Question

Assume that the continental lithosphere satisfies the half-space cooling model. If a continental region has an age of \(1.5 \times 10^{9}\) years, how much subsidence would have been expected to occur in the last 300 Ma? Take \(\rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}, \kappa=1 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\), \(T_{m}-T_{0}=1300 \mathrm{~K},\) and \(\alpha_{v}=3 \times 10^{-5} \mathrm{~K}^{-1}\). Assume that the subsiding lithosphere is being covered to sea level with sediments of density \(\rho_{s}=2500 \mathrm{~kg} \mathrm{~m}^{-3}\).

Short answer

The expected subsidence over the last 300 million years is approximately 41.47 meters.

Step-by-step solution

Identify given parameters

The problem gives us several parameters: \( \rho_{m} = 3300 \ kg/m^3 \) (density of the mantle), \( T_{m} - T_{0} = 1300\ K \) (temperature difference), \( \kappa = 1 \ mm^2/s = 10^{-6}\ m^2/s \) (thermal diffusivity), \( \alpha_{v} = 3 \times 10^{-5} \ K^{-1} \) (coefficient of volumetric thermal expansion), and the sediment density \( \rho_{s} = 2500 \ kg/m^3 \). The age of the region is \(1.5 \times 10^9\) years, and we are interested in the time frame of the last \(300 \ Ma = 300 \times 10^6\) years.

Calculate heat diffusion timescale

Convert timescale from years to seconds: \[ t = 300 \times 10^6 \text{ years} = 300 \times 10^6 \times 365.25 \times 24 \times 60 \times 60 \text{ seconds} \approx 9.46 \times 10^{15} \text{ seconds}. \]

Calculate lithosphere thickness change

The change in thickness due to subsidence \( \Delta L \) can be calculated using the formula for subsidence in the half-space cooling model: \[ \Delta L = \sqrt{\pi \kappa t} \times \alpha_{v} \times (T_{m} - T_{0}). \] Substituting the given values: \[ \Delta L = \sqrt{\pi \times 10^{-6} \times 9.46 \times 10^{15}} \times 3 \times 10^{-5} \times 1300 = \sqrt{2.971 \times 10^{10}} \times 3.9 \approx 171.2 \text{ meters}. \]

Determine apparent subsidence

Convert the actual subsidence to apparent subsidence because the layer is submerged under sediments. The adjustment is done using the densities of the mantle and sediments: \[ \text{Apparent Subsidence} = \Delta L \times \left( \frac{\rho_{m} - \rho_{s}}{\rho_{m}} \right). \] Using the values provided: \[ \text{Apparent Subsidence} = 171.2 \times \left( \frac{3300 - 2500}{3300} \right) \approx 171.2 \times \frac{800}{3300} \approx 41.47 \text{ meters}. \]

Key concepts

Half-Space Cooling Model

The half-space cooling model is a concept used to describe how heat dissipates over time in the Earth’s lithosphere. Imagine the lithosphere as an infinite plane with heat flowing from a warmer interior to the cooler surface. Over geological time, this process cools the lithosphere, causing it to contract. This model assumes heat diffuses away similarly to how heat escapes from a cooling pie.
This cooling causes subsidence, or "sinking" of the lithosphere. This is because as the lithosphere loses heat, it also loses volume, creating a lower topographical height.

• For example, in the context of our problem, the continental lithosphere cools over millions of years, leading to expected subsidence levels.
• The model is important in predicting how landscapes evolve over geological timeframes, including the gradual sinking of the lithosphere under its own weight as it cools.

Understanding this model helps explain why certain regions might have sunken over millions of years.

Thermal Diffusivity

Thermal diffusivity is a material property that measures how well it conducts heat relative to its store of heat—essentially how fast heat spreads through a material. Its symbol is often represented as \( \kappa \). For example, in the provided problem, \( \kappa = 1 \, \text{mm}^2/s = 10^{-6} \, \text{m}^2/s \).
This factor is essential in the half-space cooling model to estimate how quickly heat diffuses through the lithosphere.

• Materials with high thermal diffusivity transfer heat quickly, cooling faster. In contrast, those with low thermal diffusivity retain heat longer, cooling more slowly.
• In our context, thermal diffusivity determines how quickly the lithosphere will lose heat, contributing to the subsidence calculation.

Changes in heat distribution due to this property significantly affect subsidence rates, influencing the pace at which regions settle further into the Earth’s crust.

Volumetric Thermal Expansion

Volumetric thermal expansion refers to how a material's volume changes with temperature. The coefficient of volumetric thermal expansion, denoted as \( \alpha_v \), indicates how much a material expands per degree of temperature increase. For our specific problem, \( \alpha_v = 3 \times 10^{-5} \, \text{K}^{-1} \).
In geological terms, when the lithosphere cools, it contracts, leading to subsidence.

• A higher expansion coefficient would mean more significant volume changes for equivalent temperature changes.
• By measuring how much the lithosphere contracts, we can predict the amount of subsidence over time—this is vital in understanding landscape changes over millions of years.

The relationship between temperature changes and volume changes is critical for estimating the subsidence impact over the period as specified.

Sediment Density

Sediment density refers to the mass of sediment per unit volume, measured here as \( \rho_s = 2500 \, \text{kg/m}^3 \). This density is crucial when calculating how subsidence will be observed on the surface.
The sediments overlay the subsiding lithosphere, reducing the apparent effect of lithospheric sinking.

• When the denser mantle material sinks, the presence of less dense sediment on top modifies the apparent subsidence.
• In our calculations, the true subsidence must be adjusted for the buoyancy effect created by this sediment layer.

Thus, understanding sediment density is critical in adjusting apparent subsidence, providing a real-world perspective of how subsidence impacts land levels and outlining geological formations created over extensive periods.