Thermal diffusion in the Earth's crust A classic example of a diffusion problem with a time-varying boundary condition is the diffusion of heat into the crust of the Earth, as surface temperature varies with the seasons. Suppose the mean daily temperature at a particular point on the surface varies as: $$ T_{0}(t)=A+B \sin \frac{2 \pi t}{\tau} $$ where \(\tau=365\) days, \(A=10^{\circ} \mathrm{C}\) and \(B=12^{\circ} \mathrm{C}\). At a depth of \(20 \mathrm{~m}\) below the surface almost all annual temperature variation is ironed out and the temperature is, to a good approximation, a constant \(11^{\circ} \mathrm{C}\) (which is higher than the mean surface temperature of \(10^{\circ} \mathrm{C}\)-temperature increases with depth, due to heating from the hot core of the planet). The thermal diffusivity of the Earth's crust varies somewhat from place to place, but for our purposes we will treat it as constant with value \(D=0.1 \mathrm{~m}^{2}\) day \(^{-1}\). Write a program, or modify one of the ones given in this chapter, to calculate the temperature profile of the crust as a function of depth up to \(20 \mathrm{~m}\) and time up to 10 years. Start with temperature everywhere equal to \(10^{\circ} \mathrm{C}\), except at the surface and the deepest point, choose values for the number of grid points and the time-step \(h\), then run your program for the first nine simulated years, to allow it to settle down into whatever pattern it reaches. Then for the tenth and final year plot four temperature profiles taken at 3-month intervals on a single graph to illustrate how the temperature changes as a function of depth and time.

Step by step solution

01

- Understand the Problem

The problem involves simulating heat diffusion in the Earth's crust over a period of 10 years. The surface temperature varies sinusoidally over a year, and the goal is to compute the temperature distribution up to a depth of 20 m.

02

- Initialize Parameters

Determine the parameters given in the problem: \[ \tau = 365 \text{ days}, \quad A = 10^{\circ} C, \quad B = 12^{\circ} C, \quad D = 0.1 \text{ m}^2 \text{ day}^{-1} \]. Define the temperature at depth 20 m as \( T_{20 m} = 11^{\circ} C \).

03

- Set Up Spatial and Temporal Grid

Choose an appropriate number of grid points and time step: \[ \text{depth intervals} = z_0, z_1, ..., z_N \text{ with } z_N = 20 \text{ m} \]. \[ \text{time intervals} = t_0, t_1, ..., t_M \text{ with } t_M = 365 \times 10 \text{ days} \].

04

- Initial Conditions

Initialize the temperature at each depth to \(10^{\circ} C\) except the boundaries: \[ T(z, 0) = 10^{\circ} C \text{ for all } z eq 20 \text{ m} \]. At the surface (z=0): \[ T(0, t) = A + B \sin \left( \frac{2 \pi t}{\tau} \right) \]. At 20 m depth (z=20): \[ T(20 m, t) = 11^{\circ} C \].

05

- Update Equation

Use the diffusion equation: \[ T(z, t + \Delta t) = T(z, t) + D \Delta t \left( \frac{T(z - \Delta z, t) - 2T(z, t) + T(z + \Delta z, t)}{(\Delta z)^2} \right) \]. Run this update for all points excluding boundaries for each time step.

06

- Run Simulation

Write a program to implement the update equation and simulate temperature changes over 10 years. Allow the first 9 years to let the pattern stabilize.

07

- Plot Temperature Profiles

For the tenth year, extract and plot four temperature profiles at intervals of three months (0 days, 91 days, 182 days, 273 days). Display these profiles on a single graph to show how temperature varies with depth and time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

heat diffusion

Heat diffusion is a fundamental process where thermal energy spreads through a medium. In the Earth's crust, heat is transferred from the hot core to the cooler surface layers. This phenomenon follows Fourier's law of heat conduction, which states that the heat flux is proportional to the negative gradient of temperature. This means that heat moves from areas of higher temperature to areas of lower temperature. The speed and extent of heat diffusion depend on the thermal conductivity and diffusivity of the material.
In the given problem, we are simulating the diffusion of heat from the surface of the Earth, where temperature varies with the seasons, deeper into the crust. The parameter \(D = 0.1 \text{ m}^2 \text{ day}^{-1}\) represents the thermal diffusivity of the Earth's crust, which determines how quickly temperature changes spread through the material.

boundary conditions

In the context of heat diffusion, boundary conditions are essential to define how the temperature behaves at the edges of the system we are studying. They determine the values of temperature at specific points and can greatly influence the results of a simulation.
For our problem, there are two key boundary conditions:

• At the surface (\(z=0\)), the temperature varies sinusoidally over the year, given by \( T_{0}(t)=A+B \sin \frac{2 \pi t}{\tau}\), where \( \tau=365 \text{ days}\), \(A=10^{\circ}C\), and \(B=12^{\circ}C\).
• At a depth of 20 meters (\(z=20\)), the temperature is approximately constant at \( 11^{\circ}C\). This takes into account the heat coming from the Earth's core.

These boundary conditions ensure that our model accurately reflects how temperature varies in reality, providing more reliable simulation results.

temperature profile

The temperature profile is a representation of how temperature varies with depth and time. In thermal diffusion in the Earth's crust, this profile helps us understand how heat from the surface propagates into deeper layers over time.
Initially, we start with a uniform temperature of \(10^{\circ}C\) throughout most of the crust, except at the boundaries where specific conditions are applied. As the simulation runs, the temperature at each depth is updated based on the diffusion equation, reflecting how heat moves from one layer to the next. By the end of the simulation period, we can plot these temperature values at different depths and times to see the seasonal variation and the overall temperature distribution.
This kind of analysis is important for understanding natural processes such as geothermal energy flow, the hibernation patterns of animals, and agricultural planning.

numerical simulation

Numerical simulation involves using computational methods to solve physical problems that might be difficult or impossible to solve analytically. In this case, we use a numerical approach to simulate heat diffusion in the Earth's crust.
We choose appropriate grid points for depth and time, and then use the diffusion equation to update the temperature values at each point iteratively. This process requires careful selection of grid spacing and time steps to ensure stability and accuracy.

• Initialization: The initial temperatures and boundary conditions are set up as described earlier.
• Update Equation: The key equation we use is \( T(z, t + \Delta t) = T(z, t) + D \Delta t \left( \frac{T(z - \Delta z, t) - 2T(z, t) + T(z + \Delta z, t)}{(\Delta z)^2} \right) \), applied at each time step.
• Iteration: The simulation runs over the desired time period (10 years), allowing the system to reach a steady-state pattern over the first 9 years. The final year's data is then used to plot the temperature profiles at different times to visualize the heat diffusion.

Numerical simulation provides a powerful tool for predicting temperature changes in the Earth's crust, helping scientists and engineers make informed decisions about geothermal energy, mining, and other earth-related applications.