The knot a flow cannot untie

Euler’s equations conserve several things and most of them are recognisable. Energy is energy. Circulation round a material loop is a circulation, and what survives being wound up tests it by advecting a loop until its shape is unrecognisable and finding the number unmoved. Impulse is a momentum with the divergence mended.

They also conserve

$$H = \int \mathbf{u}\cdot\boldsymbol\omega \, dV,$$

which is not recognisable at all. It has the dimensions of a velocity squared times a volume, it is a pseudoscalar — it changes sign under a reflection — and its value is a topological count.

What it counts #

For vorticity confined to thin tubes with circulations $\Gamma_i$ and no internal twist,

$$H = 2\sum_{i<j}\mathrm{Lk}(C_i, C_j)\,\Gamma_i\Gamma_j + \sum_i \mathrm{Wr}(C_i)\,\Gamma_i^2,$$

where $\mathrm{Lk}$ is the Gauss linking number of two closed curves and $\mathrm{Wr}$ the writhe of one.

So the helicity of two unlinked rings is zero however close they are brought, and the helicity of two linked rings is $2\Gamma_1\Gamma_2$ however far apart they are pulled. It is not a measure of how much swirl there is or how tangled the picture looks. It is a count.

That is the refutation this essay carries and it is worth testing on a drawing before trusting it.

The Gauss integral, which has no reason to be an integer #

The linking number is computed by a double integral over the two curves:

$$\mathrm{Lk} = \frac{1}{4\pi}\oint\oint \frac{(\mathbf{r}_1 - \mathbf{r}_2)\cdot(d\mathbf{r}_1\times d\mathbf{r}_2)}{|\mathbf{r}_1 - \mathbf{r}_2|^3}.$$

Nothing in that expression is constrained to be a whole number. It is a smooth kernel integrated over two smooth curves, and its value moves continuously as either curve is moved.

It comes out an integer anyway. Evaluated on six configurations with two hundred points per curve, it returns 0, 0, 1.0002, $-1.0002$, 1.0002 and 0, with the departure from an integer being the discretisation.

The last of the six is the instructive one. A unit ring and a ring of radius 3 centred a unit away, in a perpendicular plane, look thoroughly interlocked in any drawing — and their linking number is exactly zero. The second ring crosses the first’s disc at $x = -2$ and $x = 4$, both outside it. Move to radius 1.6 and the crossings are at $-0.6$ and 2.6, one in and one out, and the pair is linked.

A picture does not show a linking number. An integral does.

How to tell without integrating #

The test that decides is simple once stated, and it is worth having because the integral is expensive and the eye is unreliable.

Take one curve and any surface spanning it — for a circle, its own disc. Count the crossings of the other curve through that surface, with sign according to direction. The linking number is the signed count, and it does not depend on which spanning surface is chosen.

For the pairs above, the second ring pierces the unit disc at $x = 1 \pm r$. Both piercings inside means zero net crossings after signs cancel; both outside means zero crossings; one in and one out means one, which is a link. So the family is linked for $r$ between 0 and 2 and unlinked for $r$ above it, and the transition is at the value where the far crossing leaves the disc.

That is exactly the calculation circulation is vorticity added up makes for Stokes’ theorem — a loop, a spanning surface, and a count of what passes through — and the independence of the surface is the same independence.

Two routes to the same helicity #

The topological statement is a statement about curves. The physical statement is about a flow, and the two have to be connected by something more than an assertion.

So the helicity is computed a second way, from Biot–Savart. The velocity field of one filament is evaluated at points along the other and integrated round it, which gives the mutual circulation, and $H = 2\Gamma_1 \oint \mathbf{u}_2\cdot d\mathbf{l}_1$.

That is a completely different calculation — a line integral of a computed velocity field rather than a double integral of a geometric kernel — and for a Hopf link with circulations 1.7 and 2.3 the two give

$$7.821287 \quad\text{and}\quad 7.821287.$$

The agreement is worth the emphasis because of what it establishes. The Gauss integral knows nothing about fluids: it is a kernel from differential geometry, and its integrality is a theorem about curves. The Biot–Savart route knows nothing about topology: it computes a velocity field and integrates it round a loop.

They agree to the last printed digit, and the agreement is what makes the topological statement a statement about a flow rather than about a drawing of one.

Deformation moves the writhe and cannot move the linking #

The second half of the decomposition is where the topology becomes visible as a constraint rather than as a coincidence.

Give one ring a growing three-lobed wave and watch both quantities. The writhe — the same Gauss integral of a curve with itself — runs from 0 to $-1.794$ as the amplitude grows. The linking number with a fixed partner stays at 1.000, to five decimal places, at every amplitude.

That is the difference between a geometric quantity and a topological one. The writhe is a property of the curve’s shape in space and changes with it. The linking number cannot change without one curve passing through the other, and a flow conserving circulation cannot make that happen — because the vortex lines are material.

The bookkeeping is Călugăreanu’s: linking equals writhe plus twist, so what absorbs the change in writhe is a twist of the tube about its own axis. That is why a ribbon and its axis are different objects and why helicity needs “no internal twist” in its statement.

Why an integral of velocity times vorticity counts anything #

It is worth seeing why the integral has this property, because it looks like an accident.

For a single thin tube of circulation $\Gamma$ following a curve $C$, the volume integral collapses to $\Gamma$ times the circulation of the velocity round $C$ — because the vorticity is concentrated on the tube, so $\int\mathbf{u}\cdot\boldsymbol\omega\,dV$ becomes $\Gamma\oint\mathbf{u}\cdot d\mathbf{l}$.

And the circulation round $C$ is, by Stokes’ theorem, the flux of vorticity through a surface spanning $C$ — which is $\Gamma_2$ times the number of times the other tube passes through it.

So the chain is: helicity, to circulation, to flux, to a crossing count. Each step is a standard identity and the last one is where the integer comes from. Nothing about it is an accident; it is Stokes’ theorem applied to a field that happens to be a bundle of curves.

Why it is conserved, in one sentence #

The proof is short enough to summarise and it explains the hypotheses.

Vortex lines are material lines in an ideal barotropic flow — that is Helmholtz’s theorem, and it follows from Kelvin’s, which what survives being wound up tests. A deformation that carries lines with the fluid cannot change how many times one bundle passes through another, because changing that requires cutting. So the count is conserved, and the count is the helicity.

Every hypothesis in that sentence is one of Kelvin’s. Break any of them — add viscosity, add baroclinicity, add a non-conservative body force — and vortex lines stop being material, reconnection becomes possible, and the helicity can change.

The flow that has as much of it as it is allowed #

There is an upper bound on helicity for a given energy and enstrophy, and there is a class of flows that attains it.

A Beltrami flow has $\boldsymbol\omega = \lambda\mathbf{u}$ everywhere: the vorticity is parallel to the velocity at every point. Its helicity density is then $\lambda|\mathbf{u}|^2$, of one sign, with no cancellation anywhere, and the relative helicity $H/\sqrt{E\Omega}$ is exactly one.

The ABC flow is the standard example. Computed on a $40^3$ grid its curl differs from itself by $1.4\times10^{-10}$ — which is the check that it is Beltrami rather than an assertion — and its relative helicity is 1.00000000.

A plane shear layer gives exactly zero, and it has to: a mirror-symmetric flow has no helicity, because helicity is a pseudoscalar and a reflection changes its sign while leaving the flow unchanged.

Inviscid does not mean irrotational is where this collection first meets Beltrami flows, as the family of steady rotational solutions of Euler’s equations. Their maximal helicity is the reason they are steady: a flow whose vorticity is parallel to its velocity has no Lamb vector, so the nonlinear term is a gradient and can be absorbed into the pressure.

What it is an obstruction to #

Helicity has a second life as an obstruction, and it connects directly to a neighbouring essay.

One number that runs out at three dimensions establishes that a plane flow is described by one scalar and a three-dimensional one is not; two are needed, and the natural pair is the Clebsch representation $\mathbf{u} = \nabla\chi\times\nabla\psi$, which exists locally.

A flow with a Clebsch representation has zero helicity, identically. So a flow with non-zero helicity has no global Clebsch representation, and the helicity integral is exactly the obstruction to finding one.

That is a satisfying place for a topological invariant to appear. The question “can this flow be described by two scalar fields?” has an answer, the answer is a number, and the number is a linking count.

What a flow with helicity looks like #

It is worth attaching the count to a picture, because “linked vortex lines” is easy to say and hard to see.

In a Beltrami flow the vortex lines are the streamlines, so a fluid particle travelling along a streamline is travelling along a vortex line. In the ABC flow those lines are chaotic over most of the volume — steady three-dimensional and mixing anyway is the essay about that, where a steady flow with no randomness in it mixes because its streamlines are chaotic — so the lines wander through the box for ever, threading through one another endlessly.

That is what maximal helicity looks like: not a tidy pair of linked rings, but a tangle whose linking is so thorough that the integral is as large as the energy and enstrophy permit.

At the other extreme, a plane shear layer’s vortex lines are straight and parallel. Nothing threads through anything, and the count is zero.

Most real flows are in between and their helicity is a number rather than an integer, because the vorticity is a continuous field rather than a set of tubes. The integer statement is exact for tubes and approximate for everything else, in the same way that a circulation round a loop is exact and a “vortex” is a matter of judgement — which is what where a vortex stops is about.

Where it fails to be conserved, which is where it matters #

The limit worth taking here is $\nu \to 0$, and helicity behaves like the energy does — which is to say, badly.

$$\frac{dH}{dt} = -2\nu\int\boldsymbol\omega\cdot(\nabla\times\boldsymbol\omega)\,dV.$$

The prefactor vanishes as $\nu \to 0$ and the integral grows, exactly as it does for the dissipation of energy in the limit that is not the value, where the gradient rises by precisely the factor the viscosity falls by and the product stands still across six decades.

Whether the helicity’s product also stands still — the helicity anomaly — is a real question with evidence on both sides, and nothing here settles it. What can be said is the structural point: an invariant that is conserved at every finite viscosity, and whose rate of change is a small number times a large one, is exactly the kind of invariant that can fail to survive its own limit.

Physically the failure is reconnection: two vortex tubes touch, viscosity lets the lines cut and rejoin, and the linking number changes by an integer. That is a discrete event, it takes a time that shrinks with the viscosity, and it is not available at $\nu = 0$ at all.

Why an integer invariant is worth having #

The practical value of a topological invariant is not that it is exact — many things are exact — but that it is rigid.

A conserved quantity taking continuous values can be traded, redistributed and moved about, and knowing its total constrains a flow only weakly. A conserved quantity taking integer values cannot change at all without something discrete happening, so knowing it forbids whole classes of evolution.

That is why helicity is used in the places it is used: as a constraint on relaxation problems in magnetohydrodynamics, where the same integral is conserved for magnetic field lines and where the minimum-energy state at fixed helicity is a Beltrami field; and as a diagnostic in atmospheric flows, where the storm-relative helicity of a wind profile is a predictor of rotation in a thunderstorm.

The atmospheric use is worth a caveat: a storm-relative helicity is an integral over a limited volume with an arbitrary reference frame, so it is not the topological invariant discussed here. It is a correlation between shear and updraught wearing the same name.

What it does not measure, restated #

The refutation at the top is worth repeating in the light of everything above, because the misreading is natural.

Helicity is not a measure of swirl. A single straight vortex tube of any strength has zero helicity. A single ring has zero helicity. A pair of parallel tubes, however strong, has zero helicity. What is needed is one bundle of lines passing through another, and that is a condition on the arrangement rather than on the intensity.

Nor is it a measure of complexity. Two rings can be brought into a configuration that is visually as complicated as anybody likes and still have a linking number of zero, as the radius-3 case above shows.

And it is not positive definite. A left-handed tangle and a right-handed one cancel, so a flow with enormous amounts of linking of both signs has a helicity near zero — which is one reason the quantity is less used as a diagnostic than its elegance would suggest.

Limits recorded rather than smoothed over #

Everything here is filaments. The decomposition $H = 2\mathrm{Lk}\Gamma_1\Gamma_2 + \mathrm{Wr}\Gamma^2$ is for thin tubes with no internal twist. A real vortex has a finite core and a twist distribution inside it, and the twist contributes.

Conservation is not tested here. The linking number’s invariance under a geometric deformation is computed; its invariance under a flow is not, which would need advecting two linked tubes through a solution of Euler’s equations. That would be the direct test and it is not attempted.

The Beltrami calculation is a quadrature. The ABC flow’s helicity, energy and enstrophy are integrals over a periodic box, computed on a $40^3$ grid with the curl by finite differences. The relative helicity being exactly one follows algebraically from $\boldsymbol\omega = \lambda\mathbf{u}$, so what the quadrature checks is that the field is what it claims to be.

And the helicity anomaly is stated rather than computed. It needs a turbulent field, which this site’s grid cannot resolve, and the site’s rule is to say so rather than to draw one.

Where else the same integral appears #

A closing note on reach, because this quantity is not confined to fluid mechanics and the reason is structural.

The same integral, with the magnetic field in place of the velocity and the vector potential in place of the velocity, is the magnetic helicity, and it is conserved in ideal magnetohydrodynamics for exactly the same reason: field lines are frozen into the fluid, so their linking cannot change without reconnection.

That conservation is the basis of Taylor’s relaxation theory, which predicts that a turbulent plasma allowed to relax at fixed helicity settles into a Beltrami state — the same maximal-helicity field this essay computes for the ABC flow. It is one of the more successful predictions in laboratory plasma physics and it rests on the integer being rigid.

Fluid helicity has no comparable success to point to, and the reason is worth being honest about: a fluid’s vortex lines are frozen only in an ideal barotropic flow, whereas a plasma’s field lines are frozen under much weaker conditions. The invariant is the same and its grip on the physics is not.

The residue #

The limits here are geometric ones, and they behave unusually.

Pull two linked rings apart: every geometric quantity in the problem goes to zero — the mutual induced velocity, the interaction energy, the force each exerts on the other — and the helicity does not move at all.

Push two unlinked rings together: every geometric quantity diverges, and the helicity is still zero.

What survives is not a residue of anything that was removed. It is a quantity that was never a continuous function of the configuration in the first place, and that is the whole reason to have it.