L5 - Convection and Magnetism

Planet or Vectors: Ocean Lithosphere and Age of Ocean Lithosphere

Kelvin's Estimate of Earth's Age

Kelvin proposed that the Earth formed from a molten state and solidified outwards. He suggested estimating the Earth's age based on the time it takes to reach the present-day geothermal gradient, assuming the Earth initially had a uniform high temperature (around $4000K$ or Celsius). As the surface cools, a steep thermal gradient develops, which gradually becomes less steep over time. Measuring this gradient at the surface could provide an age estimate.

Geothermal Gradient Measurements

The average geothermal gradient is about $25\degree C / km$. Kelvin's initial estimate was $24$ to $54$ million years. Adjusting for modern conductivity values gives an estimate between $24$ and $100$ million years.

Discrepancy and Explanations

This estimate is far shorter than the actual age of the Earth. One explanation for this discrepancy is heat generation within the Earth due to radioactive decay.

Even with heat generation included, the estimated geothermal gradient curve doesn't change significantly because conduction dominates near the surface.

Another explanation involves convection. Including the effects of convection yields much older age estimates because the surface gradient isn't solely due to conductive cooling. Convection is more important in this case.

Convection

Heat Transfer Mechanisms

The crust and inner core are dominated by conductive heat flow, while the mantle and outer core experience convection as the primary heat transfer mechanism.

Heat Equation

The heat equation includes a convective term that accounts for fluid motion:

$\frac{\partial T}{\partial t} = \text{conductive term} + \text{heat production term} - v \cdot \nabla T$

Where:

• $T$ is temperature.

• $t$ is time.

• $v$ is the velocity of the fluid.

• $\nabla T$ is the spatial gradient in temperature

Buoyancy and Density Variations

Hot material is generally more buoyant and rises, while cold material sinks. Convection requires horizontal gradients in buoyancy, often caused by denser cold material sinking. Density variations can be thermal or compositional.

Temperature Profiles in Convecting Systems

In a fluid layer cooled from the top and heated from below, convection leads to a homogeneous temperature in the bulk of the layer. Thin thermal boundary layers form at the top and bottom, where conduction dominates.

Internal heating shifts the temperature profile, resulting in a steeper thermal gradient at the top and a higher average temperature.

Sometimes the basal layer is missing, but convection continues due to cold, dense material sinking.

Rayleigh Number

The Rayleigh number ($Ra$) is a dimensionless number that indicates whether a system will convect. It represents the ratio of buoyancy forces to resistive forces:

$Ra = \frac{\text{buoyancy forces}}{\text{resistive forces}} = \frac{\alpha \Delta T \rho g D^3}{\eta \kappa}$

Where:

• $\alpha$ is thermal expansivity.

• $\Delta T$ is the temperature gradient.

• $\rho$ is density.

• $g$ is gravitational acceleration.

• $D$ is depth.

• $\eta$ is viscosity.

• $\kappa$ is thermal diffusivity.

If $Ra$ exceeds a critical value (approximately $10^3$), convection occurs. The Rayleigh number can characterize both planetary and laboratory systems.

Mantle as a Viscous Fluid

Fluidity vs. State of Matter

Fluidity is a substance's ability to deform under stress and isn't limited to gases or liquids. Solids can also behave as fluids over long timescales.

Dislocations

Imperfections in the crystal structure of rocks, known as dislocations, allow material to deform under stress. These dislocations move within the crystal structure, enabling flow over long periods.

Viscosity

The mantle has a high viscosity, typically between $10^{20}$ and $10^{23}$ Pascal seconds. Viscosity is temperature-dependent, decreasing as temperature increases. This allows the mantle to flow, particularly in hotter regions.

Mantle Convection Scenarios

The most realistic scenario for mantle convection involves heating from the core at the bottom, along with some internal heat production. Simulations show active upwellings from the base of the mantle.

Modeling Mantle Convection

The mantle has a complex geological structure and material behavior. The lithosphere acts as a rigid lid on the convective system. Models must consider these complexities in three dimensions.

Whole Mantle vs. Layered Convection

It's debated whether the whole mantle convects as one layer or if there is layered convection between the upper and lower mantle.

The mantle transition zone (400-660 km depth) may limit convective flow due to changes in composition and mineralogy. Phase transitions, such as Olivine to Wadsleyite to Ringwoodite, and then to Bridgmanite and ferroperriclase, also influence convection.

Seismic data suggest material exchange across the transition zone, conflicting with geochemical data indicating isolated reservoirs.

Thermal vs. Thermochemical Convection

Thermal convection is driven by temperature variations, while thermochemical convection also involves compositional variations. Thermochemical convection can be characterized by dimensionless numbers.

Mantle Flow Cartoons

Various cartoons depict different types of mantle convection, including plumes, compositionally distinct domains, and variations in flow patterns.

Convection and Plate Tectonics

Past mantle convection studies often used plate reconstructions to drive flow in models. Current models allow the mantle to convect freely and examine the resulting plate boundaries.

Core Convection

Outer Core Characteristics

The outer core is very hot (approximately $9000$ K) and has a low viscosity, potentially as runny as water. It convects turbulently.

Self-Sustaining Dynamo

The Earth's rotation organizes flow in the core into columns, giving rise to a self-sustaining dynamo that generates the Earth's magnetic field. This involves a metal disc rotating in a magnetic field, with charge flowing back to the center, inducing a magnetic field.

Magnetic Field Models

Models can reproduce a dipole field, with inward field lines in the Northern Hemisphere and outward field lines in the Southern Hemisphere, matching observed fields.

Magnetic Field Observations

Satellites, such as Swarm, measure the Earth's magnetic field and its variations. Rock samples also record past magnetic field directions and intensities.

Magnetic Field Reversals

The Earth's magnetic field has reversed polarity in the past. Simulations model these reversals. Analysis of rocks leads to magnetic polarity scales, showing periods of normal and reversed polarity.

Plate Tectonics and Magnetic Stripes

Magnetic stripes on the ocean floor, with alternating normal and reversed polarity, provided crucial evidence for seafloor spreading and plate tectonics.

Dating Oceanic Crust

Magnetic anomalies help estimate the age of the oceanic crust and are important for understanding heat flow and isostasy.