The three that never converge

Three results in this collection are exact in a limit and are approached so slowly that nothing ever gets there. They were written up separately, in three different fields, as three separate disappointments.

The flow with no solution records that Oseen’s drag coefficient for a cylinder is still moving 4.8 per cent per decade at a Reynolds number of $10^{-20}$. A limit nothing reaches records that the overlap layer’s apparent power-law exponent is still 0.102 at a friction Reynolds number of a million. The range a real Reynolds number does not have records that the enstrophy range’s slope needs forty-eight octaves above the forcing scale to reach $-3.01$.

This essay is the arithmetic they have in common, which is one function.

The function #

Write $F(\epsilon) = 1/\ln(1/\epsilon)$ for the size of the correction. Then

$$\frac{d\ln F}{d\ln\epsilon} = \frac{1}{\ln(1/\epsilon)} = F,$$

exactly: the local exponent of a logarithmic correction is the correction itself.

That single identity carries the whole essay. It says a logarithmic correction looks like a power law at every scale — the log-log slope is a definite small number — and that the number is different at the next scale down. A power law’s local exponent is a constant, which is what a power law is; a logarithm’s shrinks, for ever, at the rate at which it is itself shrinking.

After a hundred and twenty-eight decades the square root is at $10^{-64}$ and the logarithm is at 0.0034 — a third of a per cent, which is the correction and not the answer.

Because $F$ reaches $x$ only at $\epsilon = e^{-1/x}$, one per cent needs $1/(0.01\ln 10) =$ forty-three decades of the small parameter and a tenth of a per cent needs four hundred and thirty-four. A square-root correction reaches one per cent in four decades and a tenth of a per cent in six.

Forty-three decades is not a hard experiment. It is a number of decades that nothing in physics spans.

Where the logarithms come from #

They are not an accident of three particular problems. All three arise the same way, and knowing the mechanism is what lets a reader anticipate the next one.

A logarithm appears in an asymptotic expansion when the leading and next terms are the same order in the small parameter — when the expansion parameter one would like to use turns out not to work, because the two regions being matched overlap in a way that produces a $\ln$ rather than a power.

In Oseen’s problem the reason is stated in the flow with no solution: the Stokes equations are wrong beyond a radius $a/Re$, however small $Re$ is, so the limits $Re \to 0$ and $r \to \infty$ do not commute. Matching an inner solution to an outer one across a region whose size is itself a function of the small parameter is what generates the logarithm, and it is the same construction that one formula for both ends is about, with the same warning about an overlap region that may not exist.

In the wall layer the logarithm is the answer rather than a correction — the velocity profile is a logarithm, by the classical matching argument — and what is logarithmic here is the rate at which a local power-law fit to it approaches zero exponent.

And in the enstrophy range the logarithm comes from a constant flux of enstrophy through a range in which the eddy turnover time is scale-independent, so the cascade time accumulates logarithmically rather than converging.

Three different mechanisms, and all three produce a correction of the form $1/\ln$, because a logarithm is the one function that is neither a constant nor a power — it is what appears whenever a dimensional argument comes out with an exponent of exactly zero and something has to fill the gap.

The three cases #

Creeping flow past a cylinder. Lamb’s drag is $4\pi\mu U/[\tfrac12 - \gamma - \ln(Re/8)]$, whose denominator is a logarithm of the small parameter. So the fractional change per decade is $\ln 10$ over that denominator, and it falls as $1/\ln(1/Re)$ — 15 per cent per decade at $Re = 10^{-3}$ and still 4.8 per cent at $10^{-20}$.

Wall turbulence. A logarithmic velocity profile has a local power-law exponent of $1/(\kappa u^+)$, and $u^+$ grows only as $\ln y^+$. At a friction Reynolds number of a thousand the exponent is 0.15; at a million, 0.10; at $10^{12}$, 0.06. A measured profile over any accessible range therefore fits a power law with a non-zero exponent — and the exponent depends on where the fit was made, which is why the power-law-versus-log dispute in that field has outlived several generations of measurements.

Two-dimensional turbulence. Kraichnan’s correction makes the enstrophy-range spectrum $k^{-3}[\ln(k/k_f)]^{-1/3}$, whose local slope is $-3 - (1/3)/\ln(k/k_f)$. Two octaves above the forcing scale the slope is $-3.24$; sixteen octaves, $-3.03$; forty-eight, $-3.01$. No simulation and no atmosphere provides forty-eight octaves.

Plotted against $1/\ln(\text{scale})$ each is a straight line through the origin with its own slope — one third exactly for the enstrophy range, near one for the wall layer, and near $\ln 10$ for Oseen’s per-decade measure. Three fields, three literatures, one function.

The lines are not exactly through the origin, because each correction is really $1/(\ln(\text{scale}) + c)$ with its own offset. Dividing by $1/\ln(\text{scale})$ leaves $(\ln + c)/\ln$, which tends to one and approaches it — logarithmically, of course. Oseen’s offset is the largest and its ratio moves 53 per cent from end to end while settling perfectly well, so the check that matters is whether the ratio is converging rather than whether it is flat.

How much of the range each case has #

It is worth putting the three cases on a common footing, because the ranges available to them differ by a great deal and that is what decides how visible the problem is.

The creeping-flow case has essentially unlimited range in principle and none in practice. Reynolds numbers of $10^{-6}$ are reachable with a microscope and a viscous fluid; $10^{-20}$ is not reachable by anything, so the drag coefficient’s convergence has never been observed and never will be.

The wall-layer case has about four decades. Laboratory pipes reach $Re_\tau$ of a few thousand, the largest facilities a hundred thousand, and the atmospheric surface layer perhaps a million. Over that the apparent exponent falls from 0.15 to 0.10 — a change of a third, which is measurable and is exactly what the dispute in that field is about.

And the enstrophy case has about six octaves. A simulation with a forcing scale and a dissipation scale separated by six octaves is a large one, and at six octaves the slope is $-3.08$ rather than $-3$ — an eight per cent error in an exponent, which in a field where exponents are the whole currency is a great deal.

So the same arithmetic produces three quite different experiences: a limit nobody has approached, a limit approached slowly enough to argue about, and a limit approached badly enough that the leading term is not what is measured.

The diagnostic #

There is a test, and it needs no theory.

Measure the local exponent at two well-separated scales. A power law keeps its exponent; a logarithm loses it. Every one of the three cases falls, and by roughly a factor of two over a decade or so of scale — which is exactly what $F$ does.

That test is worth having because the alternative is arguing about a fit. A dataset spanning a decade can be fitted with a power law and the fit will be good, and the only way to find out whether the exponent means anything is to fit again somewhere else.

It also cost this essay a case. The fitted inertial-range exponent was originally the third example here — the essay that measured the inertial band watched it drift from $-1.64$ towards $-5/3$ — and applying the test to it shows its bias falls as roughly the square root of the scale separation. That is a power law. It converges. It does not belong, and the test found it.

What the identity says about fitting #

There is a corollary worth extracting, because it explains a common frustration rather than merely describing one.

Suppose a quantity has a logarithmic correction and a power law $A\epsilon^p$ is fitted to it over a decade of $\epsilon$. The identity says the fitted $p$ will come out near $1/\ln(1/\epsilon)$ at the middle of the range, and that a fit over the next decade down will return a smaller $p$.

So two careful experiments in the same field, done at different scales, will report different exponents, both with small error bars, and will disagree. Neither is wrong; neither is measuring an exponent, because there is no exponent. The disagreement is a property of the function and not of either laboratory, and it is exactly the situation the number that is not a number describes for a different quantity that is also not what it is being asked to be.

The way out is not more careful measurement. It is to plot the local exponent against the scale and see whether it is constant — one extra derivative of data already in hand, and it settles the question that a hundred power-law fits cannot.

A fourth case, and a fifth #

Once the pattern is visible it turns up elsewhere in this collection, and two more are worth naming because they show the range.

The energy of a two-dimensional vortex. It diverges as $\ln(R/a)$ — the same function with the sign of the argument flipped — so a vortex’s kinetic energy is not large, it does not exist, and every finite value quoted for one is a statement about where the counting stopped. That is the energy a vortex cannot have, and it is a divergence where the three above are convergences, but the arithmetic is identical: equal increments per decade, for ever.

And the added energy a circulation costs in a domain with a hole, which grows by exactly $\ln 10/4\pi$ per decade of outer radius — measured in how much more than the least as 0.1832 per decade over sixteen decades, with no sign of stopping.

Divergence and convergence are the same phenomenon here. A quantity that grows as a logarithm has no value; a correction that falls as one has no limit that is ever reached. In both cases the giveaway is that the rate per decade is constant, and the appropriate response is to quote the range rather than the limit.

What to do about it #

Three things follow, and none of them is to stop using asymptotics.

Quote the scale with the result. “$-1.64$ at $Re_\lambda = 60$” is a statement; “$-5/3$” is a limit nobody has been to. That is a discipline rather than a discovery, and it is broken constantly.

Do not extrapolate a logarithm. Fitting a power law over the range in hand and extending it is the standard move, and with a logarithmic correction it is wrong in a specific direction: the fitted exponent is too large, so the extrapolation overshoots. The size of the overshoot is computable from the identity above and it is not small.

Say which of the two it is, in the paper. A correction’s functional form is usually known to the person deriving it and almost never stated in the result, so a reader meets “the asymptotic value is $-5/3$” and has no way to tell whether that is a limit reachable at $Re_\lambda$ of a few hundred or one that would need forty-three decades. One sentence fixes it, and the sentence is cheaper than the figure that would otherwise be argued over.

And expect the argument to be unresolvable. When two theories differ by a logarithm — a power law against a log law in the overlap region, say — the difference between them over any accessible range is of the size of the correction, which is exactly the size of the experimental scatter. That is not a failure of the experiments. It is what the arithmetic says, and it is why those arguments last decades.

The general rule, stated once #

Every asymptotic result in this collection can be sorted by one question: how does the correction depend on the small parameter?

If it is a power $\epsilon^p$, the result becomes accurate at a rate that can be computed and reached. Blasius’ profile is exact in the limit of large Reynolds number and its correction is $O(Re^{-1/2})$, so at $Re = 10^6$ it is a tenth of a per cent, and the limit is a useful description of a real flow.

If it is a logarithm, the result is exact in a limit and describes nothing. It should be quoted with its scale attached, its extrapolation should be refused, and the argument about whether it is the right limit should be expected to remain open.

And there is a third case worth knowing, which none of these three is: a correction that is exponentially small, $e^{-1/\epsilon}$. Those are invisible in any expansion — every term of the power series is zero — and they are the reason some asymptotic results are far better than they have any right to be, and some phenomena are entirely missed by them.

Three behaviours, three quite different practical situations, and the diagnostic that separates the first two is one measurement of a local exponent at two scales. It is the cheapest piece of methodology in the subject and it is almost never done.

The third case, which is invisible rather than slow #

The exponentially small correction deserves more than a mention, because its practical character is unlike either of the other two.

A term like $e^{-1/\epsilon}$ has every coefficient of its power series equal to zero. So an expansion in $\epsilon$ cannot see it at any order, however many terms are taken, and a calculation that is correct to all orders can still be missing it entirely. That cuts both ways: it is why some asymptotic results are far more accurate than their error estimates suggest, and why some phenomena are absent from a theory that looks complete.

The fluid-mechanical importance of such terms is that they select. There are problems whose power series admits a continuous family of solutions — a finger of one fluid pushing into another can, to every algebraic order, have any width at all — and what picks one member out is a term exponentially small in the surface tension. To any expansion the width looks like a free parameter; it is not, and the thing determining it is beyond all orders.

Which puts a different reading on the free constants this collection keeps meeting. A quantity that an expansion declines to fix is sometimes genuinely free, and is sometimes fixed by something the expansion was structurally unable to represent.

What is not claimed #

The three cases are corrections to three different kinds of thing. One is a coefficient, one is an exponent and one is a spectral slope, and putting them on one axis required each to be reduced to a dimensionless departure from its own limit. That reduction is a choice; a different one would move the slopes of the three lines and not their common form. The site’s account of how much a choice of non-dimensionalisation can move an answer is what “of order one” is worth.

Not every slow convergence is a logarithm. A power-law correction with a small exponent converges slowly too, and the diagnostic above distinguishes them: the exponent stops falling. The essay’s claim is that these three are logarithmic, which is checked, and not that slow convergence implies a logarithm.

The offsets are not computed. Each of the three corrections has a constant added to its logarithm, and the values of those constants are not derived here — what is shown is that the ratio settles, which is a statement about the leading behaviour rather than about the constants.

The wall-layer case uses a stated log law. $\kappa = 0.41$ and $B = 5$ are put in, and the apparent exponent that comes out is a property of the assumed profile. It does not settle whether the real overlap region is logarithmic, which is the dispute it is about — it says what the exponent would look like if it were.

Three cases are not a survey. They were picked because this collection had already recorded all three, in three different fields, without noticing they were the same thing. There are certainly more, and there is no claim here about how common the pattern is — only that when it occurs, it occurs the same way, and that a site with a hundred and eighty essays had three instances of it filed apart.

And Oseen’s per-decade measure is not a correction to anything. The drag coefficient of a cylinder at low Reynolds number has no limiting value to subtract, since it diverges; the quantity plotted is its own rate of change, which is the natural analogue and is not the same object as the other two.