How far a parcel gets

A puff of smoke released into a turbulent wind spreads. How fast it spreads is the question every model of dispersion has to answer, and the answer is not a constant: near the chimney the plume is a narrow wavering ribbon, and a kilometre downwind it is a broad diffuse cloud. Something changes as the smoke travels, and what changes is not the wind.

Taylor settled it in 1921 with a single integral, and the integral is worth knowing because it contains the whole of turbulent transport and one entire class of modelling error.

One integral, and the two things it contains #

Let a parcel be carried by a velocity $u(t)$ with zero mean, variance $u^2$ and autocorrelation $R(\tau) = \langle u(t)u(t+\tau)\rangle/u^2$. Its displacement is the integral of its velocity, so the mean square displacement is a double integral of the correlation — and after one integration by parts,

$$\langle X^2\rangle = 2u^2\int_0^t (t-\tau)\,R(\tau)\;d\tau.$$

That is the whole theory. No closure, no assumption about the flow’s structure, nothing about Reynolds numbers: given a correlation function, the spreading follows.

Its two limits are the interesting part.

Short times. For $t$ much less than the correlation time, $R \approx 1$ throughout the integral and $\langle X^2\rangle = u^2t^2$. The parcel travels in a straight line at whatever velocity it happens to have, and its displacement grows linearly with time — ballistic, not diffusive. There is no diffusivity to speak of; the spreading is a spread of velocities, not a random walk.

Long times. For $t$ much greater than the correlation time, the integral saturates at $T = \int_0^\infty R\,d\tau$, and $\langle X^2\rangle = 2u^2T\,t$. That is the diffusion law, with an eddy diffusivity

$$K = u^2 T,$$

a velocity variance times a memory. Neither factor is a property of the fluid.

The numerical check is worth stating because it costs nothing and is the site’s habit: for an exponential correlation the integral has a closed form, $2u^2T^2[t/T - 1 + e^{-t/T}]$, and the integrated version agrees with it to 0.017 per cent at the resolution used, with the ballistic limit recovered to $7\times10^{-4}$ and the diffusive one to $8\times10^{-8}$.

The diffusivity is the end of a measurement, not a property #

The most useful figure in this essay is not the displacement but the running diffusivity $\langle X^2\rangle/2t$ — what an observer would infer from a measurement that had lasted a given time.

That curve is the reason two published values of a turbulent diffusivity for the same flow can differ by a factor of two without either being wrong. They are answers to different questions: one measured over three eddy turnovers, one over thirty.

It also settles what a gradient-transport model — the assumption that a flux is $-K\,d\bar{c}/dy$ — is actually assuming. It assumes that the parcels carrying the flux have travelled far enough to have forgotten their velocities, and therefore that the mean concentration varies slowly over the distance a parcel travels ballistically. Near a source, that is false: the plume’s width is smaller than the ballistic range, and no diffusivity describes it.

The third curve there is not a curiosity. A parcel in a stratified fluid does tend to come back: buoyancy restores it, its vertical velocity correlation oscillates and changes sign, and its net vertical spreading is far less than $u^2T$ suggests. The same happens to a parcel in a rotating flow, and to one in a wave field. In each case the integral is doing the work and the number $u^2T$ is not.

When the integral comes out negative #

The third correlation in the figure above — the one that changes sign, so that a parcel tends to come back — was introduced as a case where the diffusivity is smaller than $u^2T$ suggests. Follow it a little further and something more useful happens: if the negative area exceeds the positive one, the integral is negative, and the eddy diffusivity is a negative number.

That is not a pathology of a made-up correlation. It is a measured feature of one of the most studied flows in the atmosphere.

In a convective boundary layer — a sunny afternoon over land, heated from below — the mean potential temperature profile is well mixed through the middle and slightly increasing with height over the upper third, while the measured heat flux there is still upward. Flux up, gradient up: a gradient-transport model requires $K < 0$ to reproduce it, and a code that clamps $K$ positive reproduces a flux of the wrong sign over a third of the layer.

The mechanism is exactly what Taylor’s integral describes and gradient transport assumes away. Transport in that layer is not done by small eddies making short hops; it is done by thermals that rise coherently from the surface all the way to the inversion, carrying the properties of the ground into air a kilometre above it without exchanging much on the way. Such a parcel’s velocity correlation stays high for the whole traverse, so the ballistic regime extends across the entire layer and the parcel never forgets anything. The flux at a given height is then set by conditions at the surface, which is a place with a different temperature and a different gradient, and no local quantity can express it.

Gradient transport is a local model and the transport is non-local, which is a stronger failure than an inaccurate coefficient. The repair is not a better $K$ but a different form: mass-flux schemes, which carry an explicit plume alongside the mean field and let it exchange, or transilient schemes, which replace the diffusivity with a matrix saying how much fluid moves between every pair of levels. Both are attempts to write down the memory that the diffusivity integrated away.

The general test falls out of the same integral and is worth carrying. A diffusivity is trustworthy when the distance a parcel travels ballistically is small compared with the distance over which the mean profile changes. Divide the one by the other and the ratio is the thing to check; it is small in a pipe, small in a boundary layer away from its edge, and of order one in a convective layer, in a plume near its source, and in any flow whose transport is done by structures as big as the flow itself. Where that ratio is not small, the answer is not a corrected coefficient. There is no coefficient.

Where the memory comes from #

Taylor’s integral takes the correlation as given, and a fair question is what sets it. For a parcel in turbulence the answer is the large eddies: $T$ is the time over which the biggest structures in the flow persist, so $u^2T \approx u L$ with $L$ the integral scale. That gives the familiar estimate

$$K \approx u L,$$

which is the same combination as an eddy viscosity and for the same reason — an eddy viscosity is a velocity times a length, and here the velocity and the length are the ones the correlation contains.

The correspondence is not exact and the difference is instructive: momentum and a scalar are carried by the same eddies but not equally well, and the ratio of the two diffusivities — the turbulent Prandtl number — is about 0.85 in most shear flows and is another measured constant. That there should be a ratio at all is the same observation the thermal layer’s essay makes for molecular transport, where the Prandtl number separates two layers by a factor of forty.

The estimate is why turbulent diffusivities are so enormous. In a room, molecular diffusion moves a smell a few centimetres a minute; a draught with $u = 0.1$ m/s and $L = 0.5$ m gives $K \approx 0.05$ m²/s, which is five orders of magnitude larger than the molecular value. The mechanism is not different in kind — both are random walks — and the numbers differ by the step size.

There is a converse worth stating alongside it, because it is the reason gradient transport survives at all. Wherever the transporting structures are much smaller than the region they act in — a boundary layer’s inner part, a pipe away from its centre, a stably stratified layer whose eddies buoyancy keeps short — the ratio above is genuinely small, the memory is genuinely brief, and a diffusivity is not merely convenient but very nearly exact. The model’s reputation suffers because it is applied uniformly across flows where the ratio varies by two orders of magnitude, and the places it fails are the places anybody would photograph: a plume leaving a chimney, a thermal over a field, a jet in its first few diameters. The failures are visible and the successes are not, which is a poor basis for judging a model and a common one.

A chimney, with the numbers in #

The three regimes are easiest to believe with a case attached. Take a stack fifty metres tall in a wind of five metres per second, with a turbulence intensity of ten per cent — so $u = 0.5$ m/s — and an integral scale of about the stack height, giving a correlation time of $T \approx L/u = 100$ seconds.

Immediately downwind the plume is in the ballistic regime: for the first hundred seconds, which is half a kilometre of travel, its width grows linearly with distance, and what a photograph shows is a narrow ribbon wandering as a whole rather than a spreading cone. Only past two or three kilometres is the plume properly diffusive, with a width growing as the square root of distance and an effective diffusivity of $u^2T = 25$ m²/s.

Every number in that paragraph is an estimate rather than a solve, and the reason is the one the arithmetic of resolution gives: a plume at those scales contains some $10^{15}$ degrees of freedom, and nothing computes it directly.

The regulatory models used for stack design are written in the ballistic-to-diffusive transition, which is why they are given as empirical width-against-distance curves rather than as a diffusivity. The curve is Taylor’s integral with a measured correlation in it, and the reason it cannot be replaced by a single number is the figure above.

Two parcels, and why a plume is not a cloud #

Everything above concerns one parcel wandering relative to a fixed origin, which is called absolute dispersion. A plume’s width is a different quantity: it is the separation between pairs of parcels released together, and the two behave differently in a way that matters.

When two parcels are much closer together than the smallest eddies, they are carried by nearly the same velocity and separate slowly. When they are further apart than the largest eddies, their velocities are independent and their separation grows like absolute dispersion. In between — inertial-range separations — the eddies that matter are the ones of the parcels’ own size, whose velocity difference goes as $(\varepsilon r)^{1/3}$, and the separation then grows as $\langle r^2 \rangle \propto \varepsilon t^3$.

That $t^3$ law is Richardson’s, from 1926, and it predates Kolmogorov by fifteen years: Richardson found it empirically from balloon and volcanic-ash data and wrote the diffusivity as growing with the separation itself. A plume near its source therefore grows faster than any diffusion, and a model with a constant $K$ gets its early spreading wrong in the other direction from the ballistic error described above — which is why dispersion modelling is a subject rather than a formula.

A parcel’s path is not a streamline #

Everything in this essay is a statement about the path an individual parcel takes, which is a Lagrangian object, and almost everything else in fluid mechanics is written in terms of the field at fixed points. The two coincide only in steady flow, and turbulence is the least steady flow there is — so the distinction that this collection insists on from its first field is not pedantry here, it is the reason the integral has to be taken along a trajectory.

That difference has a practical edge. A hot wire measures the Eulerian correlation in a fraction of a second; measuring the Lagrangian one means following particles for many correlation times, and the number of experiments that have done it well is small. Most quoted eddy diffusivities are therefore built from an Eulerian time scale multiplied by a factor taken from a model — which is one more place where a number that looks measured is partly assumed.

What the picture cannot show #

Nothing here computes a correlation function. $R(\tau)$ is an input in every figure, and the three shapes drawn are stated models. Measuring a Lagrangian correlation requires following actual fluid parcels, which is experimentally difficult — it needs particle tracking or neutrally buoyant floats — and the resulting curves are noisy and few.

The Lagrangian correlation is not the Eulerian one. What a fixed probe measures is the correlation of velocity at a point in time, and what Taylor’s integral needs is the correlation along a moving parcel’s path. The two have different time scales, related by a factor that is itself a modelling assumption, and conflating them is a standard error with a standard name.

The velocity field is stationary and homogeneous in every figure. A real plume rises through a shear, so the statistics change along its path, and the correlation is not the same function at the hundredth second as at the first. Nothing here handles that, and the standard treatments handle it by dividing the trajectory into pieces and assuming stationarity within each.

And the parcel is a point with no inertia. A real particle of finite size and density lags the flow, which is a separate calculation with its own number in it, and its dispersion differs from the fluid’s — heavy particles fall out of eddies and disperse less, buoyant ones are trapped by them and disperse more.

What the integral is really saying #

The general lesson is worth separating from the fluid mechanics, because it is the phase’s own argument in miniature. A diffusivity is an average of a product — a velocity multiplied by a displacement, correlated over time. Taking that average requires a time long enough for the correlation to have decayed, and the answer at any shorter time is a different number rather than a noisy version of the same one.

It also explains why a flow can mix beautifully with no randomness in it at all. Two velocities applied in turn, each perfectly deterministic, give a parcel a correlation that decays because the parcel keeps meeting a different velocity — and the blinking vortex is exactly that construction, with a stretching rate that can be measured.

The same structure appears wherever a transport coefficient is defined by a correlation integral: the Green–Kubo relations in statistical mechanics give viscosity, conductivity and diffusivity as integrals of exactly this form. Taylor’s paper is the fluid-mechanical member of that family and predates the rest of it by thirty years.

Who found it, and when #

Taylor’s Diffusion by continuous movements appeared in 1921, twenty years before Kolmogorov’s theory and thirteen before his own work on the statistical theory of turbulence. It is four pages long and contains the integral, both limits and the identification of $u^2T$ as the eddy diffusivity. Richardson’s four-thirds law followed in 1926 from atmospheric data, and Batchelor connected the two to the inertial range in 1950.

The surprising connection is with the closure problem at the beginning of this field. The Reynolds decomposition leaves an unknown correlation behind, and every model of it is an attempt to write that correlation in terms of the mean field. Taylor’s integral is what the correlation is — a memory, integrated — and it says that the missing term cannot be a function of the local mean gradient unless the memory is short compared with everything else in the problem. Gradient transport is not a bad model of turbulence; it is a model of turbulence with no memory, and the amount of memory a flow has is exactly what the integral measures.

Where the ladder goes next #

Beside this rung sits the question of what a probe records when the fluid it is in is only sometimes turbulent — an average over two states rather than one — which is the other way an average can quietly stop meaning what it says. Above it lies pair dispersion and Richardson’s law, which needs an inertial range and therefore needs the machinery this field has already built.