Question

The mantle rocks of the asthenosphere from which the lithosphere forms are expected to contain a small amount of magma. If the mass fraction of magma is 0.05 , determine the depth of the lithosphere-asthenosphere boundary for oceanic lithosphere with an age of 60 Ma. Assume \(L=400 \mathrm{~kJ}\) \(\mathrm{kg}^{-1}, c=1 \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}, T_{m}=1600 \mathrm{~K}, T_{0}=275 \mathrm{~K},\) and \(\kappa=1 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\).

Short answer

The depth of the lithosphere-asthenosphere boundary is approximately 100 km.

Step-by-step solution

Understanding the Problem

We are tasked with determining the depth of the lithosphere-asthenosphere boundary. The variables given include the mass fraction of magma (0.05), latent heat \( L \), specific heat capacity \( c \), melting temperature \( T_m \), surface temperature \( T_0 \), and thermal diffusivity \( \kappa \). The lithosphere age is 60 million years (Ma).

Convert Age from Millions of Years to Seconds

Convert the lithosphere age from millions of years to seconds to match the units of thermal diffusivity. Since 1 Ma = \( 10^6 \) years and 1 year = approximately 31.536 million seconds, the age in seconds is:\[ t = 60 \times 10^6 \times 31.536 \times 10^6 \text{ seconds}\]

Apply the Diffusion Equation for Depth

The depth \( z \) of the lithosphere-asthenosphere boundary can be calculated using the diffusion equation:\[ z = 2 \sqrt{\kappa t} \]Substitute \( \kappa = 1 \times 10^{-6} \text{ m}^2/\text{s} \) and the calculated value for \( t \).

Calculate the Depth

Plug the values into the equation and solve: \[ z = 2 \sqrt{1 \times 10^{-6} \times 60 \times 10^6 \times 31.536 \times 10^6} \]Simplify and calculate the numerical value of \( z \).

Verify Considerations

Ensure that all assumptions, such as ignoring heat production in the lithosphere or frictional heating, are acceptable for this approximation of the lithosphere-asthenosphere boundary depth calculation.

Key concepts

Lithosphere-Asthenosphere Boundary

The lithosphere-asthenosphere boundary (LAB) is the zone separating the rigid, cooling lithosphere from the hotter, more ductile asthenosphere beneath. This boundary is not fixed but varies with factors like temperature and pressure, impacting tectonic processes. It is crucial in geodynamics as it influences the convection currents driving plate movements. In an oceanic context, where the lithosphere is thinner than its continental counterpart, the LAB can be determined by understanding the thermal evolution of the oceanic plate over time. With cooling, the lithosphere thickens. Its depth is often calculated using thermal diffusion equations to model how heat disperses through rocks.

Mantle Dynamics

Mantle dynamics refer to the movements within Earth's mantle caused by internal heat dissipation. These dynamics are characterized by convection currents, crucial for driving plate tectonics. The mantle's movement is a slow but continuous process, circulating heat from Earth's core to the crust. This circulation helps in understanding how continents drift and ocean basins expand. - Mantle convection influences the creation and subduction of lithospheric plates. - The lithosphere sits on the asthenosphere, a part of the mantle that allows it to drift. - Gravitational forces and varying temperatures affect the mantle flow, thus impacting the surface geology.

Thermal Properties of Rocks

The thermal properties of rocks involve understanding how heat is transported through and stored within Earth's materials. Key properties include: - **Thermal Conductivity**: The ability of rocks to conduct heat. Higher conductivity means heat moves more easily through the material. - **Specific Heat Capacity**: The amount of heat required to change the temperature of a rock by one degree. Rocks with high specific heat capacity can store more thermal energy. - **Thermal Diffusivity**: A measure of how quickly a rock can conduct thermal energy relative to its ability to store it. This value is crucial in calculating depth boundaries like the LAB. These properties are essential in modeling Earth's thermal structure and help explain phenomena like volcanic activity and geothermal gradients.

Oceanic Lithosphere Age

The age of the oceanic lithosphere plays a significant role in determining its properties and behavior. As the oceanic lithosphere ages, it undergoes cooling, becoming denser and thicker. Older oceanic lithosphere shows distinct characteristics compared to newer formations: - **Thermal State**: Older lithosphere is generally cooler, affecting its density and buoyancy. - **Thickness**: With age, the lithosphere thickens due to ongoing thermal contractions and sediment accumulation. - **Strength and Rigidity**: Increasing age enhances the lithosphere's mechanical strength making it more brittle. Age is a vital parameter in calculating the lithosphere-asthenosphere boundary depth, as thermal diffusion over millions of years affects the LAB's positioning.