Gravity & the Geoid
Gravity & the Geoid
Earth's gravitational field deviates from a perfect sphere due to rotation, topography, and internal density variations. The geoid — the equipotential surface coinciding with mean sea level — is the reference surface for heights and ocean circulation.
Definition
Gravity anomaly \(\Delta g = g_{obs} - g_{ref}\) where \(g_{ref}\) is computed for a reference ellipsoid. Bouguer anomaly corrects for topographic mass; free-air anomaly corrects for elevation only.
Key Result
Isostasy: large topographic loads are compensated by buoyancy. Airy isostasy assumes uniform density with variable crustal thickness; Pratt isostasy assumes variable density at uniform depth. Both predict near-zero long-wavelength Bouguer anomalies.
Example 1
GRACE satellites (2002–2017) measured gravity changes at monthly intervals, revealing: groundwater depletion in the Central Valley, ice loss in Greenland (−280 Gt/yr) and Antarctica (−150 Gt/yr), and post-glacial rebound in Scandinavia.
Example 2
The geoid undulates up to ±100 m from the reference ellipsoid. The deep geoid low in the Indian Ocean (−106 m) corresponds to a cold, dense slab in the mantle; the Hudson Bay low records residual ice loss since the last glaciation.
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Practice
- What is the difference between the geoid and the reference ellipsoid?
- Why do mountain ranges have negative Bouguer anomalies?
- Explain post-glacial rebound and its mathematical description.
- How are satellite-derived gravity data used to monitor sea-level change?
Show Answer Key
1. The geoid is the equipotential surface of Earth's gravity field (mean sea level extended under continents). The reference ellipsoid is a mathematical best-fit smooth shape. Geoid undulations (deviations from the ellipsoid, up to ±100 m) reflect density variations in the mantle and crust.
2. Mountains have excess topographic mass above the surface but are compensated by low-density crustal roots extending into the denser mantle (isostasy). After removing the effect of the visible topographic mass (Bouguer correction), the remaining anomaly is negative because the root's mass deficit dominates.
3. After ice sheet removal, the mantle slowly flows back and the crust rebounds (viscoelastic relaxation). Modeled as $h(t) = h_0 e^{-t/\tau}$ where $\tau = 2\eta/(\rho g \lambda)$ depends on mantle viscosity $\eta$ and wavelength $\lambda$. Current uplift rates (e.g., ~1 cm/yr in Scandinavia) constrain mantle viscosity.
4. GRACE/GRACE-FO satellites measure time-varying gravity by tracking inter-satellite distance changes. Mass redistribution (ice sheet melting, groundwater depletion) changes the geoid. Monthly gravity maps reveal ice mass loss in Greenland/Antarctica and ocean mass gain, providing independent sea-level rise estimates.