A boundary that only exists over a window

Where a vortex stops is this collection’s account of the instantaneous criteria — the ones built out of the velocity gradient at a point and an instant, which disagree about where a vortex ends and mostly turn out to be one criterion wearing four names. It ends by observing that the thing a reader usually wants is not in the velocity field at all.

This essay is about what is there instead, and about the price of it. The object a reader is usually after is a boundary: a curve that fluid on one side of does not cross, so that the flow is divided into regions with different fates. Such curves exist, they can be computed, and they are properties of a stretch of time rather than of a moment. Change the stretch and the boundary moves.

What the field is #

Take two neighbouring parcels and follow both for a fixed length of time. At a window of eight the largest stretching anywhere in the box is e to the power 3.49, which is a factor of 33; the median point is stretched by e to the power 1.46, a factor of 4.3. If they end up much further apart than they began, the fluid between them has been drawn out; if they end up as close as they started, it has not. Doing that in every direction at once is a question about the gradient of the flow map, and the largest stretching available is the largest singular value of that gradient.

The finite-time Lyapunov exponent is the logarithm of that stretch, divided by the length of the window. It is a rate, it has a window attached to it, and it is defined at every point of the flow. The field it forms is not smooth-looking: it has ridges, and the ridges are the interesting part, because a ridge is a place where fluid on the two sides of it went to very different places.

That is what makes a ridge a barrier. Two parcels straddling one separate exponentially; two parcels on the same side of it do not. Nothing is forbidden to cross it — it is not a wall — but the fluid that does cross has to be pulled out into a filament to do so, and on the time scale of the window there is not enough of it to matter.

It is not a blurred snapshot #

The obvious hypothesis is that this field is a smoothed version of something instantaneous — that the window is doing an average and the underlying structure is already visible in the strain rate at one moment. It is not: over a 49-by-25 grid the correlation between the finite-time field at a window of eight and the instantaneous strain rate is 0.281, and the two peaks are of different sizes as well as in different places — 0.437 against 1.957.

It is not. The rate of strain of the same flow at the moment the window opens has its own structure, drawn on the same grid at the same shading, and the two fields correlate at 0.28 — which is not zero, because both are large where the flow is fast, and is nowhere near enough for one to be a version of the other.

The reason is arithmetic rather than physical. The instantaneous strain rate is a local property: it is a derivative of the velocity field at a point. The finite-time exponent is a property of a trajectory: it depends on every velocity gradient the parcel met along the way, multiplied together in the order they were met. Those products do not commute, which is the subject of two strainings, and the order they came in, and the failure to commute is exactly the difference between the two pictures here.

So the crest is not where the strain is largest now. It is where the product of the strains along a path was largest, and the two questions have different answers.

Change the window and the boundary moves #

The window is not a numerical parameter to be converged. It is part of the question.

Over two units of time the field is a broad smear with a single crest near one gyre’s edge. Over twelve it is a set of thin filaments, and the brightest place is elsewhere: the crest has moved by 1.04 in a domain two units wide, which is half the tank.

Six windows on the same flow:

The peak falls by a factor of 5.0 across the sweep, which is most of what a logarithm divided by a lengthening window does; what does not follow from that is where the ridge goes. The crest moves 1.0425 units between the shortest and the longest window, across a domain two units wide — half the box. Neither answer is a converged version of the other, and asking which window is correct is asking the wrong question. The correct window is the one over which the reader’s problem happens. A contaminant that decays in two time units is separated by the two-unit barrier — peak exponent 1.024 — and does not care about the twelve-unit one, whose peak is 0.291 and whose crest sits somewhere else entirely. A larva that lives for twelve is governed by the other.

The exponent itself falls as the window lengthens — from 0.95 at one unit to 0.22 at sixteen — for the plain reason that an exponent is a rate averaged over its own window, and a long window averages the quiet stretches in with the violent ones. That fall is not a sign that the structure is weakening; it is the definition doing what it says.

Forward and backward are different curves, and both are real #

The window can be run backwards, and doing so gives a different field with a different crest, correlating with the forward one at −0.04: statistically unrelated.

The two have distinct meanings and it is worth keeping them apart, and they are not the same field: computed over the same grid at the same window magnitude, the forward and backward exponents correlate at −0.0418, which is nothing at all. The forward field’s ridges are places where fluid is being pulled apart, so they act as repelling barriers: fluid arriving at one is sent to either side. The backward field’s ridges are places where fluid is being squeezed together, so they act as attracting structures: they are the curves along which material collects, and they are what a dye filament or a line of foam actually lies along.

Anybody looking at a photograph of a flow and seeing a sharp line of tracer is looking at an attracting structure, which is the backward field’s ridge. Anybody computing where a spill will divide wants the forward field’s. Using one for the other is a common and expensive error, and the −0.04 is the measurement of how expensive.

What the solver computed, and how it was checked #

The flow is the unsteady double gyre, the same one carrying a scalar is a record of where its fluid was, and the same one whose area-preserving properties are checked in the area that must not move.

The flow map is computed once per grid point — 1,225 of them, on a 49-by-25 grid — with fourth-order Runge-Kutta, and the map’s gradient is taken by differencing neighbouring grid points rather than by integrating four extra trajectories per point, which makes the cost linear in the grid instead of five times it, and is what makes a six-window sweep affordable at build time.

Three checks stand behind the numbers. The field converges: quadrupling the number of integration steps moves it by 8·10⁻⁷, so the crest is a property of the flow rather than of the stepper. The crest is a crest: it is required to stand at least twice its own field’s median, and reads 2.4 times it, so there is a ridge rather than a slowly varying hump with a maximum somewhere. And the comparison with the instantaneous field is made on the same grid at the same moment, with the correlation computed rather than judged by eye.

One thing was got wrong first and is recorded because the correction is a general one. The crest was originally taken over the whole domain, and at very short windows it landed in a corner, where the gyre’s own boundary does the stretching and there is no structure to find. The crest is now taken over the interior with a stated margin, because a maximum on a boundary is a fact about the boundary.

What this is not #

It is not a streamline. In an unsteady flow a streamline is an instantaneous picture and nothing material follows it, which is the point of streamlines are not the paths particles take. The ridge is a material curve to the accuracy of the window: fluid does not cross it, in the sense that crossing requires being drawn into a filament.

It is not frame-independent for free. The stretching field inherits whatever frame the trajectories were computed in, and adding a uniform translation leaves it unchanged while adding a rotation does not. That is better than the instantaneous criteria manage — the picture belongs to whoever is watching is the account of how badly they fare — but it is not the objectivity a barrier deserves, and the literature’s more careful constructions exist for exactly that reason.

And it is not a fluid-mechanical object. Nothing above used the momentum equation, a viscosity, a Reynolds number or a pressure — the whole calculation is 1,225 flow maps and their gradients, and the median exponent over the box at a window of eight is 0.182 against a peak of 0.437. It is a statement about a map, so it applies equally to a velocity field from a solver, from a radar, from a model or from a made-up function — which is a strength when the flow is measured and a hazard when the measurement is poor, because the exponent’s sensitivity to the trajectories is exponential by construction.

What a barrier of this kind is worth #

The reason this construction is used rather than admired is that a great many practical questions are questions about a division of the fluid, and have no instantaneous answer.

Where a spill goes. A surface slick released on one side of a repelling ridge is carried to one coast and on the other side to another, and the ridge is often a few hundred metres wide in a current field known on a ten-kilometre grid. The answer is a boundary in the flow rather than a boundary in the data, which is why the sensitivity is exponential and why an ensemble is the honest form of the answer.

Where things collect. Foam lines, sargassum, plastic, and the plankton that follow them lie along attracting structures, which is the backward field’s ridges. A photograph showing a sharp line of material is a photograph of a curve that has been assembling for as long as the material has been in the water — its window is set by the tracer’s own age, not by the observer’s.

Where mixing is fast and where it is not. The regions between ridges are the ones that do not exchange fluid with one another on the window’s time scale, so a reactant confined in one of them reacts with itself. That is the mechanism by which a perfectly deterministic laminar flow can be a poor mixer at one time scale and an excellent one at another, and it is why a residence-time distribution measured in a stirred vessel has more than one hump in it.

And where a measurement is representative. A sample taken inside a region is representative of that region and of nothing else. Ridges are where a sampling strategy has to be dense, and they are not where the gradients in the measured field are steepest, which is the usual place a sampling strategy puts its stations.

The exponent has a horizon of its own #

There is a trap in taking the window longer in the hope of a cleaner answer, and it is worth stating because the arithmetic runs the opposite way to intuition.

The exponent field’s structure gets finer as the window lengthens, not coarser. Filaments of high stretching fold and interleave, and their spacing falls exponentially at the same rate the exponent measures. So a grid that resolves the two-unit field will not resolve the twelve-unit one, and the sixteen-unit field on the same grid is a picture of the grid.

That is a hard limit rather than a computational one. Resolving a window twice as long needs a grid finer by the exponential of the exponent times the window — for the numbers here, about a factor of nine per additional four time units — so the cost of a longer window grows as fast as the structure it is trying to see. The practical consequence is that the window is chosen by the problem and then checked against the grid, and a field whose crest sits at the grid scale is reporting the discretisation.

The same arithmetic, run in the other direction, is a scalar is a record of where its fluid was’s horizon: a decade of accuracy buys a fixed amount of extra window, and no more.

Why the instantaneous criteria were ever expected to work #

It is worth asking why the local criteria are used at all, since they are cheaper and were invented first.

In a steady flow they are correct. A streamline is a pathline, a stagnation point is a permanent feature, and the separatrices joining stagnation points are genuine barriers with no window attached. Everything in the instantaneous toolkit was built in that setting and works there exactly.

The trouble is that a flow which is steady in one frame is unsteady in another, and a flow that is nearly steady is not steady at all for the purpose: the separatrices of a slowly varying flow do not merely wobble, they open into a tangle whose fine structure is what no randomness, and it mixes anyway is about. The instantaneous criteria degrade discontinuously rather than gradually as unsteadiness is added, which is why they can look excellent in a test case and be useless in the flow they were bought for.

What the picture cannot show #

Three things, and the first is the one most likely to mislead.

The shading is a rate, not a barrier. A bright cell means the fluid there was stretched; the barrier is the ridge those cells form, and picking it out is a separate operation that this figure does not perform. A reader who traces the brightest cells by eye will draw a curve that is close to the right one in the middle of the domain and wrong near its ends, where the field is broad.

The two panels of the window comparison have different shading scales. Each is scaled to its own maximum, because the exponents differ by a factor of four and a common scale would render the long window almost blank. That is the honest choice for comparing shapes and it makes the panels useless for comparing magnitudes, which is what the separate plot of peak against window is for.

And the grid is coarse. Forty-nine by twenty-five is enough to see the crest move and is not enough to see the filaments at the longest windows, for the reason the section above gives. Nothing here should be read as a picture of the fine structure; the fine structure is real and is below this resolution.

Who found it, and when #

The idea that a barrier in an unsteady flow is a finite-time material object came from dynamical systems in the 1990s and reached fluid mechanics through oceanography, where the practical question — where will this spill go — has no instantaneous answer and a satellite-derived surface current field to work with. Haller’s papers from around 2000 gave the finite-time exponent field its standing as a diagnostic and, more importantly, gave the criteria for when a ridge in it really is material rather than merely bright.

The distinction between a bright ridge and a material one is the part of the subject still under active repair, and it is worth knowing that the field computed here is the diagnostic rather than the theorem: a ridge is strong evidence of a barrier and is not a proof of one.

Limits recorded rather than smoothed over #

The ridge here is the crest of each column, not a properly extracted ridge in the second-derivative sense. That is enough for the question asked — whether the crest is in the same place when the window changes — and it is not enough to draw the barrier as a curve, which is why no figure here draws one.

The exponent is the largest one. A two-dimensional incompressible flow has a second exponent equal and opposite, so nothing is lost; a three-dimensional or compressible flow has a spectrum, and the largest member of it does not describe the geometry on its own.

One flow, one set of parameters. The double gyre at the standard settings is a model flow chosen because it is the smallest unsteady flow that is genuinely chaotic. The numbers are its; the shape of the conclusions is not.

And the correlations are Pearson correlations over a grid, which is a crude comparison between two fields with very different distributions. A rank correlation gives a different number and the same conclusion, and neither is a substitute for the two pictures side by side.