A Cold Snap

Following a series of relatively warm days, we had a sudden (over a period of a few hours) drop in temperature and things then stayed very cold for several days.  This led me to wonder how fast the ground would cool under such circumstances.

For the purposes of getting a simple, quick feel for the effect of the air temperature change, we’ll look at a somewhat simplified version of the actual events.  Of course this result won’t exactly match what happened in real life, but it will provide a reasonable estimate.
For the simplified case, we will assume that the ground is semi-infinite and initially all at a uniform temperature.  We’ll also assume that the temperature at the ground surface changed instantaneously even though it actually took a few hours.  Finally, we’ll assume that the cold surface temperature remained uniform even though it fluctuated a little bit in the ensuing days.

With those simplifications, the transient heat transfer solution is really simple:

Where T_i is the (uniform) initial temperature of the ground, T_s is the (constant) temperature of the surface, x is the depth, t is the elapsed time since the temperature change, and α is the thermal diffusivity of the ground.

Using a thermal diffusivity of 0.1 mm²/s, and assuming T_i=25 deg C, and T_s=-5 deg C, the temperature under the ground would look like this:

After an hour, the jump in surface temperature would have been felt to 6 or 7 cm deep and by 12 hours it would have been down to around 25 cm.

While the value of 0.1 mm²/s for the thermal diffusivity is typical for ground, it will vary quite a bit depending on the makeup and moisture content of the soil.  To get some feel for the effect of thermal diffusivity, this figure shows the same profiles as the first figure, but for a thermal diffusivity of 0.7 mm²/s which would be a typical value for concrete.