Questions: Thermal Time Constants and Lithospheric Cooling

Question 1

Oceanic lithosphere is ~100 km thick with κ ≈ 10⁻⁶ m²/s, giving a thermal time constant of roughly 100 million years. If the lithosphere were only 50 km thick, what would the thermal time constant be?

• A. ~50 million years — halving the thickness halves the time constant
• B. ~25 million years — the time constant scales with the square of thickness, so halving d reduces τ by a factor of four
• C. ~100 million years — thermal diffusivity, not thickness, controls the cooling rate
• D. ~200 million years — thinner lithosphere retains heat longer because there is less surface area to radiate

Answer: (B)

τ = d²/κ — the time constant scales with the *square* of thickness. Halving d reduces d² by a factor of 4. The most tempting wrong answer (option A) assumes a linear relationship, which would be true if heat simply 'flowed through' the slab at a constant velocity. But diffusion slows as it progresses: the thermal front penetrates a distance proportional to √(κt), so the time to diffuse through a given thickness d is proportional to d²/κ. This quadratic dependence is the key conceptual insight.

Question 2

Why does heat flow at the ocean surface decrease as 1/√(age) rather than decreasing linearly as the plate cools?

• A. Volcanic activity at the ridge decreases exponentially as the plate moves away
• B. The thermal boundary layer thickens as √(κt), deepening the hot interior; as the temperature gradient at the surface decreases proportionally, so does heat flow
• C. Older oceanic crust develops higher thermal conductivity through metamorphism, letting heat escape more efficiently
• D. The mantle beneath older plates cools faster because it is further from the ridge heat source

Answer: (B)

Heat flow at the surface is proportional to the temperature gradient there (Fourier's law: q = −k dT/dz). As the oceanic plate ages, the cooled thermal boundary layer grows as √(κt) — the hot mantle is progressively deeper below the surface. The shallower temperature gradient means less heat is conducted to the surface per unit time. Since boundary layer depth ∝ √t, the surface gradient ∝ 1/√t, and heat flow follows the same relationship. This is a direct consequence of the diffusion equation, not a material property of the crust.

Question 3

The thermal time constant of a rock layer depends linearly on its thickness — doubling the thickness doubles the time required for a thermal perturbation to diffuse through it.

• T. True
• F. False

Answer: False

The thermal time constant is τ = d²/κ — it scales with the *square* of thickness. Doubling the thickness quadruples the time constant, not doubles it. This counterintuitive result follows from the diffusion equation: the distance a thermal front penetrates grows as √(κt), so the time to reach depth d is t = d²/κ. Diffusion slows as it progresses — it cannot be described by a constant propagation velocity, which would give linear scaling. The quadratic dependence is why a 10,000× increase in linear scale produces a 10⁸× increase in thermal equilibration time.

Question 4

The enormous difference in cooling timescale between a 5-meter lava flow (days) and 100-km-thick lithosphere (100 Myr) is primarily due to differences in the thermal diffusivity of different rock types.

• T. True
• F. False

Answer: False

Rocks of similar composition have similar thermal diffusivity — κ ≈ 10⁻⁶ m²/s is a reasonable value for most crustal rocks. The extraordinary difference in cooling timescale is almost entirely due to the difference in thickness. The 100 km lithosphere is 2 × 10⁷× thicker than a 5-meter flow, and because τ = d²/κ, this translates to a time constant roughly (2 × 10⁷)² = 4 × 10¹⁴ times longer. Physical scale — not material properties — is the dominant control on geological thermal timescales. This is what makes τ = d²/κ such a powerful conceptual tool.

Question 5

Explain why continental collision zones remain thermally elevated and produce metamorphism and granitic melts for tens of millions of years after active shortening has ceased.

Model answer: Continental collision thickens the crust from ~35 km to ~65–70 km by stacking one crustal slab onto another. Because τ = d²/κ, doubling the thickness quadruples the thermal time constant — the thickened crust requires roughly 100–160 million years to reach a new thermal equilibrium, far longer than the shortening event itself. When shortening stops, the deep crust contains material that was rapidly buried to depths where ambient temperatures are hundreds of degrees higher than before collision. This hot, deeply buried rock takes a geological epoch to cool, and during that time elevated temperatures drive prograde metamorphism and can cause crustal melting, generating granitic magmas that intrude upward. The thermal time constant explains why the thermal and magmatic consequences of a collision outlast the collision itself by tens to hundreds of millions of years.

The same logic applies in sedimentary basins: source rocks for petroleum must spend sufficient time in the 'oil window' (~60–120°C). The thermal time constant of the basin determines whether burial history has been long enough for organic matter to mature. The τ = d²/κ formula connects the timescale of heat diffusion to the thickness of the system, providing a simple order-of-magnitude test for whether thermal equilibration has had time to occur in any geological setting.