Reversible, and unusable

Euler’s equations have no arrow of time in them. Reverse every velocity and the flow runs backwards through exactly the states it came from, because the equations are unchanged under that substitution. There is no dissipation to break the symmetry and nothing has been thrown away.

That is a strong statement about memory: an ideal flow has kept everything. The past is not merely recorded somewhere, it is recoverable in principle by running the machinery in reverse.

This essay is about the distance between in principle and in practice, measured. The reversal works — a blob of tracer particles goes out, comes back and lands on itself to a part in a thousand million. And it is unusable, for a reason that has nothing to do with the quality of the implementation.

The round trip #

A hundred and twenty tracer particles are arranged in a blob and carried by four point vortices for four time units, in 1,600 steps. The blob is stretched and folded into something unrecognisable. Then the sign of the time step is reversed and the same number of steps are taken. The worst particle lands 1.43·10⁻⁹ from where it started and the mean miss is 5.68·10⁻¹¹.

The particles return. The worst of them is 1.4·10⁻⁹ from where it began and the average is 5.7·10⁻¹¹, against a blob of radius 0.18 — so the recovery is good to eight or nine significant figures.

The errors are not uniform across the blob. Sorted, they span two decades — the mean is 25 times smaller than the worst, and the spread is the flow’s own stretching showing up in the arithmetic: a particle that was pulled further out and folded more times has had more opportunities for the last bits of its coordinates to be lost.

Two things are worth noting about that number before going on. It is not exactly zero, because floating-point arithmetic is not exactly reversible — the intermediate quantities in a Runge-Kutta step are rounded, and rounding is not an operation that undoes itself. And it is very much smaller than anything anybody could measure in a real fluid, which is the point of doing it in a solver.

The experiment that matters #

The recovery above assumes the reversal is exact: the same vortices, in the same places, with the same strengths. That is the assumption to attack, because it is the one no experiment and no measurement can satisfy.

So: run the same four time units — twelve, in the measurement below, for a longer window — and then, at the turning point, move one vortex by a small amount before running back. The tracers are left alone. What is being perturbed is the state of the flow, which is what anybody attempting to reverse a real or measured flow would get slightly wrong.

A displacement of 10⁻⁸ returns the worst tracer 4.9·10⁻⁶ from home. That is a gain of 486: the error in the answer is nearly five hundred times the error in the state.

Above about 10⁻⁹ the response is clean and linear — double the nudge and double the miss — which is what says the mechanism is amplification rather than anything more complicated. Below that the round trip’s own arithmetic takes over at 2.2·10⁻⁷ and the nudge disappears into it, so there is a floor: improving the knowledge of the state past a certain point buys nothing at all, because the reversal itself is no longer the accurate operation.

How the gain grows #

The gain is a property of the window, and measuring how it grows is what turns the observation into a budget.

Over windows from four time units to sixteen the gain rises from 17 to 915, as the 2.87 power of the window. That is algebraic rather than exponential, and the reason is worth stating because it is this flow’s rather than a general result: four point vortices in these positions move quasi-periodically. A perturbation to their state grows roughly linearly, and the tracers inherit a power law from it.

A flow whose own state is chaotic behaves worse, and much worse. A scalar is a record of where its fluid was measures the same quantity in the unsteady double gyre and finds exponential growth at a rate of 0.281 per unit time, which gives a horizon that improves only logarithmically with precision. A power law is the friendly case, and even the friendly case costs three decades of accuracy for a factor of four in the window.

What it would take to reverse a real flow #

Putting the numbers the other way round makes the practical statement.

To recover an initial position to a part in a thousand over twelve time units in this flow, the state at the turning point must be known to about 2·10⁻⁶. That is not a hopeless requirement for a solver and it is an impossible one for a measurement: it is the position of every vortex to two parts in a million, in a system with no measurement noise at all.

Five nudges, one window of twelve:

The gain is 486 — a displacement of one part in a million in one vortex’s position becomes a miss of five parts in ten thousand — and it is flat across four decades of nudge, so it is a linear amplification rather than a threshold. Below 2.18·10⁻⁷ the round trip’s own arithmetic floor takes over and the nudge is invisible. And that is the friendly case, in a flow with four degrees of freedom, over a window of twelve, with a gain rate of 0.516 per unit time and a growth in the window that goes as its 2.87 power rather than exponentially. A real flow has an enormous number of degrees of freedom, an exponential rather than a power law, and measurements with error bars.

So the honest summary is: the information is present, it is not destroyed, and it is not retrievable. Those are three different statements and this collection has now met all three separately.

Where the amplification comes from #

The gain is not mysterious and it is worth deriving in words, because the derivation says which flows have a large one.

Displacing a vortex changes the velocity field everywhere. A tracer somewhere else therefore moves slightly differently from then on, and the difference accumulates along its path. That much would give a gain proportional to the elapsed time and to the sensitivity of the field to the vortex’s position — a linear effect, and a modest one.

What makes the gain larger is that the tracer’s own separation from where it would have been is then amplified by the flow’s stretching, in exactly the way two strainings, and the order they came in describes: the difference is carried by a product of deformation gradients rather than by a sum of increments. So the gain is a product of two effects — how much the field moved, and how much the flow then magnified the resulting displacement — and the second is what makes the exponent nearly three rather than one.

That decomposition says which flows to worry about. A flow with a strong stretching field amplifies a small error in its own state far more than a gentle one, whatever the size of that error, and the places where the amplification is worst are the ridges a boundary that only exists over a window locates. Reversing a flow is hardest exactly where its structures are sharpest, which is where anybody would want to do it.

Why this is not an argument about the second law #

It would be easy to read the above as a demonstration of irreversibility, and it is not. Nothing here dissipates anything, no entropy is produced, and the exact reversal works. That is worth insisting on, because the two ideas are routinely confused.

Thermodynamic irreversibility is a statement about which states a system will actually visit, and it comes from the overwhelming number of ways of being disordered compared with the number of ways of being ordered. It applies to a system with a great many degrees of freedom and it makes no reference to precision.

What is measured here is a statement about sensitivity: a flow that is exactly reversible amplifies an error in the state, so a reversal performed with imperfect knowledge fails. It applies to a system with four degrees of freedom and it is entirely a statement about precision.

The two meet in practice, because a real fluid has both. But the demonstration here would work identically in a system with no thermodynamics at all, and the demonstration of the second law works in systems with no sensitivity to initial conditions. Neither implies the other.

What the solver computed, and how it was checked #

Four point vortices of unequal strength, desingularised inside a small core, with tracer particles carried by the field they induce. Vortices and tracers are advanced together with the same fourth-order scheme, and the reversal is performed by taking the same number of steps with the sign of the step changed — which is the arithmetic reverse of the forward run, not an independent backward integration.

Four checks. That the round trip works, to better than 10⁻⁶, which is what makes the rest of the essay about the perturbation rather than about the integrator. That the gain exceeds a hundred, so the amplification is being measured rather than asserted. That the floor is above 10⁻⁹, so that at least one nudge in the sweep is genuinely below the round trip’s own error and the floor is visible rather than inferred. And that the gain grows faster than the square of the window, which is the statement that the amplification is not simply proportional to the time elapsed.

One decision is worth recording. The nudge is applied to the vortices and not to a tracer. An early version perturbed a single tracer, which leaves the field it is carried by exactly right, and measured a gain of 12 — a number about one trajectory in a correct flow rather than about the flow being wrong. Perturbing the state is the question anybody reversing a measured flow is actually asking.

The same shape, three times in this collection #

And the object being reversed is idealised twice over: what a point vortex is not records what the singularity leaves out, which is a second reason not to read the round trip as a statement about a real fluid.

Forwards. No randomness, and it mixes anyway is the statement that a deterministic flow separates neighbouring parcels, which is this essay run in the other direction.

Backwards. A scalar is a record of where its fluid was is the same amplification applied to the question of where fluid came from, and produces the same logarithmic horizon.

And in time. An hour for every tenfold is the forecasting version: a decade of accuracy buys a fixed interval of predictability, whichever way the interval runs.

The three are one arithmetic statement, and the reason it keeps arriving is that a flow map is a composition of many transformations and its sensitivity is a product of theirs.

Why this makes reversal useful anyway #

Having said the reversal is unusable for recovering an initial condition, it is worth saying what it is used for, because the answer is a great deal.

As a check on a solver. A round trip that fails to return is a solver with a bug or a scheme with no time-reversal symmetry, and the test is cheap and unusually sharp: it exercises the whole machinery and compares against an exactly known answer.

As a measurement of sensitivity. The gain measured here is a diagnostic of the flow, obtainable without any reference to a true solution, and it is exactly the number a data-assimilation scheme needs in order to know how far back its observations are worth propagating.

And as a way of building a stirring protocol. A flow whose reversal is easy is a flow that has not mixed, and the mixing literature uses precisely this: the reversibility experiment in a stirred vessel, where the dye comes back if the stirring has been gentle and does not if it has not.

What a reader should take to a real problem #

Three transferable statements, none of which needs point vortices.

Reversibility is not recoverability, and a code that demonstrates one has not demonstrated the other. A round-trip test is a good test of an integrator and says nothing at all about whether a backward calculation on measured data will produce anything.

The useful number is the gain, and it is measurable without a truth. Perturb the state, run, and see how far the answer moves. That number sets how far back a reconstruction is worth attempting, and it can be computed from the model alone.

And there is a floor. Every implementation has a level of self-consistency below which improvements to the input are invisible, and finding it is a matter of sweeping the perturbation down until the response stops changing. A reconstruction that claims a precision below that floor is reporting its own arithmetic — which is the same reading the drift was the instrument makes of a conservation error.

What the picture cannot show #

The blob drawn in the first figure is three sets of points on one pair of axes: where they started, where they got to, and where they came back to. The first and third are drawn over one another and are indistinguishable at any print resolution, which is the result and is also why the figure cannot show the result — the whole finding is that the two sets differ by a part in a thousand million.

Nothing in it shows the intervening motion either. The blob does not travel out and come back along the same path in any simple sense; it is stretched, folded and reassembled, and a picture of the intermediate stages would be a picture of a filament rather than of a blob.

What the vortices are doing while this happens #

One more measurement is worth reporting because it separates two ideas that are usually run together.

The vortices in this flow move regularly. Four of them, in these positions, trace out a quasi-periodic motion: perturb one and the perturbation grows roughly linearly rather than exponentially, which is what makes the gain a power law. Their motion is essentially predictable over the windows used here.

The tracers carried by that regular motion are not. They are stretched, folded and interleaved, and two of them starting a thousandth apart end up on opposite sides of the domain. That is chaotic advection: the field is simple and the trajectories in it are not, which is the whole subject of steady, three-dimensional, and mixing anyway in its steady form.

So the flow’s state is predictable and its fluid is not, and the two questions have different answers in the same system. Anybody reversing a flow wants the second, and anybody validating a solver is usually measuring the first — which is a mismatch worth noticing before quoting either number as evidence about the other.

Who found it, and when #

The reversibility of the inviscid equations is as old as the equations. The demonstration that reversibility and predictability are different properties is Lorenz’s, from 1963, in a system with three variables rather than four.

The fluid-mechanical demonstration people remember is Taylor’s dye-in-glycerine experiment, filmed in 1967 — corn syrup in a rotating annulus, a blob of dye smeared out by many turns of the inner cylinder and reassembled by turning it back. That experiment is the reversible case, and its power is in how surprising it is; the measurement here is what happens when the reversal is not quite exact.

Limits recorded rather than smoothed over #

Four point vortices are not a fluid. They are an exact solution of Euler’s equations with the vorticity concentrated at points, and their motion is a four-body problem rather than a field. The gains measured here are that system’s.

And that system is quasi-periodic, which is the friendly case and is stated as such. The exponents reported are algebraic; a chaotic vortex configuration gives exponential growth, and the essay’s conclusions get stronger rather than weaker.

The reversal is exact by construction. Taking the same steps with a negative step size is not the same as integrating a separately posed backward problem, and a code that solved the backward problem independently would accumulate its own error rather than partly cancelling the forward one. The 1.4·10⁻⁹ would be larger.

And the floor is this arithmetic’s. It is set by double precision and by the number of steps, and it moves with both. What does not move with them is the gain, which is a property of the flow.