The stretching rate that is not one number

Longer with nothing pulling it establishes the basic fact: a material line in a flow gets longer, on average, whatever the flow is doing, because stretching and compression do not cancel when they are multiplied rather than added.

That essay measures a rate. This one asks whether there is a rate, and the answer is that there is one for the typical line, a different one for the average line, and a different one again for every moment of the distribution.

The computation #

A line element carried by a flow satisfies $d\boldsymbol\ell/dt = (\nabla\mathbf{u})\cdot\boldsymbol\ell$ — the deformation gradient, integrated along a trajectory.

The flow is the ABC flow with $A = \sqrt3$, $B = \sqrt2$, $C = 1$: steady, three-dimensional, exactly solenoidal to machine zero, and an exact solution of Euler’s equations. Nothing in it is random and nothing depends on time. Steady three-dimensional and mixing anyway is the essay about why its streamlines are chaotic despite all that.

Eight hundred elements are released at random positions with random orientations and integrated for fifteen time units. Their length ratio is written $\ell = e^S$, with $S$ accumulated as the logarithm of the renormalisation factor at each step — which is the only way to hold a quantity reaching $e^{20}$ and one reaching $e^{-5}$ in the same array.

The typical element and the average element #

The Lyapunov exponent is the rate of the typical element: $\langle S\rangle / t$, which converges by Oseledets’ theorem for almost every element.

Measured here it is 0.157.

The average length grows at a different rate: $\ln\langle e^S\rangle/t$, which is 0.273. That is 1.74 times larger, and the difference is not an error in either. They are the rates of two different quantities.

If $S$ were exactly Gaussian with variance $2Dt$, the two would differ by exactly $D$: $\ln\langle e^S\rangle = \langle S\rangle + \tfrac12\mathrm{var}(S)$. Measured, $D = 0.064$ and the gap is 0.116, so even the Gaussian estimate is out by half — which is the first sign that the distribution is not one.

Why the two rates differ at all #

The mechanism is worth stating because it is elementary and it explains everything that follows.

Consider a hundred elements, ninety-nine of which double in length and one of which grows by a factor of a thousand. The typical element has doubled: $\langle\ln\ell\rangle = \ln 2$ very nearly. The average length is $(99\times 2 + 1000)/100 = 11.98$, so $\ln\langle\ell\rangle = 2.48$ — three and a half times the typical logarithm.

Exponentiate before averaging and the largest member dominates; average before exponentiating and it counts once. Since $\ell$ is a product of many stretching factors along a trajectory, its logarithm is a sum, and sums have well-behaved averages while products do not.

That is the whole of it. The Lyapunov exponent is the average of a sum and converges nicely; the average length is the average of a product and is set by the tail.

Everything else in this essay is that observation carried to higher moments and measured.

A rate for every moment #

Generalise. Define

$$\lambda_p = \frac{1}{pt}\ln\langle \ell^p\rangle,$$

which is the Lyapunov exponent as $p \to 0$ and the average-length rate at $p = 1$.

Measured over the ensemble:

A Gaussian $S$ would make that a straight line, $\lambda_p = \lambda + pD$. It is not: the measured values depart from that line by up to thirty per cent, which is the same multifractality that the exponents that stop being thirds finds in velocity increments — arrived at from the geometry of the flow rather than from its statistics.

The average is a measurement of the tail #

The reason the moments disagree is that the distribution is broad, and the practical consequence is worth quantifying.

At $t = 15$, the longest 0.5 per cent of the elements carry 47 per cent of the total length, and half the total is in the top 0.75 per cent.

So quoting the average stretching rate describes a set of material elements that almost none of the fluid is in. The typical parcel is stretching at 0.157 and the average is being computed from a handful of parcels that found the most vigorous corner of the flow.

What a curved $\lambda_p$ means #

The departure from a straight line has a name and a meaning worth extracting.

A straight $\lambda_p = \lambda + pD$ corresponds to a Gaussian distribution of $S$, which is what a sum of many independent increments would give. Curvature means the increments are not independent: an element that has been stretching fast is more likely to keep stretching fast, because it is in a vigorous region of the flow and it stays there for a while.

That correlation is the same thing intermittency is in the turbulence essays — a quantity whose fluctuations are clustered rather than scattered — and it produces the same signature, which is a non-linear dependence of exponents on the order of the moment.

The formal object is the Cramér function: the large-deviation rate function of $S/t$, whose Legendre transform is $p\lambda_p$. A parabolic Cramér function gives a straight $\lambda_p$; anything else gives a curve. Nothing here computes the Cramér function, and the curvature measured is the evidence that it is not a parabola.

And the distribution is two populations #

The shape of the distribution explains the rest.

There is a broad chaotic population, stretching steadily, and there is a spike near zero. Six per cent of the elements have $S/t < 0.02$ after fifteen time units — they have not stretched at all.

Those are the elements released inside regular islands. The ABC flow’s phase space is mixed: chaotic regions threaded with tori on which the motion is quasi-periodic, which is the generic situation for a steady three-dimensional flow and not a defect of this one. An element on a torus is stretched algebraically rather than exponentially, and after fifteen time units algebraic growth is indistinguishable from none.

The measurable consequence is the width. A purely chaotic ensemble gives a Gaussian $\ln\ell$ whose width grows as $\sqrt t$, because the accumulated logarithm is a sum of many nearly independent increments. This one grows as $t^{0.250}$, because the ensemble is two populations and the trapped one does not spread at all.

That is the same statement one level up: the ensemble does not have a rate either.

What Oseledets’ theorem actually promises #

The theorem is worth reading carefully, because it is often remembered as more than it says.

It says that for almost every initial element, $(1/t)\ln\ell$ converges to a definite number as $t \to \infty$. That is a statement about almost every element and about the limit, and both clauses are load-bearing.

“Almost every” excludes the measure-zero set of elements aligned with a contracting direction, and in a mixed phase space it is a statement within each ergodic component: the chaotic sea has one exponent and each island has another, which is zero. The ensemble average over a mixed phase space is not a Lyapunov exponent of anything.

And “the limit” is the problem this essay is about. The convergence is in probability, so at any finite time the ensemble is spread over a range that grows as $\sqrt t$ while its mean grows as $t$ — which means the spread relative to the mean shrinks, slowly, and the moments do not follow the mean at all. The moment $\langle\ell^p\rangle$ is dominated by the part of the ensemble the limit describes least well, and it is dominated by it more strongly the larger $p$ is.

Which rate a reader actually wants #

This is not an academic distinction and the right answer depends on the question.

For how long a piece of dye becomes, the answer is the average, $\lambda_1 = 0.273$ — because that is what the length is.

For what a typical parcel experiences, the answer is the Lyapunov exponent, $\lambda_0 = 0.157$ — because that is what almost every parcel does.

For how fast the flow mixes, the answer is neither and is closer to the Lyapunov exponent, because mixing is completed when the slowest regions have been stretched enough, not when the fastest have. A flow with a few extremely vigorous filaments and a large quiet region mixes slowly, and its average stretching rate is large.

And for the surface area of an interface, the answer is a third exponent again. A material surface grows at the sum of the two largest Lyapunov exponents, and in an incompressible flow the three sum to zero, so the surface rate is $-\lambda_3$ — which exceeds $\lambda_1$ whenever the middle exponent is positive. Interfaces grow faster than lines.

Why this matters for mixing #

No randomness and it mixes anyway establishes that chaotic advection stretches and folds a blob until molecular diffusion can finish the job in a reasonable time, and the stretching rate is what sets how long that takes.

The distinction here says the estimate is not a single number. The regions that stretch fastest are mixed almost immediately; the trapped regions are not mixed at all on the same time scale, and a dye released into a mixed phase space shows exactly that — a well-mixed background with sharply defined unmixed islands in it, persisting indefinitely.

That is a familiar picture in every laboratory that has done the experiment, and its explanation is a distribution of stretching rates rather than a rate.

The same shape in a different subject #

The pattern of “a limit that returns a number while the moments keep a distribution” is worth recognising because this collection meets it three times in neighbouring essays.

In the exponents that stop being thirds the limit is over moments of velocity increments and the residue is a curved $\zeta_p$. In a flux that runs both ways the limit is a time average and the residue is a flux whose scatter exceeds its own mean and whose sign reverses a tenth of the time. Here the limit is $t \to \infty$ and the residue is a spread in $\ln\ell$ that grows without bound in absolute terms.

In all three the limit exists, is correct, and describes an object that no realisation resembles. The mean is not the flow is the collection’s general statement of the hazard, and these are three sharp instances of it.

What a dye experiment shows, and why it looks like this #

The picture everybody has seen is worth connecting to the arithmetic, because the connection explains a feature that is usually described as a curiosity.

Release a blob of dye into a steadily stirred three-dimensional flow and after a while the container looks uniformly coloured — except for sharply defined regions that have stayed clean. They persist, they have smooth boundaries, and stirring for longer does not remove them.

Those are the regular islands. The dye is confined to the chaotic sea, the islands are invariant regions that no trajectory enters or leaves, and the boundary between them is a barrier that advection cannot cross. Only molecular diffusion crosses it, which on a laboratory scale takes a very long time indeed.

Six per cent of the elements in the ensemble above are inside such regions. In a dye experiment that six per cent is what is left uncoloured, and it is why “well mixed” in a laminar flow means “well mixed except for the islands”.

No randomness and it mixes anyway is the essay about the part that does mix; this is the account of the part that does not, and both are consequences of the same phase space.

The rate that decides a mixing time #

If a single number is wanted for how long a flow takes to mix, it is closer to the smallest stretching rate in the ensemble than to any average.

Mixing is finished when the slowest region has been stretched enough for diffusion to finish the job, so the completion time is set by the worst-performing part of the flow rather than by the typical or the average part. A flow with a few vigorous filaments and a large sluggish region has a large mean stretching rate and mixes slowly.

That inverts the usual intuition, which is that a larger stretching rate means faster mixing, and it is the practical reason the distinction in this essay matters. The average is the number that is easy to compute and the wrong end of the distribution to be looking at.

Limits recorded rather than smoothed over #

The flow is not turbulent. The ABC flow is a steady exact solution with chaotic streamlines, chosen because it is the simplest thing with the property being examined. A turbulent flow has a stretching distribution too and it is not this one.

Eight hundred elements is a small ensemble for a fourth moment. $\langle\ell^4\rangle$ is dominated by the few longest, and the value quoted would move at the second figure with a different seed. The Lyapunov exponent and the average rate are much more stable.

Fifteen time units is not the limit. Every exponent quoted is a finite-time estimate, and the finite-time Lyapunov exponent has a distribution of its own that narrows as $t^{-1/2}$ — slowly.

And the trapped fraction depends on where the elements were released. Six per cent is the fraction of a uniform random sample that landed in islands, which is a property of this flow’s phase space at these parameters. It is not universal, and the exponent $t^{1/4}$ that follows from it is not either.

The third rate, which is a surface’s #

There is one more exponent in the problem and it is the one an interface cares about.

A material surface grows at the sum of the two largest Lyapunov exponents, because a surface element is spanned by two vectors and both are being stretched. In an incompressible flow the three exponents sum to zero — volume is conserved — so the surface rate is $-\lambda_3$, the magnitude of the most negative one.

That is larger than $\lambda_1$ whenever $\lambda_2$ is positive, which it usually is in a three-dimensional chaotic flow. So interfaces grow faster than lines, and the gap between them is the middle exponent.

The consequence is the one that matters for mixing: what has to be created for diffusion to finish the job is interfacial area between the two fluids, not the length of any line, and the area is growing at the faster rate. That is a piece of good news in an essay otherwise full of caveats, and it is why chaotic mixing works as well as it does.

Nothing here computes $\lambda_2$ or $\lambda_3$, which would need the full deformation gradient rather than a single carried vector, and the statement is recorded as an implication of incompressibility rather than as a measurement.

Where the numbers come from, in one paragraph #

A note on the computation, because its simplicity is worth advertising.

Each element requires integrating six ordinary differential equations: three for the position along a trajectory, and three for a vector carried by the local velocity gradient. The vector is renormalised at each step and the logarithm of the factor accumulated, which is what keeps a quantity spanning $e^{\pm 20}$ inside double precision.

There is no field to store, no grid, no linear system and no boundary condition. Eight hundred elements for fifteen time units at a step of $0.02$ is six hundred thousand steps of a six-dimensional system, which is under a second.

That cheapness is why finite-time Lyapunov exponents are computed routinely for real flows — from weather models, from ocean surface velocities, from experimental velocity fields — and why the distribution rather than the mean is what those computations report. The mean was never the expensive part.

The residue #

Oseledets’ theorem is the limit. It says $(1/t)\ln\ell$ converges to a single number, and it is true.

What survives it is the entire distribution: a width growing without bound, a tail carrying half the length in under one per cent of the elements, a population that never stretches at all, and one rate for every moment anybody cares to take.

The limit produces a number. The flow keeps a distribution, and every quantity a reader is likely to want is a moment of the distribution rather than the number.