Questions: Conduction Models and Thermal Equation Solutions

Question 1

A borehole in a granitic region shows a steeper geothermal gradient near the surface than at depth, even though the rock's thermal conductivity is uniform with depth. The most likely explanation is:

• A. The mantle contributes more heat near the surface than at depth
• B. Radioactive heat production concentrated in the upper crust creates a parabolic temperature profile with steeper gradients near the surface
• C. Recent climate warming has elevated surface temperatures, creating a transient anomaly near the surface
• D. Denser rock at depth has lower thermal diffusivity, causing heat to pile up near the surface

Answer: (B)

The steady-state heat equation with internal heat production (∂²T/∂z² = −A/κ) yields a parabolic temperature-depth profile. Because radiogenic elements (U, Th, K) are concentrated in granitic upper crust, heat is added throughout that upper layer — not just at the base. Each successive depth interval must conduct both the heat from below and the locally generated heat, causing the gradient to be steepest at the surface and decrease with depth. Option A is backwards; option C is a real transient effect but doesn't explain a systematic depth trend; option D confuses diffusivity with conductivity.

Question 2

According to the half-space cooling model for oceanic lithosphere, if you compare ocean floor at age T to ocean floor at age 4T, you expect the deeper seafloor (age 4T) to be deeper by a factor of:

• A. 4 — depth is proportional to plate age
• B. 2 — depth is proportional to the square root of plate age
• C. 16 — depth is proportional to the square of plate age
• D. 1.4 — depth grows only logarithmically with age

Answer: (B)

The half-space cooling model predicts that lithospheric thickness (and therefore bathymetric depth) scales as √(κt), where κ is thermal diffusivity and t is plate age. So if age quadruples (4T vs T), depth increases by √4 = 2. This square-root dependence on age is one of the model's key testable predictions, confirmed by bathymetric surveys of young oceanic crust. Option A (linear) would apply if the lithosphere cooled at a constant rate, which it does not.

Question 3

In the steady-state conduction model with no internal heat sources, the geothermal gradient (temperature change per unit depth) is the same at all depths within a layer of uniform thermal conductivity.

• T. True
• F. False

Answer: True

With no heat production and steady state, the heat diffusion equation reduces to ∂²T/∂z² = 0, which means the second derivative of temperature with respect to depth is zero. This implies a linear temperature-depth profile — the gradient is constant. The surface heat flow equals thermal conductivity times this constant gradient. Any deviation from linearity (curved profile) signals either internal heat production or a transient, time-dependent process.

Question 4

Thermal diffusivity and thermal conductivity are the same physical quantity expressed in different units.

• T. True
• F. False

Answer: False

They are distinct properties. Thermal conductivity (k, units W m⁻¹ K⁻¹) measures how readily a material conducts heat. Thermal diffusivity (κ, units m² s⁻¹) measures how quickly a temperature disturbance propagates through a material, and equals k / (ρ Cₚ), where ρ is density and Cₚ is specific heat capacity. A material can have high conductivity but low diffusivity if it also has high heat capacity per unit volume. Both appear in the heat equation, but they govern different aspects of thermal behavior.

Question 5

Explain why adding radioactive heat production to the steady-state conduction model causes the geothermal gradient to be steeper near the surface than at depth, even when thermal conductivity is uniform.

Model answer: In steady state with heat production A (W m⁻³), the 1D heat equation is k ∂²T/∂z² = −A. This means the temperature profile is parabolic, not linear. Physically, heat generated throughout the upper crust must all ultimately flow upward to the surface. The shallow depths must conduct not only the heat arriving from below, but also all the radiogenic heat produced above them. Because more total heat flux passes through shallower depths, the gradient (heat flux / conductivity) is steeper near the surface. At the base of the heat-producing layer, only the deeper heat flow (from the mantle) remains, so the gradient steps down to a lower value.

This explains why surface heat flow varies between geological provinces with similar mantle heat flux. Granitic continents generate more crustal heat (high U, Th, K content) than oceanic or mafic settings, producing higher surface heat flow and steeper upper-crustal gradients even with the same mantle input.