Questions: Oceanic Crustal Cooling and Age Relationships

Question 1

Two sections of ocean floor are measured: one 25 Ma old and one 100 Ma old. According to the half-space cooling model, what is the approximate ratio of their seafloor depths below the ridge crest?

• A. 4:1 — the 100 Ma crust is four times deeper, because depth scales linearly with age
• B. 2:1 — the 100 Ma crust is about twice as deep, because depth scales as the square root of age
• C. 10:1 — the 100 Ma crust is ten times deeper, because older crust has lost far more heat
• D. 1:1 — depths are nearly equal, because most seafloor subsidence occurs in the first 10 Ma

Answer: (B)

The half-space cooling model predicts depth ∝ √(age). The ratio of depths is √100 / √25 = 10 / 5 = 2. The 100 Ma crust is about twice as deep as the 25 Ma crust. Option A (4:1) would be correct for a linear relationship. The square-root scaling is the hallmark of diffusive cooling: the thermal boundary layer thickens as √(κt), and the resulting isostatic subsidence follows the same dependence. This prediction was one of the first quantitative confirmations of plate tectonics — observed ocean depth profiles across multiple basins matched the √t curve.

Question 2

Very old oceanic crust (>80 Ma) is consistently observed to be shallower than the half-space cooling model predicts. The best explanation for this systematic deviation is:

• A. Hydrothermal circulation cools old crust more efficiently than young crust, paradoxically causing anomalous uplift
• B. The lithosphere approaches a finite equilibrium thickness because of heat supplied from the underlying asthenosphere, preventing it from cooling indefinitely
• C. Old crust becomes less dense over millions of years as iron is lost through seafloor weathering and alteration
• D. The seafloor stops subsiding once it reaches isostatic equilibrium with the surrounding mantle at about 80 Ma

Answer: (B)

The half-space model assumes the lithosphere cools into an infinite half-space with no lower boundary. But old lithosphere is systematically shallower and warmer than this predicts — the cooling curve flattens. The plate model explains this by adding a lower thermal boundary condition: the base of the lithosphere is maintained at approximately mantle temperature by heat from the underlying asthenosphere (possibly via small-scale convection or radiogenic heating). This prevents the plate from growing thicker than ~125 km and causes the age-depth relationship to flatten for crust older than ~70–80 Ma. The plate model replaces the half-space model for old ocean floor.

Question 3

Both seafloor heat flow and ocean depth can be predicted from crustal age alone using the half-space cooling model, because both observables arise from the same underlying thermal diffusion process.

• T. True
• F. False

Answer: True

The half-space model provides a unified thermal picture connecting multiple independent observables to a single variable: age. Surface heat flow = κ × (temperature gradient) ∝ 1/√(age) — highest at the ridge, declining steadily. Ocean depth ∝ √(age) — shallowest at the ridge, deepening as the lithosphere cools, contracts, and isostatically subsides. Both predictions follow from the same one-dimensional heat conduction solution with identical parameters. This predictive unity — one physical model, two independent observables both confirmed — was enormously compelling evidence for plate tectonics in the 1960s–70s.

Question 4

Under the half-space cooling model, the oceanic lithosphere (thermal boundary layer) grows thicker at a rate proportional to age — so lithosphere that is 4× older is 4× thicker.

• T. True
• F. False

Answer: False

The thermal boundary layer grows as the square root of age: thickness ∝ √(κt). Lithosphere that is 4× older is only 2× thicker (√4 = 2). This square-root dependence is the universal signature of diffusive heat transport — heat spreads as √(Dt) in one dimension because diffusion slows as the temperature gradient decreases over time. Linear growth (4× older → 4× thicker) would require a constant rate of thickening, which is impossible by diffusion. Recognizing the √t signature distinguishes diffusive from advective processes and is the key to interpreting seafloor cooling observations.

Question 5

Why does the half-space cooling model predict that seafloor depth increases as the square root of crustal age, and what physical process underlies this relationship?

Model answer: As oceanic crust moves away from the mid-ocean ridge, it cools by conductive heat loss to the overlying ocean. The thermal diffusion equation governs this cooling: the depth to which cooling penetrates grows as √(κt), where κ is thermal diffusivity and t is time since crust formed. This thickening, cooler lithosphere is denser than the underlying asthenosphere, so it isostatically subsides — the denser column sinks deeper. Because the thermally controlled density excess ∝ thickness ∝ √t, the depth below the ridge also grows as √t. The square-root dependence is not specific to geophysics; it is the universal signature of one-dimensional diffusion.

Connecting the observation (bathymetric depth increasing with age) to the physical mechanism (diffusive cooling creating a dense boundary layer that sinks isostatically) is the key conceptual step. The √t scaling appears in every diffusion problem — heat conduction, mass diffusion, chemical diffusion — whenever a boundary condition is applied at one face of a semi-infinite medium. Recognizing this universality allows the geophysical result to be placed in a much broader physical context.