The column that boils from the middle

A boiling point is a pressure took the familiar hundred degrees and showed it to be one point on a curve: water boils at whatever temperature its vapour pressure equals the pressure on it, 71 °C on Everest and nearly 120 °C in a pressure cooker. That essay changed the pressure by changing the altitude or sealing a lid. A column of water changes it by itself. Every metre of depth adds the weight of a metre of water, nearly a tenth of an atmosphere, and so every metre of depth has a boiling point of its own.

In most water that is an idle fact, because most water is nowhere near boiling. In a geyser it is the whole mechanism. Robert Bunsen went to Iceland in 1846, lowered thermometers into the twenty-three-metre tube of the Great Geysir between eruptions, and found something that is still the first thing to understand about any geyser. The water was hottest at the bottom, as it should be in a tube heated from below. But the water closest to boiling was not at the bottom. It was in the middle, because the boiling point there was lower than at the bottom by more than the water was cooler.

A boiling point for every depth #

The first figure draws the curve for a conduit at the elevation of Yellowstone’s Upper Geyser Basin, 2,240 metres up, where the air’s pressure is 77.6 kilopascals and open water boils at 92.7 °C. Going down, the pressure grows by 9.4 kilopascals per metre and the boiling point rises with it, from the saturation curve that every steam table is built on. It rises fast at first — eleven degrees in the first four metres — and more slowly below, only sixteen degrees over the last twelve. The flattening is the Clausius–Clapeyron relation at work. The slope of the vapour-pressure curve, $dp/dT = L/T\Delta v$, grows steeply with temperature because the vapour’s density does, so each additional kilopascal of pressure buys less and less additional boiling temperature. A column of water is a stack of pressures in equal steps, and the boiling points it reaches are steps that shrink.

Water heated from below does not follow that curve. Hot water enters the bottom of the conduit, rises, and mixes with what is there, and the column below a few metres is stirred nearly uniform by convection: in the figure it warms by only 0.8 °C per metre of depth, far less than the boiling curve’s one and a half to three. At the top the pool loses heat to the air and to evaporation, and the uppermost few metres are cooler still. The profile is therefore a bent line: steep near the surface, gentle below.

As heat keeps arriving, the whole profile moves to the right. Where it first touches the boiling curve is decided by geometry, not by where the heat comes in. The gap between the two is narrowest at the bend, the foot of the cooler layer, because above it the water is too cool and below it the boiling point outruns the water. In the figure the first boiling happens at four metres down, at 103.6 °C. The bottom water, at 118 °C, is the hottest in the tube and the furthest from boiling, 13.7 °C short. Bunsen’s mid-tube reading was this geometry, measured.

Unloading the column #

Boiling at four metres makes bubbles, and bubbles take up room. The water above them is lifted, and some spills over the rim of the pool: the swelling and splashing that geyser watchers call preplay. Whatever spills is gone, and the pressure everywhere beneath has fallen by its weight.

That is the step the second figure follows. Each parcel of water below has the temperature it had a moment before, and it now sits under less water. Its boiling point has dropped, and where it has dropped below the parcel’s temperature the parcel is superheated: liquid at a temperature where it would rather be vapour. Throw out half a metre and the layer around four metres is superheated by about a degree. Throw out two metres and the superheated layer reaches seven metres, by up to five degrees. Throw out five metres and everything from five to twelve metres down is superheated, by up to twelve degrees.

Superheated water does not stay liquid for long in a geyser’s tube. The barrier a new phase has to climb found that a bubble must reach a critical size before it can grow, and that in clean water a superheat of many tens of degrees can be held; but a conduit’s walls are rough, its water carries dissolved gas, and the bubbles already rising from the first boiling seed the rest. The superheated layers flash.

Steam that makes more room than it was given #

How much a superheated parcel flashes is fixed by energy alone. Its excess temperature ΔT is heat that the liquid can no longer hold, and boiling a mass $m_v$ of it takes $m_v L$. So the fraction that turns to steam is

$$x = \frac{c_p\,\Delta T}{L},$$

which for water is about one per cent for every five degrees of superheat. That small fraction is the mirror image of the one the supercooled water that only partly freezes found: water cooled below its freezing point turns only a fraction $c_p\Delta T/L_f$ to ice before its own latent heat warms it back to the melting point. Here the latent heat cools the water back to its boiling point, and only a fraction becomes vapour. The same arithmetic, with the sign of every quantity reversed.

What makes a geyser violent is not the mass of steam but its volume. Steam at a few tenths of an atmosphere above atmospheric is about two thousand times less dense than the water it came from, so one per cent of the mass becomes twenty times the volume. The third figure adds that up over the whole column: for each amount u of water thrown out of the top, how much steam the superheated layers beneath produce, compared with the room u that was made for it.

For small unloadings the steam falls short. A centimetre of water thrown out superheats only a thin layer by a fraction of a degree, and the steam it releases fills six per cent of the room: the column settles back, the bubbles collapse into the cooler water above them, and the pool subsides. This is preplay, and it can go on for many minutes. But the steam released grows faster than the water removed — nearly as its square, because a deeper unloading superheats both a thicker layer and by more. The two curves cross at about fourteen centimetres. Past that, every slug of steam takes more room than the water it displaced, throws out more water, superheats more beneath it, and the column runs away. A tenth of a cubic metre of steam per square metre becomes eight cubic metres by the time a metre has gone. There is no small eruption beyond the crossing. Once it is passed, the geyser empties to the depth at which the stored heat runs out.

The violence grows as the column empties #

The runaway accelerates for a second reason, which the fourth figure isolates. The same superheat makes more steam, by volume, the lower the pressure at which it flashes, because low-pressure steam is less dense. Ten degrees of superheat flashing at the surface fills thirty-nine times the water’s volume; at four bar, about thirty-three metres down, it fills nine. As an eruption proceeds the column above every parcel shortens, the pressure at which the parcel flashes falls, and each degree of superheat it carries produces more steam than it would have done earlier. An eruption that begins with water spilling over the rim ends, seconds later, with a two-phase jet of steam and droplets leaving the vent at tens of metres a second, fed by a column that is boiling from the top down faster than it can drain.

This also explains the shape of the eruption in time. Old Faithful’s typically lasts two to five minutes. The water jet comes first, while the conduit is full of flashing liquid; then, as the water runs out, it gives way to a steam phase, roaring and less visible, while the deeper and less superheated water boils off. The eruption ends when the column has been brought back down to its boiling curve everywhere — when the water that remains, at its new lower temperatures, is no longer above the boiling point of its new, lower pressure.

The same arithmetic in a mug and a boiler #

A geyser is the largest natural example of an explosion that is common at every other scale: a liquid held above its boiling point by pressure, and then relieved of the pressure. The smallest version happens in a kitchen. Water heated in a smooth cup in a microwave oven can reach several degrees above its boiling point without boiling, because nothing in the cup provides the rough surface or the trapped gas from which a bubble can start; the oven heats the water throughout rather than at a hot base where bubbles normally form. Drop in a spoonful of sugar or tap the cup and the superheat flashes: a fraction $c_p\Delta T/L$ of the water becomes steam in a fraction of a second, a few millilitres of liquid making several litres of vapour, and the cup erupts. Laboratory flasks do the same, which is why chemists boil liquids over chips of porous stone that release a stream of small bubbles before any superheat can build.

The largest versions are industrial. A steam boiler holds water far above its atmospheric boiling point, at 180 or 250 °C, kept liquid only by the pressure of the vessel. If the vessel fails, the whole charge is suddenly at atmospheric pressure with a hundred or more degrees of superheat. By the arithmetic of the fourth figure, between a sixth and a third of the water flashes to steam at once, and that steam occupies hundreds of times the volume of the water that made it. Nineteenth-century boiler explosions demolished buildings this way, and the same mechanism, called a boiling-liquid expanding-vapour explosion, is what makes a ruptured tank of liquefied gas so destructive even before anything burns.

What distinguishes the geyser is that it makes its own rupture. No vessel fails. The lid is the water itself, and the column removes it in instalments, each instalment uncovering more superheated water beneath. That is why a geyser can erupt over and over from the same plumbing for centuries, and why its eruptions build up over seconds rather than happening in a single blast.

The same column at other heights #

The last figure moves the column up and down the planet. At sea level the curve starts at a hundred degrees; at El Tatio, a geyser field on the Altiplano at 4,300 metres, it starts at 86 °C. Thirty metres down the three curves have nearly met, because thirty metres of water weighs three times as much as the whole atmosphere above even the lowest of them. The differences that matter for a geyser are near the surface: a high field’s water reaches its boiling curve at a lower temperature, and the curve is steeper there, so a given unloading superheats the water near the top by more. That steepness is what the runaway depends on, so a geyser’s elevation shapes how it erupts.

The water’s depth is no less important than its temperature, and the conduit’s shape decides both. Old Faithful’s conduit has been explored with cameras lowered down it: a fissure that narrows to a constriction roughly ten centimetres wide a few metres down and widens below it. The constriction slows the exchange between the deeper water and the pool, which keeps the upper column cooler and sets where the first boiling happens. Geysers whose tubes are wide and open, like Iceland’s Strokkur, erupt every few minutes from a shallow pool; geysers with deep, constricted plumbing and side chambers erupt less often and more violently. Every one of them works by the same arithmetic of a boiling curve and a column that unloads itself.

Where the model holds #

The figures treat the conduit as a straight vertical tube of uniform water, heated by a profile chosen to resemble what is measured, and they treat the flashing as instantaneous and complete: each superheated layer turns exactly the fraction $c_p\Delta T/L$ to steam at its new pressure. The real process has a rate. Bubbles must nucleate and grow; steam must separate from water or carry it along; and the time this takes, set against the time the column takes to unload, decides whether an eruption is a series of surges or a single sustained jet. The figures also ignore the gas dissolved in geyser water, chiefly carbon dioxide, which comes out of solution as the pressure drops and adds its own bubbles to the steam’s, lowering the threshold for runaway; and they ignore the cooling of the conduit walls, which absorb heat from the flashing water and stop the process sooner.

The density of water is held fixed at its value near a hundred degrees. Hot water is about four per cent less dense than cold, which shifts the boiling curve by a fraction of a degree over the depths drawn, and real geyser water is a dilute silica solution, slightly denser than pure water. Neither changes the shape of the argument.

What the curves cannot show #

The curves show temperatures and volumes, not motion. They cannot show the plumbing that a real geyser’s behaviour depends on: side chambers that trap steam and act as bubble traps, as seismic and video studies have found beneath geysers in Yellowstone and Chile, and the narrow places that make a column flash in stages. They cannot show the interval between eruptions, which is the time the heat inflow takes to bring the column back up to its curve. That interval is what geysers are famous for, and it is predicted from the length of the previous eruption rather than from the physics drawn here: a long eruption empties more water and more stored heat, and the recharge takes longer. Nor can they show why most hot springs never erupt at all. A spring whose water can circulate freely enough to stay below its boiling curve, or whose conduit is too wide to hold a column, simply boils quietly at the top.

Still open: what keeps a geyser regular #

Old Faithful is regular enough to be predicted to within about ten minutes, and its intervals cluster around two values, about an hour and about an hour and a half. Other geysers are periodic, irregular or chaotic, and some change their behaviour after earthquakes, which alter their plumbing and their supply of heat. Records from instruments placed in geyser conduits show that the column often comes close to eruption several times and backs off, in the preplay that the third figure describes as a disturbance below the runaway threshold, before one disturbance passes it. What sets how many such near misses a geyser makes, why some settle into two intervals and others into one, and whether the pattern of intervals carries information about the shape of the plumbing below — and so could be used to map it — are questions being pursued with seismometers, pressure sensors and cameras lowered into the tubes. Each geyser so far has turned out to have its own answer.

The habit worth carrying away is to ask what a quantity depends on before asking where it is largest. Water heated from below is hottest at the bottom of a geyser’s tube, but depth raises the boiling point faster than the water warms — from 92.7 °C at the surface to 131.8 °C at 22 m at Yellowstone’s elevation — so the column first boils partway down, and throwing out the water above superheats everything beneath: 5 m removed superheats the water to 12 m by up to 11.8 °C, and once more than about 14 cm has gone, the steam takes more room than the water it displaced and the eruption feeds itself. A geyser does not erupt because it gets hot enough. It erupts because it can unload.