How long a fluid takes to forget it was not rotating

The layer that stops at a depth is this collection’s account of the Ekman layer: a boundary layer that does not grow, because rotation supplies a frequency and the balance against diffusion supplies a length. Its most useful property is a transport that depends on the stress at the surface and not on the viscosity underneath it.

This essay is about what that transport does when nobody is asking it to do anything, which turns out to be the fastest route by which a rotating fluid changes its mind.

Two clocks #

A container of fluid at rest is set spinning about its axis. The walls now move and the fluid does not, and the question is how long the fluid takes to catch up.

Diffusion is the obvious route. Viscosity carries momentum inwards from the walls, at a rate the diffusion equation sets, so the time is the depth squared over the viscosity. For a container ten centimetres deep in water that is 10,000 seconds — 2 hours 47 minutes. The Ekman route does it in 100.

The Ekman layers are the fast route. The thin layer on each end wall is in a balance between rotation and viscosity, and in that balance it carries a radial transport: fluid in the layer spirals outwards. Mass conservation then requires fluid to be drawn down from the interior into the layer, carried outwards, and returned up the sidewall — a secondary circulation that transports angular momentum from the walls into the bulk without waiting for diffusion to get there.

The time that takes is the depth divided by the square root of the viscosity times the rotation rate. For the same container, a hundred seconds.

The advantage is exactly the reciprocal root of the Ekman number #

The ratio of the two times is worth writing out because it is exact rather than approximate:

$$\frac{H^2/\nu}{H/\sqrt{\nu\Omega}} = \frac{H}{\sqrt{\nu/\Omega}} = E^{-1/2},$$

where $E = \nu/(\Omega H^2)$ is the Ekman number — the same group the Ekman layer’s own thickness is built from, and the ratio of the layer’s thickness to the container’s depth squared.

So the fast route beats the slow one by the reciprocal square root of a number that is small in every rotating flow anybody cares about. In a laboratory tank at an Ekman number of 10⁻⁴ that is exactly a hundred — 100 seconds against 10,000. In a centrifuge rotor, where the rotation rate is 500 radians a second and the Ekman number falls to 8·10⁻⁷, it is 1,118: the fluid is spun up in 2.24 seconds where diffusion would have taken 42 minutes.

What is being remembered, exactly #

It is worth being precise about the quantity, because “the fluid forgets it was not rotating” is a phrase and the thing being carried is angular momentum.

A fluid at rest inside a spinning container has the wrong angular momentum for its position, and the walls have the right one. Bringing the two into agreement means transporting angular momentum from the boundary into the interior, and the two routes above are two ways of doing that.

The reason it is worth calling a memory is that the interior’s state during the process is a record of how much of it has been dealt with. There is no local quantity that says how far along the process is — the interior’s angular velocity does, but only because it is the accumulated result — and two containers at the same interior angular velocity with different histories have different secondary circulations in them.

That is the same distinction what a mean profile cannot tell anybody draws: a first moment does not determine the state, and here the state includes a circulation that the angular velocity does not contain.

Where these systems sit #

A teacup, a laboratory tank, a centrifuge rotor, the ocean and the atmosphere all sit at Ekman numbers between 10⁻⁷ and 10⁻³, and the two geophysical cases only reach the upper end because they are quoted with an eddy viscosity rather than a molecular one.

Five containers, across fourteen decades of Ekman number:

The ocean’s spin-up time comes out at 54.2 days and its diffusion time at 50.7 years. The atmosphere’s is 4.28 days against 116 days. The last column is the inverse square root of the column before it in every row, to the digits shown. Both are the time scales on which those systems adjust to a change in the forcing at their boundaries, and in both cases the number that matters is the short one.

Why a thin layer beats a thick one #

The result looks paradoxical — a layer a millimetre thick sets a metre of fluid rotating faster than a metre of fluid can diffuse — and the resolution is worth stating carefully.

Diffusion is slow because it is a random walk: momentum spreads a distance proportional to the square root of the time, so covering a distance costs its square. Advection is fast because it is directed: fluid carried at a speed covers a distance proportional to the time.

The Ekman layer’s contribution is to turn one into the other. It cannot carry much fluid — it is thin — but what it carries, it carries radially, at a speed set by the rotation rather than by the viscosity. The interior is then drained through it, and the draining time is the container’s volume divided by the layer’s volume flow rate, which is the depth over the square root of the viscosity times the rotation.

So the mechanism is a pump rather than a diffuser, and its rate is set by how fast the fluid can be cycled through a thin fast layer rather than by how fast momentum can crawl across a thick slow one.

What the solver computed, and how it was checked #

The two times are closed forms and the approach curves are first-order relaxations at each of them, so this is a comparison of scalings rather than a solution of the spin-up problem — which is genuinely harder and is a matched-asymptotic calculation rather than a formula.

Three checks. That the ratio of the two times equals the reciprocal square root of the Ekman number, to a part in 10⁹, which is an algebraic identity and is checked because getting a power wrong in a scaling is the standard way to be plausibly wrong. That the advantage is at least a factor of ten, so the essay’s claim has content. And that every one of the five cases spins up faster than it diffuses, which would fail if any of them were at an Ekman number above one — a fluid so viscous or a rotation so slow that the layers are the whole container.

What is actually moving #

It is worth being explicit about the flow, because “spin-up” sounds like the fluid gradually starting to turn and it is not what happens.

The interior does not accelerate uniformly. It is drained: fluid is pulled down into the Ekman layer at the bottom, flung outwards through it, carried up the sidewall, and returned across the top. Each parcel that makes that circuit comes back with the wall’s angular momentum, and the interior’s rotation rises as more and more of it has been round.

That has a consequence that is easy to test and often surprising. The interior’s angular velocity rises exponentially towards the final value rather than linearly, because the rate at which fluid is being processed is proportional to how much of it is left unprocessed — and the exponential’s time constant is the spin-up time.

After one spin-up time 63.2 per cent of the difference has been removed, after two 86.5, after three 95.0 and after five 99.3 — so a tank left for eight minutes at an Ekman number of 10⁻⁴ is rotating with its container to better than a per cent, while a tank relying on diffusion would be at 4.9 per cent of the way there.

It also means the fluid ends up stratified in its history: at any moment during spin-up some of the fluid has been through the layer and some has not, and they are at different angular velocities. The interior is not in solid-body rotation on the way to solid-body rotation.

The same pumping, doing other work #

The Ekman layer’s transport is a general-purpose device and spin-up is only its simplest use. Three others are worth naming because they are the same calculation with a different thing being pumped.

Tea leaves in a stirred cup. The leaves collect at the centre, which is the opposite of what a centrifugal argument suggests. The bottom Ekman layer carries fluid inwards — the sign is the other way round here, because the interior is spinning faster than the cup rather than slower — and the leaves are swept along with it. Einstein wrote it up in 1926, in an essay about river meanders.

Ocean upwelling. Wind blowing along a coast drives an Ekman transport at right angles to it, and where that transport is away from the shore, water is drawn up from below to replace it. That is the mechanism behind the most productive fisheries in the world, and it is the same layer with a wind stress instead of a wall.

And the mixing in a rotating machine. A gas centrifuge separates isotopes by rotation, and what carries the enriched fraction from one end to the other is a secondary circulation driven in exactly this way rather than any imposed pumping.

In all three the layer is thin, weak and decisive, which is the standing lesson of the thin layer.

What this does to an experiment #

A rotating-tank experiment has to be waited out. Any measurement made before several spin-up times have passed is measuring the transient, and the transient contains a secondary circulation that the final state does not. That circulation is small and it is exactly the sort of thing that contaminates a careful measurement.

And the waiting time is short enough to be surprising. The reason rotating-tank experiments are practical at all is that spin-up takes minutes rather than hours: at the diffusion time nobody would run them. The Ekman layers are what make the laboratory analogue of a geophysical flow possible.

The transient contaminates a closed-box measurement too. A container in which a wave field or a mean flow is being established has the same problem in a different quantity — the drift a closed box will not allow is the version of it for a wave tank, where the return flow the walls force is established over a time nobody usually quotes.

Spin-down is not the same problem. Stopping the container reverses the layers’ transport and the secondary circulation runs the other way, but the interior is now rotating faster than the walls and the layer’s structure is different. The time scale is the same and the flow is not, which is why a container brought to rest still has fluid turning in it in a way a spun-up one does not.

Which memory this is #

Placed among the others in this collection, spin-up is a case where the fluid has two available memories and takes the shorter.

The diffusive one is the kernel of the wall the fluid is listening to: heavy-tailed, no characteristic age, and reaching the middle of the container only after a very long time. It is present here and it is irrelevant, because something faster got there first.

The advective one has a clean exponential with a definite time constant, because it is a pumping process with a rate. Rotation has converted a diffusive memory into a relaxation memory, and shortened it by two orders of magnitude.

That conversion is the general lesson. Wherever a flow provides a directed transport — a secondary circulation, a mean flow, a convection — the diffusive memory stops being the one that matters, and the system acquires a time constant it did not have. The same substitution is what makes turbulent mixing fast, and it is why a millionth is enough finds a mixing rate almost independent of the diffusivity. It is also the mechanism behind the fast route in a duct that forgets everything but one number, where convection rather than rotation supplies the directed transport.

The mechanism it beats is drawn by the same solver two essays back.

What sets the layer’s own thickness #

Since everything above rests on the layer being thin and fast, it is worth saying where its thickness comes from, because the argument is short and it is the reason the Ekman number appears at all.

Inside the layer, the Coriolis acceleration is balanced against the viscous one. The first is the rotation rate times the velocity; the second is the viscosity times the velocity divided by the thickness squared. Setting them equal gives a thickness of the square root of the viscosity over the rotation rate, with no reference to the container at all.

That is the property the layer that stops at a depth is named for: the layer does not grow, because its thickness is fixed by two rates rather than by how long it has had. Every other boundary layer in this collection thickens with distance or with time, and this one does not.

Divide that thickness by the container’s depth and the square of the result is the Ekman number. So the Ekman number is not an abstract group: it is the fraction of the container that the fast layer occupies, squared — and the advantage the layer confers is the reciprocal of its square root because the interior has to be cycled through a layer that thin.

What the picture cannot show #

The two approach curves are first-order relaxations drawn at the two time scales, and the real spin-up is not a first-order relaxation: it has a short initial phase while the layers themselves form, which takes about one rotation period, and a long tail while the last of the unprocessed fluid is dealt with. Neither is in the figures.

Nothing here shows the secondary circulation either. It is the mechanism the whole essay is about and it is drawn only through the borrowed figures of the layer that carries it — because computing it requires solving the spin-up problem rather than comparing its time scales.

The number to carry #

The transferable statement is short and it is worth separating from the fluid mechanics.

When a system has two routes to the same end state, its memory is the shorter one. The diffusive route is always available and is almost never the answer, because anything that provides directed transport beats a random walk over any distance large compared with a step.

And the advantage is a power of the ratio of scales. Here it is one half — the reciprocal square root of the Ekman number — because the layer’s thickness is the geometric mean of the container’s depth and the viscous length. Elsewhere the exponent differs and the structure does not.

That is why estimating a response time by identifying the slowest mechanism is nearly always wrong. The correct procedure is to enumerate the routes and take the fastest, and the fastest is usually the one that has found a way to convert a diffusion into an advection.

Who found it, and when #

The Ekman layer is Ekman’s, from 1905, and was worked out to explain why drifting ice moves at an angle to the wind. The spin-up problem is Greenspan and Howard’s, from 1963, and their result — that the time is the geometric mean of the rotation period and the diffusion time — is what is computed here.

The geophysical importance came later and is the reason the result is well known: the ocean’s response to a change in wind stress is governed by exactly this mechanism, and its being a month rather than a lifetime is what makes the ocean’s circulation adjustable on human time scales at all.

Limits recorded rather than smoothed over #

A scaling comparison, not a solution. The two times are closed forms and the approach curves are first-order fits to them. The actual spin-up is a matched-asymptotic problem with three stages, and nothing here solves it.

No sidewalls in the arithmetic. The transport is returned up a sidewall in the description and the time scale quoted is the one for a container whose depth is comparable with its radius. A shallow wide container behaves differently and a tall narrow one differently again.

Eddy viscosities for the geophysical cases. The ocean and atmosphere numbers use eddy viscosities of 10⁻² and 10 m²/s, which are conventional and are not measurements. Both times move linearly and inversely with the square root of that choice, so the fifty-four days is good to a factor rather than to a digit.

A single relaxation, drawn. Both curves in the first figure are exponentials, chosen so that the two time scales can be compared on one axis. Neither is the real approach, which is why the ledger quotes times rather than shapes.

And no stratification. A stratified fluid spins up differently, because buoyancy resists the vertical motion the secondary circulation requires — which is most of why the ocean’s real adjustment is more complicated than the number above.