7.4 Geoid & Dynamic Topography

Gravity, Shape & Mantle Flow

The Earth is not a perfect sphere or even a perfect ellipsoid. Its gravitational field reflects the three-dimensional distribution of mass in its interior, and its surface topography is influenced not only by crustal structure and isostasy but also by the dynamic stresses exerted by the convecting mantle below. Two key observables — the geoid and dynamic topography — provide windows into mantle density structure and flow patterns that complement seismic tomography.

The geoid is the equipotential surface of the gravitational field that coincides with mean sea level over the oceans. Its undulations (deviations from the reference ellipsoid) reach ±100 m and correlate with deep mantle density anomalies. Dynamic topography is the component of surface elevation caused by mantle flow stresses pushing the surface up or pulling it down, with amplitudes of ±1–2 km. Together, these observations constrain the mantle's density and viscosity structure in ways that seismology alone cannot.

The Geoid: Earth's Gravitational Shape

The geoid is defined as the equipotential surface of gravity — the surface on which a hypothetical ocean would settle in the absence of tides, currents, and weather. Over the oceans, the geoid corresponds closely to mean sea level; over the continents, it is a mathematical construct. Deviations of the geoid from the best-fitting reference ellipsoid (the WGS84 ellipsoid) are called geoid anomalies or geoid undulations.

The geoid anomaly produced by a density anomaly within the Earth can be expressed as:

\(\Delta N = -\frac{2\pi G}{g_0} \int \Delta\rho(z) \cdot z \, dz\)

where ΔN is the geoid height anomaly, G is the gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), g₀ is the reference gravity (~9.81 m/s²), Δρ(z) is the density anomaly at depth z, and the integral is taken over the depth range of the anomaly. The factor z means that deeper anomalies contribute more to the geoid per unit mass — a counterintuitive result that arises because deeper masses affect a broader area of the surface gravitational field.

The long-wavelength geoid (spherical harmonic degrees 2–10, corresponding to wavelengths >4000 km) is dominated by deep mantle structure. Key features include:

Geoid Highs over Subduction Zones

The western Pacific and Indonesian subduction zones coincide with a broad geoid high. Cold, dense subducted slabs sitting in the lower mantle produce a positive mass anomaly that raises the geoid. The largest geoid high (~80 m) is centered on the western Pacific.

Geoid High over Africa

The African superswell is associated with a broad geoid high linked to the African LLSVP (Large Low-Shear-Velocity Province) in the deep mantle. Despite being a low-density (thermally buoyant) structure, the LLSVP produces a geoid high because the associated dynamic topography on the CMB and surface partially compensates the negative density anomaly, and the deep location amplifies the gravitational effect.

Indian Ocean Geoid Low

The deepest geoid low (−106 m) lies south of India. Its origin remains debated but may involve a combination of remnant cold material from ancient subduction and the specific mantle viscosity structure in this region. This is the largest geoid anomaly on Earth.

Dynamic Topography

Dynamic topography is the deflection of Earth's surface caused by vertical stresses (normal tractions) exerted by the convecting mantle on the base of the lithosphere. Where mantle flow is directed upward toward the surface, the radial stress pushes the surface up; where flow is directed downward (above a downwelling), the surface is pulled down. The dynamic topography h is related to the radial stress by:

\(h_{\text{dyn}} = \frac{\sigma_{rr}}{\Delta\rho \cdot g}\)

where σ_rr is the radial (normal) stress at the surface from mantle flow, Δρ is the density contrast between the mantle and the overlying material (air for continents, water for oceans), and g is gravitational acceleration. For a mantle flow velocity of ~1 cm/yr and viscosity of ~10²² Pa·s in the lower mantle, the resulting stresses produce dynamic topography of order ±1–2 km.

The radial stress itself can be estimated from the viscous flow solution:

\(\sigma_{rr} = \eta \frac{\partial v_r}{\partial r} \bigg|_{\text{surface}}\)

This expression shows that dynamic topography depends on both the vigor of mantle flow (through v_r) and the mantle viscosity (η). A stiffer (higher viscosity) lower mantle produces larger radial stresses at the surface for the same flow velocity.

Positive Dynamic Topography

Regions above mantle upwellings experience surface uplift. The Southern African Plateau (~1 km above sea level, but with thin, low-density crust) is a prime example: the African LLSVP beneath generates an upward flow that dynamically supports the high elevation. Iceland and the East African Rift are other regions with significant positive dynamic topography.

Negative Dynamic Topography

Regions above downwellings (subduction zones, cold mantle anomalies) are dynamically depressed. Parts of eastern North America and the interior of Australia are ~200–500 m lower than predicted by isostasy alone, attributed to mantle downwellings. The anomalous depth of certain old ocean basins also reflects dynamic depression.

Residual Topography

In practice, dynamic topography is estimated by computing the residual topography: the observed surface elevation minus the elevation predicted from isostatic (crustal) equilibrium. If the lithosphere were in perfect isostatic equilibrium with no mantle flow effects, the observed topography would match the isostatic prediction exactly. Any systematic departure is attributed to dynamic support from below.

For oceanic lithosphere, the procedure involves subtracting the predicted depth from a thermal plate model (which accounts for lithospheric cooling with age) from the observed bathymetry. For continental regions, crustal thickness and density from seismological models are used to compute the isostatic prediction. The resulting residual topography maps reveal:

Southern African Superswell

Southern Africa sits ~1 km above the elevation predicted by its crustal structure alone. This ~1000 m anomaly is the largest coherent residual topography signal on the continents and is attributed to dynamic uplift from the African LLSVP, a thermochemical structure extending ~1000 km above the CMB beneath southern Africa. The LLSVP may represent primordial material enriched in iron, which is intrinsically dense but thermally buoyant.

Western Pacific Superswell

The seafloor in parts of the western Pacific (French Polynesia) is 500–1000 m shallower than predicted for its age, forming the “Pacific superswell.” This is associated with the Pacific LLSVP and numerous hotspot volcanoes in the region (Tahiti, Samoa, Marquesas, Cook-Austral).

The amplitude and pattern of residual topography provide constraints on mantle viscosity structure, since the dynamic topography produced by a given density anomaly depends sensitively on the viscosity profile. Models with a factor ~30–100 viscosity increase from upper to lower mantle best reproduce the observed residual topography patterns.

Free-Air & Bouguer Gravity Anomalies

Gravity anomalies at tectonic features provide complementary constraints on subsurface density structure. Two principal types of gravity anomaly are used:

Free-Air Anomaly (FAA)

The free-air anomaly corrects the observed gravity only for the elevation of the observation point above the reference ellipsoid. It does not correct for the mass of topography. Over isostatically compensated features, the FAA is near zero because the topographic mass is compensated by a density deficit (crustal root) at depth.

Large positive FAA values over ocean trenches and their flanks indicate that these features are not in isostatic equilibrium — they are dynamically maintained by the strength of the subducting slab bending into the trench.

Bouguer Anomaly (BA)

The Bouguer anomaly corrects both for elevation and for the gravitational attraction of the topographic mass between the observation point and the reference level. Over mountains, the BA is strongly negative because the Bouguer correction removes the effect of the visible topography, revealing the mass deficit of the deep crustal root.

The BA beneath the Himalayas reaches −500 mGal, consistent with a crustal root extending to ~70 km depth as predicted by Airy isostasy.

Geoid-to-Topography Ratio & Satellite Missions

The geoid-to-topography ratio (GTR) is defined as the ratio of geoid anomaly to topographic anomaly at a given wavelength. This ratio is diagnostic of the depth of isostatic compensation:

\(\text{GTR} = \frac{\Delta N}{h} \approx -\frac{2\pi G \, \Delta\rho \, D}{g_0}\)

where D is the compensation depth and Δρ is the density contrast at the compensation level. Shallow compensation (e.g., Airy isostasy with a crustal root at ~30–40 km) produces a small GTR (~2–6 m/km), while deep compensation or dynamic support from the deep mantle produces a larger GTR (~6–10+ m/km). Measuring the GTR at different wavelengths reveals the depth distribution of density anomalies supporting the topography.

Modern satellite gravity missions have revolutionized our ability to measure the geoid and its relationship to topography:

GRACE (2002–2017) & GRACE-FO (2018–)

The Gravity Recovery and Climate Experiment measures temporal changes in Earth's gravity field by precisely tracking the distance between two co-orbiting satellites. While primarily designed for monitoring ice mass loss, ocean circulation, and water storage, GRACE provides the time-averaged static gravity field to spherical harmonic degree ~120 and has confirmed mantle density models used in geodynamic simulations.

GOCE (2009–2013)

The Gravity field and steady-state Ocean Circulation Explorer used a gradiometer to measure the gravity gradient tensor (second derivatives of the gravitational potential). GOCE resolved the static geoid to an accuracy of ~1 cm at a spatial resolution of ~80 km (spherical harmonic degree ~250), enabling detailed studies of lithospheric structure, crustal thickness variations, and the transition from isostatic to dynamic compensation.

The combination of high-resolution gravity data from GOCE and GRACE with seismic tomography and topographic/bathymetric data provides a powerful toolkit for constraining mantle convection models. Joint inversions of these datasets yield self-consistent models of mantle density, viscosity, and flow that predict plate velocities, geoid anomalies, and dynamic topography simultaneously.

Key Concepts Summary

±100 m

Geoid undulation range

±1–2 km

Dynamic topography amplitude

~1 km

African superswell dynamic uplift

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