Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2015 / Paper 1 / 3

Seismic surface-wave dispersion. Rayleigh waves combine vertical and longitudinal motion in the vertical plane of propagation; Love waves have horizontal transverse motion and can be guided by a low-shear-speed surface layer. A homogeneous elastic half-space supports a Rayleigh wave with a frequency-independent speed in the ideal elastic limit, but depth-dependent Earth structure makes the surface-wave eigenfunctions and their propagation speeds depend on frequency. This structural dispersion can occur even when each constituent material is itself nondispersive.

Distinguish phase velocity $c=\omega /k$ from group velocity $U=d\omega /dk$, which controls a narrow-band envelope in the appropriate weakly attenuating limit. Longer periods generally have deeper sensitivity, while shorter periods emphasize shallower structure. In a profile whose shear speed increases with depth this often makes long-period waves faster, but low-velocity zones, anisotropy and higher modes can change that trend. Frequency dependence need not be monotonic in every Earth model.

Measure phase delays or wave-packet arrival times across known source–receiver paths for many periods, accounting for source phase, cycle ambiguity, multipathing and mode identification. Compute predicted dispersion from trial elastic profiles and compare with observations. Linearization expresses the sensitivity schematically aswhere $\alpha ,\beta$ are compressional and shear speeds. A regularized inverse problem then estimates depth structure; many crossing paths additionally provide surface-wave tomography. Joint Love wave and Rayleigh wave measurements help distinguish shear-wave polarizations and constrain radial seismic anisotropy. Finite bandwidth, depth averaging and correlated sensitivities mean that one dispersion curve does not uniquely determine a detailed Earth profile.

Seismic ray theory, the eikonal equation and Fermat's principle. In a smoothly varying isotropic medium, insert a high-frequency ansatz $u=A(\mathbf{x})e^{i\omega [T(\mathbf{x})-t]}$ into the scalar wave equation. The leading powers of frequency giveThe travel-time surfaces are wavefronts and rays are their normal trajectories, with slowness vector $\mathbf{p}=\nabla T$. The next-order equation transports amplitude, including geometrical spreading; attenuation is an additional material effect.

Fermat's principle says that the ray makes $\int ds/c$ stationary under path variations with fixed endpoints. Stationarity, rather than always the global minimum, permits multiple arrivals. Its Euler–Lagrange equation, with arclength $s$ and unit tangent $\hat{\mathbf{t}}$, isFor horizontal layering, the horizontal slowness $p_h=\sin i/c$ is conserved, giving Snell's law at interfaces and smooth refraction between them. Here $i$ is measured from vertical. A turning point satisfies $p_{h}c(z)=1$; increasing speed with depth bends a downgoing ray back upwards. Travel-time tomography uses these ray integrals to infer spatial slowness variations.

The approximation requires wavelength to be small compared with structural variation scales. Caustics, turning-point amplitudes, diffraction, scattering and finite-frequency sensitivity require wave-based corrections. Thus seismic ray theory gives an economical geometric description and useful travel times, while surface-wave dispersion retains essential interference and boundary-mode physics.

Homologous temperature and deformation. Homologous temperature is $\Theta =T/T_m$, with both temperatures in kelvin and $T_m$ the appropriate pressure- and composition-dependent melting temperature. It measures proximity to melting more meaningfully than an absolute temperature alone. Pressure, composition and water can alter both $T_m$ and the flow law, so equal $\Theta$ does not imply identical rheology.

A useful thermally activated material creep law iswith stress $\sigma$, grain size $d$, activation energy $E^{\ast }$ and activation volume $V^{\ast }$. Water activity can add another factor. Increasing temperature raises the rate at fixed stress when the activation enthalpy is positive, greatly lowering effective viscosity. Solid material can therefore flow on geological timescales well below melting.

Diffusion creep is approximately linear in stress: bulk-diffusion Nabarro-Herring creep has grain-size scaling $d^{-2}$, while grain-boundary Coble creep has $d^{-3}$ in their ideal regimes. Dislocation creep usually has a stress exponent greater than one and weaker grain-size dependence. Its effective viscosity scales as ${\eta }_{\mathrm{eff}}\propto {\sigma }^{1-n}$ at fixed temperature, so stronger forcing changes the apparent viscosity. At lower homologous temperature, brittle/frictional deformation or high-stress dislocation mechanisms can become important. The dominant process also depends on pressure and confinement; low temperature is not by itself a guarantee of brittle failure deep in a planet.

A deformation mechanism map compares the competing rates or strengths as functions of stress and homologous temperature at specified grain size, pressure and composition. Fine grains favour diffusion creep; larger stress favours nonlinear dislocation creep. There is no universal temperature dividing all these regimes. This sensitivity helps explain a strong outer lithosphere above a slowly flowing solid mantle and why modest thermal or water variations can reorganize deformation.

Determining the actual planform of mantle convection. The task is to infer the geometry of upwellings, downwellings and connecting flow in the present Earth, rather than to draw a preferred laboratory convection cell. Plate velocities and the distribution of subduction zones directly constrain surface motion and likely cold downwellings. Ridges, volcanism and hotspots provide further clues, but a ridge need not mark the centre of a deep plume, nor is every hotspot uniquely explained by one plume geometry.

Seismic tomography supplies three-dimensional velocity structure from body-wave travel times, waveforms and surface-wave dispersion. Fast slab-like structures can often be traced beneath subduction zones into the lower mantle. The 1997 global P-wave study reported long narrow lower-mantle structures connected with convergence and interpreted some slab segments as reaching the deepest mantle; that is evidence for deep circulation, not proof that every slab follows one route.^[ref] Temperature affects seismic speed, but composition, phase changes, water and anisotropy also matter, so velocity is not itself a flow map.

Gravity and the geoid constrain the spatially integrated density field and its induced boundary deformation. Dynamic topography supplies information about vertical stresses, after separating crustal structure, isostatic loads and thermal subsidence. Seismic anisotropy can constrain deformation and fabric orientations, subject to a mineral-physics relation between fabric and flow. Discontinuity topography and the mantle's heat budget provide further tests. Each observation has a different sensitivity to density, temperature and viscosity, making their combination more useful than any one image.

Compare these observations with three-dimensional geodynamic models that solve creeping-flow force balance, mass conservation and thermal transport, including buoyancy from temperature and composition, realistic plate motions, viscosity structure and phase boundaries. Predict seismic observables, gravity, geoid and dynamic topography from the same model, and assess data coverage and resolution before interpreting a feature. Whole-mantle exchange, impeded or deflected slabs near transition zones, and broad deep anomalies can coexist; the evidence does not require identical stacked or single-layer cells everywhere. The actual planform is constrained by joint seismic, gravitational, surface-kinematic and thermal evidence, with substantial nonuniqueness in its detailed flow and history.