Lecture 21 — Isostasy & Lithospheric Flexure

By the end of this lecture, you should be able to…

• [LO-21.1] Apply Archimedes — compute crustal-root thickness (Airy) and column density (Pratt).
• [LO-21.2] Distinguish local (Airy/Pratt) vs. regional (flexural) compensation.
• [LO-21.3] Interpret a Bouguer profile as evidence for compensation; describe the isostatic-residual.
• [LO-21.4] Estimate post-glacial rebound timescale and connect to mantle viscosity.
• [LO-21.5] Use the flexural parameter $\alpha$ to predict forebulge wavelength.

Course objectives addressed: LO-1, LO-2, LO-3, LO-4.

A puzzle from Lecture 19

The Bouguer anomaly over a mountain range is broadly negative, not positive.

→ The mountain is not an uncompensated excess mass.

Beneath every large topographic feature there is a mass deficit at depth — a root of light material — that almost exactly balances the topographic excess.

The mountain is floating.

Archimedes for mountains

Hydrostatic equilibrium at the depth of compensation:

$$\int \rho_{\text{column,1}} \, g \, dz \;=\; \int \rho_{\text{column,2}} \, g \, dz$$

Below the depth of compensation, the asthenosphere flows freely on geological timescales.
Any pressure imbalance there drives lateral flow that restores equilibrium.

→ Two end-member ways to satisfy this: Airy and Pratt.

Airy and Pratt — two end-member compensation styles

• Airy: same density, variable thickness. High mountain → deep root.
• Pratt: same depth, variable density. High mountain → low-density column.

Both satisfy hydrostatic equilibrium but make different deep-Earth predictions.

Airy — the root formula

$$\rho_{c} h = (\rho_{m} - \rho_{c}) r \quad\Rightarrow\quad r = \frac{\rho_{c}}{\rho_{m}-\rho_{c}} \, h$$

For $\rho_{c} = 2700$, $\rho_{m} = 3300$ kg m⁻³:

$$r/h = 4.5$$

A 1-km mountain → 4.5-km root.

The Tibetan Plateau ($h \approx 5$ km, Moho $\approx 70$ km) is approximately Airy.

Pratt — the density formula

$$\rho_{c} H_{\text{ref}} = \rho_{\text{block}} (H_{\text{ref}} + h) \quad\Rightarrow\quad \rho_{\text{block}} = \rho_{c} \, \frac{H_{\text{ref}}}{H_{\text{ref}} + h}$$

For $H_{\text{ref}} = 35$ km, $h = 5$ km: $\rho_{\text{block}} = 0.875\,\rho_{c} \approx 2362$ kg m⁻³ (12% lower).

Mid-ocean-ridge crests, hot spots, and oceanic plateaus are approximately Pratt.

Real Earth: a mixture

Neither model is solely correct. Earth applies both.

• Airy dominates large-scale crustal features (continents, plateaus).
• Pratt-type density variations dominate at the oceanic-continental boundary and near hot spots.

→ The choice is a scale and setting question.

The isostatic-residual anomaly

$$\Delta g_{\text{iso}} = \Delta g_{\text{Bouguer}} - \Delta g_{\text{compensation prediction}}$$

• Zero residual → load is fully compensated.
• Positive residual → uncompensated excess mass; load supported elastically (e.g. recent volcano).
• Negative residual → uncompensated deficit; load not yet equilibrated (e.g. active glacier).

Transient residuals are diagnostic of active geodynamic processes.

Time-dependent compensation — post-glacial rebound

Ice sheet removed → lithosphere rebounds toward equilibrium.
Rate set by mantle viscosity:

$$\tau \sim \frac{4\pi\eta}{\rho_{m} g L}$$

For Fennoscandia ($L \sim 1000$ km, $\eta \sim 10^{21}$ Pa s): $\tau \approx 4\!-\!5$ ka — uplift continues at ~9 mm/yr today.

Inverse problem: get viscosity from rebound

Two pieces of data:

• Total post-glacial uplift (Fennoscandia: $\sim 300$ m).
• Present-day uplift rate ($\sim 9$ mm/yr).

→ Ratio gives $\tau$ → equation gives $\eta$.

This single inverse problem has driven mantle-viscosity research for 50 years (Whitehouse 2018 review, CC BY 4.0, https://doi.org/10.5194/esurf-6-401-2018).

Beyond local compensation — flexure

Real lithosphere has elastic strength. A vertical load at one point produces regional deformation that extends 100s of km.

The thin-plate equation:

$$D \, \frac{d^{4}w}{dx^{4}} \;+\; (\rho_{m}-\rho_{w}) \, g \, w \;=\; q(x)$$

with flexural rigidity $D = E T_{e}^{3} / [12(1-\nu^{2})]$
and flexural parameter $\alpha = \bigl[ 4D / (\rho_{m}\!-\!\rho_{w}) g \bigr]^{1/4}$.

What flexure looks like

Load → central depression + forebulge at $x \approx \pi\alpha$ + zero crossing.
Increasing $T_{e}$ widens and reduces the deflection.
Inverting the observed forebulge geometry → estimates $T_{e}$.

When does each model apply?

A continental craton has $T_{e} \approx 80$ km; an oceanic spreading centre has $T_{e}$ near zero.

Worked example — Mount Olympus (PNW)

Olympic Mountains: $h \approx 2.4$ km. Treat as Airy-compensated.

Predicted root: $r = (\rho_{c}/(\rho_{m}-\rho_{c}))\, h = 4.5 \times 2.4 = 10.8$ km.
Predicted Bouguer (slab approx of root):

$$\Delta g_{\text{root}} \approx -2\pi G (\rho_{m}-\rho_{c}) r \approx -272 \text{ mGal}$$

Free-air anomaly should remain small ($<50$ mGal) — root and topography roughly cancel.

→ The Olympics sit on the Cascadia accretionary prism — active loading. The isostatic residual is the signal of that active process.

Course connections

• Backward: L19 (Bouguer construction) and L20 (regional vs residual).
• Forward: L22 — rheological structure of the lithosphere; viscoelasticity.
• Forward: Module 7 — dynamic topography, vertical motion driven by mantle convection.
• Cross-link: present-day GIA is a key correction to GPS-measured vertical land motion → tide gauges → sea-level reconstruction.

Research horizon

• Whitehouse (2018) — GIA modelling review (CC BY 4.0). Key open question: depth profile of mantle viscosity.
• Forte & Rowley (2022) — much of long-wavelength Bouguer / geoid is dynamic topography, not local isostasy.
• Antarctic / Greenland mass balance — disentangling present-day ice loss from underlying GIA is the largest single uncertainty in 21st-century sea-level projections.

Societal — Cascadia subsidence and the tide gauge

PNW tide gauges show slower sea-level rise than the global mean.

This is not measurement error — it is inter-seismic uplift from Cascadia locking.

In the next M9 megathrust event:

• Coastal Washington will drop 0.5–2 m in minutes.
• Tide-gauge baseline resets.
• Locally, "sea level" is redefined.

→ Vertical land motion (this lecture's mechanics, on a transient timescale) directly informs Cascadia hazard.

USGS Cascadia resource (public domain): https://www.usgs.gov/programs/earthquake-hazards/science/cascadia-subduction-zone

AI Literacy — Prompt Lab

Three prompts that test whether AI handles this lecture's reasoning.

P1. Compare Airy and Pratt for the Tibetan Plateau. Which fits observations and why?
→ Good: derives both, computes amplitudes, notes empirical mixture. Poor: qualitative-only.

P2. Estimate mantle viscosity from Fennoscandian uplift (9 mm/yr; total 300 m).
→ Good: $\tau \approx 33$ ka → $\eta \approx 10^{21}$ Pa s. Common error: confuses rebound amplitude with load amplitude.

P3. Why do oceanic seamounts produce forebulges but individual continental peaks rarely do?
→ Good: invokes $T_{e}$ contrast and notes that continental peaks usually sit within larger compensating structures. Either side alone is incomplete.

Concept Check

1. Two ranges of identical surface elevation produce Bouguer anomalies of $-100$ mGal and $-200$ mGal. Which is more nearly locally compensated, and what does the difference suggest about lithospheric strength?
2. A continental margin shows a forebulge 250 km offshore of a deltaic load. With $\rho_{m}-\rho_{w} = 2270$ kg m⁻³, $E = 70$ GPa, $\nu = 0.25$ — estimate $T_{e}$.
3. Fennoscandia: $\sim 9$ mm yr⁻¹ uplift, total $\sim 300$ m post-glacial. Estimate $\tau$ and $\eta$. Are they in the canonical range?