Plate Tectonics

1.1 Seafloor Spreading and Magnetic Anomalies

The Vine-Matthews-Morley hypothesis (1963) provides the key evidence for plate tectonics. New oceanic crust forms at mid-ocean ridges, and as magma cools below the Curie temperature, magnetic minerals record the ambient geomagnetic field. Because Earth's magnetic field reverses polarity on timescales of $10^5$ to $10^6$ years, the result is a symmetric pattern of magnetic anomalies flanking the ridge axis.

The Magnetic Anomaly Pattern

A marine magnetometer towed behind a ship measures the total magnetic field intensity. The anomaly is defined as the deviation from the expected dipole field:

where $B_{\text{IGRF}}$ is the International Geomagnetic Reference Field. The anomaly arises from the magnetization of the oceanic crust. For a uniformly magnetized layer of thickness$h$ and magnetization $\mathbf{M}$, the magnetic potential is:

Derivation: Spreading Rate from Anomaly Spacing

The key quantitative relationship is straightforward. If a magnetic reversal boundary at time $t$ in the past is observed at distance $d$ from the ridge axis on one flank:

The full spreading rate is twice this value:

where $d_1$ and $d_2$ are distances to the same reversal on opposite flanks. For the Mid-Atlantic Ridge, typical half-spreading rates are ~1.2 cm/yr (slow), while the East Pacific Rise achieves ~8 cm/yr (fast). Asymmetric spreading occurs when $d_1 \neq d_2$, indicating ridge migration:

The geomagnetic polarity timescale (GPTS) is calibrated using radiometric dating of volcanic rocks on land. Key reversals include the Brunhes-Matuyama boundary (0.78 Ma), the Gauss-Matuyama boundary (2.58 Ma), and the Gilbert-Gauss boundary (3.58 Ma).

1.2 Plate Velocity from Euler Poles

Euler's theorem states that any rigid-body motion on a sphere can be described as a rotation about an axis passing through the center of the sphere. The point where this axis intersects the surface is the Euler pole.

Derivation: Velocity on a Rotating Sphere

For a plate rotating with angular velocity vector $\boldsymbol{\omega}$, the velocity at any point $\mathbf{r}$ on the plate surface is:

The magnitude of this velocity depends on the angular distance $\theta$ from the Euler pole. If $\omega = |\boldsymbol{\omega}|$ is the angular speed and $R$ is Earth's radius:

This shows that plate velocity is zero at the Euler pole ($\theta = 0$) and maximum at $\theta = 90°$ (the equator of the Euler pole). Transform faults lie along small circles centered on the Euler pole, while spreading ridges are approximately perpendicular.

Triple Junctions and Velocity Triangles

At a triple junction where three plates A, B, C meet, the relative velocities must form a closed triangle:

This is the velocity closure condition. Each relative velocity is given by:

where $\boldsymbol{\omega}_{AB}$ is the relative angular velocity of plate B with respect to plate A. The closure condition in terms of angular velocities becomes:

This allows determination of an unknown plate motion from two known ones. McKenzie and Morgan (1969) used this principle to reconstruct global plate motions. The stability of a triple junction depends on the types of boundaries (RRR, RTF, TTT, etc.) and their orientations relative to the velocity vectors.

1.3 Subduction Dynamics: Forces on Plates

Plate motions are driven by a balance of forces. The two dominant driving forces are slab pull (the gravitational pull of the cold, dense subducting slab) and ridge push (the gravitational potential energy of the elevated ridge).

Derivation: Slab Pull Force

Consider a subducting slab of length $L$ (measured down-dip), thickness $h$, and density contrast $\Delta\rho$ with the surrounding mantle. The slab pull force per unit length along the trench is:

The density contrast arises from thermal contraction. As oceanic lithosphere cools, its density increases according to:

where $\rho_m \approx 3300 \text{ kg/m}^3$ is mantle density, $\alpha_V \approx 3 \times 10^{-5} \text{ K}^{-1}$ is the thermal expansion coefficient, $T_m \approx 1350°\text{C}$ is mantle temperature, and $T_s$ is the average slab temperature. For a typical $\Delta\rho \approx 80 \text{ kg/m}^3$,$L = 600 \text{ km}$, $h = 80 \text{ km}$:

Derivation: Ridge Push Force

Ridge push arises from the elevation of the mid-ocean ridge above the surrounding seafloor and the lateral density gradient. The force per unit ridge length is derived by integrating the pressure difference:

Evaluating this integral using the half-space cooling model gives:

where $\kappa$ is thermal diffusivity and $t$ is plate age. Ridge push is typically an order of magnitude smaller than slab pull, confirming that subduction is the primary driver of plate motions. Resisting forces include basal drag, slab resistance (viscous drag on the slab), and transform fault friction.

1.4 Heat Flow and Thermal Models

Fourier's Law of Heat Conduction

The foundation of geothermal analysis is Fourier's law. Heat flux (energy per unit area per unit time) is proportional to the temperature gradient:

where $k$ is thermal conductivity (typical values: 2-4 W m$^{-1}$ K$^{-1}$ for rocks) and $z$ is depth (positive downward). The minus sign indicates heat flows from hot to cold.

Derivation: Oceanic Heat Flow from Half-Space Cooling

The oceanic lithosphere is modeled as a cooling half-space. A semi-infinite slab at initial temperature $T_m$ is suddenly cooled to $T_s$ at its surface ($z = 0$) at time $t = 0$. The 1D heat diffusion equation is:

where $\kappa = k/(\rho c_p)$ is thermal diffusivity ($\approx 10^{-6}$ m$^2$/s). The solution is:

where $\text{erf}(\eta) = \frac{2}{\sqrt{\pi}} \int_0^{\eta} e^{-s^2} ds$ is the error function. The surface heat flow is obtained by evaluating Fourier's law at $z = 0$:

This gives the celebrated result:

Heat flow decreases as the inverse square root of plate age. With typical values ($k = 3.1$ W/m/K, $T_m - T_s = 1300$ K, $\kappa = 8 \times 10^{-7}$ m$^2$/s), we get $q \approx 480 / \sqrt{t}$ mW/m$^2$ where $t$ is in Ma.

Continental Geotherm

In continental crust, radiogenic heat production $A$ (from U, Th, K) adds a source term:

For exponentially decreasing heat production $A(z) = A_0 e^{-z/h_r}$ with characteristic depth $h_r \approx 10$ km, the steady-state temperature profile is:

where $q_m$ is the mantle heat flow at the base of the crust. The surface heat flow follows the Birch relationship: $q_s = q_m + A_0 h_r$.

1.5 Isostasy: Gravitational Equilibrium

Isostasy is the principle that Earth's lithosphere "floats" on the denser asthenosphere, achieving gravitational equilibrium. Two classical models describe how this is achieved.

Airy Isostasy: Variable Root Depth

In the Airy model, topography is supported by crustal roots of variable depth extending into the mantle. All columns have the same density but different thicknesses. Consider a reference column of crustal thickness $t_c$ at sea level. For a mountain of height $h$, the root depth $r$ below the reference level satisfies:

Solving for the root depth:

With $\rho_c = 2800$ kg/m$^3$ and $\rho_m = 3300$ kg/m$^3$:

A 5 km mountain requires a ~28 km root. The compensation depth $D_c$ is the depth below which pressure is hydrostatic and uniform:

Pratt Isostasy: Variable Density

In the Pratt model, all columns extend to the same compensation depth $D$, but have different densities. For a column with elevation $h$ above sea level:

where $\rho_0$ is the reference density at sea level. Higher topography corresponds to lower density material. For an ocean basin of depth $d_w$:

In reality, Earth uses a combination of both mechanisms. Oceanic lithosphere is better described by a thermal (Pratt-like) model, while continental mountains are better described by the Airy model with crustal roots observed seismically.

1.6 Plate Boundaries and Kinematic Framework

Types of Plate Boundaries

Three fundamental types of plate boundaries are distinguished by the relative motion of the plates they separate:

Absolute Plate Motions: The Hotspot Reference Frame

While relative plate motions are well constrained, absolute motions require a fixed reference frame. The hotspot reference frame assumes that mantle plumes (e.g., Hawaii, Iceland, Yellowstone) are approximately stationary in the deep mantle. The velocity of plate A in this frame is:

The Hawaiian-Emperor chain provides a classic example: the bend at ~47 Ma records a major change in Pacific plate motion direction from roughly northward to northwestward. The age-distance relationship along the chain gives:

The Wilson Cycle

J. Tuzo Wilson recognized that ocean basins open and close in a cyclic pattern. The complete Wilson Cycle encompasses:

1.7 Lithospheric Thermal Structure

The Plate Model vs Half-Space Model

The half-space cooling model (Section 1.4) works well for young oceanic lithosphere but overpredicts the depth of old ocean basins. The plate model (Parsons & Sclater, 1977) introduces a finite plate thickness $a$ with a fixed basal temperature:

For $t \to \infty$, the temperature profile becomes linear: $T = T_s + (T_m - T_s)(z/a)$, and the heat flow approaches a constant value:

With $a = 125$ km, this gives $q_{\infty} \approx 33$ mW/m$^2$ for old oceanic lithosphere, in good agreement with observations for plates older than ~80 Ma.

Bathymetry-Age Relationship

Seafloor depth increases with age because cooling lithosphere contracts and becomes denser. Isostatic balance gives:

where $d_r \approx 2500$ m is the ridge crest depth. Numerically, this simplifies to the Parsons-Sclater relationship:

For old lithosphere ($t > 80$ Ma), the plate model predicts a flattening to$d_{\max} \approx 6400$ m, matching observations from the North Pacific.

Thermal Boundary Layer Thickness

The mechanical lithosphere is conventionally defined by the depth to a specific isotherm (often 1300°C). From the half-space model:

For $\kappa = 8 \times 10^{-7}$ m$^2$/s: lithosphere beneath 100 Ma old seafloor is approximately $z_L \approx 100$ km thick.

Computational Simulations