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Tectonics

A comprehensive graduate-level course on the physics and dynamics of Earth's lithosphere. From plate kinematics described by Euler's theorem to mantle convection driven by the Rayleigh number, from seismic wave equations to lithospheric flexure — explore the quantitative foundations of how our planet reshapes itself.

Course Overview

Mathematical Framework

Rigorous treatment of plate kinematics via Euler's rotation theorem, continuum mechanics of the lithosphere, thermal diffusion and convection equations, elastic and viscous rheology, seismic wave theory, and gravitational potential. Every concept is grounded in the governing partial differential equations.

Computational Geophysics

Interactive Python and Fortran simulations that run directly in your browser: finite-difference thermal convection, seismic wave propagation, lithospheric flexure, Euler pole velocity computations, Gutenberg–Richter analysis, and 2D mantle flow solvers.

Observational Evidence

Paleomagnetic data, GPS geodesy, seismic tomography, heat flow measurements, gravity anomalies, ocean floor bathymetry, and volcanic geochemistry — the full arsenal of data that constrains our models of Earth's dynamics.

Tectonic Processes

Subduction dynamics, mid-ocean ridge processes, continental collision and orogeny, mantle plume interactions, rift-to-drift transitions, transform fault mechanics, and the coupling between surface tectonics and deep mantle flow.

Fundamental Equations of Tectonics

Conservation of Momentum (Stokes Flow)

The mantle flows at extremely low Reynolds numbers (Re ~ 10⁻²⁰), so inertial terms vanish. The Stokes equation governs creeping viscous flow in the mantle:

\[-\nabla P + \nabla \cdot \left[\eta\left(\nabla \mathbf{v} + (\nabla \mathbf{v})^T\right)\right] + \rho \mathbf{g} = 0\]

where P is dynamic pressure, η is dynamic viscosity (10²⁰–10²³ Pa·s for the mantle), v is velocity, ρ is density, and g is gravitational acceleration.

Heat Equation (Thermal Evolution)

The temperature field in the Earth's interior evolves according to the advection-diffusion equation with internal heat generation from radioactive decay:

\[\rho C_p \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T\right) = k\nabla^2 T + H\]

where T is temperature, C_p is specific heat capacity (~1250 J/kg·K), k is thermal conductivity (~3.3 W/m·K), κ = k/(ρC_p) ~ 10⁻⁶ m²/s is thermal diffusivity, and H is radiogenic heat production (~5 × 10⁻¹² W/kg for the upper mantle).

Euler's Rotation Theorem (Plate Kinematics)

Any displacement of a rigid body on a sphere can be described as a rotation about a fixed axis (the Euler pole). The velocity of a point on a plate at position r is:

\[\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}\]

where ω = (ω_x, ω_y, ω_z) is the angular velocity vector. For a pole at latitude Φ, longitude Λ with angular speed |ω|, the surface velocity magnitude varies as v = |ω|R sin(Δ), where Δ is the angular distance from the Euler pole and R is Earth's radius (6371 km).

Rayleigh Number (Convective Instability)

The Rayleigh number determines whether convection occurs. For the Earth's mantle, Ra >> Ra_cr ~ 10³, confirming vigorous convection:

\[\text{Ra} = \frac{\alpha \rho g \Delta T d^3}{\kappa \eta}\]

where α ~ 2 × 10⁻⁵ K⁻¹ is thermal expansivity, ΔT ~ 2500 K is the temperature drop across the mantle, d ~ 2890 km is mantle depth, κ ~ 10⁻⁶ m²/s is thermal diffusivity, and η ~ 10²¹ Pa·s is viscosity. For the whole mantle, Ra ~ 10⁷.

Half-Space Cooling Model

The temperature structure of oceanic lithosphere created at a mid-ocean ridge and cooling by conduction as it moves away:

\[T(z,t) = T_s + (T_m - T_s)\,\text{erf}\!\left(\frac{z}{2\sqrt{\kappa t}}\right)\]

where T_s ~ 0°C is surface temperature, T_m ~ 1350°C is mantle temperature, z is depth, t is plate age, and erf is the error function. The thermal boundary layer (lithosphere) thickness grows as z_L ~ 2.32√(κt), predicting ocean depth: d(t) = d_r + 2ρ_mα(T_m−T_s)√(κt/π)/(ρ_m−ρ_w).

Lithospheric Flexure

The elastic lithosphere bends under applied loads (volcanic islands, sediment, subducting slabs). Modeled as a thin elastic plate on a fluid (asthenosphere) foundation:

\[D\frac{d^4 w}{dx^4} + (\rho_m - \rho_{fill})g\,w = q(x)\]

where D = Eh³/[12(1−ν²)] is flexural rigidity (E ~ 70 GPa Young's modulus, ν ~ 0.25 Poisson's ratio, h is elastic thickness), w(x) is deflection, ρ_m is mantle density, ρ_fill is infill density, and q(x) is the applied load. The flexural parameter α = [4D/((ρ_m−ρ_fill)g)]^(1/4) controls the wavelength of bending (~100–250 km for oceanic lithosphere).

Seismic Wave Equation

Elastic wave propagation in the Earth is governed by the equation of motion for an isotropic elastic medium:

\[\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} = (\lambda + 2\mu)\nabla(\nabla \cdot \mathbf{u}) - \mu\nabla \times (\nabla \times \mathbf{u})\]

This separates into P-waves (compressional, v_P = √[(λ+2μ)/ρ] ~ 5–14 km/s) and S-waves (shear, v_S = √[μ/ρ] ~ 3–7 km/s). The ratio v_P/v_S = √[(λ+2μ)/μ] ≈ √3 for a Poisson solid. Rayleigh and Love surface waves also emerge from the boundary conditions.

Gutenberg–Richter Frequency–Magnitude Relation

The statistical distribution of earthquake magnitudes follows a power law over many orders of magnitude:

\[\log_{10} N(\geq M) = a - bM\]

where N is the cumulative number of earthquakes with magnitude ≥ M, a describes productivity (total seismicity rate), and b ~ 1.0 (typically 0.8–1.2) describes the relative abundance of large vs. small events. A b-value of 1 means that for each unit increase in magnitude, earthquakes become 10× less frequent. This is equivalent to a Pareto distribution in seismic moment: P(M₀ > x) ∝ x^(−2b/3).

Byerlee's Friction Law

The shear stress required for frictional sliding on a fault surface depends on normal stress:

\[\tau = \begin{cases} 0.85\,\sigma_n & \text{for } \sigma_n < 200\text{ MPa} \\ 50 + 0.6\,\sigma_n & \text{for } \sigma_n > 200\text{ MPa} \end{cases}\]

This empirical law, derived from laboratory experiments by Byerlee (1978), is remarkably independent of rock type. It sets the strength envelope for the brittle upper crust and defines the brittle-ductile transition depth where frictional strength equals viscous flow strength.

Isostasy (Airy & Pratt Models)

Gravitational equilibrium requires equal pressure at the compensation depth. Airy isostasy assumes constant density with varying crustal thickness; Pratt assumes constant thickness with varying density:

\[\textbf{Airy: } \rho_c h + \rho_m r = \rho_c H + \rho_m R \qquad \textbf{Pratt: } \rho_c(h+r) = \rho_0 D\]

For Airy: h is elevation, r is root thickness, H and R are reference values. The root r = hρ_c/(ρ_m − ρ_c), typically giving ~7× the topographic elevation for continental crust (ρ_c ~ 2700 kg/m³, ρ_m ~ 3300 kg/m³). For Pratt: ρ_0 varies laterally, D is the compensation depth.

Omori's Law (Aftershock Decay)

The rate of aftershocks following a mainshock decays as a power law in time:

\[n(t) = \frac{K}{(t + c)^p}\]

where n(t) is the rate of aftershocks at time t after the mainshock, K is a productivity parameter proportional to mainshock magnitude, c is a small time offset (~0.01–0.1 days) regularizing the early-time divergence, and p ~ 1.0–1.2 is the decay exponent (modified Omori law).

Constitutive Law for Mantle Creep

The mantle deforms by thermally-activated dislocation creep and diffusion creep. The strain rate depends exponentially on temperature:

\[\dot{\varepsilon} = A\,\sigma^n\,d^{-m}\exp\!\left(-\frac{E^* + PV^*}{RT}\right)\]

where A is a pre-exponential factor, σ is differential stress, n is the stress exponent (n = 1 for diffusion creep, n ~ 3.5 for dislocation creep), d is grain size, m is the grain-size exponent, E* is activation energy (~500 kJ/mol for olivine), V* is activation volume, P is pressure, R is the gas constant, and T is absolute temperature.

Navier–Stokes with Boussinesq Approximation

For mantle convection modeling, density variations are small enough to apply the Boussinesq approximation: density is constant everywhere except in the buoyancy term:

\[\nabla \cdot \mathbf{v} = 0, \qquad -\nabla P + \eta \nabla^2 \mathbf{v} + \rho_0 \alpha (T - T_0)\,g\,\hat{\mathbf{z}} = 0\]

The first equation enforces incompressibility. The second balances pressure gradients, viscous stress, and thermal buoyancy. Non-dimensionalized, the single control parameter is the Rayleigh number.

Course Structure

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Part 1: Introduction to Plate Tectonics

The historical development from Wegener's continental drift hypothesis through Hess's seafloor spreading to the plate tectonic paradigm. Paleomagnetic evidence, magnetic anomaly stripes, the Vine–Matthews–Morley hypothesis, and the geometry of tectonic plates.

  • • Continental drift evidence: fossil correlations, geological matching, paleoclimatology
  • • Seafloor spreading: magnetic reversals, ocean floor age patterns
  • • Modern plate boundaries: 15 major plates, GPS velocity fields
  • • Wilson Cycle: ocean basin opening and closing
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Part 2: Earth's Interior & Rheology

The layered structure of the Earth revealed by seismology: crust, lithospheric mantle, asthenosphere, transition zone, lower mantle, outer core, inner core. Elastic and viscous behavior at different timescales, and the concept of effective viscosity.

  • • P-wave and S-wave velocity profiles (PREM model)
  • • Mohorovičić, Gutenberg, and Lehmann discontinuities
  • • Maxwell relaxation time: τ = η/μ (~10³ years for the mantle)
  • • Postglacial rebound as a viscosity constraint: η ~ 10²¹ Pa·s
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Part 3: Plate Kinematics

The mathematical description of plate motions on a spherical Earth using Euler's theorem. Angular velocity vectors, relative and absolute plate motions, hotspot reference frames, triple junction stability, and the construction of plate circuits.

  • • Euler pole determination from transform fault azimuths and spreading rates
  • • NUVEL-1A and MORVEL global plate motion models
  • • Triple junction classification (McKenzie & Morgan, 1969)
  • • Finite rotations and plate reconstruction stage poles
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Part 4: Divergent Boundaries & Rifting

The physics of lithospheric extension and mid-ocean ridge processes. Passive vs active rifting, decompression melting, magma chamber dynamics, hydrothermal circulation, and the transition from continental rift to seafloor spreading.

  • • Ridge morphology: fast vs slow spreading (EPR vs MAR)
  • • Decompression melting: solidus crossing at ~60 km depth
  • • Hydrothermal heat flux: ~30% of ridge-axis heat loss
  • • East African Rift: a modern continental rift example
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Part 5: Convergent Boundaries & Subduction

The mechanics of subduction zones — where cold, dense oceanic lithosphere sinks into the mantle. Slab dynamics, Wadati–Benioff zones, arc magmatism driven by slab dehydration, accretionary prism mechanics, and the role of subduction in driving plate motions.

  • • Slab pull: the dominant plate driving force (~3.3 × 10¹³ N/m)
  • • Thermal structure: T(z) along the slab surface and dehydration reactions
  • • Critical taper theory for accretionary wedges: α + β = f(μ_b, λ)
  • • Flat slab subduction and its effects (Andes, Laramide)
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Part 6: Transform Boundaries

The kinematics and mechanics of transform faults: conservative plate boundaries where lithosphere is neither created nor destroyed. Oceanic transforms offsetting ridge segments, continental transforms like the San Andreas, and transpressional/transtensional deformation.

  • • Wilson's insight: transforms differ fundamentally from transcurrent faults
  • • Stress accumulation and earthquake cycle on locked faults
  • • Fault creep, aseismic slip, and slow slip events
  • • Dead Sea Transform, Alpine Fault (NZ), North Anatolian Fault
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Part 7: Mantle Convection & Driving Forces

Thermal convection as the engine of plate tectonics. Rayleigh–Bénard instability, boundary layer theory, the relative contributions of ridge push, slab pull, and basal drag, mantle plume dynamics, and constraints from seismic tomography and geoid data.

  • • Force balance: slab pull (~3.3 × 10¹³ N/m) vs ridge push (~2.5 × 10¹² N/m)
  • • Whole-mantle vs layered convection debate
  • • Mantle plumes: Hawaii, Iceland, Yellowstone — deep vs shallow origin
  • • Dynamic topography: ±2 km from mantle flow pressure
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Part 8: Earthquakes & Seismology

The physics of earthquake generation and wave propagation. Reid's elastic rebound theory, seismic moment and moment magnitude, focal mechanisms and beach balls, earthquake statistics, and seismological imaging of Earth's interior structure.

  • • Seismic moment: M₀ = μAD̄ where A is rupture area, D̄ is average slip
  • • Moment magnitude: M_w = (2/3)log₁₀(M₀) − 6.07
  • • Double-couple source mechanism and focal mechanism solutions
  • • Seismic tomography: velocity anomalies reveal slabs and plumes
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Part 9: Volcanism & Magmatism

The generation, transport, and eruption of magma in tectonic settings. Decompression melting at ridges, flux melting at subduction zones, mantle plume partial melting, magma ascent dynamics, eruption styles, and the link between volcanism and plate motions.

  • • Clausius–Clapeyron: melting point decrease with decreasing pressure
  • • Fractional crystallization and Bowen's reaction series
  • • Explosive volcanism: magma viscosity and volatile content control
  • • Large Igneous Provinces: Deccan Traps, Siberian Traps, CAMP
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Part 10: Orogeny & Mountain Building

The mechanics of continental collision and mountain building. Lithospheric flexure under orogenic loads, isostatic compensation, thrust and fold belt mechanics, crustal thickening, and the interplay between tectonics and surface processes (erosion, sedimentation).

  • • Himalayan-Tibetan collision: double crustal thickness, ~70 km
  • • Foreland basin formation: flexural subsidence from orogenic loading
  • • Critical Coulomb wedge theory: α + β governed by friction and pore pressure
  • • Orogenic plateau formation: gravitational potential energy vs tectonic force

Key Geophysical Parameters

ParameterSymbolValueContext
Earth radiusR6371 kmMean radius
Mantle viscosityη10²⁰–10²³ Pa·sUpper mantle to lower mantle
Thermal diffusivityκ10⁻⁶ m²/sLithospheric mantle
Thermal expansivityα2 × 10⁻⁵ K⁻¹Upper mantle olivine
Mantle densityρ_m3300 kg/m³Upper mantle average
Continental crust densityρ_c2700 kg/m³Granitic average
Mantle temperature dropΔT~2500 KCMB to surface
Slab pull forceF_sp~3.3 × 10¹³ N/mPer unit trench length
Ridge push forceF_rp~2.5 × 10¹² N/mPer unit ridge length
Plate velocity (typical)v2–15 cm/yrGPS-measured, Pacific fastest
Elastic thicknessT_e5–100 kmOceanic (thin) to cratonic (thick)

Interactive Tectonic Simulations

Run Python and Fortran geophysical models directly in your browser: thermal convection, lithospheric flexure, seismic wave propagation, Euler pole calculations, earthquake statistics, and more. Edit the code, adjust parameters, and visualize results instantly.

Launch Tectonic Programs

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