Question

Suppose that the \(660-\mathrm{km}\) density discontinuity in the mantle corresponds to a compositional change with lighter rocks lying above more dense ones. Estimate the minimum decay time for a disturbance to this boundary. Assume \(\rho=\) \(4000 \mathrm{~kg} \mathrm{~m}^{-3}, \Delta \rho=100 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(\mu=10^{21} \mathrm{~Pa} \mathrm{~s}\).

Short answer

The minimum decay time is approximately \(1.02 \times 10^{14}\) seconds.

Step-by-step solution

Identify Parameters

We are given that the density of the mantle is \( \rho = 4000 \ \text{kg/m}^3 \), the change in density across the boundary is \( \Delta \rho = 100 \ \text{kg/m}^3 \), and the viscosity is \( \mu = 10^{21} \ \text{Pa} \cdot \text{s} \). These values will be used to calculate the minimum decay time for a disturbance in the boundary.

Formula for Decay Time

The decay time \( T \) for a disturbance in a fluid with a density discontinuity can be estimated using the formula: \[ T = \frac{1}{g} \times \left( \frac{\mu}{\Delta \rho \cdot d} \right) \] where \( g \) is the acceleration due to gravity (approximately \( 9.81 \ \text{m/s}^2 \)) and \( d \) is the depth of the boundary (assumed to be \( 660 \ \text{km} \) in our context, converted to meters as \( 660,000 \ \text{m} \)).

Substitute Given Values

Substitute the given values into the formula: \( \Delta \rho = 100 \ \text{kg/m}^3 \), \( \mu = 10^{21} \ \text{Pa} \cdot \text{s} \), \( d = 660,000 \ \text{m} \), and \( g = 9.81 \ \text{m/s}^2 \). Thus, the formula becomes: \[ T = \frac{1}{9.81} \times \left( \frac{10^{21}}{100 \times 660,000} \right) \]

Calculate the Decay Time

Perform the calculations: \[ T = \frac{1}{9.81} \times \left( \frac{10^{21}}{66,000,000} \right) \] \[ T = \frac{1}{9.81} \times 10^{14.803} \] \[ T \approx 1.02 \times 10^{14} \ \text{seconds} \] This is the estimated minimum decay time for a disturbance to this boundary.

Key concepts

Viscosity

Viscosity is a property that describes a fluid's resistance to flow. Think of it as the thickness or "stickiness" of a fluid. For instance, honey has a higher viscosity than water because it is thicker and flows more slowly.
In the Earth's mantle, viscosity is crucial because it affects how materials move within. High viscosity means any disturbances take longer to adjust. Here, we assume the viscosity \( \mu \) of the mantle is \( 10^{21} \mathrm{~Pa} \cdot \mathrm{s} \). This is a high value, indicating that the mantle material flows very slowly over time. Understanding viscosity helps predict how changes in the mantle affect movement and stability.

Density Change

Density change at the boundary, like the 660-km mantle discontinuity, significantly influences the movement within the mantle. Density refers to how much mass is packed into a given volume. When there is a density difference (\( \Delta \rho \)) between two layers, it can lead to interesting dynamic effects.
In the given exercise, the lighter rocks on top and the denser rocks below create a scenario where disturbances can occur at this transition. Here, \( \Delta \rho \) is \( 100 \ \text{kg/m}^3 \), showing a slight change from an overall density of \( 4000 \ \text{kg/m}^3 \). This density change could influence how gravity interacts with these layers at the discontinuity.

Decay Time Estimation

Decay time tells us how long it takes for a disturbance, like a shift or wave, within the mantle to settle back into equilibrium. Estimating decay time is crucial to understanding how the Earth's interior stabilizes after seismic or volcanic activities.
To estimate the decay time \( T \), we use the formula: \[ T = \frac{1}{g} \times \left( \frac{\mu}{\Delta \rho \cdot d} \right) \]This formula calculates how quickly a disturbance will diminish based on viscosity, density change, and the depth of the boundary affected. In this exercise, the calculation results in \( T \approx 1.02 \times 10^{14} \ \text{seconds} \), implying that these processes can take incredibly long periods, reflecting the slow dynamics of Earth's mantle changes.