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Earth Sciences

A comprehensive graduate-level course spanning the full breadth of Earth system science. From the deep interior revealed by seismic waves to the atmosphere's radiative balance, from isotope geochronology to natural hazards — explore the quantitative foundations of how our planet works as an integrated system.

Course Overview

Mathematical Framework

Rigorous treatment of continuum mechanics, fluid dynamics in the mantle and oceans, seismic wave theory, heat transport, radiative transfer, and isotope systematics. Every concept is grounded in the governing partial differential equations and their geophysical boundary conditions.

Computational Geoscience

Interactive Python simulations running directly in your browser: mantle convection solvers, seismic ray tracing, climate energy balance models, groundwater flow, radioactive decay chains, and geochemical mixing calculations.

Observational Foundations

Seismic tomography, satellite geodesy, paleomagnetic records, ice core data, ocean drilling, geochemical tracers, and remote sensing — the full arsenal of observational constraints that test and refine our models of Earth's past and present.

Applied Earth Science

Natural hazard assessment, groundwater resource management, isotope dating techniques, environmental geochemistry, and the geoscientific basis for understanding climate change — bridging fundamental physics to real-world applications.

Fundamental Equations of Earth Science

Navier–Stokes Equation (Mantle & Ocean Flow)

The fundamental equation governing viscous fluid motion in both the mantle (creeping flow, Re ~ 10⁻²⁰) and ocean/atmosphere (turbulent flow). With the Boussinesq approximation for the mantle:

\[\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla P + \eta\nabla^2\mathbf{v} + \rho\mathbf{g}\]

where P is pressure, $\eta$ is dynamic viscosity, v is velocity,$\rho$ is density, and g is gravitational acceleration. For the mantle, inertial terms vanish (Stokes limit); for oceans, Coriolis and wind stress dominate.

Seismic Wave Equation

Elastic wave propagation through Earth's interior is governed by the equation of motion for an isotropic elastic medium, yielding P-waves and S-waves:

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

This separates into compressional P-waves ($v_P = \sqrt{(\lambda+2\mu)/\rho}$ ~ 5–14 km/s) and shear S-waves ($v_S = \sqrt{\mu/\rho}$ ~ 3–7 km/s).$\lambda$ and $\mu$ are the Lamé parameters.

Clausius–Clapeyron Relation

Governs the pressure-temperature slope of phase boundaries — critical for understanding mantle phase transitions, magma generation, and atmospheric moisture capacity:

\[\frac{dP}{dT} = \frac{\Delta S}{\Delta V} = \frac{L}{T\,\Delta V}\]

where $L$ is latent heat, $T$ is temperature, and $\Delta V$ is the volume change across the phase transition. For the olivine–spinel transition at 410 km depth, $dP/dT > 0$ (positive Clapeyron slope); for the atmosphere, it controls the exponential increase of saturation vapour pressure with temperature.

Radiative Transfer Equation

The fundamental equation of atmospheric radiation, governing the absorption, emission, and scattering of electromagnetic radiation through the atmosphere:

\[\frac{dI_\nu}{ds} = -\kappa_\nu\,\rho\,I_\nu + j_\nu\,\rho\]

where $I_\nu$ is spectral radiance, $\kappa_\nu$ is the mass absorption coefficient,$j_\nu$ is the mass emission coefficient, $\rho$ is density, and $s$ is path length. At thermal equilibrium, Kirchhoff's law gives $j_\nu = \kappa_\nu B_\nu(T)$ where $B_\nu$ is the Planck function.

Darcy's Law (Groundwater Flow)

The constitutive equation for flow through porous media, fundamental to hydrogeology, oil reservoir engineering, and magma migration through partially molten rock:

\[\mathbf{q} = -\frac{k}{\mu}\left(\nabla P - \rho\,\mathbf{g}\right)\]

where q is the Darcy flux (volume flow per unit area), $k$ is permeability (m²),$\mu$ is dynamic viscosity, $P$ is pore pressure, and $\rho\mathbf{g}$ is the gravitational body force. Combined with the continuity equation, this yields the groundwater flow equation.

Radioactive Decay Law

The basis of radiometric dating — geochronology's most powerful tool for determining the ages of rocks, minerals, and geological events:

\[N(t) = N_0\,e^{-\lambda t}, \qquad t = \frac{1}{\lambda}\ln\!\left(1 + \frac{D^*}{N}\right)\]

where $N$ is the number of parent atoms, $N_0$ is the initial number,$\lambda$ is the decay constant, $D^*$ is radiogenic daughter atoms, and $t_{1/2} = \ln 2/\lambda$ is the half-life. Key systems include ²³&sup8;U–²&sup0;&sup6;Pb (4.47 Gyr), &sup8;&sup7;Rb–&sup8;&sup7;Sr (48.8 Gyr), and ¹&sup4;C (5730 yr).

Course Structure

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Part I: Solid Earth

The foundations of solid Earth geology: plate tectonic theory and its evidence, crystal chemistry and mineral classification, the origin and evolution of igneous rocks from partial melting to crystallisation, and the processes of sediment transport, deposition, and lithification.

• Plate tectonics: plate kinematics, Euler poles, driving forces
• Mineralogy: crystal structure, symmetry operations, optical properties
• Igneous petrology: phase diagrams, Bowen's reaction series, MORB vs OIB
• Sedimentary processes: Stokes settling, turbidity currents, diagenesis

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Part II: Geophysics

The physics of Earth's interior probed by seismic waves, gravity, magnetics, and heat flow. Structural geology provides the geometric framework, while seismology, deep-Earth studies, and geodynamics reveal how the planet deforms on all timescales.

• Structural geology: stress tensors, Mohr circles, strain analysis
• Seismology: P & S waves, surface waves, seismic tomography
• Earth interior: PREM model, core–mantle boundary, inner core anisotropy
• Geodynamics: mantle convection, Rayleigh number, postglacial rebound

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Part III: Climate & Oceans

Earth's climate system as an energy balance problem: solar forcing, greenhouse radiative transfer, atmospheric dynamics, thermohaline ocean circulation, and the geological record of past climate states from ice cores to deep-sea sediments.

• Climate science: energy balance, feedbacks, climate sensitivity
• Atmospheric physics: radiative transfer, convection, general circulation
• Ocean circulation: Ekman transport, Sverdrup balance, thermohaline flow
• Paleoclimatology: Milankovitch cycles, ice ages, proxy records

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Part IV: Applied Earth Science

Quantitative geochemistry and its applications: trace element partitioning, isotope systematics for dating and tracing, groundwater flow and contaminant transport, and the science of earthquakes, volcanic eruptions, landslides, and floods as natural hazards.

• Geochemistry: partition coefficients, REE patterns, thermodynamic modelling
• Isotope geology: U–Pb, Rb–Sr, Sm–Nd, stable isotope fractionation
• Hydrogeology: Darcy's law, aquifer testing, contaminant transport
• Natural hazards: seismic hazard analysis, volcanic risk, slope stability

Key References

Core Textbooks

• Press, Siever, Grotzinger & Jordan: Understanding Earth — the standard introduction to Earth system science, integrating geology, geophysics, and geochemistry
• Turcotte & Schubert: Geodynamics — the definitive quantitative treatment of heat transfer, mantle convection, gravity, and rock mechanics

Advanced References

• Fowler: The Solid Earth: An Introduction to Global Geophysics — comprehensive coverage of seismology, gravity, geomagnetism, and plate kinematics
• Hartmann: Global Physical Climatology — radiative transfer, atmospheric dynamics, ocean–atmosphere coupling, and paleoclimate