A vortex pair carries eddies no snapshot can see

The picture belongs to whoever is watching shows that streamlines and the patterns they draw depend on the frame they are drawn in, and every snapshot says vortex, and every particle leaves builds the sharpest example of what that costs: a rotating saddle in which the velocity gradient has complex eigenvalues at every point and every instant — a vortex by every criterion that reads a snapshot — while every fluid particle is flung away. The snapshot criteria saw a vortex where there was none.

That essay ends on a flow that is not uniform and is not built to trap anything: the neighbourhood of two co-rotating point vortices, whose strain turns at the pair’s orbital rate. It asks what the snapshot criteria report there and what the fluid actually does. This essay computes both and finds the opposite failure. Round a co-rotating pair the criteria miss two large eddies that go round with the pair for ever, and they miss them for the plainest possible reason: the eddies have no vorticity.

The pair and its frame #

Two equal vortices, each of circulation $\Gamma$, two units apart, move each other round their midpoint at a steady rate $\Omega = \Gamma/8\pi$, as vortices do. Here $\Gamma = 2\pi$, so each vortex induces a speed of one over its distance, and the pair turns at a rate of one half, completing an orbit in $4\pi$. Each vortex has a small Rankine core of radius 0.2, a disc of uniform vorticity, so that the velocity is finite everywhere; outside the cores, which is nearly everywhere, the flow is exactly that of two point vortices.

In the fixed frame nothing is steady: the vortices move, and the streamlines of each instant are a snapshot of a pattern that is turning. In the frame that turns with the pair, the flow is steady. Its stream function is the lab-frame one plus the rotation’s,

$$\psi_{\text{rot}} = -\ln r_1 - \ln r_2 + \tfrac14 r^2,$$

with $r_1$, $r_2$ the distances to the two vortices and $r$ the distance from the centre. Fluid in that frame moves along the contours of $\psi_{\text{rot}}$, and every contour closes — so in the turning frame, every particle goes round a closed loop for ever.

Five points where the turning fluid stands still #

The first figure draws the turning frame, and its structure is set by the points where the turning fluid stands still. On the line through the vortices there are three: a saddle at the centre, and two saddles further out, where the pair’s induced speed exactly matches the frame’s rotation. On the axis the vortices induce $2x/(x^2 - 1)$ and the frame moves at $x/2$, so the outer saddles are at $x^2 = 5$ — at $\sqrt 5$ half-separations, which the calculation finds to within rounding.

Off the line there are two more, and they are not saddles. At $(0, \pm\sqrt3)$ the stream function has two minima. These points are exactly two half-separations from each vortex, so each makes an equilateral triangle with the pair. Round each minimum the turning fluid circulates in closed loops: an eddy, bounded by the streamline through the two outer saddles, which the figure shows spreading above and below the pair across most of its width.

So the turning frame has four regions. Round each vortex there is a lobe, bounded by a figure-eight through the centre saddle. Round both lobes there is a band. Above and below are the two eddies. And outside everything is fluid that the pair leaves behind.

What the snapshot sees #

The second figure is what every criterion built from the velocity gradient reports. The shading is $Q$, the second invariant of the gradient — half the difference between the squared rotation rate and the squared strain rate — which is positive where rotation beats strain and is the most widely used test for a vortex; $\lambda_2$, the swirling strength and the Okubo–Weiss parameter are close relatives, and in two dimensions they all agree on its sign. $Q$ is positive only inside the two cores. Outside them the flow is irrotational: the vorticity is exactly zero, the gradient is pure strain, and $Q$ is negative everywhere — its largest value outside the cores, on a grid of four hundred points, is −0.028. The instantaneous streamlines show two small circulations and a set of closed curves round both, the shape a snapshot of any vortex pair has.

By every snapshot criterion, then, the pair is two small vortices of area 0.25 — the cores’ — in a straining flow, and there is nothing else there. The eddies of the first figure are invisible to all of them, because the fluid in them carries no vorticity: it is irrotational fluid that happens to go round with the pair.

What the fluid does #

The third figure asks the fluid. Three small blobs are released — one at the centre of an eddy, one in the band close to the centre, one outside — and followed in the fixed frame for three orbits of the pair, then drawn in the turning frame. The blob released in the eddy is still a compact patch within 0.32 of where it started: in the fixed frame it has gone round the centre three times, exactly with the pair. The blob in the band has been drawn out along the band into a thin arc round both lobes. The blob released outside has been sheared into an arc that has fallen behind the pair, spread over 6.2 half-separations.

The fourth figure makes that quantitative, for fluid starting anywhere on the line through the two eddies. The mean rate at which it goes round the pair’s centre, as a multiple of the pair’s own rate, falls into three plateaux. Close to the centre, in the band, fluid circles both vortices in the same sense as they turn and overtakes the pair: its rate peaks at 2.07. From 0.455 to 2.904 half-separations out — the eddy, from its inner edge to its outer — the rate is 1.000 to three figures: the fluid goes round with the pair exactly. Beyond 2.904 it is left behind, at 0.39 of the pair’s rate by 3.3.

The fluid round each vortex, in the lobes, also goes round at exactly the pair’s rate, as it must, since it is carried by its vortex. So the fluid that travels with the pair is the two lobes and the two eddies. The lobes contain the only vorticity in the flow. The eddies contain none.

What goes on inside an eddy #

The eddies are not solid lumps carried round with the pair. In the turning frame their fluid circulates round the ghost point, clockwise — against the pair’s own sense of turning — and the rate can be read off the stream function. At the ghost point its curvatures along and across the line through the vortices are exactly $\tfrac14$ and $\tfrac34$, with no cross term, so small loops round it are ellipses traced at an angular rate $\sqrt{\tfrac14\cdot\tfrac34} = \sqrt3/4$. A parcel near the centre of an eddy therefore goes round it once every $8\pi/\sqrt3 = 14.5$ time units, a little longer than the pair’s own orbit of $4\pi = 12.6$. Loops further out take longer — 16.0 for a parcel starting 2.5 half-separations from the centre, 24.8 at 2.88 — and the time grows without limit at the eddy’s edge, where the loop runs into the outer saddles and the fluid lingers.

How can fluid with no vorticity go round in loops? Because the loops are in the turning frame. A frame turning at $\Omega$ adds a uniform vorticity of $-2\Omega$ to everything it sees, so irrotational fluid, seen from the pair’s frame, has a vorticity of $-1$ and turns backwards. Watched from the fixed frame, a parcel in an eddy is carried once round the centre per orbit of the pair without spinning — like a car on a Ferris wheel, which goes round with the wheel and stays upright. The eddy is a statement about where the fluid goes, not about how it spins, and that is the whole of why a criterion that measures spin cannot find it.

The frame the flow picks out #

It would be easy to take the lesson as “use the turning frame”, and for this flow that is right: it is the one frame in which the flow is steady, and steadiness is what makes its streamlines the paths of the fluid. The frame is not an arbitrary choice made by the observer. It is picked out by the flow itself, as the picture that belongs to the watcher finds for a body moving through still air: there is one frame in which the pattern stands still, and in it the streamlines mean something.

Most flows have no such frame. A pair of unequal vortices, or three, or a pair near a wall, never looks steady from any frame, and the eddies they carry — if they carry any — have to be found by following the fluid, as the blobs and the rates do here. That is the reason definitions built on fluid trajectories over a stretch of time exist at all: they reduce, for a flow with a steady frame, to reading that frame’s streamlines, and they still work when no frame will do.

How the eddies could be seen #

Dye released into a laboratory vortex pair from above, or tracer particles photographed in a sequence, would show the eddies as the blue blob does: a patch that travels round with the vortices without being wound into them. A particle-image velocimetry measurement of the instantaneous velocity would not, however it was processed into $Q$ or $\lambda_2$, because what it measures is the gradient at an instant. The measurement has to follow the fluid, or be transformed into the turning frame before its streamlines are drawn. The comparison is a clean test of any method that claims to find coherent structures: this flow has two, of known shape and size, in a place where the velocity gradient is pure strain.

Seventy times the area the criteria see #

The fifth figure compares the two answers by area. The eddies together cover 17.5 square half-separations; the lobes round the vortices 3.6; the region the snapshot calls vortex, 0.25, which is the area of the two cores to within the counting grid. The eddies are seventy times as large as what the criteria see. And the eddies do not depend on the cores at all. They are set by the flow outside the cores, which is the flow of two point vortices whatever the cores are, so they are the same for cores of radius 0.1 and 0.4 — while the snapshot’s vortex is simply the cores, and shrinks with them.

That independence is the cleanest statement of what the criteria cannot do. As the cores shrink towards points, the snapshot’s picture of the pair shrinks to nothing, and the fluid that goes round with the pair stays the same.

Why a third point turns with the pair #

The places the eddies are centred on are not an accident of this pair. Three is the most that can be predicted records that three point vortices at the corners of an equilateral triangle turn rigidly about their centre of circulation, whatever their three strengths — a result Gröbli found in 1877. The strengths enter only the rate of turning. Let the third vortex have no strength at all, and it is a fluid particle. So a particle at the third corner of an equilateral triangle with the pair turns rigidly with it, at the pair’s own rate: it is a point where the turning fluid stands still, and its neighbourhood is the eddy.

The same geometry places the Lagrange points $L_4$ and $L_5$ of two equal masses in orbit, where a third, light body can travel with the pair; the vortex version needs no inertia at all, because a fluid particle is carried by the velocity rather than accelerated by a force. The eddies are the fluid’s Lagrange points, and like $L_4$ and $L_5$ they are wide regions of loops rather than single points.

Two failures, one cause #

This essay and the one before it find the snapshot criteria wrong in opposite directions. The rotating saddle had rotation beating strain everywhere, and no particle stayed; the co-rotating pair has strain everywhere outside its cores, and a large body of fluid stays. Both failures have the same cause. The criteria read the velocity gradient at an instant, and the gradient is the same in a fixed frame and a frame moving at constant velocity but not in a turning one. When the pattern of the flow turns, what the fluid does over time is set by the flow seen in the turning frame, where the velocity has an extra $\Omega \times r$ and the effective vorticity is shifted by twice the frame’s rate — and in that frame, the eddies’ fluid, irrotational in the lab, is circulating.

The frames question is therefore not a philosophical one. A definition of a vortex that is to mean “fluid that goes round together” has to be objective — unchanged by a turning frame — and has to follow the fluid over time rather than read one instant. Spin is not going round makes the same distinction for a single parcel: vorticity is the local rate of spin, and going round a centre is something else. The pair’s eddies are fluid that goes round without spinning.

What was checked #

The sixth figure lists the checks. The outer saddle is at $\sqrt5$ to rounding, and the turning fluid’s speed there vanishes. $Q$ never becomes positive outside the cores, which is the statement that the flow there is irrotational. The three zones’ mean rates are 1.00, 2.07 and 0.39 at points chosen in the lobe, the band and outside. And the blob released at an eddy’s centre stays within half a separation of it for three orbits while the blob outside spreads by more than two.

What the picture cannot show #

Point vortices with rigid cores. The cores are uniform discs that never deform. Real co-rotating vortices strain each other’s cores into ellipses, and a strained vortex holds only until the strain is too strong; close enough, the two merge.

An eternal orbit. The pair turns for ever at a constant rate. A real pair diffuses and eventually merges, and the eddies’ boundaries, which are separatrices, are the first thing any unsteadiness breaks: fluid then leaks across them in lobes, a process the steady picture cannot contain.

Two dimensions. The pair is two parallel lines; in three dimensions co-rotating vortices bend and wrap round each other.

The convention the numbers depend on #

Lengths are in half-separations, so the vortices sit at $x = \pm1$; each circulation is $2\pi$, so the pair turns at a rate of one half. $Q$ is $\tfrac12(|\Omega|^2 - |S|^2)$ with $\Omega$ and $S$ the antisymmetric and symmetric parts of the velocity gradient. A zone’s mean rate is the average rate at which its fluid’s angle about the centre advances in the fixed frame, over two hundred time units, as a multiple of the pair’s own. Areas are counted on a grid of 500 or 800 points a side.

Who found it, and when #

Gröbli’s analysis of three point vortices, with the rigidly rotating equilateral triangle, dates from 1877. The criteria are Okubo’s and Weiss’s, from 1970 and 1991, and Hunt, Wray and Moin’s $Q$ from 1988. The distinction between the snapshot and the fluid’s behaviour over time, and the requirement that a vortex be defined objectively, are central to the Lagrangian coherent structures developed by George Haller and others since the early 2000s.

Still open: a pair that merges #

The eddies exist because the pair turns steadily. Two real vortices close enough to merge do not: as they approach, their orbit speeds up, the outer saddles move in, and the eddies’ boundaries break. The next calculation follows a merging pair of finite patches with contour dynamics and asks what becomes of the fluid that was in the eddies — whether it is swept into the merged vortex, becoming part of it though it carried no vorticity, or thrown out into the filaments a merger sheds.