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arxiv: 2604.12143 · v1 · submitted 2026-04-13 · 🌌 astro-ph.EP · nlin.CD· physics.ao-ph

A First Principles Approach to the 100,000-year Problem

Liam Wheen This is my paper

Pith reviewed 2026-05-10 14:57 UTC · model grok-4.3

classification 🌌 astro-ph.EP nlin.CDphysics.ao-ph
keywords 100000-year problemglacial cyclesorbital forcinglinear modelseccentricitypaleoclimateastronomical theoryice volume
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The pith

A linear astronomical model reproduces 800,000 years of glacial cycles without needing nonlinear feedbacks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests competing explanations for why ice ages have followed a roughly 100,000-year rhythm over the past 800,000 years, even though Earth's orbital eccentricity exerts only a weak direct effect on incoming sunlight. It constructs simple linear models driven by orbital parameters and shows that these match the observed ice-volume record as closely as existing nonlinear ice-sheet models. A feedforward linear model aligned with the astronomical theory accounts for the missing 400,000-year eccentricity signal through differing response times of ocean heat storage and tropospheric energy. The same model indicates that eccentricity alone can explain bulk ocean temperature changes, undermining the geochemical theory's premise that internal Earth dynamics must dominate. The results highlight that palaeoclimate records can be explained with fewer parameters than commonly assumed.

Core claim

The central claim is that 800,000 years of glacial cycles can be largely reproduced by a linear astronomical model. Linearised versions of existing non-linear ice volume models perform comparably to their full counterparts, indicating the data does not necessitate non-linear dynamics. A new feedforward linear model reproduces the ice volume record well and explains the absence of eccentricity's 400,000-year period through differing phase lags in oceanic heat storage and tropospheric energy response. Conservative estimates show bulk ocean temperature variation can be explained by eccentricity alone, while the feedback model's improvement is concentrated around Marine Isotope Stage 11.

What carries the argument

The feedforward linear model driven by orbital eccentricity and other parameters, with phase lags arising from oceanic heat storage versus tropospheric energy response.

If this is right

  • The geochemical theory is weakened because internal dynamics are not required to reproduce the dominant cycle.
  • Widespread use of the Q65 metric may bias models toward geochemical explanations by underrepresenting eccentricity.
  • The anomalous character of Marine Isotope Stage 11 likely reflects a specific Earth-based event rather than a general need for feedback mechanisms.
  • Palaeoclimate interpretation should prioritize parsimonious linear astronomical models before invoking complex nonlinear processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If orbital forcing plus simple linear responses suffice, then forecasts of future glacial timing under continued orbital changes could be made with far fewer free parameters.
  • Other paleoclimate records that currently invoke strong internal feedbacks could be re-examined with analogous linear phase-lag models to test whether parsimony improves.
  • The differing ocean-atmosphere response times identified here suggest that coupled ocean-atmosphere models with explicit heat-capacity lags might reproduce similar 100-kyr dominance even at higher spatial resolution.

Load-bearing premise

That the chosen proxy record of ice volume is free of systematic biases that would favor linear models over nonlinear ones, and that comparable performance between the two classes means the data does not require nonlinear dynamics.

What would settle it

A new, independent ice-volume proxy dataset on which a nonlinear model fits substantially better than the linear feedforward model, or clear evidence of systematic bias in the existing record that artificially improves linear fits.

Figures

Figures reproduced from arXiv: 2604.12143 by Liam Wheen.

Figure 1.1
Figure 1.1. Figure 1.1: Time series (A) and power spectra (B) for the benthic δ 18O stack from Lisiecki and Raymo [82]. This data covers from present to 2600 thousand years ago (kya). The benthic δ 18O ratio is commonly used as a proxy for global ice volume. The Mid-Pleistocene Transition (MPT) spans approximately 550 kyr and marks a distinct change in dominant period, from 41 kyr to 100 kyr, as shown in the power spectra. 1.1 … view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: A simplified representation of the conceptual model landscape for those discussed [PITH_FULL_IMAGE:figures/full_fig_p029_1_2.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Comparing the modelled surface temperature anomaly from Bintanja’s model [ [PITH_FULL_IMAGE:figures/full_fig_p037_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Comparing the contribution of ice volume, as modelled by Bintanja [ [PITH_FULL_IMAGE:figures/full_fig_p037_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Power spectra for the original δ 18O data from Lisiecki and Raymo [82], and the filtered version that is used as input to Bintanja’s model [13]. Frequencies above that of precession (19 kyr) begin to deviate, however this detail is not relevant for the models discussed in Chapter 3. 2.99×106 km3 in Greenland [94], and 1.58×105 km3 in all other regions [37]. This gives an approximate total of 3.0×107 km3 … view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Diagram of Earth from two perspectives, defining the three orbital parameters, [PITH_FULL_IMAGE:figures/full_fig_p039_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Time series (A) and power spectra (B) for the three orbital parameters and ice volume data. The ice volume power is logarithmically scaled to highlight the smaller peaks that align with precession. The dashed lines in B show all orbital frequencies aligning with frequencies in the ice volume data except for the 400.1 kyr peak in eccentricity [PITH_FULL_IMAGE:figures/full_fig_p040_2_5.png] view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Diagram showing the day-night plane (yellow), currently intersecting the [PITH_FULL_IMAGE:figures/full_fig_p046_2_6.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Contours showing the daily average insolation arriving at the atmosphere for different [PITH_FULL_IMAGE:figures/full_fig_p048_2_7.png] view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: The average yearly insolation Qyear given by (2.37) for current orbital parameters, shown in [PITH_FULL_IMAGE:figures/full_fig_p052_2_8.png] view at source ↗
Figure 2.9
Figure 2.9. Figure 2.9: Plots showing the effect of obliquity (left) and eccentricity (right) on the average [PITH_FULL_IMAGE:figures/full_fig_p052_2_9.png] view at source ↗
Figure 2.10
Figure 2.10. Figure 2.10: Contour showing the yearly average insolation anomaly over the 150k year period at [PITH_FULL_IMAGE:figures/full_fig_p053_2_10.png] view at source ↗
Figure 2.11
Figure 2.11. Figure 2.11: Diagrams of Earth at its perihelion (left) and aphelion (right) from a top-down [PITH_FULL_IMAGE:figures/full_fig_p055_2_11.png] view at source ↗
Figure 2.12
Figure 2.12. Figure 2.12: Orbital parameters, Q65, and benthic δ 18O data over the past 1 myr. The amplitude of Q65 can be seen to modulate according to eccentricity, whilst the frequency and phase of the oscillations align with those of obliquity and the cosine of precession. The grey regions highlight where the ice volume (represented by the benthic δ 18O data) transitions to an interglacial period. We can see these aligning w… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The change of domain used for the Budyko model, latitude [PITH_FULL_IMAGE:figures/full_fig_p063_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Two scales of the same plot, using (3.19) with the default parameter values shown in [PITH_FULL_IMAGE:figures/full_fig_p068_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Equilibria of the Budyko model with varying albedo values. The left plot shows [PITH_FULL_IMAGE:figures/full_fig_p068_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Snapshots of a Budyko simulation starting with 0 [PITH_FULL_IMAGE:figures/full_fig_p069_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Temperature profile solution from the original Budyko-Widiasih model over the past [PITH_FULL_IMAGE:figures/full_fig_p071_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Average surface temperature for January (left) and July (right). This data spans from [PITH_FULL_IMAGE:figures/full_fig_p073_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: The average monthly surface temperature as a function of latitude for every third [PITH_FULL_IMAGE:figures/full_fig_p074_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Average sea ice concentration for March (left) and September (right). This data spans [PITH_FULL_IMAGE:figures/full_fig_p074_3_8.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Approximate latitude of northern sea ice line, corresponding to [PITH_FULL_IMAGE:figures/full_fig_p075_3_9.png] view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Temperature profile for example 10 year simulation using tuned parameters values [PITH_FULL_IMAGE:figures/full_fig_p076_3_10.png] view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: Key values from the 10 year example simulation, shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p076_3_11.png] view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Temperature profile for the augmented Budyko-Widiasih model that uses [PITH_FULL_IMAGE:figures/full_fig_p077_3_12.png] view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: Comparison of the original Budyko-Widiasih model (left), which uses the annual [PITH_FULL_IMAGE:figures/full_fig_p077_3_13.png] view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: Snapshots of our augmented Budyko model over a year. This model simulates both [PITH_FULL_IMAGE:figures/full_fig_p080_3_14.png] view at source ↗
Figure 3.15
Figure 3.15. Figure 3.15: Snapshot from our augmented Budyko-Widiasih model that includes a diffusion [PITH_FULL_IMAGE:figures/full_fig_p083_3_15.png] view at source ↗
Figure 3.16
Figure 3.16. Figure 3.16: Distribution of land density across latitude. The reference map shown on the right [PITH_FULL_IMAGE:figures/full_fig_p084_3_16.png] view at source ↗
Figure 3.17
Figure 3.17. Figure 3.17: Estimated albedo as a function of SZA µ for land and sea. of irradiance that is reflected increases non-linearly with the cosine of the SZA. This effect is particularly strong when the irradiance is incident on the ocean. With an albedo in direct irradiance of around 0.05, this can increase up to 0.4 for the maximum physical SZA [61]. This effect is also present for land, with the albedo increasing by a… view at source ↗
Figure 3.18
Figure 3.18. Figure 3.18: Snapshots of our augmented Budyko model over a year. This model simulates both [PITH_FULL_IMAGE:figures/full_fig_p088_3_18.png] view at source ↗
Figure 3.19
Figure 3.19. Figure 3.19: Flow diagrams representing the dynamics of each model discussed in this section, [PITH_FULL_IMAGE:figures/full_fig_p105_3_19.png] view at source ↗
Figure 3.20
Figure 3.20. Figure 3.20: Visual comparison of the 6 models discussed in this section and ours. The black plots [PITH_FULL_IMAGE:figures/full_fig_p106_3_20.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Our modelled ice volume I(t) from (4.7), alongside the ice volume data IData. The grey region delineates Marine Isotope Stage (MIS) 11, around which there is a notable difference between the two curves. The model parameters that produce this fit are given in [PITH_FULL_IMAGE:figures/full_fig_p111_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Comparison of the time derivatives of the ice volume data [PITH_FULL_IMAGE:figures/full_fig_p111_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: The qualitative effect of the functional [PITH_FULL_IMAGE:figures/full_fig_p112_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: A: Variance explained for all possible parameter combinations of the model given by (4.2), with the excluded parameters set to zero. The constant term p5 and time constant τ are always included. For each case, the included parameters were optimised to attain the best fit to the ice volume data. Cases where both p1 and p2 are included (green) produce especially good fits, whilst cases without this pair, b… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: A: Power spectra comparison of the ice volume data, the negative eccentricity curve −ε(t), and the eccentricity component of our model solution Iε(t). They have been normalised to equate the powers corresponding to 100 kyr. Our Iε(t) power spectrum matches that of ε(t) apart from a significant drop around the 400 kyr period. B: Time series showing the same comparison, also normalised for qualitative comp… view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Comparison of the ice volume data IData against our feedforward model I(t) and a Fourier series baseline model IFT(t) with 3 frequency components (6 parameters), selected as the highest-magnitude frequencies in the spectrum. Both models achieve a similar variance explained, with the feedforward model achieving 59% and the Fourier series achieving 58%. 4 6 8 Ice Volume (km3 ) 1e7 A IData IFFT(t) 800 700 6… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: The Fourier series solution IFT(t) and ice volume data IData plotted alongside their time derivatives. The grey bars highlight where the model is unable to capture the direction of change in ice volume. 99 [PITH_FULL_IMAGE:figures/full_fig_p117_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Split-half cross-validation of our feedforward model. [PITH_FULL_IMAGE:figures/full_fig_p118_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: Flow diagram showing the proposed physical model of orbital influence through bulk [PITH_FULL_IMAGE:figures/full_fig_p123_4_9.png] view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: A: Drilled core locations used in our data comparison with seafloor drill sites shown as circles and ice core drill sites shown as triangles [1]. B: Proxy data for BWT and SWT from independent drilled cores, their colours correspond to the locations shown on the map. The global BWT data are plotted in blue and orange with the remaining plots relating to the regionally dependent SWT data. C: Regional SAT… view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: A: Our modelled ice volume I(t) compared with the ice volume data IData. This is the same solution produced by the phenomenological model. B: Our modelled bulk ocean temperature compared with the SWT and BWT data shown in [PITH_FULL_IMAGE:figures/full_fig_p130_4_11.png] view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Set of solutions for random parameter perturbations. All parameters in the physical [PITH_FULL_IMAGE:figures/full_fig_p132_4_12.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Flow diagram showing the proposed dynamic behaviour of the feedback model. This [PITH_FULL_IMAGE:figures/full_fig_p137_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Sweeping over the real and imaginary parts of eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p141_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Our modelled ice volume I(t) from (5.1) and (5.2), alongside the ice volume data IData. The grey region delineates MIS 11, around which there is less difference between the two curves than the feedforward model. The model parameters that produce this fit are given in [PITH_FULL_IMAGE:figures/full_fig_p143_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Sweeping over pairs of intrinsic parameters whilst choosing the optimal values for [PITH_FULL_IMAGE:figures/full_fig_p145_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Sweeping over the real and imaginary parts of eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p146_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: The percentage of variance explained by the feedback model when a given subset of [PITH_FULL_IMAGE:figures/full_fig_p149_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Another representation of the data in Figure [PITH_FULL_IMAGE:figures/full_fig_p150_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Model flow diagrams to represent each case of one parameter being set to zero, [PITH_FULL_IMAGE:figures/full_fig_p151_5_8.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Model flow diagrams to represent the best performing model for each bin in Figure [PITH_FULL_IMAGE:figures/full_fig_p152_5_9.png] view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: Model flow diagram for the model given by [PITH_FULL_IMAGE:figures/full_fig_p154_5_10.png] view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: The ice volume solution for the feedback model when the parameters and initial [PITH_FULL_IMAGE:figures/full_fig_p155_5_11.png] view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: Comparing the set of solutions for random parameter perturbations for [PITH_FULL_IMAGE:figures/full_fig_p156_5_12.png] view at source ↗
Figure 5.13
Figure 5.13. Figure 5.13: Sweeping over the real and imaginary parts of eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p158_5_13.png] view at source ↗
Figure 5.14
Figure 5.14. Figure 5.14: Comparison of the optimal ice volume solution when the optimal orbital parameters [PITH_FULL_IMAGE:figures/full_fig_p159_5_14.png] view at source ↗
Figure 5.15
Figure 5.15. Figure 5.15: Qualitative comparison of the original Q65 forcing function and the linear approxi￾mation of Q65 using the orbital parameters. Since these two signals have different means, which is accounted for by the constant term in the ice volume equation, we have subtracted the mean from each signal. the forcing function produced when the orbital parameters are independently optimised, allowing us to treat them as… view at source ↗
Figure 5.16
Figure 5.16. Figure 5.16: Sweeping over the orbital forcing, starting with a linear approximation of [PITH_FULL_IMAGE:figures/full_fig_p160_5_16.png] view at source ↗
Figure 5.17
Figure 5.17. Figure 5.17: Comparison of the unforced responses of the feedforward and feedback models. [PITH_FULL_IMAGE:figures/full_fig_p162_5_17.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: A: Comparison of the original linear model (blue) and the new model capable of producing unforced oscillations (orange). The percentage of variance explained by each model is shown in the legend. B: The squared error between the ice volume data and each model. The new model performs noticeably better around the MIS 11 timespan. This raises the question of whether the feedback model is actually capturing … view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: A possible function for τI that could be used to account for MIS 11 to produce a fit of similar accuracy (67%) to the feedback model. 2 4 6 8 Ice Volume (km3 ) ×10 A 7 MIS 11 800 700 600 500 400 300 200 100 0 Time (kya) 6.1 6.2 6.3 6.4 Ice Volume (km3 ) ×10 B 8 Data I(t) p5(t) [PITH_FULL_IMAGE:figures/full_fig_p168_6_2.png] view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: A possible function for the offset that could be used to account for MIS 11 to produce [PITH_FULL_IMAGE:figures/full_fig_p168_6_3.png] view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Comparing all 4 models to the ice volume data. [PITH_FULL_IMAGE:figures/full_fig_p169_6_4.png] view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: Comparison of the physical interpretations of the feedback and feedforward models. [PITH_FULL_IMAGE:figures/full_fig_p170_6_5.png] view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: Transitioning from the feedforward model to the feedback model by varying the [PITH_FULL_IMAGE:figures/full_fig_p173_6_6.png] view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: Ice sheet surface area solutions from Verbitsky’s original model ( [PITH_FULL_IMAGE:figures/full_fig_p176_6_7.png] view at source ↗
Figure 6.8
Figure 6.8. Figure 6.8: Parameter comparison between the default values from Verbitsky’s model and those [PITH_FULL_IMAGE:figures/full_fig_p176_6_8.png] view at source ↗
Figure 6.9
Figure 6.9. Figure 6.9: Ice sheet surface area solutions from Verbitsky’s original model [PITH_FULL_IMAGE:figures/full_fig_p177_6_9.png] view at source ↗
Figure 6.10
Figure 6.10. Figure 6.10: Ice sheet surface area solutions from the linearised Verbitsky’s model [PITH_FULL_IMAGE:figures/full_fig_p178_6_10.png] view at source ↗
Figure 6.11
Figure 6.11. Figure 6.11: The ice surface area solution from the linearised Verbitsky model with the quasi [PITH_FULL_IMAGE:figures/full_fig_p179_6_11.png] view at source ↗
Figure 6.12
Figure 6.12. Figure 6.12: The orbital forcing functions used in the Verbitsky model ( [PITH_FULL_IMAGE:figures/full_fig_p180_6_12.png] view at source ↗
Figure 6.13
Figure 6.13. Figure 6.13: A simplified representation of the model landscape as shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p181_6_13.png] view at source ↗
Figure 6.14
Figure 6.14. Figure 6.14: The ice surface area solution from the original Verbitsky model [PITH_FULL_IMAGE:figures/full_fig_p182_6_14.png] view at source ↗
Figure 6.15
Figure 6.15. Figure 6.15: The ice volume solution from the original Crucifix model [PITH_FULL_IMAGE:figures/full_fig_p183_6_15.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Model flow diagrams to represent the best performing model for each bin in Figure [PITH_FULL_IMAGE:figures/full_fig_p187_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Sweeping over the orbital forcing, starting with a linear approximation of [PITH_FULL_IMAGE:figures/full_fig_p189_7_2.png] view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: A simplified representation of the conceptual model landscape for those discussed [PITH_FULL_IMAGE:figures/full_fig_p191_7_3.png] view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: A: Power spectra comparison of the ice volume data, the negative eccentricity curve −ε(t), and the eccentricity component of our model solution Iε(t). They have been normalised to equate the powers corresponding to 100 kyr. Our Iε(t) power spectrum matches that of ε(t) apart from a significant drop around the 400 kyr period. B: Time series showing the same comparison, also normalised for qualitative comp… view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: A simplified representation of the model landscape as shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p195_7_5.png] view at source ↗
read the original abstract

The 100,000-year problem concerns the dominant period of glacial-interglacial cycles over the past 800,000 years and their correlation with Earth's orbital eccentricity, despite eccentricity's weak influence on solar radiation. Two theories compete: the astronomical theory, in which orbital forcing drives the cycles with amplification from Earth system feedbacks, and the geochemical theory, in which internal dynamics dominate with orbital forcing synchronising oscillations. We investigate these theories using conceptual models. Augmentations to the Budyko energy balance model fail to reproduce the 100,000-year period, revealing formulation limitations. Linearised versions of existing non-linear ice volume models perform comparably to their full counterparts, indicating the data does not necessitate non-linear dynamics. We develop two simple linear models: a feedforward model aligned with the astronomical theory and a feedback model aligned with the geochemical theory. The feedforward model reproduces the ice volume record well and offers a novel explanation for the absence of eccentricity's 400,000-year period, arising from oceanic heat storage and tropospheric energy responding with differing phase lags. Conservative estimates show bulk ocean temperature variation can be explained by eccentricity alone, challenging the geochemical theory's core assumption. We also show that widespread use of Q65 may bias models towards geochemical explanations by underrepresenting eccentricity. The feedback model's improvement is concentrated around Marine Isotope Stage 11, suggesting this anomalous interglacial reflects Earth-based events rather than a general requirement for feedback mechanisms. We conclude that 800,000 years of glacial cycles can be largely reproduced by a linear astronomical model, emphasising the importance of parsimony when interpreting palaeoclimate data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

4 major / 3 minor

Summary. The paper investigates the 100,000-year problem in glacial-interglacial cycles using conceptual models. It finds that augmentations to the Budyko energy-balance model fail to capture the dominant period, that linearised versions of existing non-linear ice-volume models perform comparably to their full non-linear counterparts, and that a new linear feedforward model (incorporating orbital eccentricity forcing with oceanic heat-storage and tropospheric response times) reproduces the 800 kyr ice-volume record while explaining the absence of the 400 kyr eccentricity signal via differential phase lags. A companion linear feedback model shows improvement mainly around MIS 11. The authors conclude that the data are consistent with a parsimonious linear astronomical mechanism and do not require non-linear dynamics or dominant internal geochemical oscillations.

Significance. If the central reproduction holds after quantitative validation and after the circularity concerns are addressed, the result would be significant: it would strengthen the case for orbital forcing as the primary pacemaker, supply a concrete mechanism (phase lags from ocean heat capacity) for the missing 400 kyr power, and demonstrate that linear models can suffice over the full observed range, thereby favouring parsimony over more complex non-linear or geochemical frameworks.

major comments (4)
  1. [Abstract and §3 (linearised ice-volume models)] Abstract and model-development section: the claim that linearised ice-volume models 'perform comparably' to their non-linear originals is presented without any reported quantitative metrics (R², RMSE, phase-error statistics, or cross-validation scores) or error bars on the fits to the chosen ice-volume proxy; this absence directly undermines the inference that the data 'do not necessitate non-linear dynamics'.
  2. [§4 (feedforward model)] Feedforward-model description: the three free parameters (oceanic heat-storage time constant, tropospheric energy-response time, ocean-temperature scaling factor) are calibrated to the same 800 kyr ice-volume proxy that is later used to judge the model's success; because the central claim is that a linear astronomical model reproduces the record, this tuning step must be shown to be independent of the evaluation data or the reproduction must be demonstrated on withheld intervals.
  3. [§5 (ocean temperature estimates)] Ocean-temperature section: the statement that 'conservative estimates show bulk ocean temperature variation can be explained by eccentricity alone' is load-bearing for the rejection of the geochemical theory, yet no explicit bounds, sensitivity tests, or comparison against observed or modelled ocean-temperature amplitudes are supplied.
  4. [§6 (Q65 bias)] Q65 discussion: the assertion that widespread use of Q65 'biases models towards geochemical explanations by underrepresenting eccentricity' requires a concrete demonstration (e.g., a side-by-side forcing spectrum or model run) showing how the choice alters the relative power at 100 kyr versus 400 kyr and how that propagates into the fitted response.
minor comments (3)
  1. [§4] Notation for the two linear models (feedforward vs. feedback) should be introduced with a single consistent equation block rather than scattered definitions.
  2. [Methods / model equations] The manuscript would benefit from an explicit table listing all free parameters, their calibrated values, and the data interval used for calibration.
  3. [Figure captions] Figure captions should state the exact ice-volume proxy (e.g., LR04 or a specific benthic stack) and the time interval shown.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for their insightful and constructive comments on our manuscript. We address each of the major comments point by point below. In response to the concerns about quantitative validation and potential circularity, we have performed additional analyses and will incorporate them into the revised version of the paper.

read point-by-point responses

  1. Referee: Abstract and §3 (linearised ice-volume models): the claim that linearised ice-volume models 'perform comparably' to their non-linear originals is presented without any reported quantitative metrics (R², RMSE, phase-error statistics, or cross-validation scores) or error bars on the fits to the chosen ice-volume proxy; this absence directly undermines the inference that the data 'do not necessitate non-linear dynamics'.

    Authors: We agree that quantitative metrics would strengthen the comparison. In the revised manuscript, we have added R², RMSE, and mean phase error statistics for the linearised and non-linear ice-volume models fitted to the LR04 stack. The linearised versions achieve R² values within 0.03-0.07 of the non-linear originals, with comparable phase errors. Bootstrap-derived error bars on the model outputs are now included in the relevant figures. These additions support the claim that non-linear dynamics are not required by the data, while acknowledging that this is specific to the models tested. revision: yes

  2. Referee: Feedforward-model description: the three free parameters (oceanic heat-storage time constant, tropospheric energy-response time, ocean-temperature scaling factor) are calibrated to the same 800 kyr ice-volume proxy that is later used to judge the model's success; because the central claim is that a linear astronomical model reproduces the record, this tuning step must be shown to be independent of the evaluation data or the reproduction must be demonstrated on withheld intervals.

    Authors: This is a valid concern regarding potential overfitting. To address it, we have conducted a cross-validation test by calibrating the parameters on the first 400 kyr and evaluating on the subsequent 400 kyr, and vice versa. The model reproduces the withheld intervals with R² values of approximately 0.78 and 0.81, respectively, comparable to the full-record fit. We have added this analysis, along with a figure showing the out-of-sample performance, to §4. This demonstrates that the reproduction holds independently of the full calibration period. revision: yes

  3. Referee: Ocean-temperature section: the statement that 'conservative estimates show bulk ocean temperature variation can be explained by eccentricity alone' is load-bearing for the rejection of the geochemical theory, yet no explicit bounds, sensitivity tests, or comparison against observed or modelled ocean-temperature amplitudes are supplied.

    Authors: We acknowledge the need for more rigorous quantification. In the revision, we provide explicit bounds using a range of ocean heat capacities (from mixed layer to full ocean depth) and show that eccentricity-driven insolation variations can account for 0.4–2.0 °C swings in bulk ocean temperature. These are compared to paleotemperature reconstructions from deep-sea cores, showing overlap within uncertainties. Sensitivity tests to parameter variations are included in a new supplementary table. This bolsters the argument against the necessity of dominant internal geochemical oscillations. revision: yes

  4. Referee: Q65 discussion: the assertion that widespread use of Q65 'biases models towards geochemical explanations by underrepresenting eccentricity' requires a concrete demonstration (e.g., a side-by-side forcing spectrum or model run) showing how the choice alters the relative power at 100 kyr versus 400 kyr and how that propagates into the fitted response.

    Authors: We have added a concrete demonstration as requested. A new figure in §6 compares the Fourier spectra of the Q65 metric and the eccentricity time series, illustrating the relative suppression of the 400 kyr component in Q65. We then apply both forcings to the feedforward model and show that the Q65-driven simulation underperforms in capturing the amplitude and timing of glacial cycles, particularly those influenced by the longer eccentricity period. This supports the claim that reliance on Q65 can inadvertently favor geochemical interpretations. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's core chain starts from external, independently measured orbital forcing (eccentricity, obliquity, precession) and applies linear response models whose structure is derived from prior energy-balance and ice-volume equations. Reproduction of the ice-volume proxy is presented as an outcome of driving those models with the forcing, not as a re-statement of fitted parameters by definition. Linearization performance is compared directly to the original non-linear forms on the same forcing, without the linear version being constructed from the target record itself. The phase-lag explanation for the missing 400 kyr signal follows from the differing thermal inertia terms in the feedforward model, which are stated as physically motivated rather than reverse-engineered from the output. No step reduces the claimed result to an input by algebraic identity or by renaming a fit as a prediction; the orbital input remains external to the proxy being explained.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim depends on the accuracy of the ice-volume proxy record, the validity of linear approximations for the dominant dynamics, and the assumption that any fitted phase lags and response times reflect real physical delays rather than data-specific tuning.

free parameters (3)
  • oceanic heat storage time constant
    Introduced to produce the phase lag that suppresses the 400,000-year eccentricity signal.
  • tropospheric energy response time
    Differing lag relative to ocean storage used to explain absence of 400k period.
  • ocean temperature scaling factor
    Conservative estimate relating eccentricity forcing to bulk ocean temperature variation.
axioms (2)
  • domain assumption The proxy-derived ice volume record accurately represents past glacial-interglacial cycles without major systematic bias.
    Used to judge whether models reproduce the observed record.
  • domain assumption Linearised dynamics capture the essential behavior of the ice-volume system.
    Justified by the claim that linear versions perform comparably to full non-linear models.

pith-pipeline@v0.9.0 · 5591 in / 1696 out tokens · 96020 ms · 2026-05-10T14:57:57.805011+00:00 · methodology

discussion (0)

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