The deserts of New Mexico can get blazingly hot and bone-chillingly cold, extremes of temperature familiar to the outdoorsman in that kind of terrain. A few hundred feet below the ground in Carlsbad Caverns, the temperature is essentially stable in the mid-50s no matter how scorching or frigid the air a few feet above happens to be.

You can think of the earth's surface in New Mexico as subject to two superimposed sinusoidal periodic heat pulses. One has a period of 24 hours and corresponds to the heat rising and falling over the day/night cycle. The other has a period of 1 year and corresponds to the average heat changing through the seasons. Clearly this approximation is quite rough, but we can take it as a starting point for some mathematical spelunking of our own. There's some math, but you can skim over it if you wish without missing the result.

We'll need the heat conduction equation as our first piece of caving equipment. It says that the rate of change of temperature at a given point is proportional to how different it is from the average temperature of its surroundings. We also have the proportionality constant k to measure the ease with which temperature flows:

To get the temperature below ground level we also need to give the boundary condition - ie, the temperature at ground level. We assume it's just the temperature T0 multiplied by an oscillating function. (T0 is the size of the oscillation.) We're going to take the real part of an oscillating exponential for mathematical convenience:

Now this is a differential equation and boundary condition, and we'd like to solve it as painlessly as possible. One of the best ways is to start with an educated guess - try it out, and if it works you're done. If it doesn't, try something else. Let's assume that the solution giving the temperature as a function of time and depth is an oscillating function of time multiplied by some function of depth z.

Jam that into our original differential equation and see what happens:

This is what we in the business call an "ordinary" differential equation, and they're easy to solve if you've done it a few times. Teaching it from scratch would take us too far afield, so let's just say we did a few more lines of work and arrived at the solution:

Where for convenience we've defined delta as:

This has to match the boundary condition given at the start, so we end up seeing that A = T0. which leads to a final answer:

Ok, let's interpret this. We have an oscillating term, which indicates that the subsurface temperature goes up and down as the temperature outside cycles. But it's damped by an exponential factor - so the further down you go, the smaller the variations about the average are. There's a "typical depth" given by delta which says how far the variations reach. At that point the variations are only about 37% of their surface magnitude. At twice that depth the variations are only 14%, and very quickly falling off from there.

Further, this depth is dependent on the frequency of the oscillations. The depth increases with decreasing frequency. In other words, the daily changes will not penetrate as far as the seasonal changes. Even more weirdly, there's a phase factor in the cosine term - there will be a lag between the maximum temperature at the surface and the maximum temperature at depth. But let's plug in some numbers to get a feel for what actually might be happening under the ground at Carlsbad. The thermal diffusivity of limestone is about 1e-6 m^2/s. Plugging in for the daily variation, delta is about... 16 centimeters. Below even that small amount of rock, the temperature is pretty constant all other things being equal. It's one of the reasons burrowing animals can survive in the inhospitable desert.

The yearly variation has a much smaller frequency and thus penetrates deeper. I calculate the penetration depth is about 3.16 meters. Below that however, as many caves are, neither the daily nor seasonal variation makes a dent in that otherwise constant average temperature. Incidentally, the phase delay at 3.16 meters is about 2 months for the yearly variation - the warmest time of the year at that depth is actually about 2 months later than the warmest time of year above ground.

(Note: This fascinating example taken mostly from the excellent textbook by Fetter and Walecka)

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The main reason for the phase lag is that temperature will increase as long as the surrounding temperature is higher than that in the cave. The rate of temperature increase will be maximum when the difference is the largest, but even as the difference gets smaller, warming will continue at a slower rate.

For the same reason, surface temperatures are typically hottest about 3 hours after noon, and daily highs peak a month or two after the summer solstice.

Decimal trim! Add ~0.1 watt/m^2 diffusing up from below, ~10^(-4) solar input from above.

http://en.wikipedia.org/wiki/Geothermal_gradient

An interesting comparison could be made to seasonal and daily variations in ocean temperatures.

Fascinating, especially the phase lag. Thanks!

I thought an ordinary differential equation is one with no partial differentials, but you've notated it as having partials.

Nice. I'm interested in cooling Venus. If I said the outside went from present day Venus temperatures to something around 0Â°C in a hundred years, could you pretend that curve was 1/4 of a 400-year cycle, and tell me in turn how deep the cooling had penetrated in that time?

If that's not the right approach, what would be your estimate of the depth that long-term cooling penetrates to in a century?

In some parts of the country, you can observe this phenomenon by merely going down into the basement. Basements are common in areas where average winter temperatures fall below freezing but year-round average temperatures are well above freezing; the purpose is to put your foundation below the typical winter frost depth, so that it is not so vulnerable to frost heaves. (Where you live, frost does not penetrate beyond the typical thickness of a concrete slab, so you usually find basements only in buildings that are large enough to need a deeper foundation anyway.) The side effect is that the basement stays pleasantly cool in the summer even without air conditioning, and pleasantly warm in the winter even without heating.

#5, you're right, it should be a standard d rather than the partial d. However, since a partial derivative of a single-variable function is the same as the total derivative, the underlying mathematics is unaffected. As such I think I'm going to leave it because it's a royal pain to change the equation after the fact.

(Are you reading this, ScienceBlogs tech staff? Put in LaTeX!)

#6, using that approach you get around 63ish meters. However, I wouldn't be too confident about the validity of using this sinusoidal estimate for a step-function boundary condition. I'll write up the math at some point next week - my guess is that the actual penetration depth will be significantly greater.

Matt,

What do you use for the equations? I notice they are images, but do you write them in LaTeX and save as image, or in Mathematica or what?

Regards.

I write in LaTeX, generate the images as PNG (I used to do this locally, but honestly it's easier to use one of the online sites), and upload them to the site. It's pretty, but unfortunately it makes it a royal pain to go back and change anything. I've been agitating for built-in LaTeX as long as I've written here, but no dice so far. It would be great for commenters too.

Would make sense, especially for a science blog.

Since deeper depths mean longer time frames, you can figure out past surface ground temperatures by examining the temperature in a borehole. As you would expect, as you go back farther in time, the estimates of temperature are smoothed out, but it's clear that now is not the warmest the planet has been recently.

This is part of the ongoing fight between the climate modelers and the geologists.