Time averaging for tendency budget analysis

Bladwell, Christopher; Holmes, Ryan M.; Zika, Jan D.; Griffies, Stephen M.; Kiss, Andrew E.

doi:10.5194/egusphere-2026-3672

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https://doi.org/10.5194/egusphere-2026-3672

Preprints

22 Jul 2026

| 22 Jul 2026

Time averaging for tendency budget analysis

Christopher Bladwell, Ryan M. Holmes, Jan D. Zika, Stephen M. Griffies, and Andrew E. Kiss

Abstract. Climate models can be used to quantify the processes that contribute to a property's change through analysis of the tendency terms in the budget of the property. For example, a heat budget can be performed to understand the processes driving an increase in the ocean's heat content under climate change. However, conventional budget analyses in climate models are typically performed without proper consideration of their relationship to the property of interest. Specifically, most studies consider standard arithmetic time-averages of budget terms. For the tendency term, this average is related to the difference between two time instants (snapshots) of the property and thus is strongly impacted by rapid fluctuations, or "weather". Such an approach to budget analysis is thus less relevant to the "climate change" problem, that is, the difference between two long-term averages, or "epochs", of the property. Here we present a time-averaging method, referred to as the hat average, which exactly relates the processes contributing to a property's tendency budget to the epoch difference, or "climate change", of that property. We demonstrate the utility of this method by applying it to the horizontally integrated heat budget of a global ocean model. We find that the hat average removes high-frequency variability, typically associated with rapid and noisy advective processes, and so allows for a clear measure of how processes contribute to changes in climate.

Received: 23 Jun 2026 – Discussion started: 22 Jul 2026

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Christopher Bladwell, Ryan M. Holmes, Jan D. Zika, Stephen M. Griffies, and Andrew E. Kiss

Status: final response (author comments only)

RC1:
'Comment on egusphere-2026-3672', Anonymous Referee #1, 04 Sep 2026
“Time averaging for tendency budget analysis” by Bladwell et al submitted to GMD

This manuscript presents a novel time-averaging method for tendency budget analysis, the hat average. This new method differs from the standard time average widely used in the literature by considering the difference among epochs (decades over the historical period) of a particular property field (e.g. ocean heat content, as shown in this study) rather than snapshot (instantaneous) differences. Epoch differences are argued to be a better quantity for estimating the long-term net changes, as they filter out large part of the climate variability. The authors demonstrated that the hat average method provides a better representation of the long-term heat tendency, filtering out rapid fluctuations that are intrinsically included in the standard method, allowing a clearer quantification of the climate change component.

The manuscript is well written and clearly demonstrate the mathematical derivation of the method, its application to a global ocean-sea ice model, and the differences compared to the standard time-averaging method. It addresses an important issue of separating the climate change component from the “weather-like” fluctuations in ocean heat content changes and clearly deserved to be published.

However, I have a couple of questions regarding the usability and efficiency of the hat average approach.

Given that the hat average requires to save extra tendency diagnostic than the standard approach, would be good to clarify how much more expensive it is (in terms of storage and runtime). I understood that it requires every tendency term diagnostic to be sampled using both boxcar averages and rising weighted averages, making it twice as expensive for an already computationally expensive analyses.

What is the impact of the standard time average versus hat average for climate change studies (e.g. Gregory 200, Wolfe 2008, Kuhlbrodt et al. 2014)? Does the hat-average provide a different response in terms of the main processes governing ocean heat uptake and storage? Or is the main advantage for analysing the historical period, when climate variability makes a contribution to heat content changes that is comparable to the climate change signal?

Lastly, for historical analyses (such as those shown in Fig. 3), does the hat average provide a clear separation between the climate change signal and internal variability? It does not seem to be the case, given the cooling signal in the eastern tropical Pacific and in the Pacific sector of the Southern Ocean, but it would be good to clarify/quantify the extend to which these signals can be separated with this method.

Minor comments:
Fig. 1c: the rising/falling weight averages seems to smooth out the tendency in both ends of the time series. I wonder how much these contribute to the smoother maps in Fig. 3 versus truly separating the linear trend. Looking at the ocean heat content tendency maps averaged over each period (1980-1990, 1990-2000, 2000-2010) could provide some insights.

Fig. 4: Adding titles at the top of the panels A and C indicating the processes shown in each column would facilitate the interpretation of this figure.

There is any rationale for combining mean advection, mesoscale/submesoscale eddy fluxes with the neutral diffusion term, and the surface fluxes with vertical mixing for Fig. 5 panels? Several studies have addressed how these terms can be combined (e.g. Kuhlbrodt et al. 2014, Dias et al. 2020a, 2020b, Saenko et al. 2021, Savita et al. 2021) and I think these should be mentioned.

Fig. 5: it would be great to see the difference between panels A-D and B-E and how that compares with panels C and F. Does the combination of the terms mentioned in my comment above makes it harder to identify which terms are causing the differences between boxcar and hat averages?

Lines 320-325 / Fig. 6: Is the difference between boxcar and hat averages in the global-integrated heat budget anomalies dominated by the East Pacific signal, or are there contributions from other regions?

The Discussion section contains a very interesting discussion and concludes that the hat average is needed because it separates the natural variability from the long-term changes. This goes back to my point (3) above regarding how clearly this new time-averaging approach can separate these signals.

References:
Kuhlbrodt, T., J. M. Gregory, and L. C. Shaffrey, 2015: A process-based analysis of ocean heat uptake in an AOGCM with an eddy-permitting ocean component. Climate Dyn., 45, 3205–3226, https://doi.org/10.1007/s00382-015-2534-0.
Dias, F. B., C. M. Domingues, S. J. Marsland, S. M. Griffies, S. R. Rintoul, R. Matear, and R. Fiedler, 2020: On the Superposition of Mean Advective and Eddy-Induced Transports in Global Ocean Heat and Salt Budgets. J. Climate, 33, 1121–1140, https://doi.org/10.1175/JCLI-D-19-0418.1.
Dias, Fabio Boeira, Fiedler, R., Marsland, S. J., Domingues, C. M., Clément, L., Rintoul, S. R., Mcdonagh, E. L., Mata, M. M., and Savita, A., 2020, "Ocean Heat Storage in Response to Changing Ocean Circulation Processes" Journal of Climate Vol. 33, No. 21, pp 9065, 1520-0442
Saenko, O. A., J. M. Gregory, S. M. Griffies, M. P. Couldrey, and F. B. Dias, 2021: Contribution of Ocean Physics and Dynamics at Different Scales to Heat Uptake in Low-Resolution AOGCMs. J. Climate, 34, 2017–2035, https://doi.org/10.1175/JCLI-D-20-0652.1.
Savita, A., J. D. Zika, C. M. Domingues, S. J. Marsland, G. D. Evans, F. B. Dias, R. M. Holmes, and A. McC. Hogg, 2021: Super Residual Circulation: A New Perspective on Ocean Vertical Heat Transport. J. Phys. Oceanogr., 51, 2443–2462, https://doi.org/10.1175/JPO-D-21-0008.1.
C. L. Wolfe, P. Cessi, J. L. McClean, and M. E. Maltrud: Vertical heat transport in eddying ocean models, Geophysical Research Letters, Vol. 35, L23605, doi:10.1029/2008GL036138, 2008
Citation: https://doi.org/10.5194/egusphere-2026-3672-RC1
- AC1:
  'Reply on RC1', Chris Bladwell, 15 Oct 2026
  Responses to comments from reviewer 1:
  Given that the hat average requires to save extra tendency diagnostic than the standard approach, would be good to clarify how much more expensive it is (in terms of storage and runtime). I understood that it requires every tendency term diagnostic to be sampled using both boxcar averages and rising weighted averages, making it twice as expensive for an already computationally expensive analyses.
  
  The implementation proposed here requires multiplying a diagnostic field by the elapsed time, so it scales linearly with the grid resolution. The hat average requires saving two time-averaged fields; the standard average and one linearly weighted average. Line 430 mentions both linearly weighted and standard averaged tendency diagnostics must be stored. This doubles the storage requirement for the hat average budget over the standard average.
  What is the impact of the standard time average versus hat average for climate change studies (e.g. Gregory 200, Wolfe 2008, Kuhlbrodt et al. 2014)? Does the hat-average provide a different response in terms of the main processes governing ocean heat uptake and storage? Or is the main advantage for analysing the historical period, when climate variability makes a contribution to heat content changes that is comparable to the climate change signal?
  
  The main benefit of the hat average is that it relates the tendency to the epoch difference, which approximates a linear trend. However, this does not mean the conclusions of other budget analysis studies are incorrect, only that the standard-averaged budget likely includes some internal variability. Overall, the breakdown in processes in our study is not dissimilar to other ocean heat budget studies. Line 330 mentions that we found that advection played a similar role under hat averaging as has been shown in other studies. To demonstrate the difference between hat averaging and standard averaging for individual processes, we have included the difference between each averaging method for the vertically integrated budget over the upper 100m in Fig 5g-h. Fig 6 also shows horizontally integrated processes under both time-averaging methods. Since the hat average relates tendencies to the epoch difference, it can be used for both historical and future heat budget analyses.
  Lastly, for historical analyses (such as those shown in Fig. 3), does the hat average provide a clear separation between the climate change signal and internal variability? It does not seem to be the case, given the cooling signal in the eastern tropical Pacific and in the Pacific sector of the Southern Ocean, but it would be good to clarify/quantify the extend to which these signals can be separated with this method.
  
  The climate change signal shown in Fig. 3 is quantified by the epoch difference. Epoch differences are widely used to estimate climate change signals and have been shown to accurately reflect linear trends. The hat average relates physical (budget) processes to the epoch difference (like the standard average does for a snapshot difference). In the eastern/central equatorial Pacific Ocean, for example, Fig. 5 shows that there is stronger cooling from surface forcing and vertical mixing than advection and neutral mixing in the upper 100m, while in the eastern equatorial Pacific Advection and neutral mixing are stronger. This imbalance leads to the cooling signal in Fig. 5b and Fig 3a.
  Minor comments:
  Fig. 1c: the rising/falling weight averages seems to smooth out the tendency in both ends of the time series. I wonder how much these contribute to the smoother maps in Fig. 3 versus truly separating the linear trend. Looking at the ocean heat content tendency maps averaged over each period (1980-1990, 1990-2000, 2000-2010) could provide some insights
  
  The reviewer is correct that the walls of the trapezoidal filter which define the hat average contribute to the smooth pattern shown by the epoch difference in Fig 3a. The epoch difference is a widely used approach to approximate a linear trend associated with climate change (as shown in Fig 3a). Since it is a difference of averages, it smooths variability on timescales shorter than the epoch period. On the other hand, this fast internal variability affects the snapshot difference and is therefore aliased into slower modes of the boxcar averaged tendency. We have not included boxcar averaged tendencies over the periods mentioned by the reviewer as these would be subject to internal variability which may obscure the climate signal over the period.
  Fig. 4: Adding titles at the top of the panels A and C indicating the processes shown in each column would facilitate the interpretation of this figure.
  
  Fig. 4 has been updated to include titles on each panel
  There is any rationale for combining mean advection, mesoscale/submesoscale eddy fluxes with the neutral diffusion term, and the surface fluxes with vertical mixing for Fig. 5 panels? Several studies have addressed how these terms can be combined (e.g. Kuhlbrodt et al. 2014, Dias et al. 2020a, 2020b, Saenko et al. 2021, Savita et al. 2021) and I think these should be mentioned.
  
  We thank the reviewer for referencing these previous studies. We have added a reference to Kuhlbrodt et al. 2015 and Dias et al. 2015 in Ln 294. The breakdown of budget contributions used here is similar to that presented by Dias et al. 2020. Both Dias et al. 2020 and Kuhlbrodt et al. 2015 combine large scale advection, eddy-induced advection and isoneutral diffusion as the "superresidual" transport, which is similar to the advection + neutral diffusion term shown in Fig. 5b and 5e. Similarly, Dias et al 2020 note the balance between dianeutral mixing, nonlocal vertical fluxes and surface forcing terms, which is similar to our vertical mixing and surface flux terms shown in Fig 5a and 5d. While a similar balance was shown in our study, follow-up studies on ocean heat storage may revisit the balance between different processes and how they contribute to linear trends using the hat-averaged budget approach described in this work.
  Fig. 5: it would be great to see the difference between panels A-D and B-E and how that compares with panels C and F. Does the combination of the terms mentioned in my comment above makes it harder to identify which terms are causing the differences between boxcar and hat averages?
  
  Fig 5g-h have been added to show the difference, along with some additional text describing them (lines 295 and 304). The breakdown of budget terms does potentially obscure certain processes. For example, advection contains both large scale and eddy advection. While not shown here, examination of the individual contribution of advective processes showed a similar balance to that discussed in previous studies (e.g. Griffies et al. 2015, Kuhlbrodt et al 2015, Dias et al. 2020, etc.), where upward transport by eddies balanced downward transport by large scale advection. The key advantage of the hat average is to precisely relate the tendency budget to the epoch difference which closely resembles a linear linear temperature trend. A precise breakdown of budget processes under hat averaging is an area of future research which may identify a different balance of processes relating to long term heat content change.
  Lines 320-325 / Fig. 6: Is the difference between boxcar and hat averages in the global-integrated heat budget anomalies dominated by the East Pacific signal, or are there contributions from other regions?
  
  The snapshot difference shows a strong warming signal in the eastern Pacific due to our choice of time period corresponding to an ENSO event. However, there are other regions with strong warm and cool anomalies. While not explicitly shown here, in the upper 100m the largest contribution is associated with the East Pacific signal, however there are substantial contributions from outside this region as well.
  The Discussion section contains a very interesting discussion and concludes that the hat average is needed because it separates the natural variability from the long-term changes. This goes back to my point (3) above regarding how clearly this new time-averaging approach can separate these signals.
  
  We thank the reviewer for raising these points about the relationship between the climate signal and process contributions. The use of an epoch difference is a well-established approach to isolate a long-term trend in an observable variable, and in climate applications is sometimes preferred to other linear fitting approaches because it can be used with sparse or temporally inconsistent data. However, previous studies have shown that natural variability over long timescales may obscure long-term climate trends, which are often relatively small compared to the internal variability. The hat-average approach is simply a method by which process contributions to the epoch difference can be computed.
  
  Citation: https://doi.org/10.5194/egusphere-2026-3672-AC1
RC2:
'Comment on egusphere-2026-3672', Anonymous Referee #2, 08 Sep 2026

The manuscript provides a relation of epoch differences of a quantity to a weighted average of the related tendencies. They show that differences can mathematically be related to hat averaged tendencies and that those can easily be constructed from additionally saving rising or falling weighted averages. The relation is important because for climate change studies epoch differences are a popular means to improve signal-to-noise ratios.
Overall I find the articel very readable as ever detail is well explained, which leaves little to improve. Maybe one thing one could add: I find the example difference between snapshot vs differences in epochs less instructive. Usually people will no do difference between snapshot because they are not availabe. Instead they will use differences of means but still trying to relate those to tendencies during the period of change accepting a certain amount of error. I think it would be useful to mention this approach and illustrate the error. To illustrate the error, Fig. 5 could show the boxcar or the hat-average and the difference instead of two sets figures that look very similar.
Some details:
Section 2.3 First half of this section is not about frequency domain and probably fits better to the section 2.2.
l 215-116 With delta z being time dependent, does D also become time dependent and is the heat content change affected by changes of D? The contribution from D changes may become big in particular if theta is measured in Kelvin. Or do you scale the values to 100m depth?

Citation: https://doi.org/10.5194/egusphere-2026-3672-RC2
- AC2:
  'Reply on RC2', Chris Bladwell, 15 Oct 2026
  Responses to comments from reviewer 2:
  I find the example difference between snapshot vs differences in epochs less instructive. Usually people will no do difference between snapshot because they are not available. Instead they will use differences of means but still trying to relate those to tendencies during the period of change accepting a certain amount of error. I think it would be useful to mention this approach and illustrate the error. To illustrate the error, Fig. 5 could show the boxcar or the hat-average and the difference instead of two sets figures that look very similar.
  
  Fig 5g-h has been added to show the difference between the hat average and standard boxcar average for both surface forcing and vertical mixing, and advection and neutral mixing. We acknowledge the point made by the reviewer; climate studies rarely use the snapshot difference, unless they calculate a time-averaged tendency which is equivalent to the snapshot difference. We have updated the text to explicitly make reference to this point in line 70: " The instantaneous snapshot difference of [a variable] characterized by Eq. (2) is generally not available as standard model output, and is typically of less interest than the difference in time-average fields". We have also added at line 73-75: “Since instantaneous values are often not available and are subject to strong variability, they are rarely used in practice. Therefore, it is often assumed that the time-averaged tendency approximately quantifies the time-averaged difference in a variable (the left-hand side of Eq. 3 approximately equals the right-hand side of Eq. 2).” The epoch difference is the difference in means, which is widely used to quantify changes in variables associated with climate change. The key reason to present the snapshot difference is to demonstrate that the boxcar average quantifies the contributions to the snapshot difference, and how using a boxcar average for the budget may over represent particular processes like advection. The error obtained by using the standard average to compute process contributions to the epoch difference is the same as the difference between the boxcar average and the hat average of the total tendency.
  Some details:
  Section 2.3 First half of this section is not about frequency domain and probably fits better to the section 2.2.
  
  While the first half of Section 2.3 relates to temporal analysis, this was material added to introduce the notation which is used for frequency analysis and signal processing (namely window functions). Section 2.2 derives the hat average budget from the epoch difference. In ocean and climate model applications, it is uncommon to apply weighting filters when averaging, so the relationship to signal processing is mainly relevant for the frequency analysis and not overly relevant for climate model researchers. Therefore, we have chosen to keep this material in Section 2.3.
  l 215-116 With delta z being time dependent, does D also become time dependent and is the heat content change affected by changes of D? The contribution from D changes may become big in particular if theta is measured in Kelvin. Or do you scale the values to 100m depth?
  
  Yes, the reviewer is correct. D is time-varying and so we are not strictly using a fixed 100m depth layer. Instead, the volume varies with the motion of the model vertical layers. The budget accounts for all processes since the averaging was performed online at the model's time step. Yes, there will be a contribution from D changes. The model uses Conservative Temperature measured in degrees Celsius internally. To help clarify this point we have updated the text in Ln 220 to say " H_{100} is time-dependent and follows the model grid cell depth".
  
  Citation: https://doi.org/10.5194/egusphere-2026-3672-AC2

Christopher Bladwell, Ryan M. Holmes, Jan D. Zika, Stephen M. Griffies, and Andrew E. Kiss

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Short summary

To quantify climate change we compare long-term averages and not just day-to-day changes (weather). Climate simulations allow us to quantify the processes that contribute to climate change. However, the way these processes are conventionally analysed describes the change two instances of time, rather than the change in climate over long periods. We present an alternative way of quantifying how physical processes contribute to a changing climate, applying this method to an ocean model simulation.


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