The Milankovitch cycles pt II: from climate proxies to toy models
- Mark Osborne
- Jun 19
- 8 min read
Updated: Jul 1
As sure as night follows day, the solstice marks the height of summer and depths of winter we know the Earth rotates on its axis and orbits our warming sun.
Less obvious are the subtle peroidic changes in the tilt of Earth's rotation, as well as the eccentricity and "wobble" of it's orbit, which lead to large changes in climate on long geological time scales. Known as the Milankovitch cycles, their astro origins are expored in part I...
Part II dives into the fingerprint that climate cycles, through ice-ages and warm interglacials, have left in geological records and how can be used to reconstruct Earth's climate history.
From climate proxies to Fourier frequencies
Geological and paleoclimate records play a founding role in the origins of the astro view of periodic climate change, ice age and interglacial cycles.
From boulders found far from their bedrock origins in the Alps [Schimper (1835), Agissiz (1838)], to oxygen isotope stages (OIS) found in deep ocean sediments [Emiliani (1954)] and GHG gas content in Vostok ice cores [Barnola (1987)], geologists, biologists and paleontologists, have looked to explain different proxies for historic climate status [37-39].
In particular charges in O(18):O(16) isotope rato (δ18O) in sediments containing carbonates (CaCO3) from ancient benthic foraminifera shells (single cell organisms) and in the water (H2O) of ice cores, have strong correlations with ocean and atmospheric temperature through mechanisms with a common origin.
Lighter H2O(16) evaporates more readily than the heavier O18 isotopomer, but a heavier H2O(18) precipitates preferentially over its counterpart. This largely follows from the fraction of molecules with sufficient speed to form vapour, which decreases with mass in accordance with the Maxwell-Boltzmann distribution as exp(-mv2/2kT) [40].
Thus in warmer waters near the equator δ18O concentrates as H2O(16) evaporates, and is depleted in water vapour moving away from the tropics, as rains remove H2O(18). Precipitation over the poles is sufficiently depleted in O(18) to exhibit negative δ18O ratios compared to standards (the Vienna Standard Mean Ocean Water).
In a cooling atmosphere CO2 sinks, the heavier O(18) isotopomer more than O(16), and in a cooling ocean the gas and carbonate ion have increased solubilty. Availability of O(18) in carbonate is then enhanced for benthic foram to sequester in shells and δ18O rises as ocean temperatures drop (+0.22 per mill rise per 1°C fall).
The deviation in benthic δ18O measured in parts per thousand (per mill) is converted to temperatures using the empirical relationship of Epstein and Urey et. al (1953) (T = 16.5 - 4.3δ + 0.14δ^2, σ = +/-0.6°C) [41].
The 100K year cycle of the mid-to-late Pleistocene epoch (11K - 1M years past) are evident in both ice and sediment records, but are other periodicities hidden in the noise?
The Fourier transform [42], a cornerstone of signal processing, is used to convert a time series signal f(t) to a spectrum of frequencies F(ν), where a signal is decomposed into a series sine and cosine functions via the transform
For the simple signals solutions can be short, singular functions (a pulse tranforms to sinc), but the power of the process lies in the extraction of key frequencies that make up "noisy" data.
Applied to benthic δ18O records the Fourier spectrum reveals peak amplitudes at decreasing frequencies 0.0456, 0.024, 0.0099 (per kiloyears) the inverse of which correspond closely to the 21000, 41000 and 100000 year cycles of climate precession, obliquity and eccentricity.
The core frequencies and corresponding phases can now be used to give a basic reconstruction of the "astronomical" model of the periodic variations in orbital elements and insolation that map to the proxy signal.
Substituting these fequencies, phases into the reduced expression for insolation, along with amplitudes for the eccentricity and obliquity, from their maxima and minima (Table 1), generates a crude but clear reproduction of the elements controlling the climates cycles reflected in the benthic foram, δ18O, proxy signal.
Importantly, superposition of the eccentricity and precession produce the large amplitude fluctuations that correlate with large swings from interglacial δ18O "lows" associated warming climates, to isotope highs in cooling cycles.
Periods of decreasing eccentricity and obliquity tend to drive higher levels of δ18O, which correspond to deeper cooling and extended ice ages.
So what of the current climate, the influence of insolation and the Milankovitch cycles?
Making sense with toy models
A closer look at the transition from the depths of the last ice-age (-30K to -11K years) to the holocene (-11K to present), we see a lag between insolation and temperature proxies, most notably δ18O records rise as insolation falls with obliquity after post glacial warming (> -10K).
Lags and phase differences are expected from inertia in the response of land, ocean and atmosphere to changes in solar warming, due to differences in heat capacities and exchange rates between these systems.
Divergence between insolation and global surface temperatures (GST) can be reconciled to an extent via the earth-sun energy balance, which is modulated through solar absorption and scatter by atmospheric gases, land and sea mass, as well as reflection and the re-emission of radiation.
The interplay between these components and their affects on Earth's climate are complex and heavy, numerical computation of highly-coupled, heat and fluid dynamics to reveal cause and affect beyond just correlations [43]. However, insight can be gained from simple physical models.
Albedo, the fraction of reflected solar, plays a key role in the transition from snowball-Earth of the ice-age to the blue planet of our interglacial age. Snow and ice are highly reflective, with high albedo (0.9 - 0.7) compared to open water and land (0.03 - 0.2) and the extent of ice cover depends strongly on temperature, as we see from polar-ice loss and sea-level rise with global warming [44, 45, 46].
The atmosphere also adds to albedo (0.15-0.25) through scatter from gases, aerosols and clouds, as well as absorbing around 30% of incoming solar across spectral bands of H2O, ozone (O3) and CO2 in the UV, visible and near to mid-infrared.
A simple model uses historic sea level records to estimate the extent of ice cover and hence albedo from present values. Until recent declines with warming, ice has covered around 21 million km2 [45] or ~ 3-4% [47] of Earth's surface and albedo (a) averages around a ~ 30% [44].
At the base of last ice-age, sea levels were up to 120 m [46] below the present rising trend. Around 40% of the sea level rise can be attributed to expansion as waters warmed [48]. Assuming the remaning 60% of water volume define by this depth and ocean area, as 70% of Earth's surface, is redistributed as ~1 km thick ice [49], cover was around (2.6+2.1)/2.1 ~ 2.2X that of the today, around 9% of Earth's surface.
Ice depth would not be constant over the snowball to warm Earth transition, but would likely have followed a rapid thinning that reflects sea level rise, just at a 10X scale (+1 km ice down to 0 v -0.1 km sea up to 0).
![Sea level [46], ice depth & cover, -ve albedo RF, +ve GHG RFs & net sum contribution](https://static.wixstatic.com/media/40e0b0_4b830d5120a643ba8f3dfd5420089ffe~mv2.jpg/v1/fill/w_781,h_833,al_c,q_85,enc_avif,quality_auto/40e0b0_4b830d5120a643ba8f3dfd5420089ffe~mv2.jpg)
The small cover but high reflectance of ice/snow, means total albedo from varies around 10%, a ~ 0.33 to 0.30, from ice-age to present.
Reflection of solar reduces irradiance and heating at Earths surface, contributing to a negative radiative forcing (RF), which varies as the product of insolation and albedo (Qice).
The major greenhouse gases (GHGs), CO2, CH4, N2O contribute positive RF through by trapping heat that would otherwise radiate to space. Absolute heat (QGHG) is dependent on atmospheric pressure records and the absorption spectra of individual GHGs (see post on Radiative Forcing).
Water vapour adds around 32 Wm-2 of heat trapping, with H2O equally reducing incoming solar by 30% (along with ozone and CO2). Increased vapour pressure and +ve (infrared) forcing due to temperature rise is somewhat balanced by -ve (visible-NIR) forcing from solar absorption.
![Insolation (@65N, black), net with -ve RF from ice cover (blue), net with +RF from GHG, temperature reconstructions (green & orange) [51] & temperature anomaly (magenta) from net solar heating](https://static.wixstatic.com/media/40e0b0_77d83e49c16d47a681697fabe2c98be8~mv2.jpg/v1/fill/w_756,h_686,al_c,q_85,enc_avif,quality_auto/40e0b0_77d83e49c16d47a681697fabe2c98be8~mv2.jpg)
Net irradiance raising the global surface temperature (GST) is then the sum of all contributions, reduced insolation due to atmospheric absorption, reflected heat from albedo and trapped heat from GHGs. At around pre-industrial numbers add to around 0.7X467 - 139 + 14 + 32 ~ 234 W m-2.
Surface temperature is estimated from the Stefan-Boltzmann relationship between temperature T, and the radiant intensity Q, of a perfect black body emitter, and the emissivity ε, of Earth compared to the black body radiator [50].
While the Earth, land, sea and ice, is a near-perfect emitter (ε > 0.9), atmospheric absorption prevents thermal radiation escaping to space and the effective emissivity is reduced to ε ~ 0.6 [51].
Applying the toy model using sea level records to determine ice volume (km3), redistributed ice cover (km2) and change in albedo across the late ice-age to holocene period (-25K to present), reproduces the temperature anomoly records (vs 1850-1900 mean) remarkably well [52].
Temperature highs around the -6K years, the holocene climate optimum (HCO), run a little hotter than proxy reconstructions, and lows at the depths of the ice-age are lower than some records.
But with inputs to the model derived simply from literature without fitting or optimisation, the model works well to elucidate the basic physics and principle components that govern climate change.
The caveat! There's nothing new to see here. Models of this kind will used in the various reconstructions of temperature profiles from proxy data (δ18O, ice-core CO2, etc), so good alignment is not unexpected.
The aim then? A long stroll through the history and science behind our Earth's evolving climate (as per intro), an ode to the giants of astronomy and climatology, a primer for action on the key drivers of our current climate trajectory.
The take aways...
Simple physical models provide clear quantitative insights!
Solar cannot account for current trends in global warming.
Cooling, not warming would be expected from the down trend in insolation.
Even accounting for loss of ice and hence albedo, net heating from solar at Earth's surface trends down.
Insolation is the dominant driver of the transition from ice-age to the holocene climate optimum (HCO) and slow cooling up to the 19th century.
Albedo and -ve forcing from changes in ice cover dominates the overall "damping" of insolation and net irradiance of Earth's surface.
GHGs add near-constant forcing over the period of transition out of the ice-age.
But since the industrial revolution, rapidly rising levels of GHGs (1750) has driven the 1.5C uptick in GST, with recent snow and ice-loss playing a minor but non-negligiable roles.
CO2 remains the dominant GHG in radiative forcing and spiking temperatures in 2026!
Key climate controls to dial down warming:
Reduced GHGs - renewables, electrification, waste reduction/recycling, carbon dioxide removals, regenerative farming.
Enhanced albedo - re-icing the polar caps, cool (white) roofs & asphalt, SRM and stratospheric aerosol injection.
Enhanced emissivity - cirrus-cloud thinning, passive cooling through the IR window (8-14 micron).
[37] Agassiz vs Schimper https://www.jstor.org/stable/25101263?seq=1
[39] Barnola, J., Raynaud, D., Korotkevich, Y. et al. Vostok ice core provides 160,000-year record of atmospheric CO2. Nature 329, 408–414 (1987). https://doi.org/10.1038/329408a0
[40] Maxwell-Boltzmann https://www.mdpi.com/1099-4300/17/4/1971; https://sites.pitt.edu/~jdnorton/teaching/2559_Therm_Stat_Mech/docs/Maxwell_1860.pdf
[41] Epstein et al. https://pubs.geoscienceworld.org/gsa/gsabulletin/article-abstract/64/11/1315/4537/REVISED-CARBONATE-WATER-ISOTOPIC-TEMPERATURE-SCALE?redirectedFrom=fulltext
[43] Climate models https://wcrp-cmip.org/ref-launch-at-cmip2026/; https://gmd.copernicus.org/articles/19/2627/2026/
[46] Berends et al https://cp.copernicus.org/articles/17/361/2021/
[47] Ice cover % https://www.usgs.gov/faqs/how-does-present-glacier-extent-and-sea-level-compare-extent-glaciers-and-global-sea-level
[48] Expansion https://www.climate.gov/news-features/understanding-climate/climate-change-global-sea-level
[49] Ice thickness https://www.science.org/doi/10.1126/science.265.5169.195; https://www.antarcticglaciers.org/glacial-geology/laurentide-ice-sheet/laurentide-ice-sheet-at-the-last-glacial-maximum/; https://www.annualreviews.org/content/journals/10.1146/annurev.earth.32.082503.144359
[50] Stefan-Boltzmann https://babel.hathitrust.org/cgi/pt?id=uc1.a0002763670&view=2up&seq=306
[51] Emissivity Fig 7.2 https://www.ipcc.ch/report/ar6/wg1/downloads/report/IPCC_AR6_WGI_Chapter07.pdf
[52] McKay https://cp.copernicus.org/articles/18/911/2022/; Nick McKay. (2022). nickmckay/past-and-future-warming-comparison-figure: Climate of the past resubmission (resubmission). Zenodo. https://doi.org/10.5281/zenodo.5842209
*Links accessed 18/06/2026
