Flickering vortices cannot hold a settling particle up, and still make it fall faster

Inertia empties the cells that would hold a settling particle followed heavy particles through a steady lattice of vortices, the cellular flow $\psi = \sin x\sin y$ in units of the cells’ size and speed. Stommel showed in 1949 that a particle with no inertia, settling slower than the cells turn, can be held for ever on a closed path — half a cell of them at a settling speed of 0.3. Any inertia empties those traps, in about 2.3 over the Stokes number turnovers, and flings the freed particles to the cells’ edges, where they ride the down-draughts and fall on average several times faster than in still air. And a narrow band of larger inertia holds a slow particle up again, on an orbit of its own.

Every one of those results was for a lattice that never changes, and the only time in it was the turnover. Real vortices — eddies in a turbulent flow, convection cells over a heated surface, the rolls in a cloud — are born, strengthen, fade and die. The essay ended by asking what a lattice whose cells live and die does to a particle: whether the speed-up survives cells that last only a few turnovers, and whether the orbits that hold a particle up disappear as soon as the cells’ lives are shorter than the orbits’ own period.

A lattice that flickers #

The model keeps the lattice’s shape and lets its strength vary: $\psi = A(t)\sin x\sin y$, with $A$ a random signal of mean one, standard deviation $\sigma$ and a memory of $\tau$ turnovers — an Ornstein–Uhlenbeck process, the simplest signal that forgets its past at a stated rate. A cell then strengthens and fades, and at a flicker of half its mean strength it sometimes stops, or reverses its sense of rotation for a while. Every cell in the lattice flickers together, so the pattern of up- and down-draughts stays where it was while its strength wanders; that is the cleanest way to separate the effect of a changing strength from the effect of a changing pattern, and it is stated as such.

The particles are the earlier essay’s: small, heavy, dragged in proportion to their speed relative to the air, with a Stokes number — their response time over a turnover — and a still-air settling speed $W$ as a fraction of the cells’ speed. A cloud of a hundred is released over one cell and followed through two realisations of the flicker. A particle that has fallen four cells is counted free; the held share and the free particles’ mean settling speed are measured as before.

The suspension band is the first casualty #

The band of particles a steady lattice holds up by their inertia is the most fragile thing in the problem. In the earlier essay, particles of Stokes number between about two and five, falling at a tenth of the cells’ speed, could lock onto orbits that circulate round a cell without descending: the particle’s lag behind the flow and its settling balance, orbit after orbit. Twenty-two per cent of a cloud at Stokes number 3 stays aloft for six hundred turnovers in the steady lattice.

A flicker of five per cent of the cells’ strength leaves 5.5 per cent of the cloud aloft; a tenth, 1.5 per cent; a half, none. The orbits are a balance, and a balance that depends on the cell’s strength is upset by any change in it: when the cell strengthens the particle is flung outward and when it weakens it falls inward, and the orbit, which had to close exactly, now spirals. A particle on such an orbit circulates for a period of several turnovers; a cell whose strength is not steady over that period does not offer it the same orbit twice. That is the earlier essay’s guess confirmed and sharpened: the orbits disappear not when the cells’ lives become shorter than the orbits’ period, but as soon as the cells’ strength varies by a few per cent over it — which is long before they die.

Stommel’s traps drain more slowly #

The inertialess particle’s traps are less fragile. Stommel’s closed paths are the contours of $\psi + Wx$; when $\psi$’s strength changes, the contours change, and a particle on one finds itself on a different contour, which may be closed or open. With a small flicker most of the closed region stays closed under the change — a trapped particle moves to a neighbouring closed path — and only those near the region’s edge are tipped out. With a large one the closed region itself shrinks and grows by more than the particle can follow, and the trap drains.

The figure shows the traps holding 0.52 of a cloud with no flicker, 0.41 at five per cent, 0.33 at a tenth, 0.16 at a fifth, 0.03 at 0.35 and nothing at a half. That is a gradual drain, set by how much of the trapped region a change of strength moves across its own edge, and it is several times slower than the suspension band’s collapse at the same flicker: Stommel’s trap is a region, the suspension orbit a curve.

The speed-up survives #

The speed-up is the opposite case. A particle with a little inertia, flung from the cells’ cores and gathered into their down-draughts, falls 4.8 times faster than in still air through the steady lattice; through a lattice flickering by a tenth it falls 4.7 times faster, and by a fifth 4.1. Only when the cells swing by half their strength, often stopping, does the speed-up fall to 2.2 — and even then the particles fall more than twice as fast as in still air.

The reason is that the speed-up needs only a few turnovers to establish. A particle is flung to a cell’s edge within a turnover or two of meeting the cell, by the cell’s centrifuge, and it then rides the edge’s down-draught to the next cell, where it is flung again. Each step takes a turnover, and each step needs only that the cell turn the right way for that turnover — not that it turn the same way for ever. The trapping results need the opposite: orbits and closed paths that repeat indefinitely, which a changing cell cannot supply. What survives a flicker is ordered by how many turnovers it needs.

Memory matters, and a long one hurts the speed-up most #

The flicker’s memory decides which way it bites. A flicker that forgets in a third of a turnover is averaged by the particle: it jitters the cell’s strength faster than the particle can respond, and the particle sees a lattice of roughly the mean strength, losing a fifth of its speed-up. A flicker that remembers for three turnovers or ten is followed: the cells stay weak or reversed for long enough that the particles drift out of the down-draughts and have to be gathered again when the cells recover, and the speed-up is halved. At a settling speed of 0.3, where the steady lattice speeds the fall by 2.3, a long-memory flicker leaves only 1.15 to 1.26: a particle that already falls nearly as fast as the cells turn has less to gain from the down-draughts, and a flickering lattice gives it almost none.

That is a particle’s low-pass filtering applied to the flow’s strength rather than its velocity at a point: a particle with a short response time follows slow changes in the cells and is moved by them, and averages fast changes and is not. The flicker hurts most at memories comparable to the time the particle needs to be swept, which for these particles is a few turnovers.

Why a weak spell costs more than a strong one gains #

A flicker is symmetric about the mean strength, and its effect is not. When the cells strengthen above their mean, a swept particle is flung a little harder and falls a little faster, but it was already in the down-draught and there is little more to gain. When they weaken, the centrifuge that gathered the particle into the down-draught slackens, and the particle drifts back towards the cell’s core and the up-draught on its other side; when the strength recovers, the particle has to be swept again, which takes a turnover or two of falling at no better than its still-air speed. Gains are capped and losses are not, so the average over a flicker is below the steady value even when the average strength is the same.

At a flicker of half the mean, the cells are weaker than half their mean strength a sixth of the time and reversed about one time in forty. With a memory of a third of a turnover those weak spells are too brief for the particle to leave the down-draught, and it loses only a fifth of its speed-up. With a memory of three turnovers or more a weak spell lasts long enough to undo the sweeping, and the speed-up is halved. The steady lattice’s speed-up is therefore an upper bound for any lattice of the same mean strength, and it is approached only when the cells hold their strength for longer than a particle takes to be swept.

What Stommel was explaining #

Stommel’s paper was about plankton. He wanted to know how organisms slightly denser than sea water could stay near the surface in the wind-driven Langmuir circulations of the upper ocean — rows of counter-rotating cells aligned with the wind — and his closed paths gave an answer: a particle settling slower than the cells turn is held in a region of each cell, near the upwelling side, indefinitely. The answer has been taught as the mechanism ever since, and it is the mechanism the steady lattice illustrates.

Langmuir cells are not steady. They form, merge and break down over tens of minutes as the wind gusts and the waves change, which is a flicker of their full strength with a memory of a few turnovers — the regime in which, in this model, Stommel’s traps hold nothing. Plankton in Langmuir circulations are observed to gather in the convergence lines at the surface, which is a different effect — buoyant or swimming organisms collected by the surface convergence — and those that sink do sink. The traps are a property of an idealised steady cell, and the persistence of plankton near the surface needs an explanation that survives the cells’ changes: swimming, buoyancy, and the turbulent mixing that returns sinking organisms upward on average without holding any one of them.

What it says about dust in turbulent air #

The refuted claim is the common explanation for why fine particles stay suspended in turbulent air longer than their settling speed suggests: the eddies hold them up. In a steady lattice they can — Stommel’s traps and the inertial orbits are real — and the explanation is often illustrated by them. But eddies in a turbulent flow are not steady, and a flicker of a few per cent of their strength, a far smaller variation than any real turbulence has, removes most of the holding. What survives is the speed-up: particles settle faster through turbulence than through still air, by the preferential sweeping that Maxey identified in 1987 and that has since been measured in droplets, sediment and snow.

Particles do stay suspended in turbulent air for long times, but for a different reason: the turbulent diffusion of a cloud of particles, which spreads them upward against gravity in the same way it spreads a dye, is a statistical balance between settling and mixing — the Rouse profile of a suspension, which how far a parcel gets underlies — and does not depend on any particle being held. A given particle rises and falls at random and spends, on average, a long time aloft because it is mixed back up, not because it is trapped. The lattice result separates the two: trapping is what a steady picture of eddies suggests and what a flickering one removes, and mixing is what a real turbulent flow does instead.

The numbers in a cloud #

The cells’ scale sets everything’s size. For a cumulus cloud’s turbulent eddies of a metre or so turning at a metre a second, a turnover is about a sixth of a second, and a cloud droplet of twenty micrometres, with a response time of about a millisecond and a settling speed of a centimetre a second, has a Stokes number near 0.01 and a $W$ near 0.01 — far into the regime where the speed-up is large in a steady lattice, and where the flicker of real eddies, of order their own strength with a memory of a turnover or so, leaves it reduced but present. Drizzle drops of a hundred micrometres have Stokes numbers of order 0.1 to 0.3 and $W$ near a quarter, the regime where the flicker’s effect is largest. The model’s cells are an idealisation, and the numbers are orders of magnitude; what carries over is the ordering — the speed-up robust, the trapping fragile.

The convention, and the model’s simplifications #

Lengths are in the cells’ size over $\pi$, times in turnovers ($1/Uk$), speeds over the cells’ speed $U$; the Stokes number is the particle’s response time over a turnover, and $W$ its still-air settling speed over $U$. The flicker is the deviation of the cells’ strength from its mean over the mean; its memory is the Ornstein–Uhlenbeck process’s correlation time, at which the correlation has fallen to $1/e$. Held means within four cells of the start after three hundred turnovers (six hundred for the suspension band); speeds are measured over the run’s second half for the free particles. Every cell flickers together, so the pattern of up- and down-draughts is fixed; a lattice whose cells flicker independently, or drift, would also move the down-draughts and would hurt the speed-up more. The particles are points with linear drag, one-way coupled — they do not affect the flow — and gravity is the only other force. With a hundred particles and two realisations, the held shares are uncertain by a few hundredths.

Who traced particles through cells #

Stommel’s trapping is of 1949, in a paper on why plankton and raindrops can stay suspended in Langmuir circulations and convection cells. Maxey and Corrsin found the speed-up of inertial particles in the cellular flow in 1986, and Maxey in random flows in 1987; Wang and Maxey measured it in simulated turbulence in 1993. The flicker experiment here is the earlier essay’s lattice with its strength made random.

How it was checked #

The amplitude signal, generated by its exact update over twenty thousand turnovers, has a mean of 0.991, a standard deviation of 0.499 and a correlation of 0.364 one memory apart, against 1, 0.5 and $1/e = 0.368$. With no flicker the inertialess cloud is held at 0.53 by the marched paths, against Stommel’s 0.515 from the invariant, the difference being the hundred-particle sampling; and the suspension band holds 0.22, against the earlier essay’s up to 0.23. The model refuses a memory of zero, a negative flicker and a particle that does not settle.

Still open: cells that drift as well as flicker #

Every cell here flickers in place, so the down-draughts never move. Real eddies also travel: a convection cell’s rolls drift, and turbulent eddies are carried by the larger ones. The next calculation would let the lattice translate at a speed that wanders as its strength does, and ask whether a drifting lattice, which moves the down-draughts from under a particle being swept along them, hurts the speed-up more than a flickering one of the same size — and so whether the speed-up measured in real turbulence is set by how fast the eddies weaken or by how fast they move.