The further apart, the faster they part

How far a parcel gets follows one marked parcel through turbulence and finds Taylor’s integral: straight-line motion while the parcel remembers its velocity, a random walk once it has forgotten, and an eddy diffusivity that is the product of the two. That is dispersion measured from a fixed point — the spread of a plume from a chimney, averaged over a long time.

It ends by pointing at a different question, and the difference is the whole of this essay. A plume averaged over an hour is one thing; a single puff of smoke released at once is another. The puff’s size is not the distance of one parcel from where it started. It is the distance between parcels that started together, and that distance is torn at by eddies of its own size — small eddies while the puff is small, larger ones as it grows. The further apart two parcels are, the faster they separate.

A diffusivity that grows with the distance #

Lewis Fry Richardson looked at the spreading of balloons, volcanic ash and smoke in 1926 — before Kolmogorov, before the inertial range had a name — and found that the rate at which things drift apart depends on how far apart they are. He fitted it as a diffusivity for the separation growing as the separation to the four-thirds power,

$$K(r) = k_0\,\varepsilon^{1/3} r^{4/3},$$

over a range of scales from centimetres to hundreds of kilometres, and wrote the spreading of a cloud of pairs as a diffusion equation with that variable diffusivity:

$$\frac{\partial P}{\partial t} = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 K(r)\frac{\partial P}{\partial r}\right),$$

with $P(r, t)$ the probability density of the separation.

Fifteen years later Kolmogorov’s argument explained the exponent. In the inertial range the only quantities available are the dissipation rate $\varepsilon$ and the scale $r$, and the only diffusivity they make is $\varepsilon^{1/3} r^{4/3}$. The eddies of size $r$ move at speeds of order $(\varepsilon r)^{1/3}$ relative to one another — the two-thirds law in another form — and they last for a turnover time $r^{2/3}\varepsilon^{-1/3}$; their product is Richardson’s diffusivity. Richardson measured the law of the inertial range before anyone knew there was one, and his four-thirds is Kolmogorov’s two-thirds seen through the separation of a pair.

There is something odd in that formula worth pausing on. The dissipation rate $\varepsilon$ is the rate at which the smallest eddies turn kinetic energy into heat, and nothing is being turned into heat when two specks of smoke drift apart at a separation of a metre. The rate appears because the energy cascade passes through every scale at the same rate: the flux of energy from eddies of a metre to eddies of half a metre is the same $\varepsilon$ that is finally dissipated a million times smaller, so it is the one number that characterises how vigorous the eddies of a metre are. The pair law measures the cascade’s throughput at the pair’s own scale, and borrows the name of the place it ends.

The cube law, and its constant #

With the pure four-thirds law and a pair released at zero separation, the equation has an exact self-similar solution that Richardson found himself:

$$P(r, t) \propto t^{-9/2}\exp\!\left(-\frac{9\,r^{2/3}}{4 k_0\,\varepsilon^{1/3} t}\right), \qquad \langle r^2\rangle = \frac{1144}{81}\,k_0^3\,\varepsilon\,t^3 .$$

That is the celebrated result: the mean square size of a puff grows as the cube of the time, where a diffusing substance’s grows linearly. It is usually written $\langle r^2\rangle = g\,\varepsilon t^3$, and the constant $g$ — Richardson’s constant — is the number experiments and simulations try to measure. Values cluster around a half, with a spread of tens of per cent that reflects how hard it is to see the law cleanly; a half is the value used here, and it is borrowed.

The equation was solved here numerically — finite volumes on a grid uniform in the logarithm of the separation, fluxes at the cell faces, a fully implicit step that grows geometrically because the solution runs over nine decades of time — and checked against the exact solution. Probability is conserved to $10^{-14}$. The slope of $\langle r^2\rangle^{1/3}$ against time gives $g = 0.505$ against the closed form’s 0.500: the $t^3$ law and its constant come out of the diffusion equation as they should.

The shape matters as much as the width. A cloud spread by a constant diffusivity is Gaussian, and the distribution of distances between its pairs is the Maxwellian one with a kurtosis — the fourth moment over the square of the second — of 5/3. Richardson’s cloud is a stretched exponential in $r^{2/3}$ and its kurtosis is 3.76, more than twice as large. Most pairs in a Richardson cloud are closer together than the rms size suggests, and a few are much further apart, because a pair that has drifted apart is being pulled by larger eddies and drifts apart faster still, while a pair that has stayed close is still in the hands of small, slow ones. The numerical cloud sits on the exact form to within two per cent wherever the density is appreciable.

Forgetting where it started #

A pair released at a finite separation has something the exact solution lacks: a starting distance. The diffusion equation says what becomes of it.

For a while the pair barely moves: at first its separation is its starting distance, and the mean square separation sits on a plateau. Then it joins the $t^3$ law — the same law, with the same constant, whatever the starting distance was — and from then on nothing about the start is recoverable. The first figure shows three such pairs converging; at three million Kolmogorov times their mean squares agree to five parts in a hundred thousand.

How long the forgetting takes is the uncomfortable number. It scales as the eddy turnover time of the starting separation, $t_B = (r_0^2/\varepsilon)^{1/3}$ — the time Batchelor identified — but the coefficient is large: about seventy of those times before the mean square is within ten per cent of the $t^3$ line. The law of the pair describes the cloud only after the cloud has grown far beyond the size it started at, which in real flows is often after it has grown out of the inertial range altogether.

Below the smallest eddies, and why it matters #

Two parcels released closer than the Kolmogorov length are in a flow that is smooth on their scale: the velocity difference between them is the local velocity gradient times their separation, and a smooth gradient pulls a small separation apart exponentially. That is the regime of chaotic advection in a flow with no randomness in it, and it is why dye released at a point in turbulence spends its first few Kolmogorov times being stretched into filaments rather than spread into a cloud. The model here represents it by a diffusivity growing as the square of the separation, which gives exponential growth but with a rate the interpolation fixes rather than the flow.

What the exponential regime does to the pair law is set its starting point. A pair released at a hundredth of a Kolmogorov length has to be stretched to a Kolmogorov length before the four-thirds law can take over, and at an exponential rate that costs a number of Kolmogorov times proportional to the logarithm of the ratio. For molecules released together — a pair of solute molecules starting within a molecular spacing — that logarithm is large, and it is the reason a drop of ink in a stirred glass takes a noticeable time to start spreading at all and then spreads explosively. The $t^3$ law has a delayed start whose length is logarithmic in how close together the two things started, and the slope figure’s curves all begin at one Kolmogorov length for exactly that reason: starting any closer would only add time before the curves begin to rise.

The window is narrower than it looks #

The inertial range has two ends. Below the Kolmogorov length the flow is smooth and two close parcels separate exponentially, not as a power; above the integral scale the parcels are in different eddies and move independently, and their separation diffuses with a constant diffusivity. The four-thirds law holds only between, and the model here continues the diffusivity smoothly into both ends — as $r^2$ below the Kolmogorov length and as a constant above the integral scale — which is a stated interpolation rather than a result.

The figure is the reason Richardson’s law was argued about for most of the century after he found it. A $t^3$ law is a statement about a local exponent, and the local exponent reaches three only if the flow has room. A laboratory flow whose integral scale is a hundred Kolmogorov lengths never shows it: the pair leaves the dissipation range and enters the diffusive one before its exponent climbs past 2.45. At a thousand the peak is 2.81. At ten thousand, 2.94. To see the $t^3$ law as a clean plateau — above 2.85 for more than a decade of time — takes a scale ratio of a hundred thousand, which is the atmosphere’s and not a wind tunnel’s.

That is the same shape as the range a real Reynolds number does not have, for the spectrum, and for the same reason. An inertial-range law is a limit, and the approach to it is slow because the two ends of the range each eat into it by a factor. The pair’s exponent is, if anything, worse off than the spectrum’s, because the pair also has to forget its start, which costs another seventy turnover times inside the window.

A puff is not a plume #

The practical difference between single-parcel and pair dispersion is the difference between two questions a pollution forecast has to answer. Averaged over an hour, the concentration downwind of a chimney is set by the wandering of the whole plume — every eddy, large and small, has had time to push it about — and that is Taylor’s single-parcel spreading. At any instant, the concentration in the plume is set by how far apart parcels released at the same moment have drifted, which is pair dispersion, and the instantaneous plume is much narrower and much more concentrated than the averaged one. A scalar is a record of where its fluid was, and the instantaneous record and the averaged one are different records.

The figure puts numbers to that. The averaged plume is wide from the start, because it records every position the whole puff has been swept to; the puff itself starts small and grows by the pair law. Ten Kolmogorov times after release the plume is thirty-six times wider than any instantaneous puff within it, a hundred times after it is thirteen times wider, and only when the puff has grown as large as the largest eddies — around ten thousand Kolmogorov times in a flow of this size — do the two agree. A hundred Kolmogorov times after release, a concentration averaged over the plume is more than two thousand times lower than the concentration inside the puff, because concentration goes as the inverse cube of size and the sizes differ by thirteen.

To make the two descriptions meet at long times they were matched on the same largest eddies: a pair of parcels far apart separates with twice the diffusivity of one parcel, and the single-parcel model’s velocity variance and time scale were chosen to satisfy that. The matching is a stated choice; the ratio at early times, which is the point, does not depend on it.

The difference matters for anything that responds to peaks. A toxic release, a flammable cloud, a smell — each is dangerous at the concentration a person actually meets, not at the hour’s average, and the instantaneous concentration is governed by the pair law. A dispersion model that uses a single eddy diffusivity for both gets the average right and the peaks wrong, in the direction of too little, and the scalar’s own cascade is what then mixes the concentrated filaments down to the molecular scale.

What the picture cannot show #

Every curve here comes from a diffusion equation for the separation, and a diffusion equation has no memory: it treats each small change in separation as independent of the last. Real pairs do have memory — a pair that has just been pulled apart tends to keep moving apart for a turnover time — and the most careful modern treatments replace Richardson’s equation with models that carry the relative velocity as well as the separation. They agree about the $t^3$ law and disagree about the shape of the cloud and the value of $g$. The shape figure is therefore Richardson’s shape, not a measured one.

The model also has nothing to say about the time just after release, when the pair separates ballistically — each parcel keeping the velocity it started with, the mean square growing as the square of the time, with the second-order structure function as its coefficient. A diffusion equation cannot represent that regime at all. It is left out rather than approximated.

Where the model stops #

Isotropic, stationary turbulence. Real flows have shear, and a shear stretches a pair cloud into an ellipse and adds its own growth, as the cube of the time for a different reason. Near the ground the largest eddies are limited by the height, and the integral scale depends on where the puff is.

A stated interpolation at both ends. Below the Kolmogorov length the model’s exponential separation has a rate fixed by the interpolation, not by the flow’s measured stretching rate; that rate is not one number anyway. Above the integral scale the model’s constant diffusivity is Taylor’s, which is right.

A borrowed constant. $g = 0.5$ is taken from measurement and simulation, where it is known only to within tens of per cent; everything above scales with it, and nothing about the shape, the forgetting or the window depends on its value.

The convention the numbers depend on #

Lengths are in Kolmogorov lengths and times in Kolmogorov times, $\eta$ and $\tau_\eta$, which sets $\varepsilon = 1$. Separation is the distance between the two parcels, a positive scalar in three dimensions, and its density is normalised with the $4\pi r^2$ of a spherical shell. The local exponent is $d\ln\langle r^2\rangle/d\ln t$ with the time measured from release, not from any virtual origin.

Who found it, and when #

Richardson published the four-thirds law in 1926, from observations spanning scales from a few centimetres to the width of a weather system. Obukhov derived the $t^3$ law from Kolmogorov’s theory in 1941. Batchelor identified the ballistic regime and its time scale in 1950. The value of $g$ was measured in the atmosphere and the ocean for decades with scattered results, and settled into its present range only when direct numerical simulations and tracking of particles in laboratory turbulence reached large enough scale ratios in the 2000s — which is the window figure’s point made by history.

Still open: how much memory changes the cloud #

The diffusion equation has no memory, and the cloud’s shape — its kurtosis of 3.76 — is a consequence of that. A model in which pairs keep their relative velocity for a turnover time produces a cloud whose tails are governed by how long a fast-separating pair keeps separating fast, and whether that brings the shape closer to what particle-tracking experiments report is the open part.

The calculation that would follow adds the relative velocity as a second variable, with a relaxation over the turnover time of the current separation and a stated noise, and solves the resulting two-variable equation for the separation’s distribution. It asks two things: how the kurtosis changes as memory is added, and how much of the scatter in measured values of $g$ is the difference between a memoryless model fitted to a flow with memory and the flow itself. Beside it is the single-parcel wandering that this pair law is the relative of, where memory is already built in through Taylor’s correlation function and the answer is exact.