The energy a vortex cannot have

An earlier essay in this ladder established the famous half of what a diffusing vortex does. Every local measure of it falls — the peak spin, the peak velocity, the enstrophy — and the circulation round a large loop does not move at all, ever, to a part in a billion across four decades of time.

This is the other half of the ledger, and the two are in tension in a way that is worth setting out carefully.

Why the energy diverges #

Outside the core the vortex is irrotational and its swirl is $\Gamma/2\pi r$. So the kinetic energy density falls as $1/r^2$ and the area of an annulus grows as $r$, and the integrand of the energy is

$$\tfrac12\rho\left(\frac{\Gamma}{2\pi r}\right)^2 2\pi r\,dr = \frac{\rho\Gamma^2}{4\pi}\frac{dr}{r},$$

which integrates to a logarithm. Every octave of radius contributes the same amount as every other.

There is no such thing as the energy of a line vortex. Not “it is very large”; there is no number, and quoting one requires stating a cutoff, and no cutoff is physical. Aerodynamics quietly supplies one — the span of the wing, the size of the room, the distance to the next vortex — and the answer depends on which is chosen.

That is a strange property for a flow to have, and it is worth comparing with the circulation, which is the other integral quantity a vortex carries. The circulation converges: it reaches $\Gamma$ within a few core widths and stops. The energy does not converge at all. Two integrals of the same field, and one of them is a number and one of them is not.

And why the loss rate does not #

Now the part that resolves it. The total rate at which a Lamb–Oseen vortex destroys mechanical energy, per unit length, is

$$\varepsilon = \frac{\rho\Gamma^2}{8\pi t^2},$$

and there is no cutoff anywhere in it.

The absence of the viscosity from that expression deserves a note, because it is a small instance of a large idea. A thinner fluid spreads the core more slowly, so at a given time the core is smaller, the gradients in it are steeper, and the dissipation per unit volume is higher — by exactly enough to compensate for there being less of it and less viscosity to charge. The vortex loses the same energy per unit time regardless of the viscosity, which is the same structure as the dissipation anomaly in turbulence: the viscosity decides where the energy leaves and the large scales decide how much.

So the resolution of the paradox is that the rate is the physical quantity and the energy is not. Asking how much energy a vortex has is asking a question with no answer; asking how fast it is losing it has a very definite one.

Closing the books inside a circle #

The accounting is the phase’s own rule and it does not close in the obvious way.

Draw a circle of radius $R$ around the vortex. The energy inside it is falling. The dissipation inside it is positive. Those two are not equal, and the third term is the one this whole phase exists to insist on: the fluid outside is doing work on the fluid inside, across the circle, through the viscous stress.

$$\frac{dE(R)}{dt} = -\int_0^R \Phi\,dA \;+\; 2\pi R\,\tau_{r\theta}u_\theta\Big|_R.$$

Why ∫Φ is not μ∫ω² #

There is a widely used identity in two-dimensional flow: the total dissipation equals $\mu$ times the integral of the squared vorticity — the enstrophy. It is exact over the whole plane, it is genuinely useful because vorticity is usually the thing a two-dimensional calculation carries, and it is quoted over regions where it is false.

The vortex is the cleanest possible demonstration of the failure. Outside its core the fluid is irrotational: $\omega$ is exponentially small, so the enstrophy integrand is essentially zero. And it is being sheared — a circular flow with $u_\theta\propto 1/r$ has a rate of strain $-\Gamma/\pi r^2$ — so the dissipation integrand is not zero at all, and falls only as $r^{-4}$.

Over a circle of radius $R$ the two differ by exactly

$$\mu\int\omega^2\,dA \;-\; \int\Phi\,dA \;=\; \frac{\mu\Gamma^2}{\pi R^2},$$

which is the boundary term above, measured here and matching that closed form. A region with no vorticity in it can be destroying energy at any rate, and the enstrophy identity is a statement about a whole plane rather than about a place.

The identity, and how the difference was found #

The gap between the two integrals was not looked for. It turned up because this collection computes every dissipation twice, and the two numbers for a vortex refused to agree in the fourth digit.

The volume integral of Φ out to sixty core widths came to 3.97666 × 10⁻², and μ times the enstrophy integral over the same disc came to 3.97887 × 10⁻². Two parts in ten thousand apart, which is far too large to be quadrature at that resolution and far too small to be a mistake in the field.

The difference is μΓ²/πR², and at sixty core widths that is 2.21 × 10⁻⁵ — matching the gap to three figures. Working out what it was took the balance above: it is the boundary term, and the fact that the enstrophy integral has converged while the dissipation integral has not is exactly what makes them differ.

The general lesson is the one this collection keeps arriving at from different directions. Two quantities that are equal over all space are not equal over a region, and the difference is a flux through the region’s boundary. That is true of the dissipation and the enstrophy, of the drag power and the heat, and of the energy and the work — and in every case the term that is dropped is the one that makes the accounting non-local.

What the vortex is actually conserving #

Three quantities, and the contrast between them is the essay’s point.

Circulation is conserved absolutely, round any loop large enough to contain the core. That is Kelvin’s theorem surviving viscosity, and it is the subject of the essay this one is a rung above.

Enstrophy falls, as $1/t$, and it is the quantity that measures how concentrated the vorticity is rather than how much there is. Vorticity is being spread, not destroyed.

Energy is undefined and its rate of loss is exact. Which is the odd one out and is the reason this rung exists.

There is a way to see all three at once. Vorticity in two dimensions obeys a pure diffusion equation, so its integral is conserved and its concentration falls. Circulation is that integral. Enstrophy is its second moment, and diffusion always reduces a second moment. Energy is a different functional entirely — it involves the velocity, which is a non-local integral of the vorticity — and non-local functionals of a diffusing field need not converge.

The consequence for anything that trails a vortex #

An aircraft leaves a pair of them, and the practical question is how long they last. The answer “until their energy is dissipated” is unusable, because the energy is not a number.

The usable statements are about the core. The peak swirl falls as $1/t$, the core radius grows as $\sqrt{4\nu t}$, and the danger to a following aircraft is set by the peak swirl. So the decay time of the hazard is a molecular-viscosity time — and for a wake vortex from a large aircraft that is hours, which is far longer than anything observed.

What actually destroys a wake vortex is not this mechanism at all. It is an instability that brings the two vortices together and lets them reconnect, and after that a turbulent breakdown that has no similarity solution. The laminar answer is a lower bound on the persistence and is wrong by two orders of magnitude, which is exactly the sort of thing a beautiful exact solution is good at concealing.

Two vortices, and where the energy becomes a number #

Since the pair is what makes the energy finite, it is worth doing the arithmetic, because the result is the induced drag of a wing and arrives here from an unexpected direction.

Two vortices of circulation $\pm\Gamma$ separated by $b$ have a far field that cancels to leading order: at a distance $r \gg b$ the two contributions differ by a gradient, so the velocity falls as $b/r^2$ rather than as $1/r$. The energy density then falls as $r^{-4}$, the area grows as $r$, and the integral converges.

What it converges to is

$$E = \frac{\rho\Gamma^2}{4\pi}\ln\frac{b}{a} + \text{a constant},$$

per unit length, with $a$ the core radius — so the logarithm has not gone away, it has been cut off by the separation. A pair’s energy is finite and depends logarithmically on how far apart they are and how fat their cores are.

That expression is the induced drag. A wing flying a metre leaves a metre of vortex pair behind it, and the energy it has to supply to make it is exactly this number, so the drag is $E$ per unit length. Where the reaction to a wing’s lift is gives the momentum version of the same statement, and the two must agree — which they do, and the agreement is a stronger check on both than either is alone.

So the induced drag of a wing depends on the core size of its tip vortices, logarithmically. That dependence is real, is usually buried inside an efficiency factor, and is the reason the theoretical minimum induced drag of an elliptic wing is a lower bound that a real wing approaches from above rather than a number it achieves.

What the picture cannot show #

The flow is two-dimensional and unbounded. A real vortex has ends, and the energy integral over a finite length is finite for that reason — the logarithm is cut off at the vortex’s own length rather than at infinity. That does not make the divergence go away; it moves it into a dependence on a length that has nothing to do with the vortex.

Nothing here stretches. A vortex that is being stretched gains enstrophy and gains energy, and the balance is completely different — that is Burgers’ vortex, where the diffusion is held in check by a strain and a steady state exists.

And the vortex is alone. Two vortices interact, and their combined field decays as $1/r^2$ rather than $1/r$, so a pair has finite energy per unit length. That is the reason aerodynamics can talk about the energy in a wake at all: an aircraft leaves two vortices of opposite sign, and their energy is a number.

Every cutoff is something else in the problem #

The pair supplies a cutoff, and it is worth noticing that it is not the only thing that can. Put a single vortex a height $h$ above a wall and its energy per unit length is finite too, because the wall’s image is a vortex of opposite sign at $-h$ and the combined far field cancels exactly as a pair’s does. The logarithm is then cut off at $2h$, and the vortex’s energy depends on how high it is flying.

That is the general shape of it. The logarithm is always terminated by something, and the something is never the vortex. A second vortex terminates it at their separation; a wall at twice the height; a container at its own size; a wing at its span. Which of them is doing the work is a fact about the surroundings, and a calculation that quotes “the energy of the vortex” has silently nominated one of them.

What is quoted instead, when something has to be #

There is a field that must attach a number to a trailing vortex several times a day, and its answer to this essay’s difficulty is worth having, because it is the honest one.

Wake-vortex separation standards need a measure of how dangerous a vortex is. The energy is undefined. The peak vorticity is a local maximum inside a core a few metres across, and no instrument mounted on the ground or flown through the wake resolves it — a lidar measures a velocity averaged over a range gate, and the core wanders, so a peak measured that way is systematically low by an amount nobody knows.

So the quantity that gets standardised is neither. It is a circulation averaged over a stated band of radii — conventionally from five to fifteen metres, written $\Gamma_{5\text{–}15}$ — chosen because circulation converges where energy does not, because an average over a band is insensitive to the wandering that ruins a peak, and because the band is wide enough to be measurable and narrow enough to still be about the vortex rather than about the atmosphere.

Every part of that definition is a convention and the convention is written down. That is exactly what makes it usable: two laboratories reporting $\Gamma_{5\text{–}15}$ are reporting the same thing, and two reporting “the vortex strength” are not.

Which is the practical moral of the whole essay. When a quantity a subject wants turns out not to exist, the useful response is not to compute it with an unstated cutoff — that produces numbers that disagree between groups for reasons nobody can find. It is to agree a bounded surrogate, state its definition in full, and quote that. The energy of a line vortex has no value; the circulation between five and fifteen metres has one, everybody’s agrees, and the difference between those two situations is entirely a matter of somebody having written the definition down.

What it means for a number this collection quotes #

There is a place where the divergence bites on a number already in this collection, and it is worth naming rather than leaving implicit.

The speed at which a vortex ring translates depends logarithmically on the ratio of its radius to its core size — that is the cutoff appearing in a different guise. A ring with a very thin core moves faster, without limit, as the core is thinned, and the reason is precisely the reason the energy diverges: the near field of a thin vortex contains an unbounded amount of everything.

That is why a ring’s speed has to be computed from a cutoff rather than from the filament alone, and why quoting one without stating the core model is meaningless. It is the same divergence, appearing as a velocity rather than as an energy, and it is resolved the same way — by admitting that a line vortex is a limit rather than an object.

The general shape of it: every quantity that is a non-local integral of a vortex’s field carries the logarithm, and every quantity that is a local one does not. Circulation, enstrophy and peak vorticity are all local and all perfectly well defined. Energy, self-induced velocity and impulse are non-local, and all three need a cutoff.

Who found it, and when #

Oseen gave the solution in 1912 and Lamb in the 1932 edition of his book, which is why it carries both names. The logarithmic divergence of a line vortex’s energy was known to Helmholtz and is usually handled by the device of a cutoff at a core radius, which appears in every calculation of a vortex ring’s speed and is computed properly in this collection’s essay on rings.

The surprising connection is with a problem in electrostatics that has the same structure and the same resolution. The energy of an isolated line charge diverges logarithmically for exactly the reason the vortex’s does — a field falling as $1/r$ integrated over an area growing as $r$ — and the energy of a pair of opposite line charges is finite. A single vortex and a single line charge are both objects whose energy does not exist and whose interactions are perfectly well defined, and in both subjects the practical rule is the same: never compute the energy of one, always compute the energy of the system.

Where the ladder goes next #

Below this rung is what viscosity cannot take away, which establishes the conserved circulation this essay contrasts against, and the price of a gradient, which supplies the integral being computed.

Beside it is the spin that feeds itself, which is the same vortex with a strain applied and therefore with a steady state, and circulation is vorticity added up, which is the theorem that makes the conserved quantity conserved.