Question

Suppose that convection extends through the entire mantle and that \(10 \%\) of the mean surface heat flow originates in the core. If the surface thermal boundary layer and the boundary layer at the core-mantle interface have equal thicknesses, how does the temperature rise across the lower mantle boundary layer compare with the temperature increase across the surface thermal boundary layer?

Short answer

The temperature rise across the lower mantle boundary layer is 9 times that across the surface boundary layer.

Step-by-step solution

Understand the Problem

We need to compare the temperature rise across the lower mantle boundary layer to the surface thermal boundary layer, given that convection extends throughout the mantle and 10% of surface heat flow comes from the core. Both boundary layers have equal thicknesses.

Express Heat Flow Relations

The total surface heat flow is given as consisting of heat contributions from both mantle convection and the core. Let the heat from the core be \(Q_c = 0.1Q_s\), where \(Q_s\) is the total surface heat flow. Hence, heat flow due to mantle convection is \(Q_m = Q_s - Q_c = 0.9Q_s\).

Relation of Heat Flow to Temperature Gradient

For a given boundary layer, the heat flow can be expressed as \(Q = kA \frac{\Delta T}{\Delta z}\), where \(k\) is the thermal conductivity, \(A\) is the area, \(\Delta T\) is the temperature change across the boundary layer, and \(\Delta z\) is the layer thickness. Given that the heat flows are \(Q_m\) and \(Q_c\) for mantle convection and core, respectively, and using the thickness equivalence, this relates directly to temperature increases.

Set Up the Temperature Gradient Equation

Assuming equal thickness of boundary layers, \(\Delta z_m = \Delta z_c\), and the same thermal conductivity \(k\), the temperature increases \(\Delta T_m\) for the mantle and \(\Delta T_c\) for the core boundary layer are related by \(Q_m / \Delta T_m = Q_c / \Delta T_c\).

Comparing Temperature Increases

From \(Q_m / \Delta T_m = Q_c / \Delta T_c\), substitute \(Q_m = 0.9Q_s\) and \(Q_c = 0.1Q_s\). This gives \(0.9Q_s / \Delta T_m = 0.1Q_s / \Delta T_c\). Solving for \(\Delta T_m\) in terms of \(\Delta T_c\) yields \(\Delta T_m = 9\Delta T_c\).

Conclusion

The temperature rise across the lower mantle boundary layer, \(\Delta T_m\), is 9 times the temperature increase across the surface thermal boundary layer, \(\Delta T_c\), due to the proportion of heat flow from mantle convection.

Key concepts

Heat Flow

Heat flow is a fundamental process that governs how energy is transferred from the Earth's interior to its surface. It's a measure of the rate at which heat energy moves from one place to another. In the context of Earth's mantle,
heat flow occurs primarily due to two sources:

• Heat generated by radioactive decay within the mantle.
• Heat that comes from the core.

The total heat flow at Earth's surface includes contributions from both these sources. Understanding heat flow is crucial for studying mantle convection, which involves the movement of heat within the mantle. Mantle convection acts like a conveyor belt, redistributing heat through the mantle and affecting the surface conditions. In the exercise, it's noted that 10% of the surface heat comes from the core, implying the dynamic interplay between the mantle and core heat contributes to the overall energy transfer processes.

Thermal Boundary Layer

The thermal boundary layer is a thin region at the surface and at the base of mantle convection where heat transition occurs between two mediums. It is crucial in the study of heat flow because it controls the temperature gradient that drives mantle convection.
These boundary layers include:

• The surface thermal boundary layer close to Earth's crust.
• The lower thermal boundary layer, near the core-mantle interface.

In simpler terms, the thermal boundary layer acts like a filter that dictates how heat is spread out across various depths of the mantle. In the problem discussed, both surface and lower boundary layers have equal thicknesses, which simplifies comparisons between them. Their thickness affects how efficiently heat is transferred and, consequently, influences temperature differences across different parts of the mantle.

Core-Mantle Interface

The core-mantle interface, also called the core-mantle boundary (CMB), is a significant landmark inside Earth. It separates the Earth's mantle from its core. This interface is critical because it is where significant geothermal interactions occur.
Here, heat is transferred from the core upwards into the mantle layering via conduction and convection processes. At the core-mantle interface:

• The temperature is significantly high, being the source of around 10% of the surface heat flow.
• Variations in temperature and pressure at this interface can influence the convection currents in the mantle above it.
• The layer is also critical in the dynamic processes that lead to geological features seen on the surface.

Understanding this region helps scientists predict how energy passages within Earth evolve over time, enabling better understanding of processes such as volcanic eruptions and tectonic dynamics.

Temperature Gradient

A temperature gradient refers to the rate of temperature change over a particular distance. In the context of Earth's mantle, it signifies how temperature varies from the core to the surface.
In this case, it is affected by factors such as:

• Heat flow rate from the core.
• The thickness of the thermal boundary layers.
• The thermal conductivity of the materials involved.

The importance of the temperature gradient is highlighted in the problem, as it dictates the difference in temperature across boundary layers. The larger the gradient, the stronger the driving force for convection currents within the mantle. According to the exercise, the temperature rise across the lower mantle boundary layer is significantly greater than that at the surface layer. It indicates a large temperature gradient from the core upwards, promoting vigorous convective motion and influencing tectonic activities.