The plume that stops below the surface

The layer a parcel cannot leave displaced a single parcel of fluid upward in a stratified column and found it pushed back, oscillating about its home level at the buoyancy frequency $N$, the frequency set by how quickly density changes with height. A parcel is an idealisation: a blob that moves without exchanging anything with its surroundings. The fluid rising from a fire, a chimney, a volcanic vent or a leak on the seabed is nothing like a parcel. It is a continuous plume, turbulent, constantly drawing in the fluid around it and mixing it in. That one difference changes the answer to the question of how high it goes, and makes the answer depend on the source in a way no parcel calculation can capture.

The theory that settles it was written in 1956 by Bruce Morton, Geoffrey Taylor and Stewart Turner, and it rests on a single assumption about turbulence that is as simple as a fluid-dynamical assumption can be. The plume’s turbulent eddies draw surrounding fluid into it at the edge, at an inward speed proportional to the plume’s own rising speed. The constant of proportionality, the entrainment coefficient, is about one tenth. Everything else follows from conservation of volume, momentum and buoyancy.

Three fluxes and an assumption #

A plume rising from a point carries three things upward through each horizontal level: a volume of fluid per second, a momentum per second, and a buoyancy per second — the rate at which it transports its deficit of density relative to the surroundings, weighted by gravity. In a fluid of uniform density, the buoyancy flux is conserved: the plume dilutes as it entrains, but the total buoyancy it carries stays the same. The momentum flux grows, because buoyancy keeps pushing. And the volume flux grows fastest of all, because entrainment keeps adding fluid.

In a stratified fluid the buoyancy flux is not conserved, and the first figure shows why it falls. A plume rising from the bottom of a stratified layer entrains fluid at each height it passes through, and that fluid came from lower down than the fluid it is now surrounded by, so it is denser than its new surroundings. Each parcel of entrained fluid is a small dead weight carried upward, and the plume’s buoyancy flux falls steadily from the moment it leaves the source. At a height of $4.37\,F_0^{1/4}N^{-3/4}$ for an entrainment coefficient of 0.1 — where $F_0$ is the source’s buoyancy flux — the plume as a whole has the same density as the fluid around it.

It does not stop there. It is still moving upward, and its momentum, built up by the buoyancy all the way from the source, carries it on. Now it is heavier than its surroundings, decelerating, and at $5.75\,F_0^{1/4}N^{-3/4}$ it stops. What happens next is beyond the equations: the fluid at the top falls back, collapses towards the level at which it is neutrally buoyant, and spreads out sideways as a thin layer, an intrusion, travelling away from the source along a surface of constant density.

One length that nothing else can make #

The scaling in that answer deserves notice. The plume’s height depends on two quantities, the source’s buoyancy flux, with units of length to the fourth over time cubed, and the stratification’s frequency, with units of one over time. There is exactly one way to combine them into a length: $F_0^{1/4}N^{-3/4}$. Any plume, in any stratified fluid, whatever its source, rises to a height that is a fixed multiple of that length, with the multiple depending only on the entrainment coefficient. The figures check that the computed height for very different buoyancy fluxes and stratifications collapses onto the same multiple.

That is the reason the theory is so widely used. It does not need to know whether the plume is smoke in air, oil in seawater or ash in the atmosphere, or anything about the turbulence beyond the single coefficient. It needs the buoyancy flux, which can be estimated from the source’s heat output or its rate of release of light fluid, and the stratification, which can be measured with a thermometer and a salinity probe. The stratification enters through the same frequency $N$ that the layer a parcel cannot leave found governing a displaced parcel’s oscillation, and the plume’s rise time to its top, about $N^{-1}$ times a number of order one, is the time a parcel takes to feel the stratification at all.

What a plume does in uniform surroundings #

The second figure sets the stratified plume beside the same plume in a fluid of uniform density. In uniform surroundings the solution is a similarity solution that Morton, Taylor and Turner wrote down in closed form: the plume’s radius grows in proportion to height, with a spreading angle set by the entrainment coefficient — about seven degrees of half-angle — and its rising speed falls as the cube root of height, never reaching zero. Near the source the two plumes are indistinguishable, because the stratified one has not yet risen far enough for the density of its surroundings to have changed much. Above about one unit of height they part company, and the stratified plume’s speed falls steeply to zero.

It is worth seeing why the uniform plume never stops even though it slows. The buoyancy it carries is diluted into an ever larger volume, but the volume is also being pushed by all of it, and a plume whose buoyancy flux is constant has a rising speed falling only as the cube root of height. A single buoyant parcel released into uniform surroundings would accelerate forever; a plume, sharing its buoyancy with everything it entrains, decelerates forever; neither stops. Only stratification makes a plume stop, and almost every real fluid has some. Even a solar pond, which the pond that is hottest at the bottom found holding its hottest water at the bottom by a gradient of salt, is a case of the same arithmetic: there the salt’s stratification is made strong enough that the warm water at the bottom cannot start a plume at all.

The plume that rises from a single warm wire, a laminar plume governed by viscosity rather than entrainment, behaves differently again — its speed can increase with height — and the turbulent plume here is the one that applies to anything larger than a candle flame.

From a hydrothermal vent to a volcano #

The third figure applies the scaling. Because the height goes as the quarter power of the buoyancy flux, plumes from sources spanning many orders of magnitude end up at heights spanning far fewer. A hydrothermal vent on the deep seafloor, a black smoker venting water at three or four hundred degrees, rises a few hundred metres through the weakly stratified deep ocean before it spreads; oceanographers find such plumes as clouds of particles and dissolved metals hanging two or three hundred metres above the vent fields, and use them to locate vents they have not yet seen.

The most consequential application of the theory in recent decades was the 2010 blowout of the Macondo well in the Gulf of Mexico, 1,500 metres below the surface. Much of the oil and gas never reached the surface. The plume of oil droplets, gas bubbles and entrained deep water rose a few hundred metres, lost its buoyancy against the stratified deep Gulf, and left a layer of dissolved hydrocarbons spreading at about 1,100 metres depth that research vessels tracked for tens of kilometres. The theory, with the blowout’s estimated buoyancy flux, puts the trapping height in the right place. What it cannot do on its own is say what fraction of the oil escaped upward in drops large enough to separate from the plume, which depends on the drop sizes and was one of the central uncertainties of the response.

The tropopause as a lid #

In the atmosphere the relevant source is heat. A fire or an eruption heats the air it releases, and the buoyancy flux is the heat output times gravity over the air’s density, heat capacity and temperature. The fourth figure integrates the plume through an atmosphere with a realistic two-part stratification: a troposphere in which temperature falls with height, giving $N$ of about 0.01 per second, capped at around 11 kilometres by the tropopause, above which temperature stops falling and $N$ roughly doubles.

A large forest fire, a gigawatt of heat, sends its smoke a kilometre or two up; the biggest wildfires, tens to hundreds of gigawatts, can drive plumes through most of the troposphere, and when they condense their water they build thunderclouds of their own. To reach the stratosphere takes a few terawatts, and once there the plume meets fluid twice as strongly stratified, so each further kilometre costs much more heat. The great explosive eruptions, which send columns of ash twenty to forty-five kilometres high, sit at the right of the figure. Because height goes as the quarter power of heat output, a column’s height measured from a satellite or a radar gives the eruption’s mass eruption rate to within a factor of a few, and volcanologists use exactly this relation to estimate how much ash is being injected in real time — the number that decides which airspace has to close.

The same physics sets where the eruption’s products go. Ash and sulphur dioxide spread at the column’s neutral level, and an eruption that puts its sulphur into the stratosphere, above the weather that would rain it out, can cool the climate for a year or two, as Pinatubo’s did in 1991; one whose column stops below the tropopause cannot.

The overshoot rings the surroundings #

The top of a stratified plume is not a quiet place. The fluid that overshoots the neutral level is heavier than its surroundings and falls back, overshoots downward, and oscillates about the neutral level at roughly the buoyancy frequency before it settles and spreads. A stratified fluid displaced at a frequency below $N$ does not simply slosh; the wave that picks an angle found that it radiates internal waves along beams whose angle to the vertical is fixed by the ratio of the forcing frequency to $N$. A plume’s fountain-like top is therefore a source of internal waves, carrying energy away from the plume horizontally and downward through the stratification. Laboratory experiments with plumes in salt-stratified tanks show the beams clearly, and the energy they carry off is a small but measurable share of what the plume brought up.

The same overshoot is what makes the neutral level, not the top, the height that matters for where a plume’s contents end up. Whatever the plume carried — smoke, ash, dissolved gas, heat — settles out at the level of neutral buoyancy, a little below the top, and spreads there. For a seabed blowout or a volcanic eruption, that spreading layer is what is tracked and forecast.

When mixing changes what is light and what is heavy #

The theory assumes that mixing two fluids gives a fluid whose density is the average of theirs. For air, and for fresh water over the range of a lake’s temperatures, that is a good approximation. For seawater near freezing it is not. The mixture heavier than either water found that seawater’s density depends on temperature and salinity in a curved way, so that mixing two waters of equal density can produce water denser than either, cabbeling. A plume rising through cold polar water entrains surrounding water at every level, and if the mixing makes denser water, the plume loses buoyancy faster than the linear theory says and stops lower. Plumes of meltwater rising along the faces of glaciers, where fresh water from under the ice mixes into salty cold ocean water, are governed by exactly this: whether the plume reaches the surface or stops and spreads at depth, where it carries heat to the ice and melts more of it, depends on the curvature of seawater’s equation of state.

Salt and heat diffuse at different rates as well, so the boundary of a warm, salty plume entering cold, fresh surroundings can develop the fingers that the salt that sinks through a stable sea found carrying salt downward through a stably layered ocean. Over the short time a plume takes to rise, the effect on its height is small; over the long time an intrusion spends spreading at its neutral level, it can decide whether the layer holds together or slowly drains.

The single number the theory has to be told #

The theory has one empirical input, the entrainment coefficient, and the last figure shows how much rides on it. The heights fall as the inverse square root of the coefficient, so the plausible range of measured values moves the prediction by about a fifth either way. The coefficient stands for the turbulent eddies at the plume’s edge, which engulf surrounding fluid and fold it in, and nothing in the three conservation laws can predict it; it is measured, in laboratory tanks and in the field, and it depends slightly on how the plume’s profiles are defined (a top-hat profile, as here, or a Gaussian, which gives a smaller number for the same plume). Entrainment is mixing, and mixing is the irreversible step in the whole process: mixing what is already mixed found that combining two different fluids produces entropy that no later process can remove, and a plume’s intrusion, however neatly it spreads, never unmixes into the fluids it came from. Its approximate constancy over an enormous range of scales — from laboratory plumes a few centimetres high to volcanic columns tens of kilometres high — is an empirical fact about turbulence, not a consequence of any theory, and it is what makes the whole edifice useful.

Plumes in rooms and buildings #

The same theory, turned round, explains how buildings ventilate. A heat source in a room — a person, a radiator, a computer — sends up a plume that entrains room air on its way to the ceiling, spreads there, and gradually fills the room from the top with warm air, building up a stratification of its own. Paul Linden and colleagues worked out the steady states in the 1990s: with openings at the top and bottom, the room settles into two layers whose interface sits at a height set by the plume’s entrainment and the openings’ areas, independent of the strength of the heat source. The building that breathes through its doors found the pressure difference that drives that stack-effect flow; the plume theory says how the warm and cool air arrange themselves inside.

Fire engineering uses the same equations to estimate how fast a burning room’s smoke layer descends, and so how long its occupants have before the smoke reaches head height. The entrainment that dilutes the smoke also adds volume to it, which is why a ceiling smoke layer thickens far faster than the fire’s own production of gas would suggest.

Where a slender column stops being one #

The Morton–Taylor–Turner equations treat the plume as a slender column with profiles of the same shape at every height, entraining at a rate set by its local speed. They fail where those assumptions fail: very close to the source, where the plume is still forming; at the top, where it spreads sideways, a region the equations cannot describe and where the computed radius runs away; and wherever the surroundings are not quiet. A crosswind bends a plume over and greatly increases its entrainment, so that real smoke plumes on windy days rise much less than in still air; ocean currents do the same to seabed plumes. Particles and droplets that settle out of a plume, as volcanic ash and oil droplets do, change its buoyancy along the way, and a plume of gas bubbles behaves differently from one of hot water.

The atmosphere’s moisture adds another complication. A plume that rises far enough cools until its water vapour condenses, releasing latent heat, and then gains buoyancy rather than losing it. The cloud that cools more slowly than the air found the reduced lapse rate that saturated air follows; a fire plume that crosses its condensation level can turn into a thunderstorm whose height has little to do with the fire’s own heat.

Averages over the turbulence #

The figures draw averages: the fluxes of a steady plume, averaged over its turbulence and over the plume’s cross-section. They do not show the turbulence itself, the large eddies at the plume’s edge that do the entraining, nor the plume’s meandering, nor the intermittency that makes a real plume’s edge a shifting, ragged boundary rather than a smooth cone. Nor do they show the intrusion at the top, which spreads as a gravity current along the neutral level and carries the plume’s contents sideways for kilometres. The theory is a description of a steady average, and a real plume is never steady.

Still open: how much a plume entrains when it is not alone #

The entrainment coefficient is close to constant for a plume in still surroundings. It is not for plumes in turbulent surroundings, in crossflows, or when the plume carries particles or bubbles, and a plume rising from a large fire or a cluster of seafloor vents is not one plume but many, merging. How entrainment changes in each case is still being measured, with laboratory tanks, field campaigns and simulations that resolve the eddies directly, and the answers matter for predicting where oil from a deep spill will spread, how high a wildfire’s smoke will be lofted, and how much of a volcano’s sulphur reaches the stratosphere.

The habit worth carrying away is to look for the one length a problem’s quantities can make. A turbulent plume entrains its surroundings, and in a stratified fluid what it entrains from below is a dead weight above, so it stops at a height of $5.75\,F_0^{1/4}N^{-3/4}$ — overshooting the level where its buoyancy ran out, $4.37$ in the same units, for an entrainment coefficient of 0.1 — whether it is a black smoker rising three hundred metres, a seabed blowout trapped four hundred metres above its well, or an eruption column that a few terawatts lift into the stratosphere.