What viscosity cannot take away

A smoke ring, a bath vortex and the trailing vortex behind an airliner all end the same way: the sharp core softens, the spin at the centre drops, and eventually nothing looks like a vortex any more. The obvious reading is that viscosity has eaten the circulation.

It has not. There is an exact solution for a vortex left alone in a viscous fluid, and what it says is that the circulation is the one thing viscosity cannot touch.

The equation has no nonlinearity in it #

The vorticity equation is generally awful. It carries a stretching term that is the whole of three-dimensional turbulence and the reason a vortex can intensify without any external help, and a convection term that makes it nonlinear.

For a vortex with circular symmetry, both vanish identically. The flow is purely azimuthal and the vorticity depends only on the radius, so the velocity is everywhere perpendicular to the gradient it would have to convect; the flow is two-dimensional, so there is no stretching. What is left is

$$\frac{\partial\omega}{\partial t} = \nu\left(\frac{\partial^2\omega}{\partial r^2} + \frac{1}{r}\frac{\partial\omega}{\partial r}\right),$$

which is the heat equation in the plane. Starting from all the vorticity on a line, the answer is the one heat conduction gives — a spreading Gaussian:

$$\omega(r,t) = \frac{\Gamma}{4\pi\nu t}\,e^{-r^2/4\nu t}, \qquad u_\theta(r,t) = \frac{\Gamma}{2\pi r}\left(1 - e^{-r^2/4\nu t}\right).$$

This is the sense in which vorticity is the right variable for a viscous flow. Written in terms of velocity and pressure the problem is a coupled nonlinear system; written in terms of vorticity it is a heat conduction problem a first-year course can solve.

What is conserved, and what only looks conserved #

That the total circulation is constant is not an accident of this solution; it is what a diffusion equation does. Integrate $\partial\omega/\partial t = \nu\nabla^2\omega$ over the whole plane, and the right-hand side becomes a flux through a boundary at infinity where the vorticity is exponentially small. Nothing crosses. The integral is a constant of the motion.

The relation to Kelvin’s theorem is worth being precise about, because it is easy to conclude too much. Kelvin’s theorem is about a material loop — a loop of marked fluid particles — and it fails in a viscous fluid: a material loop drawn in this flow is sheared into a spiral, and its circulation does change. What is constant here is the circulation round a fixed large circle, and the reason is that the vorticity flux out of it is negligible rather than that any theorem forbids it. The two statements are different, and the second is the one this solution supports.

The core, and the number 2.2418 #

The similarity variable is $r/\sqrt{4\nu t}$ and everything collapses onto it.

Inside that radius the fluid turns nearly as a solid body; outside it, nearly as a free vortex. That two-region structure is what every vortex model in engineering is an approximation to — Rankine’s combined vortex is precisely this shape with the corner left sharp — and the exact solution says the transition happens at a radius that grows as the square root of time and at no particular speed.

Some numbers for the vortex drawn here, of circulation 1 m²/s in air. At a sixteenth of a second the core is 1.9 mm and the peak swirl is 52 m/s. At four seconds the core is 15.5 mm and the peak swirl is 6.6 m/s. And at a radius of 100 mm the swirl is 1.592 m/s at both times, and at every time in between, because that is Γ/2πr and the core has not reached there.

Three quantities fall and one does not #

The enstrophy — the integral of the square of the vorticity — is the one that connects to dissipation. In two dimensions the rate at which a flow dissipates energy is proportional to the enstrophy, so this vortex is losing energy the whole time, at a rate falling as $1/t$.

Where does that energy go, given that the circulation is fixed? Into heat, at the expense of the kinetic energy of the core. And here is an honest oddity: the total kinetic energy of a two-dimensional vortex is logarithmically divergent, because $\int (\Gamma/2\pi r)^2 2\pi r\,dr$ diverges at large r. So “the energy of the vortex” is not a finite number unless a boundary is specified, and nothing in this essay computes one. The dissipation rate is finite and the energy it comes out of is not, which is a genuine feature of two-dimensional vortex flows and not an artefact of this solution.

Why big vortices do not die of viscosity #

That figure settles a question this site has met before. An airliner’s wake vortices have cores of order a metre, and the diffusion time for a metre in air is four and a half hours. Aircraft separation is measured in minutes, so viscosity is not what disposes of them.

What does dispose of them is an instability. A pair of counter-rotating vortices is unstable to a long-wavelength sinusoidal disturbance — the Crow instability — which grows until the two tubes touch and reconnect into a chain of rings, and that takes tens of seconds rather than hours. This site does not compute it, and the honest statement is that the decay of a real wake vortex is a stability problem that the exact diffusion solution has almost nothing to say about, except to rule itself out.

The same arithmetic explains the other end. A vortex a tenth of a millimetre across in air diffuses in under a fifth of a millisecond, which is why the smallest structures in a turbulent flow do disappear by viscosity and why the Kolmogorov scale is where the cascade stops.

The impostors this test rejects #

This is the sharpest version on the site of a point that keeps recurring: conserving the right quantity is not the same as solving the equations. A core spreading as $t^{0.4}$ has the right circulation at every instant, the right shape, a plausible decay, and a residual two hundred thousand times larger than the solution’s. Nothing in a plot of it would look wrong.

The two routes to the vorticity are the other half of the check. The closed form gives $\omega$ directly; the velocity field the same solution supplies gives it again through $(1/r)\,d(ru_\theta)/dr$, differenced numerically. They must agree to a millionth of the peak, and they do — which catches the kind of error that would otherwise produce a velocity field and a vorticity field that are each individually reasonable and are not each other’s.

The Reynolds number that decides nothing #

Every other flow on this site is a story about a dimensionless number: whether it separates, whether it chokes, whether it goes turbulent, is decided by one group and the whole site is organised around that fact. A vortex has such a number — the vortex Reynolds number Γ/ν, which for the one drawn here is 67,000 — and it decides nothing whatever about this solution.

The reason is the one from the first section. Doubling the circulation doubles every velocity and every vorticity in the field, and the equation is linear, so the solution simply scales. There is no regime boundary, no critical value, no change of behaviour: the Lamb–Oseen vortex is exact at Γ/ν = 1 and at Γ/ν = 10⁷ alike.

What Γ/ν does decide is something outside the solution: whether the flow stays axisymmetric. A vortex at a large Γ/ν is easy to disturb, and the disturbances have nothing to damp them; the solution remains an exact solution and stops being the observed one. That is the same distinction the plane jet forces and it is worth naming, because a solution can fail to describe a flow in two quite different ways — by being wrong, or by being unstable, and only the first is a defect of the arithmetic.

Two vortices, and where this stops being the answer #

The solution is exact for one vortex alone in an infinite fluid, and that phrase is doing more work than it looks.

The vorticity equation is linear for this geometry only. Put two of these vortices side by side and the convective term no longer vanishes: each sits in the other’s flow, each is carried by it, and a pair moves in the way point vortices do. Worse, each is strained by the other, so the circular core becomes an ellipse and the diffusion problem is no longer one-dimensional.

Nothing on this site solves that. What can be said quantitatively is when it matters: the strain a vortex feels from a companion at distance d is of order Γ/2πd², and the core resists it as long as its own turnover is faster, which for a core of radius a means Γ/a² ≫ Γ/d², or a ≪ d. So the single-vortex solution holds as long as the core is small compared with the spacing — and since the core grows as √t while the spacing does not, every vortex pair eventually leaves the regime in which the exact solution applies.

That is a general feature of similarity solutions rather than a fault of this one. A solution whose only length grows without bound will always reach whatever fixed length its surroundings supply, and what happens then is a different problem.

What a measurement of a vortex actually reports #

The gap between the local and global measures has a practical edge, because instruments measure one and theory usually wants the other.

A probe traversed through a vortex measures the velocity profile and reports a peak swirl and a core radius. Both are local, both fall as the vortex ages, and both depend on when the measurement was made. The circulation — the quantity that is conserved, the quantity that appears in Kutta–Joukowski, the quantity a wake-vortex hazard depends on — is the integral of that profile, and reconstructing it from a traverse requires the tail as well as the peak.

That is a difficult measurement for exactly the reason this essay is about. The velocity in the tail is small, it falls as 1/r, and the circulation converges slowly: integrating out to two core radii captures 86 per cent of it and out to three, 99. A traverse that stops at the peak reports a vortex that is weakening when what it has measured is a vortex that is spreading.

The number that matters is the hardest one to see, which is the practical form of the essay’s whole argument.

What the model does not contain #

No stretching, which is the whole of the three-dimensional problem. This solution exists because the stretching term vanishes for a straight axisymmetric vortex. Tilt or curve the vortex and it does not, and the balance between stretching and diffusion produces a steady core rather than a spreading one — which is Burgers’ vortex, and the site’s own account of vortex intensification.

No axial flow. Real trailing vortices and tornadoes have strong flow along the axis, which carries its own instabilities and can reverse — vortex breakdown — and none of that is here.

No boundaries. The solution lives in an infinite fluid. A vortex near a wall generates its own image, moves, and diffuses vorticity of the opposite sign into itself from the boundary layer the wall grows, which is a different problem entirely.

The initial condition is a singularity. At t = 0 the vorticity is infinite on a line and the energy is infinite everywhere. The solution is exact for t > 0 and describes any real vortex only after enough time has passed for the initial core structure to be forgotten — the same virtual-origin caveat that a jet’s similarity solution carries.

Constant density and constant viscosity. A vortex in a stratified fluid or in a gas with strong temperature gradients in its core obeys a different equation.

Why a solution that starts from a singularity describes real vortices #

The last item in the list of limitations — that the initial condition is a line of infinite vorticity, which nothing physical is — reads as a serious objection and turns out to be the reason the solution matters. The argument is worth making, because it converts a special case into a statement about everything.

The equation being solved is linear, so its solution from a point source is a Green’s function: the diffusion of any initial vorticity distribution whatever is the superposition of Lamb–Oseen vortices, one centred on each element of the initial vorticity. That is the ordinary property of the heat kernel, and it means the solution above is not one flow among many but the building block all the others are made of.

Follow the superposition to late times and something stronger appears. Expand any compact initial patch in moments: its total circulation, its centroid, its second moment, and so on. The leading term of the diffused field is the total circulation multiplied by the Gaussian above, centred on the vorticity centroid. Every higher moment contributes a correction that decays by an extra power of $1/\sqrt{t}$ relative to it — because each carries an extra length that the spreading core outgrows.

So any compact patch of vorticity becomes a Lamb–Oseen vortex. An elliptical patch, a ragged blob, a ring of small vortices, the messy core a real aircraft or a real stirrer produces: each of them forgets its own shape and approaches this profile, with the total circulation it started with and nothing else surviving from its initial condition. The solution is an attractor, and the singularity it starts from is a convenience rather than a claim about how anything began.

That is the honest reply to the objection, and it also identifies what the solution can be trusted about. It says nothing about a young vortex, whose shape is still its own; it says everything about an old one, and it says the same thing for every old one, because they have all lost the same information. The circulation is what a vortex remembers about its birth and the rest is forgotten at a computable rate.

There is a remarkable strengthening of that statement which is worth naming even though it belongs to analysis rather than to this collection. The argument above is linear, and the full two-dimensional Navier–Stokes equations are not: a real vortex convects itself and strains itself. Gallay and Wayne proved in 2005 that the conclusion survives anyway — for any initial vorticity distribution of finite total circulation, the two-dimensional Navier–Stokes solution converges to the Oseen vortex with that circulation, at every Reynolds number, with no smallness assumption anywhere.

That is one of very few global results in nonlinear fluid dynamics, and it says the essay’s central claim in its strongest available form. In two dimensions, an isolated patch of vorticity has exactly one long-time future, it is the one drawn here, and which one it is depends on a single number that viscosity was never able to change.

Who found it, and when #

Oseen published the solution in 1912 and Horace Lamb gave it in his Hydrodynamics, which is why it carries both names and is sometimes called simply the Oseen vortex. It is the two-dimensional heat kernel wearing a fluid-mechanical hat, and Fourier had the mathematics of it in 1822.

The interesting history is what the solution replaced. The line vortex of classical hydrodynamics has an infinite velocity at its centre and an infinite energy, and nineteenth-century treatments dealt with that by simply excluding a small circle around the axis and saying nothing about what was inside. Oseen’s solution says what is inside, and it says that the exclusion was harmless: everything outside the core is exactly what the singular theory said it was.

That is the general shape of what viscosity does to an inviscid theory on this site. It does not overturn the outer answer; it supplies the inner region the outer answer could not have, and it leaves the far field alone.

Where the ladder goes next #

Every viscous exact solution so far has been in two dimensions or in a plane. The next question is what happens when the third dimension is added rather than removed — a body of revolution in an ideal stream — and the answer is that three dimensions are gentler than two in every measure: a sphere disturbs the flow less than a cylinder, and it cannot carry circulation at all.