The truncation that cannot carry three times the heat

Three numbers left of a fluid introduced the Lorenz system as what it is: a truncation of two-dimensional convection to three Fourier modes, which stopped describing convection long before the parameter at which its famous attractor appears. A millionth is enough measured the one property of it that survives the truncation. The threshold the walls decide computed when a layer heated from below begins to move at all, and the closed form for stress-free plates, 27π⁴/4, is the threshold every calculation below is measured from.

What none of them asked is the question a heated layer exists to answer: how much heat it carries once it is moving. The Lorenz system answers it in closed form, and the answer has a feature that ought to be suspicious — a ceiling. This essay computes the same convection with more of its modes kept, to find out whether the ceiling belongs to the fluid or to the three numbers.

A heat flux with a ceiling in it #

The Lorenz variables are amplitudes: $x$ the strength of the convective roll, $y$ the temperature difference between rising and sinking fluid, and $z$ the distortion of the horizontally averaged temperature profile away from the straight line of pure conduction. That last one is the heat flux. With the plates held at fixed temperatures, the heat crossing the layer is set by the temperature gradient at the plates, and the distortion $z$ steepens it.

The steady convecting state of the Lorenz equations has $z = r - 1$, where $r$ is the Rayleigh number over its critical value. Translated back into a heat flux, that gives

$$Nu = 1 + \frac{2(r-1)}{r}.$$

At onset, $r = 1$, the Nusselt number is one: nothing but conduction. Just above, it rises with slope two. And as $r$ grows without limit, it approaches three and never reaches it. A layer described by these three numbers cannot carry three times the heat of pure conduction, however hard it is heated.

That is not what real convection does. A pan of water on a stove carries many times its conductive heat flux, and measurements of convection over a wide range of Rayleigh numbers find a heat flux growing as a power of the heating. So either the truncation has thrown away something that matters, or the rolls it describes really are limited in a way the pan is not. The only way to tell the two apart is to keep more of the rolls’ modes.

The same rolls with more modes #

The calculation here takes exactly Lorenz’s problem — two-dimensional rolls between plates that hold the temperature fixed and exert no shear stress, which is the second condition chosen so that every mode is a simple sine, at the critical wavenumber $\pi/\sqrt2$ — and keeps more of its temperature field. The temperature is expanded in modes $\cos(max)\sin(n\pi z)$ with $m + n$ even and $m + n$ at most $K$. That set is what Lorenz’s modes generate when they multiply one another, so it is the natural sequence of truncations, and $K = 2$ is Lorenz’s own: the roll mode $(1,1)$ and the mean-profile mode $(0,2)$. $K = 12$ keeps forty-two modes.

The Prandtl number is taken as infinite, so that the fluid’s momentum responds instantly to buoyancy and the streamfunction follows from the temperature mode by mode. That removes the velocity as an unknown without changing what the comparison measures: the Lorenz system’s steady heat flux does not depend on the Prandtl number at all. The steady equations are then solved by Newton’s method, with the nonlinear advection evaluated on a grid and projected back onto the modes.

Two checks do not depend on the method’s assumptions. With $K = 2$ the calculation must, and does, reproduce $1 + 2(r-1)/r$ to rounding. And the heat flux leaving through the top plate must equal the heat flux entering through the bottom, which the equations imply but the Newton solution does not impose; at every truncation and every Rayleigh number here they agree to eight figures.

Exact at onset #

The first result is the one that contradicts the usual description of the Lorenz system as a crude caricature. Just above onset it is not an approximation at all. The rolls with forty-two modes and the rolls with three have the same heat flux to leading order, with the same slope of two, and they part only at second order in the distance above onset.

The reason is how the other modes are made. At onset only the roll mode $(1,1)$ grows; its square drives the mean-profile mode $(0,2)$, which is why Lorenz kept it; and every further mode is driven by products of those two and grows as a higher power of the distance above onset. So close to onset the flow really is Lorenz’s three modes, and the heat flux they carry is the leading term of the exact expansion that Malkus and Veronis wrote down for finite-amplitude convection. At r − 1 = 0.01 the missing modes add less than two parts in a thousand to the excess heat.

The modes that were dropped #

Past onset the hierarchy flattens. At twice the critical Rayleigh number the modes of order four are an eighth the size of the modes Lorenz kept and those of order eight a thousandth — a spectrum falling steeply, which is what a truncation needs. At twenty times the critical value the order-four modes are half the size of the kept ones and the order-eight modes a tenth. Nothing about them is small any more, and a truncation that throws them away is throwing away a large part of the flow.

That flattening has a physical meaning, and it is the key to the ceiling.

Why the ceiling is three #

A convecting layer carries heat by mixing its interior and conducting the heat across thin layers at the plates. The heat flux is the temperature drop across a layer divided by its thickness, and a layer carrying half the temperature difference over a thickness $\delta$ carries a Nusselt number of $1/2\delta$. Carrying more heat means making the layers thinner.

The rolls do exactly that. At twenty times the critical Rayleigh number their mean temperature is nearly uniform through the middle of the layer, at the average of the two plates, and it drops steeply through layers about a tenth of the depth thick at each wall. Drawing a profile with sharp layers takes many harmonics, which is what the flattening spectrum is.

Lorenz’s truncation has one. Its mean profile is the straight conduction line with a single sine, $\sin 2\pi z$, added, and a single sine cannot be steep near the walls without being steep in the middle too. Its gradient at the plates is $-Nu$, and in the middle of the layer it is $-1 + (Nu - 1)$. At $Nu = 3$ that middle gradient is $+1$ — the core temperature rising upwards as steeply as conduction made it fall — and the sine has bent the profile as far as it can go while keeping the walls steeper than the core. Three is the heat flux at which one sine runs out of shape, and the thinnest layer it can draw is a sixth of the depth.

Where the heat is carried, mode by mode #

The heat flux can be split exactly among the modes that make it. Only the modes with no horizontal variation — the mean-profile modes $(0,2)$, $(0,4)$, $(0,6)$ and so on — change the average temperature gradient at the plates, and the excess heat flux is the sum of their separate contributions. That makes the truncation’s failure countable rather than merely visible.

At twice the critical Rayleigh number, Lorenz’s own mean-profile mode $(0,2)$ carries 88.1 per cent of the excess heat the 42-mode rolls carry, the next one $(0,4)$ carries 11.1 per cent, and everything above that less than one per cent. The truncation is missing an eighth of the answer, and it is missing it in one mode.

At twenty times the critical value the same split is 46.7, 29.3, 13.8 and 6.2 per cent for the first four mean-profile modes, with 2.5 and 1.4 per cent in the fifth and sixth; at thirty times it is 41.0, 29.1, 15.4, 7.8, 3.8 and 2.8. Lorenz’s mode now carries well under half of the heat. The heat has migrated into the harmonics that draw the thermal layers, which is exactly where the profile figure showed it must go — and the last of those percentages, still nearly three per cent in the highest mode kept, is why the calculation at thirty times the critical Rayleigh number keeps moving as modes are added.

How many modes the answer needs #

The sequence of truncations converges, and how fast it converges depends on the Rayleigh number in the way the layers predict. At twice the critical value six modes are enough to fix the heat flux to four figures and forty-two change nothing further. At thirty times the critical value the heat flux rises by more than half between Lorenz’s truncation and six modes — from 2.93 to 4.58 — by another thirty per cent between six modes and forty-two, and is still moving by one part in five hundred between thirty and forty-two. A calculation with seventy-two modes, run to check it, changes the value at r = 30 by a further four parts in ten thousand.

That is the structure the refutation above rests on. The truncation’s error is not a fixed fraction of the answer that a cruder model has and a finer one has less of. It is the part of the heat flux the missing modes carry, it is nothing at onset, and it is most of the heat at high Rayleigh number. A truncation is exact until the flow needs the modes it discarded, and then it is not a small error but a different answer.

A power of the driving #

The local exponent — how steeply the heat flux rises with the heating, on logarithmic axes — shows the two models heading for different kinds of answer. Lorenz’s exponent falls to zero, as a quantity approaching a ceiling must. The rolls’ levels off, at 0.300 between twenty-five and thirty times the critical Rayleigh number: their heat flux is growing as a power of the driving, as layers that keep thinning produce.

That exponent is suggestive and it should not be over-read. Measured convection at Rayleigh numbers thousands of times higher grows with an exponent near a third, and a steady two-dimensional roll pinned to one wavenumber between stress-free plates is not that flow: real rolls change their wavelength as the heating rises, become unsteady and three-dimensional, and meet rigid plates. What the calculation establishes is narrower and firm. The ceiling of three is a property of three modes, and the rolls from which those modes were taken do not have it. Every figure here names its model for that reason, a habit that a vortex street no viscous stepper here would shed made a rule: what is drawn is steady rolls in a truncation, not a convecting layer.

Read the other way, at a fixed heat input #

The curve can be read sideways, and the reading makes the truncation’s error larger rather than smaller. A layer is not always held at a fixed temperature difference. A layer heated by a fixed input of power — a floor heated electrically, a planet’s mantle heated by its own radioactivity — is given its heat flux and has to find the temperature difference that carries it.

The heat flux in units of the conduction flux at the critical temperature difference is $Nu \times r$. For the rolls at thirty times the critical Rayleigh number that is $5.970 \times 30$, about 179. Lorenz’s truncation carries $Nu \times r = 3r - 2$, so to carry the same 179 it needs $r$ of about sixty. A three-mode model of a layer heated by a fixed power predicts a temperature difference twice as large as the rolls need, and the error grows without limit as the power rises, because the truncation’s heat flux can only grow in proportion to the temperature difference itself.

Why the Prandtl number could be dropped #

The comparison rests on one simplification that deserves its justification spelled out. The Lorenz system has a Prandtl number in it, σ, and σ controls its dynamics: whether the steady states are stable, where the attractor appears, how fast trajectories separate. It does not appear in the steady heat flux, because at a fixed point the time derivatives vanish and σ multiplies only the time derivative of the roll’s velocity.

The rolls with more modes share that property only partly. At infinite Prandtl number the momentum equation loses its advection term and becomes linear, which is what lets the streamfunction be eliminated. At a finite Prandtl number the steady rolls have velocity advecting vorticity as well as temperature, and the higher modes of the velocity field change the heat flux at high Rayleigh number. So the calculation is the fair comparison for the Lorenz truncation — the same steady problem, with more modes — and a lower Prandtl number would change the rolls’ curve without changing Lorenz’s. Water, at a Prandtl number of seven, sits much nearer the infinite case than air does.

What the roll calculation leaves out #

Stability. Every solution here is steady and two-dimensional, whether or not it would survive a disturbance. Rolls at one wavenumber lose stability to several instabilities as the heating rises, and a real layer at thirty times the critical Rayleigh number is not in this state.

The wavenumber. The rolls are held at the critical wavenumber. Real rolls broaden as the heating increases, and a broader roll carries a different heat flux; the comparison with Lorenz is fair precisely because Lorenz’s modes are pinned to the same wavenumber.

The plates. Stress-free plates let the fluid slide along them, which is what makes the modes simple sines. Rigid plates, the ones an experiment has, raise the critical Rayleigh number to 1707.8 and change the heat flux as the walls change the threshold.

Turbulence. At the Rayleigh numbers of weather and of stars the flow is turbulent, the heat flux is carried by plumes rather than rolls, and no steady solution of any truncation describes it.

Lorenz, Saltzman, and the heat flux #

Saltzman’s 1962 paper cut two-dimensional convection down to a handful of modes to study its time dependence, and Lorenz, a year later, cut it further to the three that produced the attractor. The heat flux $1 + 2(r-1)/r$ was in Lorenz’s model from the start, and its ceiling is visible in a line of algebra; Malkus and Veronis had already shown, in 1958, that the leading-order heat flux of finite-amplitude convection is exactly what three modes give. What the three modes were never meant to do was carry the heat flux of a layer far past onset, and the arithmetic says so as plainly as it says anything: a model that is exact at first order and bounded at infinity is exact where its assumptions hold and silent everywhere else.

Still open: which pattern the rolls become #

Every calculation here assumed rolls: parallel cylinders of rising and sinking fluid, all with one orientation. That is the pattern the Lorenz truncation builds in, and it is not the only one a layer heated from below can choose. Bénard’s first photographs showed hexagons, and hexagons are what appear whenever the layer’s top and bottom are not mirror images of each other — a fluid whose viscosity changes with temperature, or a free surface above it.

That is hexagons remember how the heat was turned up: three sets of rolls at 120° held together by a term that only an up–down asymmetry allows, a window of heating in which both patterns are stable, and a layer whose pattern depends on how it was brought there. Beside it sits the number that is an answer, which is what the Nusselt number is in an experiment rather than a truncation, and the other layer, the thermal layer at a wall whose thinning this essay found the three modes could not draw.