatically this should be “full coupling”
L8081: In these equations, the terms A, D ,qi and dn are shown without any explanation (until later in the manuscript). These new terms should be briefly labelled here for clarify.
L84: You could write here: “advection (A) and diffusion (D)” to partially address the point above.
L84: “in marine water” sounds strange. Why not say “in the ocean”?
L9899: “is called marine ecosystem model”. Here an article (“a” or “the”) is needed in front of “marine”
L109: “above equations”: please specify which equations you mean
L116: “refer to Kriest…”: I think you mean “refer the reader to Kriest…”
L151: “for the biogeochemistry tutorial”: this is confusing. What is “the biogeochemistry tutorial”? Do you mean this model was created for teaching purposes?
L152: “As the NDOP model” > “As in the…”
L171: Why have the equation numbers suddenly disappeared here?
L175176: Equation numbers?L188 and L190: Equation numbers?
There are many more examples where equation numbers are not included… this seems confusing to me.
L251: “Despite of such” > “Despite such”
L245257: This paragraph could use further discussion on why excluding negative concentrations is justified in the algorithm, given that negative values can occur in the reference case regardless of the accelerated time steps. On balance, it seems to me that this is a poor choice of criterion (the results are not good for Algorithm 2).
L284: I don’t understand the backslash here.
Figure 2: These 6 panels really need titles. It is cumbersome to have to refer backwards and forwards to the caption for the meaning of them.
Figure 3: As in Figure 2, the subplots need titles.
Table 3: I think an extra table, analogous to Table 3, should be added which shows the computational cost saving factor for each model, and the time step multiplication factor m at the end of the simulation.
L458: “Only in half of the simulation runs decreased the algorithm”: Grammar is wrong.
L460: “… applied the entire spinup large time steps”: grammar is wrong.
L461463: The sentence starting with “Although the algorithm…” is confusing to read. Please rephrase and clarify.
L466: “an reasonable”: typo
L492: “local error always needed two evaluations of the same time interval”: This highlights my general comment that the algorithm should not be checking the error every single time step.
L493494: “Due to negative concentrations in the approximations, the algorithm then used nearly always the smallest time step.” This suggests to me that this algorithm should not be recommended in the future.
L515524: Finishing the paper with a list of dot points is not a good way to conclude. Please rewrite this as a normal paragraph, or if you want to list these points like this, don’t make it the final statement of the paper.
P.S. Sorry for the delay in submitting my review.

AC2: 'Reply on RC2', Thomas Slawig, 19 May 2022
***First of all: Thank you very much for your report. We repeat your commets here and add our responses in italics below.
General Comments
It is striking that the step size control method (Algorithm 1) does not actually
do much continuous adjustment of the time step most of the time. While the
authors go to great lengths to enable the time step to change responsively, in
reality the step size quickly expands out to 32x the original time step, or it
stays anchored at 1x the original time step. The adjustment happens very early
in the simulations, meaning that for most of the 10,000 years the algorithm
simply creates more overhead for the simulation without any additional benefit.
This suggests that the whole process of finding the right time step could be
restricted to an appropriate initiation period (perhaps 100 years, or whatever
minimum envelope is needed), after which the time step is held fixed.
*** This was also some kind of surprise to us, maybe it should be stressed a
little bit more in our conclusions. And your recommendation is definitely right.On a related note, I thought the design of the Algorithm 1 could be significantly
improved by only running the error checks on a subset of time steps (not every
single time step). Presently, the steps 11 and 12 are computed for every time
step of the simulation to verify the accuracy of the chosen delta T. This error
check could easily be made on a subset of timesteps. Perhaps on every 10th time
step, an error check could be run, or if there needs to be a continuous block of
time steps, only do this for a limited window at periodic intervals. It seems
wasteful to me to run the error check on every time step, when most of the time
it will have no influence.
*** In fact this is controlled by the parameter n_s in the algorithm (see Alg. 1,
5th input parameter and used in the forloop in line 10). We will stress this point
even more since it seemed not to be clear enough.Algorithm 3 appears to have the most utility, since it clearly saves time in the
spinup, and achieves a reasonable approximation to the reference case (correct
me if I’ve got that wrong). Algorithm 2 fails because it appears to me that the
exclusion of negative tracers is too stringent a condition. The authors mention
that negative tracer concentrations sometimes occur in the reference case… so
this algorithm should be a nonstarter, shouldn’t it?
*** We agree and draw the conclusion that this alg. has no effect on time reduction.
However, we just wanted to show what happens if this strict criterion is matched.Overall, I think there could be more critical evaluation in the discussion and
conclusions as to which algorithms actually performed well, which are recommended
or not, and why.
*** We will put the points that you mentioned (see also below) in the discussion
section.Line Comments
L36: “parallelization… lowers the computational effort”. Not really, it just
speeds up the result. Parallelization results in more computational resources
being used not less… the benefit is in human time.
*** You are right. We will replace the term computational effort or cost by
runtime throughout the manuscript.L4345: There are a lot of different timesaving methods listed here, but there
is no evaluation of which methods are pertinent to the current study. I think
there needs to be a discussion of why one needs an explicit timestepping method
for the present study. The NewtonKrylov method is briefly mentioned, but the
authors don’t explain why they are not using that method (does it prevent the
biogeochemistry models from working properly?)
*** We will extend the discussion of this point. Besides the mentioned work of
Khatiwala on Newton's method, we have our own experience (ref. Piwonski Slawig 2016a)
with Newton's method: It was much more sensitive to the choice of the initial values
than the standard spinup. That is the reason why we concentrated on the spinup
in this paper. It might also be that choosing a different or varying time step in
Newton's method will affect its convergence behavior. Moreover, in Newton's method
only one year is computed in each iteration, whereas the spinup is more similar to the
solution of an initial value problem, for which adaptive timestepping
was designed originally. The GPU implementation will not be affected by an automatic
choice of the stepsize, its main improvement comes form the fact that the considered
biogeochemical models are watercolumn models. This structure allows effective acceleration
on GPUs, but only for the biogeochemical part, not that much for the ocean transport part.L56: “ignoring and avoiding negative tracer concentrations”. I did not understand
this sentence until I had read the whole manuscript. I think this should be
rephrased for clarity, to state more simply that a stepsize control method was
implemented with and without a condition to exclude negative tracer concentrations.
*** This is a misleading formulation, we will clarify it already at this point of the manuscript.L68: “Due to the fully coupling”. Grammatically this should be “full coupling”
*** will be corrected
L8081: In these equations, the terms A, D ,qi and dn are shown without any
explanation (until later in the manuscript). These new terms should be briefly
labelled here for clarify.
*** This will be added directly below the equations.L84: You could write here: “advection (A) and diffusion (D)” to partially address
the point above.
*** We will follow the suggestion.L84: “in marine water” sounds strange. Why not say “in the ocean”?
*** will be changed.L9899: “is called marine ecosystem model”. Here an article (“a” or “the”) is
needed in front of “marine”
*** will be added.L109: “above equations”: please specify which equations you mean
*** eqns. (1),(2) were meant, will be added.
L116: “refer to Kriest…”: I think you mean “refer the reader to Kriest…”
*** yes, will be corrected.L151: “for the biogeochemistry tutorial”: this is confusing. What is
“the biogeochemistry tutorial”? Do you mean this model was created for teaching
purposes?
*** I think this part of the sentence can be skipped.L152: “As the NDOP model” > “As in the…”
*** Will be corrected.
L171: Why have the equation numbers suddenly disappeared here?
L175176: Equation numbers?
L188 and L190: Equation numbers?
There are many more examples where equation numbers are not included… this seems
confusing to me.
*** We will add equation numbers throughout the manuscript.L251: “Despite of such” > “Despite such”
*** Will be changed.L245257: This paragraph could use further discussion on why excluding negative
concentrations is justified in the algorithm, given that negative values can
occur in the reference case regardless of the accelerated time steps. On balance,
it seems to me that this is a poor choice of criterion (the results are not good
for Algorithm 2).
*** It is definitely a very strict criterion because of the reasons you mentioned.
We included it anyway to show what effect this criterion has if it is applied
(it basically destroys the benefit of the stepsize choice, as you write).
On the other hand, we wanted to show that violation of the nonnegativity in Alg.
1 and 3 had no negative effects.L284: I don’t understand the backslash here.
*** This is a math notation for a difference of sets, we will use a clearer notation.Figure 2: These 6 panels really need titles. It is cumbersome to have to refer
backwards and forwards to the caption for the meaning of them.
Figure 3: As in Figure 2, the subplots need titles.
*** We will include titles.
Table 3: I think an extra table, analogous to Table 3, should be added which
shows the computational cost saving factor for each model, and the time step
multiplication factor m at the end of the simulation.
*** Will be added in a revised version.
L458: “Only in half of the simulation runs decreased the algorithm”: Grammar is
wrong.
L460: “… applied the entire spinup large time steps”: grammar is wrong.
*** will be corrected.
L461463: The sentence starting with “Although the algorithm…” is confusing to read.
Please rephrase and clarify.
*** Meant was: In some of these cases, the algorithm temporarily decreased the time step.
However, this hardly effected the accuracy of the approximation.
L466: “an reasonable”: typo
*** will be corrected.L492: “local error always needed two evaluations of the same time interval”:
This highlights my general comment that the algorithm should not be checking the
error every single time step.
*** In fact this is controlled by the parameter n_s in the algorithm (see Alg. 1,
5th input parameter and used in the forloop in line 10). We will emphasize this point
even more.
L493494: “Due to negative concentrations in the approximations, the algorithm
then used nearly always the smallest time step.” This suggests to me that this
algorithm should not be recommended in the future.
*** This definitely is a reasonable recommendation. We will clarify this in the
conclusion section, together with the remark that already the reference runs
sometimes produce small negative values.L515524: Finishing the paper with a list of dot points is not a good way to
conclude. Please rewrite this as a normal paragraph, or if you want to list
these points like this, don’t make it the final statement of the paper.*** We will change this and remove the bullet points.

AC3: 'Reply on RC2', Thomas Slawig, 19 May 2022
Small correction to my last response: Computational effort and runtime are both affected by the step size reduction, whereas parallelization only affects runtime, not effort, as you mentioned. We will make this difference clearer by the choice of the two terms "Computational effort" and "runtime" in the whole manuscript.

AC2: 'Reply on RC2', Thomas Slawig, 19 May 2022
Markus Pfeil and Thomas Slawig
Markus Pfeil and Thomas Slawig
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