A flux that runs both ways

The cascade is the central picture of the subject and it is worth stating exactly what it asserts.

Where the energy goes derives it: energy enters at the large scales, passes down through a range where nothing but the flux is available, and leaves at the small ones. The flux is constant through the range and equal to the dissipation, and the spectrum follows.

The one exact result is the version that comes from the equations rather than from dimensions: $\langle\delta u^3\rangle = -\tfrac45\varepsilon r$, with no adjustable constant.

Both are statements about averages. Neither says a word about what the transfer is doing at any instant, and the difference turns out to be large.

Why this needs a model, and what kind #

Measuring an instantaneous flux needs a system in which the flux is defined and computable at every moment, and the Navier–Stokes equations at a Reynolds number where a cascade exists are out of reach for the reason the grid nobody can build gives.

So a shell model. Replace Fourier space by a geometric ladder $k_n = k_0\lambda^n$, keep one complex amplitude per shell, and couple each shell only to its neighbours:

$$\frac{du_n}{dt} = i k_n\!\left(u^*_{n+1}u^*_{n+2} - \frac{\delta}{\lambda}u^*_{n-1}u^*_{n+1} - \frac{1-\delta}{\lambda^2}u^*_{n-1}u^*_{n-2}\right) - \nu k_n^2 u_n + f_n.$$

This is the GOY model with $\lambda = 2$ and $\delta = 1/2$. The coefficients are chosen so that the nonlinear term conserves two quadratic quantities — the energy and a signed one playing the part of helicity — for the same reason the Navier–Stokes nonlinearity does, which is that it only moves them about.

It is not turbulence. There is no space in it, no vortices, no pressure and no geometry. What it has is a genuine nonlinear cascade with a fluctuating transfer, and that is the one property being examined. Everything below is offered as an existence proof rather than as a measurement of a fluid.

Defining the flux, rather than remembering a formula #

The transfer out of the first $n$ shells is

$$\Pi_n = -\sum_{m\le n}\mathrm{Re}\left(u^*_m N_m\right),$$

where $N_m$ is the nonlinear part of $du_m/dt$ and nothing else. That is a definition: the rate at which the energy of those shells changes because of the nonlinearity, computed from the equations being integrated.

Writing it that way rather than quoting a closed form has a practical advantage. Summed over all shells it must vanish identically, because the nonlinearity conserves the total — and that identity is the check that the coefficients are right. It holds to $1.9\times10^{-16}$ of the largest flux in the sum.

What the model conserves, and why that is the whole design #

The coefficients in that equation look arbitrary and are not. There are three of them and two constraints, so there is one free parameter, and the constraints are the two conservation laws.

Requiring the nonlinear term to conserve $\sum|u_n|^2$ fixes one combination. Requiring it to conserve $\sum(-1)^n k_n|u_n|^2$ — the model’s analogue of helicity, which is the quantity the knot a flow cannot untie is about — fixes another. What is left is $\delta$, and $\delta = 1/2$ with $\lambda = 2$ is the choice that makes the second invariant have the right dimensions.

That is the entire content of the model’s design. It is not fitted to any turbulence data and it has no adjustable constants beyond the ones the conservation laws leave free, which is why its intermittency is interesting: nobody put intermittency in.

The conservation check run below is therefore not a formality. If the coefficients were wrong the model would not be in the class it claims to be in, and every conclusion drawn from it would be about something else.

The mean is the cascade #

With the model integrated for four hundred time units past a transient, the mean transfer is what the cascade says it should be.

Across shells 8 to 14 it is flat to 9.7 per cent and equal to the dissipation: $4.89\times10^{-3}$ against $5.29\times10^{-3}$. Above shell 15 it falls away as the viscosity takes over; below shell 5 it is still being fed by the forcing.

That is the picture working. A constant flux through a range, set at the top, delivered to the bottom, with a spectrum to match.

Two features of that picture are worth noting because they are the model behaving as a fluid would. The flux rises to its plateau over the first few shells, which is the forcing region; and it falls away over the last four, which is the dissipation range. Between them the plateau is what the cascade means, and its width — seven shells, or two decades of wavenumber — is the model’s inertial range.

And the instantaneous flux is a different object #

Now stop averaging.

The transfer through shell 10 swings between several times its mean and well below zero. Its standard deviation is 3.4 times its own mean, and across the inertial band that ratio runs from 2.2 to 5.5.

And it is negative — energy going up the ladder rather than down — between 15 per cent and 0.3 per cent of the time depending on the shell, nine per cent averaged across shells 5 to 15.

The variation from shell to shell is not noise; it is the GOY model’s well-known period-three structure in the shell index, which is a property of the model rather than of a fluid. That is why the honest statement is the band’s average rather than a value per shell.

What “backscatter” is and is not #

Energy going up the ladder has a name — backscatter — and it is worth separating two claims that get run together.

The claim being made here is local in scale and instantaneous in time: at a given moment, across a given cut in wavenumber, the net transfer can be upward. That is uncontroversial once measured and it is what the figures show.

The claim that is not being made is that the average flux is ever upward in three-dimensional turbulence. It is not: the mean is downward, at $\varepsilon$, and the four-fifths law fixes the sign. Backscatter is a fluctuation about a mean of definite sign, not a competing mechanism.

The distinction matters because there is a subject where the mean itself reverses. The cascade that runs backwards is two-dimensional turbulence, where the vortex-stretching term vanishes identically, enstrophy becomes a second invariant and the two conservation laws between them force the mean energy flux upward. That is a different statement about a different flow, and confusing the two is easy.

The flow whose mean runs backwards #

Every negative excursion above is a fluctuation about a positive mean. There is a case in which the mean itself is negative, and it is not an exotic one.

Two-dimensional flow has a second quantity the nonlinear term cannot change: the enstrophy, the mean square vorticity. With two invariants rather than one, the transfer is no longer free to send everything in one direction — moving enstrophy to small scales while conserving energy requires moving energy to large ones. So a two-dimensional turbulent flow has a dual cascade: enstrophy downscale, energy upscale, with the forcing scale in between.

The consequence is visible rather than statistical. Structures merge instead of breaking up, and large coherent vortices are the attractor rather than the exception — which is a large part of why geophysical flows, nearly two-dimensional at the scales that matter, organise themselves into persistent jets and long-lived storms instead of shredding into small eddies.

And it sharpens this essay’s own distinction. In three dimensions a backward transfer is a fluctuation about a forward mean, and it is a modelling nuisance. In two it is the mean, and it is a consequence of a conservation law. One word, two entirely different statuses, and telling which is which needs the count of invariants rather than a measurement.

Why it matters for anybody modelling turbulence #

The practical consequence is about large-eddy simulation, and it is the reason this measurement is made at all.

A large-eddy simulation resolves the large scales and models the effect of the small ones on them. The simplest models are eddy-viscosity models: the unresolved scales are represented as an extra viscosity, which by construction removes energy from the resolved field everywhere and always.

The instantaneous flux says that is wrong nine per cent of the time. A model that can only drain is a model that cannot represent backscatter at all, and the flows where that matters are transitional ones, where the small scales are feeding the large rather than the reverse.

The fixes — dynamic models, stochastic models, models with a negative eddy viscosity permitted — all exist to let the sign change. Whether they help is a separate question; what this essay establishes is that the thing they are trying to represent is there.

The two windows are worth putting side by side because of what they have in common, which is a statistical character and not a single event. There is no periodicity, no repeating structure, and no sense in which the record is settling down. It is a chaotic signal with a well-defined mean and a distribution much wider than that mean, and four hundred more time units of it would look the same again.

The model’s own intermittency, which nobody put in #

There is a second result in the same run and it belongs beside the neighbouring essay.

Measuring the scaling of $\langle|u_n|^p\rangle$ against $k_n$ over shells 5 to 14 gives

The sixth-order exponent is below the self-similar value, which is anomalous scaling — and it has been produced by a dynamical system with no space in it, no vortex filaments, and no dissipation field to be log-normal.

That is worth more than it looks. The exponents that stop being thirds compares four statistical models of intermittency, all of which are hypotheses about the geometry or the statistics of the dissipation. Here is a system with neither, which is intermittent anyway. The geometrical picture supplies an interpretation of the anomaly — a codimension of two, filaments — and not the anomaly itself, which a nonlinear cascade with the right conservation laws produces on its own.

What the fluctuation says about the four-fifths law #

Reading the exact result in the light of all this sharpens what it claims.

$\langle\delta u^3\rangle = -\tfrac45\varepsilon r$ is exact. It is also an equation about the third moment of a distribution whose instantaneous values are of either sign and much larger than the mean. The average being exactly determined and the instantaneous value being wild are perfectly consistent, and the first is not evidence for a steady picture of the second.

That is the same relationship the mean is not the flow records for the velocity field itself: an average that is a smooth, sensible, unrepresentative object, from which nothing can be inferred about what any realisation is doing.

What a shell model is good for, and what it is not #

Being clear about the standing of the evidence matters more here than in most essays, because the model is a long way from a fluid.

It is good for questions about what a nonlinear cascade with conserved quadratic invariants can do. Those questions include: does the flux fluctuate, does it change sign, does anomalous scaling require geometry, how do the invariants constrain the transfer. All four are answered above and all four are answers about a class of systems that includes turbulence.

It is no good at all for questions about structures, geometry or space. A shell model cannot say what a vortex filament is, where the intense events are, what the dissipation field looks like, or whether the codimension is two. Anything of that kind that appears in a shell-model paper is an interpretation laid over the arithmetic.

And it is silent about anisotropy, walls, boundaries and forcing geometry, which is most of what an engineer wants.

A note on what an average is being taken over #

There is a subtlety in the phrase “nine per cent of the time” that is worth exposing, because it is the kind of thing that makes two papers disagree while both are right.

The fraction quoted is the fraction of time, at a fixed shell, over a long record. It is not the fraction of space at a fixed instant, because a shell model has no space. And it is not the fraction of the flux that is backward, which is a different and smaller number — the backward excursions are shorter and shallower than the forward ones, which they have to be, since the mean is forward.

Those three statistics are different in any intermittent signal, and quoting one while meaning another is how a measurement of a tenth becomes a measurement of a half. The mean is not the flow is the site’s essay on the general form of that hazard.

What a physical-space measurement would show #

The shell model has no space in it, so it is worth saying what the same statement looks like in a real flow, where it has been measured.

The quantity is the subgrid energy transfer: filter a velocity field at a scale, compute the stress the unresolved motions exert on the resolved ones, and contract it with the resolved strain rate. That is the local, instantaneous flux across the filter scale, and it can be evaluated point by point in a simulation or from a multi-probe measurement.

What it shows is the same thing: a mean that is positive and equal to $\varepsilon$, and a point-by-point distribution with a substantial negative tail — typically a quarter to a third of the volume has backward transfer at any instant, which is a larger fraction than the shell model’s nine per cent because a spatial average over a plane and a time average at a shell are different statistics.

That the two disagree in magnitude and agree in kind is the right outcome for a model that has no space in it. The existence of backscatter is a property of a nonlinear cascade; its magnitude is a property of a particular flow.

Limits recorded rather than smoothed over #

The parameters are the standard ones. $\lambda = 2$, $\delta = 1/2$, twenty shells, $\nu = 10^{-6}$, forcing on the first shell. Different $\delta$ gives a model with different invariants and different behaviour; the choice here conserves the analogue of helicity, which is why it is the one everybody uses.

The transient is long and was nearly missed. From an amplitude of $10^{-3}$ the model spends about two hundred and fifty time units climbing to its attractor, and a window ending inside that reports a dissipation of $10^{-9}$ against the attractor’s $10^{-2}$ — the cascade simply has not started. The runs here start on a $k^{-1/3}$ envelope and discard three hundred time units regardless.

The averaging window is four hundred time units. The fractions quoted are stable to a per cent or so over that window and would move at the third figure over a longer one.

And the exponents are measured over ten shells, which is three decades of wavenumber — a wider inertial range than any laboratory has, and one of the reasons shell models are used for exactly this question.

Why the mean is nevertheless the right thing to build a theory on #

It would be easy to read all this as an argument against the cascade picture, and it is not.

The mean flux is what appears in every closed statement the subject has. The four-fifths law is about a mean; the spectrum follows from a mean flux; the dissipation rate that governs decay, mixing and everything else is a mean. Those results are exact or nearly exact and the fluctuations do not weaken them.

What the fluctuations do is bound what can be inferred from a single realisation. A measurement of a flux at one place and one time carries almost no information about $\varepsilon$, because its scatter is several times its mean. A measurement averaged over a few hundred eddy turnovers carries a great deal.

That is an ordinary statistical statement and it has an ordinary consequence: the number of turnovers a record contains matters more than the number of samples, which is the same trap the high-order structure functions of the exponents that stop being thirds fall into and for the same reason.

The residue #

The limit here is the time average, which is an odd thing to call a limit and behaves exactly like one.

Take it and what emerges is the cascade: a constant flux, equal to the dissipation, flat across the inertial range, with the sign the four-fifths law demands. That object is real, it is what every theory in the subject is about, and it is beautiful.

What the limit removes is the entire distribution — a quantity whose scatter exceeds its own mean, whose sign reverses a tenth of the time, and no instant of which resembles the picture the average produces.

The residue, in other words, is everything except the answer.