An hour for every tenfold

The randomness that is not in the equations makes the case that nothing in the Navier–Stokes equations is random: what is random is the observer’s ignorance of the initial data, and the flow’s habit of amplifying it. This essay measures the amplification, and the measurement has an exact form with an uncomfortable consequence.

Two particles #

Two markers started $10^{-10}$ apart in a time-periodic cellular flow — the same flow the area that must not move uses, integrated with the same area-preserving scheme. Their separation grows exponentially over nine decades, fitting a straight line on logarithmic axes at $R^2 = 0.99$, and then saturates at the size of the domain, because two points in a bounded flow cannot get further apart than the flow is wide.

The turnover is at about 376 time units. Extrapolating the exponential past that point overestimates the damage — the particles cannot get more lost than the box allows — which is worth knowing because the exponential is what everybody quotes.

The rate is the largest Lyapunov exponent, and getting it is not instant: the running average along one trajectory takes about a thousand time units to settle at 0.0940 per unit time. A quantity defined as a long-time limit takes a long time to become one, which is the first of several warnings in this essay about numbers that are exact and slow.

Why exponential, and not something else #

The exponential growth is worth deriving rather than assuming, because the whole logarithm depends on it and because it is not obvious that a separation should grow that way.

Two nearby particles are separated by a small vector $\boldsymbol{\xi}$, and to leading order it is carried by the linearised flow:

$$\dot{\boldsymbol{\xi}} = \nabla\mathbf{u}\cdot\boldsymbol{\xi}.$$

That is a linear equation with a time-dependent coefficient, and the generic behaviour of such an equation is exponential — the solution is a product of matrices along the trajectory, and the norm of a product of matrices grows or decays geometrically. The rate is the limit of $\frac1t\ln|\boldsymbol{\xi}|$, which is what the Lyapunov exponent is defined to be.

Two things follow. The growth is a property of the velocity gradient along a trajectory, not of the velocity — a fast flow with no shear separates nothing. And the exponent is a long-time average of a fluctuating quantity, since $\nabla\mathbf{u}$ changes as the particle moves, which is why the running average above takes a thousand time units to settle and why the finite-time exponents later in the essay have such a wide distribution.

It also says what would have to be true for the growth not to be exponential. A flow whose velocity gradient integrates to zero along every trajectory — a pure shear, say, with no folding — gives algebraic separation instead: two particles in plane Couette flow drift apart linearly in time, and a forecast there degrades linearly rather than logarithmically. The distinction is exactly the one a steady two-dimensional flow cannot mix is about, and the double gyre is unsteady precisely so that it can.

The exchange rate #

A forecast is useful until the initial error $\delta$ has grown to a tolerable size $\Delta$, which happens at

$$T = \frac{1}{\lambda}\ln\frac{\Delta}{\delta}.$$

Exactly logarithmic. Improving the measurement by a factor of ten buys $\ln(10)/\lambda$ more time — for this flow, 24.5 time units — and buys the same amount again for the next factor of ten, and the next. The time bought per decade is constant to fifteen decimal places across the range computed, because it is $\ln 10/\lambda$ and nothing else.

The two times a forecaster quotes — how long an error takes to double, and how much a decade of accuracy buys — are the same number in different units, and a decade is worth exactly 3.32 doubling times whatever the flow.

Starting from an initial uncertainty of $10^{-6}$: to make the forecast half as long again, the initial error must fall by a factor of 316. To double it, by a hundred thousand. To make it ten times as long, by $10^{45}$.

The arithmetic behind those is worth seeing, because it is cleaner than it looks. Since $\lambda T_0 = \ln(\Delta/\delta)$, extending the forecast by a factor $f$ needs the error improved by $(\Delta/\delta)^{f-1}$ — which does not contain the Lyapunov exponent at all. The exchange rate in time depends on the flow; the exchange rate in forecast-lengths does not, and doubling any forecast costs the same factor as its current dynamic range.

There is no amount of measurement that changes the exchange rate, because the exchange rate is not a property of the measurement. It is $\lambda$, and $\lambda$ belongs to the flow.

What that does and does not mean #

It does not mean forecasting is futile. Twenty-four time units is twenty-four time units, and the whole apparatus of numerical weather prediction is built on buying them.

For synoptic weather the error-doubling time is about a day and a half, so a tenfold better initial state buys about five days — which is a large fraction of the useful forecast and is why observational networks are worth their cost. For thunderstorm-scale motions the doubling time is tens of minutes and the same improvement buys hours.

What it does mean is that the returns are linear in the logarithm and therefore brutal. Each successive decade of observational accuracy costs far more than the last and buys exactly the same five days, and no combination of satellites and computers turns a five-day forecast into a fifty-day one.

That is a statement about a deterministic system, and it is the point of the whole essay. Nothing here involves randomness. The equations have no random term, the flow used is periodic in time and stated in closed form, and the two particles were placed by hand. The unpredictability is entirely the amplification of a known, finite ignorance — which is what a flow with no randomness in it does.

The misconception this is about #

The claim being refused is not that forecasting is hard. It is a specific and very common inference: that unpredictability implies randomness, and therefore that a better model or a finer grid will eventually remove it.

Three separate ideas get conflated, and it is worth pulling them apart.

Determinism. The equations have no random term. Given the state exactly, the future is exactly determined. That is true of the Navier–Stokes equations, of the flow computed here, and of the weather.

Predictability. Given the state approximately, the future is determined only for a while, and the while is $\ln(\Delta/\delta)/\lambda$. That is a statement about the amplification of an error, and it has nothing to do with whether the equations are random.

And randomness. Turbulence is described in the language of statistics — means, spectra, probability distributions — and none of that language appears in the equations it is a description of. The statistics are a description of an ensemble of ignorances, not a property of the fluid.

The mistake is to run those three together and conclude that a turbulent flow “is random”, which then suggests that it might be made non-random by better technique. Nothing can be made non-random that was not random, and nothing can be made predictable whose Lyapunov exponent is positive — beyond the logarithmic increment above, which is all there is.

The site’s own account of the first and third of those is the randomness that is not in the equations; this essay supplies the second, with a number.

A number worth having beside the horizon #

There is a companion quantity that makes the arithmetic concrete, and it is the one an engineer actually needs.

Ask not “how long is the forecast useful” but “how accurate must the initial state be for a forecast of length $T$”. Inverting the relation gives $\delta = \Delta e^{-\lambda T}$, so the required accuracy falls exponentially with the forecast length — which is the same statement upside down and is much more alarming in that orientation.

For the flow here, at a tolerance of a tenth of the domain: a forecast of 100 time units needs the initial state to a part in $10^4$; 200 units, a part in $10^8$; 400 units, a part in $10^{16}$, which is double precision. Beyond about 400 time units this flow cannot be forecast on a computer at all, whatever the measurement, because the arithmetic cannot represent the required initial condition.

That is a hard limit of a kind worth distinguishing from the practical ones. It is not that the observations are inadequate or the model imperfect; it is that the number of digits needed exceeds what the machine has. Halving the step size does not help, a finer grid does not help, and the only thing that would is arithmetic with more digits — which buys, per digit, one more $\ln 10/\lambda$.

And the exponent is not one number #

Over a fixed window of twenty time units, 288 starting points in the same flow give exponents from 0.0005 to 0.3352 — a factor of six hundred — with a median of 0.139 and seventeen per cent of them below half the median. So “the” horizon is a statistic of a population, and quoting one number for a flow is quoting a mean over parcels whose fates differ by orders of magnitude.

Drawn where they were measured, the fast-stretching points lie along curves — the boundaries between the flow’s two gyres, where fluid is repeatedly split and sent two ways — and the slow ones fill the gyre cores, where a parcel circulates without meeting anything new.

Predictability is a property of a place as much as of a flow. That is why an operational forecast comes with a confidence that varies from day to day and from region to region, and why the ensemble approach — running many forecasts from perturbed initial states — measures the local exponent rather than assuming a global one.

What this has in common with the rest of the phase #

The horizon is an exact result of a kind this collection keeps meeting, and it is worth naming which kind.

It is not an integral, not an optimum, and not a root. It is algebra — a rearrangement of the definition of an exponential — so the relation $T = \ln(\Delta/\delta)/\lambda$ is exact in the strongest available sense: it contains no approximation and no fitted content, and every digit of it is worth printing.

What is not exact is $\lambda$. It is a long-time limit of a fluctuating average, it converges slowly, and its finite-time version is a distribution six hundred wide. So the exactness is entirely in the form of the relation and none of it is in the number.

That split is the same one where a pure number comes from draws between a relation and its inputs, and it is the honest way to quote this result: the exchange rate is exactly logarithmic, and how much a decade buys is a measurement with its own error bars and its own dependence on where in the flow it was measured.

What actually improves a forecast #

If the initial data is subject to a logarithmic exchange rate, the returns have to come from somewhere else, and in practice they come from three places that are not measurement accuracy.

Reducing the model error rather than the data error. The formula assumes the model is exact and only the initial state is uncertain. A model with its own error contributes a growth of its own, which for weather forecasting was for decades the dominant term — so improvements in the physics bought far more than improvements in the observations, until the two became comparable.

Forecasting a different quantity. The exponent is a property of the trajectory, and a statistic of the flow is not a trajectory. Averages, spectra and probability distributions are predictable long past the horizon for individual parcels, which is the whole basis of the distinction between weather and climate — and of every turbulence closure, which forecasts moments rather than fields.

And forecasting the ensemble rather than the state. Running many trajectories from perturbed initial states does not extend any one of them; it measures the local exponent and returns a distribution, which is a different and more honest product than a single forecast with a false air of precision.

None of the three beats the logarithm. All three change what is being asked of the calculation, which is the only move available when the exchange rate is fixed.

The same logarithm, from the other side #

There is a pleasing symmetry between this essay and one earlier in the phase, and it is worth stating because the two are about opposite things.

The three that never converge is about results that are exact in a limit and approached logarithmically — so the limit is unreachable, and the practical statement is “quote the scale”. This essay is about a horizon that grows logarithmically with accuracy — so the horizon is nearly unmovable, and the practical statement is “quote the accuracy”.

In both cases a logarithm is the function standing between what is exactly true and what is available, and in both cases the reason it is so discouraging is the same arithmetic: a logarithm changes by a constant per decade, so a quantity governed by one is neither reachable nor escapable by any feasible amount of effort.

The difference is which side of the relation the logarithm is on. There, the correction is a logarithm of the small parameter and refuses to shrink. Here, the horizon is a logarithm of the small parameter and refuses to grow. Same function, opposite disappointment, and the two are the clearest pair of examples this collection has of a mathematical form dictating what an entire field can and cannot hope for.

In a flow with many scales the logarithm is generous #

Everything above assumes one exponent, which is right for a flow with one scale of motion in it. A turbulent flow has a whole ladder of them, and the answer changes in a way that is worse rather than better.

An error confined to the smallest eddies does not stay there. It saturates those eddies in about one of their own turnover times, and by then it is contaminating the next size up, which saturates in one of its turnover times, and so on up the ladder to the largest scales anybody cares about. The total predictability time is the sum of the turnover times from the scale of the initial error to the scale of the forecast.

And that sum converges. In the inertial range an eddy’s turnover time goes as its size to the two-thirds, so each rung of a halving ladder contributes about two-thirds of the one above it — a geometric series, adding to a small multiple of the largest eddy’s turnover time and no more.

Which is a much stronger statement than the logarithm. Reducing the initial error adds rungs at the bottom of the ladder, and the rungs at the bottom are the cheap ones: they contribute almost nothing to a sum dominated by its top. So a turbulent flow has a predictability horizon that is finite even for perfect initial data, set by its own largest eddies — which for the atmosphere is the origin of the two-week limit, and is not a statement about instruments at all.

What is not claimed #

The flow is a model. The double gyre is a standard chaotic advection test case, not turbulence and not a solution of the Navier–Stokes equations. What it demonstrates is the arithmetic of the horizon, which depends only on there being a positive Lyapunov exponent; the value 0.0940 is this flow’s and nothing else’s.

One trajectory pair is one sample. The fitted separation rate of 0.068 and the asymptotic exponent of 0.094 differ by a third, which is not an error in either — one pair samples one trajectory and the exponent is an average over all of them, which is exactly the point the distribution figure makes.

The exponents for real flows are quoted, not computed. The doubling times for atmospheric motions in the fourth figure come from the meteorological literature; nothing here measures them.

The Lyapunov exponent is the largest of several. A two-dimensional area-preserving flow has two, $\lambda$ and $-\lambda$, and only the positive one governs the horizon — the other governs how quickly a blob is flattened, which is what the area that must not move is about. In three dimensions there are three, and the spectrum of them carries information the largest alone does not.

Nothing here is about model error. Every calculation assumes the equations are exact and only the initial state is uncertain, which is the classical predictability problem and is not the situation any forecaster is in.

And the horizon formula assumes the error grows exponentially the whole way. It does not: it saturates, and past saturation the forecast is not merely wrong but uninformative in a different way — the prediction and the truth become two independent samples of the same climate rather than two diverging trajectories.