A cloud folds where a saddle is strong, not when

A lattice of vortices folds a cloud at the saddles’ quarter found that heavy particles in a steady cellular flow fold — their paths cross, and their concentration becomes infinite on a curve — only above a Stokes number of a quarter, where the particle overshoots the converging saddles instead of settling into them. Wobbling saddles keep the quarter moved the saddles without changing their strength and kept the threshold. A stronger strain folds a cloud only if it lasts then changed the strength itself, multiplying the whole flow by a pulsing or random amplitude, and found that particles fold at the quarter of the strain they have felt for about four strain times: a strong excursion that does not last does nothing.

In that lattice every saddle strengthened and weakened together, and a cloud below the steady quarter folded in bursts: nothing for a long time, then a large share of the cloud at once, when the single signal made a strong, lasting excursion. No real flow does that. In turbulence the strain at one place is nearly independent of the strain a few eddy sizes away, and a particle meets one strong patch after another as it moves. The question that essay ended on was what changes when each saddle has a history of its own: whether the folding becomes a steady rate, whether that rate follows an exponential in the inverse Stokes number or the Gaussian tail of the strain, and how it depends on how long a saddle’s strength lasts.

The answers, in this lattice: the folding becomes steady; it becomes rarer; its rate follows the Gaussian tail of the strain, at about a tenth of the time a saddle spends past its fold condition; and a longer-lived strain raises it only until the strain lasts a few strain times, after which nothing more is gained.

A lattice of independent saddles #

The flow is the cellular lattice, stream function $\sin x\sin y$, multiplied not by one amplitude but by a field $B(x, y, t)$ that is smooth in space. The lattice’s saddles sit at the corners of its cells, at whole multiples of π; a periodic patch of four by four of them carries sixteen independent random signals, each with a mean of one, a standard deviation of 0.3 and a correlation time of 2 in units of the mean strain’s time. Between the saddles the field is a smooth blend of the four nearest, weighted so that at each saddle $B$ is exactly that saddle’s own signal and its slope is zero. Near any saddle the flow is the steady lattice’s at that saddle’s current strength; between them it is an interpolation, and because the stream function is still a stream function, it is incompressible whatever the signals do.

When the sixteen signals are made identical the field is the previous essay’s lattice exactly, which is the first check: the velocity and its gradient reproduce the uniformly multiplied lattice to the last digit. Everything else follows the earlier essays: each particle relaxes towards the air’s velocity over its response time, carries the gradient of its own velocity field by the Riccati equation, and folds when the trace of that gradient falls through −50.

The first figure is the folded share of a cloud seeded uniformly over the whole patch, against time, at a Stokes number of 0.16, below the steady quarter, in four seeded fields. With one signal for every saddle, each cloud does nothing for a long time and then folds almost entirely in a single step, at a different moment in each field — whenever its signal first made a strong enough excursion for long enough. The step always stops at 89 per cent: a ninth of each cloud starts on a vortex centre, a point the uniformly scaled lattice holds still whatever its strength. With sixteen independent signals the curves rise steadily instead, and after 400 time units they have folded 66 to 85 per cent of their clouds.

The fold rate, and the strain’s tail #

A steady folding has a rate: the folds counted, divided by the total time the cloud’s particles spent unfolded, which is the exponential law’s own estimator. The second figure plots it against the inverse Stokes number, from a table of runs over four seeded fields at each of nine Stokes numbers, for both kinds of lattice.

The independent saddles fold less, at every Stokes number in the table. At 0.14 their rate is 2.7·10⁻⁴ per unit time against 1.3·10⁻³ for the shared signal, a fifth; at 0.16 about a half; at 0.2, 2.1·10⁻² against 3.8·10⁻². The refuted claim expected the opposite, and its reasoning was not foolish: sixteen independent saddles make sixteen times as many strong excursions as one signal does. But a particle does not fold because some saddle somewhere is strong. It folds when the saddle it is passing is strong while it passes, for long enough, and with independent saddles most of the strong excursions happen at saddles the particle is nowhere near.

The figure’s two other curves test what the rate follows. The faint dashed curve is the share of time one saddle’s strain spends above the steady fold condition — a quarter of the reciprocal Stokes number — which for a Gaussian signal is the Gaussian tail, falling as the square of the inverse Stokes number. The thinner solid curve is the rate at which one saddle makes an excursion above that condition lasting four strain times. The independent fold rate runs parallel to the first: it is 0.07 of the share of time above the condition at a Stokes number of 0.135 and 0.10 from 0.15 to 0.22. The long excursions fall much faster, by a factor of 260 between 0.2 and 0.14 where the fold rate falls by 77.

So the rate follows the Gaussian tail of the strain, not an exponential in the inverse Stokes number and not the rarer tail of long excursions. The particle samples saddles as it moves, spending a roughly fixed fraction of its time close enough to one to be folded by it, and the chance that the one it is near is past its fold condition is the time share. The near-constant factor of a tenth is that geometric fraction and the duration requirement together; it slips to 0.07 at the lowest Stokes numbers, where the excursions past the condition are shorter on average and fewer of them last long enough.

Why one signal folds more #

The shared signal’s advantage has a simple anatomy. When every saddle strengthens at once, a particle anywhere in its cell is never more than a quarter of an orbit from a strong saddle, and during a long excursion it passes several, each as strong as the last; the first lasting excursion folds nearly everyone. With independent saddles an excursion is local. A particle circling its cell visits four corners in an orbit of a few time units, and during one saddle’s strong excursion it spends perhaps a quarter of the time near that saddle. The excursion must also last four strain times of the particle’s own exposure, not of the clock, and the exposure is broken up by the orbit. Both effects cut the rate, and they cut it most at the low Stokes numbers, where only the longest, strongest excursions fold anything and where a particle therefore has to be in the right place for most of a rare event.

The independent field has one compensation the shared one does not. Its blend between saddles has slopes, and slopes push particles off the vortex centres, which the uniformly scaled lattice holds still for ever. Carried on to 800 time units, two of the first figure’s independent clouds reach 95 and 97 per cent, past the shared signal’s ceiling of 89, because their ninth that started at the centres is not trapped there: a particle placed on a centre of the patchwork field starts moving at once. The signals in each saddle are the same kind as a wandering forcing leaves the averaged four-fifths law alone used to stir a cascade, an Ornstein–Uhlenbeck process stepped exactly; there the average was protected by a linear balance, and here nothing protects it, because the fold condition is a threshold.

A parcel goes straight while the streamlines curve and steady does not mean nothing is happening are the reason the place matters at all: a parcel’s acceleration is what the flow does to it along its own path, and an inertial particle answers that acceleration late. In a lattice of independent saddles, the path decides which strains the particle is offered.

Steady, not bursty #

The third figure measures how steady the folding is: the share of all a cloud’s folds that come in the two busiest of forty equal windows of the run. A perfectly steady rate would put a twentieth there. With one signal the share is one at every Stokes number: every fold in every field comes in two windows, the burst. With independent saddles it is about a quarter to a third from 0.15 to 0.18 — not a twentieth, because a field of sixteen saddles still has its own strong episodes, but spread over the run. At 0.14 and below the folds are so few, a handful per field, that the measure means little.

That is the first of the questions answered, and its consequence is practical. A droplet cloud in a real turbulent flow does not wait for the whole flow to strain hard at once; it folds steadily, a few particles at a time, wherever a strong patch meets a particle. A particle is a low-pass filter described how a particle responds to the flow it moves through; here what it responds to is set as much by where it is as by when.

The strain a particle feels forgets faster #

The fourth figure shows a quieter consequence of independence. It follows light particles, at a Stokes number of 0.05 so that none folds, and records the strain amplitude at each particle’s own position as it moves; then it measures how fast that felt strain forgets itself, as the correlation of the record against its own lag. The dial moves the saddles’ correlation time from 1 to 8.

With one signal the felt strain is simply the signal, and it forgets at the signal’s own rate. With independent saddles it forgets faster: at a correlation time of 4, integrated to its first zero, the felt strain’s correlation time is 3.0 against 5.0 for the shared signal measured the same way. (Both estimates come from records 300 time units long, and the shared signal’s, which should be 4, shows the estimator’s own scatter; the comparison between the two is like for like.) The particle carries itself from saddle to saddle round its cell, and each saddle it reaches has its own state, so part of what it feels as a change in the strain is its own motion through a patchwork. The camera needs the window’s memory, not the probe’s made the general point for measurements: the correlation that matters is the one along the path the measurement actually takes.

A longer-lived strain helps only so far #

The fifth figure answers the last question. At a Stokes number of 0.16 the independent saddles’ fold rate rises with the saddles’ correlation time, from 4.9·10⁻⁴ per unit time at 0.5 to 1.5·10⁻³ at 1, 2.8·10⁻³ at 2 and 3.5·10⁻³ at 4, and then stops: at 8 it is 3.1·10⁻³. The shared signal does the same, peaking at 2. The marker on the axis is four strain times at the fold condition, 2.6 time units, the duration the previous essay found a particle needs. While the strain’s excursions are shorter than that, lengthening them lets more of them last long enough and the rate rises; once they routinely last that long, a longer memory adds nothing, and a much longer one even reduces the number of separate excursions in a run of fixed length.

That is the answer to how the turbulent fold rate is set, in this model: by how often the strain is strong, as long as the strain is strong for long enough, which at these Stokes numbers it is once its correlation time reaches a few strain times. Real turbulence’s strain is correlated over about one Kolmogorov time, which is close to the strain time itself, so on this reading it sits on the rising part of the curve, where duration still matters.

What was checked #

With every saddle carrying the same signal, the flow reproduces the uniformly multiplied lattice exactly. The patchwork flow’s divergence is zero to rounding at points between the saddles, and at each saddle the strain is exactly that saddle’s own amplitude, which checks the blending weights. The analytic velocity gradient, including the slopes of the blend that a uniformly multiplied lattice does not have, was checked against finite differences of the velocity to better than a part in a billion. The fold rates are a stored table, since each row takes about a minute to march; one stored run was marched again and reproduced its folds and its total unfolded time to machine precision. The calculation refuses an odd number of saddles, which would break the lattice’s periodicity, and a Stokes number of zero.

What the patchwork leaves out #

A lattice is not turbulence. The saddles here sit still at the cell corners and only their strength fluctuates; in turbulence the strong patches move, form and dissolve, and a particle’s encounters with them are not organised round a cell. Drag decides whether a cloud folds and its successors have kept the lattice deliberately, because it lets the fold condition be stated exactly; the price is that the geometric factor of a tenth belongs to this lattice.

Sixteen saddles. A larger patch would add saddles without changing what a single particle meets, which is a handful of neighbouring cells; the stored rates should not move, but they were not checked against a larger patch.

Gaussian signals. Real strain rates have tails heavier than Gaussian, and since the fold rate follows the time spent past the fold condition, a heavier tail would raise the rate at small Stokes numbers by exactly the factor the tail is heavier there.

Two dimensions. In three dimensions a particle can fold along a line or a sheet, and the fold condition has more than one strain in it.

Who folded the clouds #

The caustics of inertial particles — the folds of their velocity field — and their rate in random flows were described by Wilkinson and Mehlig in the early 2000s, who found the rate falling exponentially with the inverse Stokes number in short-correlated random flows, and by Falkovich, Fouxon and Stepanov, who tied the formation of caustics to the collision rate of droplets in clouds. The cellular lattice as a test bed for inertial particles goes back to Maxey’s studies of the 1980s. The comparison here between a shared and a patchwork strain, and the Gaussian tail it finds in a long-correlated flow, are this collection’s.

Still open: saddles that move #

The saddles of this lattice are fixed in place. A flow in which they drift — a lattice advected by a slow mean flow, or one whose cells merge and split — would carry strong patches to particles as well as particles to them. The next calculation would let the lattice translate at a speed comparable with the particles’, and ask whether the fold rate’s geometric factor of a tenth rises when the strong saddles come to the particles, and whether a particle swept along with a strong saddle, feeling its strain for longer than the saddle’s own correlation time, folds at the rate of the long excursions instead of the time share.