The egg and the ice age

A model of the glacial cycles that solves the heat conduction equation beneath the ice, to test how much Earth’s internal heat matters for the pacing of glaciations. Move the controls and see what changes over the last 800,000 years.

Dashed lines in the ice panel: deglaciations observed in the records (approximate dates). Red strip at the top of the same panel: Atlantic circulation shut down. Click or drag over the charts to choose the moment shown in the heat column. Sea level is the model’s, proportional to ice volume: it does not include Greenland or Antarctica, so it never rises above the present level, although the records indicate seas a few metres higher in past interglacials.

The heat column at that moment

What the tests show

In short: in this model internal heat prepares the collapse of the ice, but the orbit sets the pace, and the Sun supplies the energy for the melt.

Schematic drawings

Egg in hot water radius ≈ 2 cm heat at the centre in ~10 min a ~2 km skin: as far as the cold of one glacial cycle reaches Earth radius 6,371 km; by conduction alone, R²/κ ≈ 1 trillion years heat flowing out: 0.09 W/m² on average
Figure 1. The same equation holds for both, but time grows with the square of size. In 100,000 years, a temperature change at the surface penetrates √(κt) ≈ 2 km into the rock, 0.03% of the radius. The interior never feels the glaciation.
Start of the glaciation thin, cold ice surface at −25 °C frozen base, stuck to the rock: heat passes through tens of thousands of years Near the glacial maximum ~3 km of ice: an insulating blanket the ice slides trapped heat: base near 0 °C the crust sinks under the load (with a delay)
Figure 2. Where the idea of “stored heat” really shows up. The internal flux is far too small to melt the ice sheet (about 9 mm per year), but, trapped beneath thick ice, it brings the base to the melting point. The ice starts to slide on water, becomes lower and easier to melt when the summer Sun rises again.
Earth’s orbit Northern summer Sun Icevolume Albedo: icereflects more light CO₂: the cold oceanstores more carbon Crust sinks(delay of ~13,000 years) Base of the ice warmsand the ice slides Atlantic circulationshuts down with fresh water Internal heat melts it or lets it grow reinforcethe cold weakenthe ice slows the melt Blue: feedbacks that maintain or deepen the glaciation. Red: those that prepare the collapse.
Figure 3. How the pieces connect in the model. The orbit gives a small push. Albedo and CO₂ amplify the cold. The depressed crust and the warmed base, which act with a delay, make large ice unstable, and deglaciation happens when the summer Sun rises with the ice in that state.
Sunlight absorbed by Earth
240 W/m²
Difference between glacial and interglacial (CO₂ and albedo)
~6.5 W/m²
Melting all the glacial ice in 10,000 years
~0.1 W/m²
Earth’s internal heat
0.09 W/m²
Solar wind protons
0.0003 W/m²
Figure 4. Energy sources as global averages, on a logarithmic scale: each division is ten times the previous one. Internal heat and the energy needed for deglaciation are of the same order of magnitude, but internal heat is constant and spread over the whole planet, most of it beneath the oceans.

The equations

The egg. In the column of rock and ice, the temperature T(z, t) obeys conduction, plus the cold ice that moves down from the surface with the accumulated snow:

∂T/∂t = κ ∂²T/∂z² − w(z) ∂T/∂z     base of the rock: −k ∂T/∂z = G     top: T = Ts(t)     base of the ice: Tb ≤ Tmelt

The ice. The volume V (0 to 1) grows when the “summer warmth” Θ stays below a threshold and melts when it rises above:

dV/dt = kc (1 − V) max(0, θ₀ − Θ) − kd V max(0, Θ − θ₀)
Θ = (I − Ī)/σ − γVV + γBB + γSS V + γCFCO₂ + γA(A − 1)

The slow pieces. The crust B follows the ice with a delay, the fraction of the base near melting S comes from the thermal column, CO₂ follows the ice and the circulation A, and global temperature comes from the energy balance:

dB/dt = (V − B)/τB     S = clamp[1 + (Tb − Tmelt)/4]0…1     dC/dt = (280 − 95V − 10(1 − A) − C)/τC
ΔT = λ [5.35 ln(C/280) − 4.5 V]     A: shuts down if melting exceeds 0.03 per thousand years, restarts below 0.012
ParameterValueWhat it represents
G50 mW/m²Internal heat flux in continental shields (the global mean is ~90)
κ, krock 1.0·10⁻⁶ m²/s and 2.5 W/m·K; ice 1.2·10⁻⁶ and 2.2Thermal diffusivity and conductivity
wup to ~0.1 m/yrDownward motion of cold ice, proportional to snow accumulation
Ī, σ492 and 23 W/m²Mean and deviation of June insolation at 65°N
kc, kd0.03 and 0.38 per thousand yearsRates of ice growth and melt
γV, γB, γS3.25; 2.96; 0.30Weights of albedo, the depressed crust and basal sliding
γC, γA, θ₀0.10; 0.21; 0.30Weights of CO₂ and the Atlantic circulation; melt threshold
τB, τC, λ12.8 and 4 thousand years; 0.8 K per W/m²Delays of the crust and of CO₂; climate sensitivity

Limits of the model

References

Cited from memory, without consulting a database: check volume, pages and DOI before use.

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