The fraction that is really four thirds

The one exact result says what its title says: almost nothing in turbulence follows from the equations without a model in it, and one thing does. That is not quite the state of affairs. There are two, and the second has been in print since 1949 — Yaglom’s law, which fixes the mixed third moment of a velocity increment and a scalar increment at $-\tfrac43\varepsilon_\theta r$, exactly, with no adjustable constant, in the range where both fields are cascading.

The obvious question is why its fraction is different. Two exact derivations producing 4/5 and 4/3 invites the reading that they are separate results about separate things which happen to have landed on separate numbers, and the numbers then have to be memorised.

They have not landed separately. Both laws have the same constant, and it is 4/3. The famous four-fifths is that same four-thirds with a factor of 3/5 in front of it, and the 3/5 is a piece of tensor algebra about the velocity field that contains nothing about the cascade, the dissipation or the Reynolds number. This essay computes both halves of that — the 4/3 and the 3/5 — and neither of the two physical laws, because what is being separated here is geometry from physics and the geometry is the part that has been hiding.

Where a four-thirds comes from, and what the three is #

Both laws have the same shape, and it is worth writing that shape down before either of them.

A two-point correlation — of the velocity with itself, or of a scalar with itself — obeys a balance in which the nonlinear term appears as a divergence in separation space. The separation $\mathbf{r}$ between the two points is the independent variable, and the nonlinear term does not create or destroy the correlation: it moves it from one separation to another, which is the cascade written as a flux. In a steady state, with the forcing confined to large separations and the viscosity to small ones, the divergence of that flux over the range between them equals a constant sink:

$$\nabla_{\mathbf r}\cdot\mathbf F = -4Q,$$

with $Q = \varepsilon$ for the velocity and $Q = \varepsilon_\theta$ for the scalar. Isotropy makes $\mathbf F$ radial, $\mathbf F = F(r)\hat{\mathbf r}$, and in $d$ dimensions the radial divergence of such a field is $(1/r^{d-1})\,\mathrm{d}(r^{d-1}F)/\mathrm{d}r$. Integrating from the origin outward,

$$r^{d-1}F = -4Q\,\frac{r^d}{d}, \qquad F(r) = -\frac{4}{d}\,Q\,r.$$

So the four-thirds is four over the number of dimensions. The 4 is how the sink is shared between a pair of points; the 3 is the space. Integrating the divergence by quadrature at five dimensionalities reproduces $4/d$ to $2\times10^{-9}$, which is arithmetic and not an achievement, but it makes the shape of the argument visible: the constant is not a property of turbulence and would be 2 in a plane and 1 in four dimensions without a single change to the derivation.

One boundary condition enters, and it is the only one in the whole derivation. The constant of integration has to vanish, because any other value makes the flux diverge where the two points meet. That is what leaves the law with no free parameter — there is nothing to fit, because there was nothing to choose.

The five, which belongs to a hot wire rather than to a fluid #

For the velocity, the quantity the divergence argument delivers is the mixed third moment of the whole increment, $\langle \delta u_L |\delta\mathbf u|^2\rangle$ — the longitudinal component of the increment multiplied by the squared magnitude of all three of them. That is not a quantity anybody measures. A single hot wire in a mean flow measures one velocity component along one line and therefore measures $\langle \delta u_L^3\rangle$, the longitudinal third moment, which is a different object.

There is a second reason the longitudinal moment is the one that gets measured, and it is worth noting because it puts another approximation under the famous fraction. A hot wire records a time series at one point, and the separation in the structure function is obtained by multiplying a time lag by the mean speed — Taylor’s frozen-turbulence hypothesis, which has the pattern carried past the probe faster than it evolves. The increments so obtained are along the mean flow direction, which is to say longitudinal, by construction. So the quantity the divergence argument naturally produces is the one no single-probe experiment can measure, and the quantity every single-probe experiment does measure requires both the isotropic projection and the frozen hypothesis before it can be compared with a theory. The 3/5 is exact; the route that makes it necessary is not.

Isotropy and incompressibility relate the two exactly, with no dynamics involved:

$$\langle \delta u_L |\delta\mathbf u|^2\rangle = \langle \delta u_L^3\rangle + \tfrac13\,\frac{\mathrm d}{\mathrm dr}\Big[r\,\langle \delta u_L^3\rangle\Big].$$

Apply that to a power law. If $\langle \delta u_L^3\rangle \propto r^n$, the operator multiplies by $(n+4)/3$, so the longitudinal moment is $3/(n+4)$ times the mixed one. At $n = 1$ that is exactly 3/5, and $\tfrac43 \times \tfrac35 = \tfrac45$.

The exponent being 1 is not an extra assumption; it is what the exact law fixes. So the reasoning closes on itself neatly: the divergence fixes the mixed moment as linear in $r$, linearity fixes the projection factor at 3/5, and 3/5 of 4/3 is the number in the textbooks.

What that also establishes is that the 3/5 is not a constant of the subject. Differencing the relation numerically at four exponents gives 9/14 at $n = 2/3$, 3/5 at $n = 1$, 9/16 at $n = 4/3$ and 1/2 at $n = 2$ — a spread of 0.14 across that range, which is 24 per cent. Anyone converting between longitudinal and full structure functions at some other order, where the exponent is not 1 and is not even known, has to carry the right factor and cannot carry this one. That is a live issue rather than a pedantic one: the exponents that stop being thirds is the essay about higher moments departing from their dimensional values, and every such departure moves the projection too.

A scalar has one component and nothing to project #

Now the scalar. Its increment $\delta\theta$ is a single number: a temperature difference, a concentration difference, a dye brightness difference. There are no components to sum over and no transverse partner to eliminate. The quantity the divergence argument delivers, $\langle \delta u_L (\delta\theta)^2\rangle$, is already the quantity a probe measures.

So there is no projection, and the law keeps the 4/3 it was born with.

The three rows of that figure are the whole argument. Yaglom’s scalar law and the velocity’s mixed-moment law are both 4/3, because both are the same divergence with a different sink in it. The four-fifths is the second multiplied by 3/5. Nothing distinguishes the scalar from the velocity except how many components each has, and the constants differ by exactly that.

It is worth saying what this does not mean. It does not mean Yaglom’s law is a corollary of Kolmogorov’s, or that one was derived from the other; they are independent statements about independent fields and each needs its own balance equation. What they share is the last step, and the last step is where the number comes from.

It is worth saying where the balance equation’s constant sink comes from, since everything above rests on it. In a steady state the variance put in at the large scales has to leave at the small ones, and the rate is set at the top rather than at the bottom — which is what a cascade is and is as true of a dye as of the velocity itself.

The sign, which is the only theorem about the direction of a cascade #

The minus sign has been carried along above without comment and it deserves one, because it is the part of both laws that says the most and is quoted the least.

A negative $F(r)$ means the flux in separation space points inward, from large separations towards small ones. That is the cascade — correlation, and with it energy or scalar variance, moving from the scales where it was put in to the scales where it is destroyed — and here it is not a picture, a hypothesis or an appeal to a hierarchy of eddies. It is a sign in an exact result.

That is worth holding against how the cascade is usually argued for. Richardson’s rhyme, the hierarchy of whorls, the spectrum’s slope, the constancy of the flux: all of them are pictures or dimensional arguments or measurements. The direction of the transfer, in three dimensions, for a scalar and for the velocity alike, is the one feature of the whole cascade that follows from the equations with nothing added. A turbulence that transferred energy upward in three dimensions would have to violate a theorem rather than a belief.

And it is why the two-dimensional case is genuinely interesting rather than an exotic footnote. There the same derivation returns a flux of the opposite sign for the velocity, and the inverse cascade is not a surprising empirical behaviour that a theory has to accommodate: it is in the exact law, as a sign, for the same reason the forward cascade is in three dimensions. The scalar’s law does not change sign in a plane, so a two-dimensional turbulence carries variance down while carrying energy up — two cascades running in opposite directions in one flow, each of them with an exact law saying so.

It also supplies the sharpest available test of whether a measured range is an inertial range at all. Fitting a slope to a spectrum can be done to any record; it establishes that something is a power law. A third moment that comes out negative, linear in $r$, and equal to $\tfrac45\varepsilon r$ with an independently measured $\varepsilon$ establishes that the range is carrying a flux in the right direction at the right rate, which is a far stronger statement and one no second-order measurement can make — a spectrum cannot hold the third moment at all, because it is blind to the phases the transfer lives in.

The convention, which is where the factor-of-two disagreements live #

The scalar’s sink needs naming, because the literature quotes this law two ways and they differ by a factor of two.

Written here, $\varepsilon_\theta = \kappa\langle|\nabla\theta|^2\rangle$, the destruction rate of half the scalar variance — which is the exact analogue of $\varepsilon$, the destruction rate of half the kinetic energy per unit mass. With that convention the law is $-\tfrac43\varepsilon_\theta r$ and the parallel with the velocity is clean.

Written against $\chi = 2\kappa\langle|\nabla\theta|^2\rangle$, the destruction rate of the whole variance, the same law reads $-\tfrac23\chi r$. Both appear in print, both are correct, and quoting one constant with the other definition is the single most common way of misstating Yaglom’s law. The convention a result depends on is named here rather than implied, and for this result the convention is worth more than the constant: an experiment that measures $\varepsilon_\theta$ by the wrong definition and fits $4/3$ reports a scalar dissipation rate wrong by a factor of two, and nothing about the fit looks wrong.

Where the scalar’s law has less room than the velocity’s #

The two laws hold in ranges that are not the same range, and the difference is a dimensionless number.

The velocity’s law needs an inertial range: separations small against the integral scale and large against the Kolmogorov scale $\eta$. The scalar’s law needs an inertial-convective range — separations at which both the viscosity and the molecular diffusivity are negligible — so it ends at whichever of the two dissipation scales is the larger. The scalar’s own is the Batchelor scale $\eta_B = \eta/\sqrt{Sc}$, and the Schmidt number decides which way round they sit.

For $Sc > 1$ — dye in water at 2,000, salt at 700, heat in water at 7 — the Batchelor scale is below the Kolmogorov scale, and the scalar goes on developing structure in a region where the velocity field is smooth. That region is Batchelor’s viscous-convective range, the part of the scalar cascade with no velocity cascade under it, and Yaglom’s law does not hold there for the plain reason that it is a statement about the velocity’s nonlinear transfer, which has stopped. The inertial-convective range therefore ends at $\eta$, exactly where the velocity’s does, and the two laws have the same room.

For $Sc < 1$ — heat in air at 0.7, heat in mercury at 0.01, heat in liquid sodium at 0.004 — the Batchelor scale is above the Kolmogorov scale, the scalar’s diffusion bites first, and the scalar law’s range is shorter than the velocity’s.

The asymmetry is the awkward part. At $Sc = 0.01$ a whole decade is lost, and a decade is a large fraction of what is available at all: the range a real Reynolds number does not have is the essay about how little inertial range an achievable flow provides, and losing a decade of it makes the scalar law observable in fewer flows than the velocity law rather than more. That is the reverse of what “a second exact result” sounds like it should buy.

Two dimensions, where one law survives and the other turns round #

The dimension enters the two laws quite differently once the flow is planar, and the contrast is the sharpest test of the claim that the constant is geometry.

In two dimensions the scalar law reads $-2\varepsilon_\theta r$ and is otherwise unchanged, because a scalar still cascades to small scales in a plane: the variance has only one direction it can go, which is the point the scalar cascade essay rests on. The divergence argument runs identically and the constant moves from $4/3$ to $2$ for the arithmetic reason above.

The velocity’s law does not survive in the same form, because in two dimensions the energy goes the other way. Where the inverse cascade stops is the essay about that, and the consequence for the third-order law is a change of sign: the flux in separation space runs outward instead of inward, and the longitudinal law becomes $+\tfrac32\varepsilon r$ in the inverse-cascade range — a result quoted here rather than computed, since the projection relation used above is the three-dimensional one and this module refuses to apply it elsewhere rather than applying it and hoping.

So a plane has one exact law that is unchanged in character and one that has reversed, and the scalar’s is the robust one. For a subject whose exact results are as scarce as this one’s, that is a reason to take Yaglom’s law more seriously than its relative obscurity suggests, rather than less — and it is one more place where a scalar is the tractable half of a turbulent problem.

What a second exact law is actually good for #

An exact result is worth having in proportion to what can be done with it, and Yaglom’s law can do two things the velocity law cannot.

The first is that it measures a dissipation rate. $\varepsilon_\theta$ is the rate at which molecular diffusion destroys scalar variance, and measuring it directly means resolving $|\nabla\theta|^2$ down to the Batchelor scale — micrometres in water, and a measurement of a squared gradient, which is the most resolution-hungry quantity in the subject. Yaglom’s law replaces that with a third moment measured in the inertial-convective range, at separations orders of magnitude larger, where nothing needs resolving — the same saving that makes a grid nobody can build affordable to measure around rather than to simulate. The same trick is what makes the four-fifths law the preferred way of measuring $\varepsilon$, and it transfers to the scalar unchanged.

The second is that the ratio of the two laws’ outputs is a quantity that models cannot derive. Every closure carrying a scalar needs the ratio of the scalar’s destruction time to the velocity’s, usually written as a constant near a half and fitted to whatever flow the model is being calibrated on. Both numerator and denominator of that ratio are available, in the same flow at the same time, from two exact laws with no closure between them. That the fitted constant has survived fifty years without being measured that way says more about how hard the mixed moment is to measure than about the value.

Which is the honest limitation of both. The quantities are odd moments — third moments of differences — and an odd moment is a small residual between two large cancelling contributions. The second-order structure function converges in a few thousand samples; the third-order one needs hundreds of thousands, and the mixed one $\langle\delta u_L(\delta\theta)^2\rangle$ needs two simultaneous probes whose separation is known to better than the separation being measured. That is why the law is nearly a century old and still not a routine instrument, and why the argument of this essay — that its constant is the same constant as the famous one — is worth making: a reader who has learned that 4/5 is the exact law and 4/3 is some other thing has one fewer reason to reach for the second.

What this derivation cannot supply #

Neither balance equation is derived here. The statement that the nonlinear term appears as a divergence in separation space with a constant sink is taken as given for both fields, and it is the part of each derivation that contains the physics. What is computed on this page is what an isotropic radial field does to such a statement, which is why every claim above is about geometry.

Nothing here is measured against a turbulent flow. There is no field, no simulation and no data. The quadratures check closed forms against other closed forms, which is worth what it is worth: it establishes that the arithmetic relating the two fractions is right, and it establishes nothing about whether either law is observed.

The isotropy is exact and no real flow’s is. Both the divergence argument and the projection relation assume it, and the projection assumes incompressibility as well. A flow with residual anisotropy at the separations being measured satisfies neither exactly, and the size of that failure is the subject of the essay above rather than of this one.

And the range figures assume one scale each. The integral scale is put in as one metre and the dissipation rate as one watt per kilogram, so the decade counts are illustrative of the difference between the two ranges rather than a statement about any particular apparatus. What survives the choice is the shape: the scalar loses room below $Sc = 1$ and gains none above it.

Still open: whether the two laws can be measured against each other #

The two exact results hold in the same flow at the same time, and combining them has a use neither has alone.

The velocity law turns a measured third-order structure function into a value of $\varepsilon$ with no closure in it; Yaglom’s law turns a measured mixed moment into a value of $\varepsilon_\theta$ the same way. Their ratio is $\varepsilon_\theta/\varepsilon$, which is the inverse of the time scale ratio that every scalar-variance closure has to assume and that none of them can derive — the quantity usually written as a constant near 0.5 and fitted. Measured through the two exact laws it would not be fitted at all. Whether the mixed moment can be measured well enough for that at the Reynolds numbers where both ranges exist, and whether the two determinations of the ratio agree with the fitted value, is a calculation with real consequences for every model that carries a scalar.

Beside it is the question the projection raises. The relation between longitudinal and full structure functions is exact, kinematic, and testable without any appeal to Kolmogorov — it needs only isotropy and incompressibility. Applying it at second order rather than third gives a relation between the longitudinal and transverse structure functions which measurements are known not to obey, and asking what that disagreement measures is the next calculation.