A scalar is a record of where its fluid was

A velocity field is a complete description of a flow’s present. Handed to a solver it determines the next instant, and the instant after that, for ever: the equations of motion need nothing else. It is a state in the strict sense, and everything about the flow’s future is in it.

It contains nothing whatever about the flow’s past.

That is not a limitation of any particular solver. Two flows that arrive at the same velocity field by completely different routes are, from that moment on, the same flow — and yet the dye in them is arranged differently, the pollutant in them came from different places, and the temperature they carry records different histories. Something in the flow is holding the past, and it is not the velocity.

This essay is about what is holding it, which is the flow map: the function that says, for each point now, which point its fluid occupied then.

What a conserved scalar actually is #

Take a scalar that is carried without being created, destroyed or diffused — a dye concentration in the limit of no diffusion, a temperature in a flow with no heating, a tag on a parcel of fluid. Its governing equation says that its material derivative is zero: following the fluid, it does not change. Steady does not mean nothing is happening is the collection’s account of that derivative and of how a quantity can be constant along a path while varying everywhere at once.

The solution of that equation is a sentence rather than a formula. The value here now is the value there then, where “there” is the place the fluid here now was occupying then. Nothing is computed; something is looked up.

So a conserved scalar field is not really a field with its own dynamics. It is the initial field, composed with a map. Everything interesting in a dye photograph — the filaments, the striations, the places where two colours interleave without mixing — is a property of that map and not of the dye.

The picture above is built by asking every point on a grid where its fluid came from, six time units back, and reading the answer off a stripe pattern. No transport equation was solved and no scalar was stepped forward. The whole of the left panel is the right panel, seen through a map — and the map reproduces the scalar to 2.9·10⁻⁷, which is the integrator’s error and nothing else.

The map is not a function of the field #

The temptation, and it is a strong one, is to think the map can be reconstructed from the velocity that is visible now — that the present field, being a complete state, must somehow encode the displacement that produced the present pattern.

It cannot, and the cheapest way to see it is to try. The natural guess is that a parcel arrived at its present position by travelling at the velocity found there, for the elapsed time: displace the initial pattern backwards by the local velocity times the interval and read it off. On a scalar running between −1 and 1 that guess is wrong by 0.575 on average and by 1.886 at worst — which, on a range of two, is a mean error of 29 per cent and a worst error that has the sign inverted and the magnitude nearly saturated. The map’s own error over the same field is 2.9·10⁻⁷, which is six and a half million times smaller than the guess’s worst.

Over sixteen hundred points in an unsteady double gyre, six time units on, that guess is wrong by 1.89 at its worst, on a scalar whose entire range is 2 — which is to say it can be exactly as wrong as it is possible to be — and by 0.58 on average, which is more than a quarter of the range. The flow-map reconstruction over the same points moves by 3·10⁻⁷ when the number of integration steps is halved, so it is exact to the accuracy of the arithmetic.

The reason the guess fails is not that six units is a long time. It fails because the velocity at a point changes as the parcel travels, and the whole displacement is an integral of velocities the parcel met and the present field never saw. Streamlines are not the paths particles take is the same statement made about curves: the instantaneous picture and the accumulated one are different objects in any unsteady flow, and in this one they are unrecognisably different.

What the map costs to carry #

If the map is the memory, the practical question is how expensive it is to keep — and the answer is that it is much larger than the field.

A velocity field on a grid is two numbers per cell — 2,450 numbers on the 49-by-25 grid used here. A flow map from now back to a chosen earlier time is also two numbers per cell, but there is one of them for every pair of times, and the map from now to six units ago cannot be obtained from the map from now to three units ago without composing it with another map. A code that wants the origin of every parcel at arbitrary depth into the past must either store the trajectories or recompute them.

That is why particle tracking exists as a separate discipline from field solving, and why no randomness, and it mixes anyway has to integrate paths rather than read a field: the quantity of interest there — how much interface a stirring protocol has created — is a property of the map and is invisible in any snapshot of the velocity.

Why nothing else in the flow can carry the past #

It is worth being precise about the claim, because it is stronger than it first sounds. The assertion is not that the flow map is a convenient way to store the history. It is that in an ideal incompressible flow there is nothing else available.

The velocity field is a state, and a state by definition determines the future without reference to the past. Pressure is not independent: in an incompressible flow it is the solution of a Poisson equation driven by the velocity field at that instant, so it carries exactly what the velocity carries and no more — which is pressure has no speed. Vorticity is a derivative of the velocity. Density, in a constant-density flow, is a constant.

So every Eulerian quantity is a function of the present velocity field, and any two histories arriving at the same field are, in every Eulerian variable, identical. The difference between them lives entirely in the labels the fluid is carrying — which parcel is where — and the object that records those labels is the flow map. Adding a conserved scalar to the calculation adds a memory precisely because the scalar is a label rather than a field with dynamics of its own.

The exception, and it is the interesting one, is a fluid whose stress depends on its own deformation history: a polymer solution, a suspension, a viscoelastic melt. There the map is not merely a record kept by the tracer, it is a term in the momentum equation, and the flow’s past acts on its present. That is the fluid that has not finished its last deformation and it is the one case in this collection where the memory pushes back.

The memory has a horizon, and it is logarithmic #

The map is exact, and it is also unstable, which is the second half of the story and the more useful half.

Two points a thousandth apart now had origins that were much closer than a thousandth or much further apart, depending on where they sit; over a long enough window the typical pair separates exponentially. Measured over twenty-four starting points in the same double gyre, the widest pair’s origins separate by a factor of 29 in twelve time units — a rate of 0.281 per unit time — while the median pair separates by a factor of 2.3.

The consequence is a limit on how far back a position can be believed. If the present position is known to a part in ten thousand, and errors grow at 0.281 per unit time, then the origin is known to a tenth of the domain for about 8.2 time units and to nothing at all after about twenty. Improving the present measurement by a factor of ten buys a further 8.2 units and no more, because the growth is exponential and the accuracy enters through its logarithm.

Twenty-four pairs of parcels released 0.001 apart, integrated backwards, end 29.1 times further apart on average and 2.285 apart typically — a separation rate of 0.281 per unit time, which puts the horizon at 24.6 time units for a measurement good to one part in a thousand. Improving that measurement by a decade buys 8.2 more time units, and no more.

So the flow’s memory is perfect and unreadable. The map exists, it is exact, and recovering it from measurements degrades logarithmically in the measurement quality — which is the same arithmetic that governs forecasting, run backwards. An hour for every tenfold is this collection’s account of the forward version, and the two are the same statement about the same exponent: a decade of accuracy buys a fixed interval of time, in either direction.

Which direction the instability runs #

The two directions are not equivalent, and confusing them is a common error in reading dye pictures.

Running forward, neighbouring parcels separate, so a compact blob of dye becomes a long filament. Running backward, the same instability means that a compact patch of present dye came from a long filament in the past — so a small sample taken now is a mixture of fluid from a wide region then, and the finer the filament, the wider the region.

That has a practical reading for anybody interpreting a tracer measurement. A concentration measured at a point is not a sample of one place upstream; it is a weighted average over an upstream region whose size grows exponentially with how far back the question is asked. The region is the map’s inverse image of the sample volume, and it is a fractal set of filaments rather than a blob.

What the solver computed, and how it was checked #

The flow is the unsteady double gyre in its standard parameters, chosen because it is the smallest unsteady flow that is genuinely chaotic and because the same flow carries the area that must not move, which is this collection’s check that an integrator preserves what it must.

Trajectories are integrated with the ordinary fourth-order Runge-Kutta rule rather than the implicit midpoint rule used there, and deliberately: the question here is where a parcel went, not what a stepper conserves, and the explicit rule at three hundred steps is more accurate on the trajectory — it recovers the scalar to seven decimal places over six time units.

Three checks stand behind the numbers. The reconstruction is converged: halving the step count moves the reconstructed scalar by 3·10⁻⁷ of its range of two, so the map has been computed rather than approximated. The local guess is genuinely bad rather than badly implemented: its error is compared against the same converged answer over the same grid, and the check refuses a claim that its worst error exceeds the scalar’s entire range, which would be arithmetically impossible. And the separation is measured over an ensemble rather than at one point, because a single pair starting in the wrong place separates at a negative rate — reverse time contracts as often as it stretches — and the first version of this measurement did exactly that, read −0.117, and was right and useless.

The refusals are exercised: each check is offered a tolerance of zero and must refuse it.

Where this changes an answer #

Source attribution. Asking where a pollutant came from is asking for the inverse flow map, and the answer degrades logarithmically with the quality of the wind or current field. Beyond the horizon, an attribution is a statement about a distribution over the whole basin rather than about a place, and quoting it as a place is quoting a number the map cannot support.

Data assimilation. A scheme that corrects a forecast by nudging the velocity field towards observations is correcting the state and not the memory. A scalar field carried by that flow keeps the signature of the trajectory it actually took, so the tracer and the velocity can disagree after assimilation even when both are individually good.

Reading a dye photograph. The most common error in interpreting a visualisation is to read the present pattern as a statement about the present flow. It is a statement about an interval, and about which interval depends on how long the dye has been in the water. Two photographs of the same flow taken with dye injected at different times show different patterns, and neither is more correct than the other; they are answers to different questions, both of them questions about the past.

And any measurement of mixing. The rate at which a scalar’s gradients steepen is set by the flow map’s stretching, and the eventual homogenisation is set by diffusion acting on those gradients. A model that gets the velocity statistics right and the trajectories wrong will get the first qualitatively right and the second wrong by an exponential, which is why steady, three-dimensional, and mixing anyway insists that mixing is a Lagrangian question asked about an Eulerian object.

Who found it, and when #

The idea that a conserved scalar is the initial field composed with a map is as old as the method of characteristics, which is Lagrange’s, and the fluid-mechanical statement of it is Lagrange’s too — the Lagrangian description is defined by carrying labels rather than fields, and the flow map is exactly the change of variables between the two descriptions.

What is recent is treating the map as an object to be computed and looked at. That arrived with chaotic advection in the 1980s, when it became clear that a smooth, slow, entirely deterministic flow could stretch material lines exponentially, and that no statistic of the velocity field predicted how fast. The instruments that followed — finite-time exponents, Poincaré sections, the finite-time material structures that a boundary that only exists over a window is about — are all instruments for reading the map rather than the field, and they exist because the field was measured first and found to be silent.

What the picture cannot show #

The reconstructed field above is drawn on a grid of finite cells, and that is a real limit rather than a cosmetic one. By six time units the true scalar field has structure on scales far below the cell size — filaments the map has drawn out and folded — and the picture shows the value at the cell centre, not the average over the cell.

Diffusion, in a real fluid, is what stops that regress. It acts on the steep gradients the map creates and erases structure below a scale set by the balance between stretching and diffusion, so a real dye field has a finest filament and the idealised one here does not. That scale, and the fact that the mixing rate ends up nearly independent of the diffusivity, is the subject of a millionth is enough.

Limits recorded rather than smoothed over #

Conserved means conserved. Everything above is for a scalar with no source, no sink and no diffusion. A reacting scalar is not a record of where its fluid was; it is a record of where its fluid was and what happened to it on the way, which needs the whole path rather than the endpoints of the map.

One flow, two dimensions. The double gyre is a model flow with an analytic velocity field, so its map is exact to the integration’s accuracy and nothing has been interpolated. A measured flow field is known on a grid, and interpolating it introduces its own error into the trajectories — which is usually the largest term in any real reconstruction and is absent here by construction.

The separation rate is this flow’s. The 0.281 is a property of the double gyre at these parameters and over this window; another flow has another number, and a flow with closed streamlines has none at all, because its trajectories do not separate. What transfers is the shape of the horizon, not its value.

And the local guess was chosen to be the natural one, not the best one. Displacing by the instantaneous velocity is what a reader’s intuition offers and is what the essay refutes. A better approximation — displacing by the time-averaged velocity along the path — is closer and is still not the map, because it is the map that decides which velocities to average.