Physical Processes in Geology

Lab 2: Introduction to Thermal Modeling Due Date: The objective of this lab is to apply basic equations of conductive heat transfer to geologic problems. This is practice for next week, when we will use this technique on real data. Background: Temperature diffuses in the same way chemicals diffuse or landscapes diffuse; these are all described by a simple differential equation: 6T 2 K 82T 70? (922 There are 2D and 3D versions of this equation, but we will only consider temperature in 1D (depth). The constant K: is called the thermal diffusivity, which describes how rapidly temperature diffuses in a material. It is a function of a material’s thermal conductivity, density, and heat capacity. Notice that if the second derivative with respect to 2 on the right hand side is large (i.e. if there is a sharp peak or trough in temperature), the left hand time derivative will also be large, so the temperature will change rapidly at that location. The effect of this is that temperature spikes will tend to be smoothed over time, and the larger the spike, the faster the temperature change. Exact solutions to the thermal diffusion equation involve the error function, but we will use a numerical approximation scheme commonly employed for differential equations, called the finite difference method. The finite difference equation we will use for this lab exercise is: T2105) 2 T2¢(t _ 1) + A[T2i+1(t _ 1) + TZi-1(t_ 1) _ 2TZ1 (t _ 1” where A = [565-532 Although this looks imposing, it is quite simple. It says that the temperature at time step t and spot 2,- (T z, (t)) is the temperature in the same spot at the previous time step (T z, (t - 1)) plus an adjustment factor that is an approximation to the second derivative. To solve for the entire behavior of the system, we need initial and boundary conditions, such as the temperature distribution at time zero. We could optimize the speed of the finite difference method by programming it in Matlab or another programming language, but it is more illuminating (and requires no programming knowledge) to do the calculations in Microsoft Excel. In the Excel template you can download from ANGEL, the rows (from top to bottom) are the times, and the columns (from left to right) are the depths.