The salt that sinks through a stable sea

In 1956 Henry Stommel, Arnold Arons and Duncan Blanchard published a short paper under the title An oceanographical curiosity: the perpetual salt fountain. Their thought experiment was a long thin pipe, open at both ends, pushed vertically down through the warm upper layer of a tropical ocean into the cold water beneath. Pump some deep water up the pipe to start it off, they said, and then stop pumping. The water will keep rising for ever.

The reasoning took four lines. The deep water is cold and fresh; the surface water is warm and salty. Both effects make surface water light and deep water heavy — no, only one of them does. Warmth makes water light and salt makes it heavy, so the surface water is light because of its temperature and heavy because of its salt, and in the tropics the temperature wins. The water column is stable. But a metal pipe conducts heat and does not conduct salt. Deep water rising slowly inside it takes on the temperature of the water outside at each level while keeping its low salinity, so at every height it is as warm as its surroundings and fresher than them, and therefore lighter. It rises. Nothing about the column’s stability is violated, and a fountain runs on it with no pump.

The layer a parcel cannot leave is built on the argument the pipe breaks. A parcel lifted in a stratified fluid keeps its own properties, finds itself denser than its new surroundings and falls back; the column is stable if its density increases downward, and the frequency of the swing is $N$. That argument assumed the parcel kept everything it started with. Four years after the curiosity was published, Melvin Stern pointed out that the ocean does not need a pipe. The difference between how fast heat and salt diffuse is a property of water, and a thin enough tongue of water is its own pipe.

A column with both ingredients the wrong way up #

The profiles are an idealisation shaped like the central water of the subtropical North Atlantic, where evaporation leaves the surface warm and salty and the water below is fed from colder, fresher sources. Temperature and salinity both fall with depth. Each on its own would say something different about stability: the temperature gradient alone would make the column very stable, and the salinity gradient alone would make it top-heavy. Density sees both, weighted by the thermal expansion coefficient $\alpha$ and the haline contraction coefficient $\beta$, and the third panel shows the result increasing steadily downward — a column that, by the parcel argument, should sit still.

The number that matters is in the fourth panel. The density ratio

$$R_\rho = \frac{\alpha\, \mathrm{d}T/\mathrm{d}z}{\beta\, \mathrm{d}S/\mathrm{d}z}$$

says how many times more the temperature gradient does for stability than the salinity gradient does against it. Below one the column overturns outright. Above one it is statically stable, with a buoyancy frequency set by the difference $N^2 = g(\alpha T_z - \beta S_z)$. In this column the ratio is about two everywhere, which means the heat is doing twice what the salt undoes — and that leaves salt carrying half as much potential energy in the wrong direction as the heat carries in the right one. Where that stored energy can go, when nothing about the density says it can go anywhere, is the question.

The thin parcel that keeps sinking #

The integration follows a parcel’s displacement together with its temperature and salinity excesses, and lets each excess leak into the surroundings at its own diffusive rate — $\kappa_T k^2$ for heat and $\kappa_S k^2$ for salt, where $k$ is $2\pi$ over the parcel’s width. The difference between the two rates is all that changes between the curves.

Pushed down, the parcel arrives in water that is colder and fresher than itself, so it is warmer (lighter) and saltier (heavier) than its neighbours. Initially the warmth wins, as it does for the whole column, and the parcel is pushed back up. That is the first half-swing of every curve in the drawing, including the thin one.

What happens next depends on how long the parcel keeps its warmth. A parcel two metres across is too wide for heat to leave it in any time that matters; it oscillates at the buoyancy frequency, exactly as the parcel in the layer it cannot leave does. A parcel six centimetres across loses its temperature excess in about eleven minutes, while its salt excess lasts about eighteen hours. By the time it has come back up to its starting level and overshot, its warmth has gone, its salt is still there, and it is heavier than the water around it. It sinks — and every centimetre it sinks brings it into water fresher still, which renews its salt excess while diffusion quietly removes the heat excess it keeps acquiring. The motion feeds itself.

Molecular diffusion is doing this sideways, over centimetres, and the reason the two rates differ so much is molecular. Heat in water is carried by molecular collisions and has a diffusivity of $1.4\times10^{-7}$ square metres per second. Salt has to be carried by the ions themselves, jostling through the solvent as a pollen grain does, and its diffusivity is about a hundred times smaller. The ratio is the Lewis number, and for heat and salt in water it is close to a hundred. The walk that underlies diffusion covers a distance growing only as the square root of time, so a factor of a hundred in diffusivity is a factor of ten in the distance covered in the same time — and a factor of a hundred in the time needed to cross the same distance.

The width a finger has to be #

The parcel picture suggests that thinner is better, and up to a point it is. The complete linear problem for vertical fingers in uniform gradients is three equations — for the vertical velocity, the temperature excess and the salinity excess — each with its own diffusion term, and viscosity damps the velocity at $\nu k^2$. Assuming everything grows as $e^{\lambda t}$ gives a cubic for $\lambda$:

$$(\lambda + \nu k^2)(\lambda + \kappa_T k^2)(\lambda + \kappa_S k^2) + N_T^2(\lambda + \kappa_S k^2) - N_S^2(\lambda + \kappa_T k^2) = 0,$$

with $N_T^2 = g\alpha T_z$ and $N_S^2 = g\beta S_z$. A real positive root is a finger that grows.

Each curve rises from zero at small widths, peaks and falls away slowly. At small widths the fingers are killed by viscosity, which is momentum diffusing sideways and damps the motion as quickly as diffusion removes heat — water’s viscosity is seven times its thermal diffusivity — and by salt diffusion, which begins to erase the salt excess that drives them. At large widths heat cannot leave, and the parcel behaves as a wide one does, oscillating rather than sinking. In between there is a preferred width, and the curve says what it is: about six centimetres for a sinking and rising pair, three for each finger, in a thermocline with a temperature gradient of two degrees per hundred metres.

That width is built from the diffusivities and the gradient and nothing else. The standard estimate is $(\kappa_T \nu / g\alpha T_z)^{1/4}$ times a factor of order $2\pi$, and it is a length the column chooses for itself, as the angle a heap of sand chooses belongs to the grains rather than to the heap. No container, no forcing and no initial disturbance appears in it. A tank of warm salty water poured over cold fresh water in a laboratory produces fingers a few millimetres across, because the gradient across a thin interface is very steep; the ocean’s gentler thermocline produces fingers of a few centimetres, and microstructure probes dropped through the tropical Atlantic have recorded temperature signals with exactly that spacing.

Heat leaks out of a finger and salt stays in #

The drawing takes apart the fastest-growing finger. The sinking water is the shaded band, and it carries two anomalies inherited from above: extra salt, which makes it heavy, and extra heat, which makes it light. If the two had diffused equally the heat would still offset twice the salt’s weight — the density ratio of two is exactly that statement — and the water would rise, as the parcel’s first half-swing did. It does not, because in the twenty-six minutes the finger takes to grow by a factor of $e$, heat diffuses about a centimetre and a half across it, and salt about a millimetre and a half. Across a finger three centimetres wide, the first is most of the way across and the second is a thin fringe at its edge.

So the salt anomaly survives almost intact and the heat anomaly is shaved down to 58 per cent of the salt’s effect on density. The rising fingers between are the mirror image: fresher, colder and lighter than they should be. Each finger is a Stommel pipe with its walls made of the adjacent fingers, handing heat sideways and keeping salt to itself.

A related trick is played by a mixture of three gases in the gas that flows towards more of itself, where one species diffuses against its own concentration gradient because another species is dragging on it. In both cases a rule that holds for a single diffusing substance — a gradient relaxes; a stable column stays put — stops holding when two substances with different diffusivities share the job. There are no new forces in either. There is a second rate, and a second rate is enough.

Unstable up to a ratio of a hundred #

The most surprising thing about the cubic is where its instability stops. Setting $\lambda = 0$ and asking when the left-hand side is negative for some $k$ gives

$$\nu\kappa_T\kappa_S k^4 < N_S^2\kappa_T - N_T^2\kappa_S,$$

which can be met by a small enough $k$ whenever the right-hand side is positive — that is, whenever $R_\rho < \kappa_T/\kappa_S$.

The static criterion says the column is safe the moment $R_\rho$ exceeds one. The diffusive criterion says it is not safe until $R_\rho$ exceeds a hundred. Between the two lies every column in which heat does more than salt for stability, by any factor up to a hundred, and every one of them is unstable to fingers. The instability becomes very slow towards the top of the window — at a ratio of fifty the fastest finger takes days to grow by a factor of $e$, and any turbulence in the water would erase it first — but the edge of the window is exact, and it contains nothing but the two diffusivities.

The strength of the gradients does not move that edge. Doubling both gradients rescales every growth rate on the curve and leaves the window where it is. This is the same shape of result as the disturbance that grows instead of travelling, where a gas cloud’s own gravity beats its pressure above one wavelength: a restoring force that works at every scale is beaten, at some scales, by a destabilising one that the restoring force’s own mechanism does not reach. In a self-gravitating cloud the scale is a length; here the scale is a width across which one substance can escape and the other cannot.

The real ocean sits mostly near the bottom of the window. Across much of the subtropical thermocline the density ratio is between about 1.5 and 2.5, where the fastest fingers grow in tens of minutes, and it is in those waters that fingering signatures are measured most often. Where the ratio climbs towards three or four the fingers are slow enough that ordinary turbulence from breaking internal waves, including the tide’s own, competes with them and usually wins.

Fingers carry weight downward #

An instability has to be paid for, and for fingers the bill is itemised in this ratio. Sinking fingers carry salt down and heat down; rising fingers carry fresh cold water up. In density terms the salt flux moves weight downward and the heat flux moves it upward, and the ratio of the two decides the net. If it were one, the column’s centre of mass would stay put and the fingers would have nothing to run on. It is between 0.56 and 0.66 across the range where fingers grow fast, so every finger moves more weight down than up and releases potential energy as it does so — the potential energy stored in the salt’s inverted half of the stratification, which the static criterion never counted because it was outweighed.

Two consequences follow. First, fingers make the ocean more stable as they work: they remove salt from the upper layer faster than heat, so the density ratio of the water they leave behind rises. Second, the flux ratio has been measured. Laboratory experiments with warm salty water over cold fresh water, and with a sugar solution over a salt solution — where salt plays the fast role, diffusing about three times faster than sugar — give flux ratios close to the values in this drawing, which is some of the best evidence that a linear theory of the fastest-growing finger captures what fingers actually transport.

The layers the fingers leave behind #

In the tropical Atlantic east of Barbados, profiles taken in the 1980s showed the thermocline arranged as a staircase: uniform layers ten to thirty metres thick, separated by thin sheets a metre or two thick in which the temperature and salinity jumped, and in which fingers were found. Similar staircases were seen beneath the Mediterranean outflow and in the Tyrrhenian Sea. A tracer released into the Barbados staircase in 2001 spread vertically faster than the weak turbulence measured there could account for, and the measured rate was about what fingers across the sheets would supply.

Why the column arranges itself into steps is not in the linear theory. The fingers’ flux depends on the local density ratio in a way that makes a uniform gradient unstable to a second, much larger-scale disturbance: a region where the ratio is slightly lower fingers harder, which lowers it further. The steps are that secondary instability grown to completion, and theories of it — in which the flux ratio’s dependence on $R_\rho$ plays the same role here as the cubic’s instability plays for the fingers — reproduce steps of about the observed thickness without explaining in detail why they persist for years.

The same instability upside down #

Reverse the arrangement — cold fresh water over warm salty water — and the column is again stable if heat’s effect is smaller than salt’s, but now it is the temperature that is destabilising. A displaced parcel loses its heat quickly and keeps its salt, and the result is not a runaway finger but an oscillation that grows: the parcel overshoots a little further on each swing, because heat exchange with its surroundings lags behind its motion. The staircases this produces — the diffusive-convection regime — are found beneath the Arctic sea ice, where the cold fresh surface water lies over the warmer, saltier Atlantic layer, and in some East African crater lakes heated from below by volcanic springs. Heat leaks through the staircase from the Atlantic water towards the ice, and how fast it does so is one of the quantities in estimates of the Arctic’s heat budget.

The surprising connection goes much further afield. In a red giant star, the burning of helium-3 in a shell above the core produces a region where the mean molecular weight decreases with depth — the composition is heavy on top — while the temperature gradient keeps the region stable. Heat there is carried by photons, many orders of magnitude faster than the nuclei can diffuse, so the window of unstable density ratios is enormous, and fingering in stars is invoked to explain why the surface carbon and lithium of red giants change in ways that standard convection cannot account for. The stellar version was worked out by analogy with Stern’s ocean, with a star for a tank.

What the linear finger leaves out #

Unbounded uniform gradients. The cubic describes fingers in a column that goes on for ever with the same gradients. Real fingers grow across interfaces of finite thickness, where the gradients change with depth, and the fastest width is then a compromise between the interface thickness and the length in the drawing.

Infinitesimal amplitude. Every growth rate here is for a finger so weak that it does not change the gradients it grows in. A real finger grows until its own flux of salt and heat erodes the gradient that drives it, and it saturates — in theory by the fingers becoming unstable to shear between neighbours, which breaks them into shorter pieces. The fluxes of saturated fingers are measured, not derived.

Two dimensions. The fingers are drawn as sheets. In three dimensions the planform that grows is closer to a checkerboard of square cells, and in a sheared ocean the fingers are tilted into sloping sheets aligned with the current. Both change the numbers by factors of order one, not the window.

A linear equation of state. Seawater’s expansion coefficient $\alpha$ depends strongly on temperature, halving between 20 °C and 5 °C, so the density ratio of a real column is not just a ratio of gradients. The instability criterion survives with local values of $\alpha$ and $\beta$.

What the fingers do not show #

Every figure here is a single finger or a single number describing one. What none shows is a finger field — thousands of centimetre-wide tongues side by side, in a thin sheet between two layers, with the layers themselves stirred by the fingers into uniformity. That is what a microstructure probe crossing a staircase sees, and it is what a shadowgraph of a laboratory tank photographs: a forest of fine vertical stripes. The linear mode is the reason the forest exists and the reason its spacing is what it is; it is not a picture of the forest.

Still open: how much of the ocean’s mixing the fingers do #

In the staircases the case is made: fingers carry heat and salt across the sheets, and the transport has been measured by tracer. Outside the staircases, across the large parts of the subtropical thermocline where the density ratio is favourable but no steps are seen, the question is much harder. Turbulence from breaking internal waves — the beams a stratified sea carries — is intermittent and patchy, fingers grow in the quiet intervals between patches, and whether the two add, or whether one suppresses the other, depends on how strong the turbulence is compared with the fingers’ growth rate. Estimates of the global contribution of fingers to the mixing of heat and salt through the thermocline range from minor to substantial, and the answer matters for climate models, which mostly represent the process with a parameterisation of its dependence on the density ratio rather than resolving it.

The habit worth carrying away is to ask what a stability argument assumes about the thing that is displaced. A column is stable only against disturbances that keep their properties, and when two properties decide the density and they leak at different rates, the thinnest disturbances do not keep them. The static criterion counts the stored energy of the stable property and the unstable one together; diffusion can reach one and not the other, and the energy stored the wrong way up is then available to anything thin enough to take it.