The summer that reaches the cellar in December

The diffusion equation is usually met by releasing something into a medium and watching it spread. Driving it at its boundary instead — holding the surface at a temperature that goes round a cycle, for ever — produces a solution with a quite different character and one very useful number in it.

The surface of the ground does this. It is warm in July and cold in January, roughly sinusoidally, and it has been doing so long enough that whatever transient followed the last ice age has died away and the ground is in a periodic steady state.

What that state looks like a few metres down is not obvious, and the answer contains a fact worth having: at a particular depth the ground is coldest in August.

One length, doing two jobs #

Look for a solution to $\partial T/\partial t = D\,\partial^2 T/\partial z^2$ in which everything oscillates at the driving frequency $\omega$. Writing $T \propto e^{i\omega t - kz}$ and substituting gives

$$k^2 = \frac{i\omega}{D},$$

so $k = (1+i)\sqrt{\omega/2D}$ — a wavenumber whose real and imaginary parts are equal. That equality is the whole structure of the answer, and it comes from the equation being first order in time and second order in space.

Writing $\delta = \sqrt{2D/\omega}$ for the length that appears, the solution is

$$T(z, t) = T_0 + A\,e^{-z/\delta}\cos\!\left(\omega t - \frac{z}{\delta}\right).$$

The amplitude falls by a factor $e$ every $\delta$, and the phase lags by one radian every $\delta$. One length, two effects, and no freedom to have one without the other: a medium cannot damp a temperature oscillation without delaying it in exactly this proportion, and the exchange rate is fixed by the equation rather than by the material.

The generator checks the claim rather than asserting it. The written solution is substituted back into the diffusion equation by finite differences at several depths and phases, and the residual is reported — a decaying cosine is easy to write down and easy to write down slightly wrong.

The numbers, for damp soil #

Take $D = 0.5$ mm²/s, which is a reasonable figure for damp soil and varies by a factor of two or three either way with moisture and texture. For the annual cycle, $\omega = 2\times10^{-7}$ s⁻¹ and

$$\delta = \sqrt{\frac{2D}{\omega}} = 2.24\ \text{metres}.$$

At one metre the swing is $64$ per cent of the surface’s and lags by $0.85$ months. At two metres, $41$ per cent and $1.7$ months. At $7.0$ metres — which is $\pi\delta$ — the lag is exactly half a year and the swing is $4.3$ per cent of what it was: a fifth of a kelvin on a twelve-kelvin surface signal, in antiphase with the sky.

That depth is a cellar, and the inversion is what a cellar is for. It is also why a water main below about a metre and a half does not know that the surface froze last week, why permafrost has an active layer of a metre or two that thaws each summer over ground that never does, and why the temperature of a deep cave is the annual mean of its region to within a fraction of a degree — a fact used to reconstruct past climates from cave deposits.

Reading the profile as an instrument #

Because the amplitude and the phase are governed by the same length, either one measures the diffusivity — and measuring both is a consistency check that catches everything wrong with the assumptions.

Bury two thermometers at known depths and record for a year. The ratio of the amplitudes gives $\delta$ from $\ln(A_1/A_2) = (z_2-z_1)/\delta$. The difference of the phases gives $\delta$ again from $\Delta\phi = (z_2-z_1)/\delta$. If the two agree, the ground is behaving like a uniform conductor and the diffusivity follows as $D = \omega\delta^2/2$. If they do not, something in the list at the end of this essay is happening: water is moving, or the diffusivity varies with depth, or the surface condition is not what was assumed.

The amplitude method usually gives a larger $\delta$ than the phase method in real soil, and the discrepancy is a standard diagnostic for downward water flow — moving water carries heat down faster than conduction would, which attenuates the signal less than it delays it. The two-thermometer measurement therefore reports three things: a diffusivity, whether the model applies, and if not, roughly what is wrong.

The daily wave is the same solution #

Nothing about the derivation cared what $\omega$ was. Putting in the daily cycle instead gives

$$\frac{\delta_{\text{annual}}}{\delta_{\text{daily}}} = \sqrt{\frac{\text{year}}{\text{day}}} = \sqrt{365} = 19.1,$$

so the daily wave penetrates $11.7$ centimetres where the annual one penetrates $2.24$ metres.

That single square root explains a great deal of ordinary experience. A spade’s depth of soil is thermally a different place from the surface on the scale of a day and the same place on the scale of a year. A slab floor a hundred millimetres thick is thick compared with the daily wave in concrete and thin compared with the annual one, which is exactly why it evens out a day and does nothing for a season. Root systems a few centimetres down experience a daily cycle attenuated by two-thirds; at thirty centimetres they experience none.

The square root is also why thermal mass is such an awkward design variable. Doubling the thickness of a wall does not double the delay it introduces; it does so only if the wall was thin compared with $\delta$ to begin with, and past about one $\delta$ the extra material contributes almost nothing because the oscillation has already been extinguished.

What a building does with all this #

The same arithmetic decides whether a wall is worth building thick.

A heavy wall delays and attenuates the daily temperature swing, which is useful in a climate where the days are hot and the nights are cold: the peak arrives inside when the outside has cooled, and the amplitude that arrives is small. Both effects are the same $\delta$, which for dense concrete at the daily period is about $12$ centimetres.

That number is the design rule. A wall much thinner than $12$ cm tracks the outside; a wall of about $12$ cm delays by one radian, or nearly four hours, and passes about a third of the swing; a wall of $40$ cm passes three per cent and delays by half a day, which inverts the cycle and is what a thick adobe or stone building does. Making it thicker than that buys nothing at all, and this is why traditional heavy construction in hot dry climates converges on a similar thickness in places that never communicated.

Against the annual swing the same wall is thin — $\delta$ for the year is nineteen times larger, about $2.3$ metres — so no wall anybody builds moderates the seasons. That job is done by the ground, and it is why buildings that use seasonal storage put it several metres down.

It is not a wave, and the difference is testable #

The solution looks like a wave and the phrase “thermal wave” is standard, but the resemblance is superficial and the difference has consequences.

An ordinary travelling wave has a shape that moves and a speed at which it moves, and the temperature oscillation underground has neither. Its amplitude falls exponentially with depth, so no shape is preserved; its “speed” is a phase velocity that depends on frequency, so no disturbance travels at it. The resemblance between the two pictures is a resemblance between two formulae — both carry $e^{i(kx - \omega t)}$ — rather than between two phenomena.

A wave has a front. A disturbance started at one place is exactly zero beyond a certain distance until the front arrives, and the arrival is an event that can be timed. The diffusion equation has no such thing: a change at the surface produces a nonzero change at every depth immediately, exponentially small at large depth but not zero, so nothing arrives at any moment.

The testable difference is the front. A wave equation’s response to an impulse has a sharp leading edge: nothing happens at a point until the disturbance arrives, and then it does. A diffusion equation’s response has no front at all — the solution is non-zero everywhere the instant the impulse is applied, infinitesimally so at distance, which is unphysical in the limit and correct for every purpose short of it. Something that travels has an arrival time; something that spreads does not.

The quantity $\omega\delta$ has the dimensions of a speed and is not one. It depends on the period — halve the period and it rises by $\sqrt2$ — so it is not a property of the medium, and no signal moves at it. What it describes is the rate at which a phase lag accumulates with depth for a particular driving frequency, which is a statement about a steady state rather than about anything propagating.

The underlying reason is the microscopic picture: heat is being carried by a random walk, and the walk’s width grows as a square root, and a random walker has a small but nonzero probability of having gone a very long way. A wave’s front exists because every part of the disturbance moves at the same speed; a diffusive spread has no front because the carriers do not.

The square root in the penetration depth is the same square root that governs a random walk. A walk’s mean square displacement grows linearly with time, so its width grows as $\sqrt{t}$ — and that is precisely what puts the annual signal $\sqrt{365} = 19$ times deeper than the daily one. Not 365 times; nineteen. Diffusion is a poor way to travel and an excellent way to smooth.

The same three lines in three other subjects #

The structure — a first-order-in-time equation driven periodically, giving a complex wavenumber with equal parts — is not about heat.

In a conductor, an alternating magnetic field obeys $\partial B/\partial t = (1/\mu\sigma)\nabla^2 B$, which is the diffusion equation with $D = 1/\mu\sigma$. The penetration depth is $\delta = \sqrt{2/\mu\sigma\omega}$ — the same expression — and it is the skin depth, $9.3$ mm in copper at $50$ Hz and $2$ µm at $1$ GHz. The current in a wire is confined to that skin, which is why a conductor’s interior stops carrying anything, the phase of the current lags with depth, and both facts are the two halves of one complex wavenumber.

Skin depth falls as the inverse square root of frequency, which is the same power law as the thermal case and holds for the same reason: the length is a diffusivity divided by a frequency, under a square root. Copper at 50 Hz gives 9.3 mm and at 1 GHz gives 2 µm — five decades of frequency buying three and a half decades of depth, exactly as the exponent requires.

In a viscous fluid, where momentum diffuses sideways, a plate oscillating in its own plane drags a layer of fluid with it, and the velocity profile is a decaying, lagging oscillation with $\delta = \sqrt{2\nu/\omega}$. That is the Stokes layer, and its thickness is why a tuning fork’s damping depends on frequency and why an oscillating body in a fluid feels a force in phase with neither its displacement nor its velocity.

And in an alternating chemical reaction at an electrode, the concentration of a reactant obeys the same equation with a molecular diffusivity, giving a diffusion layer whose thickness sets the impedance of the electrode and whose $\sqrt{\omega}$ dependence is the signature that impedance spectroscopy looks for.

And in a body, the same equation with a metabolic source in it governs how deep a thermal insult reaches. A cold surface cools the skin to a depth of millimetres over minutes and does not reach the core, for the same reason a frost does not reach a water main — and the penetration depth of a thermal treatment through tissue is computed from exactly this expression before anybody switches anything on.

Four subjects, one solution, one square root. The material constants differ by fourteen orders of magnitude and the arithmetic does not change.

Why the two effects have to be equal, and not merely both present #

It is worth asking whether the equality of the decay rate and the phase gradient is a fact about heat or about the form of the equation, because the answer decides how far the pattern travels.

It is about the form. Any equation whose time derivative appears once and whose space derivative appears twice gives $k^2 \propto i\omega$, and taking the square root of $i$ is what puts equal parts into the real and imaginary components — the square root of $i$ is $(1+i)/\sqrt2$, and the $1$ and the $i$ are what become the attenuation and the lag.

Change the equation and the relation changes. A wave equation has two time derivatives, so $k^2 \propto -\omega^2$ and $k$ is purely real: propagation with no attenuation. Add a small damping term and $k$ acquires a small imaginary part, so a slightly damped wave has a large phase gradient and a small decay — the reverse of the diffusive case, where the two are equal. A telegrapher’s equation, with one of each, interpolates between them and its penetration depth crosses over from one behaviour to the other at a frequency where the two terms balance.

So “attenuation equals phase gradient” is a fingerprint. Measuring both in an unfamiliar system and finding them equal is evidence that the governing equation is diffusive, and finding them unequal says how far from diffusive it is — which is how the crossover between ballistic and diffusive heat transport is identified in materials where the mean free path is not small.

Where it stops #

The surface temperature is not actually imposed. The real boundary condition is a heat balance: sunlight in, infrared out, evaporation, convection to the air. What is imposed is a flux, and the surface temperature is whatever makes the flux balance. Solving it properly changes the amplitude at the surface and shifts the phase there by some weeks, and leaves everything below unchanged in form — which is why the idealisation is useful despite being wrong at the one place it is stated.

The diffusivity is not a constant. Soil dries in summer and wets in winter, and dry soil has perhaps half the diffusivity of damp. The wave therefore travels through a medium whose properties vary with the season it is carrying, which makes the problem nonlinear and introduces harmonics that a linear solution cannot contain.

Freezing is worse than that. Where the ground freezes, the latent heat of the water in it is absorbed or released at a fixed temperature, so the surface of the freezing front is a moving boundary at which heat is consumed without changing a temperature. That is a Stefan problem, not a diffusion problem, and the depth of frost penetration it predicts is not given by anything above.

Ground that freezes spends energy at constant temperature, and the linear solution has no way to account for it. A phase change absorbs heat without any temperature change at all, so the 0 °C isotherm of the diffusion solution is not the frost line: the real front lags it, stalled while the latent heat is paid. That is why frost depth is tabulated by region rather than computed from a diffusivity, and why the tables are about soil water content as much as about climate.

Nor is the ground semi-infinite. Everything above assumes the medium extends downward for ever, which for the annual wave in soil is true enough — four penetration depths is nine metres, and the signal is gone by then. For a wall or a slab it is emphatically false: a layer of finite thickness reflects the wave off its far boundary, the reflection interferes with the incoming one, and the result depends on what is on the other side. That is the difference between a wall backed by air and one backed by more of the same material, and it is why the design rule above applies to a heavy wall and not to a thin panel with insulation behind it.

And soil is not still. Water moves through it, carrying heat with it, and a flow of a few millimetres a day is enough to compete with conduction at these depths. Where groundwater moves rapidly, the temperature profile measures the flow rather than the diffusivity — which is used deliberately: measuring how much a river bed’s temperature signal is distorted is a standard way of gauging how much water is exchanging with the aquifer beneath.

And $D$ itself has been treated throughout as a given number. It is an average over a great many collisions — a mean free path times a mean speed, divided by three, for a gas — so every result here is a statement about a material’s microscopic transport wearing a macroscopic coat. A soil’s diffusivity varies by a factor of three with its water content, which is a larger uncertainty than anything else in the calculation.

The problem Fourier solved, and what Kelvin did with it #

This is not an application of the diffusion equation; it is very nearly the reason there is one.

Fourier’s Théorie analytique de la chaleur of 1822 treats the periodic heating of the ground at length, works out the attenuation and the lag, and notes the inversion of the seasons at depth. The series that carry his name were developed as the tool for exactly this class of problem — decompose an arbitrary surface history into sinusoids, propagate each with its own $\delta$, and add them up. Because $\delta$ depends on frequency, the ground is a low-pass filter: a sharp cold snap is smoothed away within centimetres and the annual cycle survives to metres, which is the frequency-dependence made physical.

Kelvin then used the same mathematics on a much larger problem. If the Earth began molten and has been cooling by conduction ever since, the present temperature gradient at the surface fixes how long it has been cooling. He got between 20 and 400 million years, and refined it downward to 20–40, and used it against Darwin and the geologists.

The arithmetic was right and the model was wrong in a way nobody could have known: there was no radioactivity in it, so no internal heat source, and no clock inside the rock, and no mantle convection, so heat was assumed to move by the mechanism above when in fact most of it is carried by flow. Both discoveries came later, and the lesson is the one this collection keeps meeting — the calculation was checkable, the assumption underneath it was not, and the assumption was where the error lived.

The ladder from here #

Later rungs on this anchor: the Stefan problem, where a phase boundary moves and the depth of frost penetration is the answer; the inverse problem of borehole climatology, in which past surface temperatures are reconstructed from the residual transient in a deep temperature profile; the thermal response of a layered ground, where each layer has its own $\delta$ and the reflections matter; and the coupled transport of heat and water, where the two diffusivities are not independent and the equations no longer separate.

The neighbouring ladders are the equation that only runs forwards, which is this equation met as a transient; how far a field gets into metal, which is the same solution with a magnetic field in it; and the jiggle that proved atoms, which is what the diffusivity is an average over.