Question 7 of 7: Airy and Pratt Isostasy Models

Question 7: Airy and Pratt Isostasy Models (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Isostasy in general

Isostasy is the gravitational equilibrium of the rigid, buoyant lithosphere "floating" on the denser, ductile asthenosphere, analogous to an iceberg floating in water: at some depth of compensation, the weight (pressure) of overlying rock must be equal in every vertical column, regardless of what combination of thickness and density makes up the column above it. Two classical end-member models explain how the crust achieves this equal-pressure condition while producing observed topography, and both matter directly to sedimentation because whichever mechanism compensates a given load also governs the vertical motion (subsidence or uplift) of neighbouring areas, and hence how much accommodation space is created for sediment to fill.

Airy model — constant density, variable thickness (root)

The Airy model assumes the crust has uniform density everywhere and that topography is compensated by variable crustal thickness: a topographically high area (mountain range) is underlain by a thick, low-density crustal root projecting down into the denser mantle, exactly like a tall iceberg having a deep keel. Local (Airy) compensation of a topographic load of height h, with crustal density ρc and mantle density ρm, requires a root of depth:

$$r = h\,\dfrac{\rho_c}{\rho_m - \rho_c}$$

For an illustrative 2 km topographic load with ρc = 2800 kg/m³ and ρm = 3300 kg/m³:

$$r = 2000\ \text{m} \times \dfrac{2800}{3300-2800} = \boxed{11{,}200\ \text{m} \approx 11.2\ \text{km}}$$

— the root (11.2 km) is more than five times deeper than the load stands high, which is the Airy model's characteristic signature: mountain ranges have deep crustal roots (confirmed seismically beneath most orogenic belts), and, conversely, a thinned crust (rifted margin, ocean basin) stands topographically low.

Pratt model — constant depth of compensation, variable density

The Pratt model instead assumes all crustal columns extend down to a common depth of compensation and that topography is compensated entirely by lateral variation in column density: a topographically high column is simply made of less dense material than a low-standing column, with no crustal root at all. Equal pressure at the depth of compensation D requires, for a reference (zero-elevation) column of density ρ0 and an elevated column of height h and density ρblock:

$$\rho_{block} = \rho_0\,\dfrac{D}{D+h}$$

For the same 2 km elevation, a compensation depth D = 100 km and ρ0 = 3300 kg/m³:

$$\rho_{block} = 3300\ \text{kg/m}^3 \times \dfrac{100{,}000}{100{,}000+2000} = \boxed{3235.3\ \text{kg/m}^3}$$

— a modest ~2% density reduction relative to the reference column is enough to support 2 km of extra elevation with no root at all, illustrating why the Pratt model is favoured for broad, low-relief thermal/compositional swells (mid-ocean ridges, some continental plateaus) where a discrete crustal root is not observed.

Role in sedimentation accommodation space

Both models predict that removing or adding load changes a column's equilibrium elevation, which is precisely how isostasy generates and destroys accommodation space for sediment: progressive sediment/water loading of a subsiding basin (itself often initiated by crustal thinning or thermal contraction, Airy-style) pushes the loaded column further down, creating additional accommodation beyond the basin's initial tectonic subsidence (a positive feedback that lets thick sedimentary basins accumulate far more sediment than the initial subsidence alone would allow); conversely, unloading by erosion of a mountain belt or by deglaciation causes isostatic rebound (uplift), reducing accommodation and driving regression, exactly the postglacial rebound invoked in Question 2(5)'s quick-clay origin. Distinguishing which model (or, more realistically, what combination of both) applies in a given setting is therefore directly relevant to predicting subsidence history and basin-fill geometry, not merely an academic distinction in the origin of topography.