The Largely Linear Response of Earth&rsquo;s Ice Volume to Orbital Forcing

Wheen, Liam; Benjamin, Oscar; Gernon, Thomas; Hall, Cameron; Wright, Jerry

doi:https://doi.org/10.5194/egusphere-2023-2893

Preprints

https://doi.org/10.5194/egusphere-2023-2893

Preprints

20 Dec 2023

| 20 Dec 2023

The Largely Linear Response of Earth’s Ice Volume to Orbital Forcing

Liam Wheen, Oscar Benjamin, Thomas Gernon, Cameron Hall, and Jerry Wright

Abstract. Orbital forcing plays a key role in pacing the glacial-interglacial cycles. However, the mechanistic linkages between the orbital parameters — eccentricity, obliquity, and precession — and global ice volume remain unclear. Here, we investigate the effect of Earth’s orbitally governed incoming solar radiation (that is, insolation) on global ice volume over the past 800,000 years. We consider a simple linear model of ice volume that imposes minimal assumptions about its dynamics. We find that this model can adequately reproduce the observed ice volume variations for most of the past 800,000 years, with the notable exception of Marine Isotope Stage 11. This suggests that, aside from a few extrema, the ice volume dynamics primarily result from an approximately linear response to orbital forcing. We substantiate this finding by addressing some of the key criticisms of the orbitally forced hypothesis. In particular, we show that eccentricity can significantly vary the ocean temperature without the need for amplification on Earth. We also present a feasible mechanism to explain the absence of eccentricity’s 400,000 year period in the ice volume data. This requires part of the forcing from eccentricity to be lagged via a slow-responding mechanism, resulting in a signal that closer approximates the change in eccentricity. A physical interpretation of our model is proposed, using bulk ocean and surface temperatures as intermediate mechanisms through which the orbital parameters affect ice volume. These show reasonable alignment with their relevant proxy data, though we acknowledge that these variables likely represent a combination of mechanisms.

Received: 01 Dec 2023 – Discussion started: 20 Dec 2023

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Liam Wheen, Oscar Benjamin, Thomas Gernon, Cameron Hall, and Jerry Wright

Status: final response (author comments only)

CC1:
'Comment on “The Largely Linear Response of Earth’s Ice Volume to Orbital Forcing”', Mikhail Verbitsky, 21 Dec 2023
Comment on “The Largely Linear Response of Earth’s Ice Volume to Orbital Forcing”
In the last few decades, phenomenological models have become a popular tool to entertain hypotheses of Pleistocene glacial rhythmicity. It is unfortunate though that in most cases, after a phenomenological model has been tuned to successfully reproduce empirical data, little efforts are made to prove physical similarity of the model with Nature. From this perspective, it is indeed commendable that the authors of this paper attempted to create a physical version of their phenomenological model. Unfortunately, the presented model does not have physical similarity with Nature.
The ice sheet model as it is described by equation (19) is not a physical model. The fundamental mass balance of an ice sheet is dV/dt =AS (Equation C1 thereafter). Here t (s) is time, V (m³) is ice volume, S (m²) is the area of an ice sheet, and A (m/s) is the mean over the entire area mass influx (accumulation minus ablation and calving). Equation C1 tells us that the changes in ice volume are equal to mass influx integrated over the entire area. Since V is proportional to S ^5/4 (see, e.g., Verbitsky et al (2018), where this scaling law has been derived from the simple scaling considerations, or, Bahr et al (2015) for most rigorous reasoning), the non-linearity of ice physics is obvious and it is not something to be ignored because it is the essence of the viscous flow of ice media.

Verbitsky and Crucifix (2020, 2023) have been arguing that the ratio of amplitudes of positive and negative feedbacks is the similarity parameter that largely defines the properties of a dynamical system. Since the model described by equations (19) – (21) does not have positive feedbacks, this ratio is equal to zero, and therefore there is no physical similarity of this model with the mass conservation law C1 where positive feedbacks are most evident: when V increases, S also increases (because an ice sheet is made of viscous “fluid” that spreads), and A is being accumulated on a larger area.

The ocean model does resemble the energy-balance but the cornerstone of it, the timescale τ = 15 kyr, may be a reason for a concern. If we take the timescale for the deep ocean to be closer to the more conventional value of about 1-5 kyr, the entire concept of the model will collapse because there will be no justification to splitting orbital forcing into “fast” obliquity and precession components and “slow” eccentricity components. All these components will be “slow” relative to the ocean.

Conclusion
There are two ideas in this paper. The first one is that the ice-climate system may be linear. This idea is indefensible because the non-linearity and a positive feedback are encoded in the mass conservation law of spreading ice.
The second idea, that the eccentricity may play a bigger role in forming ice ages if the orbital forcing is somehow delayed, is more productive. In Verbitsky et al (2018) the ocean temperature provides two positive feedbacks, the fast feedback affecting the temperature on the ice sheet surface and slow (delayed) feedback due to vertical temperature advection from the ice sheet surface to the bottom. The timescale of the vertical temperature advection in the ice sheet is the same as the timescale of ice sheet growth. The orbital forcing is part of these feedbacks because the ocean temperature is forced by ice area that responds to the orbital forcing. When these positive feedbacks are strong enough, the system exhibits fluctuations with the period of 400 kyr (Verbitsky et al, 2018, paragraph 4.4). It is possible that it is the eccentricity period.
References
Bahr, D. B., Pfeffer, W. T., and Kaser, G.: A review of volume-area scaling of glaciers, Rev. Geophys., 53,95–140, doi:10.1002/2014RG000470, 2015.
Verbitsky, M. Y. and Crucifix, M.: π-theorem generalization of the ice-age theory, Earth Syst. Dynam., 11, 281–289, https://doi.org/10.5194/esd-11-281-2020, 2020.
Verbitsky, M. Y. and Crucifix, M.: Do phenomenological dynamical paleoclimate models have physical similarity with Nature? Seemingly, not all of them do, Clim. Past, 19, 1793–1803, https://doi.org/10.5194/cp-19-1793-2023, 2023.
Verbitsky, M. Y., Crucifix, M., and Volobuev, D. M.: A theory of Pleistocene glacial rhythmicity, Earth Syst. Dynam., 9, 1025–1043, https://doi.org/10.5194/esd-9-1025-2018, 2018.
Citation: https://doi.org/10.5194/egusphere-2023-2893-CC1
- AC1: 'Reply on CC1', Liam Wheen, 20 Feb 2024
  
  Thank you for taking the time to read through our work and provide this feedback. We will aim to respond to your individual points in turn, and hope to clear up any miscommunication that may have arisen from the phrasing in the first draft of our paper.
  1. The ice sheet model as it is described by equation (19) is not a physical model....the non-linearity of ice physics is obvious and it is not something to be ignored because it is the essence of the viscous flow of ice media
  The model we have presented is a phenomenological model, which is designed to investigate the dynamic response of ice volume to orbital forcing based on the data. Many different physical or dynamic non-linear mechanisms have been proposed in different models in the literature. The benefit of having a linear model here is that we can explore what seem to be the most important aspects of the dynamic response to forcing based on as few assumptions as possible.
  There are certainly non-linear aspects of ice sheet dynamics that are not captured here but the question is whether those non-linearities are important in determining the dynamic response. A non-linear system near its stable equilibria can still show dynamics that are well approximated by a linear response to any forcing. This is the case for the model of Verbitsky et al (2018), which shows forced dynamics around its principal equilibrium. Linearising the model about this equilibrium and fitting to the ice volume data gives very similar behaviour for both linear and non-linear models.
  2. The ice model is missing any positive feedbacks
  It is true that the model can only represent the net effect of positive and negative feedbacks. By design, it is a feed forward model, that can only show a damped response to forcing. This is because the intended purpose of the model is to explore the input-output relationship between orbital forcing and ice volume, as seen in the data for the last 800 kyr. There are of course a number of different positive and negative feedbacks present in the ice-climate system, but we find that the salient features of the ice volume data can be captured by a simple feed forward model.
  3. The ocean timescale is too large
  It is true that the timescale used for ocean temperature is larger than the conventional value. However, as we discuss in the paper, what we are referring to as the ocean temperature could be a combination of mechanisms, including ocean heat and chemical transport. Crucifix (2011) puts the ocean alkalinity and ice sheet adjustment timescale on the order of 10 kyr. If we consider that both of these factors may contribute to the variable we are referring to as the ocean temperature, then a timescale of 15 kyr is not unreasonable.
  We do also accept that this variable could instead be something like the vertical temperature advection across the ice sheet, as in Verbitsky et al (2018). The focus of this paper was more on underlying mechanics of the ice volume system, rather than the intermediate physical components that facilitate the response to orbital forcing.
  The 15 kyr timescale comes out as the best fit to the data if the timescales for ice volume and ocean temperature are assumed to be the same. We have also tried using different timescales such as 10 kyr for ice volume and 5 kyr for ocean temperature, and found that the model gives similar results if other parameters are adjusted to compensate.
  The model assumes that ocean temperature is only affected by eccentricity making it the driver of the slow ocean variable. This comes firstly from what is seen in the data, which is that eccentricity has the strongest relationship to ice volume, but that its effect cannot be captured by an instantaneous dependence whereas the effect of obliquity and precession can. Physically a justification for this is that eccentricity is the only orbital parameter that affects total global insolation. If orbital forcing is envisaged to have an effect such as slowly changing ocean temperature, then to first order it is reasonable to assume that eccentricity would be the main driver of this.
  Conclusion
  The data shows the existence of a 100 kyr cycle in the ice volume. There are three main possible explanations for this: this period exists intrinsically in the ice-climate system, it is driven by the 100 kyr period in eccentricity, or it results from a combination of the obliquity and precession cycles.
  Most modelling of the ice volume data uses the Q65 signal as a representation of orbital forcing but doing so removes the possibility of a 100 kyr period arising from direct forcing. In this case the dynamics of the model must somehow generate a 100 kyr period that lines up with the ice data. For example, in the model from Verbitsky et al (2018), the unforced dynamics of the system show a damped 100 kyr oscillation. This arises from the complex eigenvalues of the system which (with default parameter values) have an imaginary part of 0.0665, corresponding to a period of 2pi/0.0665 = 94.5 kyr.
  Here we explore eccentricity as a direct driver of ice volume to see whether this can be an alternative explanation for the 100 kyr cycle observed in the data. We find that this can account for the observed data without positive feedbacks or other mechanisms that might generate a 100 kyr period. We have also considered whether a significant direct forcing effect from eccentricity could be physically plausible. We find in Section 2 that the variation of insolation due to eccentricity involves enough energy to account for a significant warming of the oceans.
  We are glad that you consider it plausible that eccentricity could play a bigger role than is usually assumed. We feel that the main result of this paper is that eccentricity's minimal power in the Q65 signal is not representative of the magnitude of its effect on the climate system relative to obliquity and precession. However, we accept that our proposed mechanism for this lag (ocean temperature) may not be accurate, and we are open to suggestions for alternative mechanisms.
  We appreciate you raising these points as it has shown there is a need for clarity when describing our model as linear. We also appreciate being introduced to your work. There are certainly similarities between our models, and we will include a discussion of these in our revised manuscript.
  
  [1] M. Y. Verbitsky, M. Crucifix, and D. M. Volobuev, "A theory of pleistocene glacial rhythmicity," Earth System Dynamics. 2018
  [2] M. Crucifix, "How can a glacial inception be predicted?", The Holocene. 2011
  
  Citation: https://doi.org/10.5194/egusphere-2023-2893-AC1
CC2:
'Comment on egusphere-2023-2893, regarding the assumptions built into LR04', Bryan C. Lougheed, 04 Jan 2024

In Figures 1 and 2, the Wheen et al. carry out a power spectra analysis on the LR04 benthic stack.
From this, they state in the introduction:
"There is a clear link between Earth’s orbital parameters and ice volume dynamics, as demonstrated by the power spectra in Fig. 2. "
As documented by Lisiecki and Raymo (2005), the LR04 benthic stack is not independently dated. For all parts 135 ka and older, the age of the benthic oxygen isotopes is determined by Lisiecki and Raymo (2005) by tuning (pattern matching) the oxygen isotopes themselves to a tuning target derived from the Imbrie and Imbrie (1980) non-linear model of ice volume forced by the mean irradiance of the day of the summer solstice at the latitude 65° north.
Firstly, since the LR04 benthic stack already incorporates assumptions regarding the relationship between Earth's astronomical parameters and global ice volume to assign age to the benthic oxygen isotopes, can it then be independently used for a power spectra analysis regarding the precise links between the Earth's astronomical parameters and global ice volume? Secondly, assuming for a moment that it would indeed be possible to apply the power spectra analysis independently, Wheen et al claim a clear link between astronomical parameters and ice volume by interpreting a power spectra analysis on an benthic oxygen isotope dataset that incorporates a non-linear model of global ice volume. Would this then not lend support to a non-linear global ice volume response?

Citation: https://doi.org/10.5194/egusphere-2023-2893-CC2
- AC2:
  'Reply on CC2', Liam Wheen, 20 Feb 2024
  
  Thank you for taking the time to read our submission and bring up these points. In regards to your first question, about the validity of using orbitally tuned data to justify orbital dependence. This is a valid point, and perhaps should be more fully addressed in the paper. Although we use the benthic stack collected by Lisiecki and Raymo (2005) (via Bintanja, 2005), this is not the only source of climatic variation that could be used to demonstrate orbital dependence.
  The Dome C ice core data collected by Parrenin et al. (2007) measures deuterium levels in the ice which can be used to infer temperature. This data closely correlates with the benthic stack and uses a time/depth conversion that utilises a combination of age markers (such as volcanic ash layers) and a glaciological model. Despite not relying on orbital tuning, this data shows the same orbital frequencies as the benthic stack.
  In regards to your second question, about whether a linear model can be used if it is fit to data that utilises a non-linear model for tuning. We believe that this is an acceptable approach. We acknowledge that the climatic system as a whole, as well as the movement of glacial media, is non-linear. However, the linear model we present here is to show how much of the global ice volume variation can be explained by a linear dependence on the orbital parameters. Because we have demonstrated that the broad dynamics of ice volume can be produced from this feed-forward model of the orbital parameters, we can use this as a baseline from which to assess more complex models. For example, Imbrie's 1980 model is non-linear because of the inclusion of a switching mechanism that changes the accumulation rate versus the ablation rate of the ice volume variable. This is a common mechanism in models of this type, with the sawtooth shape of the ice volume data often cited as justification. Our model is able to produce a similar sawtooth shape without the need for such a mechanism, calling into question the necessity of such a mechanism.
  [1] L. E. Lisiecki and M. E. Raymo, “A Pliocene-Pleistocene stack of 57 globally distributed benthic δ18O records,” Paleoceanography, vol. 20, no. 1, 2005.
  [2] R. Bintanja, R. S. Van De Wal, and J. Oerlemans, “Modelled atmospheric temperatures and global sea levels over the past million years,” Nature, vol. 437, no. 7055, pp. 125–128, 2005.
  [3] F. Parrenin, J. M. Barnola, J. Beer, T. Blunier, E. Castellano, J. Chappellaz, G. Dreyfus, H. Fischer, S. Fujita, J. Jouzel, K. Kawamura, “The EDC3 chronology for the EPICA Dome C ice core. Climate of the Past,” 2007
  [4] J. Imbrie and J. Z. Imbrie, “Modeling the climatic response to orbital variations,” Science, vol. 207, no. 4434, pp. 943–953, 1980
  
  Citation: https://doi.org/10.5194/egusphere-2023-2893-AC2
  - CC3: 'Reply on AC2', Bryan C. Lougheed, 29 Feb 2024
    
    Thanks for your reply and discussion.
    The EPICA Dome C chronology is largely orbitally tuned in Parrenin et al. (2007), as are the more recent chronologies developed for EPICA Dome C (AICC2012, AICC2023).
    I'm not so knowledgeable regarding the ins and outs of your global ice volume model, but I do find it interesting that you can develop a sawtooth output with linear modelling. Perhaps a direct comparison of your output to the Imbrie and Imbrie (1980) output would be more illustrative, as opposed to LR04?
    
    Citation: https://doi.org/10.5194/egusphere-2023-2893-CC3
    
    AC3: 'Reply on CC3', Liam Wheen, 01 Mar 2024
    
    Thanks for your response.
    I apologise if I've misinterpreted the chronology method used in Parrenin et al. (2007), but it appears that, at least for the first 400kyr of the record, the chronology is estimated using "a combination of various age markers and a glaciological model". I acknowledge that the paper employs an additional step for the later portion of the record: "the age scale is corrected in the bottom ∼500 m (corresponding to the time period 400–800 kyr BP), where the model is unable to capture the complex ice flow pattern". This appears to use the orbital tuning conducted by Dreyfus et al. (2007) to correct this second half of the chronology. However, the first 400 kyr of the record that appears to have been produced independently of orbital tuning still displays the orbital frequencies we are identifying in the LR04 data. That being said, I appreciate this should be made more clear in the paper, as we are using LR04, which does of course rely on orbital tuning throughout.
    With regards to your second point, I'm pleased that you find this result of interest, and we appreciate the suggestion about a direct comparison with Imbrie 1980. Although not included in the first draft of our paper, we have performed a range of comparisons with similar models such as Imbrie 1980. Attached is a figure showing the independently optimised solutions from Imbrie 1980, Verbitsky 2018, and Paillard and Parrenin 2004, alongside our linear model. The legend shows the percentage of variance that each optimised solution explains in the ice volume data (Bintanja 2005). Although we do not suggest that quality of fit is the only measure by which to assess a model's usefulness, it shows our model to be comparable with non-linear models, suggesting that the data does not necessitate a complex non-linear mechanism.
    
    Citation: https://doi.org/10.5194/egusphere-2023-2893-AC3
RC1: 'Comment on egusphere-2023-2893', Anonymous Referee #1, 22 Mar 2024

See attached report.

Citation: https://doi.org/10.5194/egusphere-2023-2893-RC1
RC2: 'Comment on egusphere-2023-2893', Anonymous Referee #2, 09 Apr 2024

Report on "The Largely Linear Response of Earths Ice Volume to Orbital Forcing" by Wheen et al.
This study used the zero-dimensional model to reproduce the past glacial-interglacial cycles and examine the mechanism of the observed 100,000-year ice volume of the Earth to orbital forcing. However, I would not recommend accepting the article in its current form because of several concerns in models and their evaluations as paleoclimate studies. Here are the reasons:
1. Model Evaluation

A quantitative evaluation of the glacial cycle model is needed when introducing Figure 2. What is the correlation coefficient between the data? For example, in a previous study, a minimal model of the glacial cycle can have a correlation coefficient of 0.81 (Ganopolski et al. 2024, Figure 3). The authors discuss the model can adequately reproduce observed ice volume variations for most of the past 800,000 years with the exception of Marine Isotope Stage (MIS) 11, but it seems MIS 5 and MIS 1 are weakly simulated than the data. If the ice volume during MIS2 is not maximum as in data, the model might have a limited ability to simulate glacial maximums.

The sea-level data of this study use simple calculations presented in Appendix A. I think it would be helpful to use published and frequently referred sea-level reconstruction datasets instead (e.g. Spratt and Lisiecki (2016)).
2. Section of Introduction for previous studies, particularly with conceptual models

Although many studies have proposed models of glacial-glacial cycles), only a few are introduced in the introduction. Examples of review articles can be referred to Paillard (2015) or Ganopolski (2024). The introduction reads that one primary motivation of the article is to clarify whether the glacial cycle is driven by the IOFE mechanism or the OFPA mechanism. It seems that IOFE and OFPA are newly introduced terms in this article, but their definition is not necessarily clear in the sentences, and how they can apply to other studies not referenced in this article.

Paillard (2015) summarizes physically based models as follows:

"Clearly, in order to account for observed large glacial cycles that are phase-locked to eccentricity, some non-linear mechanism is needed. Suggested concepts are stochastic resonance (Benzi et al., 1982), internal oscillations (eg. Kallen et al., 1979; Saltzman and Moritz, 1980; Saltzman et al., 1981; Gildor and Tziperman, 2000), combination tones (eg. Ghil and Le Treut, 1981; Le et al., 1983) or chaotic systems (eg. Saltzman and Maasch, 1990; for some parameter settings, see; Mitsui and Aihara, 2014)."

A significant portion of these studies were not referenced in the introduction. That makes it unclear why clarifying whether the glacial cycle is driven by mechanism IOFE or mechanism OFPA would contribute to understanding the physics of the glacial cycle.

Another issue is discussion on Q65 (L525-L534). The paragraph seems to miss the knowledge from glaciological and climate system studies in that the summer insolation in the Northern High latitudes impacts the summer temperature, critically affecting ice sheets' ablation and mass balance. The majority of the Quaternary ice sheets existed in the Northern high latitudes.
3. Concern about physics in the model (energy of the climate system)

In section 4, the authors present the model as a physical model, but the presented model seems to have several limitations, as some studies use a 3-dimensional climate-ice sheet model (e.g. Abe-Ouchi et al., 2013; Ganopolski and Brovkin 2017; Willet et al., 2019). These 3-D models have the ability to calculate the dynamics of the ice sheet and climate, including atmosphere energy balance, accumulation of ice, and melting of the ice sheet. On the contrary, the model of the present study does not have an interaction between surface temperature and ocean temperature, and the dissipation of heat in ocean temperature (Equation 5). These assumptions need validity. In this setting, incoming solar radiation can contradict the energy of the Earth's climate system, including the ocean, atmosphere and ice sheet.

As there is extensive heat exchange between the atmosphere and the ocean, it is likely that the temporal evolution of ocean temperature will remain the same. Reconstructed mean ocean temperature, which is a proxy for deep ocean temperature, does not lag much as glacial cycles change (Shackleton et al., 2021). The time series in simulated ocean temperature (Figure 10) seem to have major differences between BWT data; particularly, the simulated ocean temperature has minimum values in the interglacial.
4. Concern about sensitivity to eccentricity in the model

According to the estimated sensitivity of eccentricity in the model based on Table 2 (p1 and p2), the eccentricity can significantly impact ice sheet volume. However, it is questionable from the view of the radiation forcing of the Earth. As in L111, the eccentricity can induce the radiative forcing of ~0.6 W/m2. Given that the radiative forcing of the Last Glacial Maximum was estimated to be ~8 W/m2 (IPCC AR4 Chapter 9, 2017), the eccentricity forcing is small enough to cause significant ice sheet expansions without climate feedback.
References:

Paillard (2015): Quaternary glaciations: from observations to theories, Quaternary Science Reviews, https://doi.org/10.1016/j.quascirev.2014.10.002

Ganopolski (2024): Toward generalized Milankovitch theory (GMT), Clim. Past, https://doi.org/10.5194/cp-20-151-2024

Spratt and Lisiecki (2016): A Late Pleistocene sea level stack, Clim. Past, https://doi.org/10.5194/cp-12-1079-2016, 2016.

Abe-Ouchi et al. (2013): Insolation-driven 100,000-year glacial cycles and hysteresis of ice-sheet volume, Nature, https://doi.org/10.1038/nature12374

Ganopolski and Brovkin (2017): Simulation of climate, ice sheets and CO2 evolution during the last four glacial cycles with an Earth system model of intermediate complexity, Clim. Past., https://doi.org/10.5194/cp-13-1695-2017

Willeit et al. (2019): Mid-Pleistocene transition in glacial cycles explained by declining CO2 and regolith removal, Sci. Adv., 5, eaav7337, https://doi.org/10.1126/sciadv.aav7337

Shackleton et al. (2021): Evolution of mean ocean temperature in Marine Isotope Stage 4, Clim. Past, 17, https://doi.org/10.5194/cp-17-2273-2021

IPCC AR4 Chapter 9 (2007): https://archive.ipcc.ch/publications_and_data/ar4/wg1/en/ch9s9-2-1-3.html

Citation: https://doi.org/10.5194/egusphere-2023-2893-RC2

Liam Wheen, Oscar Benjamin, Thomas Gernon, Cameron Hall, and Jerry Wright

Model code and software

Wheen 23 Liam Wheen https://github.com/liamwheen/Wheen_23

Liam Wheen, Oscar Benjamin, Thomas Gernon, Cameron Hall, and Jerry Wright

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Short summary

We look at the impact of Earth's orbital positioning on its ice volume over the past 800,000 years. We first justify that the temperature change observed in ocean temperature proxy data could be explained by variation in the ellipticity of Earth's orbit. From there, we present a simple linear model of global ice volume. We find this to explain the data well in most regions, suggesting that the majority of global ice volume dynamics can be explained by a linear response to orbital variations.


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