# How Cold is the Ground?

It snowed in Ithaca a few weeks ago. Which sucked. But fortunately, it had been warm for the previous few days, and the ground was still warm so the snow melted fast. Aside from letting me enjoy the absurd arguments against global warming that snow in April birthed, this got me thinking: How cold is the ground throughout the year? At night vs. during the day? And the corollary: How cold is my basement? If I dig a deeper basement, can I save on heating and cooling? (I’m very cheap.)

Well, we want to know the temperature distribution

For our purposes,
we can treat the Earth as flat (I don’t plan on digging a basement deep
enough to see the curvature of the Earth), so we can assume the
temperature only changes with the depth into the ground

where

Let’s measure our time

plugging this in Eq. (1),
rearranging, and calling

Now let’s say we didn’t know how to or didn’t want to solve
this equation. (Don’t worry, we do & we will). From rearranging this
equation, we see right away there is only one “length scale” in the
problem,

Tables | Are | Cool |
---|---|---|

col 3 is | right-aligned | $1600 |

col 2 is | centered | $12 |

zebra stripes | are neat | $1 |

blloop

Material | |||
---|---|---|---|

Polycrystalline Silica (glass, sand) | 0.83 | 27 | 5.1 |

Crystalline Silica (quartz) | 1.4 | 35 | 6.6 |

Sandstone | 1.15 | 32 | 6.0 |

Brick | 0.52 | 21 | 4.0 |

Soil | 0.3-1.25 | 16-33 | 3.1-6.3 |

So we would expect that the temperature of the ground doesn’t change much on a daily basis a foot or so below the ground, and doesn’t change ever about 15-20 feet into the ground. Just to pat ourselves on the back for our skills at dimensional analysis, a quick check shows that permafrost penetrates 14.6 feet into the ground after 1 year. So our dimensional estimates looks pretty good! In the next few days I’ll solve this equation exactly and throw up a few pretty graphs, and maybe talk a little about PDE’s and Fourier series in the process.

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