The depth past which it must sink

A diver trimmed to neutral buoyancy at ten metres is not stable there — unlike a parcel of air in a stably stratified layer, which is pushed back. Rise a metre and the ascent accelerates; sink a metre and the descent does. The condition has to be re-established continuously by breathing, and every training course teaches it as a skill rather than as a state. The reason is not skill and not turbulence: it is that the equilibrium is a maximum of the potential energy rather than a minimum.

The principle, and the thing it does not say #

Archimedes’ principle is exact and is not in question. The upward force on a submerged body is the weight of the fluid it displaces, and it comes from the pressure on the bottom exceeding the pressure on the top by exactly the amount the depth difference requires.

The principle gives a force at a depth. Stability is a question about the derivative of that force with depth, and the principle is silent on it — which is why a correct statement of Archimedes can be held together with an entirely wrong expectation about what a neutrally buoyant body does next.

Differentiating the principle #

Write the net upward force on a body of total volume $V$ and weight $W$ at depth $z$:

$$F(z) = \rho_w g V(z) - W.$$

The weight does not vary with depth. So

$$\frac{\mathrm{d}F}{\mathrm{d}z} = \rho_w g\,\frac{\mathrm{d}V}{\mathrm{d}z},$$

and everything is in the sign of $\mathrm{d}V/\mathrm{d}z$: whether the body shrinks as it descends. If it does, the force falls with depth, and an equilibrium is unstable. If it does not shrink at all, the force is constant and there is no equilibrium at any depth except the one where it happens to be zero, at which the body is neutrally stable everywhere. If it shrinks less than the water does, the force rises with depth and the equilibrium is stable.

A body that is part solid and part gas shrinks a great deal. Boyle’s law gives the gas volume as $V_g(z) = V_g(0)\,p_0/(p_0 + \rho_w g z)$, so ten metres down the pocket is half its surface size and thirty metres down it is a quarter.

Nothing about the shape of that potential is peculiar to fluids. It is the same picture as a body at the top of a rise, or an axis of a spinning body that will not hold: a stationary point which is a maximum rather than a minimum, so a displacement in either direction grows. What makes the diver’s case vivid is that the instability is in depth, and depth is the variable the operator is controlling.

The numbers, for a diver #

It is worth putting sizes on it, because the instability is often described as though it were delicate and it is not.

A diver of eighty kilograms displacing eighty litres is neutral at the surface. Suppose six per cent of that volume — about five litres — is gas: lungs, drysuit, buoyancy compensator. At ten metres the pressure has doubled, that five litres has become two and a half, and the displaced volume has fallen by 2.5 litres. The lost buoyancy is the weight of 2.5 kg of water, which is 25 newtons: a downward force of about three per cent of body weight, appearing over a descent of ten metres.

Three per cent of body weight does not sound alarming and it is, because nothing opposes it. Twenty-five newtons on eighty kilograms is an acceleration of 0.3 m/s², which over ten seconds of unchecked descent is three metres per second — and the deeper the diver goes the larger the imbalance becomes. What actually limits it is drag, which at those speeds is substantial, and attention, which is why the buoyancy compensator has a hose to the mouth.

The same arithmetic run upward is worse, because the gas expands and the volume ratio is unbounded. A diver who becomes positively buoyant at thirty metres and does nothing arrives at the surface having accelerated the whole way, with lungs that have expanded by a factor of four if the breath was held. That is not a buoyancy problem any more, and it is the first thing any diving course teaches.

The Cartesian diver, and what it is demonstrating #

The desk toy is a sealed bottle of water containing a small inverted vessel with a trapped bubble. Squeeze the bottle and the diver sinks; release it and the diver rises. The usual explanation stops there — squeezing raises the pressure, the bubble shrinks, the diver becomes denser — and misses the more interesting half.

Trim the diver so that it hangs motionless in the middle of the bottle and it will not stay. The slightest disturbance sends it to the top or the bottom, and the demonstration is therefore always performed with the diver against one end or the other. The instability is not an imperfection of the toy; it is the same instability a diver at ten metres has, made visible in twenty centimetres of water because the fractional pressure change is what matters and a squeezed bottle supplies a large one.

That contrast is the whole of the subject in two figures. Floating is stable and hovering is not, and the difference is which quantity depends on displacement: at the surface, the displaced volume; below it, the body’s own.

Where the same instability is solved by biology and engineering #

Fish with swim bladders face this exactly. A bladder trimmed for neutral buoyancy at one depth is unstable, so the fish must add or remove gas continuously as it changes depth — through the gas gland, at a rate that limits how fast it can ascend. A fish brought up quickly from depth suffers the runaway directly: its bladder expands, its buoyancy grows, and it is carried to the surface faster than it can vent.

Some fish avoid the problem entirely by having no gas at all. Sharks use a liver full of squalene, an oil of density about 860 kg/m³, and oil is very nearly as incompressible as water — so a shark’s buoyancy hardly changes with depth and its equilibrium is nearly neutral in the second sense: no restoring force, but no runaway either. Deep-sea fish frequently do the same, and the deepest have no swim bladder.

The engineering answer is the same. A submarine’s ballast tanks set its average density, and its depth is held by hydroplanes — control surfaces that generate a force from forward motion — precisely because ballast alone cannot hold a depth against an unstable equilibrium. Autonomous ocean floats, which must hold a depth for years without power, are built as oil-filled bladders with a pump: compressibility matched to seawater deliberately, and depth adjusted by moving oil between an internal and an external bladder.

A floating body faces the other stability question and gets a different answer. Heel a hull and its centre of buoyancy moves sideways, generating a couple that rights it — so a ship is stable against tipping for a reason that has nothing to do with depth. The two questions are independent, and a vessel can be perfectly stable in roll and unstable in the sense this essay is about.

The same argument in air, with the sign reversed #

An atmosphere is a fluid whose density falls with height, and a parcel of it rising expands.

The same argument runs in air and the answer flips. A rising parcel expands as the pressure around it drops, so it becomes less dense — and whether it keeps rising depends on whether it becomes less dense faster than its surroundings do. That comparison is the lapse rate, and the atmosphere is stable or unstable according to it. In water the compressibility is small enough to ignore and the instability comes from the body; in air it comes from the fluid.

If the surrounding air’s density falls faster with height than the parcel’s does, a nudged parcel keeps going and the atmosphere is unstable — which is convection, and which is what makes weather. If the surroundings’ density falls more slowly, the parcel returns, and it overshoots: the atmosphere then supports oscillations, at a frequency set by the mismatch, which are the internal waves visible as rows of cloud.

The rule is one comparison in both media: an equilibrium is stable when the parcel is the less compressible of the two. Water and a gas-filled body give one sign; air and a rising air parcel usually give the other.

The extreme case is a body that is nothing but gas. A bubble has no solid part at all, so its volume follows the pressure completely and its buoyancy falls as fast as the depth increases it — which means a bubble has no equilibrium depth whatever. It rises, expands, rises faster, and the runaway is the same one the diver has, with nothing at all to slow it.

The stability condition, written once #

All the cases above are one inequality, and it is worth extracting it from the examples.

Let the body have compressibility $\kappa_b = -(1/V)\,\mathrm{d}V/\mathrm{d}p$ and the fluid $\kappa_f$. At an equilibrium the body’s density equals the fluid’s, and descending by $\mathrm{d}z$ raises the pressure by $\rho g\,\mathrm{d}z$, so the body’s density rises by $\rho\kappa_b \rho g\,\mathrm{d}z$ and the fluid’s by $\rho\kappa_f\rho g\,\mathrm{d}z$. The net upward force per unit volume is the difference, and

$$\frac{\mathrm{d}F}{\mathrm{d}z} \propto \kappa_f - \kappa_b.$$

Stable when the body is the less compressible; unstable when it is the more. Every case in this essay is a reading of that line. A gas-filled body has $\kappa_b \approx 10^{-5}$ per pascal against water’s $4.5\times10^{-10}$: unstable by five orders of magnitude, which is why the effect is so pronounced. An oil-filled float has $\kappa_b$ within tens of per cent of water’s, so its stability is a matter of careful matching and can be given either sign deliberately. A solid steel object has $\kappa_b$ about a hundred times smaller than water’s and is therefore stable — but it is also far too dense to be in equilibrium anywhere, so the stability is unavailable.

Which is the general lesson hiding in the arithmetic: the condition for an equilibrium to exist and the condition for it to be stable involve different properties of the same body, and satisfying both at once is a design problem rather than a matter of trimming.

The boat that gets heavier as it goes down #

The stability condition is a comparison of two compressibilities, and a submarine is the case where the comparison is closest and the consequences are operational.

A pressure hull is not a solid block of steel; it is a shell, and a shell under external pressure contracts by an amount set by its geometry and its elastic response rather than by the bulk modulus of the metal. The result is that a submarine’s volume falls measurably with depth — by an amount comparable with the water’s own contraction over the same range, which at three hundred metres is a little over a tenth of one per cent.

Comparable is not the same as equal, and for a real boat the hull loses the contest: the submarine compresses more than the water does, so it becomes progressively negatively buoyant as it descends. The condition derived above says it is on the unstable side, and it is.

Two further contributions push the same way. The free-flooding structure outside the pressure hull — superstructure, casing, vented ballast tanks — traps pockets of air that compress by half in the first ten metres, which is the gas-pocket instability of this essay in its pure form. And the sea itself usually gets denser with depth, which helps, though not by enough.

So depth is not held by trimming. A submarine holds a depth the way an aircraft holds an altitude: with control surfaces and forward motion. Hydroplanes generate a hydrodynamic force that has nothing to do with buoyancy, and the boat is flown rather than floated. Losing way at depth is therefore a serious matter, and a boat that must stop uses its trim and compensating pumps to chase an equilibrium it can never quite sit in.

The operational vocabulary reflects it exactly. A crew speaks of the boat being “heavy” or “light” and pumps water in or out of compensating tanks to correct, continuously, as depth and the water’s density change. That is the same continuous correction a diver makes by breathing, at a different scale and with a pump instead of a lung, and for exactly the reason the potential curve above has a maximum in it.

The buoyancy that changes without any gas at all #

Everything so far has varied the body’s volume and held the water’s density fixed. The ocean does not cooperate, and for an instrument that must hold a depth for years the water’s variation is as large a problem as the body’s.

Seawater’s density depends on pressure, on temperature and on salinity, and the last two vary far more strongly with depth than pressure does over the top few hundred metres. Fresh water is 1,000 kilograms per cubic metre and seawater about 1,025, which is why a diver moving between a lake and the sea must change ballast by about two and a half per cent of displaced volume — two kilograms for an eighty-litre diver, and enough to be immediately obvious.

Temperature does the same thing more subtly. Seawater’s volumetric thermal expansion near twenty degrees is about two parts in ten thousand per kelvin. An aluminium float’s is about seven parts in a hundred thousand, three times less. So a float descending through a thermocline into water ten degrees colder finds the water around it contracting more than the float does, which makes the float relatively lighter — an effect of about a part in a thousand, which is the same size as everything else in this essay.

That is the real difficulty in building an instrument that holds a density surface. It is not enough to match the water’s compressibility; the thermal expansion must be matched as well, and no single material does both. The usual solution is a composite: a hull of one material with a deliberate volume of a second, chosen so that the combination’s response to pressure and to temperature both land where they are wanted.

The profiling floats that now cover the ocean sidestep the problem rather than solving it. Instead of seeking a stable equilibrium, each one pumps oil between an internal reservoir and an external bladder to change its own volume on command — descending to a set depth, drifting, then rising to the surface to report. It never sits at a stable point, because the physics of this essay says there is not one worth relying on; it flies a programmed profile and corrects continuously.

The general point is the one the stability condition made and is worth restating with the extra variables in it. An equilibrium between two things requires matching a value; a stable equilibrium requires matching every derivative that matters. Here there are three — pressure, temperature and salinity — and an instrument that matches only the first will hold station until the water changes.

Where the model stops #

The water is treated as incompressible. It is not: seawater’s compressibility is about 4.5 × 10⁻¹⁰ per pascal, so a cubic metre at four kilometres’ depth is about two per cent smaller than at the surface. That is small against a gas pocket’s fifty per cent at ten metres, and it is not small against an oil-filled float’s, which is why the deep-ocean instrument problem is a genuine matching exercise rather than an obvious one.

The temperature is held fixed. Boyle’s law assumes it. A rapidly compressed bubble warms, so its volume falls less than isothermally in the short term and settles afterwards — which makes the instability slightly slower to develop than the figures show and does not change its sign.

One assumption underlies everything above and is worth stating at the end rather than losing: hydrostatic pressure depends only on depth, not on the shape of the vessel or on how much fluid sits above. Every force computed here is an integral of that pressure over a surface, so every result inherits the assumption — and it fails wherever the fluid is moving, which is why none of this applies to a diver being carried by a current.

Nothing here moves. The forces are hydrostatic, and a body that is actually rising or sinking also feels drag, which grows with speed and limits the runaway to a terminal rate. The instability decides the direction and the drag decides how fast, and only the first is in these figures.

What the pictures cannot show #

The force curves are drawn for a body whose gas fraction is fixed at the surface, and a real diver changes it constantly by breathing. The state of such a body is not a point on one of these curves but a walk between curves, and no static figure represents it.

Nor does anything here show the timescale. An unstable equilibrium says nothing about how quickly the departure grows, and for a body a few per cent off neutral in water the answer is a few centimetres a second — slow enough to correct, which is why diving is possible at all.

And the potential-energy figure has arbitrary units on its vertical axis. The quantity is a work per unit of the body’s displaced weight, and putting real joules on it would require a mass the argument deliberately does not have.

The instability as an instrument #

An unstable equilibrium is usually a nuisance, and this one has been turned into a measurement.

Because the depth at which a body balances depends on how much gas it contains, and because the balance is unstable, a body released slightly off neutral accelerates in a direction that reports the sign of the error. That is the principle of the isopycnal float used in oceanography for half a century: a float trimmed to a chosen density finds the water layer of that density and stays with it, drifting horizontally along a surface of constant density and reporting where that surface goes.

The trick is that the float has to be less compressible than seawater for the equilibrium to be stable, which means building it out of glass or aluminium with an internal volume that shrinks less than water does. Get the compressibility wrong by a few per cent in the wrong direction and the instrument sinks to the bottom or rises to the surface on its first excursion, which is precisely what the earliest ones did.

The modern profiling float goes further and exploits both signs deliberately. It pumps oil into an external bladder to become buoyant and rise, vents it to sink, and holds a depth by alternating — depth control by a sequence of small deliberate instabilities rather than by finding a stable point, because there is not one to find.

Where the ladder goes next #

Buoyancy began as the weight of the fluid that is not there, continued with the geometric stability that rights a hull, and has now been asked the third question a floating body faces: not how much force, and not whether it rights itself, but whether it stays where it is put. The rungs beyond are stratified fluids proper — where the surrounding density varies as well as the body’s — and the oscillation frequency that a stably stratified fluid supports, which is the buoyancy equivalent of a spring constant.

The habit worth carrying away is what to do with a principle that gives a force. Archimedes gives a force at a depth, and every question about whether an arrangement holds together is a question about a derivative. A correct expression for a force is one differentiation short of a statement about stability, and the sign that comes out of that differentiation is frequently the opposite of what the force alone suggests.