The moment a spectrum cannot hold

The most reproduced measurement in turbulence is a straight line on logarithmic axes with a slope of $-5/3$. It has been found in wind tunnels, in tidal channels, in the atmosphere, behind grids and in the wakes of ships, over ranges from a fraction of a decade to five decades, and its appearance is taken — reasonably — as the signature of Kolmogorov’s cascade.

This essay builds a field that has that spectrum exactly and has no cascade in it whatever.

What a spectrum is, exactly #

A spectrum is not an independent measurement of a flow. It is the cosine transform of the two-point correlation,

$$R(r) = \langle u(x)\,u(x+r)\rangle, \qquad E(k) = \frac{1}{\pi}\int R(r)\cos kr\;dr,$$

and the relation runs both ways. That is Wiener and Khinchin’s theorem, from around 1930, and the consequence for anybody measuring a flow is stark: a spectrum and a correlation carry exactly the same information, so measuring both is measuring one thing twice.

The check is worth doing rather than quoting. A field is built here as a sum of Fourier modes whose amplitudes are chosen from a stated $E(k)$ and whose phases are drawn from a deterministic generator; the correlation is then computed two ways — by summing $u(x)u(x+r)$ over the samples, and by transforming the amplitudes — and the two agree to $2\times10^{-15}$ of the variance at every separation on the grid.

Then the same amplitudes, with the phases changed #

The construction has an obvious lever. Keep every amplitude, draw a fresh set of phases, and the result is a completely different signal with an identical spectrum. That is not an approximation: the amplitudes are the same numbers, so the spectrum is the same numbers.

Since the correlation is the transform of the spectrum, the correlation is unchanged too. Since the second-order structure function is

$$S_2(r) = \langle [u(x+r) - u(x)]^2\rangle = 2\,[R(0) - R(r)],$$

that is unchanged as well — and so are the integral scale, the Taylor microscale, the variance, and every other quantity built from two points at a time.

Two fields, then, that agree on every second-order measurement in the subject and look nothing like each other. The upper trace in the hero figure has structures in it; the lower has the same energy at every scale and no structures at all. The difference is entirely in the phase relationships between scales, and no second moment is a function of those.

Where the cascade actually lives #

The cascade is a flux: energy passing from large scales to small at a rate $\varepsilon$. A flux has a direction, and a direction cannot be carried by a quantity that is symmetric under changing the sign of the fluctuation — which every second moment is.

The quantity that does carry it is the third moment. Kolmogorov’s exact result of 1941, which this collection treats as the one exact statement in the subject, says that in homogeneous isotropic turbulence

$$S_3(r) = \langle [u(x+r) - u(x)]^3\rangle = -\tfrac{4}{5}\,\varepsilon r,$$

and the minus sign is the cascade’s direction. Negative $S_3$ means that a velocity difference taken across a separation is more often a deceleration along the flow than an acceleration, and that asymmetry is what transports energy downscale. A field with symmetric statistics has no flux, and a field with a flux is not symmetric.

For the synthesised field, $S_3$ is zero in the ensemble. It is not zero in any single realisation, and that is the useful part of the experiment.

A single record cannot tell the two apart, and an ensemble can. The phase-random field’s third moment is sampling noise and averages away; a real flow’s is a property of the flow and does not. In a laboratory that distinction is the difference between a measurement repeated over many independent realisations and a measurement of one long record, and it is not a distinction that a plot of the spectrum makes visible.

The reason the noise is as large as it is deserves a note, because it looks at first like a defect of the construction. A $-5/3$ spectrum puts most of its variance in the largest few modes: the effective number of independent contributions to a third moment is therefore small — a handful, not the two thousand samples — and the standard error of a skewness estimated from them is a fifth rather than a fiftieth. Real turbulence measurements have exactly this problem, which is why published third-order statistics come with convergence tests and second-order ones usually do not.

What this does not say #

It does not say that the $-5/3$ spectrum is uninformative, and it does not say that the cascade is in doubt: the cascade is measured, in the four-fifths law and in filtered energy fluxes, and it is one of the better-established facts in the subject. A measured spectrum that is not $-5/3$ in a flow expected to have an inertial range is strong evidence that something is wrong — that the Reynolds number is too low for a range to exist, that the probe is too large, that the flow is stratified or rotating or two-dimensional. The spectrum is a good filter and a poor fingerprint.

Nor does it say that the cascade is unmeasurable. It says the measurement is a third-order one — the four-fifths law, or the equivalent flux computed from a filtered field — and that this is substantially harder than plotting a spectrum, which is why it is done less often.

There is a second thing a spectrum is genuinely good for, and it is the one this collection leans on hardest: it says how many decades of scale a flow has, and therefore how much of it could ever be computed. That count is the whole content of the arithmetic that governs this field, and it needs the amplitudes and nothing else.

The case that settles it: the same slope, the other way #

The strongest evidence that a spectral slope does not determine a flux is not synthetic at all. In two dimensions, turbulence has an inertial range with a $-5/3$ spectrum and the energy flux through it runs upscale — from small eddies to large — because two-dimensional flow conserves enstrophy as well as energy and the two conservation laws together forbid energy from going down. Above the forcing scale the slope is $-3$ instead, and energy does not flow through that range at all.

So a measured $-5/3$ is compatible with a downscale cascade, an upscale cascade, and no cascade whatever. Which of the three is happening is decided by the third moment’s sign, and by nothing that appears on a log-log plot of energy against wavenumber. The essay on the backwards cascade computes the constraint that produces the reversal, and it is worth reading beside this one precisely because its spectrum looks like the ordinary case.

How the measurement is actually made, and what that adds #

A spectrum in a laboratory is almost never measured in space. A hot wire at a point records a time series, and the series is converted into a spatial cut by Taylor’s hypothesis — that the turbulence is carried past the probe faster than it evolves, so time at a point stands in for distance along the flow. That is an assumption with a small parameter, the turbulence intensity, and it is one more thing between the measurement and the claim.

It also has a consequence for this essay’s argument. A time series and a spatial cut have the same kind of information in them, and the transformation between them is a rescaling, so nothing about the phase argument changes: a probe measuring for an hour measures a very long one-dimensional cut, its spectrum converges beautifully, and its third moment converges as slowly as the figure above suggests. The instrument is not the limitation. The number of independent large eddies that went past is.

That number is the same quantity that decides how long a measurement of a turbulent diffusivity has to run, and it is usually the binding constraint on both. Both are asking for a converged average of a quantity whose independent samples arrive at the rate of one per large-eddy turnover, and an hour of data in a metre-scale flow at ten metres a second contains a few thousand of them.

Measuring the flux itself, and what it looks like close up #

There is a second way to measure the cascade, mentioned above in a clause, and it is worth opening because what it shows changes how the four-fifths law should be read.

Filter a velocity field at a scale $\Delta$ and the equation for the energy of what remains carries a term that exchanges energy with what was removed. That term is the subgrid flux, $\Pi = -\tau_{ij}\bar{S}_{ij}$, built from the stress the unresolved scales exert on the resolved ones and the strain rate they exert it on. It is a field, defined at every point and every instant, and its average over a homogeneous flow is the cascade rate $\varepsilon$ — the same number the four-fifths law reports, arrived at without any separations, any third moments or any isotropy.

Compute it point by point in a resolved simulation and the picture is not the one the word cascade suggests. $\Pi$ is enormously variable: its fluctuations are several times its own mean, its distribution has long tails, and — the part that matters — it is negative over a large fraction of the volume, typically a third to a half depending on the filter. Energy is going from small scales to large, locally and constantly, at rates comparable with the mean flow in the other direction.

That reverse transfer has a name, backscatter, and it is not a rare event or a numerical artefact. The mean flux is the small residue of two large and nearly cancelling contributions, which is why it is so hard to measure and why the four-fifths law needed decades of data to confirm. It is also the exact reason the essay’s ensemble argument bites so hard: a quantity that is the difference of two large numbers converges at the rate its own fluctuations dictate, and its fluctuations are bigger than itself.

And it is why the standard closure cannot be right in detail. An eddy viscosity multiplies the strain by a positive coefficient, so the model’s $\Pi$ is positive by construction, everywhere, at every instant. A model of that form can reproduce the mean transfer exactly and can never reproduce the half of the field where the transfer runs backwards — which is a structural statement about the model rather than a matter of tuning its constant. Large-eddy simulation works anyway, for the reason the dissipation anomaly supplies: what the large scales need from the model is the right mean removal rate, and they are largely indifferent to how it is distributed. Where they are not indifferent — near a wall, in a transitioning layer, wherever the resolved field is close to laminar and a spurious dissipation kills it — the purely dissipative model is known to fail, and the repairs all consist of letting the model give energy back.

So the honest picture of the cascade has two levels, and this essay’s argument applies to both. The mean flux is real, downscale, and equal to $\varepsilon$. The instantaneous flux is a violently fluctuating field of both signs whose mean is a small fraction of its own magnitude. A spectrum records neither, and even the third moment records only the first — so the gap between what is measured and what is happening is wider than the gap this essay opened, by one more level.

Which is worth carrying as a general caution about any transport statement. A flux that is described as a direction is almost always a mean over a quantity that goes both ways, and how much of the traffic runs each way is a separate measurement from the net.

The same lesson, from three other directions #

Phase information carries the structure everywhere. The same demonstration works on images: an image with the amplitude spectrum of a photograph of a face and random phases is noise, and an image with random amplitudes and the phases of a face is recognisably the face. Nothing about the argument is fluid-mechanical.

A summary statistic can miss the mechanism entirely. The clearest neighbouring case in this collection is the stability of shear flows, where the eigenvalues of the linearised operator — the natural summary — all have negative real parts while disturbances still grow by a factor of thousands. What the eigenvalues cannot see is the non-normality of the operator, which is a relationship between modes, exactly as the cascade is a relationship between scales.

And structure does not require randomness. A flow can be complicated, well mixed and possessed of a broad spectrum while being entirely deterministic and, in fact, exactly reversible; the blinking vortex is the demonstration this collection uses, and its spectrum broadens while nothing stochastic happens anywhere in it.

What the picture cannot show #

The synthesised field is not a flow. It is a sum of Fourier modes on a line, it satisfies no equation of motion, and it is not incompressible, three-dimensional or evolving. Nothing about it supports a claim about what turbulence is — it supports only a claim about what a measurement can distinguish, which is a claim about the measurement.

The construction is one-dimensional. In three dimensions the correlation is a tensor, the spectrum has a directional structure, and isotropy has to be assumed or measured; that adds information a one-dimensional cut throws away, and none of it is phase information.

A spectrum is silent about anisotropy unless it is measured in several directions. A stratified layer and an ordinary shear layer can present similar one-dimensional spectra along the flow while differing entirely across it, and what separates them — the number that decides whether the mixing happens at all — is a ratio of two quantities that a single spectrum does not contain.

A one-dimensional spectrum is not the three-dimensional one. What a probe measures is the one-dimensional spectrum $E_{11}(k_1)$, which is an integral of the full three-dimensional spectrum over the two transverse wavenumbers. For isotropic turbulence the relation between the two is exact and the $-5/3$ slope survives it, but the constants differ by a factor that has to be carried through carefully — and a figure that plots one and quotes the other’s constant is making an error this collection’s habit of stating the model is designed to catch.

And the phases here are independent by construction. Real turbulence has phase correlations that are not merely non-zero but organised — sharp fronts, shear layers, vortex tubes — and the third moment is the crudest possible summary of them. Everything past the four-fifths law in this subject is an attempt to say more about that structure, and none of it is settled.

Who found it, and when #

Wiener and Khinchin established the transform pair independently around 1930; Taylor brought it into turbulence in 1938, in the paper that also introduced the frozen-turbulence hypothesis that lets a time series be read as a spatial cut. Kolmogorov’s four-fifths law is from 1941 and its third-order character was understood at once — Kolmogorov’s own papers are careful that the $-5/3$ spectrum is a consequence of the dimensional argument and not the content of it.

The phase-randomisation argument is standard practice in signal processing, where surrogate data with matched spectra are the standard null hypothesis for tests of nonlinearity, and it arrived in turbulence from that direction rather than from within.

The surprising connection is with what averaging costs at the beginning of this field. The Reynolds decomposition leaves six unknown correlations behind because the nonlinear term does not average away; those correlations are second moments, and this essay says that even knowing all of them exactly would not tell a modeller which way energy is going. The closure problem is not only that the second moments are unknown. It is that the second moments are not enough, and every model that computes a spectrum and calls it turbulence is standing on that gap.

Where the ladder goes next #

The rung above is the exact law that the third moment obeys — the one result in this subject that follows from the equations with no model in it — and what its constant turns out to be. Beside it lies the question of what a single long record can establish, which is the theory of ergodicity, and which this collection has not yet written.