A Cold Snap, Part 2

In the last post, we talked about the ground temperature during a cold snap.  Today, we’ll look at the same situation from two different viewpoints.
Recall that we were interested in the below-ground temperature following a sudden change in the surface temperature, and we used a solution for a step change in temperature on an initially isothermal semi-infinite body to approximate the real effect.

In the last post we showed the temperature as a function of depth at three different times.  We might also be interested in the temperature as a function of time at various depths:

Notice that in this picture, the time only goes out to 25 hours.  Over that time span, at 5 cm deep, the temperature has dropped below 5 degrees C, but at 30 cm deep, it has barely budged off the initial temperature.
We could also look at the time required for a certain temperature to reach any depth as in this figure:

Even after almost 10 days, the 0 deg C line is less than 10 cm deep.

As in the previous post, we can look at the same snapshots using an approximate value for the thermal diffusivity of concrete which is about seven times as large as the value that we were using for soil:

Of course, the higher thermal diffusivity means that the cold temperatures get deeper for the same amount of time, or alternatively, that the ground gets colder for a given depth.