The geyser that remembers one eruption

The column that boils from the middle found how a geyser erupts. Water in its conduit is heated from below, and because the boiling point rises with depth faster than the water’s temperature does, the column first boils part-way down; a little water thrown out at the top lowers the pressure on everything beneath, the superheated water flashes to steam, and the eruption feeds itself until the conduit is spent. A boiling point is a pressure supplied the curve that makes this happen. That essay ended with what it could not explain: why a geyser such as Old Faithful erupts so regularly, and why its intervals cluster around two values rather than one.

The answer turns out not to need the details of the eruption at all. It needs only to count water.

What the rangers noticed #

For more than half a century the park rangers at Yellowstone have forecast Old Faithful’s eruptions with a simple rule: time the eruption that has just happened, and the next will follow after an interval that depends on it. Short eruptions, of two minutes or so, are followed by short waits; long ones, of four or five minutes, by long waits. The forecast is posted at the visitor centre and is usually right to within ten minutes. A handful of other geysers in the same basin are forecast too, from the timing of their own previous eruptions or of their neighbours’, with windows of an hour or more; Old Faithful is the one whose rule is tight enough to plan a day around. When statisticians took up the geyser’s record as a textbook example in the 1980s and 1990s, the relation they fitted was close to a straight line, an hour and a half of wait for a long eruption and about an hour for a short one, with roughly ten to eleven extra minutes of wait for every extra minute of eruption.

The first figure reproduces that pattern with a model built from nothing but a water budget, and the model is worth taking apart because it says why the rule works and what it is measuring.

A conduit that must refill #

The model treats the geyser as a container. Each eruption throws out a volume of water — the figures use the volumes reported for Old Faithful, about fourteen cubic metres in a short eruption and about thirty-two in a long one — at a roughly steady rate, so an eruption’s duration is proportional to the volume it discharges. Then the conduit refills from below, from the hot water seeping in through the surrounding rock, at a rate the figures set at six-tenths of a cubic metre a minute. Once it is full, it needs a further time to be heated back up to the boiling curve, here about half an hour, before the next eruption can start.

The second figure follows the conduit through six cycles. After each eruption the water missing from it falls steadily as it refills, and a long eruption leaves a deep hole that takes nearly an hour to fill while a short one leaves a shallow one filled in twenty minutes. Then comes the heating, the same each time, and the next eruption. The interval after an eruption is therefore the refill time plus the heating time,

$$I = I_0 + \frac{V}{Q},$$

linear in the volume $V$ the eruption discharged, and since an eruption’s duration is proportional to its volume, linear in the duration too. The slope is the ratio of the rate at which the geyser throws water out to the rate at which it takes water back: here about ten and a half minutes of wait for each minute of eruption, close to what the rangers’ rule uses. The bottle that pours at one rate however full it is found a vessel whose outflow is set by a pressure that does not change; the geyser’s inflow is like that, steady and independent of how empty the conduit is, which is what makes the refill time proportional to the hole.

Memory of one eruption, and of nothing else #

The rule’s strangest feature is its direction. The duration of an eruption predicts the interval after it and says nothing about the interval before it. In the model this is exact, because whether an eruption is short or long is drawn at random each time, but the refill is fixed by what has already happened. The first figure’s last number shows the asymmetry: plotted against the eruption that follows instead, the intervals show a correlation of four hundredths, which is nothing.

That makes the geyser an unusual kind of clock. Its state after each eruption is described by a single number, the volume of water it has to replace, and that number is reset completely when the conduit is full. Nothing in it carries information further back. A record of eruptions is a sequence in which each interval depends on one preceding event and is otherwise independent, and analyses of Old Faithful’s record have found it behaving roughly like that, its durations and intervals depending chiefly on the eruption just before.

This is the property of a relaxation oscillator, the kind of oscillator that charges slowly to a threshold and discharges suddenly: a dripping tap, a neon lamp flashing on a capacitor, a heart. The clock that is pulled into step studied oscillators that keep their own rhythm and can be locked to a drive. A relaxation oscillator keeps no rhythm of its own between discharges. The slowest one ever timed, the Queensland pitch drop of the drop that takes a decade, is one too: each drop falls when enough pitch has crept down into the hanging bulb to break it away, and the next interval is set by how fast the funnel refills the bulb, with no memory of the drops before. It forgets everything at each threshold, and its regularity is only the regularity of its refill.

The heat behind the water #

The water budget has a heat budget under it, and the numbers give a sense of scale. A long eruption throws out about thirty-two tonnes of water close to boiling, and to warm that much water from the temperature of the snowmelt that feeds the system to the boiling point at the vent takes about ten billion joules. Supplied over an interval of an hour and a half, that is a heat flow of about two megawatts, the output of a small power station, delivered by the rock and the deep water around a conduit a few metres across. It is a tiny fraction of the heat Yellowstone’s volcanic system releases, and it is what the geyser’s regularity rests on: a steady supply of heat and water, delivered faster than the surface can lose it, and released in pulses because water can be superheated.

That last condition is essential, and it is the link to the earlier physics. A pot on a stove does not erupt; it boils steadily, because bubbles form freely on its hot floor and carry the heat away as it arrives. A geyser’s conduit, its walls lined with smooth silica deposited by the geyser itself, has few places for bubbles to start, and the barrier a new phase has to climb found that without such sites boiling has to overcome an energy barrier, so water can be heated well past its boiling point before it flashes. The part of the curve no fluid follows traced how far a liquid can be pushed into that metastable state. The geyser stores heat in superheated water between eruptions and spends it all at once, and the store is refilled by the steady supply. Without the superheating there would be no pulses, only a hot spring.

Two waits, not one #

The third figure counts the intervals. They fall into two humps, one for the waits after short eruptions and one for those after long ones, and the humps are separated by a valley. The average of all the intervals falls in the valley. A geyser of this kind is regular — its waits are confined to two narrow ranges — without being periodic, and the most misleading number to publish about it is its average interval. Old Faithful’s record shows exactly this bimodal shape, and its two humps have shifted over the decades as its plumbing has changed.

Why the eruptions themselves come in two kinds is a different question, and the water budget does not answer it. In the model the choice is a coin weighted to come up long about two times in three. In the geyser it is decided by how the eruption proceeds: whether the boiling front that runs down the conduit when the eruption starts reaches the deep, hot water or stalls part-way, leaving a short eruption that empties only the upper part. The earlier essay found the eruption to be a runaway that feeds itself once enough water has been thrown out; whether it runs to the bottom depends on how hot the deep water is when the runaway begins, and that varies from cycle to cycle with the heat supplied from below. Instruments lowered into the conduit find the column coming close to erupting several times before it does, and an eruption that starts from a near-miss in a cooler state is more likely to stall.

A forecast that uses everything there is #

The fourth figure compares two forecasts. Announcing the average interval every time is wrong by sixteen minutes on a typical day, because the average is a value the geyser rarely takes. Using the duration of the eruption just watched cuts the error to about five and a half minutes. What remains is noise the model puts in deliberately, variations in how much water each eruption throws out and in how fast the conduit refills and reheats, which nothing at the surface reveals. In the real geyser the forecast’s error is similar, about ten minutes either way, and the model says why it cannot be much better: once the last eruption’s volume is known, there is nothing more to learn from the past, and the rest of the uncertainty is in processes underground that change from one cycle to the next.

This is a general point about forecasting a relaxation oscillator. Its whole state is reset at each discharge, so the best predictor of the next discharge is the most recent one, and a long record adds almost nothing to a single observation. The rangers’ rule is, in that sense, optimal, and the improvements made to it over the years have been adjustments for the geyser’s slow changes, not more elaborate uses of its history.

What an earthquake changes #

The last figure shows how the rule responds when the geyser’s supply changes. Geysers are sensitive to earthquakes, which open and close cracks in the rock around them and alter how fast hot water reaches the conduit. In the model, cutting the refill rate by a quarter leaves the eruptions as they were but stretches every wait in proportion to the water it has to replace, so the long waits lengthen more than the short ones and the slope of the rule steepens. Old Faithful’s average interval has lengthened after large earthquakes in the region, notably after the Hebgen Lake earthquake of 1959 and the Borah Peak earthquake of 1983, and the rangers have had to recalibrate their rule each time. In the water-budget picture, the slope of the rule is a measurement of the geyser’s refill rate, and a change in the slope after an earthquake reads how much the earthquake has narrowed the channels feeding it.

The reading is indirect, and that is its value. Nobody can see the cracks a few tens of metres below Old Faithful through which its water seeps back, and the rule’s slope, measured from the surface with a stopwatch, measures the rate at which they deliver it.

Where the water budget stops #

The model is a caricature of a geyser and is designed to be one. It treats the refill rate and the heating time as constants with a little noise, when both depend on the temperature and pressure of the water around the conduit and on how much heat the last eruption drew from the surrounding rock. It takes the choice between short and long eruptions as a weighted coin, when it is decided by the physics of the eruption itself. It ignores the bubble traps — side chambers that collect steam and release it into the conduit — that seismic studies have found beneath several Yellowstone geysers, and which can make a geyser’s behaviour depend on more than its water level. And its volumes and rates are set to match Old Faithful’s reported discharges and intervals rather than derived from anything; what the model shows is the structure of the rule, not a prediction of the numbers.

Other geysers do not obey such a rule. Some erupt at intervals with no relation to their previous eruptions, some are periodic with a single interval, some are chaotic, and some erupt in clusters. A geyser whose eruptions always empty its conduit to the same level has one interval; one whose eruptions are triggered by heat arriving in bursts from below behaves irregularly whatever its water budget. Old Faithful’s two kinds of eruption and steady refill make it a particularly clean case.

Geysers that run on something other than heat #

The water-budget argument never used the fact that Old Faithful is hot. It used a conduit that empties in a discharge and refills at a steady rate, and a threshold at which the next discharge begins. Anything with those three parts should keep the same one-eruption memory, and some geysers that are not driven by boiling at all show how general that is.

A cold-water geyser erupts because of dissolved gas rather than heat. Groundwater that has passed through rock giving off carbon dioxide holds the gas in solution under the pressure of the column above it, as a sealed bottle of soda water does. When enough gas has accumulated for bubbles to form near the top, the bubbles lighten the column, the pressure below drops, more gas comes out of solution, and the column runs away into an eruption in exactly the way a superheated column runs away into boiling — the threshold is a saturation pressure rather than a boiling point. The tallest, at Andernach on the Rhine, throws water about sixty metres high roughly every two hours. What it must replace between eruptions is both the water and the dissolved gas, and the wait is set by whichever takes longer to come back.

Further away still, the plumes of ice and vapour from cracks near the south pole of Saturn’s moon Enceladus are driven from an ocean beneath its ice shell. They are not intermittent in the way a geyser is: they erupt continuously, but their brightness rises and falls over each 33-hour orbit, brightest when the moon is furthest from Saturn and the tidal stretching opens the cracks widest. That is a discharge regulated by a clock outside the system rather than by a refill inside it, and the difference shows up in exactly the statistic the figures here are built on. Enceladus’s plume follows its orbit, whatever it did last time; Old Faithful’s next eruption follows its last, whatever the clock says.

What an interval record cannot reveal #

The figures draw times and volumes and leave out the eruption itself, the heating of the water and the structure of the conduit, all of which the earlier essay was about. An interval record is a time series of discharges, and many different plumbings could produce the same series: a deep reservoir slowly heated, a shallow one quickly filled, a chamber that traps steam. The rule’s slope measures a ratio of rates, not the rates themselves, and the intercept, the heating time, lumps together everything that happens after the conduit is full. The noise that warns a tube is about to sing found that fluctuations before an instability carry information about how close it is; a geyser’s preplay, the near-eruptions before the real one, carries similar information that an interval record discards.

Still open: what decides a short eruption from a long one #

The water budget explains the rule that links an eruption to the next interval; it does not explain why the eruptions come in two kinds. Instruments placed in and around Old Faithful’s conduit, measuring temperature, pressure and the seismic noise of boiling, have recorded the column before and during eruptions and have shown that the heating is uneven and that eruptions start from a range of states. Whether a short eruption is a long one that stalled, and what sets the probability of stalling — the temperature of the deep water, the amount of steam in a side chamber, the timing of a pulse of heat from below — is not settled, and each instrumented geyser has given a somewhat different answer.

The habit worth carrying away is to look for what a system has to replace. A geyser that throws out 14 or 32 cubic metres of water in an eruption and takes it back from below at a steady rate waits $I_0 + V/Q$ before it can erupt again, so its next interval is fixed by its last eruption — about an hour after a short one and an hour and a half after a long one, ten and a half minutes of wait per minute of eruption — and by nothing earlier, which is why a single stopwatch reading forecasts it to within minutes while its average interval is one it rarely keeps. A system that is reset at every discharge remembers only what the last discharge took from it.