Heat conduction with advection

We now have all the tools to implement a “full” one-dimensional heat transfer equation where we consider

  • heat conduction,
  • heat advection,
  • heat generation, and
  • variable physical paramaters.
(1)\[\rho(x) C_p(x) \left( \frac{\partial T}{\partial t} + v_x \frac{\partial T}{\partial x} \right) = \frac{\partial}{\partial x} \left( \alpha(x) \frac{\partial T}{\partial x} \right) + H\]

Exercise

Discretize equation (1).

Again, use forward difference in time and central difference in space for the diffusion term. For the advection term, however, you can try using either forward, backward or even central difference. This choice is easy to change later once your code is working.

We will implement a crustal scale heat transfer model, where we consider the heat diffusion from the moho towards the surface, heat generation within the crust, and upward heat advection caused by the removal of crustal material at the surface by erosion.

Exercise

Take your solution to the one of the previous exercises where we solved the heat equation with variable physical parameters but without heat advection (heat_diff_var.py or start from the already working script heat_diff_var_answer.py). Save this file with a new name (e.g. heat_diff_adv.py).

Modify the script to include heat advection:

  1. Define a variable vx that holds the advection velocity.

  2. Modify the line T[ix, it] = ... so that you include the advection term. Use the following discretized version of the equation

    \[T_n^{i} = \left( \left. \left( \alpha_n \frac{T_{n+1}^{i-1} - T_n^{i-1}}{\Delta x^2} - \alpha_{n-1} \frac{T_n^{i-1} - T_{n-1}^{i-1}}{\Delta x^2} + H_n \right) \middle/ \rho_n C_{p,n} \right. - v_x \frac{T_{n+1}^{i-1} - T_{n-1}^{i-1}}{2 \Delta x} \right) \Delta t + T_n^{i-1}\]

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