Three numbers left of a fluid

Convection in a layer heated from below is a partial differential equation with infinitely many degrees of freedom. Barry Saltzman, in 1962, expanded it in Fourier modes and kept a handful. Edward Lorenz, reading that paper, noticed that three of the modes stayed active while the others died, kept those three, and integrated them.

The result is the best-known system of equations in the modern history of physics, and this site owns it for a specific reason: convection is fluid-flow’s ground, so the truncation of convection is a fluid-mechanical object and the price of the truncation is a fluid-mechanical question.

The price is the subject of this essay.

What the three variables are #

The truncation keeps three quantities, and it is worth knowing what each one meant while it still meant something.

x is the amplitude of the convective overturning — the strength of the roll. Positive is one direction of rotation, negative the other.

y is the amplitude of the temperature difference between the rising and descending parts of the roll.

z is the departure of the vertical temperature profile from the linear conduction profile: how much the overturning has flattened the mean stratification in the middle of the layer.

The equations are

$$\dot x = \sigma(y - x), \qquad \dot y = x(r - z) - y, \qquad \dot z = xy - \beta z$$

with σ the Prandtl number, β a geometric factor set by the cell’s aspect ratio, and r the Rayleigh number in units of its own critical value — so r = 1 is exactly the threshold the previous rung computes.

The standard parameters are σ = 10, β = 8/3 and r = 28. The first is a plausible Prandtl number for water; the second follows from the critical wavenumber, since β = 4/(1 + a_c²) and a_c² = π²/2 gives 8/3 exactly; the third is not plausible for anything, and that is the essay’s subject.

What is checked, and what it establishes #

Two of the system’s properties are exact and the site computes both.

The fixed points. For r > 1 there are three: the origin, and a pair at

$$x = y = \pm\sqrt{\beta(r-1)}, \qquad z = r - 1$$

The build evaluates the right-hand side at each of them and requires the result to vanish to within 10⁻⁹, which refuses a point misidentified as a fixed point — a defect that would leave every picture on the page looking exactly as it does.

The Hopf threshold. The two off-centre fixed points lose stability at

$$r_H = \frac{\sigma(\sigma + \beta + 3)}{\sigma - \beta - 1}$$

which for the standard parameters is 24.7368. Above it, the system has three fixed points and settles on none of them.

What the equations say when read as a fluid #

Read line by line while they still mean something, the three equations are a compact and rather beautiful statement about a convecting layer.

The first, ẋ = σ(y − x), says the roll is driven by the temperature contrast between its rising and descending sides and damped by viscosity. σ is the ratio of the two rates, which is the Prandtl number, and it appears exactly where a Prandtl number should.

The second, ẏ = x(r − z) − y, says the temperature contrast is generated by the roll carrying warm fluid up and cold fluid down — the term xr — and is reduced by two things: conduction, the −y, and the flattening of the mean profile that the roll has already caused, the −xz. That last term is the feedback that limits the motion.

The third, ż = xy − βz, says the mean profile is flattened by the correlation between the roll and the temperature contrast, which is the convective heat flux, and relaxes back towards conduction at a rate set by the geometry.

So the nonlinearity — the only nonlinearity, appearing twice — is the convective transport of heat. That is a genuine piece of fluid mechanics and it is what makes the system worth reading rather than merely computing. What is lost in the truncation is not the physics of the terms that remain; it is everything the discarded modes would have done with the energy the retained ones hand them.

Where it stopped being convection #

Here is the claim the essay exists to make, stated as precisely as it can be.

The truncation keeps three modes and discards infinitely many. That is legitimate exactly as long as the discarded modes are small, and they are small in a definite regime: just above onset, where the amplitude of the convection is small and the nonlinear coupling that feeds the higher modes is weak.

How far above onset? The standard estimate is that the neglected modes are no longer negligible by r of about 5, and the truncation certainly cannot be trusted past r ≈ 10. The chaotic regime it is famous for begins at r = 24.74.

So the picture at the top of this essay is drawn at five times the r at which the model stopped describing the fluid it came from.

That is not a criticism of Lorenz, who was entirely explicit about it. His 1963 paper describes the system as a set of equations obtained by truncation and studies them as such; the identification of the attractor with atmospheric convection is a later habit of other people’s.

Three further reasons the equations do not describe a convecting layer at r = 28.

Two dimensions. The truncation is of a two-dimensional roll. Real convection at that Rayleigh number is three-dimensional, and the three-dimensional instabilities have thresholds far below it.

Free-free boundaries. The expansion uses the stress-free case, for the same reason the closed form exists there — no experiment has those boundaries.

A single cell width. β is fixed by the critical wavenumber, which pins the horizontal scale to the one that is marginally stable at onset. A layer well above onset does not maintain that scale.

Any one of those would be enough. Together they mean the system at r = 28 is a mathematical object with a fluid-mechanical ancestry.

Why it is still worth this site’s attention #

If the equations stopped describing convection at r ≈ 5, why does a site about fluid mechanics keep them?

Because the truncation is a fluid-mechanical act. How much of a flow survives being reduced to three numbers is exactly the question this site asks of every model it draws, and here the answer can be given precisely rather than gestured at.

Because the regime where they do describe convection is real and interesting. Below the Hopf threshold the system spirals into a fixed point, which is the three-equation version of the layer convects steadily in rolls — and that is the observed behaviour just above onset. The model earns its keep in the range it was derived for and is then used far outside it, which is the commonest failure mode of any model and is worth having one clean example of.

Because what happened next is a genuine lesson about fluids. Lorenz’s discovery that a deterministic system with three variables could be unpredictable in practice is what killed the hope that better measurements and bigger computers would make weather forecasting a solved problem. That conclusion is right, it is about fluids, and it does not depend on the truncation being faithful — which is the subject of the next rung.

Because the same equations are worth two different essays. Another site in this fleet owns the same three equations as a piece of mathematics — the behaviour class, studied with no physical system named anywhere in it. This site owns them as the truncation of a flow, which means it owns the question of what the truncation cost, and the essay is written to that line.

The one exact statement about the attractor, and it is a fluid one #

There is a third property of the system that is exact, and unlike the fixed points and the Hopf threshold it is a statement about the whole flow in phase space rather than about particular points in it. Take the divergence of the right-hand side:

$$\nabla\!\cdot\!\mathbf f = \frac{\partial}{\partial x}\big[\sigma(y-x)\big] + \frac{\partial}{\partial y}\big[x(r-z)-y\big] + \frac{\partial}{\partial z}\big[xy-\beta z\big] = -(\sigma + 1 + \beta).$$

It is a constant — independent of position, and independent of $r$ — so a volume of initial conditions contracts at a fixed exponential rate everywhere in the space. For the standard parameters that rate is $-13.667$, which shrinks any blob of starting states by a factor of a million in about one time unit.

The three terms are the three dampings, one from each equation: $\sigma$ is the viscous damping of the roll, the $1$ is the conductive damping of the temperature contrast, and $\beta$ is the relaxation of the mean profile back towards conduction. The rate at which phase-space volume disappears is the sum of the truncated fluid’s dissipations, which is as direct a fluid-mechanical reading as the system offers.

Two consequences follow, and the second answers a question this essay raises at its end.

The attractor has zero volume. Whatever the trajectories settle onto occupies no volume at all, which is why the picture at the top can be a bounded object that a trajectory never leaves and never fills. It is also why there is no periodic orbit to find: the set is not a curve, not a surface, and not a point.

And it fixes the exponents. The three Lyapunov exponents must sum to the divergence, exactly. One of them is zero — displacement along a trajectory neither grows nor shrinks — so measuring the largest, as the figures here do at 0.8982, determines the third with no further computation: about $-14.6$. Two directions contract violently and one stretches gently, and that lopsidedness is what makes the attractor nearly a surface.

Put those together with the standard dimension formula and the attractor’s dimension comes out near 2.06 — barely more than a surface, and definitively not an integer. That number is the one this essay’s closing section is careful about: it is a genuine measure of how many degrees of freedom this system really uses, and the open question is whether the corresponding number for a turbulent flow is finite, large, and worth knowing.

What “chaotic” is and is not #

Some care about words is worth taking here, because the popular vocabulary around this system is loose.

Not random. The equations are deterministic. Given the state exactly, the future is determined exactly. Nothing stochastic appears anywhere.

Not complicated. Three variables, seven terms, two of them nonlinear. The system is simpler than almost anything else in this field.

Not turbulent. Turbulence is a statement about a velocity field with structure across many decades of scale. This system has three degrees of freedom. It is chaotic and it has no scales in it at all, and conflating the two is the single most common error made about this picture.

What it is: a system whose trajectories separate exponentially, so that finite knowledge of the present gives knowledge of the future only for a finite time. That is the property the next rung measures.

The general shape: what a truncation costs #

The Lorenz case is unusually clean, and the general lesson is worth extracting because this site is full of truncations that are not labelled as such.

A truncation replaces a system with infinitely many degrees of freedom by one with finitely many, and it is justified when the discarded degrees of freedom are inactive. Two things then have to be true, and only the first is usually checked.

The discarded modes must be small at the moment of truncation. This is a statement about the initial condition and is easy to verify.

They must stay small. This is a statement about the dynamics, and in a nonlinear system it is usually false, because the retained modes feed the discarded ones through exactly the term that made the system nonlinear. In convection the retained roll pumps energy into finer structure, and above r ≈ 5 that structure is no longer negligible.

This site is built on truncations of one sort or another and each of them is named. Ideal flow discards viscosity, and the discarded term is decisive in a thin layer. Boundary-layer theory discards streamwise diffusion, and the discarded term matters at separation. Thin-aerofoil theory discards thickness and gets the lift right and the leading edge wrong.

Every one of them has the same structure: excellent inside its regime, and silent about where the regime ends. The Lorenz system is the case where the regime’s end can be stated as a number and the famous picture lies well outside it, which makes it the best teaching example the subject has.

Where the model stops #

The honest inventory, since this essay is largely about being honest about a model.

Everything drawn here is an integration of three ordinary differential equations, by fourth-order Runge–Kutta at a step of 0.002. The integrator matters more than usual: the whole subject is that trajectories separate exponentially, so a first-order stepper’s error grows at the same rate as the phenomenon and the two become indistinguishable.

No claim is made that any of it happens in a fluid above r ≈ 5.

The Lyapunov exponent is measured rather than quoted, by renormalising a nearby pair, and comes out at 0.8982 against the published 0.9056 — a difference attributable to the finite integration length rather than to anything of interest.

Nothing about the attractor’s structure is proved here. That it is a genuine attractor with the properties usually attributed to it was an open question until Tucker’s computer-assisted proof in 2002, thirty-nine years after Lorenz’s paper.

What a reader should take from the picture #

The attractor is one of the most reproduced images in science, and it is worth saying what a reader is entitled to conclude from having seen it.

That a deterministic system with three variables can be unpredictable in practice. This is the real and durable conclusion, it does not depend on the truncation being faithful, and it changed how several fields think about prediction.

That simple equations can have complicated solutions. Also true, also durable, and the reason the picture became famous outside its own subject.

That the shape has meaning. Only in the sense that it is the shape of this system’s attractor. The two lobes correspond to the two directions of rotation of the roll, and a trajectory switching lobes corresponds to the convection reversing — which would be a fine physical statement if the model still described the fluid at the parameters where the switching happens.

That turbulence looks like this. No. This is the conclusion the picture most often invites and it is wrong for the reason given above: three degrees of freedom against the Re^(9/4) that a turbulent flow needs. A chaotic system and a turbulent flow share unpredictability and share almost nothing else.

The distinction matters because a genuine question sits behind it — whether turbulence can be understood as a dynamical system with a very large but finite number of degrees of freedom, and whether the attractor’s dimension is a useful measure of it. That question is open, it is respectable, and it is not settled by the resemblance of two pictures.

Who found it, and when #

Saltzman’s truncation is from 1962 and Lorenz’s paper, Deterministic Nonperiodic Flow, from 1963. It was cited about a dozen times in its first decade.

The story of the rounding — restarting a run from a printout that carried three decimals where the machine held six, and finding the new trajectory diverging from the old — is from 1961 and is genuine, though it is usually told as though the divergence were the discovery. The discovery was that the divergence was a property of the equations rather than of the computer.

Ruelle and Takens introduced the term strange attractor in 1971, in a paper about turbulence that proposed a route to it quite different from the classical one.

Tucker’s proof that the Lorenz equations do possess a strange attractor, in the strict mathematical sense, appeared in 2002 and was Smale’s fourteenth problem.

Where the ladder goes next #

The next rung takes the one property of this system that transfers to real fluids regardless of the truncation, and measures it: a millionth is enough is about sensitive dependence, the exponent that quantifies it, and what it means for the practical business of predicting a flow.