# How to calculate the heat flux within a snowpack

It sounds simple but requires at least some basic knowledge of partial differential equations. You also need to know about discretisation so that you can solve the equation

$c_p \rho_w\frac{\partial T}{\partial t} = K \frac{\partial^2 T}{\partial x^2}$

on your computer. Most commonly used are finite difference schemes, for example, explicit (forward time stepping), implicit (backward time stepping), or the Crank-Nicolson (both, forward and backward time stepping) schemes.

**Update:**

Here you see an example of a vertical temperature profile of a snowpack of 5m thickness.
On top (left, 0m) the model is forced with 10 years of surface temperature at location of the *JAR* automatic weather station as modelled by *MAR*.
You can see the alternating summer and winter heat and cold waves entering the snowpack.
The annual cycle is the strongest because a slower surface signal (i.e., longer time scales) lead to a larger penetration depth ($l$)

$l = \sqrt{2\,\frac{K}{c_p \rho_w} \Delta t}$

Because of the (almost) periodic forcing the signal at depth is phase shifted and decays exponentially.