Questions: Airy Isostasy and Crustal Thickness Variation

Question 1

Seismic imaging beneath a 2 km plateau in East Africa shows that crustal thickness is nearly uniform — the same as the surrounding lowlands. What does this imply about the mechanism of isostatic compensation?

• A. The Airy model correctly explains the plateau: uniform crust is consistent with surface elevation
• B. The plateau must have an unusually deep root that the seismic survey failed to detect
• C. Compensation is likely achieved through lateral density variations (Pratt-style), not crustal thickness
• D. Isostasy does not apply to plateaus; only mountain ranges achieve gravitational equilibrium

Answer: (C)

The Airy model predicts that elevation differences are compensated by crustal thickness — a high plateau should have a deep root. If seismic evidence shows no root, the compensating mass deficit must come from lower-density material at constant crustal thickness, which is the Pratt model's mechanism. This is a real-world situation in the East African Rift, where hot, low-density mantle material provides isostatic support without varying crustal thickness.

Question 2

In the Airy model, why does a 1 km mountain require roughly 5 km of crustal root rather than a 1:1 ratio?

• A. Because the crust is approximately 5 times denser than the mantle, so 1 km of mountain displaces 5 km of mantle
• B. Because the density contrast between crust and mantle is small relative to crustal density, so a large volume of crust must displace mantle to compensate for the added surface mass
• C. Because mountain erosion removes surface material faster than roots can grow, requiring a larger initial root
• D. Because the compensation depth is defined as exactly 5 times the surface elevation by convention

Answer: (B)

The formula r = h × ρ_c / (ρ_m − ρ_c) gives root depth. With ρ_c ≈ 2,700 kg/m³ and ρ_m ≈ 3,300 kg/m³, the density contrast is only 600 kg/m³. The root displaces dense mantle and replaces it with lighter crust, so a large volume of crustal root is needed to offset a relatively small surface elevation. Numerically: r ≈ h × 2,700 / 600 ≈ 4.5h. The ratio is determined entirely by the density contrast, not by convention.

Question 3

In the Airy model, a mountain range with twice the elevation of another has approximately twice the crustal root depth.

• T. True
• F. False

Answer: True

The Airy formula r = h × ρ_c / (ρ_m − ρ_c) is linear in surface elevation h — root depth scales directly with elevation. Doubling the surface load doubles the required subsurface mass deficit, which in the Airy model means doubling the root thickness. This linear prediction is broadly confirmed by seismic studies comparing mountain ranges of different heights.

Question 4

The Airy isostasy model explains topographic compensation through lateral variations in crustal density.

• T. True
• F. False

Answer: False

This describes the Pratt model, not Airy. The Airy model assumes uniform crustal density everywhere and explains all elevation differences through variations in crustal thickness — mountains have thick roots, basins have thin crust, but the density of the crust is the same everywhere. The Pratt model holds crustal thickness constant and varies density. Both achieve isostasy through different means; confusing them reverses the core assumption of each model.

Question 5

What physical principle requires that all vertical columns must weigh the same in Airy isostasy, and what would happen if they did not?

Model answer: The principle is equal pressure at the compensation depth: at a horizontal surface deep in the mantle, pressure must be the same everywhere. Pressure at depth equals the weight of the overlying column per unit area. If two adjacent columns had different total weights, they would exert different pressures at the compensation depth, creating a horizontal pressure gradient. Since the mantle behaves as a viscous fluid over geological timescales, this gradient would drive lateral flow from the high-pressure column toward the low-pressure one, redistributing mass until pressures equalized. Isostatic equilibrium is simply the state at which this flow has ceased.

The analogy is hydrostatics: pressure differences in a fluid drive flow until equilibrium. The mantle is solid on short timescales but flows on timescales of thousands to millions of years. Isostasy describes the equilibrium toward which this slow flow converges.